text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
{"url":"http:\/\/www.angusgriffith.com\/2015\/04\/08\/optimal-betting.html","text":"# Parimutuel betting\n\nParimutuel betting is a system in which all bets of a particular type are collect together in a pool. The payout is rewarded by sharing the pool among all winning bets. This differs from fixed-odds betting in that the payout amount is not determined until the pool closes.\n\nA fraction of all bets placed are removed by the house. This quantity is called the takeout and differs by market. Takeout values between 5% and 30% are typical.\n\n# Formalism\n\nSuppose there are $n$ runners in a race each with amount $\\{a_i\\}_{i=1}^n$ invested on them. For each dollar invested on runner $i=1,\\dots,n$ the payout should that runner win would be\n\nwhere $T \\in [0,1)$ is the takeout. There are more complicated bet types like trifectas, but for simplicity we focus on the the win case.\n\nThe division of funds among the various runners determines what the market thinks the probability of each runner winning is. That is the market thinks that the probability of runner $i$ winning is $1 \/ a_i$. If one is able to determine the true probabilities better than the market can there is an opportunity.\n\nIf the true probability $p_i$ of runner $i$ winning is greater than what the market believes it to be then betting on runner $i$ is likely to be successful (ignoring the takeout). If this happens we say that runner $i$ is an \u2018over\u2019.\n\nGiven bets $x \\in [0, \\infty)^n$ the expected profit is thus\n\nsince the total investment on runner $i$ will be diluted to $a_i + x_i$ by betting and $\\sum_{j} x_j$ is the cost of placing such bets.\n\n# Constrained nonlinear optimisation\n\nClearly, we have a function $f$ to maximise. We are constrained by the fact that in parimutuel betting one can only buy bets and not sell them. We thus have $n$ constraints,\n\nConsider an example where there are two runners and runner 1 is large favourite to win $p = (0.7, 0.3)$ and the public agree $a = (100, 15)$, and suppose the takeout is modest $T = 0.10$. We can exploit the fact that the public has not invested enough on runner 2.\n\nBy investing \\$10 we expect to make a profit of around \\$3.50.\n\nIn general, the problem is to choose $x$ such that $f(x)$ is maximised. The simplest way to solve this optimisation problem is with the gradient descent method. We start at $x^0 = 0$ and compute $x^{i+1} := x^i - \\lambda \\nabla f (x^i)$ until $f(x^i)$ and $f(x^{i+1})$ are close. Any time we fall outside the allowed region we project back onto it.","date":"2023-03-27 13:32:05","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\": 0.5140862464904785, \"perplexity\": 672.3816532728514}, \"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-2023-14\/segments\/1679296948632.20\/warc\/CC-MAIN-20230327123514-20230327153514-00673.warc.gz\"}"} | null | null |
package org.keycloak.example.ws;
import javax.xml.ws.WebFault;
@WebFault(name = "UnknownProductFault")
public class UnknownProductFault extends Exception {
private org.keycloak.example.ws.types.UnknownProductFault unknownProductFault;
public UnknownProductFault() {
super();
}
public UnknownProductFault(String message) {
super(message);
}
public UnknownProductFault(String message, Throwable cause) {
super(message, cause);
}
public UnknownProductFault(String message, org.keycloak.example.ws.types.UnknownProductFault unknownProductFault) {
super(message);
this.unknownProductFault = unknownProductFault;
}
public UnknownProductFault(String message, org.keycloak.example.ws.types.UnknownProductFault unknownProductFault, Throwable cause) {
super(message, cause);
this.unknownProductFault = unknownProductFault;
}
public org.keycloak.example.ws.types.UnknownProductFault getFaultInfo() {
return this.unknownProductFault;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,150 |
Donald A. "Donnie" Keller
By Editor | March 7, 2022
Donald A. "Donnie" Keller, 82, Maywood, Mo., passed away at 9:02 a.m., Friday, March 4, 2022, at the University of Missouri Hospital in Columbia, Mo.
Donnie was born March 21, 1939, in Taylor, Mo., to James W. and Mary Francis Elizabeth Foreman Keller.
Survivors include his many nieces, nephews, extended family and friends.
Donnie was preceded in death by his parents; two brothers, Howard Keller and James F. Kelle; one sister, Dorothy O'Bryan; one nephew, Rodney O'Bryan; and great-nephews, Shane Tyler Huckaby and Jason O'Bryan.
Donnie was a farm boy through and through, working the family farm his entire life. He took pride in taking care of every aspect of the farm.
Besides spending time on the farm, Donnie was an avid sports fan. He enjoyed softball, baseball and the KC Chiefs. He loved coaching softball and did so for many years with the local girls' team. He faithfully listened to the St. Louis Cardinals at every opportunity.
He was an enthusiastic bird and raccoon hunter and was always thrilled to hunt with his family and friends. Donnie loved to play pitch, pinochle and cribbage with all the local guys at the card room in Palmyra.
Donnie could often be found fishing and mushroom hunting at his special spots. He loved to be "in the know" and was always interested in what was going on in the community.
He spent many years taking care of his father, James W. Keller, and the two of them were inseparable. Donnie will be missed by his family and the many people who loved him.
Funeral services will be held at 11 a.m., Wednesday, March 9, at Lewis Brothers Funeral Chapel. Rev. Roger Stevens will officiate. Burial will follow at Greenwood Cemetery in Palmyra.
Pallbearers will be Larry O'Bryan, Kent O'Bryan, Kevin O'Bryan, Jeff O'Bryan, Travis Keller, Chad Keller, Mike Keller, Brad Keller, Todd Keller and Scott Keller.
Honorary pallbearers will be Dwane "Dink" DeGrant, Jim Prather, Bob Tiemann, Gordon Smyser, Albert Corey, Splinter Hoehne, Donald Bock and Dick Brackett.
Friends and family are invited to Donnie's life celebration at a visitation that will be held from 9 a.m. until the time of service on Wednesday, March 9, at Lewis Brothers Funeral Chapel.
Memorials may be made to the American Cancer Society.
Online condolences may be shared, and video tribute viewed at www.lewisbrothersfuneralchapel.com.
Sharon K. Bainter
Jacquelyn "Jackie" Johnston
Delbert Lee Coleman
Frances I. Cassidy
Murray M. Bier | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,207 |
namespace Microsoft.Azure.Management.HealthcareApis.Models
{
using Microsoft.Rest;
using Microsoft.Rest.Serialization;
using Newtonsoft.Json;
using System.Linq;
/// <summary>
/// IoT Connector FHIR destination definition.
/// </summary>
[Rest.Serialization.JsonTransformation]
public partial class IotFhirDestination : LocationBasedResource
{
/// <summary>
/// Initializes a new instance of the IotFhirDestination class.
/// </summary>
public IotFhirDestination()
{
CustomInit();
}
/// <summary>
/// Initializes a new instance of the IotFhirDestination class.
/// </summary>
/// <param name="resourceIdentityResolutionType">Determines how
/// resource identity is resolved on the destination. Possible values
/// include: 'Create', 'Lookup'</param>
/// <param name="fhirServiceResourceId">Fully qualified resource id of
/// the FHIR service to connect to.</param>
/// <param name="fhirMapping">FHIR Mappings</param>
/// <param name="location">The resource location.</param>
/// <param name="provisioningState">The provisioning state. Possible
/// values include: 'Deleting', 'Succeeded', 'Creating', 'Accepted',
/// 'Verifying', 'Updating', 'Failed', 'Canceled', 'Deprovisioned',
/// 'Moving', 'Suspended', 'Warned', 'SystemMaintenance'</param>
/// <param name="systemData">Metadata pertaining to creation and last
/// modification of the resource.</param>
public IotFhirDestination(string resourceIdentityResolutionType, string fhirServiceResourceId, IotMappingProperties fhirMapping, string location = default(string), string provisioningState = default(string), SystemData systemData = default(SystemData))
: base(location)
{
ProvisioningState = provisioningState;
ResourceIdentityResolutionType = resourceIdentityResolutionType;
FhirServiceResourceId = fhirServiceResourceId;
FhirMapping = fhirMapping;
SystemData = systemData;
CustomInit();
}
/// <summary>
/// An initialization method that performs custom operations like setting defaults
/// </summary>
partial void CustomInit();
/// <summary>
/// Gets or sets the provisioning state. Possible values include:
/// 'Deleting', 'Succeeded', 'Creating', 'Accepted', 'Verifying',
/// 'Updating', 'Failed', 'Canceled', 'Deprovisioned', 'Moving',
/// 'Suspended', 'Warned', 'SystemMaintenance'
/// </summary>
[JsonProperty(PropertyName = "properties.provisioningState")]
public string ProvisioningState { get; set; }
/// <summary>
/// Gets or sets determines how resource identity is resolved on the
/// destination. Possible values include: 'Create', 'Lookup'
/// </summary>
[JsonProperty(PropertyName = "properties.resourceIdentityResolutionType")]
public string ResourceIdentityResolutionType { get; set; }
/// <summary>
/// Gets or sets fully qualified resource id of the FHIR service to
/// connect to.
/// </summary>
[JsonProperty(PropertyName = "properties.fhirServiceResourceId")]
public string FhirServiceResourceId { get; set; }
/// <summary>
/// Gets or sets FHIR Mappings
/// </summary>
[JsonProperty(PropertyName = "properties.fhirMapping")]
public IotMappingProperties FhirMapping { get; set; }
/// <summary>
/// Gets or sets metadata pertaining to creation and last modification
/// of the resource.
/// </summary>
[JsonProperty(PropertyName = "systemData")]
public SystemData SystemData { get; set; }
/// <summary>
/// Validate the object.
/// </summary>
/// <exception cref="ValidationException">
/// Thrown if validation fails
/// </exception>
public virtual void Validate()
{
if (ResourceIdentityResolutionType == null)
{
throw new ValidationException(ValidationRules.CannotBeNull, "ResourceIdentityResolutionType");
}
if (FhirServiceResourceId == null)
{
throw new ValidationException(ValidationRules.CannotBeNull, "FhirServiceResourceId");
}
if (FhirMapping == null)
{
throw new ValidationException(ValidationRules.CannotBeNull, "FhirMapping");
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,045 |
desc 'monolingual' do
latest = github('IngmarStein', 'Monolingual').first
version = latest.tag_name.gsub(/^v/, '')
latest.assets.select! { |e| e.name.match(/dmg/) }
url = latest.assets.first.browser_download_url
Latest.new version: version, url: url
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,294 |
Alexei Navalny Released from German Hospital after 32 Days
In this photo published by Russian opposition leader Alexei Navalny on his Instagram account on Monday, Sept. 21, 2020, Russian opposition leader Alexei Navalny and his wife Yulia pose for a photo in a hospital in Berlin. (Navalny Instagram via AP)
BERLIN — The German hospital treating Russian opposition leader Alexei Navalny for poisoning said Wednesday that his condition improved enough for him to be released, and suggested a "complete recovery" from the nerve agent was possible.
Navalny, 44, spent 32 days in treatment in Berlin's Charite hospital, 24 of which were in intensive care, before doctors deemed his "condition had improved sufficiently for him to be discharged from acute inpatient care."
The hospital said that based on Navalny's progress, treating physicians believe that "complete recovery is possible," but added that it "remains too early to gauge the potential long-term effects of his severe poisoning."
Navalny, the most visible opponent of Russian President Vladimir Putin, was flown to Germany two days after falling ill on Aug. 20 on a domestic flight in Russia.
German chemical weapons experts have determined he was poisoned with the Soviet-era nerve agent Novichok — findings corroborated by labs in France and Sweden.
During his convalescence, Navalny has in recent days been posting regular photos from the hospital on Instagram, first showing him sitting up in his bed surrounded by his family, then up and about in the building.
In a wry post Tuesday night accompanied by a close-up photo, he scoffed at reported comments by Putin suggesting he might have intentionally taken poison himself.
"Good theory, I believe it deserves the most careful attention," Navalny wrote in Russian.
"Cooked Novichok in the kitchen. Took a sip from a flask on the plane. Fell into a coma."
He sarcastically said then the "ultimate aim of my cunning plan" was to die in Siberia, where the cause of death would be "lived long enough."
"But Putin outmaneuvered me. You can't fool him," Navalny wrote. "As a result, I lay in coma for 18 days like a fool, but didn't get my way. The provocation failed!"
The nerve agent used in the attack was the same class of Soviet-era agent that Britain said was used on former Russian spy Sergei Skripal and his daughter in Salisbury, England, in 2018, and Chancellor Angela Merkel and other world leaders have called for Russia to fully investigate.
Navalny was kept in an induced coma for more than two weeks as he was treated with an antidote. Members of his team accused the Kremlin of involvement in the poisoning, charges that Russian officials have vehemently denied.
Russia has bristled at the demands for an investigation, saying it needs Germany to share medical data or compare notes with the Russian doctors who said they found no trace of poison in his system while he was at a hospital in the Siberian city of Omsk.
Germany has noted that Navalny was in Russian treatment for 48 hours, and that Russia has its own data.
Germany has also enlisted the Hague-based Organization for the Prohibition of Chemical Weapons for technical assistance in the case.
Last week, the international agency said its experts had " independently collected biomedical samples from Mr Navalny for analysis by OPCW designated laboratories"
Results have not yet been announced.
The Charite statement was released in consultation with Navalny and his wife, and the hospital would not comment further on whether he would continue to receive outpatient care there.
Navalny's team has said he eventually plans to return to Russia, but had no immediate statement after his release from the hospital.
WASHINGTON — President Joe Biden put the full weight of his presidency behind voting rights action last week, heading to Capitol Hill in an effort to push Democrats to change Senate rules to pass legislation.
Sinema, Manchin Slammed as Senate Begins Voting Bill Debate | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,205 |
Q: Check for internet availability I am trying to ping a if there is internet connection, hence I have decided to ping google as follows:
Private Sub Timer1_Tick(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Timer1.Tick
Try
If My.Computer.Network.Ping("www.google.com", 500) Then
TextBox4.Text = "Internet Available"
Else
TextBox4.Text = "No Internet avilable"
End If
Catch ex As Exception
TextBox4.Text = "Cable disconnected"
End Try
End Sub
It is working fine, but the challenge is when i disconnect the cable, the application seems to freeze, maybe its due to the time wch makes the system continously ping. My delay time for the timer is 2000. Is there a better way to solve this problem. I will apreciate if u give a code or link
A: *
*What Thread is your timer running in? These kinds of diagnostics need to be put in a worker thread seperate from the one running your UI. If you have not been using a thread, then do so now for your ticker timer.
*You may also use the exposed Win 7 / XP API for checking Internet connectivity, just in case.
I hope this directs you in the right path.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,891 |
The annual Journey Prize anthology selects the year's best fiction from Canadian literary magazines and later awards one the $10,000 Writers' Trust of Canada/McClelland & Stewart Journey Prize. Now in its 17th edition, the series boasts Yann Martel, Lisa Moore, Douglas Glover, Charlotte Gill, Timothy Taylor and M. G. Vassanji among past contributors.
Half of these 14 varied stories truly compete for one's attention (and, one hopes, the enabling prize), and all but one involve the myriad trials of romantic love. Most courageous is Neil Smith's Scrapbook, which uses the pressures of a campus shooting (suggestive of the 1989 massacre at Université de Montréal) to test the limits of a young couple's relationship.
Smith, an alumnus of two previous Journey Prize anthologies and Coming Attractions '04, wisely avoids sweeping social condemnation or diffuse commentary, finding instead all the relevant and necessary fears, inadequacies and impatience in the growing strain of a young man who fled the shootings, and his worried, impatient and sometimes nervous female partner. Smith's numerous talents collapse the distance between maniacal violence and the insecurities and inadequacies beneath the surface of daily life.
Equally impressive, and also devoted to love and loss, is Emily White's Various Metals. Mined again here is the always fecund material of a thirtysomething woman without a man, drowning in the unwanted romantic advice of her (divorced) parents, her friends, her customers and of, well, anyone who sees her.
This tender, knowing story seductively flashes its plot of unexpected seduction with just the right pace, camouflaging its more ambitious emotions as care or attention one minute, then revealing an awkward advance the next. Humour and a good eye for physical detail take us toward the story's tender, wise conclusion.
Filial collisions and the bursts and contractions of the romantic heart also surface in Krista Bridge's lively and dynamic A Matter of Firsts. Capitalizing on the simultaneous distance and intimacy of second-person narration ("Your father's New York mistress was the one you met"), Bridge brilliantly captures the kind of non-consummated crush an adolescent girl can have for an older, wiser woman, while simultaneously initiating her young heroine into the yearnings and hypocrisies of an adult's romantic life.
Very different stories from Craig Davidson and Matt Shaw both make impressive use of voice, albeit with mixed success. Davidson's alcoholic narrator intriguingly confesses, "Okay, yeah, I'll cop to hoisting a few," but the story itself descends into a predictable and limited terrain of booze glorification (it cites Charles Bukowski repeatedly) and media angst. Matt Shaw's Matchbook for a Mother's Hair confronts the reader thematically with a tale of a mother abusing her developmentally challenged adolescent son, and again stylistically with the decision to narrate that abuse through the son's limited perception.
While Shaw's is undoubtedly the most technically ambitious story ("Her lighter was the colour of a fire engine like my chair"), its focus on technique limits how much and how deeply the story can evolve.
The less successful stories generally fail due to sins of selection, either within the anthology as a whole or within each story. Two stories are probably included for subject, not execution (or wisdom, or inventive language).
In one, the potentially explosive combination of three generations of women trying to live under one roof fizzles out in a formless sea of domestic details. Undirected detail also derails two stories that frankly confuse research with storytelling, parading, in one case, the unshaped facts and observations of a girl's school trip and, in another, an ill-fitting string of historical details that attempts to translate the life of architect Le Corbusier into (a) fiction and (b) short fiction.
Two (mercifully short) stories from younger journals seem selected solely to add urban grit to the collection. The annual anthology's new and regrettable inclusion of artist's statements about each story self-destructs entirely with the entries for these two non-narrative language events, and for the stories begun with homework, not heart. At its best, this anthology and its predecessors know that a good story explains itself.
More prudently devoted to the timeless narrative rewards of the fickle human heart are the honest and confident stories of Edward O'Connor and Pasha Malla. Vanity and ambition pervade the former, while the bittersweet mixture of loss and humour bubbles through the latter.
Tomorrow's short lists for the Giller and the Governor-General's prizes invariably start with today's Journey Prize Stories.
Darryl Whetter is the author of A Sharp Tooth in the Fur, and a former professor of creative writing and English. His most recent story will appear in 05: Best Canadian Stories. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,363 |
I have been treated for bi-polar II for almost 15 years but I'm almost always depressed. My psychiatrist says that I'm treatment resistant. I have always worked but long for as normal of a life as I can possibly have.
There have been several posts on ECT---one just a few days ago---where people provided detailed descriptions of their experiences. Search the sub for ECT.
I had 23 ECT sessions this winter to treat refractory depression. The last session went wrong and fried my brain. I am still trying to recover. It damaged both retrograde and anterograde memory. I am still recovering from it. It was partially effective. I now have new problems in addition to depression.
I cannot recommend any treatment other than unilateral. Bitemporal is too damaging and dangerous. I do not recommend that anyone still functional enough to work undergo ECT. The cognitive side effects could easily leave you unable to return to work.
It's very traumatic. Its therapeutic effects are temporary in many cases which means you have to keep going back. It can be very expensive even with insurance.
I wouldn't rule it out completely, but can't recommend it until you tried everything else and are in bad shape.
Read the other posts. I hope you reconsider.
Your situation is exactly why I refuse to do ECT. I barely have a functioning memory as it is. I'm not sure what would be left if I got zapped wrong.
It turns out to be a somewhat deceptively complex procedure. Many factors go into determining the anesthetics, probe placement, system settings. If they assume your threshold will be high, and it's not, I think there's a risk of overshooting and causing damage--which I believe happened to me. If they can't get you under, they'll pile on more anesthetics which will raise your seizure threshold and possibly void the whole procedure.
I keep hoping that hole in my head that is now dead and blank will come back to life. That same region was left dry searing hot pain for a month after my last procedure and nothing has been right since. I know that brain pain is not real, but wherever the pain came from it hurt like hell.
I go for my 6th treatment tomorrow. So far no issues, and I'm getting bilateral which is the most likely to screw your memory. I can remember perfectly up til being knocked out and then when I wake up again. I'm not sure if its working yet, but I haven't needed my anxiety meds since I started.
Be advised. The nature and extent of your memory losses won't be evident for at least a few months. The problem is you cannot remember what you do not remember. It takes many different attempts to remember different things to discover that some memories have been smudged. I had 23 sessions this winter. The last one had an adverse outcome. I'm still recovering.
23 treatments is a lot, I don't think I'm going to keep going for that long. I was inpatient my first week but they really screwed up my diabetes so I checked myself out and switched to outpatient. There were a few people in there who were back for maintenance treatments after being symptom free for 3 years. They had a few more treatments, felt better again, and left. I'll be happy to get even a year in between treatments.
I have GAD too and it makes going out in public while syptomatic a Nightmare. ECT really helped w/ the anxiety? What about depression?
It seems to be helping but it hasn't 'stuck' yet. I think I'll be able to tell better when school starts and I don't have to deal with screaming kids all day though. I've been told I'm smiling more.
Thanks for being honest. : ) I will talk to my Pdoc about it at out next meeting.
Depression wise, are you feeling better? Has your doc indicated that you will have to remain on your antidepressant and how many treatments you will need? It sounds as if you really don't see much of a difference.
Also, on more thing, does an antheisologist present?
I am starting to feel better. Yes, I will remain on my medication. Usually treatment is 6-12 sessions. I go for the 6th tomorrow, so I'm on the lower end right now. I've had people come up to me and tell me I look happier and that they were worried about me before because I looked so depressed, so I'm taking it as a good sign.
Yes, theres an anesthesiologist. They knock you out and use a muscle relaxer so that you don't physically seize. they put a mask over your mouth for breathing since your lungs won't work.
I haven't personally had it, however I'll share a story I saw on a doco about bipolar because it was one of the worst stories of ECT I have seen and I believe it helps give a picture of what could happen. This woman had ECT and it completely ruined her anterograde and retrograde memory. She couldn't remember simple things such how to get to the local store or how to drive to certain places. Even worse was that there was so much of her past she couldn't remember, special occasions or things that were important to her and her family/friends. But the biggest thing she struggled with was feeling as though she couldn't identify with her daughter. At times she struggled to remember her daughters name.
So take that anecdote as you will. I just think it's important to know that whilst some people don't have bad side effects, other people do. I guess you have to weigh up the risks with the potential benefits.
Yes. All those simple things are what you forget: how to drive anywhere, how to cook, what you used to like to cook, vocabulary, ... , it's damn disconcerting.
I'm fairly convinced I would have many fewer side effects, had they stayed with unilateral. That variant works for many and felt might lighter to me.
Did her memory issues persist after the ECT had concluded? For how long afterwards?
My last treatment was June 28th 2010. I still have difficulty forming new memories and have never regained any of the memories I lost. I lost entire years.
They persisted after she finished treatment, yes. At the time of the doco that I saw she was still having memory problems. I can't recall how long ago it was in the doco.
I had it this past winter and am able to go back to school this fall. I can barley remember the last two years, and some memories are just gone, but that might not be completely a bad thing.
Are your cognitive and memory functions up to college standards? I had 23 sessions this winter and mine sure as hell are not. Be sure to ask for accommodations like extra time for exams and so on.
I had an 8 month long experience with ect and it was awful. I have permanent memory loss of about 1.5 years and a hard time making new ones.
My grandpaps tried. It helped him. Bumping this thread. I have been asked if I wanted to try it, but I have always beem too afraid.
I had ECT about 4 years ago, have a diagnosis of Bipolar II, as well as PTSD and (the dreaded) borderline personality disorder. I had 15 treatments in total and I have some memory issues, particularly with forgetting words and struggling to place things in my life on a timeline. When did something happen? It was a "while" ago. It's not terrible all the time, but gets worse when I am put on the spot for an answer.
The actual procedure was kind of scary, but i was at a place in my life that I saw it as self-punishment. I was in and out of the hospital so often I was being threatened with long-term commitment. I can say now that it has been 3 years (this past May) since I have been inpatient. It helped. It really did. It took the edge off the depression, they are still playing around with my meds but its more of a fine tuning at this point.
It is a big decision, go into it with your eyes open. Know that it can take some time to see the results. I hope it helps your depression. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,382 |
Creative direction, art direction, visual design and communication are my team leadership rolls. However as a micro studio we provide a wide range of web and print services to both business clients and other developer/business teams on a per protect and ongoing basis. Every year brings creative changes as we set new goals taking on hybrid web apps this year and a couple of long term projects that include products. Exciting!! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,568 |
Scott #1s was the title of a series of sixty-one articles that ran in the American philatelic publication Stamps between January 1999 and October 2003. Quite simply, it told the story behind the first stamp issued by every country in the world, according to the Scott series of catalogs – ultimately, the series ended some way short of its target, a victim of the magazine's faltering circulation and ultimate cancellation.
The philatelic history of Aden, a small British enclave on the southern tip of the Arabian peninsula, is inextricably tied up with that of India. For, not only was Aden considered a part of British India for close to 100 years, from 1839 to 1937, it was also serviced with Indian stamps, arguably establishing Aden's number one as simply the earliest known Indian stamp to bear the correct postmark.
The first shipment of these, contemporary half and one anna stamps, were placed on sale in Aden on October 10, 1854; two months later, a supply of 2 and 4 anna stamps arrived. The earliest known usage, however, dates from some five months later, in March, 1855. Aden was the Bombay Presidency's post office #124, and this numeral will be present on all Aden mail dated prior to 1872. Thereafter, a new series of cancellations saw Aden designated B_22 (22 alone from 1878), before the numbering system was finally abandoned in 1887.
Indian stamps remained in use in Aden until 1937; British stamps were also accepted, primarily for military mail (carried free until 1854, these now cost 8 pies, or one penny). Emissions of neighboring Yemen also appear to have been tolerated for a period following their introduction in 1926. In 1937, however, Aden finally became a Crown Colony within the British Empire, and on April 1 that year, it was granted its own first stamps, in the name of the newly crowned King George VI.
Recess printed by De La Rue of London, and denominated in Indian coinage of pies, anna and rupees, Aden's first issue was a set of twelve engraved definitives, picturing a native dhow, or sailing vessel, within an ornately designed frame, which itself includes two ornamental daggers, one on either side of the central frame. All twelve stamps in the series were perforated 13×11.5, and carried the multiple crown and script C.A. (Crown Agents) watermark which is common to so many Empire stamps of the period.
The colony's name appeared in English above the design, and in Arabic below, with the value expressed (in English numerals) in the two lower corners. Aden number one itself, then, is the lowest denomination in the series, a half anna stamp printed in light green. As such, it is also the lowest valued stamp in the entire set, with a current catalog price of 35c mint, 25c used. There also exists a considerably scarcer, and much sought after specimen example, with the word SPECIMEN perforated into the stamp. It is, however, rarely seen separated from the full set of twelve.
ÜjÜŒ | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,145 |
Park Narodowy "Kalewalskij" (ros. Национальный парк «Калевальский») – park narodowy położony w Republice Karelii w północno-zachodniej europejskiej części Rosji. Znajduje się na granicy z Finlandią, w rejonie miejskim Kostomuksza, a jego obszar wynosi 743,43 km². Park został utworzony dekretem rządu Federacji Rosyjskiej z dnia 30 listopada 2006 roku. Zarząd parku znajduje się w mieście Kostomuksza.
Opis
Park został utworzony w celu ochrony naturalnych lasów sosnowych. Drzewostany z sosną na ponad 80% powierzchni Parku mają ponad 120 lat, w wielu miejscach ich wiek przekracza 300 lat, przy czym pojedyncze okazy osiągają 400–500 lat. Oprócz dziewiczych lasów znajdują się tu rzadkie ekosystemy bagienne i jeziorne (bagna to około 19% powierzchni parku). Sieć hydrograficzna obejmuje około 250 rzek i strumieni oraz 400 jezior o łącznej powierzchni około 9 tysięcy hektarów. Największymi rzekami są Sudno, Wierch i Kuito, a największymi jeziorami Kenas, Lewi i Maria-Szeleka.
Fauna i flora
Flora i fauna jest typowa dla północnej tajgi. Występują tu 333 gatunki roślin naczyniowych. Jest 160 gatunków mchów i 167 porostów.
W parku zamieszkuje 37 gatunków ssaków. Jest tu dużo wiewiórek pospolitych (Sciurus vulgaris), kun leśnych (Martes martes), rosomaków (Gulo gulo) niedźwiedzi brunatnych (Ursus arctos) i reniferów tundrowych (Rangifer tarandus). Żyją tu też w niewielkiej liczebności rysie euroazjatyckie (Lynx lynx) wilki szare (Canis lupus), norki amerykańskie (Mustela vison), bobry kanadyjskie (Castor canadensis).
Teren parku zamieszkuje 143 gatunki ptaków, w tym 127 gniazdujących. Są to m.in.: głuszec zwyczajny (Tetrao urogallus), jarząbek zwyczajny (Tetrastes bonasia), cietrzew zwyczajny (Lyrurus tetrix), nur czarnoszyi (Gavia arctica), łabędź krzykliwy (Cygnus cygnus), gęś zbożowa (Anser fabalis), mewa żółtonoga (Larus fuscus), rybołów zwyczajny (Pandion haliaetus), bielik (Haliaeetus albicilla) i kania czarna (Milvus korschun). Bagna zamieszkuje 12 gatunków ptaków, m.in. żuraw zwyczajny (Grus grus), pardwa mszarna (Lagopus lagopus) i siewka złota (Pluvialis apricaria).
Z ryb występują tu m.in.: sieja pospolita (Coregonus lavaretus), sielawa europejska (Coregonus albula), pstrąg potokowy (Salmo trutta m. fario), lipień pospolity (Thymallus thymallus), szczupak pospolity (Esox lucius) i okoń pospolity (Perca fluviatilis).
Klimat
Obszar charakteryzuje się mroźnymi i długimi zimami (pokrywa śnieżna 170–180 dni). Średnia temperatura lipca wynosi około +14,5 °С, a stycznia około -12,5 °С.
Przypisy
Parki narodowe w Rosji
Karelia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,785 |
Q: fine-tuning from an existing checkpoint: meaning of "steps" In this example of fine-tuning an InceptionV3 model on the Flowers training set, there are two parts which say:
# Fine-tune only the new layers for 1000 steps.
after which an evaluation is run.
Then,
# Fine-tune all the new layers for 500 steps.
after which a second evaluation is run.
What does this mean in context of fine-tuning? I'm not sure what this concept of "steps" means or why they need to do evaluation twice.
A: A step means a gradient descent step on a minibatch of examples. The example tunes for 1000 steps and then for 500 further steps to show how the performance improves after fine tuning.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,655 |
{"url":"https:\/\/math.stackexchange.com\/questions\/2396104\/find-all-positive-integers-such-that-mnmn-2016","text":"# Find all positive integers such that $m+n+mn=2016$\n\nWhat I have till now is that $2016= 2^5\\cdot3^2\\cdot 7$.\n\nAlso, because $m+n+mn=2016$ then $m$ and $n$ must be even. For the rest my idea is to use congruence module $3$, and $7$ to see all cases. Do you have a better idea? Because there are a lot of cases. How would you find the solutions?\n\n\u2022 The way to approach this would indeed be to attempt to factorize the LHS as suggested below. You'd perhaps see this if you tried to factorize as much as possible each time. $m+n+mn=m(n+1)+n=m(n+1)+n+1-1=(m+1)(n+1)-1$ \u2013\u00a0Shuri2060 Aug 16 '17 at 20:52\n\u2022 Very similar in spirit and solution tohttps:\/\/math.stackexchange.com\/questions\/1267670\/find-p-q-r ! \u2013\u00a0Robert Lewis Aug 16 '17 at 21:04\n\nIt's $(m+1)(n+1)=2017$ and $2017$ is a prime number.\nThus, $m+1=1$ and $n+1=2017$ or $m+1=2017$ and $n+1=1$, which says that our equation has no solutions.\nHint $\\$ This type of diophantine equation is solvable by a generalization of completing the square. Namely, completing a square generalizes to completing a product, using the AC-method, viz.\n$$\\begin{eqnarray} &&axy + bx + cy\\, =\\, d,\\ \\ a\\ne 0\\\\ \\overset{\\times\\,a}\\iff\\, &&\\!\\! (ax+c)(ay+b)\\, =\\, ad+bc\\end{eqnarray}\\qquad\\qquad$$\nSo the problem reduces to checking which factors of $\\,ad+bc\\,$ have above form, a finite process.\nIf you don't spot the factorisation straight away, you should try to separate the variables e.g. try isolating $m$ by writing $m(n+1)=2016-n$ or $$m=\\frac {2016-n}{n+1}$$ Where the right-hand side is an integer. Then divide through so you leave a fraction where the numerator has lower degree than the denominator $$m=\\frac {2017}{n+1}-1$$ and you see that if $m$ is an integer, $n+1$ is a factor of $2017$","date":"2019-05-22 02:36:45","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\": 0.8119158744812012, \"perplexity\": 181.6808328359788}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232256724.28\/warc\/CC-MAIN-20190522022933-20190522044933-00443.warc.gz\"}"} | null | null |
AG Grewal Joins Lawsuit Seeking to Block Federal Rule that Sanctions Healthcare Discrimination, Threatens Funding to the States
Submitted by RLS Staff on May 21 2019 - 1:14pm.
Attorney General Gurbir S. Grewal today joined a multi-state lawsuit that seeks to stop the U.S. Department of Health and Human Services (HHS) from implementing a so-called "conscience protection" (or "refusal-of-care") rule that would allow health care institutions to deny medically necessary care to patients based on "religious, moral, ethical or other reasons."
According to reports, New Jersey received $11.8 billion in federal health care funding last fiscal year – including money to support children's healthcare and wellness for older adults, as well as disease prevention, public health programs, opioid addiction treatment, and federally-qualified health centers.
Officials say that under the HHS rule – published today -- all of New Jersey's annual federal healthcare funding could be terminated if HHS determines the State has failed to comply with the rule's requirements. The rule could also conflict with New Jersey laws that require the provision of medically necessary health care, and that balance respect for religious freedom with the rights of patients.
"This refusal-of-care rule represents an unprecedented and unlawful overreach by the federal government, and we're proud to stand against it," said Attorney General Grewal. "New Jersey will not be strong-armed into accepting a federal rule that is unconstitutional, morally wrong and potentially harmful to some of our most vulnerable populations."
"Our state already has a framework of laws that carefully balance respect for religious freedom with the rights of patients to access health care," Attorney General Grewal said. "This rule could leave us unable to enforce those laws, and we don't intend to let that happen."
Filed in U.S. District Court for the Southern District of New York, reports say that today's lawsuit alleges that the proposed HHS rule extends far beyond the existing federal health care statutes it purports to enforce, and conflicts with existing federal laws concerning emergency health care, religious accommodations, and comprehensive family planning services.
In addition, according to reports, the complaint claims violations of the federal Administrative Procedures Act, as well as both the Establishment Clause and the Spending Clause of the U.S. Constitution.
The lawsuit asserts that "communities of color and other vulnerable populations will bear a disproportionate burden of the harms" caused by the refusal-of-care rule's implementation.
"Patients reliant on federal funding for the provision of health care are disproportionately non-white compared to the overall population," the complaint notes.
In addition, the complaint asserts that "women and LGBTQ individuals who are already stigmatized in obtaining access to health care will be further hindered in obtaining the lawful medical services they need."
The proposed rule at issue was put forth by HHS's Office of Civil Rights, which historically has protected patients from unlawful discrimination based on race, color, national origin, disability, age or sex in the provision of medical care, officials say.
Today's lawsuit contends that, in essence, the HHS rule threatens to upend the delicate balancing of patients' rights to care with conscience protections achieved by states like New Jersey. Instead, officials say that the HHS rule simply favors the right of doctors and health care providers to discriminate against patients.
Among other issues, the complaint asserts concerns about the rule's potential to conflict with state laws regarding the provision of emergency health care -- including laws addressing the treatment of sexual assault victims, and the requirement to provide information regarding emergency contraception. (New Jersey has such a law.)
The complaint also raises concerns about potential conflict with state laws prohibiting the abandonment -- or medically inadequate treatment of – patients by healthcare providers exercising their "conscience" rights under the rule. (New Jersey law requires an "appropriate, respectful and timely transfer of care" in such instances, and requires that "the patient is not abandoned or treated disrespectfully.")
Reports say that a total of 23 states and other jurisdictions are party to today's lawsuit, which is led by New York. In addition to New Jersey, parties to the lawsuit include Colorado, Connecticut, Delaware, the District of Columbia, Hawaii, Illinois, Maryland, Massachusetts, Michigan, Minnesota, Nevada, New Mexico, Oregon, Pennsylvania, Rhode Island, Vermont, Virginia, and Wisconsin. Also joining the lawsuit are the City of New York, the City of Chicago, and Cook County, Illinois.
Nj laws | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,623 |
Q: how to find modal has scrolled to the end -EmberJS i'm newbie to emberJS and i want to enable a button when it comes to the scroll end in a modal. so i tried several ways but it didn't work
View
<div class="modal--dialog--body">
<div class="app_tour--section--terms_container" id="message-container">
<ul class="app_tour--section--ul">
<li></li>
//so many lists to scroll
<li></li>
</ul>
</div>
Controller
import Ember from 'ember';
export default Ember.Controller.extend({
didTransition() {
Ember.run.later('afterRender', () => {
let objDiv = document.getElementById("message-container");
if(objDiv.scrollTop == objDiv.scrollHeight)
console.log(objDiv.scrollTop)
}, 100);
return true;
},
actions: {
close: function() {
this.send('closeModal');
}
}
});
A: Welcome to getting started with Ember! Like many things, there's a pretty useful addon that will make your life simpler for this: https://github.com/alphasights/ember-scrollable
That provides a scrollable-container and exposes an action when you've reached scroll bottom.
{#ember-scrollable onScrolledToBottom=(action "close")}}
add terms and consitions here
{{/ember-scrollable}}
A: I used basic jQuery to get the DOM's element I was targeting and listen to the scroll event with the help of Rechardo and it worked.
you can check it via her post on medium
https://medium.com/@futoricky/ember-js-detect-scrolling-events-on-html-element-using-component-ac6b2630021a
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,272 |
Andrelton A. Simmons (ur. 4 września 1989) – pochodzący z Curaçao baseballista występujący na pozycji łącznika w Los Angeles Angels.
Przebieg kariery
Simmons został wybrany w 2010 roku w drugiej rundzie draftu przez Atlanta Braves i początkowo grał w klubach farmerskich tego zespołu, między innymi w Mississippi Braves, reprezentującym poziom
Double-A. W Major League Baseball zadebiutował 2 czerwca 2012 w meczu przeciwko Washington Nationals. W marcu 2013 wystąpił w reprezentacji Holandii na turnieju World Baseball Classic.
W sezonie 2013 po raz pierwszy otrzymał Złotą Rękawicę. W lutym 2014 podpisał nowy, siedmioletni kontrakt wart 58 milionów dolarów. W listopadzie 2015 w ramach wymiany zawodników przeszedł do Los Angeles Angels of Anaheim.
Nagrody i wyróżnienia
Przypisy
Baseballiści z Curaçao
Holenderscy baseballiści
Baseballiści Atlanta Braves
Baseballiści Los Angeles Angels of Anaheim
Uczestnicy World Baseball Classic 2013
Uczestnicy World Baseball Classic 2017
Urodzeni w 1989 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,747 |
TLB (do inglês Translation Lookaside Buffer) é um dispositivo de hardware implementado a partir de uma pequena memória associativa que fica integrada na Unidade de Gerenciamento de Memória de um processador. Destina-se a facilitar a tradução de endereços lineares em endereços físicos, evitando a consulta à tabela de páginas localizada na memória.
Introdução
Visando o compartilhamento seguro e eficiente da memória entre diferentes programas ao mesmo tempo, foi desenvolvida a memória virtual que funciona basicamente gerando endereços lógicos de memória, que são posteriormente traduzidos para endereços físicos pela MMU ou unidade de gerenciamento de memória. Esta tradução é feita percorrendo uma tabela de páginas que liga o endereço virtual com o endereço físico da memória. Esta tradução acaba gerando um grande número de acessos a memória (três acessos em média), de forma que surge a necessidade de utilizar algum tipo de mecanismo para agilizar o processo, e aqui entra a finalidade da Translation Lookaside Buffer ou TLB. Quando a memória virtual é acessada a CPU busca na TLB pelo número da página virtual que será acessada, caso encontre (Hit) será usada a entrada da tabela de páginas (Page Table Entry ou PTE) armazenada na TLB, caso não encontre (Miss) o tratador de interrupção buscara uma maneira de contornar a falta deste endereço na TLB. Em resumo TLB opera com o princípio similar ao da memória cache, reduzindo acessos a memória principal sempre que possível.
Estrutura
A estrutura básica de uma TLB é formada pelos seguintes campos:
Outros campos podem ser implementados dependendo da arquitetura e da maneira como esta é implementada em hardware e tratada em software.
O TLB incluído no processador Intel i486DX era constituído por uma memória associativa em grupos de 4 vias (4-way) com 32 posições.
Tratamento de interrupção
O tratamento de interrupções (falhas ao buscar entradas nas TLB) é realizado a partir de duas formas diferentes. Dependendo da arquitetura da CPU ocorrerá por hardware ou por software.
Hardware: A CPU automaticamente busca pela tabela de páginas uma entrada (PTE) válida para o endereço virtual acessado. Quando a PTE não for encontrada é gerado um "Page Fault Exception". Uma vez assinalada a exceção, o sistema operacional fica encarregado de resolver a falta da página.
Software: A CPU informa o sistema operacional não ter encontrado a entrada. Então este ativa seu tratador de interrupções (TLB miss handler) e inicia uma busca em software pela página faltando, se uma entrada válida for encontrada, a nova tradução é inserida da TLB, se não for localizada o tratador de interrupção passada a tarefa para o PFH (Page Fault Handler) que deverá buscar pela página correta.
Seja por hardware ou por software, o resultado será uma busca na tabela de páginas e se a entrada for encontrada e validada, a TLB será atualizada com a nova tradução. A maioria das CPUs baseadas em CISC como a x86, realizam tratamento por hardware enquanto tecnologias baseadas em RISC como ARM o fazem por software.
O tratamento por hardware normalmente oferece maior agilidade no tratamento, porém sem a flexibilidade do tratamento por software. Por vezes perde-se performance por má compatibilidade do hardware com as necessidades do sistema operacional. Desta forma já vem sendo feito trabalhos positivos em desenvolver um técnica híbrida para o tratamento de interrupções, como na arquitetura IA-64.
Política de substituição
Todo sistema que opera com o princípio de uma cache de dados busca fornecer o maior número possível de dados, porém, visto o pequeno espaço de armazenamento, definir que dados armazenar torna-se crucial para seu bom desempenho. A TLB precisa armazenar aquelas entradas para tabela de página que estejam sendo mais utilizadas, logo, quando necessário incluir uma nova entrada, é necessário substituir ou sobrescrever a entrada.
Para definir qual entrada será sobrescrita existem diferentes tipos de política de substituição, baseadas em algoritmos como LRU (Least Recently Used), que visa descartar os registros que estão a mais tempo sem serem utilizados ou NRU (Not Recently Used) que descarta registros que tenham ultrapassado um período longo (pré-definido) que não tenham sido reutilizados.
Políticas de substituição, assim como tratamento de interrupção, podem ser implementados em hardware e software. Soluções em hardware tendem a utilizar políticas mais simples como NRU, enquanto políticas de software optam por técnicas como LRU e outros modelos mais complexos.
A política de substituição funciona em alinhamento com o tratamento de interrupção, de modo que se este for empregado em hardware, a política de substituição obrigatoriamente deve ser implementada em hardware. Caso contrário, tratamentos por software, mais flexíveis, permitem políticas tanto de hardware quanto de software.
Troca de contexto
Outra situação que precisa ser prevista em sistemas com a implementação da TLB são as trocas de contexto. Quando mais de um processo estão sendo executados, a CPU deve dividir seus ciclos de clock entre estes, de modo que todos os processos possam ser concluído no menor tempo possível. A alternância destes processos na CPU da-se o nome de troca de contexto.
Esta ação pode gerar algumas situações possivelmente complicadas, como o uso de entradas incorretas e o descarte de entradas úteis. Quando uma troca de contexto é realizada, as entradas armazenadas na TLB são pouco úteis, pois referem-se a endereços de memória usados pelo processo anterior, porém a CPU não "sabe" disso, logo é necessário ter um cuidado especial para evitar que entradas de processos anteriores sejam utilizadas incorretamente.
Para ilustrar como estas situações podem ocorrer observemos o seguinte exemplo. Temos o processo P1 que teve sua 10º página mapeada para o frame 100 da memória principal, então temos uma troca de contexto com o processo P2 e na 10º página deste processo foi mapeado o frame 170. Possuindo ambas entradas na TLB a situação seria a seguinte:
Na situação ilustrada o VPN 10 pode ter seu PFN traduzido tanto para 100 como para 170, logo, sem algum tipo de controle, não há como diferenciar uma entrada da outra. Uma das maneiras para solucionar este problema é a utilização o campo ASID (Address Space Identifier), um pequeno campo geralmente de 8 bits, este campo tem um funcionamento similar ao PID (Process Identifier) utilizado pelo sistema operacional, e tem sua finalidade ajudar o hardware a verificar a quais processos cada entrada pertence.
TLB multinível
Outra característica similar ao que se encontra na memória cache principal da CPU é a implementação de múltiplos níveis de armazenamento. Tipicamente os processadores atuais implementam três níveis em sua memória cache, referenciadas geralmente por L + nível (L1, L2, L3,...) , algumas arquiteturas como Intel Nehalem, implementam TLBs de dois níveis.
No caso da arquitetura Nehalem a organização destes níveis funciona com um nível L1 DTLB (Chamado também de micro-TLB), uma pequena TLB utilizada apenas para leituras, constituído com 16 entradas para páginas de 4KB e 16 entradas para páginas maiores, cada bloco de 16 entradas utilizando uma associação 4-way.
Para o nível 2 ou L2 DTLB, a configuração é de 256 entradas de 4KB e 32 entradas para páginas maiores (2MB/4MB), e ambos os blocos com associação 4-way. Quando ocorre um TLB miss no L1 DTLB, a CPU automaticamente busca na L2 DTLB, antes de iniciar outras técnicas para o tratamento de interrupção pelo TLB miss.
Ver também
Unidade de gerenciamento de memória
Gerenciamento de memória
Palavras, frases e expressões em inglês | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,781 |
Q: print long list in air desktop as3 I have a long scroll-able list that has like over 800 records i wanted to print the whole list with one print job but it prints only the viewable items only here is my function:
function printMovieClip(clip:List) {
var printJob:PrintJob = new PrintJob();
var numPages:int = 0;
var printArea:Rectangle;
var printHeight:Number;
var printY:int = 0;
if ( printJob.start() ) {
/* Resize movie clip to fit within page width */
if (clip.width > printJob.pageWidth) {
clip.width = printJob.pageWidth;
clip.scaleY = clip.scaleX;
}
/* Store reference to print area in a new variable! Will save on scaling calculations later... */
printArea = new Rectangle(0, 0, printJob.pageWidth/clip.scaleX, printJob.pageHeight/clip.scaleY);
numPages = Math.ceil(clip.height / printJob.pageHeight);
/* Add pages to print job */
for (var i:int = 0; i < numPages; i++) {
printJob.addPage(clip, printArea);
printArea.y += printArea.height;
}
/* Send print job to printer */
printJob.send();
/* Delete job from memory */
printJob = null;
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 321 |
\section{Introduction}
Charm particles, quarks and/or mesons, are produced abundantly in double- \cite{Luszczak:2011zp,Maciula:2013kd,Cazaroto:2013fua,vanHameren:2014ava,Maciula:2016wci}
or multiple- \cite{Maciula:2017meb,Cazaroto:2013fua} parton scattering. The cross section for double
charm production was shown to grow considerably with the collision
energy \cite{Luszczak:2011zp}. In our previous papers we have explained that the LHCb double charm data \cite{Aaij:2012dz} cannot be explained without inclusion of double-parton scattering.
Many processes in association with charm quarks or mesons are possible
and can be studied at the LHC. Recently we discussed inclusive production
of single jet associated with $c \bar c$ or charmed mesons \cite{Maciula:2016kkx}.
Quite large cross sections were found there. This reaction was discussed in the presented talk, however will be not
considered in the following due to the limited form of the proceedings contribution.
Here we discuss inclusive production of dijets in association with $c \bar c$ production.
We wish to include both single parton scattering (SPS) and
double parton scattering (DPS) mechanisms and check whether the process can be used to extract the
so-called $\sigma_{eff}$ parameter which governs the strength of double
parton scattering. Therefore we need to focus on how to disantangle single and double parton scattering
contributions for a simultaneous production of $c \bar c$ (or charmed
mesons) and dijets.
\section{Formalism}
\subsection{Single-parton scattering}
Within the $k_T$-factorization approach \cite{kTfactorization} the SPS cross section for
$pp \to c\bar c + \mathrm{2jets}\, X$ reaction can be written as
\begin{equation}
d \sigma_{p p \to c\bar c + \mathrm{2jets}} = \sum_{ij}
\int d x_1 \frac{d^2 k_{1t}}{\pi} d x_2 \frac{d^2 k_{2t}}{\pi}
{\cal F}_{i}(x_1,k_{1t}^2,\mu^2) {\cal F}_{j}(x_2,k_{2t}^2,\mu^2)
d {\hat \sigma}_{ij \to c \bar c + \mathrm{2part.} }
\; .
\label{cs_formula}
\end{equation}
In the formula above ${\cal F}_{i}(x,k_t^2,\mu^2)$ is a unintegrated
parton distribution function (uPDF) for a given type of parton $i = g, u, d, s, \bar u, \bar d, \bar s$. The uPDFs depend on longitudinal momentum fraction $x$, transverse momentum squared $k_t^2$ of the partons entering the hard process,
and in general also on a (factorization) scale of the hard process $\mu^2$.
The elementary cross section in Eq.~(\ref{cs_formula}) can be written
somewhat formally as:
\begin{equation}
d {\hat \sigma}_{ij \to c \bar c + \mathrm{2part.} } =
\prod_{l=1}^{4}
\frac{d^3 p_l}{(2 \pi)^3 2 E_l}
(2 \pi)^4 \delta^{4}(\sum_{l=1}^{4} p_l - k_1 - k_2) \times\frac{1}{\mathrm{flux}} \overline{|{\cal M}_{i^* j^* \to c \bar c + \mathrm{2part.}}(k_{1},k_{2})|^2}
\; ,
\label{elementary_cs}
\end{equation}
where $E_{l}$ and $p_{l}$ are energies and momenta of final state particles. Above only dependence of the matrix element on four-vectors of incident partons $k_1$ and $k_2$ is made explicit. In general all four-momenta associated with partonic legs enter.
The matrix element takes into account that both partons entering the hard
process are off-shell with virtualities $k_1^2 = -k_{1t}^2$ and $k_2^2 = -k_{2t}^2$.
We take into account all 9 channels of the $2 \to 4$ type contributing to the cross section at the parton-level:\\
\begin{center}
$\#1 = g \; g \to g \; g \; c \; \bar{c}$ $\;\;\;\;\;\;$ $\#2 = g \; g \to q \; \bar{q} \; c \; \bar{c}$
$\;\;\;\;\;\;$ $\#3 = g \; q \to g \; q \; c \; \bar{c}$\\
$\#4 = q \; g \to q \; g \; c \; \bar{c}$ $\;\;\;\;\;\;$ $\#5 = q \; \bar{q} \to q' \; \bar{q}' \; c \; \bar{c}$
$\;\;\;\;\;\;$ $\#6 = q \; \bar{q} \to g \; g \; c \; \bar{c}$\\
$\#7 = q \; q \to q \; q \; c \; \bar{c}$ $\;\;\;\;\;\;$ $\#8 = q \; q' \to q \; q' \; c \; \bar{c}$
$\;\;\;\;\;\;$ $\#9 = q \; \bar{q} \to q \; \bar{q} \; c \; \bar{c}$.
\end{center}
The calculation has been performed with the help of KaTie \cite{vanHameren:2016kkz}, which is a complete Monte Carlo parton-level event generator for hadron scattering processes. It can can be applied to any arbitrary processes within the Standard Model, for many final-state particles, and for any initial-state partons on-shell or off-shell.
\subsection{Double-parton scattering}
According to the general form of the multiple-parton scattering theory (see \textit{e.g.} Refs.~\cite{Diehl:2011tt,Diehl:2011yj})
the DPS cross sections can be expressed in terms of the double parton distribution functions (dPDFs).
These objects should fulfill sum rules and take into account all the correlations between the two partons. The theory of dPDFs is well established but still not fully applicable for phenomenological studies.
Instead of the general form, one usually follows the assumption of the factorization of the DPS cross section.
Within this framework, the differential DPS cross section for $pp \to c\bar c + \mathrm{2jets}\; X$ reaction can be expressed as follows:
\begin{equation}
\frac{d\sigma^{DPS}(c \bar c + \mathrm{2jets})}{d\xi_{1}d\xi_{2}} = \sum_{i,j} \; \frac{1}{\sigma_{eff}} \cdot \frac{d\sigma^{SPS}(g g \to c \bar c)}{d\xi_{1}} \! \cdot \! \frac{\sigma^{SPS}(i j \to \mathrm{2jets})}{d\xi_{2}},
\label{basic_formula}
\end{equation}
where $\xi_{1}$ and $\xi_{2}$ stand for generic phase space kinematical variables for the first and second scattering, respectively.
When integrating over kinematical variables one recovers the commonly used pocket-formula:
\begin{equation}
\sigma^{DPS}(c \bar c + \mathrm{2jets}) = \sum_{i,j} \;
\frac{\sigma^{SPS}(g g \to c \bar c) \! \cdot \! \sigma^{SPS}(i j \to \mathrm{2jets})}{\sigma_{eff}}\; .
\label{basic_formula}
\end{equation}
The effective cross section $\sigma_{eff}$ provides a proper normalization of the DPS cross section and can be roughly interpreted
as a measure of the transverse correlation of the two partons inside
the hadrons. The longitudinal parton-parton correlations should become far less
important as the energy of the collision is increased, due to the increase in the parton multiplicity. It is belived that for small-$x$ partons and for low and intermediate scales the possible longitudinal correlations can be safely
neglected (see \textit{e.g.} Ref.~\cite{Gaunt:2009re}).
In this paper we use world-average value of $\sigma_{eff} = 15$ mb provided by
several experiments (see e.g. Refs.~\cite{Astalos:2015ivw,Proceedings:2016tff}).
The cross sections for each step of the DPS mechanism are calculated in the
$k_T$-factorization approach, that is:
\begin{eqnarray}
\frac{d \sigma^{SPS}(p p \to c \bar c \; X_1)}{d y_1 d y_2 d^2 p_{1,t} d^2 p_{2,t}}
&& = \frac{1}{16 \pi^2 {\hat s}^2} \int \frac{d^2 k_{1t}}{\pi} \frac{d^2 k_{2t}}{\pi} \overline{|{\cal M}_{g^{*} g^{*} \rightarrow c \bar{c}}|^2} \nonumber \\
&& \times \;\; \delta^2 \left( \vec{k}_{1t} + \vec{k}_{2t} - \vec{p}_{1t} - \vec{p}_{2t}
\right)
{\cal F}_{g}(x_1,k_{1t}^2,\mu^2) {\cal F}_{g}(x_2,k_{2t}^2,\mu^2),
\nonumber
\end{eqnarray}
\begin{eqnarray}
\frac{d \sigma^{SPS}(p p \to \mathrm{2jets} \; X_2)}{d y_3 d y_4 d^2 p_{3,t} d^2 p_{4,t}}
&& = \frac{1}{16 \pi^2 {\hat s}^2} \sum_{ij} \int \frac{d^2 k_{3t}}{\pi} \frac{d^2 k_{4t}}{\pi} \overline{|{\cal M}_{i^{*} j^{*} \rightarrow \mathrm{2part.}}|^2} \nonumber \\
&&\times \;\; \delta^2 \left( \vec{k}_{3t} + \vec{k}_{4t} - \vec{p}_{3t} - \vec{p}_{4t}
\right)
{\cal F}_{i}(x_3,k_{3t}^2,\mu^2) {\cal F}_{j}(x_4,k_{4t}^2,\mu^2). \nonumber \\
\end{eqnarray}
The numerical calculations for both SPS mechanisms are also done within the KaTie code.
\section{Numerical results}
\subsection{$\bm{D^{0} + \mathrm{2jets}}$}
We start with the predictions for single $D^{0}$ meson production in association with exactly two jets.
In this analysis, the $D^{0}$ meson is required to have $|y^{D^{0}}| < 2.5$ and $p_{T}^{D^{0}} > 3.5$ GeV and the rapidities of both associated jets are $|y^{jet}| < 4.9$, which corresponds to the ATLAS detector acceptance. In Table~\ref{tab:cross sections_D} we collect the corresponding integrated cross sections for inclusive $D^{0}+\mathrm{2 jets}$ production in $pp$-scattering at $\sqrt{s} =$ 13 TeV for different cuts on transverse momenta of the associated jets, specified in the left column. We found large cross sections, of the order of a few, and up to even tens of microbarns, depending on the cuts on transverse momenta of the associated jets. The cross sections are dominated by the DPS mechanism with the relative DPS contribution at the level of $70 - 80 \%$.
\begin{table}[tb]%
\caption{The calculated cross sections in microbarns for inclusive
$D^{0}+\mathrm{2 jets}$ production in $pp$-scattering at $\sqrt{s} =$ 13 TeV for different cuts on transverse momenta of the associated jets. Here, the $D^{0}$ meson is required to have $|y^{D^{0}}| < 2.5$ and $p_{T}^{D^{0}} > 3.5$ GeV and the rapidities of the both associated jets are $|y^{jet}| < 4.9$, which corresponds to the ATLAS detector acceptance. }
\label{tab:cross sections_D}
\centering %
\begin{tabularx}{1.0\linewidth}{c c c c}
\\[-1.ex]
\toprule[0.1em] %
\\[-3.ex]
\multirow{1}{7.5cm}{experimental jet-$p_{T}$ mode} & \multirow{1}{2.cm}{SPS} & \multirow{1}{2.cm}{DPS} & \multirow{1}{2.cm}{$\frac{DPS}{SPS+DPS}$} \\ [+0.1ex]
\bottomrule[0.1em]
\multirow{1}{7.5cm}{both jets $p_{T} > 20$ GeV} & \multirow{1}{2.cm}{3.74} & \multirow{1}{2.cm}{18.49} & \multirow{1}{2.cm}{$\;\;\;\;$83 \%} \\ [-0.2ex]
\multirow{1}{7.5cm}{$p_{T}^{lead} > 35$ GeV, $\; p_{T}^{sub} > 20$ GeV} & \multirow{1}{2.cm}{1.76} & \multirow{1}{2.cm}{4.52} & \multirow{1}{2.cm}{$\;\;\;\;$72 \%} \\ [-0.2ex]
\multirow{1}{7.5cm}{$p_{T}^{lead} > 50$ GeV, $\; p_{T}^{sub} > 35$ GeV} & \multirow{1}{2.cm}{0.43} & \multirow{1}{2.cm}{1.25} & \multirow{1}{2.cm}{$\;\;\;\;$74 \%} \\ [-0.2ex]
\hline
\bottomrule[0.1em]
\end{tabularx}
\end{table}
In Fig.~\ref{fig:ptD} we show the differential cross section as a function of transverse momenta of the $D^{0}$ meson (left panel) and as a function of the azimuthal angle $\varphi_{D^{0}\mathrm{\textit{-jet}}}$ between the $D^{0}$ meson ($\overline{D^{0}}$ antimeson) and the leading jet (right panel). The DPS (dashed line) and the SPS (dotted line) components are shown separately together with their sum (solid line). We observe that in the region of $D^{0}$ meson transverse momenta $p_{T} < 10$ GeV the DPS mechanism significantly dominates over the SPS one.
We also see that the presence and the dominant role of the DPS component leads to a significant enhancement of the cross section and to a visible decorrelation of the azimuthal distribution in contrast to the pure SPS-based predictions.
In the left panel we plot in addition the typical uncertainty bands of the pQCD calculations for both, the SPS and the DPS components. The shaded bands represent the uncertainties related to the choice of renormalization/factorization scales and charm quark mass, summed in quadrature. We vary the charm quark mass $m_{c} = 1.5 \pm 0.25$ GeV and the scales $\mu^{2}$ by a factor 2, which is a rather standard procedure. The calculated uncertainties are about $\pm 45 \%$ for the SPS and $\pm 65 \%$ for the DPS mechanism. These levels of uncertainty also apply for the integrated cross sections.
\begin{figure}[!h]
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{dsig_dpT_kTcut_both20_D0_ATLAS_uncert.eps}}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{dsig_dphid_kTcut_both20_D0_ATLAS.eps}}
\end{minipage}
\caption{
\small The transverse momentum (left) and azimuthal angle $\varphi_{D^{0}\mathrm{\textit{-jet}}}$ (right) distribution of the $D^{0}$ meson for SPS (dotted) and DPS (dashed) mechanisms for the ATLAS detector acceptance. The solid line represents a sum of the two components. Details are specified in the figure. For example in the left panel we show explicitly theoretical uncertainties, see discussion in the text.
}
\label{fig:ptD}
\end{figure}
\subsection{$\bm{D^{0}\bar{D^{0}} + \mathrm{2jets}}$}
Now we also consider the case of production of the $D^{0}\overline{D^{0}}$-pair in association with two jets.
Both, $D^{0}$-meson and $\overline{D^{0}}$-antimeson are required to enter the ATLAS detector acceptance.
The corresponding theoretical cross sections are collected in Table~\ref{tab:cross sections_DD}. Here, the predicted cross sections for $D^{0}\overline{D^{0}}+\mathrm{2 jets}$ are slightly
smaller than in the case of $D^{0}+\mathrm{2 jets}$ production (see Table~\ref{tab:cross sections_D}) but still large (in the best scenario, of the order of a few microbarns).
Also the relative DPS contribution is somewhat reduced and varies at the level of $50 - 70 \%$.
\begin{table}[tb]%
\caption{The same as in Table~\ref{tab:cross sections_D} but for inclusive $D^{0}\overline{D^{0}}+\mathrm{2 jets}$ production. Here both, $D^{0}$ meson and $\overline{D^{0}}$ antimeson are required to enter the ATLAS detector acceptance.}
\label{tab:cross sections_DD}
\centering %
\begin{tabularx}{1.\linewidth}{c c c c}
\\[-1.ex]
\toprule[0.1em] %
\\[-3.ex]
\multirow{1}{7.5cm}{experimental jet-$p_{T}$ mode} & \multirow{1}{2.cm}{SPS} & \multirow{1}{2.cm}{DPS} & \multirow{1}{2.cm}{$\frac{DPS}{SPS+DPS}$} \\ [+0.1ex]
\bottomrule[0.1em]
\multirow{1}{7.5cm}{both jets $p_{T} > 20$ GeV} & \multirow{1}{2.cm}{1.10} & \multirow{1}{2.cm}{2.35} & \multirow{1}{2.cm}{$\;\;\;\;$68 \%} \\ [-0.2ex]
\multirow{1}{7.5cm}{$p_{T}^{lead} > 35$ GeV, $\; p_{T}^{sub} > 20$ GeV} & \multirow{1}{2.cm}{0.55} & \multirow{1}{2.cm}{0.58} & \multirow{1}{2.cm}{$\;\;\;\;$51 \%} \\ [-0.2ex]
\multirow{1}{7.5cm}{$p_{T}^{lead} > 50$ GeV, $\; p_{T}^{sub} > 35$ GeV} & \multirow{1}{2.cm}{0.15} & \multirow{1}{2.cm}{0.14} & \multirow{1}{2.cm}{$\;\;\;\;$52 \%} \\ [-0.2ex]
\hline
\bottomrule[0.1em]
\end{tabularx}
\end{table}
In the case of the $D^{0}\overline{D^{0}}+\mathrm{2 jets}$ final state we also find a very interesting correlation observable that may be useful to distinguish between the DPS and SPS mechanisms.
Figure~\ref{fig:phiDDbar} presents the distributions in azimuthal angle $\varphi_{D^{0}\overline{D^{0}}}$
between the $D^{0}$ meson and $\overline{D^{0}}$ antimeson in the case of $D^{0}\overline{D^{0}}+\mathrm{2 jets}$ production.
One can observe an evident enhancement of the cross section in the region of $\varphi_{D^{0}\overline{D^{0}}} > \frac{\pi}{2}$ caused by the presence of the DPS mechanism.
\begin{figure}[!h]
\begin{center}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{dsig_dphi34_kTcut_both20_D0D0_ATLAS.eps}}
\end{minipage}
\caption{
\small The azimuthal angle $\varphi_{D^{0}\overline{D^{0}}}$ distribution for SPS (dotted) and DPS (dashed) mechanisms for the ATLAS detector acceptance. The solid line represents a sum of the two components. Details are specified in the figure.}
\label{fig:phiDDbar}
\end{center}
\end{figure}
More details can be found in our original paper \cite{Maciula:2017egq}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 487 |
Q: Convert database from Antislaed to Drupal whether is there a converter database from AntiSlaed to Drupal? And if not, how quickly convert the base from Antislaed to Drupal?
Thank you very much. Sorry for bad English and noob-question. (:
A: If this software uses MySQL or some database that can be converted to MySQL tables then you should be able to use drupal's Table Wizard and Migrate modules to import your content into drupal.
This tutorial is a good place to get started.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,078 |
Lodge L24 – Gull is lodge located on the South Terrace. With amazing views across the neighbouring fields and out to sea, particularly from the decking. Bedrooms: 1 x Double and 2 x Twin. Parking is on the gravel drive way right outside with a ramp access to the front door. Recommend for anyone with mobility issues. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,253 |
var expect = require('./spec-helper').expect;
var minim = require('../lib/minim');
var Namespace = require('../lib/namespace');
describe('Minim namespace', function() {
var namespace = new Namespace();
var ArrayElement, NullElement, ObjectElement, StringElement;
beforeEach(function() {
namespace.elementMap = {};
namespace.elementDetection = [];
namespace.useDefault();
ArrayElement = namespace.getElementClass('array');
NullElement = namespace.getElementClass('null');
ObjectElement = namespace.getElementClass('object');
StringElement = namespace.getElementClass('string');
});
it('is exposed on the module', function() {
expect(minim.Namespace).to.equal(Namespace);
});
it('gets returned from minim.namespace()', function() {
expect(minim.namespace()).to.be.an.instanceof(Namespace);
});
describe('default elements', function() {
it('are present by default', function() {
expect(namespace.elementMap).to.not.be.empty();
});
it('can be created empty', function() {
expect((new Namespace({noDefault: true})).elementMap).to.deep.equal({});
});
it('can be added after instantiation', function() {
var testnamespace = new Namespace({noDefault: true});
testnamespace.useDefault();
expect(testnamespace.elementMap).to.not.be.empty();
});
});
describe('#use', function() {
it('can load a plugin module', function() {
var plugin = {
namespace: function(options) {
var base = options.base;
// Register a new element
base.register('null2', NullElement);
}
};
namespace.use(plugin);
expect(namespace.elementMap).to.have.property('null2', NullElement);
});
});
describe('#register', function() {
it('should add to the element map', function() {
namespace.register('test', ObjectElement);
expect(namespace.elementMap.test).to.equal(ObjectElement);
});
});
describe('#unregister', function() {
it('should remove from the element map', function() {
namespace.unregister('test');
expect(namespace.elementMap).to.not.have.key('test');
});
});
describe('#detect', function() {
var test = function() { return true; }
it('should prepend by default', function() {
namespace.elementDetection = [[test, NullElement]];
namespace.detect(test, StringElement);
expect(namespace.elementDetection[0][1]).to.equal(StringElement);
});
it('should be able to append', function() {
namespace.elementDetection = [[test, NullElement]];
namespace.detect(test, ObjectElement, false);
expect(namespace.elementDetection[1][1]).to.equal(ObjectElement);
});
});
describe('#toElement', function() {
it('should handle values that are ElementClass subclass instances', function() {
var myElement = new StringElement();
var converted = namespace.toElement(myElement);
expect(converted).to.equal(myElement);
});
it('should allow for roundtrip conversions for values', function() {
namespace.register('foo', StringElement);
// Full version
var fullVersion = namespace.fromRefract({ element: 'foo', meta: {}, attributes: {}, content: 'test' }).toRefract();
expect(fullVersion).to.deep.equal({ element: 'foo', meta: {}, attributes: {}, content: 'test' });
// Compact version
var compactValue = namespace.fromCompactRefract(['foo', {}, {}, 'test']).toCompactRefract();
expect(compactValue).to.deep.equal(['foo', {}, {}, 'test']);
});
it('should allow for roundtrip conversions for collection elements', function() {
namespace.register('foo', ArrayElement);
var fullRefractSample = {
element: 'foo',
meta: {},
attributes: {},
content: [
{
element: 'string',
meta: {},
attributes: {},
content: 'bar'
}
]
}
var compactRefractSample = [
'foo', {}, {}, [
['string', {}, {}, 'bar']
]
]
// Full version
var fullVersion = namespace.fromRefract(fullRefractSample).toRefract();
expect(fullVersion).to.deep.equal(fullRefractSample);
// Compact version
var compactValue = namespace.fromCompactRefract(compactRefractSample).toCompactRefract();
expect(compactValue).to.deep.equal(compactRefractSample);
});
});
describe('#getElementClass', function() {
it('should return ElementClass for unknown elements', function() {
expect(namespace.getElementClass('unknown')).to.equal(namespace.BaseElement);
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,029 |
{"url":"http:\/\/openstudy.com\/updates\/560de3c9e4b0331c08e8afe0","text":"## anonymous one year ago Find the x-intercepts (if any) for the graph of the quadratic function. 6x2 + 12x + 5 = 0 Give your answers in exact form. Show your work.\n\n1. anonymous\n\n@Hero @dan815\n\n2. anonymous\n\n@Jhannybean @satellite73 @Loser66 @freckles @EclipsedStar @King.Void. @sleepyjess can someone help me please?\n\n3. sleepyjess\n\nDo you have an idea on how to start?\n\n4. anonymous\n\nno not at all\n\n5. sleepyjess\n\nDo you know what the quadratic formula is?\n\n6. anonymous\n\nwhat you mean?\n\n7. sleepyjess\n\n$$x=\\dfrac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$\n\n8. sleepyjess\n\nThat is the quadratic formula\n\n9. sleepyjess\n\nIt may look intimidating, but it's not too bad once you know how to work it :)\n\n10. anonymous\n\nOkay how do I work it lol..\n\n11. sleepyjess\n\nA quadratic equation is ALWAYS \"Ax^2 + Bx + C = y\" To use the quadratic formula, you will want to substitute 0 for y, which has already been done. Can you figure out what A, B, and C are in the quadratic equation you have?\n\n12. anonymous\n\nA= 6x^2 B= 12x C= 5\n\n13. sleepyjess\n\nClose, but we just want 6, 12 and 5\n\n14. anonymous\n\nOhh lol\n\n15. sleepyjess\n\nSo now we just substitute in 6 for a, 12 for b, and 5 for c then work out the equation\n\n16. anonymous\n\nI got down to $x= \\frac{ -12\\pm \\sqrt{24} }{ 12 }$ now what do I do?\n\n17. anonymous\n\n@sleepyjess\n\n18. carolinar7\n\nNow simplify what you have there\n\n19. anonymous\n\nHow?\n\n20. carolinar7\n\n$x=\\frac{ -12-+4\\sqrt{6} }{ 12 }$\n\n21. carolinar7\n\nCan you simplify any further?\n\n22. anonymous\n\nWhy 4 sqrt7 instead of sqrt24=4.89....?\n\n23. anonymous\n\nI meant sqrt6 sorry\n\n24. carolinar7\n\nSorry let me correct myself it is 2sqrt6\n\n25. carolinar7\n\nIs this Alg. 1\n\n26. anonymous\n\nHow!? And no its pre-calculus\n\n27. carolinar7\n\nwhat grade ar you in\n\n28. anonymous\n\n29. carolinar7\n\nme too im in pre-calc in 9th grade\n\n30. sleepyjess\n\noh goodness, I forgot about the square root simplifying >_<\n\n31. anonymous\n\nHow do I do that?\n\n32. sleepyjess\n\n@carolinar7 simplified it for you\n\n33. carolinar7\n\nwhat do you mean you simplified it for me\n\n34. sleepyjess\n\nyou simplified the square root already\n\n35. anonymous\n\n2 sqrt6?\n\n36. carolinar7\n\nIt is not completely simplified though, you can simplify further too\n\n37. carolinar7\n\n$x=\\frac{ -6+-\\sqrt{6} }{ 6 }$\n\n38. anonymous\n\nso where do I go from $x= \\frac{ -12 \\pm 2\\sqrt{6} }{ 12 }$\n\n39. anonymous\n\nOhhh how did you get that?\n\n40. carolinar7\n\nYou simplify it and divide by 2 because that is what they share in common\n\n41. anonymous\n\nOhhh okay soo now I do -6\/6?\n\n42. carolinar7\n\n|dw:1443753373282:dw|\n\n43. carolinar7\n\nno you cant simplify any further\n\n44. anonymous\n\nso thats the exact form?\n\n45. carolinar7\n\nWait let me do it on a paper\n\n46. carolinar7\n\nYes that is the most you can simplify it\n\n47. anonymous\n\nOkay thank you!\n\n48. carolinar7\n\nyup","date":"2017-01-23 04:47:07","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\": 0.6894639730453491, \"perplexity\": 6248.967966449308}, \"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-2017-04\/segments\/1484560282110.46\/warc\/CC-MAIN-20170116095122-00166-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
En tipografía, una columna es uno o más bloques verticales de contenidos posicionados en una página, separados por medianiles (espacios en blanco verticales) o corondeles (líneas delgadas, en este caso verticales). El uso más común de las columnas es para romper los grandes cuerpos de texto que no caben en un solo bloque de texto en una página. Adicionalmente, las columnas se utilizan para mejorar la composición y la legibilidad de la página. Los diarios usan con frecuencia diseños complejos de muchas columnas hasta fragmentar diferentes artículos y cuerpos más largos de textos dentro de un artículo. De un modo más general, la columna también puede referirse a las delineaciones verticales creadas por un sistema de retícula tipográfica que pueden ser posicionados el tipo y la imagen. En maquetación, se conocen como márgenes al espacio en blanco en el exterior de la página (limitando la primera y última columna); el vacío entre dos páginas enfrentadas también es considerado como un medianil, puesto que hay columnas en ambos lados. (A los márgenes interiores o de lomo puede llamárseles también «medianiles», pero los márgenes exteriores o de corte y los de cabecera y pie no son medianiles.)
Estilo tipográfico
Tradicionalmente, los tipógrafos llamaban medida al ancho de columna. Para obtener una mejor legibilidad, los manuales de tipografía sugieren que las columnas deben ser lo suficientemente anchas como para contener aproximadamente 60 caracteres por línea. Una fórmula sugiere multiplicar el tamaño en puntos de la fuente por 2 para alcanzar el ancho de una columna que debe estar en picas, lo cual resulta en un ancho de columna de 24 cuadratines. Estas pautas generalmente favorecen múltiples columnas estrechas frente a una columna ancha única. Históricamente, los libros que contienen predominantemente texto suelen tener alrededor de 40 líneas por columna. Sin embargo, esta regla no se aplica a textos complejos que contienen varias imágenes o ilustraciones, notas al pie, comentarios, encabezados, folios y etiquetas.
El contraste de columna se refiere al color general o luminosidad de gris establecida por columna, y puede ajustarse de diversas maneras. Una es ajustar la relación entre la anchura y la altura de la columna. Otra manera es mediante ajustes a la familia tipográfica, escogiendo una fuente concreta, valor, estilo, medida e interlineado. El contraste de columna puede utilizarse para establecer jerarquía, para balancear la composición de página, y para activar visualmente áreas de la página.
Diseño web
En diseño web, las columnas se utilizan a menudo para separar el contenido primario del secundario y terciario. Por ejemplo, una disposición común de dos columnas puede incluir una columna izquierda con enlaces de navegación y una columna derecha para texto de cuerpo. Un método para crear columnas para la web es para colocar texto dentro de un elemento de tabla HTML, a menudo con el borde puesto a cero. Aun así, para algunos este método está considerado obsoleto. Otro método incluye el uso de CSS para flotar o posicionar el texto correspondiente. Estos métodos no eran tan sencillos como el uso de tablas HTML, lo que hizo que un diseño de tres columnas sin tablas fuera una especie de santo grial una vez que estas técnicas fueron descubiertas a principios de los años 2000. Los niveles más recientes de CSS han abordado los comportamientos de columna, y el soporte de navegador de la web para estos comportamientos continúa mejorando.
Notas
Referencias
Carter, Rob. Day, Ben. Meggs, Philip. Typographic Design: Form and Communication 2.ª ed. John Wiley & Sons: 1993. pp. 51–53, 90-91.
Romano, Frank J. The TypeEncyclopedia. R.R. Bowker Company: 1984. pp. 86–86.
Enlaces externos
Tipografía
Diseño de página | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,977 |
Q: Estilização de HR Tenho uma linha em um layout que estou fazendo, e não queria criar uma div só pra ela.
Gostaria de continuar a hierarquia que criei. Então decidi usar o <hr />
Exemplo:
.empresa hr {
width: 100%;
height: 1px;
background-color: #dddddd;
}
O que acontece, é que, olhando com zoom, tenho a impressão de que a altura da linha não fica em 1px, tenho a impressão de que já uma borda.
Isso é normal? O <hr /> ainda é usual?
A: A hr também possui bordas. Tente fazer o seguinte, eliminar as boras laterais e a borda inferior, por exemplo.
.empresa hr {
width: 100%;
height: 1px;
border: 0px;
border-top: 1px solid red;
background-color: #dddddd;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,484 |
\section{Introduction}
\label{intro}
The Evershed flow \citep[EF;][]{1909MNRAS..69..454E} is a distinct property
of sunspot penumbrae which exemplifies their filamentary structure
\citep[][and references therein]{2003A&ARv..11..153S}. In the inner
penumbra the EF starts as upflows \citep{2006A&A...453.1117B, 2006ApJ...646..593R,
2009A&A...508.1453F} that turns into downflows in the mid and outer penumbra
\citep{1997Natur.389...47W, 1999A&A...349L..37S, 2003A&A...410..695M, 2004A&A...427..319B}.
In addition to the EF, other types of mass motions exist in the
penumbra, as reported recently by \citet{2010A&A...524A..20K} using {\em
Hinode} observations. They detected small-scale downflowing patches
with velocities of $\sim$1~km~s$^{-1}$ which have the same polarity
as the parent sunspot. Some of them also appear to be co-spatial with
chromospheric brightenings. Based on their physical properties,
\citet{2010A&A...524A..20K} inferred that these weak downflows are different
from the Evershed flow returning to the photosphere, which sometimes
happens well within the penumbra
\citep{2004A&A...427..319B,2007ApJ...668L..91B,2008A&A...481L..21S}.
\begin{figure}[t]
\centering
\includegraphics[width=11cm,angle=90,bb=-7 100 595 792]{./louis_fig1.eps}
\caption{Continuum image of NOAA AR 10923 at 630~nm. The white dashed
square represents the small region chosen for analysis and whose
magnified image is shown in the inset. The white arrow points to disk
center.}
\label{cont_image}
\end{figure}
\citet{2011ApJ...727...49L} observed a new type of downflows which are supersonic and
occur at or near the umbra-penumbra boundary of sunspots. These
downflowing patches are conspicuously large in size with areas ranging
from 1.6--6 arcsec$^2$. They have the same polarity as the sunspot and
occur along bright penumbral filaments. Their properties are different
from those of the Evershed flow and the downflows reported by
\citet{2010A&A...524A..20K}, which indicates a different physical
origin. The strong downflows possibly represent energetic and dynamic
processes occurring in the inner penumbra which also affect the
chromosphere. In this paper we restrict our discussion to the
supersonic downflows observed in NOAA AR 10923 and briefly describe
the photospheric and chromospheric activities associated with them.
\section{Observations}
\label{data}
High resolution spectro-polarimetric observations of NOAA AR 10923
were carried out using the Solar Optical Telescope
\citep[SOT;][]{2008SoPh..249..167T} on board {\em Hinode} \citep{2007SoPh..243....3K}
on November 10, 2006 when the sunspot was located at a heliocentric
angle of $50\deg$ (Figure~\ref{cont_image}). The AR was
mapped by the {\em Hinode} spectro-polarimeter
\citep[SP;][]{2001ASPC..236...33L,2008SoPh..249..233I} from 16:01 to 17:25~UT in the
normal map mode with an exposure time of 4.8~s and a pixel size of 0\farcs16.
The four Stokes profiles of the neutral iron lines at 630~nm were recorded
with a spectral sampling of 21.55~m\AA\/ at each slit position. In addition to the
Stokes spectra, G-band and \ion{Ca}{ii} H filtergrams acquired by he Broadband Filter
Imager (BFI) close to the SP scans were also employed. The filtergrams had a
sampling of 0\farcs055 with a cadence of 30~s. These data were
recorded from 13:00 to 14:00~UT.
\begin{figure}[t]
\centering
\includegraphics[width=6.5cm,angle=90,bb=100 00 526 842]{./louis_fig2.eps}
\caption{{\em Left:} LOS velocity derived from SIR combining one- and
two-component inversions. {\em Right:} The fill fraction of the fast
component. The black contour depicts the umbra-penumbra boundary. Both
images have been scaled as shown in their respective color bars. The
black cross corresponds to the pixel exhibiting supersonic
velocity. The Stokes profiles emerging from this pixel are shown in
Figure~\ref{stokes}.}
\label{velo}
\end{figure}
\section{Results}
\label{res}
\subsection{Supersonic Downflows from SIR Inversions}
\label{downflows}
The detection of supersonic downflows was carried out by constructing
red and blue wing magnetograms at $\pm$34.4 pm from the line center of
the \ion{Fe}{i} 630.25~nm line as described by \citet{2011ApJ...727...49L}. The
far wing magnetograms are useful to detect pixels with strong
downflows but do not yield the magnitude of the velocities present in
those locations. In order to determine the velocities, the observed
Stokes profiles were subject to an inversion using the SIR code
\citep[Stokes Inversion based on Response Functions;][]{1992ApJ...398..375R}.
Two sets of inversions were carried out. In the first run, a single
magnetic component was assumed in each pixel with the vector magnetic
field (field strength, inclination and azimuth) and LOS velocity
remaining constant with height. The resulting LOS velocity map is
shown in Fig.~2 of \citet{2011ApJ...727...49L}. Pixels exhibiting velocities
greater than 2~km~s$^{-1}$ were then selected and subject to a second
set of inversions in which two magnetic components were assumed to
co-exist in a single resolution element, both of which having
height-independent physical parameters. The one and two component SIR
inversions also retrieved height-independent micro- and
macro-turbulent velocities as well as the fraction of stray light in
each pixel. The left panel of Fig.~\ref{velo} shows the LOS velocity
combining the two inversion sets, while the fill fraction of the fast
component is depicted on the right.
\begin{figure}[t]
\centering
\vspace*{-.5em}
\includegraphics[width=6.5cm,angle=90,bb=-11 -22 621 842]{./louis_fig3.eps}
\includegraphics[width=6.5cm,angle=90,bb=-11 -15 621 850]{./louis_fig4.eps}
\vspace*{-.5em}
\caption{{\em First and second rows:} Observed (black) and best-fit (red)
Stokes profiles of the pixel marked in Fig.~\ref{velo} using a
two-component model. {\em Third and fourth rows:} Height-independent
stratifications of the physical parameters corresponding to the two
components (solid and dashed lines, respectively).}
\label{stokes}
\end{figure}
Figure~\ref{velo} reveals the existence of supersonic downflows of up
to $\sim$8~km~s$^{-1}$ in AR 10923 located nearly $3\arcsec$ from the
umbra-penumbra boundary. To the best of our knowledge, these are the
largest downflows ever detected in the inner penumbra close to the
umbral boundary. The right panel of the figure indicates that a large
fraction of the resolution element is dominated by the stronger
downflowing component. We estimate the typical size of the downflowing
patches to be $\sim$1.6 arcsec$^2$. The strong downflowing zones are
surrounded by upflows measuring 2 km~s$^{-1}$ which can be identified
with the Evershed flow.
The Stokes $V$ profiles emerging from the downflowing regions exhibit
a satellite in the red lobe while the Stokes $I$ profiles have highly
inclined red wings, as illustrated in Fig.~\ref{stokes} for the pixel
marked with a cross in Figure~\ref{velo}. The above spectral
characteristics are reproduced satisfactorily by the two-component
inversions. The two magnetic components could either reside
side-by-side in the same resolution element or could be stacked one on
top of the other. While the exact configuration remains uncertain,
supersonic velocities exist in the presence of very strong magnetic
fields as indicated by the third row of Figure~\ref{stokes}. In
addition, the polarity of the strong downflowing component is the same
as that of the parent sunspot, which rules out the possibility of them
being Evershed flows returning to the solar surface. The strong fields
exceeding 2~kG should also inhibit convection, suggesting that the
downflows are likely to be caused by an alternative mechanism.
\subsection{Photospheric and Chromospheric Brightenings}
\label{activity}
We now turn our attention to the photospheric and chromospheric
brightenings associated with the supersonic downflows. These
brightness enhancements are observed to lie in close proximity to the
downflowing patches and can have intensities comparable to the quiet
Sun. The penumbral filament near one of the downflowing regions has an
intensity of 0.9$I_{\textrm{\tiny{QS}}}$ in the continuum at 630~nm
(see inset of Figure~\ref{cont_image}). Higher up in the chromosphere,
these brightenings are $\sim$77\% more intense than the penumbral
microjets (MJs), which appear to be in the decay phase
\citep{2008ApJ...676.1356R}. While the brightness enhancements observed
near the supersonic downflows appear as isolated blobs on the filaments, MJs
are oriented nearly perpendicular to the filament (see Fig.~4 of
\citealt{2008ApJ...676.1356R}).
\begin{figure}[t]
\vspace{-15pt}
\centerline{
\includegraphics[width=0.67\textwidth,angle=0]{./louis_fig5.eps}
}
\vspace{-15pt}
\caption{G-band ({\it{bottom}}) and Ca ({\it{top}}) event maps depicting
locations with intensity close to the quiet Sun photosphere. Blue
contours of LOS velocity greater than 2 km~s$^{-1}$ have been overlaid
on the filtergrams. The black contour corresponds to the continuum
intensity at 630 nm. The images have been scaled as shown by the
horizontal color bar.}
\label{bright}
\end{figure}
To determine if the proximity of the enhancements to the downflows
endures with time, event maps were constructed in the photosphere as
well as the chromosphere using the G-band and Ca filtergram time
sequence. The procedure has been described in \citet{2011ApJ...727...49L} with
threshold values of 0.85 and 0.9 being employed for the photosphere
and chromosphere respectively. The resulting event maps are shown in
Figure~\ref{bright}.
At both heights we find a large number of events concentrated near the
downflows resembling blobs that were seen in the individual
filtergrams. While the brightenings in the photosphere persist for
the entire 1 hour sequence, in the chromosphere they last for only
about one third of the time. More importantly, the enhancements appear
very similar in shape and are nearly co-spatial. The use of a large
threshold in the chromospheric event map removes all signatures of
the relatively weaker and transient MJs.
\section{Discussion}
\label{orient}
The downflows that are associated with the EF can sometimes be supersonic
in the outer penumbra \citep{2001ApJ...549L.139D,2004A&A...427..319B} or even beyond the
sunspot boundary \citep{2009ApJ...701L..79M}. Such a configuration represents mass flux
returning to the photosphere and has a polarity opposite to that of the sunspot.
The supersonic downflows we have observed have the same polarity as the parent sunspot
and so they cannot be related to the Evershed downflows. One could assume that
these strong downflows are the photospheric counterpart of some kind of inverse
Evershed flow seen in the chromosphere. However, it is not clear how
such a chromospheric phenomenon could produce supersonic downflows close to the
umbra-penumbra boundary in the photosphere.
The orientation of the filaments P1 and P2, bifurcating at the strong
downflowing patch (Fig.~\ref{unsharp}), resembles the
post-reconnection configuration illustrated in Fig.~5c of
\citet{2008ApJ...686.1404R}, suggesting that the origin of the downflows is
the slingshot effect associated with the reconnection of the
filaments. The bisecting angles shown by the solid green lines were
estimated to be $51\deg$ and $46\deg$.
According to \citet{2008ApJ...686.1404R}, the unwinding of filaments in a
cork screw fashion can lead to reconnection, transient brightenings
and twists in the penumbral filaments. This model was proposed as a
possible mechanism for producing penumbral MJs. \citet{2010ApJ...715L..40M} investigated
the above scenario using numerical simulations and concluded that
MJs occur in the intermediate region between nearly horizontal flux tubes and the
relatively vertical background field of the penumbra. In this model,
only parts and not the entire penumbral filament participate in the reconnection process.
Slingshot reconnection may be a possible mechanism for producing the supersonic
downflows and the photospheric as well as chromospheric brightenings,
although there is no strict one-to-one correspondence between the two phenomena.
The above process has to be different from the one producing MJs since
their intensities and lifetimes are much smaller than the
events described in this work.
\begin{figure}[t]
\centering
\vspace{-2pt}
\hspace{20pt}
\includegraphics[width=0.55\textwidth,angle=90]{./louis_fig6.eps}
\vspace{-25pt}
\caption{Continuum image that has been unsharp masked using a
$3 \times 3$ pixel boxcar. The red arrow indicates the location where
the filaments P1 and P2 appear to intersect each other. The solid
green lines refer to the bisecting angles between the filaments P1 and
P2. The blue contour has been drawn for LOS velocities greater than 2
km~s$^{-1}$.}
\label{unsharp}
\end{figure}
\section{Summary}
\label{summary}
High resolution spectro-polarimetric observations of NOAA AR 10923
taken with {\em Hinode} reveals supersonic downflows near the
umbra-penumbra boundary. The downflows are observed in large patches
having an area of 1.6 arcsec$^2$. Using a two-component model
atmosphere the SIR code retrieves supersonic values of
8~km~s$^{-1}$. These are the largest velocities ever detected at those
locations in a sunspot. Frequent occurrences of strong downflows at
the border of umbrae without penumbrae have been reported by
\citet{2008ApJ...680.1467S}.
The strong velocities are associated with 2 kG magnetic fields which
have the same polarity as the sunspot. This would imply that the
downflows are not related to the Evershed flow, although the latter
could be separately present at those locations. We find intense and
long lived chromospheric brightenings near the strong photospheric
downflows extending over an area of 1--2 arcsec$^2$. Furthermore,
photospheric brightenings nearly as intense as the quiet Sun are also
present in the downflowing regions or close to them. These downflows
may have lifetimes of up to 14 hours (or more) as found from several
consecutive {\em Hinode}/SP scans
\citep{2011ApJ...727...49L}.
The penumbral filaments in the vicinity of the strong downflows appear
to be twisted in the manner described by \citet{2008ApJ...676.1356R} that
would arise from a reconnection process. Such a process could produce
the transient penumbral microjets. These events are ubiquitous in the
penumbra and the photospheric downflows associated with them are
typically 1 km~s$^{-1}$, as reported by \citet{2010A&A...524A..20K}. The
chromospheric brightenings described in Sect.~\ref{activity} however,
are stronger, bigger, and longer-lived than the microjets. The
supersonic downflows that we have observed are an entirely new
phenomenon. They are possibly driven by dynamic and very energetic
processes occurring in the inner penumbra which produce highly intense
and long duration brightenings in the photosphere as well as in the
chromosphere. A suitable theory is yet to be formulated for explaining
these events and will be a challenge for future numerical models of
sunspots.
\acknowledgements
We sincerely thank the {\em Hinode} team for providing the high
resolution data. {\em Hinode} is a Japanese mission developed and
launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC
(UK) as international partners. It is operated by these agencies in
co-operation with ESA and NSC (Norway). We thank the organizers for a
wonderful meeting and for their warm hospitality.
| {
"redpajama_set_name": "RedPajamaArXiv"
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\section{Introduction}
Collisions of highly charged ions provide a unique tool
for tests of relativistic and quantum
electrodynamics (QED) effects
in the scattering theory~\cite{Eichler_95,Shabaev_02,Eichler_07}.
Investigations of such
processes can also give an access
to QED in supercritical
fields, provided the total charge of the colliding nuclei is
larger than the critical one, $Z_c=173$ (see, e.g.,
Ref.~\cite{Greiner_85} and references therein).
One of the most attractive way for indirect observation of the
supercritical field created in the collision
is to investigate the
dynamics of inner-shell electrons that can be
rather sensitive to the collision parameters.
The most favorable conditions for studying the electron dynamics
in the supercritical field regime
correspond to the projectile energy of about
the Coulomb barrier \cite{mul_76}. In case of U$-$U collision this requires
the energy of about $6$~MeV/u that means a low-energy collision.
One of the key processes in such collisions is the charge transfer of
electrons (see, e.g., Ref.~\cite{Bransden_92} and references therein).
A systematic approach to relativistic calculations of the charge-transfer
and electron-excitation
probabilities in low-energy heavy-ion collisions was developed
in our previous paper~\cite{Tupitsyn_10}, where the consideration was
restricted to collisions of H-like ions with bare nuclei.
Since the experimental study of such collisions for high-$Z$ systems
is presently
rather problematic, an extention of the method to collisions
of highly charged ions with neutral atoms, that can be
studied in experiments with the current GSI and future FAIR
facilities~\cite{Hagmann_10, Hagmann_11}, is needed.
In this paper
we present the desired extention and perform calculations
for low-energy ion-atom collisions.
To examine the approach we calculate the K-K charge transfer and K-vacancy
production probabilities for low-energy collision of H-like F (F$^{8+}$)
and neutral Ne, the process which has been investigated
both experimentally~\cite{Hagmann_87}
and theoretically~\cite{Fritsch_85,Toepfer_87,Thies_89}.
The calculations are performed
at the F$^{8+}$ projectile energies $130$~keV/u and $230$~keV/u.
We also evaluate the probabilities of
the K-K charge transfer and K-vacancy production
in the Xe~--~Xe$^{53+}(1s)$ collision at
the projectile
energy of $3.6$~MeV/u. The latter processes are planned to be
studied in the nearest future in
experiments at GSI ~\cite{Hagmann_10, Hagmann_11}.
The paper is organized as follows.
In the section~\ref{subsec:Kohn-Sham} we describe the time-dependent
one-electron equation in so-called active electron approximation~\cite{Lin_81}
and the method for constructing the local Kohn-Sham
potential induced by the passive electrons. The wave function of
the active electron is expanded in terms of the Dirac-Fock and Dirac-Fock-Sturm
basis functions, which are central-field 4-component Dirac bispinors
centered at the ions.
The two-center relativistic Kohn-Sham equation in the finite basis set
is briefly discussed in the section~\ref{subsec:two-center}.
The basis functions are obtained by solving numerically the
atomic Dirac-Fock and Dirac-Fock-Sturm equations in an external field with
a special choice of the weight function, as it was proposed in
Refs.~\cite{Tupitsyn_03, Tupitsyn_05}. The external potential is spherically
symmetric Coulomb-Hartree potential of the other ion (atom) taken in the
monopole approximation. The basis set constructed in this way
and the related calculation procedures
are described in section~\ref{subsec:basis}.
Basic formulas for the K-vacancy production probability
are given in section~\ref{subsec:prob}.
In section~\ref{subsec:results} we present the results of the
relativistic calculations of the Ne K-shell-vacancy production
and K-K charge transfer probabilities in
the Ne~--~F$^{8+}$ collisions
as a function of the impact
parameter $b$ at the projectile energies $130$~keV/u and $230$~keV/u.
In this section we also present the results of the neutral
Xe K-shell-vacancy production and K-K charge transfer probabilities
in the Xe~--~Xe$^{53+}(1s)$ collision.
\section{Theory}
\label{sec:theory}
\subsection{ Dirac-Kohn-Sham equation in the active electron approximation}
\label{subsec:Kohn-Sham}
In this paper we use so-called active electron approximation~\cite{Lin_81}
to describe the ion-atom collision. In this
approximation, we consider only the active electron which participates in
the charge transfer and excitation processes, while the other passive
electrons provide a screening potential. In our
calculations the screening potential is defined by the density functional
theory (DFT) in the local density approximation (LDA). In this approach
the time-dependent wavefunction $\psi(\vec{r},t)$ of the active electron
$\psi(\vec{r},t)$ is the solution of the relativistic time-dependent
Kohn-Sham equation. In atomic units ($\hbar=m=e=1$), this equation is
given by
\begin{equation}
i \frac{\partial \psi(\vec{r},t)}{\partial t} = h_{\rm D} \,
\psi(\vec{r},t) \,.
\end{equation}
Here $h_{\rm D}$ is the two-center Dirac-Kohn-Sham Hamiltonian defined by
\begin{equation}
\hat h_{\rm D} =c (\vec{\alpha} \cdot \vec{p}) +
\beta \, c^2 + V_{AB}(\vec{r}) \,, \qquad
V_{AB}(\vec{r})= V_{H}[\rho] + V_{xc}[\rho]\,,
\end{equation}
where $c$ is the speed of light and $\vec{\alpha}$, $\beta$ are the
Dirac matrices. $V_{H}[\rho]$ and $V_{xc}[\rho]$ are the Hartree and
exchange-correlation potentials, respectively. Both of them are
the functionals of the electron density $\rho(\vec{r})$.
The Hartree potential $V_{H}[\rho]$ includes the electron-nucleus
interaction and the electron-electron Coulomb repulsion $V_C[\rho]$:
\begin{equation}
V_{H}(\vec{r}) = V_{\rm nucl}^{A}(\vec{r}_A) + V_{\rm nucl}^{B}(\vec{r}_B) +
V_C[\rho]
\,, \qquad \vec{r}_A=\vec{r}- \vec{R}_A\,, \qquad \vec{r}_B=\vec{r}- \vec{R}_B,
\end{equation}
where
\begin{equation}
V_{\rm nucl}(\vec{r}) =
\int d^3 \vec{r}^{\prime} \,
\frac{\rho_{\rm nucl}(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|} \,,
\qquad V_C[\rho] = \int \, d^3\vec{r^{\prime}} \,
\frac{\rho(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|} \,,
\end{equation}
$\rho_{\rm nucl}(\vec{r})$ and $\rho(\vec{r})$
are the nuclear and electron densities, respectively. The exchange-correlation
potential $V_{xc}[\rho]$ was taken in the Perdew-Zunger
parametrization~\cite{Perdew_81} including the self-interaction correction (SIC).
The electron density $\rho(\vec{r})$, obtained with the many-electron
wave function which is represented by a Slater determinant, is invariant
with respect to the rotations
in the occupied orbitals space. For this reason, to obtain the
electron density we can use atomic-like orbitals, localized on the both
centers
($A$ and $B$). In this case the electron density, constructed from the
orthogonal localized orbitals, can be represented as a sum of densities
$\rho_A(\vec{r})$ and $\rho_B(\vec{r})$ which are localized on
the centers $A$ and $B$.
This is not the case, however, if the orbitals localized on the different centers overlap and
are non-orthogonal. The electron density, derived from non-orthogonal
orbitals, is given by
\begin{equation}
\rho(\vec{r}) = \sum_{i,j} \psi_i^{(p)^{\ast}}(\vec{r}) \,
\left(S^{-1} \right)_{ij} \, \psi_j^{(p)}(\vec{r}) \,,
\end{equation}
where $\psi_i^{(p)}(\vec{r})$ are the atomic-like wavefunctions of the
passive electrons and matrix $S$ is the overlapping matrix. Note that the electron
density is normalized on the number of the passive electrons,
\begin{equation}
\int d^3 \vec{r} \, \rho(\vec{r}) = N-1 \,,
\end{equation}
where $N$ is the total number of electrons.
The electron density $\rho(\vec{r})$ can be divided into three parts,
\begin{equation}
\rho(\vec{r}) = \rho_A(\vec{r}) + \rho_B(\vec{r}) +
\rho^{\rm (ovlp)}_{AB}(\vec{r}) \,,
\end{equation}
if we split the summation over indices $i,j$ into the sum over $i,j \in A$,
the sum over $i,j \in B$ and the remaining overlapping part.
We can also split the overlapping density into two parts dividing the space
into two regions $(A)$ and $(B)$,
\begin{equation}
\rho^{\rm (ovlp)}_{AB}(\vec{r}) =
\rho^{\rm (ovlp)}_{A}(\vec{r})+\rho^{\rm (ovlp)}_{B}(\vec{r}).
\end{equation}
This can be done by the plane passing through
the middle of the internuclear distance (see
Ref.~\cite{Tupitsyn_10} for details).
For simplicity, let us consider
spherically-average values of
the electron densities in each region,
\begin{equation}
\overline \rho_A(r_A) = \int d\Omega_A \,
\left[ \rho_A(\vec{r})+\rho^{\rm (ovlp)}_{A}(\vec{r}) \right ]\,, \qquad
\overline \rho_B(r_B) = \int d\Omega_B \,
\left [\rho_B(\vec{r})+\rho^{\rm (ovlp)}_{B}(\vec{r}) \right] \,.
\end{equation}
This procedure does not change the normalization of the total electron
density,
\begin{equation}
\int d^3{\vec{r}} \, \overline \rho(\vec{r}) =
\int d^3{\vec{r}} \, \left [\overline \rho_A(r_A)+
\overline \rho_B(r_B) \right ] = N-1 \,.
\end{equation}
As a result, the potential $V_{AB}(\vec{r})$ can be approximated
by the sum of the spherically symmetric potentials of the two different centers,
\begin{equation}
V_{AB}(\vec{r}) \simeq V_A[\overline \rho_A](r_A) +
V_B[\overline \rho_B](r_B) \,.
\end{equation}
The overlapping densities must be taken
into account especially for the short internuclear distances. Otherwise, the Pauli
principle is violated and, as a result, the
number of electrons on the $1s$ shell of the united system
can exceed $2$.
It should also be noted that the time-dependent wave function $\psi(\vec{r},t)$
of the active
electron is orthogonalized to the wave functions $\psi^{(p)}_i(\vec{r})$
of the passive electrons.
This means that the transitions of the active electron to the states
occupied by
the passive electrons are forbidden in accordance with the Pauli principle.
\subsection{ Two-center Dirac-Kohn-Sham equation}
\label{subsec:two-center}
The two-center expansion of the time-dependent wave function $\psi(\vec{r},t)$
can be written in the form
\begin{equation}
\psi(\vec{r},t) = \displaystyle \sum_{\alpha=A,B} \, \sum_{a}
C_{\alpha a}(t) \, \varphi_{\alpha,a} (\vec{r}-\vec{R}_\alpha(t)) \,,
\label{expan1}
\end{equation}
where index $\alpha=A,B$ labels the centers, index $a$
enumerates basis functions at the given center, and
$\varphi_{\alpha,a} (\vec{r}-\vec{R}_\alpha)$ is the central-field
bispinor centered at the point $\alpha$.
In what follows, the shorthand notations
$|j\rangle \equiv |\varphi_j\rangle \equiv |\varphi_{\alpha , a}\rangle$
for states $j \equiv \alpha , a$ are used.
The expansion coefficients $C_{a \alpha}(t)$ of the time-dependent
wave function $\psi(\vec{r},t)$ can be obtained by solving
the linear system of first-order differential equations
\begin{equation}
i \sum_{k}S_{jk} \frac{dC_{k}(t)}{d t} = \sum_{k} \,
( H_{jk} - T_{jk}) \, C_{k}(t) \,,
\end{equation}
where indices $j$ and $k$ enumerate the basis functions of both centers, and
the matrix elements of $H$ and $S$ are
\begin{equation}
H_{jk} \,=\, \langle j \mid \hat h_{\rm D} \mid k \rangle \,, \qquad
S_{j k} \,=\, \langle j \mid k \rangle \,.
\label{matr1}
\end{equation}
The matrix elements of $T$ are given by
\begin{equation}
T_{jk} \,=\, i \langle j \mid \frac{\partial}{\partial t} \mid k \rangle =
T^{\ast}_{kj} + i \frac{\partial}{\partial t} \, S_{jk} \,.
\label{matr2}
\end{equation}
Obviously the matrix $T$ is non-Hermitian, if the overlapping matrix $S$
depends on time.
The functions $\varphi_{\alpha , a}$ depend on time due to two reasons.
First, the basis functions centered at the target and projectile nuclei
move together with the nuclei. Second, the basis functions depend
parametrically on the distance between the nuclei, since their radial
parts are obtained from the radial equations, where for each center
the potential of the other ion (atom) is included in the so-called monopole
approximation (see section~\ref{subsec:basis}).
Calculations of the matrix elements $H_{jk}$,
$S_{jk}$, and $T_{jk}$ were considered in detail in Ref.~\cite{Tupitsyn_10}.
\subsection{Basis functions}
\label{subsec:basis}
In our approach the basis set contains Dirac-Fock and Dirac-Fock-Sturm
orbitals. The Dirac-Fock-Sturm orbitals can be considered as pseudo-states,
which should be included in the basis to take into account the
contribution of the positive- and negative-energy Dirac continuum.
Both types of basis functions $\varphi_{\alpha a}$ are central field
Dirac bispinors centered at the position
$\vec{R}_{\alpha}$ ($\alpha=A,B$) of the corresponding ion,
\begin{equation}
\varphi_{n\kappa m}(\vec{r}) =
\left ( \begin{array}{l} \displaystyle
\,\, \frac{~P_{n \kappa}(r)}{r} \, \chi_{\kappa m}(\Omega)
\\[4mm] \displaystyle
i \, \frac{Q_{n \kappa}(r)}{r} \, \chi_{-\kappa m}(\Omega)
\end{array} \right ) \,,
\end{equation}
where $P_{n \kappa}(r)$ and $Q_{n \kappa}(r)$ are the large and small radial
components, respectively, and $\kappa=(-1)^{l+j+1/2}(j+1/2)$ is the
relativistic angular quantum number.
The large and small radial components are obtained by solving
numerically the Dirac-Fock and Dirac-Fock-Sturm equations in the central
field approximation. The radial Dirac-Fock equation is
\begin{equation}
\left(h^{\rm DF}_{\alpha} + V_{\rm ext}(r) \right) \, F_{\alpha n\kappa}(r)
=
\varepsilon_{\alpha n\kappa} F_{\alpha n\kappa}(r) \,, \qquad
F_{\alpha n\kappa}(r) = \left(
\begin{array}{c}
P_{\alpha n\kappa}(r) \\ Q_{\alpha n\kappa}(r)
\end{array}
\right ),
\label{dirac1}
\end{equation}
where $ h^{\rm DF}_{\alpha}$ is the radial Dirac-Fock Hamiltonian of ion $\alpha$
($\alpha=A,B$), $F_{\alpha n\kappa}(r)$ is the two-component radial wave function,
and $V_{\rm ext}(r)$ is a local external potential. The explicit form
of the radial Dirac-Fock equation and the description of the corresponding
computer code
are presented in Ref.~\cite{Bratsev_77}.
The radial components of the Dirac-Fock-Sturm orbitals
$\overline{\varphi}_{n\kappa m}$, which we denote by
$\overline F_{n\kappa}(r)$, are the solutions of the generalized
Dirac-Fock-Sturm eigenvalue problem,
\begin{equation}
\left(h^{\rm DF}_{\alpha} + V_{\rm ext}(r) -
\varepsilon_{\alpha n_0 \kappa} \right ) \, \overline F_{\alpha n\kappa}(r) =
\lambda_{\alpha n \kappa} \, W_{\alpha \kappa}(r) \,
\overline F_{\alpha n \kappa}(r) \,.
\label{sturm1}
\end{equation}
Here $\lambda_{\alpha n \kappa}$ can be considered as the eigenvalue of the
Dirac-Fock-Sturm operator and $W_{\alpha \kappa}(r)$ is a constant sign weight
function. The energy $\varepsilon_{\alpha n_0 \kappa}$ is fixed in the
Dirac-Fock-Sturm equation. If the weight function $W(r) \to 0$ at $r \to \infty$,
all Sturmian functions have the same asymptotic behavior at $r \to \infty$.
It is clear that for $\lambda_{\alpha n \kappa}=0$ the Sturmian function
$\overline{\varphi}_{n\kappa m}$ coincides with the reference Dirac-Fock orbital
$\varphi_{\alpha n_0 \kappa}$. In our calculations we use the following weight
function
\begin{equation}
W_{\kappa}(r) \,=\, - \, \frac{1 \,-\,
\exp(-(\alpha_{\kappa} \, r)^2)}{(\alpha_{\kappa} \, r)^2}\,.
\label{sturm2}
\end{equation}
In contrast to $1/r$, this weight function is regular at the origin.
It is well-known that the Sturmian operator is Hermitian. It does
not contain continuum spectra, in contrast to the Dirac operator.
Therefore, the set of the Sturmian eigenfunctions forms a discrete
and complete basis set of one-electron wave functions.
The external central-field potential $V_{\rm ext}(r)$ in equations
(\ref{dirac1}) and (\ref{sturm1}) is arbitrary, and, therefore, it can be chosen
to provide most appropriate Dirac-Fock and Dirac-Fock-Sturm basis orbitals.
At small internuclear distances the wave function of the atomic electron
experiences also the strong Coulomb field of the other ion (atom).
To the leading order this effect can be taken into account by
including the Coulomb-Hartree potential of the second ion (atom)
as the external potential $V_{\rm ext}(r)$ within the so-called monopole
approximation. For instance, the external central-field potential
$V^{A}_{\rm ext}(r)$ is given by
\begin{equation}
V^{A}_{\rm ext}(r) = V^{B}_{\rm mon}(r) =
\frac{1}{4 \pi} \,\int d\Omega_A \,\, V^{B}_{H}(\vec{r}-\vec{R}_{AB}) \,.
\end{equation}
where $V^{B}_{\rm mon}(r)$ is the spherically-symmetric part of the
reexpansion of the Coulomb-Hartree potential $V^{B}_{H}(\vec{r}-\vec{R_{AB}})$ of the
ion $B$ with respect to the center $A$ and
$\vec{R}_{AB}$ is the internuclear distance vector.
Calculations of the two-center integrals with the basis
functions employed were described in detail
in our previous paper~\cite{Tupitsyn_10}.
The time-dependent Dirac-Kohn-Sham equation for the active electron is
solved using the two-center basis set expansion.
The expansion coefficients are determined employing the direct evolution
(exponential) operator method~\cite{Tupitsyn_10}, which is more stable
compared to the others, such as, e.g., the Crank-Nicholsen propagation
scheme~\cite{Crank_47} and the split-operator method~\cite{Fett_82}.
To obtain the matrix representation of the exponential operator
in the finite basis set one has to diagonalize the generalized complex
Hamiltonian matrix at each time step. Since our basis set is not too large,
the diagonalization procedure is not too time consuming.
\subsection{Charge-transfer and vacancy-production probabilities}
\label{subsec:prob}
The amplitudes of the charge transfer and excitations to different
bound states of the projectile and target ions are calculated
by projecting the time-dependent wave function
of the active electron onto the atomic Dirac-Fock orbitals of the projectile
and target. The corresponding calculations for collisions
of H-like ions with bare nuclei
were described in detail in
our work~\cite{Tupitsyn_10}. That is
why here we restrict our consideration only to the new features
of the calculation procedure that occur for atom-ion
collisions within the one active electron approximation.
Consider the collision of a neutral atom $A$ (target)
with a hydrogenlike ion $B$ (projectile). We assume that before
the collision the active electron occupies the $1s$ state of the target
with spin up (in the relativistic case, with the total angular
moment projection $\mu=1/2$) and the passive electron of the hydrogenlike ion
occupies the $1s$ state with spin down. In what follows, we are interested in
two processes: the K-K charge transfer and the K-shell vacancy production.
Let $P_A(1s)$ is the probability to find the active electron in the $1s$
state of the target after the collision, $P_B(1s)$ denotes the probability
to find one active electron in the $1s$ state of the projectile or,
in other words, the probability $P_{\rm K-K}$ of the K-K shell charge
transfer of one electron. To obtain the probabilty $P_{\rm vac}$ of the K-shell
vacancy production, we introduce the probabilities $P(E_1)$ and $P(E_2)$ of
the events $E_1$ and $E_2$, when a hole is created in the $1s$ state
of the target with spin up and down, respectively. The probabilities
of these events are defined by
\begin{align}
\left \{
\begin{array}{lll} \displaystyle
P(E_1) &=& \displaystyle 1-P_A(1s) \\[3mm]
P(E_2) &=& \displaystyle 1-P_A(1s)-P_B(1s)
\end{array}
\right ..
\end{align}
Assuming the events $E_1$ and $E_2$ are independent, the
probability of production at least one hole in the $1s$ state of the target
is given by
\begin{align} \label{pvac}
P_{\rm vac}=P(E_1)+P(E_2)-P(E_1)P(E_2)=
1-P_A(1s)(P_A(1s)+P_B(1s)) \,.
\end{align}
We note that, since the sum of the probabilities $P_A(1s)$ and $P_B(1s)$ is less
than 1, the vacancy production probability satisfies the condition
$0 \le P_{\rm vac} \le 1$. It should also be noted that the K-shell vacancy production
defined by Eq. (\ref{pvac}) includes the production of two holes in the target K shell.
\subsection{Results of the calculations and discussion}
\label{subsec:results}
To test the approach we have studied the
Ne~--~F$^{8+}(1s)$ collision for low energies where
experimental and nonrelativistic theoretical results are
available~\cite{Fritsch_85, Hagmann_87, Toepfer_87, Thies_89}.
In this case the nuclear charge numbers are rather small
and, therefore, relativistic effects are negligible. We stress, however,
that our approach
can be directly applied to heavier systems where the relativistic effects
become stronger or even dominant.
In Fig.~\ref{Fig:NeF_230} we present the results of our calculations for
the probabilities $P(b)$ of the Ne
K-shell-vacancy production (solid line) and of the K-K-shell charge transfer
(dotted line) as functions of the
impact parameter $b$ for the Ne-F$^{8+}(1s)$ collision at the projectile
energy of $230$ keV/u. The results
for the Ne K-vacancy production are compared with experimental
values (circles)~\cite{Hagmann_87} and with theoretical results obtained
by Fritsch and Lin (dashed line)~\cite{Fritsch_85} and by
Thies {\it et al.} (dash-dotted line)~\cite{Thies_89}.
It can be seen that our results are in perfect agreement with
the experimental ones.
\begin{figure}[hbt]
\centering
\vspace{-0.2cm}
\includegraphics[width=10.5cm,clip]{NeF_230.eps}
\caption{\small The results of the present calculations
for the probabilities $P(b)$ of the Ne
K-shell-vacancy production (solid line) and of the K-K-shell charge transfer
(dotted line) as functions of the
impact parameter $b$ for the Ne-F$^{8+}(1s)$ collision at the projectile
energy of $230$ keV/u. The circles indicate experimental results by Hagmann
{\it et al.}~\cite{Hagmann_87}.
The dashed and dash-dotted lines present
theoretical results by Fritsch and Lin~\cite{Fritsch_85} and by
Thies {\it et al.}~\cite{Thies_89}, respectively.}
\label{Fig:NeF_230}
\end{figure}
The related results for the Ne-F$^{8+}(1s)$ collision at the projectile
energy of $130$ keV/u are presented in Fig.~\ref{Fig:NeF_130}, where
the same notations as in Fig.~\ref{Fig:NeF_230} are used.
We note that our theoretical results are in a good agreement
with the experimental ones at small impact parameters.
However, in contrast to Fig. 1, the agreement is not so good
for medium and large impact parameters, although
the theoretical predictions for the maximum and minimum
positions agree rather well with the experimental ones.
As one can see from Figs.~\ref{Fig:NeF_230} and \ref{Fig:NeF_130},
for both energies at large impact parameters
the K-vacancy production is mainly determined by the K-K-shell charge transfer,
which is indicated by the dotted line. The difference between the K-vacancy
production and the K-K-shell transfer probabilities at small impact
parameters is due to the contribution from the charge-transfer excitation
into the $2s$, $2p$, and higher vacant states of the projectile.
This is in accordance with
the experimental results of Ref. \cite{Hagmann_82},
where the K-vacancy production in the Ne~--~F$^{6+}((1s)^22s)$ collision
was studied. The calculation of the latter process is currently under way
and will be published elsewhere.
\begin{figure}[hbt]
\centering
\vspace{-0.2cm}
\includegraphics[width=10.5cm,clip]{NeF_130.eps}
\caption{\small
The results of the present calculations for the probabilities $P(b)$ of the Ne
K-shell-vacancy production (solid line) and of the K-K-shell charge transfer
(dotted line) as functions of the
impact parameter $b$ for the Ne-F$^{8+}(1s)$ collision at the projectile
energy of $130$ keV/u. The circles indicate experimental results by Hagmann
{\it et al.}~\cite{Hagmann_87}.
The dashed and dash-dotted lines present
theoretical results by Lin {\it et al.}~(taken from Ref. \cite{Hagmann_87}) and by
Thies {\it et al.}~\cite{Thies_89}, respectively.}
\label{Fig:NeF_130}
\end{figure}
In this work we also performed the related calculations for
the Xe~--~Xe$^{53+}(1s)$ collision at the projectile energy of $3.6$ MeV/u.
The experimental study of this process is planned at GSI (Darmstadt)
\cite{Hagmann_10,Hagmann_11}.
The probabilities $P(b)$ of the Xe
K-shell-vacancy production and of the K-K-shell charge transfer
as functions of the
impact parameter $b$ are plotted in Fig.~\ref{Fig:XeXe_3.6}. The solid and
dotted lines represent the vacancy production and the charge transfer, respectively.
For comparison, in the same figure we display the K-shell-vacancy production
for the Xe$^{53+}(1s)$~--~Xe$^{54+}$ collision that is indicated by
the dashed line.
We note again that at large
impact parameters the K-vacancy production is
almost completely determined by
the K-K-shell charge transfer. It can be also seen that
in the case under consideration the screening effect is rather small.
\begin{figure}[hbt]
\centering
\vspace{-0.2cm}
\includegraphics[width=10.5cm,clip]{Xe2_3.6_rel.eps}
\caption{\small
The probabilities $P(b)$ of the Xe
K-shell-vacancy production (solid line)
and of the K-K-shell charge transfer (dotted line)
in the Xe-Xe$^{53+}(1s)$ collision
as functions of the
impact parameter $b$. The dashed line indicates
the K-shell-vacancy production
for the Xe$^{53+}(1s)$-Xe$^{54+}$ collision.}
\label{Fig:XeXe_3.6}
\end{figure}
To investigate the role of the relativistic effects we performed
the same calculations for the Xe-Xe$^{53+}(1s)$ collision
in the nonrelativistic limit by multiplying the standard
value of the speed of light by the factor 1000. The obtained relativistic
and nonrelativistic results are presented in Fig.~\ref{Fig:XeXe_3.6_nonrel}.
As one can see from the figure,
the oscillatory behavior of both curves is the same but the nonrelativistic
curve is shifted toward higher impact parameters.
\begin{figure}[hbt]
\centering
\vspace{-0.2cm}
\includegraphics[width=10.5cm,clip]{Xe2_3.6_nonrel.eps}
\caption{The probability $P(b)$ of the Xe K-shell-vacancy production
in the Xe-Xe$^{53+}(1s)$ collision
as a
function of the impact parameter $b$.
The solid and dashed lines present relativistic and nonrelativistic results,
respectively.
}
\label{Fig:XeXe_3.6_nonrel}
\end{figure}
\section{Conclusion}
In this paper the method that was previously developed
for evaluation of the electron-excitation and charge-transfer processes
in collisions of H-like ions with bare nuclei has been extended
to collisions of ions
with neutral atoms. The extention is
based on the active electron approximation, in which the interaction of the
active electron with the passive electrons
is accounted for by the screening DFT potential.
The method developed has been applied to evaluate the K-vacancy production
and the K-K charge transfer in the low-energy Ne~--~F$^{8+}$ collision.
The results of the calculation
are compared with available experimental data and with theoretical
calculations by other authors.
The influence
of the relativistic effects on the K-vacancy production probability
is investigated for the Xe~--~Xe$^{53+}$ collision.
It is demonstrated that the relativistic and nonrelativistic probabilities as
functions of the impact parameter exhibit the same
oscillatory behavior at low energies but the relativistic curves
are shifted toward lower impact parameters compared to the
nonrelativistic ones.
In our further investigation we plan to continue calculations of low-energy
heavy-ion collisions that are of interest for current and nearest future experiments
at GSI and FAIR facilities in Darmstadt. Special attention will be
paid to the critical regime, when the ground-state level of the united quasimolecule
dives into the negative-energy Dirac continuum.
\clearpage
\section{Acknowledgments}
This work was supported by DFG (Grants No. PL 254/7-1 and VO 1707/1-1), by RFBR
(Grants No. 10-02-00450 and No. 11-02-00943-a), by GSI, by DAAD, by the Ministry of Education and Science of
Russian Federation (Grant No. P1334). The work of A.I.B. and I.A.M. was also supported by the Dynasty
foundation.
Y.S.K. and A.I.B. acknowledge financial support by the FAIR--Russia Research
Center.
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,452 |
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\section{Cosmic Particle Accelerators}
Our Universe is full of particle accelerators.
Cosmic particle acceleration is witnessed most directly by the presence of cosmic-ray (CR)
particles near the Earth, which are readily observed by direct measurements above
the Earth's atmosphere \cite{alcaraz00,wang02},
or by observation from the ground of their induced particle cascades in the Earth's atmosphere
\cite{becker09}.
While those measurements provide valuable information about the CR energy spectrum and
composition, they offer limited insight into the nature of the CR sources as the
particles get considerably deflected by magnetic fields on their ways from the acceleration
site to the Earth.
An observable manifestation of this deflection is the radio synchrotron emission
that arises when relativistic electrons spiral through magnetic fields.
Such radiation is seen from a large variety of objects, covering
the Sun,
nova and supernova explosions,
gamma-ray bursts (GRBs),
supernova remnants (SNRs),
pulsar wind nebulae (PWNe),
relativistic stellar binary systems,
galaxies (and in particular our own Galaxy from which widespread diffuse emission is observed),
active galactic nuclei (AGN), and
galaxy clusters.
The ubiquity of synchrotron emission is a clear demonstration that particle acceleration is
omnipresent in the Universe.
Nevertheless, we still ignore where and how the large majority of CRs are accelerated
\cite{halzen10}.
Furthermore, it remains a big mystery how nature manages to concentrate up to
$\sim10^{20}$ eV of kinetic energy into a single particle that is observed in CRs at
Earth \cite{stanev07}.
We also have still a poor understanding of how CRs propagate through the Universe
\cite{kotera08} or within our Galaxy \cite{cowsik10}, and about how they impact their
environment \cite{indriolo09,papadopoulos10,sijacki08}.
Our ignorance is partly related to the fact that synchrotron emission traces generally
only the leptonic component of CRs.
Leptons, however, constitute only about 1\% of the CR particles while the remaining
99\%, mainly protons and helium nuclei, escape observations.
Energetically, however, only the hadronic component is relevant.
Fortunately there are gamma rays.
First of all, gamma rays (photon energies $\ga511$~keV) provide a cleaner signal of
particle acceleration than do the radio signals due to the absence of competing thermal
emission processes at these high energies.
Even more important, gamma rays give access to the hadronic CR component which is
traced by a (Doppler broadened) gamma-ray line centred at $68$~MeV (in photon flux)
that arises from the decay of $\pi^0$ mesons generated in inelastic collisions of CR hadrons
with interstellar matter and radiation.
Competition arises in the gamma-ray domain, however, from
non-thermal Bremsstrahlung,
inverse Compton scattering or
curvature radiation of relativistic electrons interacting with
gas,
photon fields
or strong magnetic fields, respectively.
The drawback of this competition is that hadronic and leptonic signatures are blended.
Disentangling the various emission processes thus requires the broadest possible
energy coverage in the gamma-ray domain, eventually aided by lower energy
observations (radio, optical, infrared, X rays), to allow the spectral separation of the
components.
Having several different emission processes is, however, also an advantage: it
allows for a large variety of particle accelerators to be studied, and gives access to
information about the physical conditions of the local environment.
\section{Existing Facilities}
The study of cosmic particle accelerators is thus by nature a multiwavelength endeavour.
Only recently, a complete spectral coverage of the gamma-ray domain has become
available, thanks to the advent of new generation ground-based gamma-ray
telescopes such as H.E.S.S. \cite{aharonian06}, MAGIC \cite{baixeras04},
Milagro \cite{atkins04} and VERITAS \cite{weekes02}.
These instruments cover the energy range between a few tens of GeV up to $\sim10$~TeV
-- the Very High Energy (VHE) domain -- and rely on the detection of electromagnetic showers
created by interactions of VHE gamma rays with the Earth's atmosphere.
So far\footnote{
See http://tevcat.uchicago.edu/ for an up-to-date list of VHE sources.},
more than 100 cosmic particle accelerators have been unveiled by this technique,
comprising Galactic (PWN, SNR, gamma-ray binaries) and extragalactic objects (starburst
galaxies and AGN).
This demonstrates that current VHE telescopes have passed the critical sensitivity
threshold that enables the study of VHE particle accelerators in the Universe.
The {\it Fermi} Gamma-Ray Space Telescope, launched in 2008 and operating since then
at lower energies between 30~MeV to 300~GeV \cite{atwood09},
indicates that this is only the tip of the iceberg.
The {\em Fermi} Large Area Telescope first source catalogue \cite{abdo10a} lists 1451
sources, comprising large populations of pulsars and AGN in addition to PWN, SNR,
gamma-ray binaries, globular clusters, normal galaxies, starburst galaxies and radio
galaxies.
A comparable number of sources is also expected in the VHE domain, provided that the
detection sensitivity can be improved by a factor of $\sim10$.
With such an improvement, the required broadband energy coverage would become
available for a large number of sources, enabling a thorough study of the physics of
cosmic particle accelerators in the Universe.
\section{The CTA Observatory}
The Cherenkov Telescope Array (CTA) will provide the required increase in sensitivity,
together with improvements in angular resolution and an extension of the energy range.
CTA will consist of two arrays of Cherenkov telescopes located at sites in the northern and
southern hemispheres, allowing full-sky coverage.
The increase in sensitivity will be achieved by the deployment of large numbers (50 to 100)
of Cherenkov telescopes, while the extension of the energy range will be accomplished
by using telescopes of different sizes.
An increase of angular resolution with respect to existing facilities will be achieved by
improved imaging of the air shower, both in terms of resolution and photon statistics.
For a detailed description of the CTA design and the expected performance, the reader should
refer to \cite{hofmann10}.
CTA will be operated as a proposal-driven open observatory, with a Science Data Centre
providing transparent access to data, analysis tools and user training.
A deep survey of the Galactic plane, and possibly, a more shallow all-sky survey will
complement these observations, possibly in form of a core or key program.
The CTA project recently entered the preparatory phase, funded by the European Commission
under the Seventh Framework Programme (FP7), which will address a number of crucial
prerequisites for the approval, construction and operation of CTA.
This phase will last for 3 years and will deliver a complete and detailed implementation
plan for the CTA infrastructure.
Array deployment may then start from 2014 on, provided that funding is secured.
\section{Science prospects}
The main science themes that will be addressed by CTA are summarized in the following
sections.
For a more comprehensive discussion of the CTA science case, the reader should refer to
\cite{hofmann10}.
\subsection{Origin of cosmic rays and their role in the Universe}
The standard paradigm, mainly based on energetic grounds, is that Galactic CRs are
accelerated in the shocks generated by supernova explosions \cite{ginzburg64}.
Gamma-ray emission is indeed detected from a growing number of Galactic
SNRs, yet the nature of the underlying emission process (hadronic or leptonic) is still
under debate \cite{zirakashvili10}.
Moreover, Galactic CRs reach energies of at least several $10^{15}$ eV (PeV),
which in turn should produce VHE gamma rays with a relatively flat energy
spectrum extending to hundreds of TeV.
Sources that show such characteristics have been dubbed PeVatrons.
So far, however, not a single SNR PeVatron has been detected.
This may still be commensurate with the estimate that only $\sim10$ such
objects exist today in our Galaxy, since rapid escape of PeV particles
limits their acceleration lifetime in SNR shocks to only several hundred years.
The deep CTA survey of the Galactic plane should unveil these $\sim10$
SNR PeVatrons, provided that the standard paradigm for the Galactic CR origin
is correct.
Several PeVatron candidates have indeed already been detected by the existing VHE
facilities, but they all seem to be associated with PWNe \cite{camilo09,abdo10b}.
Also the Crab, which is considered to be the only known Galactic PeVatron
\cite{aharonian04}, is a PWN.
Conventionally, VHE emission of PWNe is interpreted to be predominantly of leptonic
origin \cite{baring10} although there are arguments suggesting the presence of
substantial amounts of relativistic protons in PWNe \cite{hoshino92}.
The observed anisotropy in the arrival directions of CRs with energies $\sim20$ TeV
\cite{desiati10} may be a hint for nearby Galactic CR sources, for which the Vela and
Geminga PWNe appear to be plausible candidates \cite{halzen10}.
The detection of a clear hadronic signal by CTA from PWNe would thus be an
invaluable piece of evidence to establish their role as Galactic CR sources.
PWNe constitute also the biggest class of identified Galactic VHE sources, and CTA
is expected to detect on the order of $\sim100$ objects, allowing for the first time
comprehensive population studies.
A comparably sized sample of Galactic SNRs should be visible to CTA, enabling studies
of particle acceleration as function of SNR age and environment.
For example, if our general picture of SNR evolution is correct, the position of the cutoff
in the VHE spectrum should depend on the age of the SNR and on the magnetic field at the
shock.
A SNR population study will allow testing this hypothesis and constraints to be placed
on the physical parameters of SNRs, in particular on the magnetic field strengths.
CTA offers also the possibility to directly observe the diffusion of CRs in the Galaxy.
While travelling from the accelerator to the target, the spectrum of CRs is a strong
function of time, distance to the source, and the local diffusion coefficient.
Depending on the values of these parameters, varying gamma-ray spectra are expected
from the environment surrounding CR accelerators, in particular from massive molecular clouds
that provide thick targets for CR hadronic interactions \cite{gabici09}.
As a proof of principle, surprisingly small CR diffusion coefficients have been recently
inferred from observations of gamma-ray emission from molecular clouds near two
Galactic SNRs \cite{gabici10}.
CTA will enable detailed mappings of VHE emission around potential CR accelerators,
enabling thus an experimental determination of the local diffusion coefficients and/or the
local CR injection spectra in our Galaxy.
At a larger scale, CR diffusion can also be studied by CTA in nearby galaxies of the Local
Group, such as the Magellanic Clouds or M31.
Recent {\em Fermi} observations of the Large Magellanic Cloud revealed
a surprisingly close confinement of CRs around star forming regions, which also may
point towards a small proton diffusion length \cite{abdo10c}.
With its superior angular resolution, CTA will be able to refine these studies by providing
a detailed mapping of the CR density in these nearby galaxies.
CTA will also enable comparative studies of external galaxies, such as the Local Group
galaxies and nearby starburst galaxies.
These comparative studies will provide important clues about the key parameters of CR
acceleration and transport \cite{lacki10}.
They also will elucidate the role of CRs in galactic feedback processes
\cite{papadopoulos10,socrates08}.
\subsection{Nature and variety of particle acceleration around black holes}
Accretion of matter onto a black hole provides one of the most efficient energy source
known in the Universe.
The most massive black holes in our Universe are presumably hosted in the centres
of active galactic nuclei (AGN), and accreting onto these black holes readily explains
the tremendous luminosity of these objects \cite{dermer09}.
AGN often show jets of relativistic plasma, and jet sources are often most luminous
at gamma-ray energies.
AGN with jets that are aligned with the line-of-sight to within a few degrees are dubbed
blazars, and those present the dominant class of extragalactic VHE emitters known so
far.
The observed fast variability of the VHE flux from blazars, down to minute time scales
\cite{aharonian07}, indicates that gamma-ray production occurs in spatially
confined region, with sizes as small as a few times the Schwarzschild radius of
the black hole.
What remains unknown is how the relativistic jets are launched, what their structure
and composition is, and by what physical mechanism the particles are accelerated to
very high energies \cite{hardee10}.
Multi-wavelength observations with high temporal and spectral resolution can help to
answer these questions.
By giving access to the VHE emission for a large population of blazars, CTA will provide
new insights into the physics that is driving these sources.
In particular, CTA will be able to probe variability time scales well below minutes, putting
constraints on acceleration and cooling times, instability growth rates, and the time evolution
of shocks and turbulences.
As VHE blazars are seen to considerable distances, their gamma rays may also
be used to study the diffuse ultraviolet to infrared radiation along their lines-of-sight.
VHE gamma rays travelling from remote sources interact with photons of the
extragalactic background light (EBL) via e$^+$e$^-$ pair production, leading to an
energy-dependent absorption of the intrinsic source spectrum.
If this intrinsic spectrum is known, the observations may provide a measure
about the integrated EBL density towards the source \cite{raue10}.
First attempts to constrain the EBL density from VHE observations are promising
\cite{aharonian06a}.
CTA will characterise a sufficiently large sample of extragalactic VHE sources that will
constrain the EBL density as function of redshift, complementing thus direct measurements
in the infrared domain that are hampered by strong Galactic and zodiacal
foreground emissions.
More locally, radio galaxies have recently emerged as a new class of VHE emitting AGN
\cite{aharonian06b}.
Jets of radio galaxies show only small -- if at all -- relativistic boosting as they make
larger angles to the line-of-sight than the jets of blazars.
This considerably reduces the jet luminosity, and thus only nearby radio galaxies are
detectable in gamma rays; on the other hand, most AGN in the local Universe
are radio galaxies, hence the number of potential sources is large.
Given the proximity and the observing geometry of the sources, detailed spatially resolved
studies of individual jet components become possible in gamma rays.
Furthermore, as the relativistic boosting is no longer a poorly constrained parameter of
the problem, more stringent constraints on the VHE emission physics and geometry can
be derived.
CTA will allow to resolve the outer and inner kpc jet structure of nearby radio galaxies,
enabling to spatially pin down the site of the emission.
With the help of simultaneous multi-wavelength observations and (temporal) correlation
studies, different sections of the jet and the core can be probed, down to the smallest
sub-pc (milli-arcsecond) scale, only accessible to VLBA radio observations.
As proof of principle, a multi-wavelength observing campaign of the radio galaxy M87
combining VHE, X-ray and VLBI radio observations has recently localised the observed
VHE emission to within a few Schwarzschild radii of the black hole \cite{acciari09}.
Relativistic jets are also the plausible origin of the gamma-ray emission that has
recently been detected from the Galactic microquasar Cyg X-3 \cite{abdo09a}.
Furthermore, a handful of VHE gamma-ray emitters are known to be Galactic binary
systems, consisting of a compact object (neutron star or black hole) orbiting a massive
star.
Whilst jet emission powered by accretion onto the compact object has been suggested
to explain the observations \cite{dermer06}, there are arguments that favour the
VHE emission to originate in the interaction of relativistic outflows from highly magnetised
neutron stars (a.k.a. pulsar wind) colliding with the wind of the massive star \cite{dubus06}.
High-mass X-ray binary population studies predict that CTA may detect a dozen new
systems of that kind \cite{cerutti09}.
But already the observations of the few known systems with improved sensitivity will
provide tight constraints on the spectral dependency of their orbital modulations, offering
deeper insights into the underlying physics \cite{dubus10}.
Furthermore, the CTA deep Galactic plane survey has the potential to
reveal more sources, and continuous monitoring of key objects (such as Cyg X-3 or Cyg X-1)
may allow to catch also flaring VHE emission that, in combination with multi-wavelength
observations, will inform about the link between accretion and particle acceleration around
compact objects.
\subsection{Physics beyond the horizon}
It is difficult to speculate about the unknown, and definitely we cannot accurately predict
how much CTA will unveil about any physics that is beyond our ``standard model'' of the
world.
Gamma rays, however, hold the potential to reveal properties of the elementary
particles that make up our Universe because photonic signatures of particle interactions, decays
and annihilations show up in this energy range.
Weakly interacting massive particles, the leading category of particle dark matter
candidates, may give rise to continuum and/or line signatures in the VHE domain that are
in principle observable by CTA \cite{bringmann09}.
An alternative dark-matter candidate are axions which are expected to oscillate
into photons (and viceversa) in the presence of magnetic fields \cite{dicus78}.
These oscillations could distort the spectra of gamma-ray sources, providing an
indirect method to detect these particles from observations with CTA \cite{sanchez09}.
Another domain of fundamental physics for which CTA may provide important constraints
is the validation of Lorentz invariance, a principle on which the theory of special
relativity is based \cite{mattingly05}.
Some theories of quantum gravity predict violation of Lorentz invariance which would
manifest as an energy dependence of the speed of light.
While the most stringent constraints on this energy dependence come today from arrival
time delays measured by {\em Fermi} in GRBs \cite{abdo09b},
CTA may provide potentially more stringent constraints from precise timing of AGN flares.
\section{Conclusions}
The CTA observatory is the logical next step in the exploration of the high-energy
Universe, and will promote VHE observations to a public tool for modern astronomy.
CTA will explore the VHE domain from several tens of GeV up to more than
10 TeV with unprecedented sensitivity and angular resolution, enabling a
comprehensive understanding of cosmic particle acceleration physics at
various scales, distances and time scales.
Major advances are expected in understanding the origin of Galactic cosmic rays,
their propagation within galaxies, and their impact on their environment.
Particle acceleration in the vicinity of black holes will be explored in a large variety of
sources, and interactions and feedback effects of the particles on their surroundings
will be explored.
CTA will also probe physics beyond the established horizon, holding promises for
a better understanding of the ultimate laws that govern the Universe.
The CTA project just started the 3-years lasting preparatory phase, and array
deployment could begin as early as 2014, with a full observatory operational
before the end of this decade.
Early science may be optimistically expected from 2015 on.
\begin{theacknowledgments}
I particularly thank G. Dubus, B. Giebels and B. Khelifi for a careful reading of the manuscript.
In addition, I want to thank all my colleagues from the CTA consortium for the tremendous
work being done during the Design Study and the now commencing Prepatory Phase.
The support of the involved National funding agencies and of the European community
is gratefully acknowledged, as is the support by the H.E.S.S. and MAGIC collaborations
and the interested parties from the US.
\end{theacknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,963 |
Q: Finding tension in a freely sliding ring on a wire A heavy small ring of weight $W$ is free to slide on a smooth surface wire of radius $a$, fixed in a vertical plane. It is attached by a string of length $l$ where
$$2a > l > a\sqrt{2}$$
to a point on the wire in a horizontal line with the centre.
Find tension in the string.
Approach :
1.
Here, If A be the point where string is attached to wire, P be the equilibrium position of string, I get Tension as
$$ \dfrac{W(l^2-2a^2)}{a\sqrt{4a^2-l^2}}$$
2.
Here, If A be the point where string is attached to wire, P be the equilibrium position of string, I get Tension as
$$ \dfrac{- W(l^2-2a^2)}{a\sqrt{4a^2-l^2}}$$
Clearly, 2nd Approach is wrong as magnitude of tension can't be negative. But why is it wrong ? Why isn't this diagram possible ?
I have verified that with given restriction on $l$, the 2nd diagram should very well be possible. Can anyone point out where am I going wrong ?
Thanks!
A: Considering first the first position
By geometric considerations the angle $\angle PAB = \alpha $ is such that
$$
2 a \cos\alpha = l
$$
Now calling
$$
\vec R = r(\cos(2\alpha),\sin(2\alpha))\\
\vec W = w(0,-1)\\
\vec T =- t(\cos\alpha,\sin\alpha)
$$
in equilibrium we have
$$
\vec R + \vec W + \vec T = 0
$$
or
$$
\left\{
\begin{array}{rcl}
r \cos (2 \alpha )-t \cos (\alpha )& = & 0 \\
-w-t \sin (\alpha )+r \sin (2 \alpha )& = & 0 \\
2 a \cos (\alpha )& = & l \\
\end{array}
\right.
$$
and solving for $r,t,\alpha$ we obtain
$$
\left[
\begin{array}{ccc}
t & r & \alpha \\
\frac{\left(2 a^2-l^2\right) w}{a \sqrt{4 a^2-l^2}} & -\frac{l w}{\sqrt{4 a^2-l^2}} & \tan ^{-1}\left(\frac{l}{a},-\frac{\sqrt{4
a^2-l^2}}{a}\right) \\
\frac{\left(l^2-2 a^2\right) w}{a \sqrt{4 a^2-l^2}} & \frac{l w}{\sqrt{4 a^2-l^2}} & \tan ^{-1}\left(\frac{l}{a},\frac{\sqrt{4 a^2-l^2}}{a}\right)
\\
\end{array}
\right]
$$
one of them is discarded.
in the second position, the string can not remain taut.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,886 |
Carl Frampton beats Kiko Martinez to become IBF world champion
By Ben Dirs
BBC Sport, in Belfast
Carl Frampton (right) lands a heavy right-hander on Kiko Martinez
Northern Ireland's Carl Frampton outpointed Spain's Kiko Martinez in Belfast's Titanic Quarter to secure the world IBF super-bantamweight title.
In front of 16,000 fiercely partisan fans in a purpose-built outdoor arena, Frampton controlled the fight throughout and floored the champion in the fifth round.
Martinez demonstrated remarkable durability to make it to the final bell but the judges scored the fight 119-108, 119-108, 118-111 - all in Frampton's favour.
The 27-year-old had knocked Martinez out last February to win the European crown, only for the Spaniard to win the IBF title six months later.
"I've got the world title!" Frampton told BBC Radio 5 live after adding the IBF belt. "I feel a bit emotional - it has been a long time coming, it has been a hard road.
"I intend to hang on to it for a very long time."
Frampton's promoter, former world champion Barry McGuigan, gives his man some last-minute advice
Frampton is managed and promoted by former featherweight world champion Barry McGuigan, who was such a unifying force during Northern Ireland's Troubles in the 1980s, and trained by Barry's son Shane.
McGuigan Sr said after the fight: "I love him like a son - he's a part of me and I know how talented he is.
"He showed us his bravery tonight and he certainly showed the skills he's got. He's really got a tremendous future ahead of him."
The victory for the Tiger's Bay native, which makes him his country's first world champion since fellow Belfast fighter Wayne McCullough in 1996, provided more cause for celebrations across Northern Ireland and beyond.
As well as the United Kingdom and Ireland, the fight was broadcast in the United States, South America, China, Japan and the Middle East.
As for McGuigan Sr, he will be relieved as well as elated, having pumped a huge amount of money into the event (it was the country's biggest gate for a boxing match by some distance) and brought Martinez over from Spain at considerable cost.
Martinez had defended his title twice since stopping Jhonatan Romero last August and promised to gain revenge over Frampton, the only man to knock him out in 35 previous fights as a professional.
Frampton's journey
Born 21 February 1987, Belfast, Northern Ireland
Family Lives with wife Christine and three-year-old daughter Carla in Lisburn
Boxing background Started boxing at Midland ABC, Tiger's Bay before moving to Holy Family ABC in New Lodge as a professional
Connections Managed by Barry McGuigan, trained by Barry's son Shane in London
Amateur record 125 fights, 114 wins, 11 defeats; two-weight Irish champion; European silver medal 2007; 12 international medals
Pro record 19 fights (13 KOs), 19 wins
Pro honours World, European & Commonwealth super-bantamweight champion
But overlooked by the famous Harland and Wolff cranes, next to where the Titanic was built and launched a little more than 100 years ago, Frampton was not to be denied.
He won a cagey first round courtesy of a couple of snappy right crosses before the fight opened up in the second, Martinez trying his luck with some swinging right hands and Frampton having success on the counter.
Frampton, fighting on the back foot for the most part, repeatedly made Martinez look clumsy in the third, suggesting that the champion was starting to unravel.
The challenger continued to control proceedings in the fourth, keeping the stalking Martinez at bay with jabs and two-shot combinations, although Frampton did mix things up with one juddering uppercut.
Martinez was lucky not to be docked a point at the start of the fifth, having hit a prone Frampton on the back of the head. But Frampton exacted sweet revenge, flooring his rival with a short, chopping right towards the end of the round.
Martinez, bleeding from the cut over his left eye, was up almost immediately and straight back at Frampton, but was picked apart some more in the sixth.
The seventh was Martinez's best round of the fight, the Spaniard landing with a couple of those looping right hands. But Frampton was back in control in the eighth, drawing Martinez on to some hurtful left-right combinations.
Carl Frampton's celebrates his famous victory
Frampton was showing a cut over the right eye at the start of the ninth, which was a difficult round to score, and the 10th round was grim stuff as both men traded toe-to-toe on the inside, although Frampton landed the cleaner shots.
A battered and bruised Martinez looked close to folding in the 11th as Frampton rained blows upon him, but the man from Alicante proved that he is as game as they come by extending the contest into the final round.
Martinez looked ready to go again midway through the 12th but was still trading blows when the final bell sounded. But the result was never in doubt.
Frampton is now one of five world champions from the United Kingdom, alongside Carl Froch, Kell Brook, Scott Quigg and Jamie McDonnell, although those last two hold lesser versions of titles.
British fight fans would now like to see Frampton fight Bury's WBA title-holder Quigg but American Chris Avalos is the IBF's mandatory challenger.
Martinez was given special dispensation to fight Frampton instead of Avalos but the 24-year-old Californian is likely to get his chance next spring, especially given that he recently teamed up with British promoter Eddie Hearn.
Quigg, 25, defends his portion of the WBA title (Cuba's Guillermo Rigondeaux is their so-called 'super' champion and regarded as the best in the division) against Belgium's Stephane Jamoye in Manchester on 13 September.
But Frampton, who emerged from his triumphant night with a badly marked face and a damaged hand, will now take a well-earned holiday with his wife Christine and daughter Carla before weighing up his options.
BBC Radio 5 live Sports Extra will be replaying commentary of the fight in full at 0900 on Sunday morning.
Frampton was heavy favourite for the fight having beaten Martinez in February last year
Frampton dominated the early rounds against Martinez
Northern Ireland's biggest ever boxing crowd were in attendance to see the fight
Frampton knocked down Martinez in the fifth with a savage right-hander
Frampton is Northern Ireland's first bona fide world champion since Wayne McCullough in 1996
England Boxing
Read more on England Boxing
Champion Frampton wants Quigg fight
Read more on Carl Frampton: World champion wants Scott Quigg fight
A beacon for peace, ready for war
Read more on Carl Frampton: The boxer following where McGuigan dared to tread
How to get into boxing
Read more on Get Inspired: How to get into boxing | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,647 |
Ours is a tale of two historic, well-established family businesses, merged together to better serve you. Our "blended family" has not only survived, but strengthened over the years and has been honored to be at your service for over a century.
The Rowland family funeral business was founded in December 1895 by Francis M. Rowland. The original funeral home was located on Jefferson Street in Philadelphia. In 1915, Thomas Mifflin Rowland, son of Francis M., received his undertaker's license and continued the family business; in 1940 he moved the business from 1908 Diamond Street in Philadelphia to 1415 West Allegheny Avenue in Philadelphia. After the death of Thomas Mifflin Rowland in 1944, his widow, Gladys B. Rowland, operated the business with a widow's license until their son, Baron Rowland, received his funeral directors license in 1950 and took over the family business.
In January of 1961, Baron Rowland and his wife Barbara S. Rowland moved the Rowland Funeral Home to Abington. In 1980, the business name changed to Baron Rowland Funeral Home, Inc. In September 1996, Joseph M. Schlupp, began his career at the funeral home by serving his internship at Baron Rowland Funeral Home and received his license the following year. He has been with the funeral home ever since.
In November of 2002, the business and related property was sold to Joseph Gritz, a licensed funeral director in Pennsylvania and New Jersey, at which time Joseph M. Schlupp was appointed Supervisor at Baron Rowland and remained on staff as Funeral Director. Two short years after proudly opening the business, Joseph Gritz died in December 2004 at the age of 50 years.
On July 1, 2005, the business and related property was sold to Bonnie Helweg-Campbell. Mrs. Helweg also owned Helweg Funeral Service in Jenkintown, PA and had been operating that family business since the death of her husband, Joseph E. Helweg, Jr., in 1985.
Joseph M. Schlupp, the current supervisor, manager, and funeral director is available 24 hours a day to respond to a family's needs. He has served the location through its changes of ownership and has been serving as Supervisor since 1997. He treats every person that walks through the door or who calls on the phone with kindness and concern. He is with the family from the first notification when the death occurred, at the arrangement conference, at the service, at the cemetery, and is available for assistance and guidance after the service. Personal service and attention is what has made our business successful and is the reason families call on us over and over again and refer us to their friends and families. We have built a relationship on honesty and trust that spans generations. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,392 |
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<section id='main' class="content source"><pre class="line-numbers"><span id="1"> 1</span>
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</pre><pre class='rust '>
<span class='kw'>use</span> <span class='ident'>rustc</span>::<span class='ident'>lint</span>::<span class='op'>*</span>;
<span class='kw'>use</span> <span class='ident'>rustc_front</span>::<span class='ident'>hir</span>::<span class='op'>*</span>;
<span class='kw'>use</span> <span class='ident'>rustc_front</span>::<span class='ident'>util</span> <span class='kw'>as</span> <span class='ident'>ast_util</span>;
<span class='kw'>use</span> <span class='ident'>syntax</span>::<span class='ident'>ptr</span>::<span class='ident'>P</span>;
<span class='kw'>use</span> <span class='ident'>consts</span>::<span class='ident'>constant</span>;
<span class='kw'>use</span> <span class='ident'>utils</span>::<span class='ident'>span_lint</span>;
<span class='macro'>declare_lint</span><span class='macro'>!</span> {
<span class='kw'>pub</span> <span class='ident'>EQ_OP</span>,
<span class='ident'>Warn</span>,
<span class='string'>"equal operands on both sides of a comparison or bitwise combination (e.g. `x == x`)"</span>
}
<span class='attribute'>#[<span class='ident'>derive</span>(<span class='ident'>Copy</span>,<span class='ident'>Clone</span>)]</span>
<span class='kw'>pub</span> <span class='kw'>struct</span> <span class='ident'>EqOp</span>;
<span class='kw'>impl</span> <span class='ident'>LintPass</span> <span class='kw'>for</span> <span class='ident'>EqOp</span> {
<span class='kw'>fn</span> <span class='ident'>get_lints</span>(<span class='kw-2'>&</span><span class='self'>self</span>) <span class='op'>-></span> <span class='ident'>LintArray</span> {
<span class='macro'>lint_array</span><span class='macro'>!</span>(<span class='ident'>EQ_OP</span>)
}
}
<span class='kw'>impl</span> <span class='ident'>LateLintPass</span> <span class='kw'>for</span> <span class='ident'>EqOp</span> {
<span class='kw'>fn</span> <span class='ident'>check_expr</span>(<span class='kw-2'>&</span><span class='kw-2'>mut</span> <span class='self'>self</span>, <span class='ident'>cx</span>: <span class='kw-2'>&</span><span class='ident'>LateContext</span>, <span class='ident'>e</span>: <span class='kw-2'>&</span><span class='ident'>Expr</span>) {
<span class='kw'>if</span> <span class='kw'>let</span> <span class='ident'>ExprBinary</span>(<span class='kw-2'>ref</span> <span class='ident'>op</span>, <span class='kw-2'>ref</span> <span class='ident'>left</span>, <span class='kw-2'>ref</span> <span class='ident'>right</span>) <span class='op'>=</span> <span class='ident'>e</span>.<span class='ident'>node</span> {
<span class='kw'>if</span> <span class='ident'>is_cmp_or_bit</span>(<span class='ident'>op</span>) <span class='op'>&&</span> <span class='ident'>is_exp_equal</span>(<span class='ident'>cx</span>, <span class='ident'>left</span>, <span class='ident'>right</span>) {
<span class='ident'>span_lint</span>(<span class='ident'>cx</span>, <span class='ident'>EQ_OP</span>, <span class='ident'>e</span>.<span class='ident'>span</span>, <span class='kw-2'>&</span><span class='macro'>format</span><span class='macro'>!</span>(
<span class='string'>"equal expressions as operands to {}"</span>,
<span class='ident'>ast_util</span>::<span class='ident'>binop_to_string</span>(<span class='ident'>op</span>.<span class='ident'>node</span>)));
}
}
}
}
<span class='kw'>pub</span> <span class='kw'>fn</span> <span class='ident'>is_exp_equal</span>(<span class='ident'>cx</span>: <span class='kw-2'>&</span><span class='ident'>LateContext</span>, <span class='ident'>left</span> : <span class='kw-2'>&</span><span class='ident'>Expr</span>, <span class='ident'>right</span> : <span class='kw-2'>&</span><span class='ident'>Expr</span>) <span class='op'>-></span> <span class='ident'>bool</span> {
<span class='kw'>if</span> <span class='kw'>let</span> (<span class='prelude-val'>Some</span>(<span class='ident'>l</span>), <span class='prelude-val'>Some</span>(<span class='ident'>r</span>)) <span class='op'>=</span> (<span class='ident'>constant</span>(<span class='ident'>cx</span>, <span class='ident'>left</span>), <span class='ident'>constant</span>(<span class='ident'>cx</span>, <span class='ident'>right</span>)) {
<span class='kw'>if</span> <span class='ident'>l</span> <span class='op'>==</span> <span class='ident'>r</span> {
<span class='kw'>return</span> <span class='boolval'>true</span>;
}
}
<span class='kw'>match</span> (<span class='kw-2'>&</span><span class='ident'>left</span>.<span class='ident'>node</span>, <span class='kw-2'>&</span><span class='ident'>right</span>.<span class='ident'>node</span>) {
(<span class='kw-2'>&</span><span class='ident'>ExprField</span>(<span class='kw-2'>ref</span> <span class='ident'>lfexp</span>, <span class='kw-2'>ref</span> <span class='ident'>lfident</span>),
<span class='kw-2'>&</span><span class='ident'>ExprField</span>(<span class='kw-2'>ref</span> <span class='ident'>rfexp</span>, <span class='kw-2'>ref</span> <span class='ident'>rfident</span>)) <span class='op'>=></span>
<span class='ident'>lfident</span>.<span class='ident'>node</span> <span class='op'>==</span> <span class='ident'>rfident</span>.<span class='ident'>node</span> <span class='op'>&&</span> <span class='ident'>is_exp_equal</span>(<span class='ident'>cx</span>, <span class='ident'>lfexp</span>, <span class='ident'>rfexp</span>),
(<span class='kw-2'>&</span><span class='ident'>ExprLit</span>(<span class='kw-2'>ref</span> <span class='ident'>l</span>), <span class='kw-2'>&</span><span class='ident'>ExprLit</span>(<span class='kw-2'>ref</span> <span class='ident'>r</span>)) <span class='op'>=></span> <span class='ident'>l</span>.<span class='ident'>node</span> <span class='op'>==</span> <span class='ident'>r</span>.<span class='ident'>node</span>,
(<span class='kw-2'>&</span><span class='ident'>ExprPath</span>(<span class='kw-2'>ref</span> <span class='ident'>lqself</span>, <span class='kw-2'>ref</span> <span class='ident'>lsubpath</span>),
<span class='kw-2'>&</span><span class='ident'>ExprPath</span>(<span class='kw-2'>ref</span> <span class='ident'>rqself</span>, <span class='kw-2'>ref</span> <span class='ident'>rsubpath</span>)) <span class='op'>=></span>
<span class='ident'>both</span>(<span class='ident'>lqself</span>, <span class='ident'>rqself</span>, <span class='ident'>is_qself_equal</span>) <span class='op'>&&</span>
<span class='ident'>is_path_equal</span>(<span class='ident'>lsubpath</span>, <span class='ident'>rsubpath</span>),
(<span class='kw-2'>&</span><span class='ident'>ExprTup</span>(<span class='kw-2'>ref</span> <span class='ident'>ltup</span>), <span class='kw-2'>&</span><span class='ident'>ExprTup</span>(<span class='kw-2'>ref</span> <span class='ident'>rtup</span>)) <span class='op'>=></span>
<span class='ident'>is_exps_equal</span>(<span class='ident'>cx</span>, <span class='ident'>ltup</span>, <span class='ident'>rtup</span>),
(<span class='kw-2'>&</span><span class='ident'>ExprVec</span>(<span class='kw-2'>ref</span> <span class='ident'>l</span>), <span class='kw-2'>&</span><span class='ident'>ExprVec</span>(<span class='kw-2'>ref</span> <span class='ident'>r</span>)) <span class='op'>=></span> <span class='ident'>is_exps_equal</span>(<span class='ident'>cx</span>, <span class='ident'>l</span>, <span class='ident'>r</span>),
(<span class='kw-2'>&</span><span class='ident'>ExprCast</span>(<span class='kw-2'>ref</span> <span class='ident'>lx</span>, <span class='kw-2'>ref</span> <span class='ident'>lt</span>), <span class='kw-2'>&</span><span class='ident'>ExprCast</span>(<span class='kw-2'>ref</span> <span class='ident'>rx</span>, <span class='kw-2'>ref</span> <span class='ident'>rt</span>)) <span class='op'>=></span>
<span class='ident'>is_exp_equal</span>(<span class='ident'>cx</span>, <span class='ident'>lx</span>, <span class='ident'>rx</span>) <span class='op'>&&</span> <span class='ident'>is_cast_ty_equal</span>(<span class='ident'>lt</span>, <span class='ident'>rt</span>),
_ <span class='op'>=></span> <span class='boolval'>false</span>
}
}
<span class='kw'>fn</span> <span class='ident'>is_exps_equal</span>(<span class='ident'>cx</span>: <span class='kw-2'>&</span><span class='ident'>LateContext</span>, <span class='ident'>left</span> : <span class='kw-2'>&</span>[<span class='ident'>P</span><span class='op'><</span><span class='ident'>Expr</span><span class='op'>></span>], <span class='ident'>right</span> : <span class='kw-2'>&</span>[<span class='ident'>P</span><span class='op'><</span><span class='ident'>Expr</span><span class='op'>></span>]) <span class='op'>-></span> <span class='ident'>bool</span> {
<span class='ident'>over</span>(<span class='ident'>left</span>, <span class='ident'>right</span>, <span class='op'>|</span><span class='ident'>l</span>, <span class='ident'>r</span><span class='op'>|</span> <span class='ident'>is_exp_equal</span>(<span class='ident'>cx</span>, <span class='ident'>l</span>, <span class='ident'>r</span>))
}
<span class='kw'>fn</span> <span class='ident'>is_path_equal</span>(<span class='ident'>left</span> : <span class='kw-2'>&</span><span class='ident'>Path</span>, <span class='ident'>right</span> : <span class='kw-2'>&</span><span class='ident'>Path</span>) <span class='op'>-></span> <span class='ident'>bool</span> {
<span class='comment'>// The == of idents doesn't work with different contexts,</span>
<span class='comment'>// we have to be explicit about hygiene</span>
<span class='ident'>left</span>.<span class='ident'>global</span> <span class='op'>==</span> <span class='ident'>right</span>.<span class='ident'>global</span> <span class='op'>&&</span> <span class='ident'>over</span>(<span class='kw-2'>&</span><span class='ident'>left</span>.<span class='ident'>segments</span>, <span class='kw-2'>&</span><span class='ident'>right</span>.<span class='ident'>segments</span>,
<span class='op'>|</span><span class='ident'>l</span>, <span class='ident'>r</span><span class='op'>|</span> <span class='ident'>l</span>.<span class='ident'>identifier</span>.<span class='ident'>name</span> <span class='op'>==</span> <span class='ident'>r</span>.<span class='ident'>identifier</span>.<span class='ident'>name</span>
<span class='op'>&&</span> <span class='ident'>l</span>.<span class='ident'>identifier</span>.<span class='ident'>ctxt</span> <span class='op'>==</span> <span class='ident'>r</span>.<span class='ident'>identifier</span>.<span class='ident'>ctxt</span>
<span class='op'>&&</span> <span class='ident'>l</span>.<span class='ident'>parameters</span> <span class='op'>==</span> <span class='ident'>r</span>.<span class='ident'>parameters</span>)
}
<span class='kw'>fn</span> <span class='ident'>is_qself_equal</span>(<span class='ident'>left</span> : <span class='kw-2'>&</span><span class='ident'>QSelf</span>, <span class='ident'>right</span> : <span class='kw-2'>&</span><span class='ident'>QSelf</span>) <span class='op'>-></span> <span class='ident'>bool</span> {
<span class='ident'>left</span>.<span class='ident'>ty</span>.<span class='ident'>node</span> <span class='op'>==</span> <span class='ident'>right</span>.<span class='ident'>ty</span>.<span class='ident'>node</span> <span class='op'>&&</span> <span class='ident'>left</span>.<span class='ident'>position</span> <span class='op'>==</span> <span class='ident'>right</span>.<span class='ident'>position</span>
}
<span class='kw'>fn</span> <span class='ident'>over</span><span class='op'><</span><span class='ident'>X</span>, <span class='ident'>F</span><span class='op'>></span>(<span class='ident'>left</span>: <span class='kw-2'>&</span>[<span class='ident'>X</span>], <span class='ident'>right</span>: <span class='kw-2'>&</span>[<span class='ident'>X</span>], <span class='kw-2'>mut</span> <span class='ident'>eq_fn</span>: <span class='ident'>F</span>) <span class='op'>-></span> <span class='ident'>bool</span>
<span class='kw'>where</span> <span class='ident'>F</span>: <span class='ident'>FnMut</span>(<span class='kw-2'>&</span><span class='ident'>X</span>, <span class='kw-2'>&</span><span class='ident'>X</span>) <span class='op'>-></span> <span class='ident'>bool</span> {
<span class='ident'>left</span>.<span class='ident'>len</span>() <span class='op'>==</span> <span class='ident'>right</span>.<span class='ident'>len</span>() <span class='op'>&&</span> <span class='ident'>left</span>.<span class='ident'>iter</span>().<span class='ident'>zip</span>(<span class='ident'>right</span>).<span class='ident'>all</span>(<span class='op'>|</span>(<span class='ident'>x</span>, <span class='ident'>y</span>)<span class='op'>|</span>
<span class='ident'>eq_fn</span>(<span class='ident'>x</span>, <span class='ident'>y</span>))
}
<span class='kw'>fn</span> <span class='ident'>both</span><span class='op'><</span><span class='ident'>X</span>, <span class='ident'>F</span><span class='op'>></span>(<span class='ident'>l</span>: <span class='kw-2'>&</span><span class='prelude-ty'>Option</span><span class='op'><</span><span class='ident'>X</span><span class='op'>></span>, <span class='ident'>r</span>: <span class='kw-2'>&</span><span class='prelude-ty'>Option</span><span class='op'><</span><span class='ident'>X</span><span class='op'>></span>, <span class='kw-2'>mut</span> <span class='ident'>eq_fn</span> : <span class='ident'>F</span>) <span class='op'>-></span> <span class='ident'>bool</span>
<span class='kw'>where</span> <span class='ident'>F</span>: <span class='ident'>FnMut</span>(<span class='kw-2'>&</span><span class='ident'>X</span>, <span class='kw-2'>&</span><span class='ident'>X</span>) <span class='op'>-></span> <span class='ident'>bool</span> {
<span class='ident'>l</span>.<span class='ident'>as_ref</span>().<span class='ident'>map_or_else</span>(<span class='op'>||</span> <span class='ident'>r</span>.<span class='ident'>is_none</span>(), <span class='op'>|</span><span class='ident'>x</span><span class='op'>|</span> <span class='ident'>r</span>.<span class='ident'>as_ref</span>().<span class='ident'>map_or</span>(<span class='boolval'>false</span>,
<span class='op'>|</span><span class='ident'>y</span><span class='op'>|</span> <span class='ident'>eq_fn</span>(<span class='ident'>x</span>, <span class='ident'>y</span>)))
}
<span class='kw'>fn</span> <span class='ident'>is_cmp_or_bit</span>(<span class='ident'>op</span> : <span class='kw-2'>&</span><span class='ident'>BinOp</span>) <span class='op'>-></span> <span class='ident'>bool</span> {
<span class='kw'>match</span> <span class='ident'>op</span>.<span class='ident'>node</span> {
<span class='ident'>BiEq</span> <span class='op'>|</span> <span class='ident'>BiLt</span> <span class='op'>|</span> <span class='ident'>BiLe</span> <span class='op'>|</span> <span class='ident'>BiGt</span> <span class='op'>|</span> <span class='ident'>BiGe</span> <span class='op'>|</span> <span class='ident'>BiNe</span> <span class='op'>|</span> <span class='ident'>BiAnd</span> <span class='op'>|</span> <span class='ident'>BiOr</span> <span class='op'>|</span>
<span class='ident'>BiBitXor</span> <span class='op'>|</span> <span class='ident'>BiBitAnd</span> <span class='op'>|</span> <span class='ident'>BiBitOr</span> <span class='op'>=></span> <span class='boolval'>true</span>,
_ <span class='op'>=></span> <span class='boolval'>false</span>
}
}
<span class='kw'>fn</span> <span class='ident'>is_cast_ty_equal</span>(<span class='ident'>left</span>: <span class='kw-2'>&</span><span class='ident'>Ty</span>, <span class='ident'>right</span>: <span class='kw-2'>&</span><span class='ident'>Ty</span>) <span class='op'>-></span> <span class='ident'>bool</span> {
<span class='kw'>match</span> (<span class='kw-2'>&</span><span class='ident'>left</span>.<span class='ident'>node</span>, <span class='kw-2'>&</span><span class='ident'>right</span>.<span class='ident'>node</span>) {
(<span class='kw-2'>&</span><span class='ident'>TyVec</span>(<span class='kw-2'>ref</span> <span class='ident'>lvec</span>), <span class='kw-2'>&</span><span class='ident'>TyVec</span>(<span class='kw-2'>ref</span> <span class='ident'>rvec</span>)) <span class='op'>=></span> <span class='ident'>is_cast_ty_equal</span>(<span class='ident'>lvec</span>, <span class='ident'>rvec</span>),
(<span class='kw-2'>&</span><span class='ident'>TyPtr</span>(<span class='kw-2'>ref</span> <span class='ident'>lmut</span>), <span class='kw-2'>&</span><span class='ident'>TyPtr</span>(<span class='kw-2'>ref</span> <span class='ident'>rmut</span>)) <span class='op'>=></span>
<span class='ident'>lmut</span>.<span class='ident'>mutbl</span> <span class='op'>==</span> <span class='ident'>rmut</span>.<span class='ident'>mutbl</span> <span class='op'>&&</span>
<span class='ident'>is_cast_ty_equal</span>(<span class='kw-2'>&</span><span class='op'>*</span><span class='ident'>lmut</span>.<span class='ident'>ty</span>, <span class='kw-2'>&</span><span class='op'>*</span><span class='ident'>rmut</span>.<span class='ident'>ty</span>),
(<span class='kw-2'>&</span><span class='ident'>TyRptr</span>(_, <span class='kw-2'>ref</span> <span class='ident'>lrmut</span>), <span class='kw-2'>&</span><span class='ident'>TyRptr</span>(_, <span class='kw-2'>ref</span> <span class='ident'>rrmut</span>)) <span class='op'>=></span>
<span class='ident'>lrmut</span>.<span class='ident'>mutbl</span> <span class='op'>==</span> <span class='ident'>rrmut</span>.<span class='ident'>mutbl</span> <span class='op'>&&</span>
<span class='ident'>is_cast_ty_equal</span>(<span class='kw-2'>&</span><span class='op'>*</span><span class='ident'>lrmut</span>.<span class='ident'>ty</span>, <span class='kw-2'>&</span><span class='op'>*</span><span class='ident'>rrmut</span>.<span class='ident'>ty</span>),
(<span class='kw-2'>&</span><span class='ident'>TyPath</span>(<span class='kw-2'>ref</span> <span class='ident'>lq</span>, <span class='kw-2'>ref</span> <span class='ident'>lpath</span>), <span class='kw-2'>&</span><span class='ident'>TyPath</span>(<span class='kw-2'>ref</span> <span class='ident'>rq</span>, <span class='kw-2'>ref</span> <span class='ident'>rpath</span>)) <span class='op'>=></span>
<span class='ident'>both</span>(<span class='ident'>lq</span>, <span class='ident'>rq</span>, <span class='ident'>is_qself_equal</span>) <span class='op'>&&</span> <span class='ident'>is_path_equal</span>(<span class='ident'>lpath</span>, <span class='ident'>rpath</span>),
(<span class='kw-2'>&</span><span class='ident'>TyParen</span>(<span class='kw-2'>ref</span> <span class='ident'>lty</span>), <span class='kw-2'>&</span><span class='ident'>TyParen</span>(<span class='kw-2'>ref</span> <span class='ident'>rty</span>)) <span class='op'>=></span> <span class='ident'>is_cast_ty_equal</span>(<span class='ident'>lty</span>, <span class='ident'>rty</span>),
(<span class='kw-2'>&</span><span class='ident'>TyInfer</span>, <span class='kw-2'>&</span><span class='ident'>TyInfer</span>) <span class='op'>=></span> <span class='boolval'>true</span>,
_ <span class='op'>=></span> <span class='boolval'>false</span>
}
}
</pre>
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OUTPOURING
BOOK SIX of THE STARLIGHT CHRONICLES
# Table of Contents
Title Page
Outpouring (The Starlight Chronicles, #6)
Warmth and Wakefulness
Ordinary
Lakeview
Spawn
Tests
"Discussions"
SWORD's Play
Friends and Family
Foundations
Findings
Prom
Last Dance
Fight of Destiny
The Void
Trial by Fire
Battle of the Heart
The Real Battle
Martha
Good-bye
THANK YOU FOR PICKING UP THIS BOOK!
Jennifer C. Sell
Amalia Chitulescu
EVERLASTING
Beginning Again
Sign up for C. S. Johnson's Mailing List
Also By C. S. Johnson
C. S. Johnson
Copyright © 2016 by C. S. Johnson.
All rights reserved. This book or any portion thereof may not be reproduced or used in any manner whatsoever without the express written permission of the publisher.
eBook ISBN: 978-1-948464-08-6
Print ISBN: 978-1-948464-07-9
For Sam. Every word I have written, I have written that you might believe, both in God and in the love and truth that sets both of us free.
I would also like to dedicate this book to my special fans in my favorite Mrs. Wong-Johnson's class, Kekoa and Marley. We are on separate halves of the world, and it is all wonder to me that my words can inspire you. Every artist has days of crippling insecurity, but thanks to you and your kindness and your enthusiasm, I have a perpetual silver lining in the clouds of doubt.
To Get Awakening (A Special Christmas Episode of The Starlight Chronicles) as a bonus for picking up this book,
Click Here
Download It At:
<https://www.csjohnson.me/awakening>
Check out Reflecting (A Dream Episode of The Starlight Chronicles), a short story that takes place before Book 5!
☼1☼
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---|---|---
# Warmth and Wakefulness
It was not the usual matter of desperation that fueled me forward, as I ran in the rain, heading toward my favorite coffee shop.
Don't get me wrong; despite the early, early morning hours, I fully expected to be greeted with a steaming, warm cup of coffee, one that was perfect for warding off the chill in the air. I knew I was going to need it to get through the day, and it was likely I was going to need the second or third cup I would leave with, too. But in recent months, coffee had become the secondary reason that I loved to stop in and sit for a while at Rachel's Café. (It was not a love easily dethroned, either.)
Coffee had been my true love, until I'd found my true true love.
I glanced up to see the soft light coming from the room on the second floor. She's awake.
I pushed open the back door to the small café and headed up the stairs, as silently as possible, and then all of a sudden there I was, standing in the doorway to her room. The echo of the rain was slightly louder, as the newly renovated wall in her room still needed some work, but the soft glow of her desk lamp was on, casting a small shadow of relief against the thunderstorm outside.
"Raiya."
She was sitting on her bed, her eyes glowing with wakefulness as she remained curled up in the warmth of her covers.
"What are you doing here, Hamilton?" she asked, her voice bracing against the subtlety of the night. There was no accusation in her tone, just surprise.
"I wanted to check in on you," I admitted, suddenly feeling dumb.
"At five-thirty in the morning?" Raiya asked. "I know I told you I've been having trouble sleeping, but it's—"
"Sorry." I scratched my head, suddenly very aware of how wet and cold I was. "I had a dream about you. I wanted to make sure you were okay."
"So you ran all the way here?" Raiya's lips curled into thoughtful smile. "You didn't want to call me?"
I considered arguing with her, which I would have delighted in, but thought the better of it. She was more beloved to me than arguing, too. "No."
"No?"
"I wanted to see you."
She pushed back her covers, allowing me a good grin at her fluffy-pants pajamas, and came over to me. "I'm glad you're here," she admitted, "and I would hug you, but you're all wet. Come on. I'll get you a towel and a cup of coffee."
"I feel like a king already," I said, although I probably looked more the part of the pauper. Even moments later, as my hands wrapped themselves tightly around my mug and a towel was draped over my shoulders, I felt more of the part of the humble and helpless, while I'd meant to be the hero.
"How's that?" she asked. "I can't imagine you're warmed up yet, but hopefully it'll help."
"You're the only Raiya sunshine I need," I assured her.
As she rolled her eyes and walked past me with a handful of creamers, I tugged on her shirt, pulling her in close. "Thank you," I said, as I finally got to kiss her again.
Raiya chuckled as she drew back. "It's my pleasure."
"No, you're my pleasure," I replied, staring at her long enough to make her blush.
She redirected me immediately; for all the brashness and boldness she had to stand up to me and my opinions, I knew and appreciated that Raiya had a modest side.
"Tell me about the dream you had. It must've been pretty bad if you're coming here this late," she prompted, as she moved to the other side of the counter. I knew she was making some tea. She loved her espresso as much as I did, but she was more of a tea drinker in the mornings, and I loved her for it. "Or should I say this early, since Rachel won't be here for another two hours?"
"Letty won't wake up, will she?" I asked, suddenly dreading the thought of Rachel's old-lady mother coming cranking down the stairs as we spent our time together.
"Not likely," Raiya said, effectively putting my shallow fears to rest. "I've been taking the morning shifts here at the café, since I dropped out of school. Aunt Letty doesn't usually wake up till noon anymore. Unless, of course, she hears me when I wake up in the middle of the night. But the rain should provide some cover tonight."
"I can't tell you how lucky you are, getting to drop out," I said. "Even if AP Gov is not the same without you to argue with."
"I imagine it's much more peaceful," she said neutrally.
"Peace might seem like an attractive offer," I said, "but I'll take arguing with you over semantics and historicity and context any day of the week."
"How is Mrs. Smithe?" Raiya asked. "Has she said anything else to you about SWORD lately?"
"Not since January," I said. "Almost two months later, and nothing in all that time."
"She's not the only one who's gone quiet," Raiya said as she sat down across from me. Her eyes fell to the seat that her grandfather, the esoteric and elusive Grandpa Odd, would sit in, and my reasons for scurrying over to see her immediately jumped to the forefront of my mind.
I reached out for her hand. "Everything will be alright," I said.
She squeezed my hand in return. "I'm not sure you know that," she replied easily enough. She sipped from her own mug with a peace I envied.
Raiya had a point, as she usually did, and it was a big one. If I truly believed things would be okay, why did I come running to see her before daybreak?
I shoved that thought aside. I loved her. I wanted to be with her. I knew we faced a considerable challenge. So, there was nothing inherently wrong with running through the rain and the dark of the night to see her.
"I'll admit, I'd feel better if we knew where Draco was hiding," I said bitterly. "I guess it didn't matter if he had his dragon skin or not. He's still terrible to try to locate."
"Agreed."
"He hasn't been here, has he?" I asked.
"No," Raiya said, shaking her head. I watched, transfixed, as some of her gingerbread hair broke free from the loosened bun at the back of her head. "Rachel and Aunt Letty were surprised to hear he went missing after the last attack near Rosemont. They haven't made much of a thorough investigation, but that's more because of the 'police' jurisdiction than anything else."
I snorted. "SWORD's going to have to think of a better cover soon."
"They've gotten away with sillier explanations," Raiya pointed out. "They've done more clean-up around the city, as far as damage goes. That's probably the reason that the assistant mayor's willing to let it slide for now."
I shrugged. "Assistant Mayor Dunbrooke doesn't seem as interested in the supernatural stuff as Stefano did."
"That's probably because he hasn't been taken over by a Sinister or a demon monster," Raiya replied.
"So far as we know." I frowned, thinking of the small, wiry man who seemed more machine than man, especially when it came to running what he referred to as "his domain." Which included me, for the three or four days a week I would go into work at City Hall.
I didn't mind that much. At least he was smart enough to leave me alone.
"True." Raiya smiled. "You have me there."
"Did I tell you that he's ordered the judiciary council to give Cheryl a deadline to produce the city superheroes?" I asked. "She has ten days to find them or the case is getting dismissed. Dunbrooke says it's costing the city time, money, and manpower."
"I'll bet your mom didn't like that."
"No," I said. "She didn't, putting it mildly. Blowing up ballistically when she got the report is more accurate."
Raiya laughed. "I would've loved to see her face. It's not often that the famous Cheryl Thomas-Dinger, the Queen of Apollo City Courtrooms, doesn't get her way."
"I'll try to get a picture of it when her time's up and she's left without us to fight in court."
"I'm assuming that your dad hasn't told her the truth about us?"
Thinking of my dad made me flinch. I shook my head. "No. He wouldn't. He knows how to keep secrets. And he's mandated to do so, with healthcare laws as they are. Or so he says. I can see him working around them if he wanted. Or," I added, "if Chery wanted."
When Raiya's grandfather revealed himself to be not only Elysian's rebellious brother Draco but also the mysterious Ogden Skarmastad, the founder of Apollo City, he gave us quite a surprise. And an unwelcome one, at that.
But finding out my father had known about SWORD and my secret superhero identity smashed through me. Since then, it was as if a chasm of secrets had suddenly pushed itself between us, damaging the ideas we had about each other irrevocably.
Mark usually came home late, left for work early—which really wasn't out of the norm—and our interaction was limited to the raw food dinners my mother's latest chef, a sushi master named Ayako, was making for us. We didn't talk much.
Raiya nodded. "I guess if he didn't tell her about me, he wasn't going to tell her about you. He loves you very much."
"Psh." I finished my coffee. "Coffee and intellectual levels, that's really all we have in common. And even with that, I'm pretty sure I'm smarter, and he likes his coffee darker."
"You really think you're smarter than your father?" Raiya arched her brow at me.
"I'm not the one who's best friends with a SWORD operative," I reminded her.
"Good point. You're making a lot of good points, despite being up this early," she observed.
"I know you're trying to get me off the original argument because you can't win," I told her, "but I'll humor you because I love you."
"I know you're just charming me because you're afraid I will come up with something better," Raiya responded. "But I'll humor you, because I love you, too."
I grinned. "Intellectual banter is so much fun with you."
"It always was for me," Raiya said. "Although I do miss you getting ticked off with me for winning before you knew who I was. That was pretty amusing."
"Ha, ha." I laughed drily. "If it makes you feel better, I'll start getting more angry when you attempt to win. But anyway, there are good reasons I'm awake and I'm here."
"Yes, you should tell me those." Raiya sipped her tea thoughtfully. "You mentioned the dream. Is there something else? Is Elysian bothering you?"
"I wish," I admitted. "He's been pretty alert and disciplined since Draco's reappearance. He probably sleeps less than you do."
"A considerable feat," Raiya said with a laugh. "Although I probably sleep more than you realize. I take naps after Letty relieves me, before you're out of school and swim practice."
"Thankfully the season's over now." I shook my head. "No new records this year, but still a lot of wins."
"Maybe you'll break some records next year," Raiya said.
"Will we be done with this mission by then?" I asked. That would be super. Absolutely perfect, actually. The sooner this is over, the happier I will be.
She shrugged. "I don't know. But there's no harm in hoping."
"I'm just hoping that I'll stop having these premonitions in the middle of the night." I sighed. "As much as I love you, and I love seeing you, Mark's already not exactly happy with me, and Cheryl's passive-aggressive enough to make me worried. I don't want to be punished for feeling like I need to come to your rescue."
"Couldn't have been that bad, even if you did run all the way here, and in the rain, no less."
"It was bad enough." I tapped my empty cup on the counter. "In my dream, I just saw you looking sad, like you were upset, so I wanted to come and rescue you."
Raiya pursed her lips. "I know that when I was in the hospital and attacked, you were scared," she said carefully, "but there's no reason to believe I was in immediate danger."
"Attacked" was the tidy way to summarize Raiya getting her heart smashed and her soul ripped out of her body just weeks ago. I clenched my fingers together, trying not to shout at her for her flippancy.
I calmed down enough before replying with, "I know."
I know, but I couldn't help it. Maybe I wanted to come more for me, than you.
"Are you sure you weren't the one who wanted me to comfort you?"
Hearing my own thoughts echoed back to me just made me more frustrated. "No," I insisted.
Raiya was smarter than that. "I know you better than you realize, you know." She laughed. "I still remember that whole issue last year with your birthday cake."
"I can't believe I apologized to you for that. I take it back."
She smirked. "It's too late, I already took it."
I stuck my tongue out at her, before sinking into silence.
"I know it doesn't help you any with Mikey still in the hospital," she added, after a while.
I still said nothing. Mikey had been my best friend, like my brother at one point. Now, he might as well have be permanently planted in the hospital bed where we could occasionally go to visit. As swim season dwindled down, I had a harder time not telling him he was going to pay for just lying around all day and night. At least the poor quality hospital food was keeping him from getting fat.
His mind and heart had seemed to heal more. He was, apparently, doing better with the tutor Central had sent over, and he seemed more like his old self when we went to see him.
Sometimes.
Then Mikey would remember he was supposed to hate me and I'd been the one who'd caused the demise of his true love, or whatever he wanted to call her, who just happened to be my ex-girlfriend.
I didn't think it was my direct fault that Gwen got her Soulfire stolen by Taygetay, one of the last of the Seven Deadly Sinisters we'd captured. If I had to make a case for it in court, I could probably make it convincing. But sometimes, when I did present the case inside my mind, it went back and forth enough between the "innocent" and the "guilty" verdict. It made me uncomfortable.
The best thing I could do, as far I as could figure—and Raiya agreed with me on it—was work to free Gwen's Soulfire.
It was probably going to take some time to destroy Draco, and that was of no comfort to Mikey, especially since he'd witnessed Gwen's pain.
I still had trouble seeing his PTSD diagnosis, but I did know that part of the reason for it was true, and the other reason was for his protection.
His estranged father, Dante, my less-than-agreeable and less-than-amicable, most-of-the-time contact from SWORD, was keeping him there. And I could appreciate it, because he was keeping him from my mother interrogating him about the identities of Wingdinger and Starry Knight—me and Raiya, respectably.
"Maybe he'll get out once the timeline on the case is over," I said. "Stefano said before only Cheryl could get to him now. Maybe once she's out of the way, the statute of limitations will be over, and Dante will allow Mark to give him a clear discharge."
"Maybe." It was Raiya's turn to shrug. She glanced outside the windows, where the rain was picking up, pitter-pattering down as it washed the world clean.
She picked up my empty cup and poured me a new one. "Let's not worry about it now."
"What?" Incredulously, I looked at her as if she'd gone crazy. "How can we not worry about this?"
"Talk to me about other things," she said. "Tell me stories of all the other girls at school terrifying you, thinking that you're not secretly in love with your coffee barista. Tell me about the swim team drama this semester."
When my mouth just dropped open, appalled at her appeal to the meaningless, she smiled. "I can tell you about some of the daytime soaps that Aunt Letty leaves on upstairs, if you can't think of something more interesting."
"Don't we have to worry about this?"
"We've worried about it for a long time," she said. "I need a break. Just a small one." She came around and sat down next to me.
Tentatively, I nodded. "Okay. I can think of more interesting things than Letty's soap operas. If you're sure you want to."
"I do," she said. "We'll go crazy trying to figure out everything right now. Let's just be normal for a bit."
"I can't argue with that," I said, and then I obliged her with stories of Poncey's latest pranks, the swim team's gluten-free swim-ghetti disaster, and Via's constant attempts to push her new boyfriend in my face, despite my eternal apathy.
I watched in wonder as she made faces and comments and more coffee.
It was a good two hours I got to spend with her, on a cold, rainy, late March morning, with nothing else to look forward to, except coming back to her at the end of the day. As Rachel came in, and customers soon after her, I wondered if I would have a "normal" life like that, where I wouldn't have to say good-bye to the "normal" parts and slink back to into the dread that accentuated my day.
☼2☼
| |
---|---|---
# Ordinary
To be fair, most of my day was still "normal." It just wasn't any real fun.
To be really fair, I don't suppose eleventh grade was really supposed to be fun. There's good reason the proverbial "they" have a special category for "angst" when it comes to teenagers.
"What do you think, Dinger?"
I glanced up from my computer screen to see Evan, better known as Poncey to me and my friends, looking at me expectantly. "What do I think about what?"
"Come on, man, were you listening at all?" Poncey sighed. "I was just telling you that Jason totally failed at asking Laura Nelson to the prom."
"Prom?"
"Uh, yeah, that huge party that we're all going to next month, remember?" Poncey reached over and poked my forehead.
I frowned at him before turning back to my work.
"It's that party you said you would even cancel your birthday party for since it's the same weekend."
It was very clear, at this point, I knew what he was talking about. I didn't know if he was doing this shtick to get me angry or not. It seemed that of late, my friends had been more determined than ever to get on my bad side.
"I get it, Poncey," I snapped. "I'm just trying to work. Give me a break."
"Calm down, dude. Anyway, Laura was flirting with him like she was just hoping that he would ask, and he did—"
"Yeah, that's nice," I grumbled.
"Hey, Mr. Gallows is out talking to the principal. You don't have to worry about getting in trouble." Poncey gestured toward the front of the computer lab, where I could clearly see Poncey was right. "Besides, it's Mr. G. He wouldn't punish us at all. He's too nice."
"Uh-huh," I muttered. "I'm working on my project."
Poncey shuffled himself in front of my screen. "No you're not. You're reading the local news!"
"You say it like it's a bad thing," I replied. "And I need some info for my project. That graphics project's not going to make itself."
"What do you need to know about"—Poncey peered closely at my screen—"the Flying Angels case?"
"I was looking for a picture," I lied.
"Well, you're going to get an A anyway. I don't know why you worry about it."
"Because I actually have to do the work to get the A," I answered. I was tempted to remind him of all the times I'd been the one who did his work so he could get an A, too, but I decided against it. I had more important things to worry about.
Such as the Flying Angels v. Apollo City case. The D. A., my mother, had filed for an extension. The courts would decide in a few days if she could have more time to produce the suspects in question. Other than that, they didn't give much information. Not that there would be a whole lot. Still, I nearly laughed at it. The media was clearly trying to appease the new mayor, since they'd taken out a lot of their commentary (they hadn't held back any chances to blister about Wingdinger and Starry Knight before, when Stefano was sane and still in good health.)
"So who are you taking to prom?" Poncey asked.
"Um ... what?" I turned back to face him after skimming through the rest of the article. "Oh, prom. I don't know. I might not take anyone."
"Come on, Drew's taking Simon's younger sister, Phoebe, and I'm going to take Felicia," he said. "Jason's been shot down, and Mikey's not likely to come, being hospitalized and all—"
"I don't know who I'm going to take," I said blandly. "I'll worry about it later."
Poncey frowned. "What's wrong with you, man?"
"What do you mean?"
"You're just acting weird lately," Poncey said. "I mean, you don't seem that interested in what's going on with all of us. Drew and Jason, and me, and all of us—we're your friends."
"Hey, I come to your parties," I objected. "I was just at Jason's last week, when he had that video game party all-nighter."
"And you sat there and just played and commented about half as much as usual." Poncey leaned back in his chair. "It just seems like you're not really paying attention to us anymore. Don't you care?"
"Of course I care," I snapped. "But I have a lot to do."
"You can tell us that," Poncey said. "We won't get upset with you for being busy, man."
"Well, I'm busy right now. I'm not going to worry about getting a prom date. Especially since it's a whole month away."
"All the good girls will be gone," Poncey warned me.
"I'll survive," I assured him. "But first," I said, trying to soften my tone some so it didn't seem like I was being so defensive, "I have a few things to take care of."
"That's right," Poncey declared, the rush of remembering in his voice making me instantly suspicious, "we have the SATs coming up."
I nearly groaned before I realized that was the perfect excuse for my moodiness. "Duh!" I said. "Why else would I be so on edge lately?"
"Yeah, I can't blame you. I signed up for that SAT class and I'm just totally freaking out," Poncey said. He continued talking to me about this while Mr. Gallows came back into the class.
When Mr. Gallows glanced my way, I gave him a rueful smile and shrugged. Mr. G was a good guy. He was one of those teachers that didn't ask for much; he didn't expect you to change the world with his imparted knowledge. He just helped you out as much as you wanted, and then let you be on your way.
He's still one of my all-time favorite teachers.
Mr. Gallows seemed to get my message that Poncey was hopeless, and there was no point in trying to get him to stop talking.
After class ended, and Poncey was still talking to me, I almost wish I'd advocated harder for Mr. Gallows to come and stop him.
"Drew and I are rounding up some people to get another season of Ultimate Frisbee up again," Poncey said. "You want to play with us more this time?"
"Well, I do love Ultimate," I agreed. "I'll pencil it in. Text me when you've gotten a time locked down."
"Sure thing, Dinger," Poncey said with a grin. "We're going to meet up next week after school, in Shoreside Park."
"Oh, cool." It was on the way to Rachel's at least. It was possible I would be able to stop by when I went to go see Raiya after school.
"Hey, Poncey, Dinger, wait up!" Jason, another one of my best friends, skipped up from behind us. "Can't believe it's not the weekend yet."
"I know, really." Poncey shook his head. "Not that this one counts for most of us, right?"
"Huh?" Jason looked confused.
"SATs, man. Where've you been?"
Poncey joined Jason in arguing over plans for the week, while I slipped out of the conversation.
The Flying Angels case's extension would be decided soon. If it was denied, I would be free of Cheryl's likely torture in court as of next week, even if I had to put up with her resulting tyrannical outbursts at home.
I prayed that it would be alright in the end. I was getting tired of that particular threat. Wanting to be a lawyer and getting caught up in the very system you want to learn how to manipulate didn't sit well inside of me.
A sudden movement caught my attention; I saw a sliver of light flashing off a coffee thermos. It was Martha.
A sudden impulse struck me, and I was unwilling to let it go.
"Hey guys, I'll see you later," I said, waving while I was already sliding away. "Gotta talk to Martha.
Martha—or Mrs. Smithe, until I graduated (got to be proper about these things, you know)—had admitted to me she knew about SWORD, and she'd used to work for them. I knew that she knew about other things, too, not the least of which was that I was Wingdinger and Raiya was Starry Knight, and that we were dating.
Maybe, I thought, maybe I can get her to help me. We need something to go on, after all.
Raiya and I were not going to succeed just waiting around for Draco to come crawling out of his hiding spot and announce himself. He'd been hiding out as Raiya's half-ancient grandfather for decades. Who knew how long it would take him to show his creepy, scaly, demonic dragon face around Apollo City again?
If anyone would know, it would be SWORD.
Maybe Martha was my ticket to finding SWORD.
"Mrs. Smithe," I said, as I inserted myself into her path. "I've been looking for you."
"What is it?" she asked, stepping around me and indicating that I should follow. "Speak quickly, because I have another class in three minutes."
"I was wondering if I could talk to you in private?"
"No." Her dark eyes were hard as steel as she looked at me. "I told you, I have a class in three minutes."
I sighed. "It's about SWORD," I said softly.
I should've known better than to ask her about that and expect her to be surprised. She narrowed her gaze even further, but her voice dropped its volume as she spoke.
"I've been expecting you to ask me questions for a while," she said. "It's about time. I would not have expected this long of a wait from you, Dinger."
I blushed, flustered. "I've been ... "
"Distracted, bored, or busy?" she asked. "It's always one or the other with you lately. I can't believe Raiya dropping out of school has affected you this much."
"It's not just that," I told her. "We've been ... " My voice trailed off as I realized we hadn't been doing that much. We discussed things, Elysian and I took turns at patrol, and Raiya kept an eye and an ear out for any whispers from where we knew we could get answers. But we were at a dead end. "Okay, we've run out of options."
"Waiting is generally the safe option," Mrs. Smithe said, not in agreement, but more out of default. "What it is you want to know?"
"Can we talk about this later?" I asked. "We both have class."
She stopped by her door, and then swiveled around. While she was shorter than me by a good deal, her gaze was sharp enough to make me straighten up. "Listen to me," she said, "and listen well, Hamilton. SWORD has a lot of resources, and they know of my position here. It's better if you don't actively seek my company quite so much, or they'll suspect you."
"They probably already do," I argued.
"That doesn't mean I want to make it worse." Martha huffed. "Now, we have a minute before class begins again. Ask your questions."
"I need to find SWORD," I said. "They've been hired by the city under the name Otherworld, Inc., but the Skarmastad Foundation is footing the bill. While I know where one of the agents is, I can't always go to him, either."
"You want to know where the business is?"
"Sort of. I'm not sure what kind of information would help."
"Those are poor research skills right there, Dinger."
"I know." I shrugged. "I'll fix it before college."
"Ha! That's a riot. But for SWORD, there's no reason I would know where they are now," she said. "I've been decommissioned, and for over a decade now."
"Do you know anything that would help us?" I asked.
"I don't know where their base is now, but I can probably find out," she said slowly. "But I need you to tell me something."
"Anything," I agreed.
"What is your goal with all of this?"
I looked at her, dumbfounded. "What do you mean?" I asked. I felt like I'd been asking that question a lot since I'd found out about my superpowers.
"I mean, what are you doing? I've seen what you and your friends do."
"We protect people."
"But what else?"
"What else?" I echoed the question like it was in a foreign language.
"Yes, what else?"
"What else do we need to do?" I asked, my tone now exasperated. "We do plenty."
"You're on the defensive. It's a losing position." Mrs. Smithe shook her head. "You need to start going on the offensive."
I suddenly understood what she was saying. "You're right," I admitted. "We don't have much of a game plan. We don't have a long-term plan, either."
"Tell you what," Mrs. Smithe said. "If I can get you the location of SWORD and its branches, then you have to do something for me."
I sincerely hoped it had nothing to do with AP Government or Mock Trial. She was already mad at me for not signing up this year. But I nodded. "Okay. What is it?"
"You should never promise something that you don't know you can fulfill," Mrs. Smithe told me in her most serious voice. I balked at it, and I was wondering if that was what she wanted me to do when she said, "I want you to stop asking me about it after this. Like I said, they might suspect you. Or me."
"Then why did you tell me about working for SWORD in the first place?"
She ignored the question. "Come back to me in a few days, just like this, in between classes when the hallways are full, and I'll let you know what I've found for you."
"Thank you," I said. I smiled up at her, grateful. "Raiya says thank you, too."
"Oh, really?" Martha smirked. "And I guess your dragon does, too?"
"Oh, yeah, you know it," I replied with small laugh.
"Is he here to deliver that in person?" Martha gestured toward the window. I leaned over, peeking through the door of her class, and saw she was right. Climbing up the waterspout was my stubborn and belligerent changeling dragon.
"If he is," I said, "I'm going to intercept."
"Good idea," Mrs. Smithe agreed, before she brushed past me, and turned her attention to her blissfully unware and waiting class just as the bell rang.
Well, it looks like once more, I will be missing my last period class.
Who was I kidding? I didn't mind.
But I did mind Elysian sneaking around the school, and I was determined to remind him of that as I headed out to meet him.
It was getting easier to cut out of class, I noticed, as I walked through the hallways toward the locker rooms. There was an entrance I knew of from my swim team insider knowledge, and the upside to using it was that it was close enough I didn't have to circle back around much.
That alone was probably the reason I managed to sneak up behind Elysian without alerting him to my presence.
"I'm going to kill you for this one of these days," I said, picking him up by his lizard-neck.
"Ha!" Elysian looped his tail around my arm and clung to me; he knew that drove me crazy. "Sometimes you don't even see me, you know. I was wondering if I was going to have to go inside today or not."
"I doubt that," I said. "Considering how you've been sitting around my house watching the news while no one is there, I'm willing to bet that's just the story you're telling to make yourself look good."
"What other kind of story would you possibly know of?" Elysian huffed. "All your stories make you look good in the end."
"That's because I'm the hero," I asserted. "And because I have a natural talent for getting out of tough spots."
"I would only agree to that because everything that's natural dies."
"Just shut up," I grumbled, finally getting tired of our argument. "Tell me why you're here."
"Your girlfriend asked that I come and get you once you were finished," Elysian said.
"I'm not done with school yet! I have one more period to go."
He shrugged. "What does it matter to me?" he asked. "You're almost done, that's close enough in my book."
"Your book must be abridged for morons." I slapped him off my arm and tucked him up behind my backpack. "I'm already too late to get away with getting back in without being noticed. I might as well leave."
"See?" Elysian sneered. "You're already trying to make yourself out to be some hero for skipping school. Starry Knight's enjoying herself at the moment, talking with Logan at the Lakeview Observatory. She'll be fine while she waits."
I paused. Dare I admit it to Elysian?
"I'm worried about her," I said a moment later. "I had a dream about her the other night. She was alone and terrified and sad."
"She usually is."
"Hey!" I whacked him over the head. "She is not."
"She doesn't seem to have a lot of friends besides us," Elysian noted. "And a member of her own family ended up betraying her."
I thought about it, and then I remembered what I'd seen before, what I'd heard before. "Adonaias is waiting for her," I said. "Surely, between us and him, that's all she needs. Raiya's much more introverted than either of us."
Elysian shrugged. "I guess so."
"She has Rachel, too," I added, thinking of the pretty redhead who ran the café with her name on it. "And Letty."
"Letty's not what I would call ideal company," Elysian said.
"I would've thought you'd like her," I said with a grin. "She seems to smoke as much as you do."
"Ha, ha, ha," Elysian grumbled. "You're hilarious."
"She's got some claws, too, come to think of it," I added, starting to laugh at my own joke as we headed out of the school and toward the hill by the marina. Elysian snarled while I snapped, as we headed toward Lakeview Observatory.
☼3☼
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# Lakeview
Elysian eventually managed to ignore me properly as we walked through Shoreside Park and then along the pier, and I promptly returned the favor with fervor, thinking of possibly asking Raiya to take a walk by the water with me in a little while. It was still early, and the sun was still out, even if it was chilly. We would have the time, and I wanted it.
Ever since she'd come back from visiting Alora, the Star of Time, and found out the truth of Grandpa Odd, Raiya hadn't had a lot of time with just me—or, maybe I should say, we didn't have enough time with each other. I know that was hard to imagine, considering I would go running to her house early in the morning, but it was true.
Raiya and I were busy, and we liked our own version of busy. She had her GED test prep and work at Rachel's in the morning, while I had work and the SATs and school. It was part of the reason we went well together. But I was determined to make her happy, and to spend as much time with her as I could. (It was limited enough as it was, really.) Especially time when we could be normal teenagers in love. As much as I loved spending time with her, I didn't really want to count trying to save the city as a bonding activity.
As we walked into the shadows of Lakeview Observatory, I pressed the four-point star on my wrist and transformed into my superhero self. Logan was on good terms with "Wingdinger." Or maybe it was Wingdinger that was on better terms with him than Hamilton. Either way, I needed to be able to get information from him, and I didn't think Logan would just up and give it to a mere acquaintance.
"Starry Knight's in the telescope room, talking with Logan," Elysian said. "Or at least she was when I left her."
"Does Logan know her real identity?" I asked. Raiya seemed pretty chummy with Rachel's brother-in-law.
"I don't think so," Elysian said. "Come on, she wouldn't have told you if you weren't going to find out, remember? Do you really think it's likely she would've told Logan?"
"I guess not," I said, as we approached the back door to the observatory. I didn't add that I was just a bit jealous of Logan. He had earned Raiya's respect since the beginning. I'd lost it several times, it seemed, even if I never lost her love.
Before I could ask Elysian what he thought she wanted, a guardsman came out. "You! Stop there," he commanded.
"Excuse me?" I shuffled a few steps back, making sure I was out of his reach. "I'm here on official business."
"You're trespassing," the guard insisted. "This is a restricted entrance." As he seemed to finally register the outfit, he gasped. (Seriously, how did anyone miss the wingdings?) "You're Wingdinger!"
"That's right," I asserted, trying to keep my attention on him, rather than his radio. The last thing I wanted was trouble.
But trouble didn't seem to ever take the hint, especially when it came to this sort of stuff.
The guard frowned and shifted his feet. I knew he was getting ready to charge. "There's a warrant out for your arrest."
He reached for the gun at his belt, and that's when I just reacted instantly, sending a jolt of power straight at him.
Before he could properly cry out, he fell over, unconscious.
I just stared at him. I'd never just attacked someone like that before, let alone a guard. It was too close to the police for comfort.
Elysian let out a guffaw. "You got him! Good aim."
"I hope I didn't hurt him."
"He was going to hurt you, with no qualms."
"Still, he was doing his job."
"Evil 'just does its job,' too, boss. We're allowed to disturb that," Elysian argued.
"Evil has to be intentional," I said, sure I was right. "I'm not sure if it's the same thing as ignorance."
"Willing ignorance is evil."
"Yeah, yeah." I waved him off. "We'll have to discuss it later; I'm not in the mood to discuss the finer points of ethics or philosophy or whatever it is." At least with Draco's revelation and betrayal, I don't have to worry about Grandpa Odd making me converse with him.
"Door's open," Elysian said after he used one of his claws to pick the lock.
"Cool."
We headed down the labyrinthine walls, the ones that somehow connected each room and closet. I peeked into the room where the meteorite had been kept prior to its theft; nothing looked any different from the last time I'd seen it.
The fuse box was still melted, and the rest of the display looked untouched.
I stopped for a moment. "Elysian," I said, "the fuse box has been melted. Would Draco be able to breathe fire like you can?"
Elysian snorted. "Of course," he said. "He's always been just as powerful as me. He never explicitly lauded it over me, at least until now. You've already seen how he can transform into people."
"Yeah," I agreed. "But he didn't breathe fire at us while we were fighting him, back when we first found out. Why wouldn't he, if he has such power?"
"Maybe I was right before, and he's not up to full power," Elysian suggested. "That would be my first guess. The second, well, you have to understand, that Draco is a fiercely proud dragon. If he didn't breathe fire when he fought us earlier, it's likely because it wouldn't have been up to his standards."
"But he could still do it?"
"Yes." Elysian nodded, his dragon head bobbing up and down.
"So he must've used the fuse box to disconnect the power," I said. "Just like Logan told me."
"It makes sense, too, because it is older technology," Raiya called out from behind me.
"Starry Knight." I greeted her with a big grin on my face. Quickly glancing over, I saw the lanky shadow next to her. "Logan."
"Nice to see you again, too," Logan said. He had the eager face that came with innocence, and some days I envied him for it. "Starry Knight was just talking about you."
"She was?" I knew the grin on my face turned into a goofy-looking one, and I didn't even care. That much.
"Logan's been able to reconfigure the radiation feed," Raiya told me. "I thought you'd like to see it."
It took me a moment to remember. (There's just so much stuff to recall about this stuff, it was a wonder I could remember any of it at all. Of course, that was where being a genius made a big difference.)
Logan told me before the meteorite had been giving off a strange radiation pattern, one that was a mirror image of the universe's. Almost like an infrablue, he said, instead of infrared. He had every reason to guess that it'd come from outside of our universe. I was more than willing to agree with his conclusion, although I didn't think it was going to help him make his case before the various scientific communities to tell him it had been from a supernova caused by Starry Knight.
In fact, I figured Logan would be laughed at. So I said nothing. It's better to let the facts speak for themselves at times like these.
The meteorite's radiation pattern had also shown up in various parts of town—specifically, in places that seemed to correlate with places where demons of all sorts attacked.
As I glanced over at the map of the city next to the computer reading, I knew all too well which attacks had been at which locations.
"So it's back up and running?" I asked. When the meteorite was stolen, the radiation tracker was disabled.
"Yeah, and it gets better. There's a collection of radiation that's growing just north of the center of the city," Logan said, waxing enthusiasm for his topic, "and the radiation isn't just different, but it's thick. There's a major flux going on, almost like a vortex, collecting the energy and perpetuating itself into ... "
I exchanged looks with Raiya as Logan went on with his scientific-sounding spiel. While I liked the guy, and I could appreciate a serious scholar not bent on world domination, this was more or less like one of Mr. Hale's lectures on science stuff I didn't care about, or one of Mr. Elm's chemistry lectures on stuff I only cared about to get me through the SATs.
When I saw what I suspected was a similar expression on Raiya's face, I almost laughed.
"Where is the center of the vortex, Logan?" I asked. "Can you tell me where I could physically see it, maybe?"
"Let me pull up the coordinates on the map," Logan offered.
When a little red blip appeared on the screen, we all focused in on a particular point. A second later, Raiya shook her head. "I should've guessed," she said. "That's where Rosemont Academy was."
"It's being cleared, since Maia more or less managed to bring it crumbling down," I said. "Why would you guess there?"
"Because," Raiya said, "The Skarmastad Foundation was a big donor in making the school as renowned as it was." She shook her head, and after glancing over at Logan, chose the rest of her words carefully. "I got a scholarship. My grandpa said it was a good move, especially since it would let me pursue my interests."
I knew she was thinking of her artwork, and she nodded. It would be a good set up, I thought. Grandpa Odd—Draco—could easily keep tabs on her.
"What happened when the meteorite struck down in front of it?" I asked.
"Grandpa was not happy," Raiya recalled. "I thought at the time it was because he was concerned for me, being upset, but I don't think that anymore."
"Are you guys going to go check it out?" Logan asked. "Do you mind if I come with you? The meteorite might be there, after all," he said. He pushed a stray lock of his black hair out of his eyes, looking at Starry Knight like a hopeful child asking his mother for permission.
"It would be best if you stayed here," I said. I was happy when Raiya nodded. "If there's a lot of radiation, it could mean more of the demons like the one who ... hurt you before." I still didn't think "possessed" was the right word. Especially after seeing Elektra and Asteropy hiding out in humans, too. Their minions seemed more like amateur puppeteers playing at power games too advanced for them.
Fortunately, Logan didn't make me explain anything. He just nodded, disappointed but secretly relieved. For a long moment, he almost reminded me of Jason on the night the meteorite crashed into the city.
Raiya patted his arm. "We'll come and check in more frequently," she said, "and we'll bring the meteorite back if we find it."
"Thanks," Logan said, cheering at her message. "The police haven't been terribly helpful, and I at least know that you have an idea of what to look for."
I laughed. "Yeah, the last thing you need is people bringing in fake space rocks for some reward."
Logan grinned. "Exactly."
"What else is going on here?" I asked. I'd failed to mention my attack on the guard to Raiya and Logan, but that didn't mean I'd forgotten it. "How are the Otherworld guards?"
"Tyrannical at times," Logan said with a shrug. "But it's expected, especially since we just got more funding."
"From the Skarmastad Foundation?"
"Yeah. They had insurance on the meteorite, if you can believe it."
"I'm surprised it was covered," Raiya retorted. "Who would cover a piece of rock?"
"It was quite a prize," Logan said, "but it wasn't really enough that the insurance companies would have noticed. The average layman would have no idea it was special."
"What was special about it, besides the radiation?" I asked.
"The metals and substances of it," Logan answered readily, making me suspect he'd thought this over before. "Most meteorites break down because of the heat and pressure from entering the atmosphere. This one did, quite a bit we suspect, but this piece is the core of it. It's awesome, and very strong."
"So basically it's pretty hard to crack?" I asked.
"Yep." Logan laughed. "You'd need some serious heat to melt it down. It's different from any other meteorite or meteor I've ever seen or read about."
"Maybe that's why the Skarmastad Foundation wanted to have it covered," I speculated.
"Maybe."
"And maybe that's why they hired extra people to guard it," I said.
"Most likely," Logan agreed. "But even then, I only found out about the composition a few weeks before it was stolen. They've had extra guards here for nearly a year, or maybe more." He glanced at Raiya. "I was hired here as the main lab coordinator once I entered my second year of graduate school. As long as I can come in and work with no trouble, I usually don't pay attention too much to what happens on a day to day basis."
"Passion has a way of clouding the mundane," Raiya replied.
Logan nodded in agreement, before glancing at his watch. "I'll say. I've got a few grant proposal sheets to fill out. You guys are welcome to stay as long as you can. I wouldn't touch anything, though," he said, looking at Elysian pointedly. "We're pretty conscientious about germs here, sorry."
Elysian grumbled to himself as Logan left. "Spoiled science brat," he muttered.
"Logan's been very helpful," Raiya shot back.
"Hopefully not enough SWORD will notice," I said. "Especially in conjunction with what happened earlier."
"What happened earlier?"
"The kid here was attacked by a guard," Elysian told her.
"He was doing his patrol and he said that there's an arrest warrant out for us," I said. "And so, when he started to look like he was going—"
"You're okay though, right?" Raiya asked.
"Yeah, of course."
"Good." Raiya sighed. "Why would there be an arrest warrant out for us, anyway? Don't you have to be proven guilty?"
"Not if there's suspect," I told her.
"Oh. I guess I should know that."
"I'm here," I reminded her. "I'll take care of the legal side of things." She grinned back at me, giving us a moment of warm comradery.
"What we should know," Elysian cut in, "is what's going on down near your old school, Starry Knight."
"I agree," she said. "But we're going to have be careful about it. Grandpa knows me really well. He'll know what to expect, and he can anticipate it."
"Don't you know him pretty well, too?" I crossed my arms over my chest.
Raiya flushed over red. "I thought I knew him," she said. "But we already know the depths of Draco's cunning. I can never be fully sure, when it comes to knowing what he is really like."
I turned to Elysian. "What do you think?"
"I think we should go and check it out before he does something with the meteorite," he said. "He's got his full power, and he in all likeliness has the meteorite, too."
"What can he do with it?" I asked. "I mean, I know he can cause trouble, but I don't have any specific idea."
"Causing trouble is the most specific we can probably get," Raiya said. "We'll need to investigate."
"Speaking of investigating," I said, "I'm trying to find Otherworld. See if we can find its hiring base or something."
"Why?" Raiya frowned. "You know that Dante works for them, but it's just a front for SWORD."
"Draco's been around for a long time," I said. "And he's old enough to have the same weakness as plenty of other older people. He's not as familiar with technology or anything remotely popular."
"What are you talking about?" Raiya asked.
"The Internet," I said. "Otherworld, Inc., as an incorporated business, has to submit documents every year, but I've checked, and there's nothing. No records. I was wondering if they were the connection to the Skarmastad Foundation."
"We already know they are the linked to the Skarmastad Foundation." Elysian huffed indignantly. "They were hired by them."
"But why?" I asked. "SWORD is a global company that's largely a shadow organization. They seek out power to control it. Why would Draco risk his plan—which we can all agree is pretty sadistic—and hire them? Why not just hire a regular bodyguard or a private police force?"
When neither of them could answer me, I said, "There has to be something linking them in more than one way."
"Like the insurance?" Elysian asked. "So he can get more money funneled into his company while still keeping the goods?"
"Exactly." I nodded.
"Maybe Draco's also the head of SWORD," Raiya suggested. "He's the founder of the foundation. Why couldn't he be in charge of SWORD, too?"
"I don't know about that," I said. "I know Dante knows the leader, and he said 'she' when he was talking about her."
"Maybe he hired them to focus on us," Elysian said. "As a distraction."
"That's a good theory," I acknowledged.
"Just be careful," Raiya said. "Grandpa—uh, Draco—told me about SWORD some. He said they were not to be trusted, even if they said they were on our side."
"And yet he hired them?" I asked.
"He's been here a long time," Elysian said. "Maybe he's picked up an additional grudge or two along the way."
I sighed. "This is not getting us anywhere. We need more information."
"Well, that's why you're trying to get it, right?" Raiya said. She nodded toward the door. "Keep us updated. In the meantime, I say we go try and get some information on the vortex Logan found for us."
Elysian frowned. "Let me go," he said. "I'll check it out by myself. Draco wouldn't expect just one of us to show up."
"I'm okay with that," I said. "But you only get to go for reconnaissance. No attacking him."
"Aw, come on," Elysian whined.
"No whining," I added. "We've got to get out of here carefully. We don't need the guards busting in on us because you're being a whiny dragon."
Fortunately, we were able to sneak out without interruption.
As Elysian took off for the Rosemont Academy remnants, Raiya and I stayed behind, watching him take off.
I squeezed her hand. "Hey," I said, glancing over at her.
"Hey, what?" she asked. There was a smile on her face as I reached over and took her hand in mine.
"So, I was talking with Drew today," I said.
"If you're going to ask me about patrolling the city while you have another game night," Raiya warned, "I'm going to tell you no. I don't—"
"It's not that," I assured her. "He was talking about the prom."
Instantly, I could see her body tighten up. "Prom?"
"Yeah," I said. "And I was thinking, it would be kind of fun to go with you. We can finally tell all my friends and the school we're dating, and it'll be the talk of the town for weeks."
"I don't know," she said.
"Come on, it'll be fun."
"Why would you think it'll be fun?" Raiya asked.
"Because you'll be there with me and all my friends and some of yours, and we'll, you know, be normal for a night."
She hesitated. I glanced closer, watching her emotions wisp off her in wavy ripples. Stress, fear, uncertainty ...
I stifled a groan. I mean, geez, all that from going to a dance? How did she manage to face down demons and fight off evil with poise and grace, but fall apart at the idea of having fun at prom?
"So are you asking me?"
Her question caught me off guard. "Oh, yes," I said. "I mean, will you go to the prom with me?"
She paused long enough for me to wonder if she would try to wiggle out of it. "Alright." She answered firmly, like she was trying to make up for the lack of enthusiasm.
Luckily, I had enough for the two of us. I took hold of her and twirled her around, before drawing her close to me. "Thank you," I said, before I kissed her soundly. "This means a lot to me. I'll do my best to make sure it's a night you'll always remember."
Raiya laughed, settling into the crook of my collarbone. I could feel how much more content she was as I put my arm around her. I was also more content, as we spent the next few hours walking through Shoreside Park together.
☼4☼
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# Spawn
As March transitioned into April, while my eighteenth birthday drew closer, and with Cheryl's deadline on death row, Raiya and I regularly met for breakfast after I was done with school. We were even finally able to study together—me for the SATs, Raiya for her GED—something I had been hoping to do for some time. For some strange reason (I hardly ever studied, I hardly ever needed to, and I didn't like to study with other people, so this was new), I was nothing short of sublimely happy.
I should've known it would come to an end.
I should've known it would come to an end.
Maybe I did know, but I was reluctant to admit it.
The first sensation my happy bubble was about to be blown away into pieces of bubble murder—not the exact feeling at the time, but more along the lines of the result—came the first weekend of April.
"What's the formula to figure out compound interest?" Raiya asked as she slumped over her test booklet.
I glanced over. "It's something with a 'p' in it, and a couple of 'n' and 't's,'" I said.
"I know that," Raiya muttered. "I can't remember the order."
"Doesn't your calculator have the automatic function for it?"
"Can't you just tell me the formula?"
"Isn't it on the page of the practice test?"
"No."
"Why do you need it then? Usually those tests give you the formulas."
"I don't see it—"
"Ugh, can't you two just stop talking, period?" Elysian snuggled into the booth we were sharing. He put his claw over his head, dramatically, as if he was going through some kind of fainting spell. "Just look it up and then shut up."
"Are we interrupting your rest, or are you upset I put a cap on your cookie limit?" I asked. "Because you can go back to my house. Or you're free to go on another round of patrols."
"Hey, I've been on enough patrols. You seem to forget who takes care of that sort of thing while you're in school, learning all those facts you'll likely only need if you wind up on a gameshow."
"Considering how you even know about gameshows tells me you've spent way too much time watching television," I said.
"They advertise it in between news reports," Elysian grumbled.
"That doesn't mean anything. You'd actually have to watch them to know what kind of questions—"
Raiya interrupted. "Elysian, stop hijacking my arguments with Humdinger. You know he can't resist arguing with you. Now, he's just going along with it so he doesn't have to admit to me he doesn't know how to find compound interest."
"I cannot believe I ever wanted to study with you," I said, half-teasing, half-exasperated. "Here, just give me your paper. I'll write it down."
Several minutes passed as I went through the test question and wrote down the formula for compound interest (which is P(1+r/n)^(nt) for those who don't know).
"There," I said.
"Thank you." Raiya gave me a smile and went to work on the next question. "Maybe one day I'll be able to return the favor."
"I'll keep that in mind." I turned back to my own work, which was less interesting than counting rocks. It was English work. "Actually, what can you tell me about The Great Gatsby?"
Raiya put aside her GED math book with a very clear sense of relief. I engaged her services as tutor/entertainer while she told me all about the tragic figure of F. Scott Fitzgerald and his hollow, glamorous life that slipped into sinking madness.
Elysian periodically let out a groan, but he didn't do much more than that.
I didn't mind (much.) Elysian hadn't been lying or even exaggerating—shocker—about going out and checking the town for himself. Ever since Logan had shown us what was happening with the radiation and its swirling signature, we'd been careful to make sure we checked every area close to the old Rosemont Academy building.
So for now, since we didn't see anything, I didn't care. More often than not, I was gratified by my semi-apathy, too, when Elysian or Raiya came back with nothing to report. I never had anything to report either, other than a headache.
True, it was a bit unnerving to worry about Draco. But as the days slipped by, and there was nothing to report, and very little to do, I was growing more sure with each passing day that there was nothing to worry about.
After all, Justice had come to Draco, and she was going to be his undoing, just like the Prince of Stars had said.
Again, I should've known better.
As I was finishing up a sample quiz while sipping the last of my mocha, pain pinched at my wrist.
Immediately, I shook it off. I mean, really, I was better off thinking it was possibly carpel tunnel, since I did type a lot, and working in the silly standardized testing booklet required more writing than I was used to.
Before I could question myself on the matter further, or maybe avoid questioning myself on the matter, Raiya distracted me.
"They're talking about the Flying Angels case on the news," she said, nodding toward the television screen over Rachel's bar.
"Don't pay any attention to it," I said. "Nothing good comes from worrying."
"They're trying to force the police to help capture us," Raiya said.
"Only informally," I said. "I mean, we're fugitives, but you don't see the police careening around the town doing raids."
"Still, it's disconcerting."
"I think it's more than fair to say Cheryl has that effect wherever she goes," I replied drily, not really wanting to worry about my mother.
Raiya's eyes lit up with laughter. "That's true enough."
"Another couple days and we'll be completely in the clear. Cheryl will have to appeal it, and Assistant Mayor Dunbrooke's already told her he has no intention of pursuing it."
"I wonder why."
"Maybe he's talked with Mayor Mills and decided getting too close to it would result in his own hospitalization."
"Or maybe it's just not popular enough of a topic to talk about anymore," Elysian said. "After all, there have been fewer attacks ever since you guys started working together. They came in random bursts, and now, with the Sinisters defeated, we only have Draco to worry about."
"They know a winning team when they see it," I declared.
Raiya arched her brow and turned her full attention to the screen.
Another burst of pain twisted against my pulse, but I once more pushed it aside. I didn't want to think about it. My SATs were this weekend, and my time to study was limited.
It was growing more limited, too, as I stopped more often than I should have to watch Raiya work.
What is it about really seeing someone? Raiya, as Starry Knight, told me once that I'd seen her, but I wouldn't know her.
Now I did know her, and for some reason it was easier to see her more clearly. As if by loving her, I knew her better, and saw her better, and loved her better because of it.
Some people talk about vicious cycles; I had to wonder what the opposite would be. A virtuous cycle, perhaps?
"What?" Raiya asked, as she glanced over at me.
I put my musings away for the meantime. "Nothing," I assured her. "Just watching you."
"Why?" Raiya glanced over at my practice test materials. "Stuck on the reading section again?"
"No, stuck on you," I told her.
"You know, that might get annoying," she said.
"What?"
"All your attempts at flirting."
"I'm not attempting if I'm succeeding."
"Still, it seems a bit forced."
"Only because I wouldn't naturally do it," I assured her.
"Sure you do," Raiya said. "I remember how you would do this all the time with Gwen."
The nerves around my wrist began to sound off like sirens. I grimaced. "I don't want to talk about Gwen," I said.
"But you flirted with her too," Raiya said. "I'm not bringing it up to make you mad, I'm just pointing out that your time with me is more important to me than your praise."
"I happen to like telling you that you're beautiful," I insisted, still trying to breathe properly while I was in pain. I cleared my throat. "Look, I might've flirted with Gwen while I was dating her, and probably before," I said carefully, realizing that even at my best, this was sounding worse, "but this time I'm with you. It's different because it's you."
She opened her mouth, probably to object, and I shook my head.
"Okay, look, everyone's out there, searching for love, right?" I held out my hands, angry I had to explain myself, but also frustrated I wasn't doing a very good job at it. "Some people try to find it with different people. It's the same with math, you know. You keep trying and adjusting things until you find the right answer."
"Does that make me your right answer then?" Raiya asked.
"Yes." I never hesitated.
She considered it a moment, and then smiled. "I like that better than the cheesy flirting," she said.
"You were the one who taught me that it doesn't count unless I suffer," I reminded her.
"Ugh, you need to write a movie or a sappy love song together." Elysian shifted in his seat. "Or maybe you need to write a book. Though I doubt a lot of people would buy it."
"Even if we included you in it?" I asked.
"Well, maybe," he said. "But you'd have to tone down my awesomeness, or I'd easily overshadow you both, especially if you have too much of that lovey-dovey mushy stuff in it."
I was about to start arguing with him when his nostrils flared. Elysian unfolded himself and slithered around the seat. "Something's wrong," he said.
"What is it?" I asked.
"Can't you tell?" He nodded toward my wrist, where what had begun as a simmer began to boil over.
"Well, um, I just thought maybe since the Sinisters were captured, I was just getting paranoid," I lied.
Raiya narrowed her eyes at me. "You should have said something."
"I didn't want to interrupt our time together," I admitted, somewhat sheepishly.
"Hamilton," she said, "this is too important."
"Can't you tell anymore?" I asked.
"Not always," she admitted. "I think it might be because of Grandpa—I mean, Draco. He knew I was sensitive to the monsters and their activity. He would be clever enough to take precautions if he could."
"So, we're both at fault."
She frowned. "Just say something if you feel something next time, alright?"
"Okay, fine," I snapped, tired of trying to impress her and please her. "Next time I'll yell it in the streets for you, too."
"We need to go," Elysian interjected. "So if you're going to fight like an old married couple, let's save it for later, shall we?"
I felt my face turn purple as angry fireballs appeared in my vision.
"We're coming," Raiya said. She shut her books and tucked them into her backpack, and then she reached for mine. "I'll put them upstairs and meet you guys on the roof," she told me.
I said nothing, still sore from her admonishment, only nodding and hurrying off after Elysian.
Elysian was in mid-transformation when I grabbed him by the neck. He choked and gasped. "What?"
"Don't make jokes like that," I scolded. "Raiya and I are not an old married couple, and I don't want her thinking that."
"You act like it sometimes," Elysian retorted. "And what's wrong with that?"
I watched as understanding dawned. "Oh," he said, trying to hold back his giggles. "I see! You're worried she won't want to—"
"Shut up," I grumbled, tossing him behind me. We turned down the alley and I pressed into my pounding mark, transforming into my superhero self.
My wings caught me as I jumped up to land on the rooftop. Raiya was already waiting for me, and I was more than pleased to see the feather I'd given her—one of my own burning flame feathers—in its familiar spot, tucked securely in her own bound-back hair.
I landed beside her, already looking around for any sign of an aura.
"There," I exclaimed, pointing off toward the marina. "There's a dull shadow coming up from the docks."
"I see it, too," Raiya said. Worry lined her face. "It's close to Lakeview Observatory."
I glanced over at her, and another aura caught my eye. I frowned and squinted, trying to see if I could discern anything else from it.
It was the vortex, I realized a moment later. It was close to the Rosemont ruin, and it was quietly humming a steady power, while the aura by the marina was erratic and fluctuating.
Before I could say anything, Elysian took off, and Raiya followed. "Hey, wait for me," I called.
"I don't know if it's Draco," Elysian rumbled as I caught up to him. "For all he is evil, he wouldn't leave this much of a demonic trail."
I didn't know what to say to that. While it was true Draco was more dragon than demon, he had captured the Sinisters, and Orpheus, too. They used their powers over their many minions to the point where, when we captured them or subdued them, their shadows still seemed to remain.
"It wouldn't surprise me if Draco was causing all the activity," I said. "Especially if he was working his way up to full power, as you surmised months ago, Elysian."
Raiya grimaced as she landed. "I don't think this is his work," she said. "This looks more like the work of a fenfleal demon."
A fenfleal demon was a rogue demon, one who gained power and worked separately from the Sinisters. I'd worked with a few of them before, so I knew they were tricky to handle. They tended to be more unusual, which, since they weren't following orders, was expected to some degree. But they were just as powerful, if not more so, because of their self-leadership.
As I landed, I tended to agree with Raiya's assessment.
There were three people on the ground, fallen over, their eyes vacant and their bodies still.
I walked over carefully to the nearest one, a man looking close to Mark's age. "Hello?" I shook his shoulder. "Hello? Can you—ugh!"
I'd glanced down to see his wrist was boiling with leaking black bubbles.
"What is it?" Raiya asked, hurrying over. "Oh, my."
"Yeah, it's gross," I said. "I don't know 'it' is, though."
"It looks like the demon used his mouth to suck out the soul," Raiya said. "He bit the man on the wrist, and he left a mark behind."
I glanced down at my own wrist, to see the Emblem of the Prince. "I guess that's not that uncommon," I said.
"The Prince and the demons don't have the same purpose or motive," Raiya reminded me. "And it's unlikely the Prince would use a pair of fangs to poison you."
"I guess so," I agreed. "I guess they can't help but imitate him in the worst possible ways."
"True enough," Elysian agreed. "You don't need to look far to see that in other areas as well."
"Well, I didn't have to look far to see it here, that's for sure," I pointed out. I dropped the man's arm and shuffled back from the burning black bubbles spitting out of his demon bite.
"Are the others the same?" I asked.
"They should be," a new voice responded.
A creepy feeling shuddered up my spine. Simultaneously, Elysian, Raiya, and I turned around.
And there it was. The demon, the fenfleal, was lounging around on the ground. It was long in body, almost like a worm or snake, with no hands and slits for its reddened eyes.
Eyes so similar to Draco's.
"Guys, that can't be ... uh, that can't be Draco, right?" I asked.
Elysian snorted as he leered down at our enemy. "No, it's not him," he said. "But he might be a spawn."
Even Raiya looked surprised. "A spawn?" she repeated.
"Yes," Elysian said. "He's been able to take a demon and overtake its sentience."
"I still remain myself," the demon objected. "I am Mahiem; but I will admit Draco's power infusion was a great resource for me."
"Draco's smart enough to figure out how to do it so they don't object, or even think anything's wrong with them," Elysian further intoned.
The worm-snake laughed. "You're not as smart as I figured you would be," he said. "Considering all I know about you from your brother."
Elysian growled impatiently.
"Wait." I pulled out my sword. "Release the Soulfire that you've stolen," I ordered.
"I'll release them," Mahiem said with an evil grin, "if you can defeat me."
"What good is the word of a monster like you?" Raiya asked, stringing an arrow in her bow.
When Mahiem only laughed, I took it as a sign that the fight was on.
"Go!" I called, rushing forward, ready to strike.
Mahiem twisted his long body out of my way, recoiling enough that he managed to strike me in turn.
I bucked against the blow, while Raiya hurried forward. Her bow's sharp edges were out as she swung.
Mahiem ducked and slapped her across the back, her wings catching most of the attack. I felt her pain, but I admired how she shrugged it off, same as she always had.
A whiplash of sparkling energy suddenly hit me hard.
"Ouch!" I grit my teeth together. Elysian wrapped his body around me and sent me a targeted look; I caught the meaning, and tepidly inched forward, out of his protection, but also out of the demon's sight.
"I'll catch you yet!" I heard Raiya cry, glancing over Elysian as she unleashed a slew of arrows at once. The arrows flew up, shining brightly, before winding around and binding him to the ground.
As Mahiem squirmed, I jumped forward from behind him; my sword raised high and then went low, cutting through him with quick integrity.
A whirl of energy whipped around, dissolving the demon's body and sending it into nothingness.
"We did it!" I cheered, watching as the Soulfire he'd consumed trickled back to the people on the deck.
"We're not finished," Raiya said. "Watch it."
"Watch what?" Elysian said. "Is Draco here?"
"No, I mean, watch it, the demon aura." Raiya pointed. "Mahiem's gone, but Draco's power remains."
I saw at once the ghost of a shimmering shadow, as it fluttered past, speeding away toward the vortex. I watched it as it disappeared under the city skyline. And then I blinked, and it was gone.
"This is disturbing," Elysian murmured. "He's using other demons."
"I wonder why," Raiya mused.
"You were the one who said he wasn't up to full power," I said, looking at Elysian. "Is it possible he's gathering more energy to refuel?"
"That's possible," Elysian said. He turned to Raiya. "What do you think?"
When she said nothing, he further pressed her for answers. "Despite all that you might think, you're still the one who knows him best."
"But he lied to me," she shot back, embarrassment flooding her cheeks.
"He didn't lie to you about everything," Elysian argued.
"No, and that makes it worse," she said. Her gaze lowered to the ground. "I can't tell you how often I've replayed it all over in my mind, trying to figure out what was true and what was a lie, and what could've been. It's horrifying."
I stepped up beside her. "Starry Knight's got a point, Elysian," I said.
"Think about what he told you regarding Alküzor," Elysian suggested. "He's unlikely to have lied that much about him, because you would have found out the truth."
"He said that he was trapped inside the fires of the earth," she said. "He said he was always trying to break free, but it would require a lot of power to do it. He also said that Alküzor wanted the power to steal this universe away from Time."
"Thank you," Elysian said. "I know it's hard, but we're going to do much better fighting him if we can anticipate him."
"We could also be playing with fire," Raiya replied. "If we focus so much on anticipating him in one regard, we'll completely miss what he has planned. And Grandpa—I mean, Draco—would jump on that."
"Maybe we can use that to our advantage," I suggested. "If we can make him think we're planning something specific, he'll think he has the advantage."
Raiya hesitated. "He might be able to guess that, too. He knows us well."
"You more than me or Elysian," I clarified.
"I don't know about that," she said. "Elysian's his brother."
"That doesn't mean anything," I assured her. "Adam and I are nothing alike."
"So far as you know," Raiya countered.
"Exactly my point," I told her. "I don't know him well enough to think that he would be like me."
Her mouth dropped open at my admission, and, I would argue, how I scored that point against her. I felt the rush of triumph, and I completely forgot about our earlier argument at Rachel's.
Elysian came to her rescue. "You're not actively trying to destroy your brother." He snorted. "And as annoying as you are, I doubt Adam feels like trying to do away with you, either."
"Fine." I gave up. There's no use trying to reason with people who refuse to see reason. "Let's assume he knows everything, then. Everything, every possibility. What would he be doing, using demons to collect power?"
"Power to set Alküzor free, power to regain his strength, power to separate the realm from Time's power." Raiya counted on her fingers. "Anything else?"
"Wouldn't it be so much easier if you could just ask me?"
At the sound of his voice, we turned. And there he was—his Santa Claus beard, his black robe, and his blood red eyes. I tightened my fingers around my sword. "Draco."
☼5☼
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# Tests
"Draco!" It wasn't long before Elysian echoed my sentiments on the matter. Elysian hissed and reared back, the horns and spikes down his back shooting up in protest.
Draco only laughed, sounding just like an egomaniac in Grandpa Odd's voice.
I felt rather than saw Raiya stiffen beside me. She straightened a second later, but from Draco's glare, I knew he'd seen her reaction.
"No need to fret," he told Elysian. He raised his hand. Seconds later, lightning flashed from his fist. "Why don't you lie down and relax for a bit?"
"Augh!" Elysian recoiled, unable to dodge the attack; he bore the brunt of it, until it bowled him over.
Raiya leapt into action, her bow out, and I followed her. We'd double-teamed enough demons in past fights, and practiced our striking points. I knew we could take him.
Boom! We met his power in mid-air, the compelling clash of our juxtaposed power hurting my ears as it sang out over the city.
"Augh!" Raiya and I both hollered as throbbing power charged through us, pushing us backward and forward at the same time, suspending us forcibly in our position.
Draco laughed as he poured out more power, keeping us close enough to provoke and far enough away we couldn't attack.
And then, all of a sudden, he thrust us back, and we went flying into Elysian.
"Come on," Elysian grumbled, shaking us off, sending us back up on our feet.
"You foolish children," Draco said, no doubt deliberately mocking me. "You're too far out of your league."
"It won't stop us!" I yelled.
"Too bad." He shuffled back, his cloak billowing. "I'll play the gentleman this time, then."
"You're leaving?" Raiya asked, surprised and appalled.
"Of course." He grinned. "But don't worry. I'll be back."
"Why?" Raiya shouted. "Why keep us in suspense? Why not finish us now?"
"For power, of course." Draco turned to her. "The Prince believes that power is ideally spread out among different people," he said. "And I'll agree with him, especially in a fallen world, for unity is a rare and dangerous thing."
"What does that have to do with you?" Raiya asked.
"The Sinisters, and Orpheus, while I am appropriately grateful for their sacrifices for me to recover my body and my power, their powers were too much for their discipline—or rather their lack of it."
"I'll say," I agreed.
"It didn't help that Time and Memory had managed to place a safeguard on their full power," Draco added.
"That's why you need more power?" Raiya continued, inching forward. I saw her plan at once; no doubt, Draco did as well.
"There's no fun, despite what the movies might say, in telling you what I'm doing," he said, glancing tauntingly at me. "Even though it would be amusing to see you ask. So I'll leave you with this for now: When it comes to collecting power and overshadowing lesser demons, you're completely right about me—and completely wrong."
And with a swish of his cloak, he disappeared into the wind.
Raiya launched out another arrow, but she was a second too late; she caught only the sound on the air as he disappeared.
We stood around where we were, all of us clearly torn between stunning confusion and harrowing indecision.
The sound of clicking cameras and approaching footsteps jolted us out of it a moment later. A few people called out our names and waved to us.
I waved back, awkwardly, my personality still drawn to appeal to the public good, while Elysian turned his back, and Raiya tightened her grip on her bow.
"I miss Aleia," I said. "Stopping time just worked so well for us."
"I wonder what Grandpa meant," Raiya wondered aloud.
"Draco," I corrected her. "And he probably just said that to confuse us."
"He had the upper hand," Elysian said. "He would tell us a half-truth like that to kick us when we're down."
"I don't think he was lying," Raiya said. "He didn't seem like he was lying. And he does enjoy a good riddle."
"What could he mean then?" I asked, beginning to back up as more people began to move over to where we stood.
"I don't know." She grimaced. "That's the inconvenient part."
"How can something be completely right and completely wrong at the same time?" I asked.
"True love can be like that," Raiya murmured softly. I almost didn't catch her words, and I knew why she'd been less than willing to share. She was no doubt thinking of Rachel when she said that, and that thought made me all the more circumspect.
True to my colors, I wrinkled my nose in disgust. "I doubt that has anything to do with that."
Raiya said nothing, only sighing as we took off, barely avoiding close encounters with the pedestrian kind.
*☼*
Later that night, as I was back in my room "studying" alone, I realized Draco had a way of frustrating us, even if we were getting better at fighting together as a team.
Maybe it was the situation or just how stressful and tiring it seemed, but there were other things that frustrated me, too.
I had the SATs this weekend. I had to get my homework done. I had to go to work. I had to do all this superhero work, and now, with Aleia back with Alora, the media were adding a lot of social pressure against me. I had my brother to worry about, my parents were increasingly distanced from me, and for the first time in my teenage life, that made me worried.
The pressure was mounting, and all I wanted to do was go running.
The door opened behind me. I was relieved when Elysian saw me at work, and then he merely grumbled and left the room.
I turned my attention to the window beside my desk. Night was coming swiftly, coloring the sky with the sharpness of its darkness. There was a violet overcast to the city lights, making the night seem more dangerous.
Of course, it was still a city, I told myself. There were still criminals, and homeless people, and people who needed help, and people who had plenty. There wasn't anything special about it, not really. Every day in town, people fell in love, people broke up, people grew apart. Others would learn, make mistakes, and try again. Some would give up. A few might make it big, while others destroyed things.
I thought about Adonaias, about how he seemed so far away and irrelevant to the ordinary person's suffering, the small voice in my heart asking if that included even my suffering.
I mean, really. How could just "having faith" and "accepting the belief" of who he was really change things?
Life was hard and unfair. That was the way of things. But how could this be the way of things? The angry, cynical part of me said that's how it was, and that was how it would always be.
But even as I turned my lights off and closed my eyes, drifting off to sleep, I knew I was not being fair. There was a lot to consider, much more—maybe too much more—than I wanted to, and I was tired.
*☼*
I was glad—really, really, really, supremely glad—that in all of Cheryl's crazy diets, she never failed to give up coffee. And there were some insane diets. Since entering high school alone, I'd had to deal with the vegan diet, the meat diet, the sugarless diet, the builder's regime, the kidney flush diet, the root diet, and many, many others. Some of them overlapped.
I was in the kitchen, making my own pot, the first time in what seemed like years, when Cheryl bustled into the room. She wearing her work heels, with her hair pulled back in a flawless knot.
Sometimes my mother was a terrifyingly precise woman. If I didn't know from my own personal experience, I would've thought she was a demon, from how perfect she always seemed to look.
"What are you doing here?" she asked, bewildered.
"I live here," I reminded her bluntly. (Not all of us woke up perfect and perky.)
"I mean, what are you doing here, making coffee?" She tensed, and I could see she was weighing her words carefully. "Don't you usually go out for coffee this early in the morning?"
"Would you prefer if I left?"
"No. I'm just ... surprised." She glanced over at the coffee machine. "It'll take less time if you push the start button."
"Oh." I pushed it. "Right."
I leaned back against the counter, while Cheryl stood there, looking at her phone and her watch.
"It's almost the last day," she muttered unhappily.
"The last day? Of what?"
"The Flying Angels case." She frowned, the lines in her face crinkling disapprovingly. "I miss Stefano," she admitted. "He would have given me more time to get those superheroes in custody."
"I'm sure," I agreed, thinking of Stefano's Sinister-influenced bloodlust. "I suppose Dunbrooke is less lenient?"
"Lenient?" Cheryl scoffed. "Ha! That's a poor choice of words, even for you, Hamilton."
"Sorry ... ?" I shrugged. "Martha told me that it wasn't likely that the city would indict them, anyway."
"It could have made my career," Cheryl snapped.
"Really?" I felt a rush of anger hit me as the coffee brewed and my mother stewed. "You would've wanted to be known as the legal form of a political tool who tried to bring people who were helping the city to so-called justice?"
Cheryl blinked, shocked and angry, blindsided by my words. "What are you saying? You wanted to help with the case."
"I did," I agreed, "and having seen it, and its near end, I can say I think it's terrible. That Dante person is terrible, and Stefano was obsessed about the whole thing. It didn't help him any, unless it made him look good to voters and his reelection campaign managers."
"Of course!" Cheryl fumed. "But it opened doors for us. For you, and for me."
"For what?" I shouted back. "So we could hold our heads up and look down on the crumbling ruins of the city?"
"We could've made a name for ourselves."
"You've already made one," I assured her. "My friends all know you. This city already knows you. You have a solid reputation."
"Don't you want one?"
"Not like this, and not like that," I said.
She calmed down, straightening her shirt. "I don't suppose this is because I didn't stay home with you as you grew up?" she asked. "Because I didn't mother you as much as you would've liked?"
"What? What are you talking about?" I cringed. "No. That has nothing to do with this—and that has nothing to do with anything. This is about right and wrong. The case was wrong from the start. It wasn't bad just because people don't want to consider the supernatural."
"It was bad because it caused a lot of damage to the city. There are still people who are affected by it. Your own friends are in the hospital because of those so-called superheroes."
"Those so-called superheroes are the only ones who were able to stop the demons from sucking out more souls," I insisted. "It's not my fault, or their fault, that any of those people are still in the hospital."
Cheryl narrowed her gaze. "You didn't think this before," she said. "Has it been that girlfriend of yours who's poisoned your mind against me?"
I didn't bother to tell her that I had thought it before, I just didn't tell her. "I'm not arguing against you," I said, throwing my hands up in exasperation. "I'm arguing against your stance and your underlying perspective." I almost roared at her, hoping to leave Raiya out of this, that she had nothing to do with this, but I knew that wasn't entirely true.
"Do you even want to be a lawyer anymore, Hamilton?" Cheryl asked me. "You're living in a fool's world. We don't live in a black and white world, where there's such thing as 'right' and 'wrong.'"
"Just because a lot of people say there is no 'right' answer and no 'wrong' answer doesn't mean that they're right," I said. "That's the majority fallacy. And anyway, as lawyers, we're only concerned with the law, and the ambiguity between theory and practice."
She stared at me for a long moment. "If you really believe that," she said, "you're going to have a hard time being a lawyer."
"Why?" I asked. "Are ethics and morality dead?"
"It's easier to do the work if you believe they are at least on life support," she said. "Idealism is never a good path, Hamilton. Disappointment can be harsh."
"Life is harsh," I said. "But there are still things worth living for."
"You're too young to know that."
"No, I'm not!"
"Yes, you are." She shook her head. "You only know what's worth living for when you've found something worth dying for."
"Dying for something is easy!" I yelled back. "It's too easy! Living for something is harder."
"You've changed, Hamilton." She sighed. "This is because of your girlfriend, isn't it?"
"You leave her out of this," I snapped. There was no way I was going to admit she was right—well, partially right, anyway. There were other reasons I'd changed, too; Raiya just happened to be one of the bigger reasons why.
"I'm only asking because I'm concerned for you," Cheryl insisted.
It always amazes me in life how the people who say they are "trying to help" are usually the ones who are doing the most damage. It is almost like a red-flag phrase.
"I'm not worried," I lied.
"What are you going to do?" Cheryl shook her head. "You want to be a lawyer at Pitt still, right?"
"Of course."
"Is she going to just move with you to college? You don't know what she wants to do, do you?"
I said nothing.
That was a mistake; I should've lied, fast and immediately. Cheryl knew the real answer at once, and she began to milk it for all it was worth.
"What about her family? Is she going to leave them to go with you? How are you going to provide for her? I know from paying all your bills, you couldn't afford standard rent and your coffee habit. And if she stays here while you go off to school, how do you know she won't find someone else? That she won't find a job here or far away from here that she'll want?"
"That's none of your business."
"I'm just trying to make sure you know what you're doing. You need good grades and high scores on your tests to get scholarships and awards and prizes, Hamilton."
"Have I not been doing that?" I asked.
"Technically, you have done that. But lately, I see you're distracted." She shrugged delicately. "You're going to need to change that if you're still working toward dual enrollment next year."
The coffee machine beeped, probably saving me from screaming at Cheryl until the rest of the neighborhood woke up.
I swore I could almost hear Elysian chuckling at the picture I made.
I poured a cup into one of the many hundreds of coffee thermoses we had around the house. "Well, I have my SATs today," I said, abruptly changing the subject. "Let me go ahead and show you how not distracted I am."
I shoved the coffee pot at her and turned away, secretly hoping I'd managed to spill some drops on the counter just to tick her off.
Getting outside the door was walking into the very essence of sweet relief, even if I was headed out to take a test on a Saturday.
The chilly morning air reminded me that there was still some talk of snow coming, even in April, but the steaming cup in my hand shut that possibility up.
At the bitter taste, I suddenly wondered why I hadn't gone to Rachel's. It was true that since it was Saturday, she wouldn't have been open until seven, the same time as my test, but I could've managed to get a cup out of Raiya.
I looked down at my cup, seeing past the coffee-colored liquid joy to see my own reflection, and I knew.
I was still feeling unsettled about yesterday.
I didn't want to fight with Raiya. At least, not to the point where I hated her for her remarks.
I didn't even know why I was bothered by our argument. Sure, maybe I should've said that I was sensing a demon attack. But my reluctance to ruin our time together should've redeemed my poor judgment in the matter.
Well, I decided silently to myself, poor judgment yesterday doesn't qualify for a follow-up today.
I decided I would go and see her immediately after my SATs were over. I had four hours to take the test, and then I would be done. Maybe I would even feel better to the point where I would welcome some demon-fighting action.
*☼*
I can't imagine sitting in a desk chair on a Saturday for a good half of a school day is something a lot of students gleefully anticipate.
SATs, the standardized college tests, were a legacy of the twentieth and twenty-first centuries. Even as I sat down for mine, I had a feeling they would either go away or they just get more complicated in time. I thought it was hard to argue for them, generally speaking, but it was harder for schools to look for another way to get a kid's parents to shell out a considerable sum in order to look good on a college admission packet.
Cheryl was right, for once, about college and dual enrollment. I needed that in order to skip through the first year of college, which I knew from my own investigations was similar to my high school classes. It was possible if I did well enough, I could graduate from law school by the time I was twenty-one.
Raiya had joked, when I told her, that it was perfect, since I could start my career the same time it was legal for me to become an alcoholic. I smiled at the memory. I wasn't worried about law school. I'd succeeded in school stuff all my life. It wasn't hard for me to learn.
It was the physical toll on me that I was less enthusiastic about, and the SATs were goading me more there than anything else. Actually succeeding into college clearly required the ability to sit and wait for an hour, and then sit, answer questions, and then wait for another hour or two, all while having no access to the internet; I'm sure making the money from the testing fees was just a perk for colleges.
Other than that, the test wasn't hard for me. I was done with each section with tons of time to spare, and no Game Pac or phone or anything to play with. I had to fight off the temptation to bother Jason, who was just a row over from me; though, in all fairness, it would've been a short-lived endeavor, since he was taking his precious time.
Poncey was a couple rows over from Jason, but he couldn't see me very well. It wasn't like he could turn around without one of the testing proctors getting angry, anyway.
My desk was next to the window, and that was seriously the only entertainment I could enjoy. And the horizontal blinds were halfway shut, making it all the more enjoyable.
[Insert sarcastic eye roll.]
It was boring and I don't remember much, only staring into space for hours, and then looking up at the clock to see approximately two minutes had passed, and then repeating this a bunch of times. I also remember glaring at the clock, and mentally yelling at Aleia to speed things up for me.
Surely she wouldn't deny me that request? She was a friend, after all.
Adonaias probably wouldn't let her, I thought an hour-long moment later.
There is absolutely nothing worse that this—
My train of thought was derailed and ran into a barn silo as searing soreness clasped around my wrist.
Again?!
I almost cried out an overly-clichéd long, drawn-out "No!" If there was any appropriate time to do just that, it was at that moment, while I was sitting with an hour to go on my SAT writing test, and demons were working in all their terrible ways.
Glancing out the windows, I could see next to nothing, other than the other wing of the school building. I was in a different classroom than my normal routine, so even Elysian would have a harder time finding me.
Raiya, I knew, had opted not to take the SATs. She decided to worry about her GED test first, and then she said, she would "worry about college when and if the time came." I assured her, in my typical, unfeeling fashion, that it was a matter of "when."
College had been a mandate on my life for forever. Now that she was a part of my life, it was one on hers, too.
There was a loud crash! that sounded from out the window.
But fighting off the demons, and all of their leaders and masters, was also a calling on my life.
Instant frustration ran through me. I felt like Draco did this on purpose, like he had been working in secret conjunction with Cheryl. Her challenge to me this morning burned into my mind, making me hate my mother and life and everything else all the more as I faced brutal reality. I could taste blood as I contemplated revenge.
I did this while I remained in my seat.
I let the pulsations of pain and suffering slither through me like a poison, seeping from my wrist down to my heart and infiltrating my mind.
Never did I feel the demand on myself so much as when I sat there, writing my essay, reading through questions, figuring out math problems—all while my friends were called to the battlefield.
I glanced at the clock.
Come on, Alora, stop time for me so I can help ... Aleia, tell Alora to help me!
When I got no response, I turned to a higher power. Adonaias, where are you? Answer me! Go help Raiya and Elysian! What could you possibly be doing that matters as much as this does right now? Why aren't you helping them?! Why are you keeping me here?!
Nothing.
I got nothing.
Minutes passed. I could hear sirens and screaming. I could hear buildings buckle. At one point, I felt the shockwave of an attack. My heart twisted in agony, and I wondered if Raiya was doing okay.
But I did nothing.
I couldn't leave. I mean, I really couldn't. Not without cancelling my scores, forfeiting my testing fees, and losing my chance for dual enrollment at Apollo City College next year! This was the last testing session they had before my application had to be in. Even if I took the test again, it would be too late to apply for the program.
There was also the matter of my mother. If I left, I would be proving her point—that I was focused on other things besides my long-term success.
Not to mention, I rationalized, I could easily draw suspicion to myself and Wingdinger, if someone were to realize the connection between my disappearances and Wingdinger's appearances. Gwen had been quick enough to pick up on it, hadn't she, months ago? And she was quick enough to use it against me, too.
So I did nothing.
All my conflicting panic stymied me, stilling me, trapping me in my chair.
I was only slightly comforted by the fact no one else who was around did anything. I could see the confusion on the faces of the testing coordinators. Some of them asked questions over their radios, but there was no change.
All of us in the room were told nothing—nothing other than what to do, how to answer questions, how to pack up our materials, and when to expect our scores.
The dam of confusion and anger broke the instant we were allowed to pack up and leave.
A hand clasped me on the shoulder. "Dinger!" Poncey called. "What do you say?"
"To what?" I snapped.
"To going to my house, kicking back, and playing some video games? I got Drew and Simon coming," he said, clearly unaware of my inner turmoil. "They're going to bring over some—"
I shook my head quickly, interrupting him. "No." I pushed him away. "I gotta go."
"But—"
"I gotta go," I repeated brusquely, emphasizing each word, like he was unable to understand me.
He had a hurt expression on his face, and I felt my frustration compound itself further inside of me.
But I brushed it off. I had to. I had to go.
*☼*
"Augh!"
I winced at the sound of Elysian's cry; I was flying over the old Rosemont Academy grounds as I saw him slump down against the ground, defeated not entirely in his body, but somewhat in his spirit.
I shook my head, which is a bad idea to do mid-flight, and tried to reassure myself it was not too late to make a difference. There was a price to be paid when it came to making difficult choices.
Raiya, transformed into Starry Knight, was fighting with Draco in his human form. His energy crashed against hers as they edged closer to the vortex.
"Hold on," I yelled, probably more to myself than to either Starry Knight or Elysian. "I'm coming!"
The aura around the battlefield was deadly dark; I could see the demonic spirits swirling around inside the core of the vortex.
I landed beside Elysian and put my hand up to his scaly cheek. "Are you okay?" I asked.
His jaw snipped at me, as angry smoke came rushing out. "What took you so long?!" he yelled.
"I had my test today. I couldn't leave."
Elysian roared at my words. He blew a string of fire out of his mouth as his body snapped back into action, his will sharpened by his anger.
Even if it was because he was angry with me, it was good to see him moving.
But it was not good to see the fire stirring. The vortex grabbed up the celestial fire he unleashed, spinning in faster circles with flames leaping up out of the mix.
"What's happening?" I called, keeping up beside Elysian as he hurried over to dodge a spurt of his own power.
"My power is mixing with the demon remains," he explained. "We need to make sure it doesn't spread!"
I glanced at the fire. How was I going to stop that? My power would likely cause more trouble for us. Would my sword work?
Doubt burned into me, as the flames grew hotter.
"It's nice of you to finally join us," Draco said to me. "You're just in time for the big finale." Calling forth his power, Draco squeezed it into his palms, gathering it up together in one big energy ball.
I stepped forward, my sword out and ready. "I'm the one who just got here, and I'll decide when the show's over—"
Draco didn't wait. He unleashed his power; it circled around me, before trapping me in a deadly spiral.
"You ... won't ... win," I said, gasping as his power met mine. The sword in my hands shook from the pressure. I could feel my power bend around me, protecting me, but I was unable to move forward against the surging tide.
I caught sight of Raiya and Elysian; both were struggling in the power fields, the same as me.
Draco's cold laughter forced my attention back to him. His power continued to grow as the vortex opened up to reveal its heart—or rather, what was lying at its heart.
A strange but familiar rock was there, glowing with a blueish shimmer.
It was the meteorite!
The same one that had smashed through the Rosemont Academy streets, the same one that had blown up as it was being moved to a lab, and the same one Logan had kept watch over for all those months at Lakeview Observatory.
I watched as the meteorite burned away, the shadow of Elysian's flames and the darkness of the demonic aura reforming its shape.
The power binding me slowed, and I was able to wriggle into a weak spot. Seconds later, I felt a rush of jubilee. "I'm out!"
"Watch out!" Raiya called to me as she was pushing through her own bonds.
I jerked my sword up just in time to stop another round of Draco's energy. "I can't hold out forever!"
Draco's red eyes pierced me through the wall of flames between us. He opened his mouth, and for a moment I thought he was going to laugh or gloat again.
I was wrong.
A burst of fiery lightning erupted out of Draco's mouth, scorching the skies and mixing with Elysian's fire in the demonic pit. Power unleashed upward, cutting its way through clouds.
It was loud and terrifying and I wished I'd had earplugs. A foreboding hum electrified through me. My skin crawled with light and chaos, as my ears closed up at the frightening force.
"No!" I saw, rather than heard, as Elysian cried.
When I could see again, the only things I could make out clearly were Draco's smile, and the newly-transformed meteorite.
It was no longer a rock, but a weapon of power and shadow; the dragon fire combined with the pressure of the vortex had forged a sword.
As the vortex disappeared, Draco reached out for his prize. He grabbed it by the hilt, and it was at once remade, colored with the emptiness of his soul. The power holding me back, the power holding Elysian and Raiya back, disappeared.
"Excellent," he hissed.
The vortex was suddenly gone; even the road seemed to fix itself, emptied of the demonic power it once held.
Fully released from Draco's power, Raiya fell forward onto her knees, clearly spent.
"Stop him before he gets away!" Raiya called to me.
I didn't even stop to think; I knew she was right. Elysian and Aleia had warned me before that objects such as the meteorite could be used to cause further damage, and its power would magnify when used to further evil's cause.
That had to be it!
He'd used the vortex as a dumping ground for calamity, and used the fire and power provided to meld the meteorite into an even deadlier weapon of choice—one that could counter our attacks and serve an even darker purpose.
Our swords met and clanged, sending a tremor through my body. Lightning flared out from our swords, as mine attempted to seal away a sword of nothingness.
"Ouch!" He managed to score a lucky swing, and I tumbled to the ground, cursing my delayed defense.
His eyes narrowed in grim pleasure as he lifted his sword over me, preparing to deal me another blow.
I braced for it, calling my power up to shield me. My eyes squeezed shut, bracing for the impact.
Nothing came; I heard nothing other than a quick intake of breath.
Glancing up, I saw him glaring at the arrow of light sticking through his shoulder.
Behind him, I saw Raiya shaking. She'd broken free, and her bow was out; she had a despondent look on her face.
Draco surprised us both by grinning. "Tsk, tsk," he told her as he jerked the arrow right out from his body. The purity of Starry Knight's power had decimated a hole right through him, but a second later he healed himself. "I thought I taught you better than that, Raiya."
Raiya looked as shocked as I felt. I knew intimately what kind of power she was capable of wielding. What happened? And why wasn't she shooting another arrow?
"But it's an improvement from the last time we met," Draco drawled on. "I'll have to work around that."
I shot out my own power at him, hoping with his attention diverted that he would be vulnerable.
No such luck, of course.
After dodging my attack, Draco laughed and faded away. His voice whispered past my ear as he left.
"Parting is such sweet sorrow, is it not, young Hamilton?"
His words echoed inside of me, taunting me, making me flustered and frustrated.
What did he mean by that? I wondered.
Of course, I had to wonder if it meant anything at all. Clearly, it was an insult of some kind, I get it, but insults don't usually have hidden meanings in them. They were just insults.
The scene of the fight cleared, and I was left, still hunched forward, my sword limply held up in my wrist, standing alone, unable to get over the meaning or the final message.
"Well," I said as I allowed myself to regroup, "that was weird."
Raiya and Elysian stepped forward. I turned to face them, knowing they would be disappointed.
"Good job, kid." Elysian snorted. "Now he's got a sword, and one that has a considerable amount of power, too."
I frowned. "It's not like I could've stopped him from leaving," I argued. "He just disappeared into nothing."
"We might have been able to stop him if you'd gotten here earlier," Raiya said.
It took me less than a second to wheel around. "Excuse me? You were the one who clearly hesitated here!"
"What do you mean? I managed to get him!"
"You told me once your power allows you to hit any target you want. Why didn't you get his heart?"
"You know why!" she shouted.
"Well then, it's both our faults at best that he got away!"
"We still might have a better chance if you'd gotten here earlier," she repeated, this time with more anger than I'd been expecting.
"I couldn't get here any sooner," I declared. "And at least I got here after the SATs finished up."
"Is that why you were late? Seriously?!" Raiya stepped forward. "I can't believe you! We just talked about this yesterday."
"Hey, it's one thing for me to keep quiet because I wanted to spend my time with you," I replied, "and it's another thing for me to ruin my chances at dual enrollment next year. I have to consider the future. Once Draco's defeated, our mission will be complete, and we'll be able to return to our regular lives."
"You know how important it is that we protect this city," Raiya argued. "There won't be a future if you don't work for it!"
"Exactly my point," I shot back. "I won't get into college if I don't take these tests seriously."
"These tests are secondary to survival," Raiya shot back. "You're risking that on the chance we actually succeed! Why are you so eager for your time as Wingdinger to be over?"
"Because I actually have a life separate from him, unlike you!"
She looked stricken, as though I'd struck her, and I knew instantly I'd made an unfair argument.
But ...
I also had a point, as painful as it was for her to admit.
"Come on, Raiya. I have a life separate from this," I said again, "and frankly, I like it."
Her eyes lowered, and I cringed.
"I don't consider you separate from my normal life," I quickly explained. "You're the best thing in my life. But don't you like just hanging out, you know, talking about school and college and the future, or even talking about stupid things, like the terrible shows on TV and how ugly someone's new haircut is it just how awful it is waiting to grow up?"
"If you have to ask me that," Raiya said, "it's clear that you haven't grown up."
"I'm not even an adult yet," I said. "It's easy for you to say, because you're eighteen."
"I'm not talking about age!" Raiya folded her arms across her chest. "I'm talking about maturity. You have seen things, things not just anyone has seen. You know better. And you still choose the temporary things over the eternal."
"That's because the temporary has an effect on the eternal!" It was my turn to glare at her. "If you can't see how this matters to me, you have some nerve saying you love me."
"Real mature," Raiya muttered. "I suppose you're going to stomp your foot and march out of here with your nose up in the air?"
"Maybe I will!" I huffed indignantly and then decided she was right.
I left.
☼6☼
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# "Discussions"
I left, but it didn't take long for the guilt to settle in. Our fight had rocked me, shaken me up to the core, and I felt the emptiness inside quickly being replaced by shame.
Ignoring things that bothered me had become a problem in the time I'd been roped into working the superhero gig. I mean, it wasn't like I could talk to just anyone about it. After all, who would allow you to complain about your duties and not report you? I doubted even the most legally conscious therapists would find a way to spill the beans if they wanted.
There is a downside to ignoring things, though, and most of it has to do with sleep.
It's much harder to ignore things when even sleep won't let you forget them.
That night, I felt terrible. I woke up from dreams too unreal, and too real to be unreal. All I could think about was Draco pulling the sword out of the space rock, like some kind of demonic King Arthur on a mission to destroy the world.
As morning came and attempts to sleep slowed, I could hear the soft drizzling of April showers as raindrops flowed down my bedroom window.
It wasn't long before all I wanted to do was go see Raiya, sit in the café, and listen to her talk while I drank coffee. But I wasn't sure she would let me in, and that doubt was enough to keep me tucked underneath my bed covers.
After all, I'd managed to insult her quite a bit, adding injury to more injury from our previous argument on the matter.
And I had let Draco get away with his new weapon, which was going to make defeating him much harder, no doubt.
But most of all, I'd shown my hand; I had allowed my normal life to take precedence over our supernatural calling.
Can you still love someone, even if they make bad choices? Break promises? Take selfish risks?
Of course you can. It's just probably not the wisest thing in the world to remain associated with them.
I was so sure Raiya would agree with that. So I avoided her.
I continued to avoid her through the rest of the weekend. She called and texted me, but I ignored her. I didn't even go to Rachel's. I went to Poncey's instead, and played video games longer and louder than the rest of my friends.
Monday came. School was less than exciting, but it kept me busy and safe from Raiya and her eventual rejection.
I was only able to wake up from my self-pity stupor when Martha caught my attention as class finished up. I turned in my reading questions on the chapter in the Supreme Court, and she whispered to me, "I found them."
A long moment passed before I remembered I'd asked her to help me find Otherworld. We had unfinished business to attend to, and I was not going to give up on it just because I didn't feel like taking care of it. "Really?"
"I wouldn't lie to you, Dinger." Mrs. Smithe scowled.
"Sorry." I shrugged. "Not used to asking for help. What can you tell me?"
"I found Otherworld," she said. "I got in touch with an old contact. Most of them are contracted workers staying around the city in hotels."
"What does that mean?" I asked. "It sounds like a lot of work, if we have to search out all the different hotels."
I thought of the time Aleia, Elysian, and I managed to find one of SWORD's black sites. It was in a hotel, too; I guess since that particular incident, they'd spread their forces out among the city.
"It's not like that," Mrs. Smithe insisted. "Otherworld, Inc. is a new company."
"That would explain why I couldn't find any tax documents online," I said.
"It's a common enough name," Mrs. Smithe told me. "It's the name of several businesses and organizations that have worked in different parts of the world. Many, if not all, have closed or have been bought, sold, and rebranded."
I recalled some of the conspiracy theories I'd seen about the company. I paused for a moment before I asked my question. (Does anyone really want to learn a truth that's world-shattering?)
"What does this mean?"
I was a bit disappointed when she didn't confirm any conspiracy theories.
"It's a front," she explained. "Its current address is a P. O. box, located in the Time Tower downtown."
"That's the headquarters of the Skarmastad Foundation," I said, as I suddenly remembered.
"The leader of this current taskforce is staying in Lake County Heights."
I groaned to myself. She had to be talking about Dante. (He was their lead guy? Really?) "I know. He's staying at the house down the street from me."
"If you knew, why did you need my help?"
Good question. "I thought it would help," I admitted. "I didn't think of trying to ambush him."
"It's better if you take him out." Martha's eyes were uncharacteristically dark as I looked at her. "Before they take you out."
"Well," I said, suddenly feeling like I had made a big, huge mistake, "I don't want to get SWORD on my complete bad side."
I decided not to mention how Dante was determined to protect me, as he saw me as the key to defeating the bad guys.
"There is no such thing as their good side, Hamilton," Mrs. Smithe said. "Morally ambiguous people have already compromised themselves. They'll ditch you once they have what they want."
"What could they possibly want from me, other than to stop the demons running around the city?" I asked, keeping my voice as soft as I could.
"There's no way to know all the time," she said. "But you can assume it's something bad."
"I'll try to keep an open mind about it."
"I'm serious." Mrs. Smithe admonished me, her lips tightening even more than usual. "They've been around for decades now. They're good at their business."
"What did you do for them?" I asked.
"I don't want to talk about it." She shook her head. "You wouldn't believe me if I told you, and I don't want you to know. They're good at keeping their secrets."
"I guess that's true." I thought of Dante's face, when I first told him I knew about Mikey. He was shocked, or so I thought. Now that I knew him a bit better, I realized he was more than shocked; he was terrified.
"You'd better keep your word."
I glanced back at Martha, uncertain for a moment of what she meant.
She looked over her glasses frames, eyeing me intently. "You can't talk to me about this again. It'll be trouble for us both."
I nodded. "Thank you for your help. I'll see what I can do to use it well."
"I'm glad to hear it." She gave me a half-smile. "Be careful out there, Dinger. I have faith in you, but you're up against a formidable enemy."
Having Martha's approval stunned me, awed me, even humbled me. As I watched her take another swig of her coffee, I felt a rush of gratitude for the small lady who suffered so much to teach me what she knew; I had a feeling she taught me more about honor and courage than history and government.
"I will do my best," I promised, and maybe for the first time, I actually meant it.
"Good. Now, skedaddle. You're going to be late for your next class."
*☼*
Despite all the warm feelings, I was unsettled by the conversation I had with Martha. So far, SWORD had seemed more like an ally than an enemy. True, Dante had a large role in Taygetay taking Gwen's Soulfire. And they had tried to capture us before ... and Dante was on Cheryl's side when it came to our legal disputes.
There was also the fact that Raiya believed them to be against us.
But wasn't that how things often worked? Two sides would come together to defeat the greater evil?
Surely even Martha would know that, considering she taught history for a living.
I respected these women in my life, and I knew they had a good record of being right. Of course, that didn't mean that they were automatically right, but their conclusions were discomforting.
On the bright side, Martha's information did force me to go and see Raiya once school was out. I'd thought about trying to avoid her for a couple more days, giving my self-righteousness some time to both calm down and strengthen itself for her fury. But Martha's insight, coupled with her pragmatic relentlessness, seemed to warrant discussion.
And I knew that, even if Raiya wanted to step back from me, she was stuck with Wingdinger as her co-defender and Starlight Warrior companion.
After the bell rang, I hurried down the hallway and headed out to Rachel's. I coasted through, my mind and mouth coordination in a dance I seemed to be born knowing; I skimmed through my friends' catcalls, parried their jabs, and echoed back their whoops. (I didn't even know what they were saying, but they were always relatively low-maintenance followers.)
By the time I got to the street, seeing Raiya again remained my only focus; she was the Ritalin to my life's ongoing ADD.
As I stepped inside the café, I suddenly had to wonder if Elysian wasn't there as well; I knew he'd spent a lot of time with Aleia while she was with us, and he seemed to like our other teammates more than me (the feeling was mutual, I assure you.) I certainly hoped he'd stayed at my house, avoiding the maids as they cleaned and Ayako while she cut up her dead fish.
Of course, that was probably a good reason why he wouldn't want to be there.
"Hamilton," Rachel called out from behind the bar. "Nice to see you again."
"Always nice to see you, too, Rachel," I said. The pretty barista's greeting calmed me as much as the coffee she gave me re-energized me.
She handed me one of her newer concoctions—a cupcake filled with a fruit and yogurt parfait mix—before nodding toward the back stairs. "Raiya's upstairs, if you're looking for her," she said. "She said she wanted to go out today, so she'll be down soon."
"Thanks. This looks great," I said. I held up the crumbly cupcake.
"How was your test?" Rachel asked. "Raiya told me that you slept all weekend after taking it."
"Uh, yeah, totally," I said. "It was exhausting."
"I guess so, if you weren't even here this morning."
I frowned. "How did you know I wasn't?" I asked.
"I've been coming in some mornings," Rachel said, "ever since Grandpa disappeared."
"Oh."
"Raiya's been distressed about it." Rachel sighed. "Of course, I can't believe he's gone, either."
I thought about Draco and his long-term undercover act; he'd been settling here for decades, waiting for her to show, and setting up a web of interests and businesses long before his grandfather persona took the stage. Before that, he played Ogden Skarmastad, the city founder and the founder of the Skarmastad Foundation, and turned out to be a sadistic form of evil incarnate, who transformed himself several times throughout his long, immortal life. Who knew how many other lives he had before that?
"Maybe he'll come back," I said, trying to be kind to Rachel, even if the inauthenticity of my tone nullified most of my efforts.
"I hope so," Rachel said. "But in the meantime, I'm trying to step up and help around here more. Jason's been here a lot more, too."
I hadn't noticed that, either.
"I know Raiya has her GED to pass, and Mom is not the most willing worker, even if it is our family income," Rachel continued.
"I could tell," I assured her, recalling Letty's perpetual pouting. I dreaded the days I would come in and she would be working. Grandpa Odd had weirded me out, but Letty's displeasure made it seem like I was an inconvenience. I felt bad she was also watching Adam for me a couple of times a week, although I didn't know if I should feel worse for Adam or her.
"That's an understatement. But I appreciate your restraint."
"Are you doing okay?" I asked. "I mean, I know you're concerned for Raiya, but you have to keep track of yourself and the others in your family now."
"Lee's been fine. He and Grandpa weren't really close," she said. "Logan's been more morose, but that's because he's still sad that the meteorite of his has gone missing and he won't be able to continue his research."
I laughed. "That sounds like Logan, from what Raiya's told me."
"She likes going there to see him," Rachel said. "I'm wondering if she won't go into astronomy, too. A lot of her paintings have a cosmic theme behind them."
I glanced at the door, where the painting Raiya had given Rachel for her wedding still hung. It depicted the scene from the legend of the Weaving Girl and the Herder Boy, who married and loved each other so much they neglected their duties. They were separated, only allowed to visit each other once a year, and they became stars in the sky, positioned across the Milky Way.
"Maybe." I turned back to Rachel. "I'm sure whatever she decides to do, she'll surprise me."
"I think it's easier when you've known what you wanted to do for a long time," Rachel replied. She gestured around the café. "I graduated college, took out a business loan, and set up shop here. And while we're still making payments, this is my home."
"Yes, it is." I nodded. "I also agree with you, about wanting to know what you want to do. I was practically born wanting to be a lawyer."
"You've been looking for justice all your life, huh?" Rachel asked. There was a smile on her face as she said it, and I knew she was teasing me.
She was right, of course. I had literally been looking for justice all my life, to the point where I'd fallen in love with its living incarnate.
"I guess so," I finally replied. I tightened my grip on my cup as I finished the cupcake. "This was great, as usual, Rachel."
"Thanks!" She glowed contentedly.
"I'm going to go check in on Raiya," I said, "but I might want seconds before I leave."
"Don't you always?" Rachel laughed and waved as she picked up my dishes and carried them into the kitchen.
I hurried up the stairs and knocked on Raiya's door. After a moment of silence, I twisted the knob, surprised to see it was unlocked, and peeked in.
"Raiya?"
It took me less than a moment to realize why she hadn't answered. She was painting as she listened to music.
I was surprised to hear her sing. Her voice was soft and deep, a comforting alto rather than a smooth soprano. Observing her even further, watching her frown and examine each stroke of her brush, and smile when she was satisfied, I couldn't help but think she would make a good mother.
I came up beside her, standing beside a familiar case, the one I recognized from the school play last year; it was, as usual, stuffed with different paints and supplies.
Leaning over, I placed a quiet kiss on her cheek. I felt a rush of guilt when she pulled back, stirring her out of her song.
"You can keep going," I said. "I'll wait for you."
"I need to stop, don't worry," she said. Raiya pushed her canvas away as she sat up, already grabbing at my coffee cup. "I have stuff I need to get done today besides this."
"I like it," I said, gesturing toward the painting. It was full of colors, separated by thick, hard lines, reminding me of a stained glass window. At the center of it was nothing but light, as darkness and shadows crinkled all around. "What is it?"
"A neo-expressionist supernova."
"You know, I was just going to guess that." I grinned. I didn't know much about art, but I knew how to tease her. "I think you've captured it perfectly."
From her expression, I knew Raiya could tell I was bantering. "Thanks, but it's far from done."
"Are you sure you want to stop?" I asked. "I know you like to do this stuff."
She shook her head, pushing her long hair back from her face as she sipped my coffee. "It's fine."
"Alright."
I watched as she put the rest of her supplies up, before she flopped down on her bed and reached for her shoes.
"What are you doing here?" She smirked. "Have you come to apologize, by any chance?"
"Not really," I said. "But I can fake one, if you'd like." I decided not to turn the tables on her by asking if she wanted to apologize to me. Her arrow might have pierced Draco, but it could've been a better shot. She would've defeated him, and we could have been on our merry way to a normal date night instead of worrying about SWORD and Draco's next moves.
She distracted me as she swirled the coffee around in the cup. "Stubborn." She sighed. "I should've guessed."
"I came to tell you about Martha," I said. "I got some information from her about SWORD."
"She told you about them?" Raiya's eyes sparked with instant surprise.
"Well, sort of. I was looking for information on Otherworld, Inc., the company Dante said he worked for, and she told me it's a new company. There's reason to believe it's a front."
"She told you this?"
"I asked her to help," I explained. "I thought it was a good idea—"
Raiya stopped me. "What?!"
"Come on. I told you she used to work for them. She told me that herself."
"Still, that doesn't mean that you have her go snooping around for us. It could be dangerous."
"I'd say it's more dangerous for them than it is for her," I remarked. "This is Martha, after all. She can be terrifying if she doesn't like you."
"But—"
"Just let me tell you what I know," I said, trying to hold in my irritation. How many times is she going to interrupt me?
"Fine." She folded her arms across her chest and looked at me expectantly.
"She told me that Otherworld was a new company, set up this year, and its mailing address is in the Time Tower, same as the Skarmastad Foundation."
"Is that it?"
"Well, she confirmed Dante's the lead on the project," I said.
"So Otherworld is a fake company," Raiya repeated, "and Dante is heading the situation here in the city? That's all?"
"That's all," I said.
"It's not that much," Raiya said.
"Hey, it's more than I've been able to find in the last several months, looking through Cheryl's files and scanning through the city payouts."
"It still doesn't seem worth it to ask Mrs. Smithe to risk her safety for."
"She agreed to it, didn't she? That's something," I argued. "Besides, now we know that Dante's leading up a smaller force here. We don't need to worry about SWORD so much."
"Just because Otherworld is a new and smaller company doesn't mean they don't have other forces stationed here," Raiya pointed out. "That's precisely why it should be a concern."
Anger washed over me, and I gave up trying to stay above its dark water. "It's hardly a concern when compared to your ancient grandfather-turned-immortal-dragon, especially since you can't take him down."
"You really want me to destroy my own grandfather?" Raiya asked.
"That's not a fair question," I said.
She shot up out of her seat. "You're not being fair, either!"
"I think it's still a legitimate question. We need to be able to make sure we can do our job."
"That's not our only concern," Raiya snapped back. "We have to make sure that the people we protect are okay, too. And that's why I'm angry about Mrs. Smithe helping us out!"
"It was her choice," I replied.
"Yes, but you can't live with some decisions!" Raiya shouted.
The door to her room squeaked open. We jumped back, stepping apart from each other.
It was Elysian.
He sighed as he saw us. "Oh, great," he said. "I've either arrived in the middle of one of your make-out sessions or you've been arguing."
Raiya's cheeks burned red, matching, I suspected, my own. "We were not making out," I grumbled.
"I was going to bet on that," he said, "given the volume of your discussion, but it's still fifty-fifty, knowing you two."
"What brings you here, Elysian?" Raiya asked. "Besides, of course, Rachel's baking?"
"Hey, Jason's getting pretty good, too." Elysian scoffed. "No one seemed to mind me taking his burnt cookies."
I glared at him. "You know, Cheryl's been getting onto me about my bills," I said. "That's partially your fault."
"It is not my fault, however, that you have a job and your mother still pays your credit card bills," Elysian replied. Before I could retaliate on a more physical level, he added, "What's the big deal? She can afford it, and it's not like you pay anything either."
"Just stop it," Raiya said, turning to me. "Rachel uses you as a test subject more often than not, and for Jason's stuff, too. Your tab here will always be manageable."
"I'd rather it be non-existent than manageable," I bit back.
"You just can't stop, can you?" She rolled her eyes at me. "You know, I've known you now for over a year, and you still do that. Don't you think that's a bit immature?"
"Not if I'm right!"
Elysian let out a loud roar before our conversation could devolve any further. "If you could stop for a moment, I have some news."
Raiya frowned. "Fine. But hurry up. I have somewhere to I need to be."
"Alright," Elysian agreed. "I was headed over here—"
"Where do you need to be today?" I asked. "Your GED's not until this weekend."
"I have an appointment," she muttered.
"Excuse me," Elysian called. "Hello? Big news, anyone?"
I gave up on Raiya; she was obviously still angry with me. "Fine. What is it?"
"Mikey's out of the hospital," Elysian said. "I saw him walking through Shoreside Park just now while I was on my way over here."
"You didn't stop him?" I asked.
"Why would I?" Elysian snorted. "He's your friend, not mine."
Raiya sighed and stood up. "Why don't you two go and talk to him?" she said. "See if you can find out why Dante discharged him."
I rolled my eyes. "Cheryl's deadline for the Flying Angels case expires soon. He probably got out because she's not going to get an extension, so he's safe from her."
"You should still go and check."
"Maybe I should," I said, "but that might put his safety at risk."
"Well, it's so good to see you're working on that maturity already, Humdinger," Raiya retorted. "I'll see you later."
She sidestepped me and waltzed through the door, determined not to say anything else.
I glanced over at Elysian, who was frowning. "She's got a point," he admitted.
"Shut up."
"See? You have a problem."
"You're going to have some problems, too, if you keep it up," I warned. "You're lucky I'm more concerned about Mikey now."
"So we're going to go see him?" Elysian asked.
"Yes. Right now."
"Aw, I was hoping for a cookie before we left."
I sighed. It was times like these that I missed Aleia. She had her soft spots, but she was good at getting us going.
No wonder Raiya thinks I have a maturity problem. Look who I'm hanging out with.
☼7☼
| |
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# SWORD's Play
It didn't take Elysian and me long to spot Mikey. Shoreside Park was just across the street and down a bit from Rachel's.
"Mikey!" I called out to him, but he didn't seem to hear me. "Geez, you'd think after all that time he spent by himself he'd be glad to see his friends."
"Are you still his friend?" Elysian asked.
"Uh, well, I ... I guess that's the question to ask," I admitted. "I haven't seen him since January."
"Nice," Elysian muttered.
"Hey, I've been busy," I reminded him. "The assistant mayor's a slave driver, school was still going on, I was studying for the SATs, and we all had to work on finding Draco."
"So tell me, how many times did you go to all-night gamer parties?"
The all-too-innocent look on his face made me frown. "Shut up."
"Come on, boss, don't be such a fascist," Elysian teased. "You need someone to keep you in line. That's why you have me. And that's why you argue with Starry Knight as often as you do."
"She's not always right," I insisted.
"It's better that you both keep each other in line, then," he said. "Personally, I don't think we would get along as well as we do if I had to do all the work by myself. It's too much work and you get too angry with people who try to correct you."
My fists clenched. "I hate you," I muttered under my breath.
"What did you say?"
"Nothing," I grunted. "Nothing that matters. We're here to see Mikey. That's what ... " My voice trailed off as I realized I didn't know where he was.
I threw up my hands in exasperation. "Great! We lost him."
"You should've been paying better attention," Elysian told me.
I grabbed him by the scruff of his miniaturized dragon neck. "Are you kidding me?" I nearly screamed. "You were supposed to be paying attention, too!"
He jostled himself free of my grip. "There's no need to punish me for your problems. Go transform," he barked. "I'll take to the skies and search for him there."
I watched Elysian take off, before I grabbed my phone and called him. There's no use doing something the hard way if you can avoid it.
I only groaned when the call went right to voicemail. "I can't believe he doesn't have his phone on him," I muttered, before I ducked into a nearby wooded area.
Once I was out of sight, I pressed the mark on my wrist and felt the power blossom out inside of me. Energy charged through me, spurting out like lightning through my blood.
My wings fluttered wide open, and I took off. Flying with wings of fire never ceased to amaze me.
Up ahead of me, Elysian was spiraling in and out of the cloud cover, beckoning me to follow. "This way!" he called, before heading down to the ground.
I hurried to follow him, but I began to slow down when I finally caught sight of Mikey again.
He walked into the Time Tower.
Considering what I'd learned about Otherworld, and how the Skarmastad Foundation owned the building, I felt a distinct sense of reluctance as I stepped down onto the earth once more.
Elysian hovered beside me, slinking down to his smaller size. "Well?" he asked as he settled onto my shoulder.
"I don't know," I said. "I can't imagine why he would be here."
"Maybe he's meeting Dante," Elysian suggested.
"That makes sense," I said. "Okay. Let's go. If nothing else, we can see what Dante knows about Draco and his connection to the Skarmastad Foundation."
As we walked inside the building, I realized there weren't a lot of people around; hardly any passersby, hardly any people nearby at all.
It's the perfect set up for Dante to meet Mikey, I thought.
"Hello there, Wingdinger."
Speaking of which ...
Dante stood, leaning casually against the far wall as he looked at me. His goatee had grown more, since the last time I saw him, I noticed.
"It's a pleasure to see you again," he intoned, and instantly I knew something was wrong.
"Boss," Elysian whispered. "This is—"
"Yep," I said. "A set up. For us."
"Where's Starry Knight?" Dante asked as more SWORD agents began to peel away from the shadows.
"She's got things to do," I said, uneasily. I was beginning to see why Raiya didn't trust them. Besides the whole "being captured" by them the first time we met thing, too. "What do you want?"
"It's not what I want," he said as he began to saunter over to us. "It's what Apollo City wants. Assistant Mayor Dunbrooke has given the D. A. a time limit, and we've been instructed to assist her in getting you to cooperate."
Dante glanced over his shoulder. "My son has been kind enough to help us in getting your attention."
"Mikey?" My gaze shot to him as I realized he'd been standing behind Dante.
Mikey stepped forward, hesitant but still determined.
What in the world is going on?!
"What did you do?" I whispered.
Mikey came and stood beside me. "Come on, give me a break. Grandpa Odd came to visit me in the hospital," he told me. "He said SWORD had reached out to you, that you could heal all the people with the sickness, including Gwen, but you weren't willing to help out."
"Oh, Mikey ... " I could've cheerfully shot him. And myself. I was the one, after all, who did not visit him, and I didn't tell him about Grandpa Odd's true identity.
Elysian snorted, sending sparks flying. "This is Draco's game," he said, voicing my thoughts.
Mikey ignored him. "When Dad came to the hospital, I asked him about it. He said we could try to get you to cooperate if I helped out."
"They needed you. They used you!" I shook my head. "All that time in the hospital and you've gone soft-headed. They tricked you."
Mikey frowned. "I'm just trying to get people back to their normal lives," he said bitterly. "I know I would feel much better if Gwen was better."
"You're an idiot," I said. "You just gave up on life because you couldn't have things go your way?" I grabbed his arm, making Dante step forward menacingly. Great. They've managed to bond over making me the bad guy.
While I dropped Mikey's arm, I didn't stop glaring at him. "I told you I would take care of things on my own. Starry Knight and I are working on it. We're so close."
"Well, these guys can get you closer," Mikey said. "That's what Grandpa told me."
"He's not your real grandpa," I hissed. "He's the enemy!"
"Well, it's too late," he said. "Dad's here, and they need your help."
"Don't you know 'needing help' is code for—wait, what?" I sputtered. "Why did you call him 'Dad?' You don't even like him."
"He's the only one who came to visit me and didn't seem to think it was a chore," Mikey said. He shrugged, suddenly very uncomfortable. I didn't even have to read his emotions to see it. "Besides, he is my dad, after all."
Oh, great.
I closed my eyes, and it was just as well that I did. I barely saw Mikey get tasered, before I was tasered, and Elysian tried to fight off the other guards.
*☼*
I woke up, probably hours later, groggy and grunting in pain. I was cold, and I was alone.
A rush of recognition and fear flooded through me. This had happened before, at SWORD's black site by the marina, where I was dragged off after I stopped Starry Knight's supernova.
Same as last time, I was still cold, and my wings were still able to cushion me from the hardwood floor. Not too many things were different this time around; I knew for certain I was in another location. Looking around at the near-empty office-like room I was in, I estimated I was in the Time Tower still.
At least last time, Raiya was with me.
Thinking of Raiya calmed me down.
I hated our fight earlier, but maybe it was good that we fought earlier, or she would have been captured, too.
She said she had somewhere she needed to be, but I was more than willing to bet Raiya made that up just so she didn't have to be around me. After I made her so upset, I could hardly blame her.
Looking around at the small, dark room I was trapped in, she would probably say that I was getting my comeuppance.
Mikey betrayed me, while SWORD captured me. Yep, that was some hardcore poetic justice.
And I didn't even know if they did anything else to me, I realized. What if they put a monitoring chip in me? Or if they're running my prints? Or if they took DNA samples to compare to my records?
All of the alien abduction shows I'd seen over the years came crashing into my mind, making one big pile-up of fear, panic, and madness. "Ugh," I groaned. They were going to find out who I was. If Mikey hadn't told them already.
My stomach rumbled. At least that was one question answered. I'd been stuck long enough I was hungry again.
Time passed slowly. It was worse than when I was stuck in the classroom for my SATs, but that might have been because I didn't have a clock to count the seconds. Of course, that might have made it go faster, too.
"You're killing me, Aleia," I muttered. "Or is this Alora's fault?"
"Blaming someone else for your problems? That's not going to help you." The door opened and a figure walked in. "It's no one's fault but your own that you're in this position."
I tensed up immediately at the sound of my mother's voice.
I'd had nightmares about this very situation. And here it was, coming to complete fruition. My mother, the talented lawyer, was here, and she was going to get the legal right to whip me.
"Where's Dante?" I asked.
Cheryl turned toward the door, where two SWORD agents appeared. "He's not your concern," she said. "You're dealing with me now."
I stood up (no need to look sloppy as I fell on the mercy of the court) and met Cheryl's gaze. She would appreciate the courage, I thought. There was nothing she could respect more than a worthy opponent, and I was determined to give her one as her offspring, even if she didn't know it.
Which brought me to my first goal: Establish what she knew, without giving away anything else. That was going to require a lot of bluffing.
"So," I said, "nice to meet you at last, formally."
Cheryl narrowed her gaze. She was going to try to break me by getting me to break me, I realized.
All I had to do was clam up. I knew I could outwait her.
Even if my stomach couldn't.
"Hungry?" she asked me as my stomach gurgled loudly.
"Just a little bit," I muttered apologetically.
"I can make arrangements."
"No thanks. I've seen that trick before."
"I can make other arrangements, if you're interested," Cheryl began. Her waiting had ended. She just wanted to get to a deal.
I remembered tonight was the deadline for her case. If I could hold out to midnight, she would have no legal right to keep me, since the assistant mayor had put an expiration on the case.
But that was only if I didn't give anything away, didn't admit to anything, and somehow managed to escape without giving her just cause to keep me for another felony.
That was going to be harder.
A small push behind my heart comforted me. Raiya is not here. We are both required to be here for any case to proceed.
Immediately I relaxed—slightly. No need to tip Cheryl off.
Ah, who could be immune to the rush of pleasure after realizing such profound logic, provided so divinely?
"I'm not interested in making arrangements right now," I assured Cheryl.
"Why not?" Cheryl asked. "You and your friends are facing massive fines, possible prison time, not to mention the more unpleasant task of facing the press."
"I've got a distinct advantage when it comes to dodging them," I said, indicating my wings.
Cheryl frowned. "There's no way you can run from the law."
"I happen to know the law says that you don't have much longer for your case to proceed," I said. "So it's not a matter of 'where,' so much as 'when.'"
"Dante was right," she growled. "You're a troublemaker."
"I'm a troublemaker who he is protecting," I said.
"It's easier to do that from a prison cell with round-the-clock surveillance," Cheryl shot back.
I tried not to let the truth of that bother me. I shrugged. "I imagine you might think so," I said.
Why was it so hard to be cool, like in the movies? But then, I supposed, I'd only been captured a few times. I needed to get more experience in before I judged my performance too harshly.
"That's all you have to say for yourself?" Cheryl asked. "After the lives you've destroyed, all the damage you've caused, and all the trouble you've brought to the city?"
"Yep."
Cheryl put her hands on her hips. She was prepared to do battle. "You seem pretty confident," she said, trying to wheedle me. She was going the route of circumvention, it looked like.
"I'm a teenager," I said. "I'm good at faking it."
"And you've done this before? You've gotten in trouble with the law?"
"I have been accused of breaking the law," I said. "It doesn't mean that I actually broke the law."
"Semantics," Cheryl assured me. "A jury doesn't want the truth; they just want to hear a good story."
"Narrative fallacy is common enough," I told her. "And it's easily debunked with the truth."
"There is no such thing as 'truth,' in this case," Cheryl countered. "It's only a matter of perspective."
"If there's no such thing as objective truth," I said, "then there wouldn't be a law for me to break, would there?"
She seemed taken aback. For a moment, I imagined my mother, a woman who would could give mannequins lessons in looking perfect, fluster as I easily parried her attacks.
"You think you're clever," she muttered, pacing the room in front of me. "Or you have a reason to believe nothing is going to happen to you. Tell me why."
"Why?" I couldn't resist teasing her. This was the most mother-son bonding we'd done in years.
"Because I'm interested. What's the point in being brilliant if you don't have an appreciative audience?"
"Is that why you want to prosecute me?" I asked. That seemed to make sense. There was no denying that Cheryl was very ambitious. Even the rest of her relatives, the little I knew of my extended family, knew not to get in her way.
With me, she'd always pushed me to get my law degree, get some practice in, and then run for public office. Maybe she was hoping, with her special D. A. appointment, she might get further assignment higher up one day, too.
It wouldn't surprise me.
"We're not here because of me," Cheryl countered. "We're here—"
"Because the city needs a scapegoat and you want to further your career." I snorted. "That's all."
"You've been accused of disturbing the peace, for damages—"
"I've heard of it," I said, silently reminding myself at the last moment not to let it slip that I helped write it. "And I know for a fact that Starry Knight is also listed on the lawsuit. You can't file it without both of us present."
"You're not willing to renegotiate terms?" Cheryl asked. "We could drop the charges against her if you wanted to take full responsibility."
I clenched my fists. "No," I said. "Starry Knight and I are in this together."
"You'd have her tried and sentenced alongside you?"
"We're allies."
"And lovers?"
My face felt hot all of a sudden. "We're together," I murmured, hoping that was enough to get her to leave that part of it alone. Superhero or not, I couldn't imagine any teenager willingly having the sex talk with his or her parents. Or his butler.
"I saw that kid's blog out there," Cheryl told me. "He seems to think that you'll confess, especially if it's to protect her."
"I suppose he was the one who told you who I am?" I asked. This is it! The moment of truth ...
I held my breath nervously.
"He said he could get you here." Cheryl tapped her foot against the floor. "He was right about that."
"He was expecting Starry Knight, too, wasn't he?" I almost laughed. "She's busy today."
"Hopefully she's not too busy," Cheryl said. "In the event that we are unable to come to an acceptable agreement here today, Otherworld, Inc. will be taking you into their full custody."
"That's fine with me," I said. "Their mission is to protect me."
Cheryl laughed, making me feel uncomfortable. "That might be part of it," she said. "But they have other plans as well. They've assured me they have a great many cases on which they could use your help."
Swallowing suddenly seemed beyond my capabilities. I thought about what Martha had told me before: SWORD would find a way to use people if they need cooperation. They weren't above murder or torture if they needed it.
And, I had to uncomfortably admit, they might be tempted to see me as another agent, like Dante, if they had the right amount of leverage.
I decided the next time I talked to Dante, I would beat him up.
SWORD was in the business of power—power for themselves first.
This has to be why Raiya's against them. They want to use us to save the world from threats like Alküzor and Draco, and then they want us to work for them.
They would likely be able to make us, too. Martha had been discharged, but not before she paid a hefty price.
What about Dante? I wondered. Was it possible he'd been coerced into service, too?
That would possibly explain his soft spot for Mikey. Of course, Mikey was actually his son.
"What do you know about Otherworld?" I asked. "Do you know they're a front company for a shadow organization?"
"That sounds like a movie," Cheryl said dismissively. "Weren't you the one that was just trying to point out the shortcomings of narrative fallacy?"
"It's true." I pushed forward. "They're a front for an organization known as SWORD."
"SWORD?"
"The, er, Special World Organization and Research Division," I said, hoping that I remembered it correctly (stupid acronyms.)
Cheryl cocked her eyebrow at me. "That's the best you've got?"
Before I could answer, there was a resounding bang! from behind the door.
I could feel the heat of Elysian's familiar fire, and I could hear Starry Knight calling out orders as they burst into the building.
I grinned as I saw familiar flickers of light casting new shadows underneath the door. The SWORD agents guarding us ran out of the room to help, and I grinned as I glanced back at Cheryl.
"No," I said, "she's the best I've got."
Cheryl muttered a string of curses as she followed the guards out the door.
Before I could escape her, she turned to me. "You stay here. You're in custody; as long as you cooperate, you'll be safe. But if you put so much as a finger out this door, we'll have no choice but to take action."
"I'm getting used to being tasered," I told her.
"I'll take that to mean you need more practice with it," Dante said. I realized he must have been waiting outside the door as I met with Cheryl.
Figures. He has to do to the whole creepy-stalker-spy thing.
"I'd hate to take up your time." I shuffled back a few steps regardless.
"Maybe Starry Knight will give me the pleasure," Dante said, pulling the weapon out of his coat pocket.
"You leave her out of this," I warned.
"We'll see." He smirked, probably cheering at the thought that he'd managed to get me riled. He would be that petty.
More calls came from outside the room. As Cheryl ran out the door, Dante stepped inside.
"Watch him," she ordered as she pushed past him.
Then I was left all alone with Dante.
It didn't take me long to start taunting him. "I see you've let Mikey out of his cage," I said.
"He called me himself," Dante replied. "He'd seen reason. He told me the truth." He pulled out a file from his pocket and unfolded it. "I'm guessing your mother doesn't know the truth, Hamilton?"
"If you didn't tell her," I said.
"I thought I would leave that up to you," Dante remarked. "After all, you're a star, aren't you? Figuratively speaking, I mean." The leer on his face tipped me off; he was trying to be funny, and it wasn't working.
He continued on when I said nothing. "Star of the swim team, top of the class, a favorite on the 'Hot List' of your high school elite, even Martha Smithe's favorite student. How quaint. I suppose you're the reason she came snooping around last week?"
Still, I said nothing.
Dante grinned. "You'll have to pardon me. As the son of Cheryl, the city's top lawyer, and Mark, my old best friend from high school, your tragedy is quite amusing to me."
"Because I was under your nose the entire time?" I asked, folding my arms across my chest. "Or because you gave away information to me without a second thought? Or," I continued, "is it because I was the one who helped Mikey go through losing his father when we were younger?"
From the expression on his face, stone silence wrapped in anger, I knew I'd hit a mark—and it stung.
"Leave him out of this," Dante hissed.
"He seems to have implicated himself already," I retorted. "Especially if he told you about me."
"He said you were smart."
"He wasn't lying."
"No, he wasn't," Dante agreed, and despite the fact he was my enemy at the moment, I felt a rush of pride. "He also told me that Starry Knight is your greatest weakness. He said you were in love with her."
"Love is not a weakness," I insisted.
"We have yet to see it as a strength."
"You told me before that you're here to protect me," I said, changing the topic. I didn't like it when he started talking about Starry Knight.
"And to do that," Dante grumbled, "we need your cooperation."
"Why are you helping my mother then?" I asked. "She'll destroy me in court."
"We have to cooperate with the city as well," he said. "We've made deals with the media and we've kept up a façade of comradery with the city. Part of that includes helping Cheryl."
"So you're not just sucking up to Cheryl and Mark?"
"No," Dante snapped.
I thought about it. Dante made me angry a lot, but he genuinely seemed concerned about Mikey, who, up until he brought me to the Time Tower, was a friend of mine. He also seemed to actually like Mark and, if nothing else, tolerate Cheryl. Maybe I could use that as a bargaining chip of sorts, to keep Raiya's involvement to a minimum.
Before I could put forth some kind of proposal, there was a beeping noise.
Dante glanced down at his watch, and I noticed that there was only silence from the other side of the room.
While he was busy, I turned and burst the door open.
Only to see Starry Knight, followed closely by Cheryl, heading toward me.
"Starry Knight," I called, excited to see her.
There was a rush of relief that radiated from inside her, one that nearly bowled me over. On the outside, she looked bored and resolute.
I had a lot to learn from Starry Knight, especially when it came to self-control.
Dante grabbed me from behind, firmly holding onto my arm. "We have a meeting arranged in here," he said. He glanced down at me. "Starry Knight has promised to surrender."
"What?!" I balked at the very thought of her at the mercy of my mother. It was nothing short of ironic to me that I was born as the Star of Mercy, because I knew Cheryl had exactly none—especially when it came to her court cases.
"Come inside," Cheryl instructed me and Starry Knight, as she gestured to Dante. He dropped me before heading over to a closet. From the shadows, two chairs suddenly appeared.
"What were you thinking?" I muttered to Raiya as she reached out and put a hand on my arm. "This is Cheryl we're dealing with."
"I'm happy to see you, too," she said, her voice hushed. "Do you realize you've been missing for a whole day?"
"No." I struggled to reorient myself to the time. It was strange that it was lost, even though seconds before I hadn't missed it.
"I could feel your pain." I noticed for the first time her hands were shaking slightly as she held onto me.
"What do you mean?" I asked. "I wasn't in any pain." Not that I could remember, anyway. "Did you think Draco had gotten to me?"
Raiya ignored my question. "I was worried for you."
"I'm worried for you now," I whispered back.
"I'm the one with the plan right now," she said, giving me a small shadow of a smile, one I could barely see in the dim light.
"Stop muttering to yourselves and sit down," Cheryl barked. "We have business to discuss."
Raiya moved swiftly around me. "Indeed we do, Mrs. Dinger," she agreed. "You said this meeting would be in complete privacy."
"Dante here is an agent assigned to protect me and support my mission to bring you to justice," Cheryl replied. "He is an extension of myself. I will not send him away."
Raiya frowned at Dante, and then glanced back at me. That was obviously not part of her plan, I realized.
Come to think of it, I mused, Mikey had told Dante who I was. There'd been no indication Mikey had revealed Starry Knight's true identity.
Only a handful of people knew Raiya and I were dating—including Mikey and Cheryl. But—but—it was possible that she could get out of this without revealing herself to SWORD.
"Fine," Raiya said, interrupting my thoughts.
"No, it's not," I said, jumping up. SWORD already knew who I was. Or at least, Dante did. There was no way to be sure of everything, including if they knew Raiya and I were together, but if there was a chance I could save her from being entangled in their operations, I had to take it.
Cheryl instantly rebounded on me. "Starry Knight is here because she agreed to surrender," she said. "In exchange, we have agreed that you will not be harmed."
"I won't have her harmed in my place," I objected.
"No one said anything about harming people," Cheryl insisted. "This is a matter of breaking the law."
"We haven't broken any laws," I argued. "And I would know that better than anyone. Even you, Cheryl ... or should I saw 'Mom?'"
"Hamilton, you're supposed to call me ... "
That was the moment when I should've had the camera ready. Once more, the opportunity passed and I was unable to take it. My mother, always precise and proper, slumped over as her mouth dropped open.
Cheryl sank into silence, before she looked over at Dante, who nodded.
"I see ... " she murmured. She turned her attention down to her phone, where she fiddled with it long enough to convince me she was struggling to find a way to respond, and short enough to convince me she was at a loss.
Probably for the first time in years, she was at a loss.
I took Raiya's hand, lacing my fingers through hers. I was desperately hoping that Cheryl would get the message and not reveal Raiya as Starry Knight.
Secrets upon secrets, I thought with a silent groan.
"I know that you promised someone once," I said, "that if there was a time when you were able to help, you would give it."
"What are you ... ?" Cheryl's voice trailed off as I tugged harder on Raiya's hand. She glanced from me to Raiya, and then back again. From her expression, I knew my plan was working; I knew she was thinking of her promise to Raiya.
She was picturing that moment in the hospital, at Adam's birth, when, after nearly losing him, he was handed back to her, his health restored thanks to Raiya's blood donation. Her emotions all corresponded to such an event; desperation, panic, the most primal sort of terror, and helplessness—all of this, before the innocent pure euphoria of something deeper than relief.
As desperate as she had once been, I found myself now.
"If you need more proof," I said quietly, "I can transform back into my normal self."
"No, that won't be necessary," Cheryl replied instantly. I think she was afraid to face the truth.
I tried to smile for her. "This should explain some of my missed curfews better," I said.
"Dante," Cheryl snapped. "Leave us."
His eyebrows raised in surprised. "Are you sure?" he asked.
"Yes. Leave."
Dante's lips tightened, but he followed orders.
The instant the door shut behind him, Cheryl stood up. "I knew you were dangerous," she said, snapping at Raiya.
"Anyone who manages to get you to owe them a favor is dangerous." I snorted. I pushed myself out in front of Raiya, protecting her from my mother's wrath.
Non-ironically, Cheryl's expression did remind me of Taygetay at the moment, as she fumed in pacing steps around the room.
"How could you do this?" she asked, turning her attention to me. "All this, and you attacked your father and brother in the hospital a few weeks ago?"
"Some of the reports were greatly exaggerated," I said. "You can talk to Mark about it later."
"Mrs. Dinger," Raiya spoke up. "I'm going to ask—"
"No!" Cheryl interrupted. "Don't say it. Don't ask that of me."
Raiya frowned. "Your son's future is on the line," she pointed out. "There is no cause on which we should agree more."
"My son has lied to me," Cheryl argued. "And he's been breaking the rules by seeing you."
"I offered to trade myself in for him," Raiya reminded her, "and I would like to make good on—"
"Hey, stop right there," I interrupted. "You're not doing that for me."
I stepped in front of her. "If you don't drop the case against her, I'll go public. I'll humiliate the family name, and I'll lose out on going to college and the presidency and every other good thing you've ever thought I would achieve."
"You don't know what you're saying," Cheryl grumbled.
"No, you're wrong. I know exactly what I'm saying," I said. I turned to Raiya, who had a surprised look on her face. I gripped her hand again.
Raiya's resolve came back at my determination. "I'm sure if anyone can find a way to throw out this case, it's you, Mrs. Dinger."
Yes! I cheered to myself. Raiya was no longer protecting me; she was fighting for us.
"Adam's life is more than worth it," I added quietly.
"I was under a lot of stress when I made that promise," Cheryl muttered.
"Come on, Mom, seriously?" I frowned at her. "That excuse hardly gives you grounds to dismiss it."
"I also promised to uphold my duty the day I was given the job of District Attorney," Cheryl shot back. "Don't you dare lecture me on duty, Hamilton."
Raiya tightened her hold on me before I could respond, silently letting me know to wait it out. In the end, I think her way was better. As silent moments passed, Cheryl only became more agitated.
At last, Cheryl spoke. "Fine." She shook her head and sat down, placing her hand on her head in frustration. "Fine."
She picked up her phone again. This time, she punched in a number and waited for a response.
It came. "Hello? Carly? I need you to cancel my press conference and reschedule it for tomorrow morning."
She paused while the girl on the other end seemed to ask questions, and I exchanged glances with Raiya.
Cheryl really can be incredulous sometimes. I can't believe she actually booked a press conference!
"Yes, yes, they got away," Cheryl said, her teeth gritted together. Her voice was strained and her eyes were hard as she looked over at us. "Yes, I know the statute of limitations is up as of midnight tonight."
A moment later, she barked, "I can't do anything else about it. We'll have to spin it to make it look like the assistant mayor's fault, or something to that effect ... "
I watched as she turned away and began pacing on the far side of the room.
"Geez," I muttered. "That's my mother for you," I whispered to Raiya.
"You didn't have to do that for me," she whispered back. "I could have made a deal with her to take away your charges."
"I know Cheryl is upset," I replied, "but she'll get over it. This is the first case in a while that's given her national coverage. When she calms down, she'll realize it was a pipe dream to begin with."
"Still—"
"Still nothing." I brought her hand up to my cheek. "We're in this together, right?"
Raiya smiled. "Yes, we are."
There was nothing in that moment that I wanted to do more than kiss her. But, given my mother was already unhappy, I decided to hold off for the moment.
"That reminds me," Raiya said. "I should probably go signal Elysian."
"What's he doing?"
"He's the one who managed to hold off a lot of the SWORD agents, or Otherworld officers," Raiya explained. "He's still out there, keeping the place at a standstill."
"Oh, good," I said. "I'm surprised he's not in here. He would've liked the chance to see me squirm."
"He might've liked it, but that wasn't going to happen while I was around," Raiya assured me. "Elysian knows that."
"Technically, one never knows when it comes to Cheryl," I said, gesturing back toward my angry mother.
I turned back just in time to see her hang up the phone. She gripped it hard; even in the low lighting, I could see her knuckles turning white.
"There," she said, facing us. "It's done. As of midnight tonight, you're free."
"Thank you," Raiya replied.
"Don't thank me," Cheryl scoffed. "But we're even now. I don't want to hear I owe you anything ever again."
"Done."
Cheryl turned to me. "And you ... you! I can't believe it. My own son has turned against me."
"I'm not against you," I said. "I've never been against you."
"Really?" Cheryl turned and narrowed her eyes at Raiya. "I can think of a few good examples."
"If you keep that up," I said, "you're going to make me change my mind."
Softening my expression, I came up to her. "You are my mother," I said. "You know me. You know we have our shortcomings and our spats, but I wouldn't try to hurt you more than staying out and breaking curfew to spite you."
Cheryl shook her head. "That's enough," she said. "Your father and I will discuss this later."
I decided not to tell her that he knew the truth, too. That was something she should hear from Mark himself, I knew.
Cheryl brushed past me, ignoring Raiya completely as she opened the door and walked out.
I saw Dante's shadow move in on her, and felt his confusion as he realized the truth of what happened.
Well, more or less the truth. He didn't need to know the exact details of the truth. I knew I could count on Cheryl to keep quiet, too—if she was the one person who could've broken Mikey, she was also the one person who wouldn't be broken.
Speaking of Mikey ...
I turned back to Raiya. "Let's go," I said.
"Where?" she asked.
"We need to find Mikey," I said. "I was captured because I saw him walking here."
"He brought you to them?"
"Yes," I admitted, feeling stupid about it now. "He told me that he revealed to Dante who I was—"
"So they do know for sure who you are?" Raiya asked. The worried tone of her voice made me falter slightly.
"It doesn't matter," I insisted, hoping that was the truth. "They don't know about you, or that we're dating." I hope.
"I'm not worried for me," Raiya told me. "I can fight off SWORD."
I thought about what Martha had said, about how we were up against a cunning opponent when it came to SWORD. "I don't think that will matter so much," I said, "especially if we can get to Mikey."
"But he was the one who told them about us."
"Exactly," I said. "They failed to get us to court. He just turned us in, and now we're free. He's going to have to find some way to be cooperative."
"I see what you mean," Raiya said, "even if I don't agree with your conclusions."
"About which part?" I asked. "The part where we can get Mikey to cooperate, or the part where it doesn't matter if SWORD knows who I am?"
"Both." She sighed. "But if he's here, we should get him. Come on, Elysian is waiting for us, too."
"Wait." I took her arm and drew her close to me.
All the worry in the world would wait while I kissed her. The instant her soft lips were pressed against mine, everything seemed to right itself. I was caught up in an everlasting moment, a moment where time had no measure, beauty had no end, and truth had no competition. I felt her pleasure, and knew it as my own as it radiated between us.
The world could be falling apart, I thought, and as long as I had her beside me, I would welcome it.
I drew back from her reluctantly, the taste of her still tingling on my lips. "There," I said, tugging her toward the door. "Now we can go."
☼8☼
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# Friends and Family
Finding Mikey—or rather, not finding Mikey—quickly made me worried.
I was briefly able to remember he'd been tasered along with me, and I feared for him once Raiya, Elysian, and I were all able to escape the Time Tower and we weren't able to find him.
A large part of me was still upset about his betrayal, and another part of me, a very, very, very small part, said that Draco was most likely behind it.
And that was the part that made me worry.
Draco vowed before he would frustrate us as we attempted to stop him. While Mikey should've known better, and I should've told him about Grandpa Odd's true identity, and I should've tried harder to visit him in the hospital ...
You know what? I'm going to stop that train of thought right there.
After hours of searching, I stared up at the sky as I was standing on the edge of the boardwalk by Lake Erie. My wings drifted softly along with the light, springtime breeze.
Even from where I was standing, I could see the Time Tower, like a great white sword, sticking up from the ground, piercing the bodies and souls of mankind as easily as it stabbed the city cloud cover.
Raiya landed beside me on the docks of the marina. "He's not over at the observatory," she said.
Elysian came up to us a moment later, his tail whipping through the wind. "No luck in the downtown area by the college," he reported.
"It's entirely possible he's moving around," Raiya reminded us. She clenched her hands into fists. "It's times like this when I miss Aleia."
"Not that you don't miss her anyway," I said.
Raiya nodded. "You're right. It was very nice to see her down here."
Elysian shuffled his claw against the wooden landing. "It's getting late," he said. "We're not going to find him while you two have a pity party."
"First of all," I said, "if we were having any kind of party, we wouldn't be inviting you. Second, we're doing about all we can right now."
"Did you try calling him?" Raiya asked, while Elysian stuck his long tongue out at me.
"Yes," I said. I held up my phone. "He hasn't answered his phone in weeks though. I wondered if his mom stopped paying on it."
"Well, we've searched through all of the likely places in the city," Raiya said. She counted off her fingers as she listed them. "We checked his house, the school, the marina, the college, some hotels, and Rachel's."
"Do you think he was still in the Time Tower when we left?" I asked.
We all looked at each other, and we all knew what we were thinking: No one really wanted to go back there. Not tonight, anyway.
"Well, I have school tomorrow," I said. "I'd better get back. Cheryl's got more reasons now than ever to chide me for slipping up."
"Be careful," Raiya said. "I know she's your mother, but I don't want you to let your guard down."
"I won't." I turned to Elysian. "I've got backup, besides."
She smiled, and then reached over and pet Elysian on the head. "That's true. I do count on you to take care of him, Elysian."
Elysian puffed. "Someone's got to do it."
"Yeah," I said, "and I do it just fine, thank you very much."
Raiya gave me a quick kiss on the cheek. "Good night," she said. "I'll be working the early shift tomorrow at Rachel's if you come by in the morning."
"What do you mean, 'if?'" I asked, playfully twirling a lock of her hair in my fingers.
"You didn't come this morning. Well, yesterday morning, but this time."
"Because I was apparently kidnapped."
"That's why I got worried," she said.
"Elysian was with me. Why didn't he tell you where I was?"
"They managed to hold me down for a bit," Elysian admitted. "And when I got free, I ran into Draco."
"You did?" Raiya and I turned on him at the same time.
"Yes." Elysian shrugged, sending a ripple down his long body. "He was close by, enjoying the scene."
"Did he see you? Did you talk to him?" Raiya asked.
"Yes, to both questions," Elysian grumbled. He raised his left wing. "And more, too."
"Ugh." I nearly puked at the sight of the wound under his wing. There was a purplish-green x-shaped gash, bubbling over with a thin layer of translucent scales.
"Are you alright?" Raiya rushed forward, but Elysian reared back.
"Don't," he said. "Don't. Dragon's blood is powerful; it could hurt you."
"It's me," Raiya said. "I don't have healing powers for nothing."
"Nothing's all that you're going to be able to do for me," he snapped. "You're a Star, and a Starlight Warrior, a defender of the earth. I'm not part of that world."
"But you're part of ours," Raiya insisted.
For a moment, Elysian's eyes glazed over. He finally shook his head. "No," he said. "No, I'm not."
He shuffled his tail and turned, clearly hurting Raiya.
Before he could go, I said, "Don't worry about it, Starry Knight. He always thought he was better than us."
Elysian flicked his tail back at me, sending me flying into the water of the marina.
I grappled with the water, surprised to see my wings still on fire, even as I pushed myself back to the other side of the water.
When I resurfaced, it was just in time to see Elysian as he took off.
Raiya reached down and grabbed me. "Are you okay?" she asked, pulling on my hand as she dragged me back from the chill of the night waters.
"I'm fine," I told her, spitting out my disgust as well as some of the lake water.
"You didn't have to insult him, you know," Raiya told me as she finished freeing me from the lake. "My feelings can take a hit."
"He was hurting me, too," I said. "And while he spoke the truth, I did, too."
"I know you and Elysian have always had a bit of a rough relationship," she said gently.
"You have a gift for understating things," I told her, pushing the wet wingdings out of my eyes. "But I'd rather talk about your other gifts right now."
She shot me a confused, suspicious look. "I'm not going to kiss you right now," she said, "even if you're pretending you were dashing and charming tonight."
"Not that gift," I said. "I meant your coffee-making skills."
"I thought you had school tomorrow?"
"But I get to be with you tonight," I said. "Throw some coffee in, and I'll be more than fine."
She pursed her lips together in thoughtful consideration.
It is time to give in, I thought. "Please, Raiya? Don't make me go home to Cheryl and Elysian. Not without proper sustenance."
Raiya grinned. "I suppose Rachel's is open for a bit longer."
"Exactly."
*☼*
It turned out to be the best thing ever, to go and hang out with Raiya for a bit after searching for Mikey.
True, we could've continued the search, but I was cold and wet, courtesy of Elysian, and I considered Rachel's a place for healing as well as comfort.
He was lucky I didn't get pneumonia.
But those weren't the only reasons it was the best thing ever to be with Raiya. I mean on top of the normal reasons, too, like how I loved her and I loved being with her and how I loved the smell of the special espresso beans Rachel ordered.
It was the best thing ever hanging out with her, because I woke up to approximately the worse thing ever.
My mother had me woken up at the crack of dawn the next day so I could head down to City Hall and help her with her new press conference talking points.
As I ran around the small auditorium at City Hall, taking orders for coffee, getting supplies, and helping with the lighting, I almost wondered if Cheryl was doing this in order to punish me. I wondered if she was going to rat me out anyway.
After a few moments and a few sips of Rachel's coffee Mayor Mills had the brilliance to stock, I figured that was as crazy as it was unlikely.
Why would she reveal my identity now, when she could hold it over my head for several years to come?
That was the Cheryl I knew.
As the news media shuffled in, minus a few of its more familiar faces, Cheryl came up beside me.
"Are you ready?" I asked her.
"No."
My eyes widened briefly at her admission. "Are you afraid?"
"It's not a matter of fear," she said, "but rather what you are afraid of, Hamilton."
"So, yes?" I guessed.
She frowned at me. "We're here because of you and your girlfriend, remember?"
"You leave her out of this," I warned. "She hasn't done anything wrong. Or illegal," I added, as Cheryl planted her hands on her hips.
"Fine," Cheryl remarked. She sighed. "I have tried to be a good mother, you know."
"You are a good mother," I told her. When she looked at me, surprised, I shrugged. "You're a good mother," I said, "but I think you're a better lawyer."
I was expecting her to frown again, but she smiled. "Well, that's good to hear, at least."
Why would that be a good thing to hear? I shook my head as my mother's name was called, and she went out to her adoring public.
I watched as the people began clapping and cheering when she walked out to the news podium. As much as I knew she was angry at me, and likely sad at her circumstances, I was proud of her. No one likes to lose, but there were a lot of people who wouldn't have been one-tenth as graceful as my mother over it.
My mother was a favorite among the crowds. As fearsome as she was in court, she was a compelling figure, and, for her age, even attractive. I could see why the public loved her and painted her out to be a true heroine among the many city prosecutors. Her ambition only endeared her more to the public.
Because, of course, the public didn't see her as I did, though in truth, they probably saw her more often than I did.
"Thank you, thank you very much," Cheryl began, smiling to the crowd.
I sighed as she talked for a bit about her many accomplishments, under the guise of thanking her public fans for all their support and votes since Stefano had taken office and used his influence to get her promoted.
Realization hit me hard and fast. No wonder Cheryl's been so focused on this case. She's trying to run for re-election.
I knew before her initial appointment was special, because the last mayor of Apollo City had been forced to resign, and several workers with the city, unwilling or unable to handle the supernatural crises, had stepped down as well, citing irreconcilable differences with the change in leadership.
Cheryl had followed Stefano's lead in prosecuting the superheroes because it translated into votes. She wanted to run for the office on her own terms.
I was cynical enough to wonder how she was going to present this case to the public, so they would see she had no choice but to call it off.
"So, let's get down to business," Cheryl said, and the crowd went wild, interrupting the last of my thoughts. "We are unable to continue any further with the Flying Angels case. Assistant Mayor Dunbrooke, who has stepped up to fill in the gaping hole left by the estimable Mayor Mills while he is recovering from his heart attack—"
It was hard not to gag or laugh while she managed to compliment and insult both leaders simultaneously.
Only my mother. I shook my head, even as I admired her for her skill.
There were some questions that followed, but Cheryl never faltered.
"Mrs. Thomas-Dinger," one called out. "What will you be focused on now that the case has been called off?"
"Thank you for your question," Cheryl said. "I have decided to step down from my position as District Attorney."
A collective gasp went through everyone, including me.
"What? Why?" I stammered.
Fortunately, someone else with a mic asked the same question (different tone.)
"I have been practicing law for over a decade now," Cheryl said. "I've had several victories, and some losses."
There were cheers and one collective "Aw" from the crowd.
"While I work hard at what I do," Cheryl continued, "I feel I have reached my potential in serving the city. I wish to redirect my efforts into other areas of my life, not the least of which is being a mother to my two sons."
She could've gotten in her car, run me over, backed up and run me over three more times, and I would've been less shocked than I was at her admission.
More questions came, and minutes passed. Cheryl thanked the crowd, told them she would continue to work hard and lead a good life, and then she exited the stage. More reporters tried to follow her, asking her other questions, but she simply waved them off.
She came back up and stood beside me. "Well?" she asked. "How was that?"
"Astonishing," I said. It was the only word I could think of. "I think I'm dreaming."
"You're not," Cheryl snapped.
"Have you talked about this with Dad?" I asked.
"Since when do you care about what your father thinks?" Cheryl shook her head. "I have worked in law for a long time, Hamilton. No vacations unless they were mandated, no time off for being sick, and no rest taken unless pills were involved. I have made enemies of different people, people who have been put away, and others who are awaiting probation."
"So?" I asked. "You've done well."
"Thank you," she said. "But you said it yourself. I'm a good mother, but a better lawyer. I've perfected my lawyering. Now I can work on the mothering."
That was heartfelt and disconcerting all at the same time.
"With Adam, I hope," I said. "You might as well consider me a lost cause."
"I don't think so, not quite yet," she said. "But you'll get there, don't worry."
I'm pretty sure she was teasing me, but after all the shock I'd received from her already, I was done. "I'm leaving," I said, turning away. "I got to go to school."
"Don't forget to stop for coffee on the way," Cheryl called.
I stopped in my tracks. Glancing back, I eyed her carefully. "Does this mean you're okay with Raiya now?"
"If I am, are you more likely to dump her?"
I grinned, despite myself. "No."
"Then it doesn't matter, does it?" She waved. "Now, I've got to go home and settle into a week-long vacation before I set up my own law firm."
I laughed as I headed out.
Of course, Cheryl will never be down for long.
But seeing her give up her dreams humbled me. As much as I didn't want to care about it, I knew she was giving up something precious for me. That took more than love and unselfishness—that took a special kind of courage.
*☼*
Raiya met me as I walked down the street toward Rachel's.
"How did you know I was coming?" I asked her as she handed me a large cup of my favorite mocha.
"I saw the press conference," she explained. "You know Rachel has the news on all the time. It's been exploding across the front page, about your mother."
"She's always been flashy about headlines," I murmured, taking a drink.
"You look nice in your dress clothes," Raiya said.
"I don't have time to change before going to school."
"Then don't go."
"Huh?" I glanced at her, surprised. Usually it was Elysian who was trying to convince me that school didn't matter so long as I had demons to destroy, and even then, I was already doing so well that a day off wouldn't hurt me at all.
"I said, don't go," she repeated.
"Why would I do that?"
"I found Mikey," she said. "He's back at the hospital. Just for a bit," she said, noting my concerned look.
"How do you know that's where he's at?"
"Your dad called and told me."
I frowned. "Okay, I feel like I am asking a lot of questions here, but why would my dad call and tell you that?"
"The day you were captured, I had to go to one of my doctor's appointments." She pointed to her heart, as if I were in second grade and didn't know where it was. "Dr. Dinger was scheduled to see me. So I went and I talked with him, and then I left."
"You didn't see Mikey there then," I said. "I was following him to the Time Tower."
"Your dad was calling me to see if I would reconsider," she said.
"Reconsider what?"
"I've decided to stop giving blood for now," she admitted. "When Grandpa—I mean, Draco—started taking me to the hospital, I don't think anyone was fully aware of my power. When your dad recognized it, he made sure I found out soon enough."
"So you did know, from pretty early on."
"Sure," she said. "It wasn't a hard guess, especially after I came into my powers. Your dad allowed me to help his research in a lot of ways."
"So I guess Adam's not the only one who's been helped by you."
Raiya nodded. "Anyway, I told him the other day I didn't want to come in and give blood anymore. There are too many confusing things between what Draco told me and what I think about it. I can always change my mind," she added quickly, "especially if there is a need for a special treatment or something."
"I'm glad you decided to stop," I said, reaching out and pulling her against me as we walked.
Tension broke around her and through her. It was strong enough I could sense it; I didn't even have to focus my power to see it. Her hands gripped me around my waist. "Me, too," she admitted quietly.
"How did my dad respond to that?" I asked. "Hopefully, it was better than how my mom took the news she wasn't going to be able to charge us."
"He was nice about it, even if he was disappointed." Raiya shrugged. "He said my heart does have some irregularities, and it would be good to still come in to get it checked. But I told him no."
"Why?"
"Because I can't do it without feeling terrible now. Even if I am helping him with his research or finding cures, I wouldn't feel right about it. You start to feel more like a test subject than a human after a while."
I considered this, and eventually agreed with her.
"I was thinking," Raiya said, "that rather than be a test subject, maybe it would be a good thing for me to still work with it. Become a doctor or nurse or something, you know?"
My head snapped around to face her.
"After all," she said with a grin, "we're going to need jobs once all this superhero business is over, right?"
At her words, I couldn't stop myself from kissing her.
"Hamilton." She gasped as I pulled her close to me again. "Hamilton, people are looking at us."
"We live in a cynical world," I said. "Let them look. It's not every day they get to see what true love really looks like."
Raiya laughed, and finally sidestepped me enough she was able to slip free. "No offense to your kisses," she said, "but I still prefer romance to be less of a public spectacle."
"I suppose Mikey's blog ruined it for you?"
"You could say that," Raiya agreed. There was a playful twinkle in her eye. "Maybe I'll remind him of that while we go see him."
"Maybe I'll remind him of that when we see him." I chuckled. "Why did my dad tell you he was here?"
"I heard him say to another nurse that Mikey was coming in this morning to pick up one of his school books he'd left behind."
She grinned. "He's not like you that much. He forgot he was on speaker phone with me when he said it."
"I've been saying the same thing for years," I assured her.
The time I spent with Raiya always seemed to make the rest of my life shimmer over with a haze of sorts, unless I was paying close attention.
I didn't pay attention a whole lot, I guess.
It seemed that I only had to blink, and we were already at the floor where Mikey had been staying as an outpatient for the past six months.
Raiya managed to get us through the barrage of nurses and assistants to his room, and I managed to put up a blockade on the door when I saw Mikey stuffing a book into his backpack on the bed.
"Mikey."
He glanced up at me, his brown eyes troubled. "Dinger."
"I'm guessing you weren't expecting me?" I asked.
"No," he said bitterly. "I wasn't. But then, you stopped visiting me months ago, so you really only have yourself to blame."
"I suppose that's why you thought it would be okay to tell your dad about me?"
"Grandpa told me that you weren't helping Gwen and the other victims, even though Dad and his company had found a way to cure them," Mikey insisted. "If I 'betrayed' you, it's only because you deserved it."
He turned toward Raiya. "You're no better either, if you didn't want to help," he said. "But I didn't give you up."
That was one question that was answered, at least, and it was a comforting answer.
"Grandpa Odd wasn't who he said he was," Raiya told Mikey, stepping forward. "I'm here to tell you that. I should've known, and when I did, I should've said something."
"He was always nice to me," Mikey said. "He was always looking out for me."
"He's always looked out for himself, like everyone else on the planet," I huffed.
"Like you are right now, I'm guessing?"
Ouch.
"I'm not lying," I said. "He's not a real person. He's a dragon, a changeling dragon like Elysian, who can change into human forms. He's been playing this town for decades, or possibly even longer."
"His real name is Draco," Raiya added. "And he has the power of immortality."
"Well, that's great," Mikey said. "Next you'll be telling me that Rachel's actually a sun goddess and Letty is the daughter that sprang out of her head or something."
"We're not lying."
"It doesn't matter, does it?" Mikey shot back. "Dad got what he wanted."
"Did you?" Raiya asked quietly.
He frowned at her, obviously displaced. "No," he finally said. "They captured him, and they captured me, and they set me free. It's over."
"Why did they want Hamilton?"
"I don't know," he scoffed. "Why don't you ask them?"
"Probably just for Cheryl," I told Raiya, thinking of what Dante had told me before while we were in the room together. "But Draco didn't see Cheryl stepping down from the case, or from her position as the D. A."
Frankly, I wouldn't have seen it, either.
"Martha said before they might want cooperation," I said. I looked at Mikey. Was it possible they would use him against me again, especially if I needed, as Martha had warned me, to be "convinced" to cooperate?
"I don't know why you're worried about me." Mikey scowled. "You're not worried about Gwen or anyone else."
"Draco—Grandpa to you, I guess—was lying about that," Raiya told him. "There's only one way to get Gwen back to normal, and that's to get her Soulfire from Draco."
"He has it?"
"Technically." She frowned. "We told you this before."
Mikey frowned. "Why should I believe you?"
"We wouldn't lie to you," Raiya insisted. "We didn't lie to you before."
I stepped forward. "There's no use in trying to convince someone who's already made up their mind," I said. "Mikey, you've been my best friend for years, and maybe—"
"Maybe I've outgrown you."
"Maybe," I admitted. "But this isn't something I would lie to you about, and you know it. If you want proof there is nothing we can do to bring Gwen back, there are only two ways to get it. And I'm not doing the one."
Raiya arched a brow. "What are you thinking?"
I'll admit, it was nice to hear her asking the questions.
"Get your dad to meet us," I told Mikey. "He knows who I am now. Get us together, and we'll do an experiment."
"What's the other way?" Raiya asked, curious.
"I'd say we could go break into Gwen's hospice care and see if we can heal her, but I don't think her parents would be too eager to see me again," I said. "Even if I did just get the charges against me dropped."
"I'll set up the meeting with my dad," Mikey said, interrupting Raiya and me.
"Fine." I moved out of the doorway. "Until then, you can go. But you need to stay away from Grandpa Odd if you see him again."
"Fine, whatever," Mikey grumbled as he pushed past me. "Just get out of my face."
"We've always tried to protect you," Raiya told him.
"For all the good it's done," Mikey snapped.
"If you want to put yourself out of misery," Raiya countered, "there are still plenty of demons hanging around. Draco's terrified a good amount of them with his power, but there are always some who stick around to see what they can get. If you want to join Gwen, you have a chance."
I stepped in front of Raiya. "You don't need to be that hard on him," I murmured to her, before I turned to face Mikey. "Look, I promise you that we're doing all we can right now. I'm sorry about what's happened. I really am."
"I'm tired of your apologies," Mikey said.
"Okay, well, how about a call for mercy here, huh?" I held my palms out to him, face-up and empty. "Look, I know things have deteriorated between us lately. I don't want that. Not really. We've been through harder times than this. Fighting over girls, fighting over trust, this isn't like us."
Mikey said nothing.
"I've been a bad friend to you," I continued. "And to be fair, you've been a bad one to me."
Mikey's mouth gaped.
"You broke the superhero creed," I said. "You're not supposed to tell anyone my real identity. According to most movies, you will die soon for doing just that."
"But—"
"You also broke the bro code," I said. "Now, I've broken it too, but what I'm trying to get at here is that you put my life in danger, you put Raiya in danger—and since I'm in love with her, that also doesn't look good, according to most movies—and you put our mission at risk."
Mikey regained his composure. "What are you getting at, Dinger?" he asked.
"I'm saying that I could easily feel justified in hurting you," I said.
"Hamilton, that's not prudent," Raiya hissed at me. "And you thought my tactics were bad."
"No, see," I said, "I have every incentive to hit you or harm you, and I haven't. I'm angry at you—oh, yes, I'm angry at you—but I'm not going to do something stupid about it. I'm going to forgive you."
"Forgive me?" Mikey spat.
"Yes. You know I've done wrong, but you know you've done wrong, too."
"That doesn't excuse your wrongdoing."
"That doesn't excuse yours."
"You're just trying to make yourself feel better. Like you're the better person between us."
"If you forgive me, if you show me some mercy," I said, "then we'll be even."
Mikey glared at me. "I'll have to think about it," he finally said, though it was more likely he said it because he had nothing else to say, rather than he was actually going to do it.
Still, it was a start. It was a small win for me.
"Fine. Now, can you tell us anything about Grandpa Odd?" I asked. "I want to know, so I can help protect you, and help protect others."
Mikey paused for a moment, and then he sighed. "No. He just came to visit, same as always," Mikey said. "He told me about what was happening at the Time Tower, that you weren't willing to work with Dad, and that Gwen was beginning to fade from some of the reports he'd heard at Rachel's."
"Nothing else?" I asked.
"Nothing." Mikey shook his head. "I'll call you and let you know about meeting with Dad." He narrowed his eyes. "I'm not going to trust you unless you show up for the meeting."
"Until then, can we at least have the benefit of the doubt?" Raiya asked.
"No," he retorted. "We're at an impasse. That's all."
I waited until he was out of sight before I said, "That's all we need right now."
"Maybe we should send Elysian to watch him?" Raiya suggested.
"No." I shook my head. "There's no point, now. Mikey can take care of himself. Or at least he can if he wants to. We should let him. Maybe after Draco tries to claw his soul of his body, he'll be more apt to fight alongside us rather than against us."
"You're right about that."
I laughed. "Well, it must be a day for miracles, if you're going to admit I'm right."
"We agree on a lot more than you make it sound like."
"Are you disagreeing with me about agreeing with me?"
"I'll let it slide for now," she said. "I should go out and do a patrol."
"What about me?" I asked.
"You need to get to school," she reminded me.
I groaned. "Come on, you didn't need to tell me that."
"You should go," Raiya said. "What's the point of getting that 2398 on your SATs if you're not going to get the glowing school record to back it up?"
"2398?" I asked. "I haven't gotten my scores yet."
"I was taking into account the English section you were worried about," she teased.
"I wasn't worried about it, per say."
"You still need to go," she said. "Enjoy your time at school. You never know when it'll be over."
"It'll be over next June," I retorted.
"I thought that too, once," she said. "Things don't always turn out the way we want them to."
I didn't reply to that. She had a point, but it was a vague one, and that didn't do much to persuade me. But I knew she wanted me to do well, and her love made up the difference. So I relented.
"Will you be okay on the patrol?" I asked.
"I can get Elysian to help me," Raiya said. "He came in to Rachel's this morning, you know."
"What was he doing?"
"Looking for breakfast."
"I am not completely surprised," I said. "But my mother's latest chef makes a lot of raw fish and stuff like that. I figured he would have liked that, as a dragon."
"Believe me," Raiya said with a laugh, "he vastly prefers cupcakes."
"Make sure you put whatever he eats on my tab."
"I already do that. But I don't mind. It's nice to have someone else watching out for Rachel and the café when I'm not around. I'm worried Grandpa would go after them, too. He especially knows what Rachel means to me."
I put my arm around her shoulder. "We're almost there," I said. "We just have to defeat Draco, and that's it."
"There are smaller demons around," Raiya reminded me.
"There will likely always be something we could do," I said. "But my mission was originally to recapture the Sinisters. Orpheus and Draco are just extras, and stopping Alküzor from ruling the universe seems like a reasonable thing to do. After that, everything will be easy."
"It's true that there will always be evil somewhere on Earth," Raiya agreed. "Many other demons and devils are trapped inside the fire of the earth. They're waiting there until their final judgement comes."
"You can't forget humans," I said. "There are plenty that do evil things and don't have any qualms over it."
"Yes, that is true." Raiya leaned into me. "The line between order and chaos is severed with a simple choice, as good and evil run through all human hearts."
"And Stars, too," I said, giving her a quick kiss on her forehead. We were coming up on the school, and it was time to say good-bye again. "We'll have to finish this conversation later, I guess."
Raiya gave me a rueful smile. "It appears so," she said, still bantering with me.
"See you later," I said, slowly letting her pull away from me, hesitant to go back into the world surrounding the school.
I knew Martha's class was waiting and my friends all likely needed some help with their homework, and the girls were, as always, just eager to see me.
I turned back to see Raiya as she pressed into the four-point mark on the underside of her wrist and transformed into Starry Knight. She waved to me and then took off, her radiant form darting across the cloudy April skies.
Sighing, I turned back to the school. "Alright," I told myself. "Time for the real battle of the day to begin."
☼9☼
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# Foundations
Meeting with Mikey left me feeling a mixture of hopeless and angry. I felt hopeless, I decided, because he was hopeless, and I was angry with him for being hopeless.
As much as I might've wished that to be true, the truth was more along the lines of I wanted Mikey to be safe, but he didn't want me to protect him. I was angry at him for making me care, and I was angry that I couldn't just stop caring either.
His stupidity was a liability.
But as the days passed, I was able to receive some relief. Mikey came back to school, and even though he wasn't in my classes (he fell behind on a few subjects while he was tutored in the hospital), I was able to better watch over him. Better yet, I was able to do it where he couldn't accuse me of hovering around him.
That relief was a relief in itself, even if I had to dance around Drew, Poncey, and Jason, as they all asked about why Mikey and I were fighting "this time."
I played it off as one of our usual spats, either shrugging it off or distracting them while I managed to get away without actually answering any of their questions.
Fortunately, I was not the only one who was feeling an extra dose of relief. With the SATs out of the way, there were two things on everybody's mind: Prom, and summer vacation.
With a week until prom, and five weeks until the end of the school year, the school was settling into its summertime routines quite nicely. Teachers were more cheerful, students were less stressed, and everyone was generally more agreeable.
Glancing around the classroom as we finished up, a wry smile made its way onto my face. What a difference hope makes, I thought. Even Brittany Taylor, who'd been one of Gwen's friends, and a friend of Samantha Carter, an irritating girl who had her soul sucked out the year before, seemed more like her normal cheerful self.
Of course, that was probably just because she was getting the chance to boss people around again, I noted. She'd been elected the head of the prom-planning committee.
It was the last period of the day when Brittany buzzed her way around the room, heading toward me. "Dinger," she called, "hang on for a sec."
"What is it?" I asked. The bell rang, signaling the end of the day, and I was more than ready to leave.
"You haven't bought your prom ticket," she said, waving a checklist of people's names who I assumed also did not buy a ticket.
"Oh," I replied. "Right."
Cheryl was supposed to fill that out and get it in.
Ugh, you just can't depend on your mother, especially right after she just vowed to be a better mother in front of the whole city.
"Okay, give me one of those papers," I said, gesturing toward the stack she carried, "and I'll bring in a check tomorrow."
"Here." She gave me the permission slip (yes, that's really a thing for prom, when most people are close enough to legal adulthood). Brittany smiled sweetly and said, "Don't forget, if you're bringing a date, you'll have to pay for her ticket, too."
"What if she's paying for her own ticket?" I asked.
"She'll need her own permission slip." Brittany handed me an extra one.
"She doesn't go to this school," I said. "She shouldn't need it."
"So you actually have a date then?"
I groaned. I knew the gossip going around, and I knew of its power. Gossip was part of the information exchange at high school, and I usually managed to pay my dues; since the end of swim season, I had nothing of any virtual importance to share with my peers.
So the Gossip Karma Queen by default had come hunting for me.
I didn't like to share information about my personal life. My public life, and even its implications, were all up for grabs. It didn't matter to me, so long as it mattered to them—and so long as I knew the truth.
Despite Brittany's unwelcome inquiry, I held my ground. "Yes, I have a date. She doesn't go to this school." I passed back one of the slips. "I'm bringing her." No matter how much she'll probably hate getting scrutinized all night by you and your cronies.
Brittany's face scrunched up. "I heard from Poncey that you don't actually have a date, and Via told me you were going to go with her."
"First of all, that's my business," I said. "Second, I wasn't aware you were paying attention to Poncey at all, since he dissed you back in elementary school."
Brittany blushed, but she glared at me as she excused herself, saying she needed to go find Guy Fitch, another social outcast, and take care of him.
Not the most genial of confrontations, I thought as she walked away. But I had another year of high school yet, to make my political prowess known and complete. So I didn't worry.
I especially didn't feel like worrying, because I was certain the battle was almost over. Draco just had to be defeated, and while his sword and his skill gave me pause, I didn't see any future in which he would not be defeated.
That left me free to hold off on worrying about silly things, like social cliques and school rivalries.
I did choose to concern myself about the prom, though. (Come on, this was the first year I was allowed to go, as a junior, and it was like a rite of passage.)
Because I concerned myself with it, I decided to concern Raiya with it, too, when I headed over to see her.
She was upstairs when I came, listening to some music as she worked on her supernova painting again, with its thick strokes and fuzzy clarity.
"Looks nice," I said, well aware I was not much of judge when it came to art, at least past announcing something was either "good" or "bad."
Raiya sighed. "It'll work for now," she said, as she put it up to dry.
"You have plenty of time to work on it," I said with a shrug. I pulled out the review book for AP Gov. We were creeping closer to May, and AP tests were coming up. "This review section, on the other hand, has to be done by tomorrow, or supposedly Martha's going to be upset."
"I can't imagine Mrs. Smithe getting upset with you." Raiya smirked as she began cleaning off her brushes.
I watched her for a moment, recalling Romeo and Juliet, the stupid play Gwen was starring in when the Sinisters started attacking Apollo City. "I was cleaning brushes," I said.
"What was that?" Raiya pushed some hair out of her face as she glanced back at me.
"I was cleaning paintbrushes the day that you came from Rosemont to work on the set for Romeo and Juliet," I said. "I remember your fight with that girl."
Raiya laughed. "I'd forgotten about fighting with Courtney," she said. "I was more surprised to see you that day."
"How did you know it was me?" I asked. Awkwardness took over. "I mean, how did you know I was, you know, the same person as Almeisan?"
"It took me a while to believe it," Raiya told me. She put her brushes up. "I don't have an exact answer for you, or at least one that would make sense to a scientist or a theorist like Logan, for example. But I know that there are things that we can't see, things that can outlast time. Who we are, as people, as creations, is one of them."
"But people change," I said.
"True." She reached out and took my hands. Her fingers were strong, comforting. Capable of making beautiful things. "But despite change, we still exist."
"I guess you have a point. We're still here, even if we have different names."
"Exactly," she said, "although I'm not sure 'Astraiya' is much different from 'Raiya.'"
"Your name is technically still 'Astraiya,'" I reminded her. "My parents gave me a completely different name."
"I think with being here, it's more like a title change of sorts," Raiya said. "I mean, you can agree with me. That happens here already. You're the one who calls your mother by her first name."
"Cheryl suits her better." I thought about my mom's decision to go back to private practice. "Maybe she'll feel more like 'Mom' now that she's going to start her own firm."
Raiya gave me a smile, and for a long moment, for unclear reasons even to me, I wondered if she wanted kids. "So you think being here, in this realm, is like motherhood?" I asked.
"The logic has parallels."
"Do you think you'd like to be a mother someday?"
Her sudden stillness answered the question, and she turned away to grab a towel to clean up. I narrowed my gaze at her, calling on my power, and I was surprised to see she was afraid.
"Are you afraid of having kids?" I asked.
"No," she snapped. "I just ... I just don't think of it very often."
I could sense her hesitancy, so I moved over to her and took a hold of her hands. I saw they were shaking slightly, like they had been the day she came to rescue me from Cheryl and Dante.
I knew at once what was wrong; she wasn't afraid of having kids; she was afraid of not having them, and losing them. Just like she was afraid of losing me again.
"Show me," I said.
Raiya looked at me quizzically, before I clarified. "I mean, show me how you feel about it," I said.
"Why?"
"Why not?" I countered. "Come on. I want to see what's in your heart."
Raiya tensed under my touch, but I met her gaze with my own, and I silently promised her that I could handle it.
Slowly, I felt her submission. Warmth trickled into my skin. I felt a tentative pulse, the forerunner to a strong flood underneath the initial hesitancy.
I felt the love she had for me, pressing past the limits of my knowledge and imagination. It burst through me, penetrating me, pushing into the core of my heart and being. It took strength to keep standing as I held her there. I heard the sharp release of my breath, and I felt my knees buckle slightly.
I closed my eyes; immediately, I saw the rush of images accompanying her power. I saw our past, our present, and our future, as they came together, splashing together in a world of wet darkness, wrapping around itself into a bundle of light and joy.
A heartbeat later, I saw the bundle as Raiya held it, as I held both of them, and felt a rush of pride and love so much I had to let go of her for fear I would be swamped by the vision—for fear I would willingly run into it and drown myself.
Raiya pulled back, letting my hands drop from hers. "I'd love to be a mother," she said quietly, allowing me time to reorient myself.
Even as I put my mind back into its usual order, I knew what she was really saying; she wanted to be a mother—as long as I was there with her.
"But I know that it's not something I want right now," she added, still moving away from me.
"Right," I said, finally finding my voice again.
"I don't think about it too often," she confessed. "I'm still getting used to thinking of the future, past this mission."
I nodded.
"Actually, if you're up for it, I thought about heading over to see Logan again," she said, grabbing a jacket from her desk chair. "We promised—what is it?"
As she passed me, I felt my hand reach out and take her arm. It was an awkward motion from my position, but I felt compelled.
"What is it?" Raiya asked.
I stared at my hand, suddenly wondering what exactly I should say.
I knew what I wanted to say. I wanted to tell her I loved that she wanted to be a mother, that she wanted to be with me. That I loved her, and I wanted that, too. That I wanted to marry her and make her my permanent home, officially, here, in this realm and in this life.
Instead of any of that, I said, "I never really wanted kids."
Before I could explain I was changing my mind about that, and about so much, really, since she'd come into my life, Elysian came into the room, walking on his hind legs.
Talk about a mood killer.
I sighed and let it go. I had a number of things to discuss with her, so I would have the chance to bring it up again.
"There you are," he said. "I was wondering where you guys were."
I dropped Raiya's arm. "What is it?" I asked.
"Dante." Elysian scowled. "He's walking toward the observatory."
"I'm just saying we should go there anyway," Raiya remarked. "Sounds like it would be a good time to check in on Logan."
"We can confront Dante in the meantime," I said. I thought about texting Mikey, but decided against it a second later. He wanted to be on his own, so I would let him. Even if he was unfit for the job in question.
Elysian, Raiya, and I all transformed in the shadows of the alley next to Rachel's, and then we took off.
We had almost arrived at Lakeview Observatory when I realized I'd forgotten to remind Raiya that the prom was this weekend. I'd gotten distracted by our talk of the future, and the vision I had of Raiya's heart.
I'll get it later, I promised myself, along with the other stuff.
☼10☼
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# Findings
Considering how I'd left Dante before, I wasn't really that excited to see him again. Of course, I don't think I was ever really happy to see him, but this time I was especially not that excited.
I was glad to see, however, that the Otherworld, Inc. guards seemed to have vanished, and we had no trouble slipping through the back door of the observatory.
Lakeview was still open to the public, so we had to be cautious as we walked around. More than once we all scurried around a corner or ducked into a nearby room. We walked through the bottom floors, and it was only when we came to the telescope room that we stopped.
"I don't see Logan anywhere," Raiya said.
I glanced at one of the clocks. "Maybe he's teaching down at the college tonight?" I wondered aloud. "Or grading papers or something there for one of his other professors?"
"That's true," she said with a sigh. I knew she was not completely convinced.
Dante stepped out of the shadows. "He stepped out for an early dinner with one of his colleagues from the college. You'll have to excuse him for the moment."
In one motion, we swiveled around.
"You've really got to stop doing that," I told him.
"It's practical," he said, in a smug, apologetic sort of way.
"It's irritating."
"Most irritating things are practical."
"Not necessarily." I folded my arms across my chest. "We could argue the specific points on that for a while, but let's just get down to business, shall we?"
"Fair enough."
"What are you doing here? Mikey didn't call you already, did he?"
"No," Dante replied. "I can't imagine why he would, either. I told him to stay away after we caught you."
"He doesn't listen," I said. "It's not surprising that he didn't call you, after he said he would. Did you really tell him I wasn't helping you help the people who had their Soulfire stolen—the people who have that sickness?"
"No," Dante said, "I didn't say that."
"Draco did," Raiya reminded me. "Dante just let Mikey go on the assumption, I'll bet."
"Life is complicated, and if you're lucky enough to live long enough, you'll get an idea of just how complicated it is," Dante said neutrally.
So Raiya was right.
"What are you going to do when he calls you and tells you that I'll go with you to help the victims?" I asked.
"I'll tell him what I told him before: Stay away from me," Dante said.
"He won't trust me again until we do."
"That's his problem, then, more than yours."
"It might not be enough to stop him from doing something stupid."
"You'll see to it, and I'll see to it," Dante said, "that he doesn't get caught up in this. That's why I wanted to speak with you."
"You mean," Raiya said, "that you, not SWORD, want to talk to us."
"Yes." He stepped forward and looked at me. "You know as well as I do, Hamilton, that we both have liabilities."
I agreed with him, even if I hated to hear him say that Mikey was a liability. "We can't trust you."
"I have a solution for that," Raiya said, as power glowed in her hand. She reached out to Dante. "This is my power, as the Star of Justice. Your word will bind you; if you lie to us, you will not escape your due."
"Charming," Dante muttered, but he shook her hand.
"Fine," I said. "Now, let's talk."
"I want you to leave Mikey out of this."
"Why?" I asked.
"Because he is my son," Dante said.
"You didn't seem to care about that before you left him," I pointed out.
"Do you think this is easy for me?" Dante's voice was hushed. "I have given my word not to lie to you, but I said nothing about hurting you."
"I'll say something about that," Raiya warned. Her bow was out in a flash of light.
Elysian snorted behind us. "There's no need to threaten us. It wouldn't make much of a difference, anyway," he told Dante. "The kid here is a bit slow when it comes to learning through pain."
I almost whacked him over the head for the remark, but since it technically discouraged an attack against me, I let it go. For the moment.
Dante wisely retreated. "Fine," he grumbled. "But I still need you to leave my son out of this as much as you can, and protect him if I can't."
"Fine," I repeated. We could agree on that.
"Even if it's me he needs protecting from," Dante continued, "you must stop me."
That took me aback a bit.
I guess he really is here without SWORD's approval.
"We agree," Raiya spoke up. Her determination, with no hesitation, startled me.
"Good," Dante said, before I could object or question him further. "Now that Mikey is off the table, I will release your identity as well."
"The rest of SWORD doesn't know who I am?" I asked.
"I have the file," he said, "and the only copy." He pulled it out of his pocket and tossed it to me. "So, no, no one else knows."
"Why are you doing this?" I asked.
"Mark is also one of my oldest friends," Dante said. "We were friends in high school. I have tried to keep him out of the loop as much as possible, but there are some things he knows, and he's in danger just from that. I think we can agree that Mark, and by extension your mother, need not be an issue between us."
"We can agree on it," I said. Glancing over at Raiya, I added, "I want Starry Knight protected, too."
"Done," Dante agreed.
Raiya looked like she was going to say something, but I shook my head. We could discuss it later.
"Now that we've come to terms," Dante said, "I'd like to hear what you know."
"About Draco, or about anything else in particular?" I asked.
"SWORD is a company that monitors power first," Dante said. "We have good resources, but there's nothing like hearing it from the other side."
So we are still on opposing sides. Right?
"It was my brother," Elysian said. "I'd tried to warn you before. Draco has been masquerading as a human for some time here in the city. He used Orpheus to weaken Time's power and to break his dragon skin free from its prison."
"He's a changeling dragon, like you?"
"One of the few," Elysian agreed. "He has an immortal life. With his power, and the power he's used from the Sinisters and the meteorite that was kept here, he's grown considerably more powerful."
"What does he want?"
"To release Alküzor and set the realm free from the Prince of Stars' power," I said. "Alküzor is trapped inside the world, but he has a lot of power, too. He wants to take over."
"I see you've started paying attention to the facts," Elysian murmured beside me.
"I always did," I said. I shot him a smirk. "I just pretended not to care to annoy you."
"Fire has a purifying effect," Dante mused. "I can see why he's been put in there, even if it would take forever to purify him."
"Draco hasn't made a move to release him," I said. "I get the feeling from when he talks to me he's holding off on it for some reason."
"Evil is better at waiting, better at hiding," Raiya said. "Goodness doesn't need to change or adapt."
"I've heard that," I said. "But it doesn't explain why he's waiting, not entirely."
"It's possible he doesn't actually want to do it," Dante said. "He's a powerful foe, and he's enjoying it. I doubt competition is something he wants."
"That might be true." I thought about what Draco had said to me before, and how he still seemed emotionally connected to Raiya. Maybe Dante had a point, but I was more inclined to think that Draco's heart wasn't completely in it.
Alora told me once that only in Time was everlasting change possible. The temporal was discarded, and the eternal was solidified. While Draco was an immortal being, was it possible that Time's power had affected him more than he realized?
"That's all we really know about Draco's intentions," Raiya said. "But we do know that he was once known as Ogden Skarmastad."
"The founder of the Skarmastad Foundation." Dante frowned.
"The guys who paid for you to be hired through Otherworld," I added.
He nodded. "But why would he play both sides?" Dante asked. "We're here to essentially stop him. SWORD's main job, right now, is to protect you." He nodded toward me, and I fought the urge to shrink back.
"Draco's always been crafty," Elysian said. He flicked his tail against the ground.
"For now," Dante said, "Otherworld, Inc. has been dropped by the city. With your mother folding on the case, and shuffling over to the private sector, we have no obligation to Apollo City anymore."
"But SWORD is still on assignment to protect and assist Wingdinger," Raiya said.
"Yes." Dante turned to me again. "You are the Star of Mercy, and you are the only one who had the power to overcome a being like Alküzor, should he get free."
"The Blood Flame," I whispered.
Dante nodded.
"So it will come down to me," I said, "if we want to save the city."
"Alküzor will still have to get through me," Raiya said.
"And Draco will not pass me," Elysian added.
I felt warmth as they surrounded me; not only were they my allies, but they were my friends. "Hopefully," I said, "it will be enough."
"It will," Dante said.
"How do you know?" I asked, wondering if he was friends with Alora, or if Adonaias had reached out to him, too.
"Fate is a funny thing, sometimes. The Skarmastad Foundation paid for SWORD to come in, not me. But I never wanted to come back here, knowing as I did the memories that awaited me once I got here. But my bosses knew I was from this area, so they arranged for me to take the lead.
"SWORD might be after power, and I can respect that, even now. But they are helpless when it comes to love. I thought power was all I needed. Turns out, power is at its most potent when it is laid down for love."
"That sounds too cliché for you to say," I said, unable to stop myself from sneering.
"Yes, well, as I said before, life is complicated," Dante grunted. "The inner lives of people are even more complex."
"Can you tell us the exact reason SWORD was hired?" Raiya asked. "You've said before they know of the Stars and other elements outside this world. Are they allied with Draco or Alküzor at all?"
"No. SWORD was hired because this is what we do—investigate and intersect the supernatural, the paranormal, etc. This is what the company has done for the past twenty years now." Dante shrugged. "We have been given a few more assignments here, and then I will depart with them."
"Assignments like what?"
"For now," he replied, "our next assignment is to help you stop Alküzor from tearing the gates of hell open and sending the universe as we know it into a black hole of some kind. You know as well as I do—perhaps even better—that the situation has escalated and a more dangerous threat has arisen. If this is a power we can't control, we need to stop it. Or exploit it."
"Whatever serves you better," Raiya grumbled.
I could understand why she was so adamant about keeping them in the "bad guy" box, but I frowned at her. There was no need to close the door completely on SWORD or Dante, especially right now. We didn't have to be friends with someone to appreciate their help, and for the moment, our goals were aligned.
"Good to know," I said. "What about Mikey? Can you talk to him and explain that there's nothing we can do about Gwen, and the others, for now?"
"Blaming Draco for his deception will likely be enough to take care of that problem," Dante said. He hardened his gaze. "Warning him away will have to be enough to prevent any other problems."
"If he's anything like the kid here," Elysian muttered, "don't count on it."
"Elysian, I swear, you need to—"
Our conversation quickly devolved into an argument. It was only when Raiya finally managed to stop us, several moments later, that we realized Dante had slipped away.
"Great." I nearly stomped my foot in frustration. "He's gone."
"He might have had to leave," Raiya said.
"You're defending him now?"
"No," Raiya insisted, even though I could see her cheeks fluster over in the dim lighting. She cleared her throat. "I heard some movement down the hallway. Maybe he had to leave so he wouldn't be seen by other SWORD agents or by any witnesses. He did tell us that he was here without their approval or direction."
"We don't know if he was here without their knowledge," Elysian said.
The door opened before I could say anything else. Logan came in, one hand holding an open book, and the other carrying a take-out bag. He glanced up at us in surprise. "Hey guys," he said.
"Hi, Logan," Raiya said. "How are you?"
"Good," he said as he gave us a smile. "You guys must've known I was thinking of you earlier."
"I'm surprised you were thinking about us at all. We heard you were out on a date," I said.
Logan grinned. "I'm not going to confirm or deny anything about that," he said. "But before I left, a couple of things happened I thought you should know about."
"What is it?" Elysian asked.
"First, the Otherworld guys are gone," Logan said. "I got a memo about it this morning. Apparently they have been assured that the insurance company has paid out for the loss of the meteorite. So they're not worried about it getting stolen anymore."
"Considering it was stolen," I said.
"Yep," Logan replied. "No need to guard something that's not around to guard."
"Saves them some money."
"The government has never really seemed to worry about that," Elysian muttered.
He's been watching too many news channels, I thought.
"What's the second thing you wanted to tell us?" Raiya asked.
"The vortex has disappeared from my radiation maps," Logan said.
Raiya, Elysian, and I all exchanged knowing glances. That was because Draco had used the vortex for its purpose—to forge the meteorite into a sword of his own.
"But I have picked up on some traces of the radiation signature," he added.
"Can you bring it up on the screen?" Raiya asked, stepping forward. She took his bag and his book as he obliged her.
Logan keyed in a few words and seconds later, he pulled up the picture of the city maps. There were dots and shapes and colors all swirling around, like a weather map mixed in with tie-dye.
"Here," he said, gesturing toward several lines slivering their way around the northern part of the city. "The blue lines here are similar radiation patterns to those of the meteorite," he said. "It's a bit different from before, and it's more faded. I didn't see it the first couple of times I looked at it."
I squinted my eyes, looking toward the lines. "It's running through Rosemont's old site," I said.
"And the hospital, and the Time Tower, and the marina," Elysian added.
"And Rachel's," Raiya whispered.
As I silently vowed to protect my coffee kingdom, I glanced over at Logan. "What do you think it means?"
"I don't know," he admitted. "I know that the meteorite has disappeared, and by all indications it's either been destroyed or it's out of the city limits. Or out of the reach of our tracking satellites."
"Thank you for showing us," Raiya said. Her tone was gracious and her eyes were kind, but I could tell she was worried.
"I don't know if it will help you," Logan said. "But now that my research has been tanked, it's back to observation for now."
"It might help us," I said, "but I'm not entirely sure how at the moment."
"It might help if you had a degree in astrophysics and aeronautics," Elysian said.
Stop being unhelpful! I wanted to scream at him. He'd already cost me quite a bit that night—stolen moments of sweetness with Raiya and a tolerable span of time to discuss things with Dante. I didn't need him bruising my ego or stoking my temper.
Logan didn't seem to notice my irritation, but kept walking us through charts and facts and figures. Soon, I began to feel like I was in one of Mr. Hale's science courses, my body stuck in one place, and my mind adrift in a sea of numbers and symbols.
"Thanks for all your help, Logan," I heard Raiya say. "I think that's all we can handle for tonight, but it's given us a good place to start."
"I'm glad to hear it," Logan said. "I know you can't tell me everything that's going on, but I know that you're the good guys. I'm happy to help."
"We're more than grateful," I assured him.
I was telling the truth. As Logan sat down with his book and his doggie bag once more, and Elysian, Raiya, and I all headed out of the building, I couldn't help but feel like Draco's days were numbered.
"That was interesting," Raiya said. "And disturbing."
"It was all good information." I reached over and knocked Elysian on the head. "We probably could've gotten more information out of Dante if you hadn't started arguing with me."
"I didn't start the argument," Elysian grumbled. "I said something and then you started arguing with me. I'd say it's more your fault than mine."
"As usual!" I shook my head. "That's what you always do."
"I'm not the only one who's doing just what he always does," Elysian shot back. "You're no different!"
"How?" I asked.
"You're back to planning out your life and just dragging people along," Elysian said to me in an accusing tone.
"I have to drag you along sometimes!" I threw up my hands. "Do you know how awful it is, trying to get Starry Knight to actually live?"
"Excuse me?" Raiya interrupted. "I'm doing just fine on my own!"
"No you're not," I said. "I practically had to twist your arm about prom."
"So I don't want to 'live' unless I want to go to prom?"
"Yeah," Elysian joined in. "Where you can parade her in front of your friends as proof that your life is better than theirs?"
"What are you talking about?" I hissed at him. I turned back to Raiya. "Come on, prom is practically right up there with getting your driver's license. It's part of growing up."
"That's the difference between us," Raiya retorted. "I already have grown up—and I didn't need to go to prom to do it!"
She turned away from me and took off before I could stop her.
I whirled around, glaring at Elysian. "You didn't have to make me fight with her."
"I didn't make you do anything," he said. "You did all the talking."
I clenched my fists angrily. "You're just terrible!"
"So what?" Elysian asked. "You still have a problem."
"I'll say! I'm looking at it."
"You know, even if we are terrible, flawed beings," Elysian said, "that doesn't mean that we are incapable of getting things right. It doesn't mean we will do the right thing. It means we know what's right and want you to be better."
"That's stupid," I said. "Clean up your own act first."
"I will!" Elysian assured me. "And you will not like it when I do."
"You must've been hanging out with Aleia more than I thought if you're going to hurl cryptic remarks," I yelled at him.
Seconds later, he huffed loudly, sending a stream of smoke out of his nostrils, and then he flew away.
☼11☼
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# Prom
I went home after fighting with Elysian and Raiya. It was easier to fight against them and feel like I was the winner; fighting against myself meant that I always lost, even when I won.
It was becoming more of an issue the longer I knew both of them.
Of course, I still loved myself more. I'd known myself longer.
Or had I? I mean, less than two years ago, I didn't know I had any supernatural powers (other than getting good grades and making it look easy at the same time.)
Maybe that was what drove me from my dreaming, leaving me largely sleepless, making me even more restless and on edge than usual.
It was a good thing I was talented; the last days of the week were tense. Raiya didn't talk to me much at all, and Elysian only seemed to sniffle at me.
I was giving them room, and I assumed they were giving me room, too—room to fume, more than anything else. Room to worry, the rest of it.
One of the things I decided not to worry about was SWORD. I would get to that later, if I worried about it at all. Even if Raiya had her reservations, I thought they were basically allies now, and that was all that mattered. We still had some philosophical differences as far as I could see, but there was nothing preventing me from dealing with them after we took care of Draco. If I had to take care of them at all. Dante had said they had a few more assignments, and then they would be stepping out.
No, I was more worried about other things.
Friday was a half-day at school, as it was prom night. The school was decorated and the students were all cheerful, myself included. Poncey was practically prancing around, while Jason was cheering about finally finding a date. I was happy to hear my friends were coming—even Simon was coming, though he was bringing someone I didn't know as his date.
Mikey was the only one of us who was rather reserved. But he had other problems to worry about. Since he'd fallen behind in his schoolwork, he wasn't able to go to the prom.
But I wasn't worried about him, either. Not really, anyway. I was more worried that Raiya was going to back out of our agreement.
I hadn't been that nice to her, I supposed. She was already not excited about going to prom; I didn't need my emotion-reading skills to know that. The fact I'd hurt her didn't seem to help.
So after school, and the obligatory shout-outs to my friends and discussions with others who wanted nothing more than to be my friends, I headed over to see her.
"Hamilton." Rachel greeted me with her usual warmth. "How are you today?"
There was no hint to suggest anything was wrong.
I cheered instantly. "Hi, Rachel." I walked over to the counter where she was cleaning. "Raiya getting ready for the prom?"
"Prom?" Rachel frowned. "That's tonight?"
"Yes. Didn't I tell you about it?"
She laughed at my frightened confusion. "Just teasing," she said. "But no, Raiya had her GED this morning. She'll be finished with it in another hour."
Shame sank into me. I'd forgotten about her test!
"I forgot," I said, half in disbelief and half in annoyance.
At least nothing happened while she was taking her test, I thought.
"I think she did, too," Rachel admitted. "I came in early this morning so she could go. She was running behind schedule. She seemed distracted." Rachel gave me a teasing glance. "She seems to be distracted a lot when you come around now."
I grinned, despite feeling bad about forgetting her.
"You really do love her, don't you?" Rachel asked.
"Of course," I said. "Even if I forget about her tests."
"Good." Rachel handed me my cup. "There's nothing like love to add magic to a special night like this. Go home and get ready for tonight. By the time you're ready, she'll be ready, too."
I agreed. I left, feeling better knowing Raiya wasn't going to leave me hanging. Rachel would see to that.
For all of Rachel's more annoying fluffy true love stuff, I was glad she was like that—and not just because it meant I could count on her to help me in certain instances like this one.
As I changed my clothes and showered, I got a voicemail from Cheryl, who, even though she wasn't working, was held up at a meeting with a potential investor for her firm.
I suppose many parents take pictures and gush over their kids at prom, but I didn't have to deal with that. Adam was the only one who seemed excited for me; he took pretend pictures of me as I came down into the living room with his toy camera, while Mark dozed in the armchair. Even Elysian seemed to be missing, but considering his fondness for sweets and his concern over Draco, I had a few ideas of where he was.
"Boy."
I turned to see Ayako poking her head out from the kitchen. She had a real camera in her hand. "Let me take picture for you."
"Alright."
A few pictures later, she slipped me a nice surprise: A corsage. "For your lady," she said with a kind smile.
I regretted, for a teeny tiny moment, that I'd been so harsh on Ayako's cooking. Just because she didn't make food I liked didn't mean she wasn't a nice person. Maybe that was true of the others, too, and I'd just been unable, or unwilling, to see it.
The corsage in itself was pretty; there was a large white bloom, surrounded by small pink blossoms, with a nice arrangement of baby's breath and small decorative inserts.
I didn't know the names of the flowers, nor did I care much.
"Thank you," I told the kind cook, giving her a quick kiss on the cheek.
"Oh, sweet boy," she said with a small chuckle. "Go have fun."
I didn't need any more prompting. I hurried outside, surprised to find Elysian, clearly waiting for me. I didn't know if it was the weather or his presence, but all of a sudden there seemed to be a chill in the air.
"What are you doing here?" I asked.
"Starry Knight asked me to come and get you," he said.
"I didn't feel anything," I said, glancing down the mark on my wrist. "Is something wrong?"
"No," he said. "She's getting ready for prom, and I was bothering her, so she sent me on this errand to get rid of me."
I laughed. "She's smart."
"She also figured this would give you a chance to apologize to me," Elysian said.
I stopped laughing. "She's diabolical," I muttered.
"So I guess that's a no-go on the apology?"
"What do I have to apologize for?"
"For your general selfishness, maybe?" Elysian shrugged. "At this point, the list is so long I don't think I could read through it all."
"If you're supposed to take me over to Raiya's," I said, my tone low and dangerous, "you'd better stop talking."
"You don't make it easy," he snapped. "Your life is always just one long to-do list, and you act like we're pitted against you when we don't think a certain way or fall into line behind your expectations."
"My expectations are reasonable," I argued, starting to walk down the road, out of the subdivision and toward the coffeehouse café.
"Your arrogance is not," Elysian muttered behind me.
"What about your arrogance?" I said. "You're the one who's judging me."
"I've seen the kind of behavior you have before," Elysian reminded me. "Draco was just like you, even before he chose to rebel along with Alküzor and the others."
"I'm not like him."
"Yes, you are." Elysian pushed back. "You're all about yourself, and you don't mind running the people who care about you over if it means you get your way."
He scurried ahead of me, and we dropped the subject along with the conversation.
I rebelled against his reasoning, but I had to wonder at him. Did Elysian just admit he cared about me?
Despite my anger over Elysian's accusations, I made a promise to myself to be nicer to him. I didn't want to apologize—seriously, the end of the world would happen before I willingly apologized to him first—but I could forgive him for being a jerk about things.
After all, I thought, Draco's betrayal had to have hit him hard. While Elysian and I weren't brothers, we were brothers in arms, and we had a mission to face together. He could have easily been hesitant to trust me because of the past.
It didn't excuse his behavior, but it explained it better.
I forgot about Draco and Elysian as we arrived at Rachel's. The corsage was in my hand, and I was surprised to feel a bit nervous.
Rachel waved to me from the back as we walked in. "Hey!" she called. "Raiya's upstairs."
"Thanks," I called back.
I turned to Elysian. "Stay down here, out of sight," I said. "I'm going to go get Raiya."
"Fine. Hurry up. I'm hungry," he muttered.
"Don't you mean, 'take your time,' then?" I asked. "Raiya already told me she puts whatever you eat on my tab."
Elysian thought this over for a moment. "Good point," he finally said, and then he slithered out of sight, already changing into a chameleon-like form as he headed toward the kitchen.
I let him go and allowed my nerves to resume their persistent hum. My nerves doubled the instant I walked up to Raiya's room and saw her.
She was wearing the dress she'd worn for Rachel's wedding. I recognized it instantly, and not just because of what it looked like; I felt the same rush of reaction, the same admiration I felt for her last summer. Of course, this time it was much more potent, because I let myself admit I was attracted to her.
Her hair was pulled back in a half-bun, no doubt a nod to her Starry Knight persona. There was even a line of small flowers in her hair where the wings on her head normally sat.
As I caught her eye, she winked at me.
I stood up straighter and made sure my mouth hadn't dropped open in shock.
She came up to me and slid her arm through mine. "Rachel wants to get some pictures," she warned me.
I pretended to groan. "We have a long night ahead of us, then."
"You could end it right now," Raiya offered. "It would spare me a lot of embarrassment in front of your friends."
"You're the one who told me everything was going to be alright," I reminded her. I showed her the corsage Ayako had given to me. "You're going to have to prove to me you know it this time."
I pinned it to her dress, careful not to poke her, and stepped back. "There," I said, leaning forward to give her a kiss on the cheek. "You're mine."
As she blushed, I felt the desire behind her eyes as she looked back at me. No amount of Starlight defender training in the world could've prepared me to resist her silent plea.
The instant I allowed myself to lean in and kiss her, I felt the rest of my resistance disappear. My hands were suddenly tangled in her hair, my mouth fused to hers, and her body pressed against mine.
She welcomed me, her hands running down my back as she tried to pull me closer. We stumbled against each other, and I barely managed to catch us against the wall. I was too intoxicated at the taste of her, the feel of her, the heat between us—all of it overwhelmed me, and I was unable to do anything more. Everything washed over me, leaving nothing behind. Nothing but her.
"Hamilton," she whispered as she began to push me away. "We're going to be late."
"It's okay," I insisted.
"But—"
"Please," I begged. "Please, just let me kiss you for a bit longer."
We'd had this discussion before, several times. I knew she was hesitant when it came to passion, and I knew she had good reason to be—she hadn't been thinking clearly when Orpheus tricked her into going supernova on the other side of Time, and the possibility of losing me again terrified her.
So when she relaxed and gave me a tremulous smile, I felt the weight of her trust and the burden of her hope as they broke through my yearning.
"Okay," she whispered, leaning forward to kiss me again. "But Rachel still wants pictures. We can't be too much longer, or she'll come looking for us."
I barely heard her as I kissed her throat. "Okay," I said, savoring her shock at the tender caress.
I wasn't sure how much time passed before we were interrupted; I only knew that it was too short of a time. I nearly fell over in surprise when Rachel knocked on the door.
"I did warn you," Raiya said with a giggle. Her arms pulled back from me, moving to straighten her hair and smooth the wrinkles out of her dress.
I was only a little disappointed to see the corsage I'd given her was crushed.
"Fair enough," I replied, trying to catch my breath as well as my balance.
Rachel came in just as I managed to steady myself.
"There you are," she said. "It's almost time for the prom to start."
"We're ready," I said, grabbing Raiya's hand.
"Great! Let me get some pictures and then you guys can head out." Rachel grinned. "Do you want a coat, Raiya? I heard on the news it's supposed to snow tonight."
"Snow?" I asked.
"It's an onion snow," Rachel explained. "They come sometimes, at the end of April or the beginning of May. The temperature's been dipping since earlier, so I thought I'd warn you."
Huh, I guess it was the weather earlier, rather than Elysian. I was surprised.
"I'm okay," Raiya said. "I don't think I'll need a coat."
"She can have mine," I said, tugging at my suit, "if she needs it."
Once I stepped outside, I realized Rachel was right. The downside of being a Star, I thought. I could no longer appropriately dress for the weather on my own.
I was comforted that I had all the warmth I would need. Raiya was beside me, and a night of adventure was ahead of us.
"Ready to go?" I asked her.
She gave me a quick smirk. "It would be my pleasure," she assured me.
☼12☼
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# Last Dance
When you are popular, especially in high school, it is necessary to have some contingency plans. You never know when your ex-girlfriend will launch herself at you, trying to convince you it was all your fault you ran into her, because you were still in love with her (to which you respond, "I was never in love with you"). You'll also likely never expect your friends to stare at your date in disbelief, making her increasingly uncomfortable, to the point where she asks you if it was time to leave just fifteen minutes after finally getting into the main room.
The prom was set up in a small conference hall close to the marina. We could see some of the ships coming in and going out as we waited in the welcoming line.
The theme was "A Night on the Sea," probably loosely based on the Titanic, guessing from some of the decorations. I didn't bother to dress up in anything nautical, and I was glad to see I wasn't the only one who hadn't bothered to learn the theme ahead of time.
But despite some of the tackier elements, there were some really nice furnishings, and the setup for the dance floor was nice. As I've said before, credit where it's due.
Raiya's hand stayed firmly in mine, but I could feel her fingertips digging into my hand every time Poncey or Drew looked at her, as if they weren't sure if I'd been telling the truth about her identity.
Jason, for his part, welcomed her, and even complimented her nicely, which set her a little more at ease. He'd seen her around Rachel's enough that he knew who she was, and I suspected he knew we'd been close for some time.
"Good job, man," he said, slapping me on the back as Brittany began chatting with Raiya about her GED.
"Thanks," I murmured back.
"I thought you might bring her," Jason added. "Rachel's been bubbly and ecstatic all week, but she wouldn't tell me who Raiya was going with to the prom."
"Well, that's Rachel for you," I remarked. "A sucker for true love if I ever saw one."
"True enough," Jason said.
"Brittany seems like an odd choice," I said.
"Hey, I complimented you."
"We're friends. I'm allowed to ask you awkward questions and say less than nice things."
Jason grimaced. "I didn't have a date, and I thought it would be a nice event. See if I couldn't smooth over her Poncey-hatred."
"Did it work?"
"I'll let you know later," he said with a grin.
I didn't want to know what he meant by that. Thankfully, I was distracted as Simon came along and met us with his date, a senior girl, but one I didn't recognize. I think she said her name was Casey or something similar.
She was nice enough, joining in with our conversations about the latest music, food, entertainment, etc. Felicia, Poncey's date, heralded us with tales of his success in their cooking elective, which was how they met, apparently.
Simon had to be on his toes some, I noticed, when his younger sister, Phoebe, arrived and met up with Drew, who was her date. He eventually calmed down some, but he made sure he was close enough to them that they never left his sight.
We talked, we ate, we took plenty of pictures, and we celebrated.
I didn't get many more questions about Raiya from the guys, so long as she was nearby. When she went to go get some punch, Drew came up to me and asked me if that was really the Raiya I'd told him all the stories about.
"I didn't hire someone to play her," I retorted. "Is it so hard to believe?"
"No," he admitted. "Just surprising."
"Why?"
"You didn't seem to like her very much. I mean, don't get me wrong, she's cute, but ... "
I half-listened as Drew started reiterating some of my more pompous arguments I'd mentioned to him, recounting some of our more tense disputes from Mrs. Smithe's class and even some from Mrs. Night's class. (How old was this information, exactly?)
As he continued to let them roll off his tongue, I glanced over at Raiya. She seemed to sense my gaze, and turned to give me one back. Her eyes were bright and her smile was immediate, as she saw me. I held her eyes until she finally turned away, her cheeks flushed over with crimson.
But I'd seen it—the look of pride on her face, the warmth and approval in her gaze.
Hearing my complaints against her almost made me laugh, half in amusement and half in horror. I was suddenly taken aback by my own selfishness. I'd wanted comfort in telling my friends about the arguments, but I knew I was wrong in some of them, at least, and my friends were all wrong to say nothing about my bias.
I saw now that having Raiya there was the best thing I could ever have, and I didn't want her to ever leave.
"You know what, Drew?" I said. "You're right. I don't like her. I love her."
I didn't have to turn my head to see the shock coming off of his expression. His surprise gave me a push of confidence, and I made my decision the moment I heard the music slow its tempo. "Now," I said, "I'm going to go dance with her. If you'll excuse me ... "
Raiya met me halfway in the middle of the dance floor. "What are you doing?" she asked.
I plucked the drink out of her hand and passed it to some other person, who, in recognizing me, mindlessly took it. "I'd like to ask you to dance," I said.
"You're not really asking if you're not really giving me a choice." There was a bit of a smile on her face, and I knew she was arguing with me in a friendly manner.
"I can't risk you making me look like a fool," I said.
"You're not risking that at all, by asking me to dance," Raiya said.
"So you'll say yes?"
"If you ask."
"Okay. Will you dance with me?"
"No." She laughed as I cocked an eyebrow. "Just kidding." She took my extended hand in hers, and while I didn't really do any dancing, I led her around the floor in a respectable-looking pattern.
"I've never really been to a dance," Raiya admitted. "Rachel has music nights sometimes at the café, but I avoid them."
"I don't really dance at these things, much," I told her. "So you're in good company."
"The best," Raiya said with a grin. I was more gratified when she took a step closer to me. Her eyes met mine, as her lips were only inches away.
"Are you nervous?" I asked.
"Why would I be nervous?" she asked, even as I knew she was, and she was only trying to put on a good face for me.
"Because people are staring at us," I said. I nodded toward the crowd of people around us, as several people—some friends, some frenemies, all busybodies—tried to look casual when they looked our way.
"Oh. Well, I'm better than what I thought I would be," Raiya replied. "Your friends aren't so bad, I guess."
"That's what I thought, too." I laughed.
I held her close for a few long moments. As the music died down, I sighed. I was all happy, except for one, nagging reminder, and I knew I had to do something about it.
"What is it?" she asked.
"I'm sorry," I said. "I'm sorry about fighting with you so much lately."
"Let's just say that we were both a little wrong. I know there's nothing wrong with having fun, but I got defensive when you pushed it on me. And I know this was important to you," Raiya told me. Her voice was quiet and steady, and she was talking to me like a mother would talk to her toddler after having a tantrum. I had a feeling I deserved it.
"It's not as important to me as you are," I insisted. "And it's definitely not as important as protecting the city from Draco."
She smiled. "Thank you." Raiya leaned over and tucked her head into the crook of my collarbone, snuggling closer to me. "It's alright."
"I don't always like fighting with you."
"I know. You lose sometimes."
"I don't mean just that," I told her. "I mean, I hate it when we argue to the point where we hate each other."
"You and I have a long history of each other," she reminded me. "Practically none of it has to do with hating each other. A lot more of it has everything to do with loving each other despite our disagreements."
I suddenly wished I could remember more of all of our lives together; I barely remembered her at all, and the little I knew of my time up on the other side of Time, it had all been through Alora and Aleia's information, and some of St. Brendan's, too.
Raiya continued. "We never had to worry about this before we came to Earth," she said. "And maybe it's a good thing, to have our affection tested."
"You think it's a good thing?" I asked. "Even when I'm arguing with you?"
"In some ways. Even truth has to be tested against the evidence in court," she pointed out. "Can our affection outlast our anger and pride?"
"Yes, it can," I affirmed.
"Then it will." Raiya leaned further up and placed a gentle kiss on my cheek. "Life will test every part of us, to find what is good and what will last. I think saying you're sorry is a good way to pass that test."
"Are you going to pass it too?" I asked.
"I'm sorry," she conceded, though I thought I saw her roll her eyes, "if you thought I was angry enough to stop loving you."
"Thank you."
I clung to her as the music's last note faded away into nothing. "I'll work on trying to be better. I don't want to keep forgetting our lives are both here, and past, and ahead of us, all at the same time."
The world is a strange place sometimes. At that moment, I could see the full arc of our story; I could see Raiya by my side as I studied at Pitt, facing down a tight schedule. I could see her coming along with me when we moved, arguing down the cost of labor and the rental fee from the moving company. I could even see her, standing beside me as we were married, facing down a lifetime of arguing over the TV remote, paying off our bills, and balancing holiday plans—and still saying "I do" an infinite amount of times.
Suddenly, the decision hit me hard and fast. But it was the right thing to do, and I knew it.
I took her hand. "Come with me," I said, leading her out of the ballroom.
I headed out toward one of the small patios off to the side. I opted for one that was close to the water, under the moonlight, reflecting the stars through the thick cloud cover.
I can't believe I am going to do this, I thought. But I was turning eighteen in just a few days, and that would make me a legal adult.
It couldn't be that bad. My dad managed to get my mom to say yes, right? And less deserving people than me married all the time. Just look at Hollywood.
I had nothing to lose in asking her, and a lifetime of everything to gain.
And it wasn't like Raiya would say no.
We reached the outside, and I saw her briefly shiver at the breeze. Even I felt its sting this time, but I decided ultimately it wouldn't bother me; I was more concerned with other matters.
I smiled at her, knowing at once it was true. We were here, together, in this place, public with our love, and we both looked good in our prom clothes.
True, I didn't have a ring, but I had something I knew she would value just as much and guard just as zealously: My pride.
Hey, she's the one who said before it doesn't count unless you suffer.
Raiya glanced up at me, a quizzical look on her face. "Why did you want to come out here?"
I took her hand and tried not to shake. I took a deep breath, hoping my years of just winging important speeches would save me once more, giving me the greatest impromptu speech I would ever muster—
Only to have the shock of my life, as my wrist burned with blackened pain.
"No." I shook my head, wanting to howl in pain and raise my fist against the unfairness of the world. "No, not now."
"What is it?" Raiya asked, and then she stilled. "Draco."
"Yes," I grumbled. "He's back."
☼13☼
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# Fight of Destiny
As we turned, another vortex formed over the city, this time encasing it in a crystalline bubble. I felt Time's pull shift dramatically away from us, as though it was tearing the fabric of space-time apart.
A boom! echoed throughout the night, and then cut off as sharp and quickly as it had come.
I watched the rumble of the world around us stop completely, and I knew we were in big trouble.
Raiya gripped me. "Are you alright?" she asked. "Can you move?"
"Yes." I looked over at her. "Why? What's wrong?"
"He's disrupted Alora's connection to this world," Raiya explained. She pointed at the bubble-shield being conjured up in the sky before us. "He's stopped her power from affecting us. Only those of us who have been born outside of time can move now."
I glanced around to see she was right. The explosion I heard had stopped, mid-boom, right where Rosemont Academy ruins once stood. Flames, stilled yet slowing and splintering, rose out of the ground.
"We need to transform."
Her statement washed over me, and at that moment, I realized how loudly she was speaking; the explosion was deafening. My ears popped open.
Without another nudge, I pressed into the mark on my wrist, watching as the blood-red mark glowed.
Seconds later, I was no longer looking at Raiya; I was staring into the violent eyes of Starry Knight, my co-defender and trusted ally.
"Wait," I said, before she could take off.
"What is it? We don't have a lot of time," she warned me.
"I know." I sighed. "When this is over, I have something I want to ask you, alright?"
She stilled, blushed, and then she nodded. "Alright." She took my hand and tugged me along as we took off.
Elysian met us in midair. "It's Draco! He's gained enough power that he's already broken through Time's power."
"We kinda figured that it was Draco," I said.
"Where is he?" Raiya asked. Her bow flashed out, and I was pulled back into the moment before, when he fought with us as he forged his sword.
Suddenly, I knew I had to stop her. "Starry Knight," I called out, as Elysian turned his attention toward the center of the vortex.
"What?" she asked. "I thought we agreed you can talk to me about other stuff later."
"It's not that," I grunted, regretting all over again Draco's timing. I pointed to her bow. "I don't want you to worry about taking Draco out."
"What?" Both Elysian and Raiya objected.
"I mean it," I said. "Look, I'm the Star of Mercy. I know your grandfather-turned-evil-dragon-puppet is still important to you. I'm going to be the one who takes him out."
Elysian snorted. "Good luck with that."
"I'll call dibs if needed," I told her. "I could use your help distracting him. I think that would work best anyway; when we fought him before, after he revealed himself, he still seemed affected by you, too."
"You really think so?" Raiya's voice was soft against the building winds.
"I can't imagine an immortal life like his is full of friendships." Some part of me couldn't believe I was actually feeling empathetic toward Draco. I felt better knowing I was using it to destroy him, preventing him from hurting Raiya and other people ever again.
And I'm working to free the remaining Soulfire, I remembered, thinking of Gwen and Mikey.
Elysian nodded. "Alright," he said. "It makes sense. He's only connected to Starry Knight and me, so the kid—"
"Boss," I interrupted, correcting him.
"—will be the better opponent for him, if we want to win."
"Of course, we want to win," I exclaimed.
"I don't think we can really win this war," Elysian said darkly, his yellow-green eyes dimming as the sky was shut off from the rest of the city.
"We have to," I yelled back.
"Then get ready," Raiya said. "He's coming this way."
I scanned my field of vision. A spark on the horizon came into view, and then there was no mistaking him: Draco was flying down toward us, his sword out, his power ready.
I pulled my sword out as well. "Alright," I said. "Let's go!"
Elysian and Raiya took off, heading to meet him from different sides.
I almost took off to join them when I caught sight of the bright light blinking at me. I squinted down onto the ground, where I saw Dante was running up from the marina, hailing me.
As Raiya and Elysian met Draco in battle, I hurried down to see Dante.
He was freezing over, his body cut off from Time's power.
"Dante," I called. "What is it?"
"You," he said. "You have to stop him."
"We know that!" At his hardly-new information, I nearly hit him for calling me away from the fight.
"No," he said. "You have to stop him."
"I know that, too," I cried. "I was just about to go and stop him when you called me down here—"
"No," Dante said. "That's not what I mean. You have to stop him from ripping the world away out of this realm and you have to stop him from freeing Alküzor. You can do this by releasing the Blood Flame."
"Okay ... " I was starting to get weirded out. "And I can do this by—"
"To release it completely, you'll have to die."
His words slammed into me. What?!
Dante breathed in deeply. "SWORD is not able to withstand this pressure," he said. "We are—I am—not able to help you."
"I have to die?" I repeated. I couldn't get past that part. I looked back up at Raiya, who clashed in the air once more with Draco, as he met her, his sword against her bow.
Elysian roared, letting the flames of his celestial fire heat up the sky, adding spirals of fire to crawl across the crystalline ceiling of Draco's bubble.
I turned back to Dante, watching as the last of his body froze over, falling away into the trap of a forever moment.
Helplessness suddenly came over me. I have to die?
"Boss!" Elysian called. "Are you coming or not?"
I turned toward him. "Coming," I called back, not sure if I would.
As helplessness fell around me, a memory suddenly called to me—the memory of the first time I met Adonaias.
I had felt a similar way; like I was going to die, but much more so that I wasn't going to live. My fear had strangled me, wrapped me up, and sentenced me to a life of slavery to the self, to the endless service of selfish fear
There had been little hope, no help.
And then, all of a sudden, there he was, calling out the power I had inside of me to prove me wrong.
I have to believe that there's still hope.
I looked at Raiya, as she faltered from one of Draco's attacks.
And then it all made sense—to me, at least. I had wished to go here, to follow Raiya to Earth, after she damned herself. Our enemies had been captured within Time for us to subdue, and now, as I faced down the greater foe, I knew I could stop him. There was a power inside of me that had been freed and tamed and strengthened through its growth and discipline. And I knew I could be saved—hadn't I already been, by Adonaias? I would be, again, by Raiya's power.
All this full circle was bound by a power driven to be overcome by love.
I saw it, and it was a thing of beauty.
My fingers tightened around the hilt of my sword, and I narrowed my wings in determination.
Raiya swept herself aside as I flew into the battle foreground. My sword came up, swinging hard and fast.
It met Draco's with a resounding clang, moving a gigantic force of energy between us. The clash went through me, pushing me back even as I refused to move.
Draco frowned underneath the sheer weight of my power. "Been practicing since last time, have you, lad?"
I couldn't answer him; my jaw was set, continuing to stream power at him, until he relented.
For a long second, I wasn't sure if he would give up or not. I wasn't sure if I would give up or not, either.
At last, a moment later, Elysian roared, sending another burning burst of fire our way.
I shouted in triumph, as Draco stepped back to avoid the flames. I pushed through a second after, holding him back again.
He managed to rebound, enough to where I knew I wouldn't be able to keep up the pace if this continued.
An idea hit me.
"Starry Knight!" I called. "I need to you shot an arrow at my sword!"
"Why?" she yelled back.
"Just do it!"
She looked frightened, and for a moment I didn't know if it was for me or for Draco. Even as far apart as I was from her, I knew the second she decided to do as I asked.
"I love you," I whispered to her as she drew back the arrow in her bow.
"I trust you," she whispered back. And then she released her arrow.
It sliced through the air, headed toward my sword; it moved through the air seamlessly, perfectly. Still concentrating on pushing back Draco's sword, I had to move quickly. I couldn't mess this up.
I only have one chance for this ...
Draco laughed, even as it came barreling toward us. I grinned. At the last second, I stuck my arm out toward the arrow. It struck me hard, slicing open my skin and sending a trickle of blood down my arm.
I saw Draco jump back from me at once. He nearly dropped his sword in surprise, and I took advantage of his momentary weakness.
"Augh!" I lunged forward, bringing my blade down on his arm.
I grimaced as my sword struck him; he was more human than the Sinisters had been. I felt the shock of hurting another person like me, and I stumbled back.
I heard Raiya call out to me, and Elysian cheering. But I didn't pay attention to them. I was too focused on Draco.
"You'll lose this battle," I declared, bringing my sword back up to the ready as I breathed in deeply, trying harder to steady myself.
Draco stepped back, and I watched, grimly, as his arm fell to the ground and his blood, black as a demon's, came rushing out.
Despite this, there was a smile on his face. "Haven't you figured it out yet, young Hamilton?" He laughed. "Evil doesn't die; even if you destroy me, you will always find evil waiting, just waiting, to rise up in a new form."
"I'll settle for a new form," I shot back. "So long as it's not you."
I swung my sword again, this time calling forth the power of my soul. I felt it mix with my blood and burn, and I knew this was the moment—this was the moment where I could seal him away and save the day.
My sword swished through the air, only momentarily clipping him on his side as he stepped just out of reach. Immediately, I pushed forward, but he surprised me by ducking out of my way and diving toward the ground.
For the split-second I saw him, I felt a rush of relief; there was nothing underneath us but the sunken ground where Rosemont formerly stood.
It has to be over now.
Raiya's cry changed all of that.
"We have to stop him!" she said, as she hurried past me.
"Watch out, boss," Elysian called, as he tackled me just before the vortex's center came reeling toward us, following Draco and Starry Knight toward the ground.
"Starry Knight!" I cried.
"She'll be fine," Elysian snapped. "I'll get her." He hurried toward the ground as I shoved my sword into its scabbard.
From the angle I was at, I could see Draco was not going to make it; he was going to fly straight into the ground, sword first, closely followed by his head.
He isn't going to survive that, I thought. Surely not.
The image of Orpheus' sacrifice soared into my mind, and I realized Raiya was right; Draco wasn't concerned with survival anymore. Now, it was only about winning.
What better way for him to win than to die setting Alküzor free?
That makes his taunting make a bit more sense.
Coming to my senses, I followed Elysian down; despite the large dragon butt in my face, I never took my focus off of Draco.
I could see him bleeding out as he thrust his sword into the heart of the darkened ground—the sword made out of meteorite rock and demon power, forged from the fires of a celestial dragon, capable of standing up to the power of my Sealing Sword.
There was a hollow ringing noise that sounded out, as though the heart of the earth had been pierced.
"No!" I cried.
It was too late. I was too late.
☼14☼
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# The Void
Elysian managed to use his tail to grab a hold of Starry Knight at the last moment, before the power of the vortex began to collapse on us. He flung her into me, and he managed to catch both of us as we felt the pressure of Draco's bubble suffocating us.
The center point of the vortex came crashing down on top of the sword; Draco let go of it with his good arm only seconds before the power rushed behind him.
The earth groaned, and I could hear its cry of anger and tiredness. I could hear its guttural call for rest, for peace, for salvation—only to be met with the crack of the voided lightning, the power behind Draco's vortex, as it further divided the world into pieces.
The ground sunk even further in as the vortex pressed the sword into the ground like a wedge. From the split halves of the ground, I saw something that looked like a black hole rising up between them.
It's the void. Alküzor is coming.
Raiya, after she unwrapped herself from Elysian's tail, came over to me. I felt her fingers dig into my skin, as she grabbed onto me. "What do we do?" she asked.
It was surprising to see she was at a loss. I stared at her blankly, until I felt the rush of her power around my arm.
"No, don't do that," I told her. "I can use my blood and my Soulfire to purify the void that's breaking through. Alküzor can't come through if there's something blocking his way."
"I'm going with you," Raiya declared. "It'll be dangerous, but we'll face it together."
Before I could tell her that was the plan, if we were both going to survive, a force from behind knocked me over and out of her reach.
"Didn't forget about me, did you?" Draco grinned as he glanced toward the sword in the ground. "Soon, Alküzor will be free, and I will be at his right hand."
"We can still stop you," I told him, drawing my sword out for battle once more.
"It doesn't matter what you do now," Draco said. "My mission has been fulfilled."
I glared at him. "We'll see about that."
"We will indeed," Draco said, as he transformed. His human-like form glowed with a bright, angry black, and seconds later, I watched as his dragon form took shape.
He was still missing an arm, and he had a gash in his side to rival the one he left on Elysian before.
I saw Raiya unleash another arrow, aiming close to his eyes; Elysian fired a ball of dragon's fire at him, and I took up my sword, flying up as close to his wound as I could.
All of us began to fight, unleashing our power and helping each other out. For a few moments, as tired and sweaty as I was getting, I felt the relief of winning a battle inside of me, and one I shared with my friends. Raiya, Elysian, and I had all had moments where we were never completely in sync with each other. The fight with Draco, while the world was tearing apart and we could hear the looming roar of a demon encased in the fires of the earth, was among our finest moments as a team.
The vortex's power continued to fall into the hole forming in the earth. I wasn't expecting Alküzor to come, or I wasn't paying attention at least, because when his arm jutted out of the ground, I cried out in shock.
"Augh! What is that thing?"
A grisly ghost of an arm, covered in fire and ash, salting the air with sulfur, reached out and lashed power enough to rival gravity's revenge.
"Watch it," Raiya called. She turned her attention to the arm, shooting several arrows. I saw she followed my example, slicing open her palms and saturating them with blood, so it would be easier to seal away the demonic powers.
"Be careful," I called back. "I don't want you to—"
Before I could finish my sentence, Draco's tail wrapped around me and squeezed. "I don't want you to worry about fighting anymore," he finished for me.
Frankly, it was a great mischaracterization of words, and I almost sighed as he said it.
But that wasn't the only thing that was mischaracterized.
Elysian shot up out of the shadows and tackled Draco. Using his teeth, he managed to pull Draco's tail away from me.
Draco roared and let out a beam of his own dragon's fire. It was entrenched in lightning, with fire swirling devilishly in between the forks of twisting light. Elysian and Draco fell into a beastly battle.
"Go help Starry Knight," Elysian called, as he managed to (briefly) tie Draco down.
I watched as Draco's bloody stump smooshed into his face. "Are you sure?" I asked.
"Go!" Elysian ordered.
There was no doubt in his voice, and I heeded his call.
Hurrying over, I saw Starry Knight digging her feet into the ground where Alküzor's various body parts and power kept exploding free; she was trying hard to get closer without getting sucked down into the void.
"Raiya," I called. She reached out for me, and I grabbed her hand eagerly. In the fiercest hub of our battle, her warmth became life-sustaining for me—and she wasn't even using her healing powers.
"We've got to close up the break," she said. "Draco's sword managed to pierce into the next realm. If Alküzor escapes, he could use it to break us completely away from Time."
"What would happen then?" I asked.
"I don't want to know!" she insisted.
"So how do we stop it?" I yelled back. The gravitational pull toward the hellish opening increased, making us dig into the ground even more fiercely.
She cried out as Elysian and Draco rolled overhead, their snakelike bodies whipping around each other in a deadly dance. I grabbed her hand and felt the trickle of oozing blood.
The Blood Flame.
I knew what I had to do all of a sudden. I grabbed my sword. "Give me your hand," I said.
It was hard to maintain my balance and coat the sword with our blood, but I managed it. I tucked my sword into my one hand, and then I grabbed one of hers with my other. "Stay with me," I called.
"I will," she said. "I promise." Her fingers wound around mine, and there was a new power that flowed between us.
"Get ready to get sucked in," I said. "On three?"
"Three's good," she said.
"One ... Two ... "
"If we don't make it out of here," she said, "I'll be waiting for you on the other side. Assuming I make it there."
"Raiya," I breathed. "There's nothing Adonaias and I want more."
"I love you."
"I love you, too," I said. "But we're going to make it out of here, just like you're going to get forgiven by the Prince. All we have to do now is believe. How hard can that be?"
"Right." She grimaced, as if she knew I was pretty sure I was going to die if any part of this went wrong.
I was about to say, "three," when it happened. I felt a new, familiar rush of power.
"There's something more you can do." A voice spoke to us, calming the winds around us.
Adonaias appeared between us. "Do not let go of each other," he ordered. "When you go in, you will be tempted to let go. But I tell you the truth, you must remain together."
For a long moment, Raiya just gaped at him. "Adonaias," she finally said, gasping out his name.
"Astraiya." He greeted her warmly, reminding me of a father who was welcoming his daughter home. "Follow his lead and stay with him."
She clung to my hand, but she also reached out for his. Her hand went right through him, as if he was a hologram or a projection. She stepped back.
"I am with you in spirit," he explained patiently.
Raiya nodded and steadied herself. "Does this mean I'll get a new wish?" she asked him.
I almost yelled at her for asking for such a trivial thing when we had Alküzor to stop and the world to repair, but I couldn't. One look at the desperate, disbelieving love on her face, and I knew I would have thrown the world away for her if it meant she was welcomed into the Celestial Kingdom again.
"No," Adonaias told her. "I have something better for you. But you will not receive it until you are willing to let go of your own wish."
"But, what—"
"Are you going to just stand there?" Elysian yelled, raking his claws down Draco's underbelly.
I guess he can't see Adonaias, I thought. It made sense; Elysian had avoided meeting him directly before. The more I thought about it, the more I could see it; Elysian wanted a peek, but he didn't want to get caught doing the peeking. Adonaias likely knew that.
Before Raiya or I could respond to Elysian's call, Adonaias vanished, and we were left back in the center of the storm of wind and shadow.
I squeezed Raiya's hand. "It'll be alright," I told her, even as I was the one who more likely had to be told that.
"Right." She nodded. "Let's—"
"Augh!" We both flinched and stalled, as Elysian howled in pain behind us. Hands together, we turned to see Draco laying him out against a hard surface, kicking his head further into the crumbling earth.
Elysian stilled, and I felt my heart stop as Draco laughed.
He turned to us. "My brother always thought he would be better than me one day," he said. "I've waited centuries, millennia, to prove him wrong."
"Get up, Elysian!" I cried. I took a step toward him, but Raiya held me back.
"We need to stop Alküzor," she said.
I didn't really want to hear her words. It took me twice as long to process them.
Draco turned on us, his long, scaly body, still dripping with blood and dust, snaking around us in unholy excitement. "There's no way you'll win," he taunted us, striking forward and snapping his fangs at us.
Alküzor roared, and a long, shady tongue launched out of the vortex's opening.
Before I could respond, I heard Elysian stir from his position on the ground.
Elysian coughed, but crawled out to do battle. "I don't need to win," he said, as he faced Draco, "I just need to make sure you lose."
With that, Elysian launched himself at Draco once more, this time grabbing onto his neck. Elysian's nose snorted out fire and blood, as his eyes turned orange with pain and power.
He managed to wrestle Draco over to the void's opening. Raiya and I watched in horror as he dived in, the darkness encasing him and Draco, as they both roared and burned with new levels of pain.
"Elysian," I called. "Wait!"
Raiya was with me this time, as I leapt forward, casting myself into the foyer room of Hell.
☼15☼
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# Trial by Fire
I coughed and sputtered and gasped, as ash and soot flooded my vision and my throat. My wings fluttered around as much as they could, shielding me and Raiya from the worst of it.
"I'm supposed to follow your lead," Raiya called from beside me.
"Don't let go," I begged her. There was something about this place that made the idea of loneliness all the more terrifying, and I was terrified enough.
All of a sudden, I saw my dream again—the one that had haunted me since Draco revealed himself. I saw Raiya, alone and unsure, and sad because there was nothing left to feel.
I tightened my grip on her.
I was almost grateful for the distraction as a loud, monstrous groan gutted out behind us.
That was the moment I got my first really good look at Alküzor. He was a demon of fire and density, collecting the matter of the world and eating up Time's residue to keep the dreariness of his pride intact.
It was then that I realized Alküzor wasn't just in Hell; he was Hell.
Or at least part of it. The fires around us burned black, and any light we had was painfully obscured—clear enough to taunt us, perhaps, but foggy enough to lose us.
On the other side of me, I could hear Elysian and Draco still fighting it out, though if they were doing so with each other or not, I couldn't see.
I breathed in as deeply as I could without polluting my lungs beyond capacity, and then drew on my center of calm. In my one hand, I still clung to Raiya; in my other, I had my sword.
It was then, as I choked and struggled to remain strong, that I noticed I could see Draco's sword below us. It was as black and burnt as the devil's word, and I had to destroy it with the sword I had been given.
If I could do that, I realized, we would still need to hold off Alküzor and Draco until the hole was closed off, before leaving ourselves. It seemed like a lot of work, and a high-risk challenge, but it was the only way I could see to make things right.
We can do it. I know we can.
"Call your power," I instructed Raiya. "We're going to launch all we have here at Draco's sword."
"That'll close the barrier," Raiya said.
"Watch out," I called, as a stream of fire and energy flew all around us. I knew at once, from the different feel and look of it, Alküzor was attacking.
But Alküzor's power moved all around us, never touching us. I was about to make fun of him for having such a lousy aim when I realized Raiya and I were glowing with a protective shield.
Was it the joint power we shared, or had Adonaias given us one last gift of protection before we went into the oven of the earth?
Raiya shuddered beside me. "I don't think your idea is going to work."
"I know it's risky. We'll have to make a run for it," I said. "Although that's not the right verb for it."
"I'm sure Mrs. Night would commend you for realizing that," Raiya called back sarcastically. "What if we don't make it?"
"We can!" I argued.
"Since when are you such an optimist?"
"Well," I said, giving up, "we have to try."
Raiya sighed. "Alright. Just do it!"
At her words, I felt our power combine, coming together, mixing with our blood, running through our hearts and our hands. I felt as though we had created our own supernova of sorts, a binary star pairing, as we poured out our power to light up the darkness of the crushing void.
When I felt it had built up enough, I gazed downward.
I had a clear shot of Draco's sword.
Nothing was stopping me, and every ounce of power I possessed guided me. I tossed my sword downward, letting it slice through the scorching clouds and flames around us.
The moment my sword left my hand, Raiya and I unleashed our power together after it, as we both yelled—me in triumph, her in frustration, and all of it tied together with the most primal and fervent prayer for this to work.
By some miracle (poor choice of words at this point), my sword flew down and cut through Draco's sword. With my power and Raiya's following swiftly behind it, Draco's sword smashed, crackling into a million and a half tiny little pieces. The earth began to return, pushing back against the remaining intrusion. I could see my sword, cast into the dirt and dust of the weary world. Its power faded, and the sword darkened.
I knew, instantly, that its job was done.
Surrounding us, our power continued to spread out, shining a bright light of hope, like a star trapped inside the world finally breaking free. I almost had to wonder if the Star of Hope, whom I'd met a couple of Christmases ago, would come down with her fairylike Star babies and dance around it.
Alküzor didn't like it; he forced himself against us, trying to drown out the light with his own fire.
Raiya cried out in pain, as a tongue of fire snaked out and cut her arm. Her blood ran out, scorching Alküzor and making him fluster more of his darkened fire.
I felt her weaken next to me. "Are you alright?" I asked.
Her energy was low, and I saw her face run pale. Her emotions flickered out to me—there were traces of desperation, weariness, a resigned quality. But there was also an angry determination, and while I was glad she was still fighting, I felt fear tug at me.
"I'm fine," she insisted, showing nothing in her expression or tone that would've made me look twice.
She'll die if she keeps fighting.
I had to do something.
With my free hand, I reached over and drew her close to me. I kissed her, hoping fervently that it wouldn't be our last, even as I feared it would be.
Dante's words came back to me. To release the Blood Flame, I had to die.
Or at least, I had to pull my Soulfire out of my body. Which was more or less the scientific equivalent of dead, right?
"What are you doing?" Raiya asked. Fear was starting to crack through her mask.
"Hold your power as much as you can," I instructed her.
Using my free hand, I pressed into the mark on my wrist, hoping my power wouldn't harm her.
*☼*
––––––––
It wasn't the most graceful of motions, as I reached inside of my heart and tried to pull it out of me.
I saw my heart, the pure-white clouds of the confusion around me and my identity, and the bridge between my regular self and my supernatural self.
Since the last time I'd seen it, and the one time I tried to dismantle it, the connections between the two had been rectified and reinforced—two halves of my life that had been splintered and shattered, but like a part of my body, they were healing up the fissures, this time with more intention and appreciation.
Awe struck me all over again at the sight. Different memories, as they hung in crystal balls like the one Aleia had carried with her throughout her time on Earth.
I was tempted to take a closer look at some of them, but I shook my head. "Later, later," I promised myself.
At the center of my heart, I saw it—my Soulfire.
I came up to it, and held out my hands to receive it.
When I first saw them, I thought all of them were like tiny balls of fire and light. Looking at it now, I realized I was both right and wrong. It was a glowing force, a mishmash of separate things bound together—the integration of emotion and intellect, reality and possibility, reason and intuition, will and wish, all into a form that did not diminish nor enhance their individual power when they came together, even as it increased its value.
It really is amazing to think everyone has one of these! And they are all different, too.
Everyone had a Soulfire. Mine was further protected by my Starlight defender identity, the Starfire and the Starsoul, which wrapped around my essence like a ghost and allowed me to freely communicate with the Celestial Kingdom (assuming I would ever figure that one out).
Gazing at it, I saw the fire within the fire; some part of it was not my own, but another's, and that held everything together in perfect harmony.
I held onto the Soulfire. For a quick second, I saw a reflection of my former self staring out at me. "Time to be brave, Almeisan," I told myself. "But you can do it. It's for Raiya."
The reflection smiled back at me, a cynical smile on his face where I would've thought a happy one would reside. I frowned back at him, or me, I guess, as I called my self out of my self, back to the battlefield.
Later on, I would recall my other self's expression and wonder if he was trying to remind me that all power had its price.
*☼*
Instantly, I was back in the middle of the battle. But my Soulfire was out, and its light was shining in full force.
"You can't do this," Raiya warned me. "I won't let you die."
"That was the plan," I told her, feeling strangely stretched with my power before me. I'd had this feeling before, once, when Elektra managed to pull my Soulfire from my body. My vision slipped all the way around me, and I could see things in more dimensions. It was hard to describe the sensation, especially in a way that didn't sound like I was on drugs.
Her power to me increased as she no doubt sensed my discomfort.
"I know you're in pain," I said, "but we only need a few moments of power to keep Alküzor and Draco in this dimension."
"Just tell me what you need," she ordered. "I can handle it."
I didn't want to. I knew she was tired from our battles, and I was rapidly tiring too. But we had to do it. It was the only way to stop.
"Supernova time," I told her.
We exchanged careful glances, as I took hold of both of her hands. "After that, it'll be alright."
"Just like last time, right?" She gave me a small smile.
I grinned back. "Exactly."
And with that, the light burst out all around us. I closed my eyes against the rush, and for the moment, I felt the peace of protection, and then the forefront of the battle pushed against us.
I heard Raiya shouting, and I heard my own shout, as we came together.
My memory flashed against my mind, and I saw this same thing as before—Raiya on the other side of Time's power, her own supernova raging against an impossible barrier, and breaking free when I crashed into her.
Opening my eyes, I could see it all; I saw, for the briefest second, into the life of the eternal.
I barely fought off the darkness as it came for me, taking the last of my energy reserves. I pushed through for Raiya, keeping my hands in hers.
"Look," Raiya said, her voice as weary as I felt. "It's working."
The resulting light was too much for Alküzor. I caught a glimpse of the terror in his piercing green eyes, as he narrowed them and then turned and fled farther into the world's burning heart. For such a large and imposing figure, made of flaming fire, he sure darted away from the light quickly. Some of the opaque clouding went away with him, and I could finally breathe relatively normal once more.
"Awesome," Raiya whispered beside me, watching as the last of the light filtered back into our world as my Soulfire sank back into the chamber of my heart.
I felt like cheering. I didn't die! The plan worked! It really worked!
I had to wonder if Raiya didn't feel some of my joy and my excitement (and if I was going to get a tirade from her, since I tried something that almost killed me and tried to hide it from her). "We need to go."
I nodded. "We're not out of this yet."
"Right."
"Elysian," I called. "Time to go!"
Draco's head popped out of the bed of flames. His teeth snapped at my wingdings, and I cringed in pain.
"Stop it!" Raiya yelled, pulling out her bow. She wasn't able to let go of me to load it, so she used the edge of it to strike him across the jaw.
He reared back in pain, and then retaliated. Raiya brought up her bow to deflect it.
"Elysian!" I called. "Where are you?"
There was nothing.
"Elysian!" I called even louder, all while trying to dodge Draco's frequent snapping.
"I'm here," he said, "holding onto Draco's tail."
"We've got to go."
"Hey," he said. "I want to say I'm sorry."
"Sorry for what?" I asked. "Let's go."
"No. Not until you forgive me. For all the judging, and all the attitude, and all that."
"What are you talking about?" I glared at him, while Raiya managed to dodge another attack. Her bow shot out, catching Draco between the eyes.
"Do you forgive me?" Each word out of Elysian's mouth had a grating undertone, one that weirded me out. Normally, I think I would have been angered by it, but I was just desperate to leave. Draco's power on the earth was gone, and our only way out was closing fast.
"Of course I forgive you," I told him. "You're my friend."
"Really? Do you really mean it?"
"Yes!" I shook my head. "Now, come on. We've got to go!"
"No," Elysian told me. "You've got to go. I've got to protect you."
"What are you—"
It was only when I heard a different kind of snapping that I went still.
My gaze swiveled to Raiya's as her bow shattered by Draco's biting power. Her mouth dropped open in pained surprise, and her eyes glazed over in disbelief.
The broken pieces of her bow sparkled, before the river of fire around us swept them up, flushing them further down into the heart of fiery darkness. I felt the last of her supernatural power break inside of her.
"Raiya!"
I was surprised that I wasn't the only one who called out her name.
I narrowed my eyes at Draco. I only stopped my attack because, for the smallest, most minute second of time, I'd heard him: Grandpa Odd. The old man's eyes pierced through the redness of Draco the dragon's.
Justice will be his undoing.
The words echoed in my mind.
Before I could make sense of it, he backed away. Raiya's tears began to slip free, and Draco allowed Elysian to grab onto him once more. I watched, unable to say anything
Immediately, I felt a pull on my own shoulder, as if my body remembered Adonaias' commands better than my brain did at that moment.
The vortex's power was gone, and the momentary, makeshift entrance to the realm inside the earth was closing.
I barely thought anything, as I pushed myself out of the hole and then turned to Raiya.
"Come on," I called back to her as I pulled her up.
"Let me go," she said.
"What? No!" I gripped her hand even harder in mine.
"My bow broke," she said.
"And I had to rip my Soulfire out of my body," I snapped back. "And Elysian made sure that the world is being protected from Draco by sacrificing himself! We've all had losses. We still have things we need to do."
"My power is gone," Raiya tried again. "My mission is over."
"Adonaias told us not to let go of each other," I yelled back.
"But we finished what we had to do."
"You'll get a new mission, then," I said. "I still need you."
"But—"
"You promised!" I shouted. "You promised me you would live for me."
She sighed. "Okay. You're right. Pull me up."
Really? I have to cajole her into staying alive? I groaned inwardly as I hoisted her out of the other realm. I felt bad that she lost her bow. I would feel lost, too, if I wasn't able to use my sword to help us get out of the vortex.
The opening between the two realms closed just seconds after we cleared it.
And then it was over.
A numbness fell over me, and even Raiya's presence beside me was hard to grasp onto as we stared at the sight before us.
The vortex had pushed a crater into the ground, much larger than the one the meteorite had. In the center of it, still stuck in the ground, was my sword.
We made it out just in time. The vortex disappeared, the skies cleared up, and Time's power—familiar enough to notice, but subtle enough to miss—resumed.
I felt the shield of protection, the one I'd felt so surely when we were in the midst of trouble, disappear. Adonaias' gift had been given and had been used for its intended task.
"Well done, good and faithful one."
The words whispered out to me from nowhere, but there was no mistaking the reality of them.
☼16☼
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# Battle of the Heart
I grappled with the end of the battle. I was glad I had done a good job, but I was still sad at saying good-bye to Elysian.
Raiya was breathing hard on the ground next to me, as the vortex died, and my sword remained stuck in the ground, in the center of the large crater where Rosemont used to be.
The night resumed quietly; I couldn't tell if it had begun while my sword struck Draco's, or if it just resumed as we'd shut the wormhole into Alküzor's realm.
I pulled Raiya over next to me. We were both heaving, laboring for breath, and covered in sweat. But neither stopped me from reaching over and peeling some of her hair away from her face, and kissing her.
Beneath all the sweat and tears and blood and dirt, I could still taste her. It was the most bittersweet kiss of my life, I decided, half in jest.
"It's over," I whispered.
"We did it," she said, her voice incredulous at the realization that we'd survived, and we'd won.
In the distance, I could hear fire alarms going off, cars honking, and the other sounds of rescue teams, as they went around to the different parts of the city that had fallen apart.
I stood up. "Come on," I said, tugging at her. "Let's get out of this hole. It's making me uncomfortable."
Raiya coughed and struggled up the hill, but I held onto her, moving her forward. We reached the top of the crater and fell onto the grass.
I breathed in the smell of the grass and wondered at it. I knew I'd been so close to never seeing this world again. There was no taking things for granted. Not anymore.
Especially since victory had come at a high cost.
"Elysian didn't make it," I said, barely hearing myself say the words. I turned to see Raiya, as she wore the same stricken look of shock on her face I'd seen when her bow broke.
I turned away and stared at the ground, hoping for a miracle. Hoping that Elysian would burst out of the ground and then chide me for worrying in the first place.
And then we could argue all night long about who got what end of the bed and which blankets smelled like human versus the ones that smelled like dragon.
Then I heard a coughing sound. "Raiya?" I asked. "Are you alright?"
She gasped in pain as she doubled over, grabbing at her chest.
"What's wrong?" I cried. "What's happening?" I caught her and held her up, cradling her against me.
"Hamilton," she whispered.
I kneeled down onto the ground, lying her carefully on her back, using my knees as a pillow. "What? What is it?" I asked, checking her temperature. I saw that her wound, the one on her arm that Draco had inflicted, was still bleeding out.
Has something happened to her heart? I wondered. A warning bell rang out in my head. She'd told me before, if I lost too much blood, there wouldn't be anything she could do for me. It was likely true for her, too.
Weakly, Raiya pulled me close to her. "I'll be waiting for you," she said, before her eyes closed.
She went limp before I could ask her what she meant.
I felt her cheeks as they went soft, and I felt her pulse as it went weak; I watched as the Emblem of the Prince, the mark we shared under his banner, disappear from her wrist.
"Raiya?" I cupped her cheek and held her close, giving her a desperate kiss, as my last attempts at coherency failed me.
The image of her bow snapping shot through my mind. Was her heart truly broken at last?
"Come back," I begged her, my voice suddenly thick and scratchy. I placed my hand over her silent heart, reaching out to her in the only way I could. "Come back to me."
Nothing happened. Nothing changed.
"Adonaias!" I yelled. "Adonaias, help me! Please, we need your help ... "
I saw him appear before us. He said nothing, only stretched out his arms. The white of his tunic fluttered in the wind of my world as he waited for me to come rushing to him.
"She needs your help," I tried to explain to him. "She's ... "
I couldn't say it. I wouldn't say it.
"You need to help her," I repeated, when he didn't move.
His eyes glittered as they looked into mine.
It was at that moment that I realized he wasn't going to help.
"No," I shouted. "No, get away. You can't have her!"
He didn't move. His arms were still outstretched, waiting for me.
"No!" I hugged Raiya's limp form to my chest even more tightly. "No, don't. Please don't."
Nothing.
"You can't have her! If she dies, you can't have me, either!" I shouted, hardly realizing my sadness and despair were quickly morphing into anger and bitterness. "I can't live without her—huh?"
I was surprised to feel his hand on my shoulder. Peace settled on me, and I could feel Raiya's soul resting peacefully, too. I looked up into Adonaias' eyes, their light piercing through my thick veil of tears.
"I make all things new," he said, softly and surely.
It was a funny choice of words for the guy. Especially as the shallow peace Adonaias had brought me was just that—shallow. The cloak of protection I'd felt before whisked away, transforming into an insubstantial shadow.
Adonaias disappeared as I turned from him, and I told myself very certainly that I did not miss him.
I cuddled Raiya into my body as I tried to move her. I have to find a hospital, I thought. It wasn't that far away. I could carry her. Couldn't I?
I felt weak and numb as I tried to lift her. My power was dwindling, my energy was being depleted, and fast.
"Somebody, help!" I called, resuming my search for help.
Time passed, and my voice became cracked as I continued to cry out, calling out for help, any help at all. I eventually remembered health class, and tried CPR. I was screaming before long.
I watched the fires of the feather in her hair, the one I gave her, as it slipped away into ash. I said plenty of words, and saw other things, but I couldn't be entirely sure of them, since my eyes were running over with water.
I thought I saw some emergency medics heading our way. I saw people moving, and I couldn't do anything about it as I sat there in the middle of the grass, finally overwhelmed and crushed by life's expectations.
The last thing I felt, before I closed my eyes, was the soft touch of snowflakes on my skin. And then, all I felt was the familiar sting of lightning as it tasered through my body.
*☼*
I woke up, some unknown amount of time later, in the hospital.
I was in one of those pitiful little rooms, where it looked more like a storage room for the older models of medical tech. I was no longer transformed, but I had on a hospital gown. I was hooked up to an IV, and I felt as though I'd been poked and prodded in places I didn't want to think about.
The smell was the worst part, but, considering I had some idea of what true pain felt like, I knew I could ignore it.
There were some things I couldn't ignore, two of which struck me right away. The first thing I saw was that the Emblem of the Prince was gone.
My mission was over. I couldn't transform into Wingdinger anymore. I assumed, anyway; I didn't bother to try, and I knew I wasn't going to.
The second thing I noticed was that my eyes were still crinkly and blotchy. Remembering what happened took a toll on me I never really measured.
My nose prickled with pressure, and my sinuses were ready to explode. I felt the same amount of fear and helplessness as I had before the battle.
This time, Adonaias didn't come. I felt his presence, but I didn't see him, didn't hear him, didn't want him. Instead, I pressed back into my pillow, trying to push back the rest of my tears.
Once more it hit me, that Adonaias just didn't fit in with normal suffering. What did he know of losing someone like Raiya? What did he know of pain like mine?
My mark was gone.
My sword was gone.
Elysian was gone.
Raiya was gone.
I wished I was gone, too.
*☼*
I guess I went to sleep again, because I woke up again, still unsure how much time had passed. I felt numb to all power, all forces. I didn't care that it was dark, I didn't care that it was cold. I didn't even care I hadn't eaten.
Even Mark's presence wasn't enough to make me worried.
"Hamilton," he said, and not for the first time, my name sounded completely alien to me. "You're awake."
I looked over at him. I wanted to scream, "Congratulations on your superb observations skills, Captain Obvious!"
But I held myself back. Just like I held my tears back. I breathed in, sharply and deeply, trying to recall I had to live, even if I didn't really want to.
"You've been in here for a couple of days," Mark told me, answering at least one of the questions I might've asked if it mattered to me anymore. "The reports have gone out, saying that you and some others were caught up in a surprise gas leak near the old Rosemont School that the repair crew missed."
I said nothing. Mark would need to go away soon enough. I doubted he was here during his off-hours. I was too old to remember if he had ever had any off-hours that he sought to really spend time with me.
Losing my will to live, even if my body insisted on dragging me through it, while death seemed unwilling to let me die, didn't seem like something Mark would change for.
Minutes passed before he spoke again.
"I don't suppose you really care about the cover story," he said.
I looked back over at him, still silent.
"Dante brought you here," he said, "after everything."
Everything.
What a quaint manner in which to discuss the death of not only Raiya, but Elysian, too. It was also a way to diminish their effort to stop Draco, to stop Alküzor, to stop the destruction of the world and the universe and everything.
"He asked me to find a private room for you," Mark went on. "He thought it would be best to keep you here until you recovered."
If I recover.
"You were pretty beat up," Mark continued. "You had a large laceration on your arm, and several other scrapes. You were in a state of shock when you woke up the first time. We kept you down for a little bit, but we stopped the morphine drip for now.
"If you're worried about your identity, don't be. Dante and I were able to cover it up."
I can't believe he thinks I would worry about that now, of all times.
As Mark continued to tell me of how he had been on duty when the attack happened, and how he'd been called right after surgery, I took the time to glance back down at my wrist.
The mark was still gone. I ran my fingers over my wrist, my hands shaking slightly. I didn't want the mark any longer; I felt used, abused, and discarded. I felt like I didn't matter at all.
How could I? Life didn't matter anymore, therefore it was ridiculous to even think I mattered.
"Your mother is hoping that you will be able to get back to school soon," Mark said. "They've talked to us about delaying your AP exams until you've been through physical therapy. You might need it for your arm, since the cut was so deep," he explained.
Again, why would I care about this?
Mark eventually dropped off into silence. I felt relieved. I didn't want the responsibility of keeping the pretense of caring. I wasn't really paying much attention, anyway, although I managed to give a silent, half-hearted, "yay," as he told me Cheryl had let Ayako go. As much as I appreciated her kindness, her culinary tastes could easily go to hell.
Which was where I was. Alone, alone, and more alone. I was living through hell.
Finally, Mark sighed. He looked over at me somberly, like he was going to say something else (I prepared myself for the inevitable cringe), when his beeper went off.
He was being called back into a heart surgery.
"Well," Mark said, "I've got to go." He rose from his chair.
I was surprised when he leaned over and kissed my forehead. He hugged me, slightly, so as not to disturb my bandages.
"I love you, son," he said, quietly but firmly, and in his own doctor/father way, I knew he was telling me to keep holding on. If I couldn't move on, if I couldn't continue on, the best thing was not to fall away.
He had only taken a step away from me when I spoke.
"What happened to Starry Knight?"
He jumped at my voice—I couldn't blame him, I sounded like some kind of monster—before he turned around.
"Please," he said, "please, don't ask me that."
I frowned. Instantly, I sat up as straight as I could. "Tell me," I commanded, hoping that he would realize not knowing was the greater pain.
I felt a tingle of fear when Mark narrowed his eyes at me, which he had never done quite so horrifically. "I can't give you the answers you're looking for. Please don't ask anything of me again, Hamilton."
And then he walked out the door after adjusting the morphine tap.
It was almost welcoming to feel the rush of numbness replaced by the desire to sleep.
Some part of me, shocked at both ends of my father's behavior, fully expected to die, even as he wanted me to survive.
When I did wake up, and I mean for real wake up, and not just to be forced to eat or change clothes or bathe, I found Dante staring down at me.
Mustering up what strength I could, hoping I wouldn't end up slurring my words, I spoke up.
"What happened to Starry Knight?"
To his credit, Dante didn't try to distract me from the truth of the matter.
"She died."
He told me, so simply. I wondered if he would have answered me in a similar manner if I'd asked what was for dinner.
My chest felt swollen and bruised, as those simple words killed me all over again.
"Do you want to know the specifics?" Dante asked, making me wonder if he wasn't getting some kind of sick pleasure out of my absolute misery.
I nodded.
"We reached you after we were able to move again," he said. "We didn't realize all that had happened, but we saw you both come up from the crater. You were both struggling, so we got the medical kits ready."
I nodded. "I remember," I admitted quietly.
"Then you'll remember that you were inconsolable at the time," Dante said. "We had to put you down."
The fire of being tasered ran through my memory. I nodded.
"We brought you both here. We tried to save her. We didn't."
"What happened to her?" I asked.
"Your father warned her before about her heart," Dante said. "She had an irregular heart. Mark even put her on the transplant list several years ago."
She never mentioned that.
But I waved the thought away a second later. Until recently, she'd planned to die fighting the Sinisters and their cronies.
"We moved her into surgery," Dante continued, "to see if we could restart her heart. Nothing worked. When we cut into her body, water came out along with blood."
I remembered that description from somewhere else. "Her heart exploded."
"Yes."
Bitter laughter echoed from the depths inside of me. "Well," I said, "it wouldn't be the first time she left me that way."
Dante said nothing about that. He changed the subject. A wise move, and one that I could appreciate, despite the fact that I still half-hated him.
"You've been placed under special care," he said, "and your parents are understandably upset, but as you can see, they are still working. Some people grieve with work, you know."
My parents have been "grieving" with work for a lot longer than the last three days.
"There was a lot of damage down to the area where the battle was," he said. "There were also other places where the explosions caused damage, including Shoreside Park, and Lakeview Observatory. No one was seriously hurt. Maybe that was a miracle."
His choice of words burned me, inflicting another round of wounds on me.
Adonaias had failed to save Raiya. He'd tried to comfort me, too, to make it worse.
Dante was still talking, taking my silence as a sign I wished for him to continue. I didn't, but I let him.
I was tired, and I felt numb, and I wanted nothing to do with anything anymore.
Maybe, if I was silent long enough, he would give me a reason to keep on fighting. Or maybe he would bore me to death and I would be saved that way.
Salvation through damnation, I thought bitterly.
He only spiked my interest, briefly, when he mentioned Mikey wanted to come visit me.
"Why?" I spat. "So he can rub everything in my face?"
"I told him no," Dante said, "if it makes you feel better."
I glared at him, my temper flaring. Only death will make me feel better.
"If you want him to visit, I'll be happy to send him the message," Dante said. "But that is something up to you, and I would not take that choice away for anyone's sake, not even my son's."
I had to stop myself from saying that it was not a surprise to him, since Dante had the gall to leave Mikey and his family in the first place. He never seemed to have trouble disappointing his family.
"Where is Starry Knight now?" I asked. "Can I ... can I at least see her ... "
In response to my question, Dante pushed the morphine tab. My IV drip began to increase, and I felt woozy only seconds later.
"You'd better hope I don't get addicted to that stuff," I murmured.
"Let's hope for the hospital's sake," Dante corrected me. "But I believe it's better for you to hear this now, when you're unable to lose it again: Starry Knight gave all she had to protect you and this world."
I felt even more numb, and not just from the morphine. I barely heard the rest of Dante's speech as he told me that even though they have to deny the supernatural aspect of this situation to the public, citing that political power is best left out of the hands of those who would seek divine right to rule, or abuse the notions which have destroyed the past civilizations, he would always consider Starry Knight a true heroine.
"But she is gone from your life, Hamilton, and there is no escaping it," he finished.
And then the darkness, the emptiness which had embraced me before, welcomed me back with mocking arms.
I had been blinded by the light inside of me, by the light of creation and power, an unparalleled spirited fire for goodness, justice, love, and mercy, too awesome to describe or even say without trembling.
But I found I was blindsided by Raiya's absence. The light inside of me was gone, and so was she. I didn't know how awful the vengeance of darkness and despair could be before that moment. But then that moment came, and I knew and it was enough for my own heart to die a star's death, and cave in on itself till there was nothing left, with the fullness of my emptiness crushing me from all sides with unimaginable strength and power.
It had to have been grace that saved me, because I could not, would not, have saved myself.
*☼*
When I woke up again, I saw a shadow shifting outside the door.
"Well, come in," I called out. "I can see you."
I wasn't that surprised to see Mikey walk through the door.
"So," he said tentatively, "you're awake."
I frowned at him.
"You've looked worse," he said. He was obviously trying to cheer me up, and possibly avoid apologizing to me. I wasn't up for either one.
"What are you doing here?" I grumbled. "I thought Dante told you to stay away."
"Well, he also told my mother they would be together until death do they part," he retorted.
"I thought he was your buddy now."
"He never really was," Mikey gave me a small smile. "I know I said that he was before, but I was just using him to see if he could help Gwen."
I cocked an eyebrow at him. Mikey didn't seem that smart.
He blushed, and I was surprised as I realized I could still read people's emotions. They were still responding to my scrutiny in color, too, just as before. His embarrassment and humiliation leapt off him as he tried to conceal it.
"Well, anyway, he told me that he's got a new project, and so he's leaving again," Mikey admitted.
"Where is he going?" I asked, surprised. Dante told me before that Otherworld had been let go, but SWORD was still an active force in Apollo City. Was it possible that they had just come to stop Draco and the Sinisters, and all them, really?
"I don't know," Mikey said. "I hope far away though. For you more than me," he added. "I know you probably don't want to see him right now."
I snorted. "If I ever see him again, it'll be too soon. He told me that I had to die to save the city, you know."
"He did?"
"Yeah."
"Did you?"
Not well enough, I thought. But instead, I shrugged. "No."
"Well, that's good." Mikey smiled at me, but I just glared at him.
"I get that it's your turn to be upset," he said. "I wanted to say I was sorry for everything, and I hope we can still be friends."
"Friends don't normally have the issues we do," I retorted. But I remembered what Raiya said before, about having relationships tested, and as I shoved her further from my mind, I relented. "But I know you'll need the help graduating on time."
"Yeah, no kidding," Mikey agreed. He seemed a bit happier. I hoped it wasn't just because I implied I'd help him with his homework.
"So," he said, "when are you getting out of here?"
"I don't know," I told him truthfully. Part of me didn't want to leave. That meant I would have to go on living, and there was no reason to do that now that Raiya was gone. The hospital had always been this depressing place of death for me. Now, it was a depressing place of stillness, where I couldn't go back, and I couldn't go forward, almost like a purgatory that smelled like cleaning alcohol.
"Can you move?" he asked.
"I suppose. Why?" I asked. I wondered if he was going to make me go down to the cafeteria with him. It was the sort of thing I could expect from Mikey.
So I was surprised when he said, "I thought you'd want to go and see Gwen and the others," he said. "They've all woken up."
"I don't know ... "
He tugged at my arm, the one with the large gash in it from Raiya's arrow, and I winced in pain. He didn't notice. "Come on. It'll be good for you. I've already talked with her some. She's just down a few floors."
I didn't care, I reminded myself, but Mikey did, and shrugging him off would likely cause me more pain, physically.
So we went down to the room, arriving just as Gwen was getting checked out.
I noticed that her auburn hair was much longer, and seemed to be darker closer to the roots.
I wished I could say that she was happy to see me.
The moment she saw me, her honey-colored eyes darkened. "Well, if it isn't my least favorite ex."
I gave Mikey a look that said, "I told you so," before turning back to Gwen. "Hi, Gwen."
"You have some nerve, showing up here," she muttered. "You let me get captured, and you didn't save me."
I didn't bother to tell her that technically, I did save her, and everyone else, too. I just let her vent. Her words couldn't hurt me anymore, and thankfully, since Dante had mentioned he was releasing a statement to say that both Starry Knight and Wingdinger died in the blast that occurred, she couldn't blackmail me anymore.
I considered it the truth anyway.
As I listened to Gwen as she continued to ramble on, and I watched Mikey as he tried to get her to stop, I realized something uncomfortably inconvenient.
I was not the only one was who in pain. I was still pretty sure that I was the only one who would be in pain for the rest of his life, though.
So I decided to try to help Gwen.
I tapped her on the shoulder. "What is it that you're really angry with me over, Gwen?" I asked. "I mean, really?"
It's hard to downplay getting your soul sucked out by a demonic being bent on destroying the universe, but I managed.
"I'm angry that you didn't love me," Gwen finally admitted, as fresh tears swelled up in her eyes. "I would've loved you forever, if you'd only have let me."
I'm glad I didn't.
I didn't tell her that, though.
"I'm sorry," I said. That's all I could say. I didn't tell her it was her fault (even though it mostly was) and I didn't tell her that I loved someone else more (which was definitely true) and I didn't tell her that I wanted to be friends or fix anything (which would have been brutal, to be honest).
So I let her know that I was worthy of her hatred more than her love, and I let Mikey take over when she needed a shoulder to cry on.
As I walked out of the room, I turned back to look at them. As terrible as the whole situation was, I wasn't jealous of them. I wished I had Raiya, but I didn't. Some part of me knew that I would have to go on without her. But all of me knew that Mikey and Gwen deserved to get their chance to be together, and I hoped that it made them happy. I silently wished them all the best as the door swung shut behind me.
☼17☼
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# The Real Battle
The following months were speckled with waking moments, where I found myself back to acting like myself.
Grieving privately, I was ashamed that I had been found to be a fool. Who was I, to think I was anything or anyone important, or that my life would leave a mark on the world in which I lived? Accomplishments I may have had, but they were nothing in the light of eternity. Actually, I don't think nothing is the best word to describe it. They were more like eclipses, shadowing the real meaning of true meaning from me even further, removing me from the truth found only in choosing to face the stark, unfiltered starlight.
There were a million, million times when I wished that the meteorite had just damaged the city when it'd struck. Easily half of those were wished after Raiya was gone.
Only once, after high school was over, did I go out to the ruins of Rosemont Academy, where my sword remained stuck. A demolition group was working on the area, trying to turn it into an underground mall. My sword would eventually become its centerpiece. I figured it was appropriate, in many ways. The meteorite had struck into the heart of the earth, unleashing the enemy's ambitions. My sword, and the power behind it, had stopped it.
My friends never said anything much to me about that year of school, and if they did, it was silenced it quickly enough.
Mikey managed to prove somewhat useful in the end, acting as my advocate more than he had to. I could tell he was never entirely certain of my position on his position in my life, but, true to his nature and our habits, he clung to me like a second shadow.
Things changed a bit when he got a swim scholarship to Ohio State. Coach Uzzy was so proud of him.
Gwen was even more hesitant, even more terrified of me. There was one time between the moments and months of time continuing where she looked at me, caught my eye, and tried to say something.
She was walking toward the auditorium of the school, no doubt going to see Mr. Lockard, the old drama teacher, who had also been awakened from his soulless state, as he came to work beside Ms. Carmichael on the school play, Pippin.
Our eyes met more by accident, and in them, I could see she wanted to ask me for something. But it was something I was not sure I wanted to give her, so when I turned away, that seemed to be the end of it. She shied away from me for the rest of the year, and several years after that as well.
Before I knew it, the school year was over. Soon after, summer was over.
My sabbatical from work was over. I went back, finishing up another month with Assistant Mayor Dunbrooke and then finishing out the rest of Mayor Mills' tenure. I never called him Stefano again, and I decided I never wanted to run for mayor.
My mother's business took off.
My dad's work stayed about the same.
They made arrangements for me to get a car for college after I got my license.
I joined the football team again, much to Jason's happiness, especially since I made good on my promise, and he was given the quarterback position.
Samantha Carter, who had been one of the more annoying persons of interest during my high school years, was crowned homecoming queen. I was playing during the game, but when I got the chance, I came up to her, smiled, and congratulated her. It seemed like a nice thing for her, especially after all the time she'd been stuck in the sleeping sickness coma.
I also congratulated the homecoming king—her boyfriend, Guy Fitch.
Some part of me was very happy that they'd grown into their own, happy that they'd gotten something they'd wished for and wanted for so long. Even if I couldn't be happy for myself, I was happy for them.
I got my SAT scores back. 2394. I'd missed a couple of questions on the English section.
I entered into Apollo City College's dual enrollment program. In addition to some CLEP exams, I had sixty credits ready to transfer by the time the University of Pittsburgh accepted me, and a 4.2 GPA.
I broke four of my previous swim records.
I skipped prom the following year. Instead, I went on a "vacation" with my mom instead, as she negotiated some contracts for some company in the Czech Republic.
I was able to keep my title of "Tetris King." Once high school ended, I never picked my Game Pac up again.
Nothing big. Nothing grand. It was a normal life, just the one I wanted, and I hated most of it.
☼18☼
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# Martha
Time continued to pass, even if my pain did not.
Even my body seemed to get the message. The golden halo effect, which had clung to me in my prime superhero days, dimmed and eventually disappeared. Even my blondish hair faded back to brown, and even seemed to get darker, slowly slipping away from me as I slipped back into the "real world."
Despite my "normal" life, I had some more surprises coming at me, as if pain demanded that it be a forever part of that "normal" adage.
First, Mrs. Smithe retired at the end of my 11th grade year.
As I walked down the halls, headed for my locker at the end of the day, she pulled me aside.
We didn't say a word as we walked into her half-empty classroom. But the moment the door shut, and the rest of the world was shut out, she told me the truth.
She was leaving me, too.
"What? Why?!" Her admission finally allowed me to give a proper vent to some of the suffocating grief I carried.
"Do you know why I told you I worked for SWORD before?" she asked.
"No." I snorted. "I don't."
"I told you," she said, "because I know what it is like to be without hope. I know what it is like to lose the people you love most, and have nothing left."
"Why are you telling me this now?" I balked.
"Because time and death are not permanent things," Martha said. "Not in the way that you think, anyway. Remember what I told you before? Your life is not about you. There must be something greater."
I watched her as she started slapping books and papers into a small box on her desk.
"Raiya's gone," I choked out, barely able to say her name. I found that it was much easier to pretend I was getting better if I didn't think about her. "I don't have anything greater."
"Her love is still real," Martha reminded me gently. "For her sake, find something greater."
I thought about Adonaias, about doing the right thing. Starlight Warrior duty was no longer an option, but even if it had been, I would've rejected it, just as I rejected Adonaias.
Rather than tell Martha the truth, I shrugged. "I don't know what it is yet."
"Extraordinary things come from ordinary places," she said. "You might just find something."
"What did you find?" I asked. "After SWORD killed your husband and son?"
I didn't mean to be mean about it, but my voice had a hard edge, one I knew was only half-intentional.
Martha came over to me and gave me a hug. It was awkward and tense, but strangely comforting in its stiffness. I could feel Martha's bony back and, despite her small frame, I felt the strength inside of her and her heart.
"I didn't find anything," she told me. "I was the one who was found."
Her cryptic remark made me jerk away. I was done with the pseudo-intellectual-spiritual-emotional crap. From that moment on, I only wanted to concern myself with what I could see and what I could touch and what I could understand.
"Thanks," I muttered, trying to hide my hurt. "I'll keep that in mind."
"See that you do," Martha said. "I expect law school will give you plenty to think about."
"You heard I got accepted to Pitt?"
"Of course." She gave me a small smile. "Your mother's very proud. She sent me a thank you note for preparing you."
"I'm surprised."
"I'm not." Martha picked up her coffee cup. "No one who knows you well would have second-guessed you."
I nodded. "Thanks." I didn't know what else to say.
"You'll do well in college, too," Martha said. "I know you will."
I nodded.
"Maybe we'll see each other again."
"Why?" I asked. "Are you going to go teach law or something in Pittsburg?"
"No." She grinned this time. "I'm retiring from teaching, and I'm going to go back to law school myself."
"I thought you already had your degree in law."
"Well, yes," she said, "but it's always been a dream of mine to go into practice. I need some more continuing education for that."
"Why are you going now?"
She glanced up at me. "Hamilton," she said, "there is a proper time for everything. I can't teach you any further, and I've reached my work requirement for my teacher's pension. For the moment, SWORD is leaving the city, and I have decided to leave, too. The time has come for me to move on."
I nodded again, more slowly this time. There was something to be said for timing, and I knew there was especially a time for retirement. If you can get one without the money worries, it was a good thing.
Reaching out, I took her hand. "Well," I said, "maybe I'll see you around Pitt, then."
"I've been accepted into Duquesne," she said. "But I'll be around in the city if you need a friend."
"A friend?" I asked, smiling for the first time in what seemed like a long time. I felt like an ancient old man, whose face was sunburned into submission, only to feel it crinkle and crack by the force of a miracle.
"Yes. I'm retiring, and you'll be eighteen next year." She sighed. "I suppose you can call me Martha now." She glanced at me over her thick-rimmed glasses. "Officially."
Shock, along with warmth, struck me in the heart, briefly breaking through the numbness that had settled in me in the last few weeks.
"Thank you," I finally replied, my voice soft.
Martha took my hand and squeezed it affectionately. "Now, skedaddle," she said. "I've got to finish cleaning out my room for the new teacher."
On some level, I knew, or at least I believed, that it would be the last time I saw her. Certainly, it would be the last time I left her classroom.
It was a moment that, like the previous moments of pain, should have broken me.
And maybe it did.
As I left her classroom, I felt the last of my teenage innocence slip away. I had worked all my life in classrooms, only to find that the real world didn't let you keep your troubles in nice, neat compartments. I'd known this before, but never realized until I walked out, full of certainty of Martha's approval, how uncertain life suddenly seemed. I was grateful for that—that small, unchanging element of life, as the rest of it all rocked around me, moved by the earthquakes of the moments and the shifting sands of the seconds.
As that chapter of my childhood closed, I felt a new one begin, one where weakness slumbered on, strength swiftly stirred.
For the moment—that moment, at least—I was comforted. I was not better, I was not unburdened, I was not past my pain. But I was comforted, and it strengthened me. It was enough to get me to the next moment, and the moment after that, and the next one after that.
It took me a lot longer to realize I could survive, and it took me much longer to realize I could even be happy again, and longer still to believe it was okay to be happy again.
☼19☼
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# Good-bye
Saying good-bye to Martha wasn't the worst, surprisingly.
Of course, I didn't think about it at the time, but leaving in general was just the worst.
I blocked out everything I had of Raiya—from our memories together to the places she lived, worked, and dreamed—and left it all, all for my own desired comfort, rather than facing the uncomfortable truths I left behind.
Still, saying good-bye to Martha was like ripping the flesh off just over my heart.
Leaving my parents behind proved to be more easy and more difficult than I expected. I don't think I ever officially said good-bye to them. In the first weeks of summer break, something changed between us.
Cheryl busied herself with her work and her new firm, but stopped working overtime. She was home when I came home from work, and she was home when I came home from school after it started up at the end of summer. I can't remember a time in my life where she deliberately sought to give me more attention, and paid much more attention to what I wanted and what I needed. And me, finally receiving her attentions without working hard or trying to please her, or even without trying to tick her off, found that her love was a very small something I could learn to depend on in a world where more of myself had disappeared.
She even gave up the weird diets.
Mark, in comparison, began ignoring me and avoiding me. He said barely anything to me over the next several months. He was there, and he did things for me, but his forlorn detachment was enough to cause a sense of relief when I finally got ready to head off to college the next summer, starting a semester earlier than originally intended. I think that the handshake he gave me as I headed out the door settled something of whatever was bothering him about me, but I didn't know for sure.
I gave him a smile, regardless, because I knew as much as my heart was broken, he had been trusted and burdened with my true identity for longer than I had known, and he had protected me as best as he could. Some people would think that was expected of a doctor, or a parent, but I knew just how hard it really was to put yourself before any other person, even if you did love them. It is something that, if expected, becomes selfish and destructive.
No, my parents weren't hard to leave.
It was Rachel who was perhaps the hardest to say good-bye to.
When I walked into her café for the last time, the place where I had called my second home for the majority of my high school years, her eyes met mine and immediately sparkled with tears.
"Hamilton."
I was a little surprised she recognized me. I didn't go back to the café until I knew it was time to say good-bye for forever. "Hi, Rachel."
She tried to smile, to play it off, but it was Letty who really saved her from losing it.
"Just get it over with," Letty said. "You'll feel better."
"Mom!" Rachel gaped at her.
"What? It's true."
"I can't believe you're being so rude."
"Oh, well, dear. Life goes on," she scoffed, before lighting up a cigarette. Rachel, for once, did not stop her. Instead, she sat down, slumped over, and closed her eyes.
When she opened them, she thanked me, for everything, and that was that. There was so much unsaid but unable to be said, and so much we knew just couldn't be said. Saying some of those things would have ended something inside of us, and I had little enough left to go on. Rachel didn't look like she had the gumption for it, either, so we said, "See you soon," and "Keep in touch," and "Thanks," and that was it.
I did not even ask for a doggie bag.
Letty was, I suppose, trying to be helpful in her own way, but I think she wasn't completely right. Time goes on, the world goes on, and even life may go on, but it doesn't mean living goes on.
*☼*
So I sat there, on the train, with my bags packed along with the pieces of my heart. A new chapter of my life was beginning. Or maybe it was ending, or both. Or maybe even neither. I didn't know.
I am a lawyer at heart, and as I saw it, there was a very big distinction between beginning and ending that started with hope and ended with expectation.
Thinking of endings made me think of my graduation day, of the speech I gave as valedictorian. I had somehow managed to find a speech inside of me devoid of several important memories, but still it managed to reveal more of myself than I ever could have guessed. My twelfth grade English teacher, Mrs. Runsallus, painstakingly prodded me to get it done and finished.
I remember making my speech. But rather than reading through the one I'd written, I made a new speech up, talking about how small things were great things, and how ordinary things became extraordinary, all thanks to life and love and other stuff that I thought sounded good. (This is the part that, should Hollywood ever get the rights to make a movie about me, they will have to make up. In all fairness, I was doing that, too, so I figured it would even out in the end.)
I didn't mention Raiya.
Maybe some part of me was trying to make up for never giving Raiya the speech she deserved. I was doing this, of course, in typical grief-stricken illogical fashion, by trying to pretend I'd never met her.
A wind had whipped by, scattering the music of heaven past me, and I found the first sparkle of hope as I looked down at my scattered speech papers, only to see the barest hint of a springtime violet anchored to the summer dirt. I picked it up, and then I heard it.
"I'll be waiting for you."
I heard the words, and nothing happened.
Nothing happened. Nothing.
It was a moment that breaks you or builds you, and you have the choice to decide which.
Later on, I knew, as the train left Apollo City, beginning its hustle toward Pittsburgh, pulling past the buildings of the city of my birth and childhood, past the places remembered, places forgotten, the foreign and the familiar, all the things which had made me who I was and promised to remain in me as I became something more ... I knew I hadn't made up my mind just yet.
C. S. Johnson is the author of several young adult novels, including sci-fi and fantasy adventures such as The Starlight Chronicles series, the Once Upon a Princess saga, and the Divine Space Pirates trilogy. With a gift for sarcasm and an apologetic heart, she currently lives in Atlanta with her family. Follow her on Twitter at @C_S_Johnson13.
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# THANK YOU FOR PICKING UP THIS BOOK!
To Get Awakening (A Special Christmas Episode of The Starlight Chronicles) for Free,
Click Here
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AUTHOR'S NOTE
Dear Reader,
Please, don't panic! This is not the end of the story. There's one more book to go yet, and to tide you over until then, Chapter 1 immediately follows my usual message.
If you have read my other work, or if you are familiar with my story, you might know I began write this series my last year of high school. It would take several rewrites and many drafts to become what I wanted, but eventually, Slumbering was born in its current form, and the rest have followed faithfully.
This book is part of that original pain. I didn't have a completely happy ending to high school. I was done, but the triumph I expected was more like a tepid pleasure, which almost makes it worse than outright disappointment. Not because I didn't finish well (I did), but because so much seemed unsettled. I expected things to be finished, but they were far from over. This realization was not only unexpected, but disappointing and depressing.
I've come to see that, while time does not heal all, it does provide important perspective. Years later, as I write this, some months before my ten-year high school reunion, I can see that. It is not only my pain I understand better, but God's hand through it all. I can look back, and I can see more glory and more grace than I could see while I was facing the fire. It is indeed an ongoing, eternal moment that continues to astound me.
In the believer's journey, we go through trials and pain. Suffering is a given in this life. As my favorite line from Cry, the Beloved Country, says, "For our Lord suffered."
As dark and deep suffering is, love still lights the way. Love carries us through it, love lifts us up out of it, and love is waiting for us, surrounding us beyond the pain.
For this reason, part of this book's theme is courage. Be courageous. Dare to suffer for what you love, for what you believe, for what is right.
Christians are called to holiness more than happiness. There is a cost to choosing that life, to committing to that life, to continuing through to the end. It is not easy.
We must be brave. Power is at its most potent when it is laid down for love. And we know this to be true, for this is what our Lord did.
So, be brave and take heart. Hamilton and I will see you soon, next time, in the last book in this series, appropriately titled Everlasting (Book 7 of the Starlight Chronicles).
Until We Meet Again,
C. S. Johnson
AUTHOR'S ACKNOWLEDGEMENTS
EDITOR
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# Jennifer C. Sell
Jennifer Clark Sell is a professional book editor and proofreader. She works from her home in Southern California. With her years of professional and personal experience, she offers several quality packages for authors. Find her at <https://www.facebook.com/JenniferSellEditingService>.
Photo Credit: Savannah Sell
AUTHOR'S ACKNOWLEDGEMENTS
COVER ILLUSTRATOR
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# Amalia Chitulescu
Amalia Iuliana Chitulescu is a digital artist from Campina, Romania. Raised in a small town, this self-taught artist has a technique which is delineated by the contrast between obscurity and enlightenment, using dark elements in a dreamy world. Her areas of expertise include the use of theatrical concepts to create a macabre and surrealistic world that still maintains a highly recognizable attachment to reality. Bridging a diaphanous environment with light elements, an eerie view, she creates a dream world of dark beauty, done with a blend of photography and digital painting. Find her at <https://www.facebook.com/Amalia.Chitulescu.Digital.Art>
Photo Credit: Amalia Chitulescu
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SAMPLE READING
Chapter 1from
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# EVERLASTING
BOOK SEVEN of THE STARLIGHT CHRONICLES
C. S. Johnson
☼1☼
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# Beginning Again
For the fourth or fifth time in less than an hour, I looked down at my legal pad, only to realize I had no idea what was going on.
My office seemed too bright. The lights seemed too sharp. I glanced around as the client in front of me continued to babble on about some financial legal nonsense.
Everything else was in its place: My various diplomas and certificates, celebrating my undergrad and graduate degrees, hung on the wall, proudly and prominently displayed right across from the entrance; my desk was off to the side, facing the wall, with the window behind me; my books were stacked in precise order on the bookshelf, tucked in with some of my awards. Among them was my newest one, the one from the Pittsburgh Law and Order Association, declaring my position as "Best Associate Lawyer of the Year" that arrived just before Thanksgiving last month.
Outside, the streets of my adopted city were hobbled with people looking for warmth and a cozy corner to cuddle up in, all while the snow continually dumped down out of the sky.
Nothing was out of place, not even the typical, dull-looking client in front of me.
No wonder I was bored.
The room was a temple to law and intellect, and I'd made sure to erase quite a bit of my heart in the process of setting it up.
No wonder that, instead of taking notes, I'd been doodling.
Despite the fact I had suddenly caught myself not paying attention to my meeting (again), I smiled thoughtfully, almost longingly. Raiya used to do this same thing in Martha's class, I recalled.
Instantly, as though I'd touched some mental flame inside of my mind, I flinched. Where in the world did that thought come from?
I didn't like to think of her anymore. Not if I could help it.
Of course, now that I was completely against thinking about her, my mind wasn't listening to me, and I wasn't sure I could help it.
I didn't have to glance at the calendar on the wall of my office to know more than seven years had passed since that day.
Seven years. Seven years.
Seven years, and I still crumbled as I remembered Raiya's body as it collapsed against mine, still sucked in my breath as the last breath of her words passed me by, still felt the dying chill of the fire-feather she'd tucked into her hair as it flicked into darkness.
Seven years, and it was still much too painful for me to acknowledge that the only person I'd ever loved more than myself, I was unable to save.
"Sir?"
I nearly jumped out of my seat as I realized that I was still in the middle of a legal hearing. "Yes?" I straightened up in my seat, trying to look nonchalant, and, as was my usual, managed to succeed enough to get out of any possible trouble.
"Are you all right? You look ... troubled," my client, Mr. Brown, muttered reproachfully.
I put on my most winning smile. "I'm perfectly fine, Mr. Brown. I am just making some extensive notes on your business concerns so we will be ready with the rebuttal at the end of the trial." I tucked the drawing up against my chest, making sure he couldn't see it, just to be on the safe side.
Mr. Brown visibly relaxed. "Oh ... well, good." He nodded. "For a moment there, I could've sworn you started daydreaming."
I almost shouted, "Lawyers don't daydream!" It took a surprising amount of self-control not to.
Instead, I laughed cordially. "No, no, sir, not at all. I assure you, I am the most capable lawyer available for handling your case. I know what is needed to get the job done."
"See to it, then," Mr. Brown said as he looked at his watch. "Well, I best be off for now. I'm to meet my wife for dinner."
"Well, don't keep her waiting on my account." I smiled. "I'll get your files pulled and we'll be ready for court as soon as the judges assign a day for it."
"Good to hear."
"Thank you for your time. Don't worry about anything. I'll have your case wrapped up in as little time as possible."
"I'll send you your retainer check in the mail." He grinned. "With a nice Christmas bonus, just to make sure you know how I appreciate your dedication."
And then Mr. Brown picked up his hat, put on his jacket, and walked out the door.
When he was gone, I just rolled my eyes. I hated representing people who weren't concerned so much with justice as with getting out of justice, but I had to do my job. I supposed.
I didn't like to think about that too much either.
When I did allow myself those small moments of reflection, I longed for another life. But I could hear a certain annoying voice in my head, chiding me for making life one big to-do list.
Then I quickly squelched my desire. I would, to this day, never admit to Elysian, my old "pet" dragon from years ago, was right. Not if I could avoid it.
I began to pack up my stuff for the day. I had my own dinner plans for tonight.
One of my best friends from my hometown of Apollo City was coming in to see me, and I wanted to have some time to prepare for the unpleasant lecture I was sure to receive.
My eyes fell to the notepad drawings I'd created while Mr. Brown was droning on and on about the unfairness of his situation, the integrity of his investment portfolio, and how his company was no doubt infiltrated with spies who had set him up to look like an embezzler.
My face softened for a moment, as the picture staring back at me was of a lovely young woman with wings fluttering out of her head. Even though the picture was not in color, I knew her eyes were the shade of the most vibrant spring violets, and her hair was the color of Christmas gingerbread. And even though the picture was not supposed to be real, the watchfulness and steadfastness of her eyes were more than mere reflections.
I sat down in my chair and looked at the picture glumly. When did I get so good at drawing? I wondered to myself.
I'd been only working as a lawyer here at Pharris & Dahlonega for ... was it a year, already? Surely, I hadn't been doodling for all that time.
And when did I start allowing myself to think of Raiya again?
Of course, it's entirely possible I never really forgot her, I mused, since she was the—No! I'm not going to think about it!
I silently chastised myself as I packed up my things. Thinking about her only made it worse, I knew.
I barely remembered graduating from high school at all. When I looked back at my pictures from that time, I could tell part of me was not there.
I didn't have any pictures from the summer after I graduated. All I could remember was looking for her and not finding her, and having to drag around this emptiness in my chest all the time.
Thankfully, I'd transferred out of Apollo City College's dual enrollment program, and actual college began soon enough. I'd whisked myself away to Pittsburgh, to a new city, to a new home, a new school, and new distractions.
But no new self to go along with it. I was still in love with a memory.
Anger and sadness pushed through me; I shoved the doodles into my briefcase and slammed it shut. I'll deal with those later. I can't think of this now. I have stuff to do ...
Once more, all of my secret longing, all of my hurt and anger, all of it was swept away under the carpet of scheduling.
*☼*
"So, you're a bonafide Steelers' fan now, huh, Dinger? That's great." The man sitting across the table from me laughed heartily.
I smiled at him; he was one of my oldest friends, Mikey Salyards. "Come on, why wouldn't I? They are among the best teams in history."
Mikey stopped laughing and turned more somber. "You've changed a lot, I guess."
"I haven't changed, Mikey. I'm still the best of best of best."
If anything, I thought, it's Mikey who's changed since high school. The awkward teenage years and the pressure of always being in my shadow had dispersed to reveal a strong, confident, and capable (looking) person underneath. It was almost a shock for me to see my old friend looking so different.
The weird-looking beard didn't help, I silently decided. It reminded me too much of his father, and Dante Salyards was a man I was more than happy to forget.
Mikey cocked an eyebrow at me. "Maybe in your field of law," he said. "But you couldn't hold a candle to me when it comes to coaching or gym class."
"I could probably handle the PE, but I doubt I would handle the sixty-some immature teenagers running around a gym after snorting sugar," I conceded after careful thinking.
"Aw, it's not that bad," Mikey said. "There're only about forty-five students. Apollo's a pretty small district, actually."
We began laughing as our dinners came.
"Thanks for treating me, Dinger," Mikey said, cutting up the medium-rare, freshly harvested, lightly seasoned teriyaki steak before him.
"No problem. I suppose it's worth it if you're going to travel six or seven hours in the car to come and see me."
"You know," Mikey said in a careful tone, "you could make the trip shorter for me if you wanted."
I flinched. I should've known that was coming.
I side-stepped the subject. "How does Gwen feel about you coming all this way out here?"
Mikey caught on pretty quick and grinned; I was still as hardheaded as always, and he knew it. "She's fine with it. I think she'd come herself if she wasn't so worried you'd throw her out or something."
"I wouldn't do that to Gwen," I said with a huff. Goodness knows I had more reasons to hate Mikey than Gwen, and I had agreed to meet with him.
"Well, she'd probably feel more than a little awkward with the whole high school thing, too," Mikey admitted. "She is doing well though."
"Really? How nice."
Mikey frowned at my tone. "I know you're still sore at her for what happened to you, but you can't keep this up, Dinger."
"She attempted to blackmail me, before she tried to stop me from ... " I shook my head, but my fists clenched. "And she blamed me for getting attacked."
"She was confused at the time."
"Please. Don't give her an excuse."
"Get over it. You didn't even love her. You were just mad she got in the way. And besides, you had—"
"Shut up. I don't want to talk about it."
"You had Raiya—"
"I told you, I didn't want to talk about it!" I slammed my fists down on the table, hard.
So much for self-control.
Mikey frowned. "Come on, Dinger! It's been years! You still can't talk about it?"
I gritted my teeth together. "I don't want to talk about it. And frankly, you're a gym teacher, not a therapist. If I wanted to talk about it, I would hire one of those to sit around and question me."
"Hey, I'm a coach, and I have to deal with my students' problems all the time," Mikey said. "I'm good at handling problems. And besides that, you should talk about it with someone. There might be a clue in your information."
I froze. "What do you mean?"
Mikey smiled; he seemed to be glad he'd finally arrived at the point he'd wanted to bring up all evening. He leaned closer and said, "Weird stuff's happening again."
"What do you mean, 'weird stuff,' Mike?" I asked carefully.
Did I really want to know? I didn't think I did. The last time "something weird" had happened, I needed to transform into Wingdinger—oh, God, how I cringed at the very thought of that stupid name—and had to save the world.
Mikey sighed. "Dad's back in Apollo City. I heard from Jason he came by, and he was asking for me. And you."
I said nothing, only remained motionless. Mikey's dad was not really a pleasant memory to either of us. If he was back in town, something was certainly up. And it certainly was going to be unpleasant.
After all, Dante Salyards was not a man who would take anything like supernatural trouble lightly. During his last stay in Apollo City, I knew Mikey's dad had seen a good extent of what trouble the supernatural could do.
But still, that selfish, peace-seeking center inside of me wanted this to be fake. So I showed no emotion or reaction as I asked, "What does he want?"
"I don't know what he wants," Mikey admitted with a shrug. "But if he's looking for both of us, I'm pretty sure it's not to give us an award or any money. Trouble's coming."
"How would you know for sure?"
Mikey frowned. "Come on, Dinger, don't get like that. We have to do something."
"You mean I have to do something, don't you?" I snorted into my drink.
"Well ... yeah. I came all the way out here to see you and talk about it."
"I can't just get up and leave, Mikey," I told him. "I have a job. I've been working with the firm for a while now, and if I needed to take time off, I would've had to put it in months ago."
"Can't you take a leave of absence?"
"I can't just go, I just told you."
"You have to!"
"Why? Why should I?"
"For Raiya."
"Shut up!" I reached across the table and grabbed a hold of Mikey's shirt. Death was staring through my eyes as I growled, "Don't mention her to me. She's dead."
"Are you sure?" Mikey asked. We'd caught the attention of quite a few people by this time.
"What? What do you mean, 'Are you sure?!'" I nearly shouted. "I was holding her as she died!" I cringed as I looked down at my hands; they tingled at the memory of Raiya as she shuddered and breathed her last breath, her blood mingling with my tears ... I felt shame-faced as I recalled asking her—begging her—to come back, to stay alive, to stay with me ... and how, all of a sudden, she was gone. We were gone.
"But ... she wasn't human, remember?" Mikey whispered uneasily as he glanced around to see that some of the people nearby were still looking at us. "Maybe she didn't die, maybe she just ... went somewhere else."
Why had I never thought of that?
I was taken aback. Was it possible? Was it true?
Maybe Raiya hadn't died. It was possible—after all, I believed more unbelievable things; I'd seen more unbelievable things.
But I shook my head as the last remnants of my daydreams melted away in the cold light of reality. "It doesn't matter. If she wasn't dead," I said slowly, "she would have come back to me."
I might have believed it, but I still felt dumb for saying it.
"Maybe she's trying, and you're just not there." Mikey raised his eyebrows, no doubt silently congratulating himself on his advanced logic.
It was enough to get me to release him.
I allowed Mikey's remark to settle in my mind. It didn't make any sense. Why would she come back now? What else was going on? I wondered this and questioned Mikey on it.
Mikey took a bite of his steak and said, "Like I said, there's a bunch of unexplained events going on ... and Dante promises it's not even half of it. I can't explain it, but you're not there and Raiya's not there—"
"You just said she wasn't dead!"
"Hey, give me a break. Rachel would be the first one to know right? And Jason hasn't said anything about her saying stuff like that."
I remembered the bright-eyed redhead who ran my favorite coffee shop when I was in high school. Years later, I still had nothing to compare to Rachel's food. Even the steaks and lobster fillets I billed to my bosses were unable to fill the longing my stomach carried since I moved.
"How is Rachel?" I asked, deciding I'd had enough of the gloomy topics.
Mikey shrugged. "She's fine. Pregnant, actually."
"Oh? Really?" Images of a recent dream popped into my head. A little girl with red hair, smiling as she chased a smaller boy with brown hair and matching violet eyes. Rachel's children? Is that whose kids those were?
It was possible, I supposed.
"It's a girl, they know that much," Mikey continued on, not realizing I was caught up in my own thoughts. "She's due in the spring; she and Lee announced it at their anniversary party this summer."
"Do you think she'd really come back?" I whispered softly.
Mikey sighed. "Let's change the subject. Here." He pulled out an envelope. "I was going to mail this, but it's okay to give it to you now."
I opened it and quickly scanned through the letter. Then I went back and read it again, properly this time, just to make sure I understood it. I groaned to myself, but put on a smile for Mike. "You and Gwen are getting married, huh?"
"Yeah." There was nothing but joy in his big, goofy grin.
So I humored him. "That's great, man."
I humored him, and he caught me. Mikey laughed. "Come on, you're going to have to do better than that."
"You can't blame me for not being excited," I snorted. "This means I'll have to go back home."
"And not to mention be my best man," Mikey added.
"Are you kidding me?" I felt oddly conflicted. Like I should agree to it, but I would rather stick needles in my eyes or puke up a pig.
"No. I'm asking you to be my best man."
"Okay, sure, I suppose." But this better not be a lot of work, I added silently.
"Great!" Mikey grinned. "Wedding's in two weeks, so I'll be looking forward to my bachelor party—"
"Two weeks? What do you mean, two weeks?" I looked down at the invitation again. Yep, two weeks. "I can't get off from work just like that."
"Ah, don't be such a sour-butt, Dinger. Everyone else is coming. Besides, you can't really like your job that much."
"Huh?"
"I saw you when you came in. You're tired and exhausted, and probably sick, too. You don't seem to care, either. I can tell. We've been friends a long time."
"That's not the truth."
"It's a good part of it."
I grumbled, caught. It had been forever since someone was so good at reading me. "That's not the whole picture."
"What else is there? Is Charlotte still giving you grief?"
I moaned at mention of her. I put my head in my hands. "You've got a point there, I suppose."
Charlotte was one of the first friends I made in college. I remembered only introducing myself to her because I'd thought, foolishly, she was actually Raiya. Charlotte had the same long, reddish-brown hair as Raiya, and looking at her from behind, I was too hopeful to be cautious.
But Charlotte had been more than gracious to me, even helping me get my current job at her father's law firm, and we were friends—but that was the problem, for her, of course.
She'd been bugging me lately with hints of how we should be dating. Or married with seventeen children and living down in the South with her mother's side of the family. I wasn't sure which she wanted, but I was terrified to discover the specifics.
"You don't have to bring her to the wedding." Mikey smirked.
"I wasn't going to. But she did get me the job at her dad's office. That's why I don't think I can just get off."
"Come on, surely he'll be okay with it."
"I'm not worried about him. It's my other boss, Pharris, who's the real piece of work."
"Piece of work" was the kindest way of putting it. How do you explain to scientists that you've found the missing link between humans and the Tyrannosaurus Rex? Everything from her meticulous, over-gelled bob to the edge of her gilded fingernails screamed dinosaur DNA. If that wasn't enough, her attitude and tone sealed the deal.
"Just ask," Mikey said. "And if you can't get off, well, just quit. You don't like it anyway."
"I'm good at it."
"So what? You need something in your life that you love."
"Look, just get off my back, will you? If a miracle happens and I can get off work, I'll do it. But short of that, you'd better call up Poncey."
Mikey grinned, and for a moment, I had to laugh; Mikey looked just like his high school self, with food stuck in his teeth, his smile wide and innocent. "If that's the best I can get from you for now, I'll take it. Besides, given our history, I'd say a miracle is right around the corner."
"Ha, ha, yeah right." I rolled my eyes. I'd forgotten how to believe in miracles.
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THE STARLIGHT CHRONICLES
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Also by C. S. Johnson
Favan & Flew
One Flew Through the Dragon Heart (Coming Soon)
Once Upon a Princess
Beauty's Curse
Beauty's Quest
Beauty's Kiss
Beauty's Gift
The Divine Space Pirates
The Heights of Perdition
The Breadth of Creation
The Price of Paradise
The Divine Space Pirates Trilogy
The Legend of Eydis
Eydis: The Island of the Dragon Bride
The Moonlight Pegasus
The Moonlight Pegasus
One Night of Moonlight
The Signs of the Stars
The Birth of Gemini
The Starlight Chronicles
Searching
Slumbering
Calling
Submerging
Remembering
Continuing
Outpouring
Everlasting
Till Human Voices Wake Us
Across the Floors of Silent Seas
Standalone
The Serpent-Bearer and the Prince of Stars (Coming Soon)
Watch for more at C. S. Johnson's site.
| {
"redpajama_set_name": "RedPajamaBook"
} | 5,272 |
Q: Convert a string to a Multi-Level Dictionary I have a a text file that looks like
tabs.main="******"
tabs.settings="******"
settings.setting1="******"
settings.setting2="******"
settings.setting3="******"
settings.setting4="******"
settings.setting5.title="******"
settings.setting5.settingoption1="******"
settings.setting5.settingoption2="******"
What I'd like to be able to do is parse this into a multi-leveled dictionary. For example, I'd have a root dictionary that looks like this Dictionary<string, object> and in that I would have tabs and settings which were both Dictionary<string, object> them selves. In a graphical form what I'd like is:
Dictionary<string, object>
-> Dictionary<"tabs", object>
-> Dictionary<"main", "*******">
-> Dictionary<"settings", "*******">
-> Dictionary<"settings", object>
-> Dictionary<"setting1", "*******">
And so on and so forth. Is this possible? And if so, could someone give me a pointer in the right direction.
A: Dictionaries aren't really designed to hold your data in that manner. You could create classes to represent your file and load the data into them instead. Based on what you've shown something like this might work.
public class File
{
private List<Setting> settings = new List<Setting>();
public Tab Tab { get; set; }
public List<Setting> Settings { get{ return settings; } }
}
public class Tab
{
public string Main{ get; set;}
public string Settings{ get; set;}
}
public class Setting
{
private List<string> options = new List<string>();
public string Value{ get; set; }
public List<string> Options{ get{ return options;} }
}
A: The problem you are going to run into is that some of your keys are going to need to be stored in the dictionary with values (strings presumably) and some of them will need to be stored with Dictionaries.
In your example you listed:
settings.setting5="******"
settings.setting5.settingoption1="******"
These two lines contradict each other. settings.settings5 would have the key settings5 store a value in the dictionary, but the next line is expecting settings5 to be a dictionary of other values.
You could write a block of code like this:
var settings = new Dictionary<string, object>();
var lines = File.ReadAllLines("...");
foreach (var line in lines)
{
var parts = line.Split(new char[] { '=' }, 2);
if (parts.Length != 2) continue;
var keys = parts[0].Split('.');
var value = parts[1];
var dict = settings;
for (int i = 0; i < keys.Length - 1; i++)
{
if (!dict.ContainsKey(keys[i]))
dict.Add(keys[i], new Dictionary<string, object>());
dict = (Dictionary<string, object>)dict[keys[i]];
}
dict.Add(keys[keys.Length - 1], value);
}
*
*It doesn't have any error checking.
*You'll find that while it does populate your objects. It is a pain to retrieve values from it. Since you have to cast each result to either a string or another Dictionary.
Either way you'll find that you will most likely want to avoid trying to creating dictionaries of dictionaries for this purpose.
A: Please consider using an object other than a dictionary, but to answer your question.
Note that this data structure does not allow both a dictionary and a string value with the same key.
static Dictionary<string,object> Parse(string contents)
{
var root = new Dictionary<string,object>();
using (var rdr = new StringReader(contents))
{
string line;
var equals = new [] {'='};
while(null != (line = rdr.ReadLine()))
{
if(!string.IsNullOrEmpty(line))
{
var keyValue = line.Split(equals, 2);
AddValue(root, keyValue[0], keyValue[1]);
}
}
}
return root;
}
static void AddValue(Dictionary<string,object> dict, string dottedKeys, string value)
{
string [] keys = dottedKeys.Split('.');
for(var i = 0; i < keys.Length - 1; i++)
{
var key = keys[i];
dict = GetOrAdd(dict, key);
}
dict[keys[keys.Length - 1]] = value;
}
static Dictionary<string,object> GetOrAdd(Dictionary<string,object> parent, string key)
{
object o;
Dictionary<string,object> childDict;
if(parent.TryGetValue(key, out o))
childDict = (Dictionary<string,object>) o; // This will throw when adding a dictionary to a value.
else
parent[key] = childDict = new Dictionary<string,object>();
return childDict;
}
static string fileContents=@"
tabs.main=""******""
tabs.settings=""******""
settings.setting1=""******""
settings.setting2=""******""
settings.setting3=""******""
settings.setting4=""******""
settings.setting5=""******""
settings.setting6.settingoption1=""******""
settings.setting6.settingoption2=""******""
";
A: I actually ended up managing to right some code myself in the time being, but as I hate to have wasted peoples time answering this question, I'll mark Jason's question as the answer as it is kinda what I came up with and he helped me write my code!
Dictionary<string, object> _dic = new Dictionary<string, object>();
using(StreamReader reader = new StreamReader(file))
{
string text;
while ((text = reader.ReadLine()) != null)
{
if (text.Equals(String.Empty))
continue;
string line = text.Trim();
string[] keyval = line.Split('=');
string value = keyval[1].Trim('"');
string[] keys = keyval[0].Split('.');
Dictionary<string, object> prevDic = _dic;
for (int i = 0; i < keys.Length; i++)
{
if (i + 1 != keys.Length)
{
Dictionary<string, object> temp = new Dictionary<string, object>();
if (prevDic.ContainsKey(keys[i]))
{
prevDic = prevDic[keys[i]] as Dictionary<string, object>; // Avoids exceptions when trying to cast the string value to a dictionary
continue;
}
else
{
prevDic.Add(keys[i], temp);
prevDic = temp;
}
}
else
{
prevDic.Add(keys[i], value);
}
}
}
}
Thanks for everyones help in this!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,758 |
{"url":"https:\/\/www.comsol.com\/blogs\/global-modeling-of-a-non-maxwellian-discharge-in-comsol\/","text":"# Global Modeling of a Non-Maxwellian Discharge in COMSOL\u00ae\n\nNovember 19, 2018\n\nGlobal modeling of plasmas is a powerful approach to study large chemistry sets. In these models, the reactions are represented by rate coefficients. In particular, the rate coefficients of electron impact collisions depend on the electron energy distribution function (EEDF), which is often non-Mawellian and can be computed from an approximation of the Boltzmann equation (BE). Here, we explain how to create a global model fully coupled with the BE in the two-term approximation using the COMSOL Multiphysics\u00ae software.\n\n### Setting Up a Global Model of a Non-Maxwellian Discharge\n\nThe equations in a global model are greatly simplified because the spatial information of the different quantities in the plasma reactor is treated as volume-averaged. Without the spatial derivatives, the numerical solution of the equation set becomes considerably simpler and the computational time is reduced. Consequently, this type of model is useful when investigating a broad region of parameters for plasmas with complex chemistries.\n\nFor a closed reactor without net mass creation at the surfaces, a mixture of k=1,\\dotsc,Q species and j=1,\\dotsc,N reactions is described by the mass fraction balance equations for Q-1 species\n\nV \\rho \\frac{d w_k}{dt} = VR_k+\\sum_{l}h_l A_l R_{surf,k,l}M_k\n\nwhere V is the reactor volume, \\rho is the mass density, w_k is the mass fraction of species k, A_l is the area of surface l, h_l is a correction factor of surface l, R_s is the surface rate expression of surface l, R_k is the volume rate expression for species k, and M_k is the molar mass.\n\nThe sum in the last term is over surfaces where species are lost or created. One of the species mass fractions is found from mass conservation.\n\nThe electron number density is obtained from the electron neutrality condition\n\nn_e=\\sum_{k=1}^N Z_k n_k\n\nwhere n_k is the number density and Z_k is the charge number. The electron energy density, n_{\\varepsilon}, can be computed from\n\ne\\frac{d n_{\\varepsilon}}{dt}=R_{\\varepsilon}+\\frac{P_{abs}}{V}\n+\\sum_l \\sum_{ions} e h_l \\frac{A_l}{V}R_{surf,k,l} N_A \\left( \\varepsilon_e + \\varepsilon_i \\right)\n\nwhere n_{\\varepsilon}=n_e \\overline \\varepsilon, \\overline \\varepsilon is the mean electron energy, P_{abs} is the power absorbed by the plasma, and e is the elementary charge. The last term on the right-hand side accounts for the kinetic energy transported to the surface by electrons and ions. The summation is over all positive ions and boundaries with surface reactions, \\varepsilon_e is the mean kinetic energy lost per electron lost, \\varepsilon_i is the mean kinetic energy lost per ion lost, and N_A is Avogadro\u2019s number.\n\nIn the equations above, the source terms R_k and R_{\\varepsilon} are computed using rate coefficients that represent the effect of collisions. In particular, for electron impact collisions, the rate coefficients depend on the EEDF, which is often a non-Maxwellian distribution and depends on the discharge conditions. In practice, the EEDF can be obtained by solving an approximation of the electron BE using fundamental collision cross-section data. Once the EEDF is known, the rate coefficients are computed by a suitable averaging of the electron impact cross sections over the EEDF.\n\n### Describing the Boltzmann Equation and Two-Term Approximation\n\nThe BE that describes the evolution of an ensemble of electrons in a six-dimensional phase space is\n\n\\frac{\\partial f}{\\partial t}+\\textbf{v} \\cdot \\nabla f-\\frac{e}{m}\\textbf{E} \\cdot \\nabla_v f=C[f]\n\nwhere f is the EEDF, \\textbf{v} is the velocity coordinates, m is the electron mass, \\textbf{E} is the electric field, \\nabla_v is the velocity gradient operators, and C is the rate of change in f due to collisions.\n\nNormally, a rather simplified BE is solved instead. It is assumed that the electric field and the collision probabilities are spatially uniform. The BE is then written in terms of spherical coordinates in the velocity space and f is expanded in spherical harmonics. The series is truncated after the second term, and the so-called two-term approximation of f is\n\nf\\left( v,cos \\theta,z ,t \\right) = f_0 \\left( v, z ,t \\right) + f_1\\left( v,z ,t \\right)cos \\theta\n\nwhere f_0 is the isotropic part of f, f_1 is an anisotropic perturbation, v is the magnitude of the velocity, \\theta is the angle between the velocity and the field direction, and z is the position along this direction.\n\nThe problem is further simplified by solving only steady-state cases where the electric field and EEDF are either stationary or oscillate at a high frequency. The last piece of simplification consists of separating the energy dependence of the EEDF from its time and space dependence using\n\nf_{0,1} \\left( \\varepsilon, z, t \\right) = \\frac{1}{2 \\pi \\gamma^3} F_{0,1} \\left( \\varepsilon \\right) n\\left( z, t\\right)\n\nwhere F_{0,1} is an energy distribution function constant in time and space that verifies the following normalization\n\n\\int_0^\\infty \\varepsilon^{1\/2} F_0 d\\varepsilon =1\n\nwhere \\gamma=\\sqrt{2e\/m} and \\varepsilon = \\left( v \/ \\gamma \\right)^2.\n\nUsing the above-mentioned approximations and after some manipulations, the equation for F_0 can be written in the form of a 1D convection-diffusion-reaction equation\n\n\\frac{\\partial }{\\partial \\varepsilon} \\left( W F_0-D \\frac{\\partial F_0 }{\\partial \\varepsilon}\\right) = S\n\n(For more details, see Ref. 1.)\n\nThis equation can be used to compute an EEDF, providing a set of electron collision cross sections and a reduced electric field, E\/N (ratio of the electric field strength to the gas number density). Depending on the operating conditions, it might be necessary to include the effect of superelastic collisions and electron-electron collisions. Quite often, the input quantity of interest is the mean electron energy. In this case, a Lagrange multiplier is introduced to solve for the reduced electric field such that the equation below is satisfied.\n\n\\int_0^\\infty \\varepsilon^{3\/2} F_0 d\\varepsilon =\\overline{\\varepsilon}\n\nOnce the EEDF is computed, the rate coefficients needed for a plasma global model are computed from\n\nk_k=\\gamma \\int_0^\\infty \\varepsilon \\sigma_k \\left( \\varepsilon \\right) F_0\\left( \\varepsilon \\right) d\\varepsilon\n\nwhere \\sigma_k is the cross section of reaction k.\n\nThe figure below plots a computed EEDF obtained for argon at \\overline{\\varepsilon} = 5 eV and a corresponding Maxwellian. Note how the computed EEDF strongly deviates from a Maxwellian and how sharply it falls above the first excitation level of argon at 11.5 eV. In the same figure, the cross section for the excitation of the lumped level (corresponding to the first 4-s levels of argon) is plotted. With the information in this figure, the rate coefficient for the excitation of the 4-s levels can be computed.\n\nAlso important to note from this figure is that the computed EEDF and the cross section vary by several orders of magnitude in the overlapping region. In consequence, a small variation in \\overline{\\varepsilon} (or E\/N) causes a large change in the rate coefficients. This example is for argon, but the same behavior is found in many other gases, and it is one of the reasons why plasmas have very nonlinear behavior.\n\nIn a practical application, the BE in the two-term approximation can be solved to provide rate coefficients to a global model. In such cases, the EEDF is computed every time the input conditions for the BE have changed.\n\n### Coupling the Global Model with the BE in the Two-Term Approximation\n\nIn this section, we show how to make a global model fully coupled with the BE in the two-term approximation using COMSOL Multiphysics. A three-step procedure is advised:\n\n1. Create a global model where an analytic EEDF is used\n2. Use the EEDF Initialization study to solve only for the EEDF\n3. Solve the fully coupled problem\n\n#### Creating an EEDF Initialization Study\n\nAfter having the global model working for an analytic EEDF (step 1), you can decide to investigate further to see if the EEDF used is suitable for your needs. You can do this by using an EEDF Initialization study to solve the BE in the two-term approximation. This study solves the BE for the electron impact cross sections provided and a choice of the reduced electric field or the mean electron energy. This procedure is exemplified in the screenshots below. First, select the Boltzmann equation, two-term approximation (linear) option \u2014 or the Boltzmann equation, two-term approximation (quadratic) option \u2014 in the Electron Energy Distribution Function Settings section.\n\nThen, set the Reduced electric field so that it\u2019s used in the solution of the EEDF.\n\nAt this stage, you can compare the computed EEDF and the rate coefficients with the ones you used in the global model in step 1 and assess if the model needs further improvements. If you decide to solve the fully coupled problem, add another study and use the solution of the EEDF Initialization study as the initial condition, as shown below. Using the solution from the EEDF Initialization study is a requirement.\n\n#### Coupling the Global Model Equations and BE\n\nThe coupling between the global model equations and the BE can happen in two different ways, depending on whether you use the Local field approximation or the Local energy approximation to define the mean electron energy in the Plasma Properties section. When using the Local field approximation, the excitation of the system is given from a reduced field. This electric field can be constant (a parameterization can be made over E\/N) or can come from a solution of an equation (e.g., a circuit equation). When using the Local energy approximation, the global model equation for the electron mean energy is solved and the power absorbed by the plasma needs to be set by the user. In this case, the E\/N is found so that the equation below is satisfied.\n\n\\int_0^\\infty \\varepsilon^{3\/2} F_0 \\left( E\/N,n_k\/N,n_e\/N,\\dotsc\\right) d\\varepsilon-\\overline{\\varepsilon}=0.\n\n#### Example: A Plasma Sustained by a Direct Current Voltage Source\n\nAs a practical example, we chose to model an argon plasma created within a 4-mm gap by a direct current (DC) voltage source of 1 kV in series with a 100-k\u03a9 resistance at 100 mTorr. This model is inspired by Ref. 2. We emphasize that the model has no spatial description and that geometrical parameters and volume-averaged quantities are used to describe the plasma in the gap. The voltage applied to the plasma, V_p, comes from the circuit equation\n\nV_p=V_{dc}-RI_p\n\nwhere V_{dc} is the applied voltage and R is the circuit resistance.\n\nThe plasma current, I_p, is computed from\n\nI_p=e A n_e \\left( \\mu N\\right) \\left(E\/N\\right)\n\nwhere A is the plasma cross-sectional area and \\mu N is the reduced electron mobility.\n\nSolving for E\/N, we obtain\n\n\\frac{E}{N}=\\frac{V_{dc}}{dN+e R A n_e \\left( \\mu N \\right)}\n\nwhere d is the gap distance between electrodes.\n\nIf we choose to use the Local field approximation, the equation for E\/N above can be used directly in the EEDF Inputs section, as shown in the screenshot below.\n\nIf we choose to use the Local energy approximation, the power absorbed by the plasma can be defined as\n\nP_{abs}=V_p I_p\n\nin the Mean Electron Energy section, as in the screenshot below.\n\nIn this model, both approaches give very similar results, since the same electron energy loss\/gain from collision events is accounted for in the BE and the mean electron energy equation, and because no energy losses to the wall are included in the mean electron energy equation.\n\nIn the figure below, the temporal evolution of the charged species and the reduced electric field are shown. Initially, there is no plasma in the gap and the electric field (black line, right axis) maintains a constant value. When breakdown starts to occur, there is a rapid increase of the charged carriers and the current flowing into the circuit, resulting in a voltage drop across the gap. After this transient regime, a steady state is reached where the plasma is sustained with a reduced electric field of only 4 Td.\n\nThe temporal evolution of the EEDF is presented below. Initially, the EEDF has a large population above 15 eV, as it is necessary to facilitate the plasma breakdown. After the plasma formation, and due to the decrease of the electric field, the electron population cools down and the EEDF develops a tail with a steeper slope. As time progresses, and with the increase of the argon excited-state density, the influence of the superelastic reactions in the EEDF becomes noticeable, with the appearance of a bump at the high-energy end. Note that the time variation is presented on a log scale in this animation.\n\n### Next Steps\n\nTo try the example featured in this blog post, click the button below. Doing so will take you to the Application Gallery, where you can download the MPH-file for the model in addition to the step-by-step documentation.\n\nYou can also read more about modeling plasma physics in the following blog posts:\n\n### References\n\n1. G.J.M. Hagelaar and L.C. Pitchford, \u201cSolving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models,\u201d Plasma Sources Science and Technology, vol. 14, pp. 722\u2013733, 2005.\n2. S. Pancheshnyi, B. Eismann, G. Hagelaar, and L. Pitchford, \u201cZDPlasKin: A New Tool for Plasmachemical Simulations\u201d, The Eleventh International Symposium on High Pressure, Low Temperature Plasma Chemistry, 2008.\n\n#### Categories\n\n##### Engr Hameed ullah\nApril 22, 2020\n\nHello Sir\nI am Engr Hameed Ullah from North China Electric Power University Beijing. can I ask something for you, sir. Kindly reply to me I will very thank full to you for this. Sir, I am reading your Ph.D. Student thesis ( Charge Transport and Breakdown Physics in\nLiquid\/Solid Insulation Systems by Joya Jadidian).\nSir can you guide me about these Simulations of Electrical streamers because I want to work on it further sir. Kindly help me, please, please\n\n##### khalid hussain\nSeptember 10, 2021\n\nHello Sir\nI am Engr Hameed Ullah from North China Electric Power University Beijing. can I ask something for you, sir. Kindly reply to me I will very thank full to you for this. Sir, I am reading your Ph.D. Student thesis ( Charge Transport and Breakdown Physics in\nLiquid\/Solid Insulation Systems by Joya Jadidian).\nSir can you guide me about these Simulations of Electrical streamers because I want to work on it further sir. Kindly help me, please, please\n\nEXPLORE COMSOL BLOG","date":"2022-07-03 06:16:33","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.833021342754364, \"perplexity\": 1090.0742505820574}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104215790.65\/warc\/CC-MAIN-20220703043548-20220703073548-00440.warc.gz\"}"} | null | null |
Somalis in the Netherlands are residents or naturalized citizens of the Netherlands who are of Somali ancestry. They form one of the larger Somali communities in Europe and amongst the second largest African foreign community in the Netherlands. The Somalis form the second largest African community in The Netherlands and are one of the fastest growing communities.
Migration history
From 1989 to 1998, the Netherlands was the second-most common European destination for Somali asylum-seekers, only slightly behind the United Kingdom and more than double the total of the next-most common destination, Denmark. However, between 2000 and 2005, there was a significant outflow of Somalis from the Netherlands to the United Kingdom, unofficially estimated to be as large as 20,000 people. Factors mentioned as driving forces behind the exodus included an increase in opposition to Muslim immigration, as exemplified by the rise of Pim Fortuyn, Somali opposition to housing policies which forced them to live scattered in small groups all over various cities rather than in a larger agglomerated community, a restrictive socio-economic environment which, among other things, made it difficult for new arrivals to find work, and the comparative ease of starting a business and acquiring the means to get off social welfare in the UK.
Demography
, Statistics Netherlands estimated the following figures with respect to Dutch people of Somali origin:
15,281 persons of first-generation background (8,831 men, 6,850 women)
6,517 persons of second-generation background (3,322 men, 3,195 women), of which:
543 persons with one parent born in the Netherlands (273 men, 270 women)
5,974 persons with both parents born out of the Netherlands (3,049 men, 2,925 women)
For a total of 21,798 persons (11,753 men, 10,045 women). This represented roughly 9% growth over the 1996 total of 20,060 persons; the composition of the population had changed slightly, with the proportion of the population of second-generation background more than doubling over that time frame. The proportion of men has typically been greater than that of women. Most men are single without dependents, while most women are single mothers with one or more children. This is largely due to being only able to send certain family members to a different country.
Religiosity
According to a 2018 report, Islam takes a central role in the lives of nearly all Somalis and in many ways their religiosity rose from the 2009 already high levels.
Notable people
Yasmine Allas, actress and writer
Hussein Suleiman, fashion designer, CEO of Daily Paper
Abdi Nageeye, athlete
Liban Abdulahi
References
Notes
Sources
Further reading
External links
Federation of Somali Associations in the Netherlands - FSAN
African diaspora in the Netherlands
Islam in the Netherlands
Ethnic groups in the Netherlands
Netherlands
Muslim communities in Europe | {
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Simple things don't have to be bland.
Usagi offers authentic Japanese food as well as a brilliant selection of great long drinks and cocktails. Various beers and constantly changing rarities, such as beautiful wine champagne or fine wines, paired with punk and wild zests are waiting for you. Ramen, Gyoza, fresh salads and many more delicious dishes want to be discovered. Since its "first come, first served" here, you better come in early! | {
"redpajama_set_name": "RedPajamaC4"
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Bevans Branham is a proud resident of Palm Springs California. Alongside his loving wife Lorie Branham, Bevans runs The Branham Companies, a company formed after graduating from the University of Colorado. Bevans and Lorie were married shortly thereafter in 1973.
Throughout their careers, Lorie and Bevans have been recipients of top-notch training in advertising, marketing & public relations, team-building, hospitality and classical French cooking. Over the years they have gained recognition for excellence as promoters, producers, and developers of award winning advertising campaigns, public relations programs, nationally recognized restaurants/clubs and "home run" venture capital projects. They have been instrumental creating and establishing a unique group of businesses celebrated for success – critically, financially and culturally.
Along with that, they are the proud and grateful parents of Lindsay Branham, a prize-winning investigative journalist & documentary filmmaker focusing on human rights and social justice issues, worldwide, and Chaz Branham, an emerging gifted entrepreneurial coach/trainer, athlete and strategic advisor in the world of health, fitness and wellness.
For over forty years, Bevans Branham has focused his considerable gifts and talents in the following main areas of expertise: Marketing/Advertising/Public Relations & Promotion, Restaurants and Clubs, Venture Capital and Strategic Advisory Services, Humanitarian & Non-Profit Organizations and Social Entrepreneurship, currently developing the premier hot, new boutique hotel group for couples, worldwide, associated websphere and institute. | {
"redpajama_set_name": "RedPajamaC4"
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Q: What should I use for the Android context here? I'm trying to use this code from "Android Recipes":
AlertDialog.Builder builder = new AlertDialog.Builder(context);
builder.setTitle("FetchAndPopTask.doInBackground exception");
builder.setMessage(e.getMessage());
builder.setPositiveButton("OK", null);
builder.create().show();
...but don't know what I should replace "context" with. I've tried the .java file's class, the immediate class, and "this" but none of them compile.
In more context, the code is:
public class SQLiteActivity extends ActionBarActivity {
private FetchAndPopTask _fetchAndPopTask;
. . .
private class FetchAndPopTask extends AsyncTask<String, String, String> {
@Override
protected String doInBackground(String... params) {
. . .
try {
. . .
} catch (Exception e) {
AlertDialog.Builder builder = new AlertDialog.Builder(this); // <= "context"...?
builder.setTitle("If I go blind, I'll use a Service Platypus (instead of a Service Dog)");
builder.setMessage(e.getMessage());
builder.setPositiveButton("OK", null);
builder.create().show();
return result;
}
I tried all of the following:
AlertDialog.Builder builder = new AlertDialog.Builder(SQLiteActivity);
AlertDialog.Builder builder = new AlertDialog.Builder(FetchAndPopTask);
AlertDialog.Builder builder = new AlertDialog.Builder(this);
...but none compile; so what does "context" need to be here?
A: AlertDialog.Builder(SQLiteActivity.this) should probably work
But take a look at this question
EDIT!!!!!
Sorry, didn't notice you're trying to show it in non-UI thread. Please place it in constructor or in onPreExecute()/onPostExecute() methods
A: doInBackground() will be run on a background thread. You cannot touch your UI on a non-UI thread. This includes displaying dialogs. Remove the AlertDialog from doInBackground(). You can put it in e.g. onPostExecute() that runs on UI thread. In there you can use YourActivityName.this to refer to the outer class this to be used as a Context.
A: You have to pass an Activity to this constructor.
You have to add this in the FetchAndPopTask class:
private SQLiteActivity context;
public FetchAndPopTask(SQLiteActivity context) {
this.context = context;
}
Then, in the SQLiteActivity class, you have to pass this context by using this keyword (as you are in an activity, it refers to it):
/*...*/
FetchAndPopTask task = new FetchAndPopTask(this);
task.execute();
/*...*/
A: new AlertDialog.Builder(this);
the this means that you are getting the pointer/reference of FetchAndPopTask so instead of using this use the pointer/reference of your SQLiteActivity by calling SQLiteActivity.this
A: Define a constructor for your class and pass the context there
private class FetchAndPopTask extends AsyncTask<String, String, String> {
private Context mContext;
public FetchAndPopTask(Context context){
mContext = context;
}
@Override
protected String doInBackground(String... params) {
. . .
try {
. . .
} catch (Exception e) {
AlertDialog.Builder builder = new AlertDialog.Builder(mContext); // <= use the variable here
builder.setTitle("If I go blind, I'll use a Service Platypus (instead of a Service Dog)");
builder.setMessage(e.getMessage());
builder.setPositiveButton("OK", null);
builder.create().show();
return result;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
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package com.hazelcast.internal.server.tcp;
import com.hazelcast.cluster.Address;
import com.hazelcast.instance.EndpointQualifier;
import com.hazelcast.instance.ProtocolType;
import com.hazelcast.internal.cluster.impl.MemberHandshake;
import com.hazelcast.internal.nio.Packet;
import com.hazelcast.internal.server.ServerContext;
import com.hazelcast.logging.ILogger;
import java.util.ArrayList;
import java.util.Collection;
import java.util.HashMap;
import java.util.Map;
import static com.hazelcast.internal.cluster.impl.MemberHandshake.OPTION_PLANE_COUNT;
import static com.hazelcast.internal.cluster.impl.MemberHandshake.OPTION_PLANE_INDEX;
import static com.hazelcast.internal.cluster.impl.MemberHandshake.SCHEMA_VERSION_2;
public class SendMemberHandshakeTask implements Runnable {
private final ILogger logger;
private final ServerContext serverContext;
private final TcpServerConnection connection;
private final Address remoteAddress;
private final boolean reply;
private final int planeIndex;
private final int planeCount;
public SendMemberHandshakeTask(ILogger logger,
ServerContext serverContext,
TcpServerConnection connection,
Address remoteAddress,
boolean reply,
int planeIndex,
int planeCount) {
this.logger = logger;
this.serverContext = serverContext;
this.connection = connection;
this.remoteAddress = remoteAddress;
this.reply = reply;
this.planeIndex = planeIndex;
this.planeCount = planeCount;
}
@Override
public void run() {
connection.setRemoteAddress(remoteAddress);
serverContext.onSuccessfulConnection(remoteAddress);
//make sure memberHandshake packet is the first packet sent to the end point.
if (logger.isFinestEnabled()) {
logger.finest("Sending memberHandshake packet to " + remoteAddress);
}
MemberHandshake memberHandshake = new MemberHandshake(
SCHEMA_VERSION_2,
getConfiguredLocalAddresses(),
remoteAddress,
reply,
serverContext.getUuid())
.addOption(OPTION_PLANE_COUNT, planeCount)
.addOption(OPTION_PLANE_INDEX, planeIndex);
byte[] bytes = serverContext.getSerializationService().toBytes(memberHandshake);
Packet packet = new Packet(bytes).setPacketType(Packet.Type.SERVER_CONTROL);
connection.write(packet);
//now you can send anything...
}
Map<ProtocolType, Collection<Address>> getConfiguredLocalAddresses() {
Map<ProtocolType, Collection<Address>> addressMap = new HashMap<ProtocolType, Collection<Address>>();
Map<EndpointQualifier, Address> addressesPerEndpointQualifier = serverContext.getThisAddresses();
for (Map.Entry<EndpointQualifier, Address> addressEntry : addressesPerEndpointQualifier.entrySet()) {
Collection<Address> addresses = addressMap.computeIfAbsent(addressEntry.getKey().getType(), k -> new ArrayList<>());
addresses.add(addressEntry.getValue());
}
return addressMap;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,937 |
São Patrício is een gemeente in de Braziliaanse deelstaat Goiás. De gemeente telt 2.144 inwoners (schatting 2009).
Gemeente in Goiás | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,852 |
\section{Introduction}
The exact solutions \cite{seiberg1, seiberg2, seiberg3}
for the IR Wilsonian effective theory of N=1
supersymmetric QCD (SQCD) reveal some surprising dynamical effects.
Most striking are the occurence of massless composite bound states (or
solitons) in the strong coupling regime. It is intriguing
whether these massless states could smoothly map to states important to
the dynamics of non-supersymmetric gauge theories.
It is highly implausible that the massless composite
fermions of SQCD can survive in the QCD limit. The
lattice arguments of Weingarten \cite{Wein} imply that
any composite states in QCD must be heavier than the pions.
Nevertheless, it is possible, for example,
that the scalar ``dual quark'' solitons might
survive in some form and be involved in some ``dual magnetic''
description of confinement in QCD.
Soft breaking terms, such as squark and gaugino masses, may be
introduced to the SQCD theories as spurion fields with non-zero
F-component vevs that explicitly break supersymmetry \cite{soft1}.
The symmetries of the enlarged spurion model
constrain how they may appear in the low energy Wilsonian
theory. In general these constraints are not however sufficient to
determine the low energy theory since ``Kahler Potential'' terms may be
constructed that are invariant to all symmetries and are hence
unknown \cite{soft2}. For the cases where a squark and/or gaugino mass are the
sole
supersymmetry and chiral symmetry breaking parameters these Kahler terms
dominate the behaviour of the potential. Some speculations as to the
behaviour of these theories were made in Refs \cite{peskin,hoker}.
In this paper we discuss these difficulties and investigate some cases in which
the effects of the soft breakings {\it can} be controlled.
We start with a model with supersymmetry preserving quark/squark masses,
and then break supersymmetry with squark and
gaugino masses resulting from spurions that occur linearly in the
superpotential. It can be shown that any possible Kahler corrections are higher
order in the soft breakings, and thus control may be retained over the
low-energy theory. The analysis is
similar to that performed on the N=2 SQCD solutions in Ref
\cite{soft2}. The
derivative (low-energy) expansion performed to obtain
the solutions of SQCD restricts the
solutions of the softly broken models to the regime where the soft
breakings are small relative to the strong interaction scale. At first
sight the resulting models appear to behave almost identically to their
supersymmetric counterparts but, as for the N=2 solutions \cite{soft3},
the models
have the additional new feature of displaying $\theta$ angle
dependence. The softly broken models distinguish the $N_c$ vacua of the
SQCD models and as $\theta$ is changed these vacua interchange at first
order phase transitions. We contrast this behaviour with that of the QCD
chiral Lagrangian.
\section{N=1 SQCD}
We begin from the N=1 $SU(N_c)$ SQCD theories with $N_f$ flavors
described by the UV Lagrangian
\begin{equation}
{\cal L} = K^\dagger K (Q^\dagger_i Q_i + \tilde{Q}^\dagger_i
\tilde{Q}_i)|_D + {1 \over 8 \pi}
Im \tau W^\alpha W^\alpha|_F + 2 Re\, m_{ij} Q_i \tilde{Q}_j|_F
\end{equation}
where $Q$ and $\tilde{Q}$ are the standard chiral matter superfields and
$W^\alpha$ the gauge superfield. The coupling $K$
determines the kinetic normalization of the matter fields. The
gauge coupling $\tau = \theta/2 \pi + i 4 \pi/g^2$ defines
a dynamical scale of SQCD:
$\Lambda^{b_0} = \mu^{b_0} exp( 2\pi i \tau)$,
with $b_0 = 3 N_c - N_f$ the one loop coefficient of the SQCD
$\beta$-function.
And, finally, $m$ is a supersymmetric mass term for
the matter fields. We may raise these couplings to the status of spurion
chiral superfields which are then frozen with scalar
component vevs. The SQCD theory
without a mass term has the symmetries
\begin{equation}
\begin{tabular}{ccccc}
&$SU(N_f)$ & $SU(N_f)$ & $U(1)_B$ & $U(1)_R$\\
$Q$ & $N_f$ & 1 & 1 & ${N_f - N_c \over N_f}$\\
$\tilde{Q}$ &1& $\bar{N}_f$ & -1 & ${N_f - N_c \over N_f}$\\
$W^\alpha$ & 1 & 1 & 0 & 1\end{tabular}
\end{equation}
The mass term breaks the chiral symmetries to the vector symmetry. The
classical $U(1)_A$ symmetry on the matter fields is anomalous and,
if there is a massless quark, may be used to rotate away the theta
angle. In the massive theory the flavor symmetries may be used to
rotate $m_{ij}$ to diagonal form and the anomalous $U(1)_R$ symmetry
under which the $Q$s have charge $+1$ may be used to rotate $\theta$ on
to the massless gaugino. Including
the spurion fields the non-anomalous $U(1)_R$ symmetry charges are
\begin{equation}\label{sym}
\begin{tabular}{cccccc}
$W$ & $Q$ & $\tilde{Q}$ & $\tau$ & $m$ & $K$ \\
1 & ${N_f - N_c \over N_f}$ & ${N_f - N_c \over N_f}$ & 0 & ${2N_c \over
N_f}$ & {arbitrary} \end{tabular}
\end{equation}
The anomalous symmetries may be restored to the status of symmetries of
the model if we also allow the spurions to transform. The appropriate
charges are
\begin{equation}
\begin{tabular}{ccccccc}
&$W$ & $Q$ & $\tilde{Q}$ & $\Lambda^{b_0}$ & $m$ & $K$ \\
$U(1)_R$ & 1 & 0 & $ 0 $ & $2(N_c-N_f)$ & 2 & arbitrary\\
$U(1)_A$ & 0 & 1 & 1 & $2N_f$ & -2 & arbitrary
\end{tabular}
\end{equation}
The $m_{ij}$ spurions also transform under the
chiral flavor group.
The solutions of the models are $N_f$ dependent. For $N_f < N_c$ the
low energy superpotential is exactly determined by the symmetries and
the theory has a run away vacuum \cite{seiberg1}. For $N_f = N_c$ the low
energy theory
is in terms of meson and baryon fields
\begin{eqnarray}
M_{ij} & = & Q_i \tilde{Q}_j \nonumber\\
b^{[i_1,...,i_N]} & = & Q^{i_1} ... Q^{i_{N_c}}\\
\tilde{b}^{[i_1,...,i_N]} & = & \tilde{Q}^{i_1} ... \tilde{Q}^{i_{N_c}}
\nonumber
\end{eqnarray}
subject to the constraint $det M
+ b \tilde{b} = \Lambda^{2N_f}$ \cite{seiberg2}. For $N_F = N_c+1$
the theory is again described
by baryon and meson fields with the classical moduli space unchanged
\cite{seiberg2}.
When $N_c+1 < N_f < 3N_c$ the theory has
an alternative description of the low energy physics in terms
of a dual magnetic theory with an $SU(N_f-N_c)$ gauge group, $N_f$
flavors of dual quarks, $q$ and $\tilde{q}$, and $N_f^2$ meson fields,
$M_{ij}$ \cite{seiberg3}.
The dual theory has the
additional superpotential term $M_{ij}q_i \tilde{q}_j$. Generally one of the
two
duals is strongly coupled whilst the other is weakly coupled (the
electric theory is weakly coupled for $N_f \sim 3N_c$, the magnetic
theory when $N_f \sim N_f+2$). In the strongly coupled variables
the low energy Wilsonian effective
theory is a complicated theory with all higher dimensional terms in the
superfields equally important (since the IR theory is in a conformal
regime the scale $\Lambda$ at which the theory entered the conformal
regime is not available to suppress higher dimension terms and similarly
the gauge coupling is order one and may not suppress these
operators). The weakly interacting theory however, has a very simple
Wilsonian effective theory of the canonical bare form. According to the
duality conjecture these two effective theories must describe the same
physics and therefore there is presumably a (complicated) mapping
between the electric and magnetic variables in the IR.
\section{Soft Supersymmetry Breaking}
Soft breaking interactions terms which explicitly break supersymmetry
may be included in the UV theory by allowing the spurions to acquire
non-zero $F$-components.(These are the terms that can be induced by
spontaneous supersymmetry breaking and hence may be included
perturbatively while inducing only logarithmic divergences in the
theory as a remnant of the supersymmetric non-renormalization theorems
\cite{soft1}).
We will consider
three such breaking terms, a squark mass ($F_K \neq 0$), a gaugino
mass ($F_\tau \neq 0$) and a squark mass mixing ($F_m \neq 0$).
The dependence of the IR effective
theory on the spurion fields is
determined in the N=1 limit by the
dependence on their scalar components, the couplings and masses.
The exact solutions of Seiberg, however, do not provide
sufficient information to take the soft breakings to infinity limit and
obtain results for models with completely decoupled superpartners since
the solutions are only low energy derivative expansions. Higher
dimension terms are suppressed by the strong coupling scale $\Lambda$
and hence in the non-supersymmetric theories there are unknown soft
breaking terms of higher order in $F_S / \Lambda^2$.
A second problem is that squark masses are only generated through the
Kahler potential (the spurion $F_m$ generates a squark mass mixing but it
is unbounded without additional contributions to the masses from the
Kahler sector) via such terms as $|F_S|^2 |Q|^2$ with $S$ a general
spurion. There are no symmetry constraints on these terms so we do not
know whether they occur in the low energy theory or if they do, their
sign. We note that the sign of these terms relative to the sign of the
equivalent terms in the UV theory is crucial.
As a particular example consider theories close to $N_f = 3N_c$ where the
electric theory has a very weak IR fixed point and the magnetic theory a
strongly coupled IR fixed point. We are interested in what happens when
we introduce squark and gaugino masses in the UV magnetic theory. We can
consider the case where these soft breakings are small relative to the
scale $\Lambda$ at which the theory enters it's strongly interacting
conformal phase. We
expect a conformal phase down to the soft breaking scale but can we say
anything about the theory below that scale? The dual squarks in the
weakly coupled IR description only
acquire masses from $F_\tau$ and $F_K$ from the Kahler terms. For
infinitesimal soft breakings we do not expect the weakly coupled nature
of the dual theory at the breaking scale to be disturbed. If these
masses are positive (as investigated in Ref\cite{peskin}) then below the
soft breaking scale the theory behaves like QCD and presumably confines
and breaks chiral symmetries at an exponentially small scale relative to
the soft breaking masses. Alternatively if the masses are negative (as
investigated in Ref\cite{hoker}) then the magnetic gauge group is
higgsed with the possible interpretation in the electric variables of a
dual Meissner type effect. The spurion symmetry arguments are not
sufficient to distinguish between these possibilities.
It should be remarked that there {\it is} a strongly coupled magnetic
theory that corresponds to the introduction of any soft breaking terms
in the electric theory. This is true since we can use the
mapping of
electric to magnetic field variables from the SQCD theory
(which is not known explicitly, but exists in principle)
to
write the soft breaking terms of the simple weakly interacting theory in terms
of the
strongly interacting variables in the IR. The result will be a
complicated mess of relevant higher dimension operators in the strongly
interacting
theory. The subtlety is that if we now run the renormalization group back to
the UV in the
magnetic variables we will, very likely, never recover a weakly
interacting theory. At each step to recover the effective theory at the
lower scale we must add important higher dimension terms. The problem
is therefore to identify which soft breaking terms in the IR electric
description correspond to canonical soft breaking terms in the UV
magnetic theory.
In the next section we shall resolve this problem for the $F_\tau$ and
$F_m$ cases after including a supersymmetric mass that determines the
squark masses at order $F^0$. Then for small soft breakings relative to
$m$ (and $\Lambda$) exact solutions may be obtained.
\section{Controlled N=0 Theories}
To obtain solutions to softly broken N=1 SQCD theories,
we begin by including a supersymmetric mass for the matter fields.
The resulting theories
have a mass gap on the scale $m$ and the induced meson $M_{ij}= Q^i
\tilde{Q}_j$ vev is determined independently of $N_f$ by holomorphy
\begin{equation}\label{Slimit}
M_{ij} = \Lambda^{{3N_c - N_f \over N_c}} (detm)^{1/N_c}\left( {1 \over
m} \right) _{ij} = |M_{ij}| e^{i\alpha}~~~.
\end{equation}
The resulting supersymmetric theories have $N_c$ distinct vacua
corresponding to the $N_c$th roots of unity, $\alpha = 2n\pi/N_c$
(as predicted by the Witten
index). Note that for the theories
with magnetic duals putting masses in for all flavors breaks the dual
gauge group completely. For simplicity henceforth we shall take $m_{ij}$
to be proportional to the identity matrix; in this basis $\langle
M_{ij} \rangle$ is also proportional to the identity matrix.
These massive theories may be softly broken in a controlled fashion.
If the spurion generating the soft breaking enters
the superpotential linearly then we may obtain desirable results when that
spurion's F-component $F \ll m \ll \Lambda$. Any D-term contributions to
the scalar potential take the form $F_X^\dagger F_Y$ with $X$ and $Y$
standing for generic fields or spurions. In the supersymmetric limit all
F-components are zero and will grow as the vacuum expectation value of the
soft breaking spurion.
These Kahler terms are therefore higher order in
the soft breaking parameter than the linear term from the
superpotential. The unknown corrections to the squark masses in the
theory are subleading to the masses generated by the supersymmetric mass
term and hence we may determine the potential minima at lowest order.
\subsection{Squark Mass Mixing}
The first model we consider includes the bare squark mixing term
\begin{equation}
Re(F_{m \, ij} ~Q_i \tilde{Q}_j)
\end{equation}
which is generated from the superpotential. Again for simplicity we will
take $F_{mij}$ to be diagonal with degenerate eigenvalues in the basis
in which $m_{ij}$ is diagonal.
The form of the effective
theory is governed by the symmetries in (\ref{sym}) which determine that the
superpotential of the theory is not renormalized. The soft breaking term
is therefore also not renormalized and
is the leading term in an expansion in
$m/\Lambda$. For $F_m \ll m \ll \Lambda$ we find that there are the
$N_c$ minima of the SQCD theory given by the values of $M_{ij}$ in
(\ref{Slimit})
and distinguished by their contribution to the potential
\begin{eqnarray}\label{Fmpotential}
-Re Tr[ F_{m} M_{ij}]& = & - N_f |F_m| |M|
\cos([\theta_0 + (N_f - N_c) \theta_m +
N_c \theta_f + 2n \pi]/N_c])\\
& = & - N_f |F_m| |M| \cos([\theta_{phys}+ 2n \pi]/N_c)~~~. \nonumber
\end{eqnarray}
Freezing the spurion $F_m$ explicitly breaks $U(1)_R$ and introduces
dependence on the $\theta$ angle.
$\theta_{phys}$ is the correct combination of phases on $m$, $F_m$ and the
bare $\theta$ angle. To see this in the bare Lagrangian we
may use the anomalous $U(1)_A$ symmetry
to rotate any
phases on $F_m$ onto $m$ and into the $\theta F \tilde{F}$ term.
Then using the anomalous $U(1)_R$ symmetry under which $Q_i$
transforms with charge 0 we may rotate the resulting phase on $m$ into
the $\theta$ angle as well. The resulting $\theta$ angle is the physical
$\theta$ angle in which the physics is $2 \pi$ periodic:
\begin{equation}
\label{phth}
\theta_{phys} = \theta_0 + (N_f - N_c) \theta_m + N_c \theta_f
\end{equation}
We can also understand the form of (\ref{phth}) as follows.
Once the $U(1)_R$ symmetry
is explicitly broken by $f_m$ a gaugino mass is generated by radiative
effects. We can think of $\theta_{phys}$ as generated by the effective
phases on the quark and gaugino masses. The gaugino mass is generated
by a perturbative graph with a quark-squark loop. The result is of the
form $F_m / m$, leading to an effective phase which is $\theta_f -
\theta_m$.The effective gaugino phase then
appears in (\ref{phth})
with an anomaly factor from $C_2 (R)$ of $N_c$ rather than $N_f$.
The equivalent effective superpotential term is of the form
\begin{equation}
ln [ m ]~ WW~ \vert_F,
\end{equation}
which yields another contribution to the potential when the
gauginos condense. Using the Konishi anomaly \cite{KA},
one can see that
this term has the same form as (\ref{Fmpotential}).
The resulting potential (\ref{Fmpotential})
distinguishes the $N_c$ vacua. For $\theta_{phys} = 0$ the $n=0$ vacua
is the true minima.
\vspace{.4cm}
$\left. \right.$ \hspace{-0.4in}\ifig\prtbdiag{}
{\epsfxsize 12truecm\epsfbox{angles.eps}} \vspace{-1.7cm}
\begin{center}
Fig.1\,: First order phase transition as $\theta_{phys}$ is
varied from 0 to $\pi$.
\end{center}
As $\theta_{phys}$ passes through $\pi$ the $n=0,N_c -1$
vacua become degenerate and there is a first order phase transition. Then
as $\theta_{phys}$ moves through (odd)$\pi$ there are subsequent first order
phase transitions at which the SQCD minima interchange.
\subsection{Gaugino Mass}
In the UV theory we may induce a gaugino mass through a non zero
F-component of the gauge coupling $\tau$
\begin{equation}
{1 \over 8 \pi} Im [ F_{\tau} \lambda \lambda]
\end{equation}
In the IR theory $\tau$ enters through the strong interaction scale
$\Lambda$ which again occurs linearly in the superpotential of the
theory. Taking $F_{\tau} \ll m \ll \Lambda$ we again may determine
the vacuum structure. The IR superpotential terms compatible with the
symmetries of the theory involving $\Lambda$ are
\begin{equation}
Re[ m M_{ij} + ({\rm det} M_{ij})^ { 1/(N_f - N_c) } \Lambda^{(3N_c-N_f)
/ (N_c-N_f)}]
\end{equation}
where the final term results from non-perturbative effects in the broken gauge
group. At lowest order in perturbation theory the vev of $M_{ij}$ is
given by (\ref{Slimit}) which also contains $\Lambda$ and hence has a
non-zero F-component. Including $F_\tau$ and performing the superspace
integral we obtain up to a coefficient the following corrections
to the potential that
break the degeneracy between the $N_c$ SQCD vacua
\begin{eqnarray}
\label{gpot}
\Delta V & = & - Re\left[ m^{N_f/N_c} i F_\tau \Lambda^{(3N_c-N_f)/
N_c}\right]\\
& = \nonumber & - \left|m^{N_f/N_c} F_\tau \Lambda^{(3N_c-N_f)/
N_c}\right| \cos[ ~ \theta_{phys}/N_c ~+~ \alpha ~]
\end{eqnarray}
where again $\alpha$ are the $N_c$th roots of unity and
$\theta_{phys}$ is the physical theta angle in which the
physics must be $2 \pi$ periodic. It may be obtained by again making rotations
with the anomalous $U(1)_A$ and $U(1)_R$ symmetries
\begin{equation}
\theta_{phys} ~=~ \theta_0 ~+~ N_c ( \theta_{F_\tau} +
\pi /2) ~+~ N_f \theta_m
\end{equation}
The factor of $\pi/2$ occurs as a
result of the discrepancy between the
phase of $F_\tau$ and that of the canonical definition of the
gaugino mass.
There is also an additional contribution to the vacuum energy
arising from the gaugino condensate. Using the Konishi
anomaly \cite{KA}, we see that it has the same form as
(\ref{gpot}).
The supersymmetry breaking contributions again break the degeneracy
between the $N_c$ supersymmetric vacua. There are again phase transitions
as $\theta_{phys}$ is varied, occurring
at $\theta_{phys} ~=~ $(odd)$\pi$.
\section{Discussion}
We have investigated some examples where controlled, low-energy descriptions of
softly broken massive SQCD may be obtained, despite the lack of supersymmetry.
The models we studied are obtained by the inclusion of
soft breaking masses from spurions occuring linearly in the
superpotential. Examples of such soft
breaking terms are gaugino masses and squark mass mixings.The soft
breaking corrections to the potential distinguish between the $N_c$
vacua of SQCD at a generic value of theta angle. At the special values of
$\theta_{phys} = $(odd)$\pi$ there are first order phase transitions at which
two of the $N_c$ vacua interchange.
This behavior can be compared
with the theta angle dependence of the QCD chiral Lagrangian \cite{chiral}
for which there are $N_f$ distinct vacua which interchange through first
order phase transitions at $\theta =$(odd)$\pi$. This difference in the
number of vacua between the softly broken theories and QCD would prohibit us
from seeing any sign of a smooth transition between the two theories
(one might hope that the $M_{ij}$ vev might smoothly map to the quark
condensates of QCD for example) even if we were able to begin to
take the squark and gaugino masses towards infinity. There is however
one conclusion for QCD that we can tentatively draw from this
analysis. In these models the form of the confined effective
theory changes smoothly with the theta angle and there is no sign of a
break down of confinement as suggested in \cite{schierholz}. This lends some
support to the assumption \cite{chiral} that the chiral Lagrangian remains
the correct discription of QCD in the IR even at non-zero theta.
\vspace{3cm}
\noindent {\Large \bf Acknowledgements}
NE would like to thank R. Sundrum for useful discussions. This work
was supported by DOE contracts DE-AC02-ERU3075 and DE-FG02-91ER40676.
\newpage
\baselineskip=1.6pt
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,660 |
{"url":"http:\/\/rin.io\/category\/camel\/","text":"## CAMEL Poster\n\nThe poster I\u2019m using to present my research creating CAMEL is finally finished\u00a0(full-size version: CAMEL_poster).\n\nIn order to condense the entirety of my paper into a viewer-friendly poster, I decided to make diagrams to describe the program\u2019s inner workings in layman\u2019s terms.\nIf you like the format of this poster:\u00a0all code used in this project is on github.\n\n#### Introduction to CAMEL\n\n\u2022 machine learning program that uses Braille as a language\u00a0platform. CAMEL is an acronym of ContextuAl MachinE\u00a0Learning.\n\u2022 uses context of unknown symbols to deduce meaning and\u00a0compress information.\n\u2022 provided the meaning of an initial set of symbols (a\u00a0dictionary, or dict). CAMEL deduces meanings of\u00a0unknowns and adds these meanings to the dict.\n\u2022 grows more acurrate as the dict increases in size andoptions. Some symbols differ in meaning depending on\u00a0their context. These translation options are stored in thedict in the form of Map[String, TranslationOptions].\n\n#### What is Grade 2 Braille\n\n\u2022 As English words are composed of letters, Braille words are composed of Braille cells.\n\u2022 Contractions are special characters used to reduce the length of words.\n\u2022 Some contractions stand for a whole word. \u00a0For example: \u2018for\u2019 = \\braille{{for}}; \u2018and\u2019 = \\braille{{and}}; \u2018the\u2019 = \\braille{{the}}.\n\u2022 Other contractions stand for a group of letters within a word. In the example below, the contraction \u2018ing\u2019 is used in the word \u2018sing\u2019 and as an ending in the word \u2018playing.\u2019 {ing} ; \u2018s\u2019 + {ing} = s{ing}; \u2018play\u2019 + {ing} = play{ing}\n\u2022 Grade 1 Braille is uncontracted Braille.\n\n#### Binary Braille\n\n\u2022 The Braille alphabet is depicted by a cell that contains six raised\/flat dots, numbered one through six beginning with the dot in the upper left-hand corner with the number descending the columns (see figure below).\n\u2022 To simplify the calculation, I let \u201c0\u201d = flat, \u201c1\u201d = raised.\n\u2022 The 3\u00d72 matrix (Braille cell) is represented as a 1\u00d76 bitstring (Binary Braille).\n\n\u2022 Thus, the letter \u201cc\u201d\n\n#### String Processing Method\n\nCAMEL deduces the complex grammar rules of Grade 2\u00a0Braille given partially translated text.\n\nCAMEL learns new symbols by taking 2 input text files (Braille text and corresponding English text), and analyzing them until all unknowns are identified, their meanings are found, and said symbols and their meanings are added to the dictionary.\n\n#### Methods of Tagging and Text Extraction\n\nCAMEL must Tag Unknowns & Compare to English(Extract Chunks) to infer symbol meaning. Four different tag types were used: end, front, mid, and full-word.\n\nBelow are examples of how these different types of tags were each used to extract meaning.\n\n#### Using Contracted Braille as a Platform\n\nAn example of this process infers the symbols that\u00a0represent en and in using the word penguin (contracted to\u00a0p{en}gu{in} in Grade 2 Braille).\n\n### Safety of Community\n\n\u2022 commercial application in\u00a0development that will\u00a0prevent future mislabeling,\u00a0such as this sign:\u00a0labeled \u201cElectrical Room\u201d the Braille translates to \u201cstairwell\u201d\n\n### Proof of Concept\n\n\u2022 1st successful automated program that learns compressed Braille\n\u2022 translation system is effective for arbitrary symbol systems\n\u2022 language platform easily changed\n\n#### Acknowledgements\n\nThe most difficult part of the poster was creating a mature acknowledgements section; I was very tempted to thank\u2026\n\n\u2022 insomnia for allowing me to code at 2 am\n\u2022 coffee for powering me through the day after coding at 2am\n\u2022 my friend for introducing me to the instant protein-rich \u201cmeal\u201d that is trail mix\n\u2022 my research partner that refused to code in C++, which forced me to learn\u00a0Python\n\u2022 Guido van Rossum for inventing said language\n\u2022 my parents for putting up with me when I immerse myself in research\n\u2022 my friends for putting up with me when I stop in the middle of a conversation to write down ideas and\/or zone-out thinking\n\nAlthough these were essential to my completion of this project,\u00a0I think it\u2019s best to not include these points in the poster.\n\n## CAMEL paper\n\nI began working on CAMEL (Contextual Machine Learning Through the Analysis and Chunking of\u00a0Partially Translated Grade 2 Braille) when I saw this atrocity: the Braille below translates to \u201cSTAIRWELL\u201d Most sighted people don\u2019t know Braille (to my dismay- the grammar is beautiful), this gave me the idea to write an optical Braille reader app for the sighted: the user holds their Android camera up to a sign\u00a0and it\u00a0automatically translates the Braille into English.\n\nAs I sat down to code\u00a0this, I realized that I\u2019d have to hard code a dictionary of Grade 2 Braille (a grammatically complex language). I have a\u00a0deep disgust\u00a0for hard coding\u00a0$\\Rightarrow$ CAMEL is the program I wrote to automate the creation of a Grade 2 Braille dictionary. All code used in this project is on github.\n\nThis program is based on\u00a0contextual machine learning, so I named\u00a0my project CAMEL (ContextuAl MachinE Learning). The title of my paper is a mouthful, because I\u2019m unsure of how to shorten it while maintaining clarity:\u00a0CAMEL\n\n## Introduction to Braille\n\nI\u2019m currently researching machine learning using Braille as my language platform. There is a Research Symposium for Mason\u2019s College of Science at the end of April.\n\nBelow is a sneak peak of my paper. I\u2019ve just finished my \u201cIntroduction to Braille\u201d section:","date":"2018-02-25 19:19:49","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\": 0.32818862795829773, \"perplexity\": 4022.6623687944398}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891816912.94\/warc\/CC-MAIN-20180225190023-20180225210023-00329.warc.gz\"}"} | null | null |
Hyalomis espia is a moth of the subfamily Arctiinae. It was described by Paul Dognin in 1897. It is found in Ecuador.
References
Euchromiina
Moths described in 1897 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,668 |
Gyula Tóth (16 April 1927 in Salgótarján – 18 March 2001 in Budapest) was a Hungarian wrestler who competed in the 1956 Summer Olympics and in the 1960 Summer Olympics.
References
External links
1927 births
2001 deaths
Olympic wrestlers of Hungary
Wrestlers at the 1956 Summer Olympics
Wrestlers at the 1960 Summer Olympics
Hungarian male sport wrestlers
Olympic bronze medalists for Hungary
Olympic medalists in wrestling
Medalists at the 1956 Summer Olympics
People from Salgótarján
Sportspeople from Nógrád County
20th-century Hungarian people | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,140 |
O divórcio (do termo latino divortium, derivado de divertĕre, "separar-se") é o rompimento legal e definitivo do vínculo de casamento civil.
Em outras palavras, o divórcio pode ser definido como sendo o rompimento legal e definitivo do vínculo de um casamento civil, formalizado através da justiça e/ou de um cartório de registro civil. Sendo o divórcio amigável e/ou litigioso, a separação do casal acontece.
O processo legal de divórcio pode envolver questões como atribuição de pensão de alimentos, regulação de poder paternal, relação ou partilha de bens, regulação de casa de morada de família, embora estes acordos sejam complementares ao processo principal.
Em algumas jurisdições, caso do Brasil atual, não é exigida a invocação da culpa do outro cônjuge. Ainda assim, mesmo nos ordenamentos jurídicos que adaptaram o sistema do divórcio "sem culpa", é tido em conta o comportamento das partes na partilha dos bens, regulação do poder paternal, e atribuição de alimentos.
Na maioria das jurisdições, o divórcio carece de ser emitido ou certificado por um tribunal para surtir efeito, onde pode ser bastante estressante e caro a litigância. Outras abordagens alternativas, como a mediação e divórcio colaborativo podem ser um caminho mais assertivo. Em alguns países, como Portugal e Brasil, o divórcio amigável pode até ser realizado numa conservatória de registo civil (Portugal) ou tabelionato de notas (Brasil), simplificando bastante o processo.
A anulação não é uma forma de divórcio, mas apenas o reconhecimento, seja a nível religioso, seja civil da falha das disposições no momento do consentimento, o que tornou o casamento inválido; reconhecer o casamento nulo é a mesma coisa que reconhecer que nunca tenha existido.
Num divórcio, o destino dos bens do casal fica sujeito ao regime de bens adotado na altura do casamento, e que geralmente em todos os países são: separação de bens, bens adquiridos, ou comunhão de adquiridos.
Os países onde mais ocorrem pedidos de rompimento do matrimônio são: Estados Unidos, Dinamarca e Bélgica, com índices entre 55% e 65%. Em contraponto, os países com menos incidência de separação são países extremamente católicos como República da Irlanda e Itália, com números abaixo de 10%. Atualmente, apenas as Filipinas e o Vaticano não permitem o divórcio em seu sistema legal. Por outro lado, em maio de 2011, Malta votou a favor da inclusão do divórcio em sua legislação por meio de um referendo não vinculante, aprovando posteriormente sua legalização no Parlamento durante o mês de julho, sendo o último país do mundo em legalizá-lo depois do Chile (que o aprovou em 2004). No Parlamento das Filipinas, entretanto, um debate sobre uma lei potencial que incorpora em seu sistema legal começou no final do primeiro semestre de 2011.
Quanto ao poder paternal (pátrio poder), ele assume cada vez maior importância no divórcio, sendo atribuído em 95% das vezes às mulheres, segundo dados oficiais de 2003 quer no Brasil, quer Portugal, Espanha, e América do Norte.
Divórcio no Brasil
O casamento introduzido no Brasil no tempo do Império era regido pelas normas da Igreja Católica e o maior dogma referia-se à sua indissolubilidade. Até mesmo nas hipóteses em que se autorizava o divortium quoad thorum et habitationem ("divórcio de cama e habitação"), não havia rompimento do vínculo matrimonial. O que ocorria era apenas a separação de corpos.
Com a República e a laicização do Estado através do Decreto 119-A, de 7 de janeiro de 1890, veio o instituto do casamento a perder o caráter confessional.
O casamento civil foi instituído no Brasil através do Decreto nº 181, de 24 de janeiro de 1890, que não tratava da dissolução do vínculo conjugal, mas previa a separação de corpos (também chamado de divórcio, contrapondo-se ao divortium quoad thorum et habitationem, que era regido pelas leis da Igreja).
As causas aceitáveis a separação de corpos eram:
adultério;
sevícia ou injúria grave;
abandono voluntário do domicílio conjugal por dois anos contínuos;
mútuo consentimento dos cônjuges, se fossem casados há mais de dois anos.
Foram apresentadas propostas divorcistas, sem êxito.
No Código Civil Brasileiro de 1916 foi introduzido o desquite (judicial ou amigável), como forma de pôr fim à sociedade conjugal. A sentença do desquite apenas autorizava a separação dos cônjuges, pondo termo ao regime de bens. Porém, o vínculo matrimonial permanecia.
A enumeração taxativa das causas de desquite foi repetida: adultério, tentativa de morte, sevícia ou injúria grave e abandono voluntário do lar conjugal (artigo 317). Foi mantido o desquite por mútuo consentimento (art. 318).
Assim, esse instituto criado em 1916 nada mais era do que o divórcio regido pelo Decreto n. 181/1890, mas com outra nomenclatura. Segundo Sílvio Rodrigues:
"A palavra 'desquite' foi introduzida no direito brasileiro com o Código Civil de 1916. O Decreto n. 181/1890, que instituiu entre nós o casamento civil, ainda utilizava a expressão divórcio, embora não o admitisse com o efeito de romper o vínculo conjugal. De forma que o Código Civil, fora modificações menores, nada inovou ao direito anterior, a não ser o nome do instituto."
O divórcio foi instituído oficialmente com a emenda constitucional número 9, de 28 de junho de 1977, regulamentada pela lei 6515 de 26 de dezembro do mesmo ano. A chamada Lei do Divórcio passou a designar o desquite como separação judicial, revogando o Capítulo I e parte do Capítulo II do Título IV do Código Civil de 1916 (artigos 315 a 328) que tratava da Dissolução da Sociedade Conjugal e Proteção da Pessoa e dos Filhos. A lei estabeleceu a modalidade de divórcio-conversão, isto é, depois de separado judicialmente por três anos, o casal poderia requerer a conversão da separação em divórcio. Abria também a possibilidade do divórcio direto, mas somente para os casais separados de fato há mais de cinco anos em 28 de junho de 1977. É importante destacar que esse divórcio era admitido somente uma única vez.
A Constituição Federal de 1988, no seu art. 226, §6º, alterou profundamente o divórcio: reduziu o prazo para conversão de três anos para um ano; admitiu o divórcio direto em qualquer época e não somente para separações de fato anteriores à EC n° 09/77; reduziu de cinco para dois anos o prazo de separação de fato e não colocou limites ao número de divórcios, que era limitado pelo artigo 38 da lei 6.515/77 a apenas uma vez. Art. 226.(...) §6º. O casamento civil pode ser dissolvido pelo divórcio, após prévia separação judicial por mais de um ano nos casos expressos em lei, ou comprovada separação de fato por mais de dois anos.
Com a lei 11.441 de 4 de janeiro de 2007, o divórcio e a separação consensuais podem ser requeridos por via administrativa, ou seja, não é necessário ingressar com um ação judicial para o efeito, bastando comparecer a um tabelionato de notas e apresentar o pedido. Tal facilidade só é possível quando o casal não tiver filhos menores de idade ou incapazes.
A Emenda Constitucional nº 66/2010 trouxe significativas mudanças ao § 6º do artigo 226 da Constituição Brasileira. Segundo a regra anterior, o divórcio só poderia ocorrer quando o casal já estivesse separado judicialmente por mais de um ano ou separado de fato por mais de dois anos. Com a emenda, o único fator imprescindível é a vontade exclusiva de um ou de ambos os cônjuges.
Um estudo do Banco Interamericano de Desenvolvimento associou a expansão do consumo de telenovelas a um aumento no número de divórcios no Brasil.
Divórcio em Portugal
O divórcio foi legalizado em 1910, menos de um mês após a proclamação da República, com o Decreto de 3 de Novembro daquele ano. Marido e mulher terão, desde então, o mesmo tratamento legal, quanto aos motivos de divórcio e aos direitos sobre os filhos. A esposa deixa de ter o dever de obedecer ao marido. O adultério é crime, mas não se distingue o cometido pela mulher ou pelo homem. Em 1911, o número de divorciados era de 2 685. Contudo, a Concordata assinada com o Vaticano em 1940 retira, dos que se casem na Igreja Católica, o direito de se divorciar - restrição que seria revogada em 1975.
Atualmente, a lei prevê duas modalidades de divórcio: o divórcio por mútuo consentimento e o divórcio sem consentimento do outro cônjuge (divórcio litigioso).
No primeiro caso, a competência para decretar o divórcio cabe, em princípio, às conservatórias do registo civil e, conjuntamente com o divórcio, são reguladas as questões conexas, como sejam o exercício das responsabilidades parentais relativamente aos filhos menores, a atribuição da casa de morada de família, a fixação de uma pensão de alimentos para o cônjuge que deles careça e poderá também ser efectuada a partilha dos bens comuns.
No caso do divórcio litigioso, a competência para o decretamento é dos tribunais e exige-se que o pedido de divórcio tenha um dos seguintes fundamentos: a separação de facto por um ano consecutivo; ou a alteração das faculdades mentais do outro cônjuge; a ausência, sem que do ausente haja notícias, por tempo não inferior a um ano; quaisquer outros factos que, independentemente da culpa dos cônjuges, mostrem a ruptura definitiva do casamento.
No que respeita aos custos do processo, o divórcio por mútuo consentimento realizado numa conservatória do registo civil paga, de emolumentos, 625 euros ou 280 euros, consoante haja ou não partilha de bens. Já nos casos da acção judicial de divórcio (litigioso), o valor mínimo da taxa de justiça a pagar é de 306,00 euros por cada parte. Este pagamento só será devido, porém, após a sentença.
Em 2016 houve em Portugal 22 649 divórcios, menos 1 037 face ao ano anterior e menos 4 411 relativamente a 2011.
Divórcios históricos
Abaixo, os valores de alguns divórcios de casais famosos. Os valores estão expressos em dólares estadunidenses.
Príncipe Charles e Lady Diana: 28 milhões
Donald Trump e Ivana Trump: 50 milhões
Kenny Rogers e Marianne: 60 milhões
Kevin Costner e Cindy: 80 milhões
Julio Bozano e Iva: 100 milhões
Steven Spielberg e Amy Irving: 89,97 milhões
Visão religiosa
Cada religião tem a sua própria maneira de encarar o divórcio. Para o catolicismo, este não é possível, uma vez que na Bíblia encontra-se a frase Quod ergo Deus coniunxit, homo ne separet (Mc 10,2-16).
No judaísmo, por sua vez, é apenas possível o divórcio por parte do homem, apoiando-se na Torah:
O islamismo reconhece, tecnicamente, o direito de ambos os parceiros de pedirem o divórcio, embora para a mulher o processo seja consideravelmente mais complicado: enquanto para o homem basta repetir três vezes "eu te repudio", para as mulheres é exigida alguma falta grave do marido (em teoria, ela poderia pedir o divórcio pelo simples fato de não querer se manter mais na casa, através da Khula, todavia isto é na prática impossível nas sociedades conservadoras).
Causas
Um estudo anual no Reino Unido feito pelo consultor de gestão Grant Thornton, estima as principais causas proximais do divórcio com base em pesquisas de advogados matrimoniais.
As principais causas em 2004 foram:
Adultério; Sexo extraconjugal; Infidelidade – 27%
Violência doméstica – 17%
Crise de meia-idade – 13%
Vícios, por exemplo, alcoolismo e jogo – 6%
Workaholismo – 6%
Outros fatores - 31%
Consequências
As consequências de uma vida conjugal arruinada vão desde o nível físico até o setor emocional, não somente do casal, mas também dos que o cercam. O casamento está estatisticamente relacionado a um ganho de peso, mas estudos dizem que o divórcio também pode aumentar significativamente o peso corporal. Um estudo realizado em Chicago e contando com a participação de 8 652 pessoas com idades entre 51 e 61 anos também detectou que os divorciados têm 20% a mais de chances de desenvolver doenças crônicas, como o câncer, do que aqueles que nunca se casaram.
Se o casal sofre psicologicamente e fisicamente, os filhos também não ficam ilesos. Portanto, consequência para as crianças existem, mais ou menos, de acordo com vários fatores, incluindo a própria resolução favorável da separação para os pais, a idade das crianças e o seu grau de desenvolvimento. Poucas crianças demonstram sentirem-se aliviadas com a decisão do divórcio. Na idade de 8 a 12 anos, em geral, a criança reage com raiva franca de um ou de ambos os pais, por terem causado a separação. Por vezes, demonstram ansiedade, solidão e sentimentos de humilhação por sua própria impotência diante do ocorrido. O desempenho escolar e o relacionamento com colegas podem ter prejuízo nesta fase. Já os adolescentes sofrem com o divórcio muitas vezes com depressão, raiva intensa ou com comportamentos rebeldes e desorganizados.
Divórcio no mundo
Na República das Maldivas registam-se mais de 10.97 divórcios por ano, por cada mil habitantes, sendo o valor mais alto do mundo.
Ver também
Venda de esposas, prática na Inglaterra medieval
Desquite
Ligações externas
The superstition of divorce : Chesterton, G. K. (Gilbert Keith), 1874-1936 : Free Download, Borrow, and Streaming : Internet Archive
Direito de família | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,429 |
from collections import namedtuple
from heapq import heappush, heappop
from itertools import cycle
import six
from six.moves import xrange, zip
from threading import Condition
import sys
from cassandra.cluster import ResultSet
import logging
log = logging.getLogger(__name__)
ExecutionResult = namedtuple('ExecutionResult', ['success', 'result_or_exc'])
def execute_concurrent(session, statements_and_parameters, concurrency=100, raise_on_first_error=True, results_generator=False):
"""
Executes a sequence of (statement, parameters) tuples concurrently. Each
``parameters`` item must be a sequence or :const:`None`.
The `concurrency` parameter controls how many statements will be executed
concurrently. When :attr:`.Cluster.protocol_version` is set to 1 or 2,
it is recommended that this be kept below 100 times the number of
core connections per host times the number of connected hosts (see
:meth:`.Cluster.set_core_connections_per_host`). If that amount is exceeded,
the event loop thread may attempt to block on new connection creation,
substantially impacting throughput. If :attr:`~.Cluster.protocol_version`
is 3 or higher, you can safely experiment with higher levels of concurrency.
If `raise_on_first_error` is left as :const:`True`, execution will stop
after the first failed statement and the corresponding exception will be
raised.
`results_generator` controls how the results are returned.
If :const:`False`, the results are returned only after all requests have completed.
If :const:`True`, a generator expression is returned. Using a generator results in a constrained
memory footprint when the results set will be large -- results are yielded
as they return instead of materializing the entire list at once. The trade for lower memory
footprint is marginal CPU overhead (more thread coordination and sorting out-of-order results
on-the-fly).
A sequence of ``ExecutionResult(success, result_or_exc)`` namedtuples is returned
in the same order that the statements were passed in. If ``success`` is :const:`False`,
there was an error executing the statement, and ``result_or_exc`` will be
an :class:`Exception`. If ``success`` is :const:`True`, ``result_or_exc``
will be the query result.
Example usage::
select_statement = session.prepare("SELECT * FROM users WHERE id=?")
statements_and_params = []
for user_id in user_ids:
params = (user_id, )
statements_and_params.append((select_statement, params))
results = execute_concurrent(
session, statements_and_params, raise_on_first_error=False)
for (success, result) in results:
if not success:
handle_error(result) # result will be an Exception
else:
process_user(result[0]) # result will be a list of rows
"""
if concurrency <= 0:
raise ValueError("concurrency must be greater than 0")
if not statements_and_parameters:
return []
executor = ConcurrentExecutorGenResults(session, statements_and_parameters) if results_generator else ConcurrentExecutorListResults(session, statements_and_parameters)
return executor.execute(concurrency, raise_on_first_error)
class _ConcurrentExecutor(object):
max_error_recursion = 100
def __init__(self, session, statements_and_params):
self.session = session
self._enum_statements = enumerate(iter(statements_and_params))
self._condition = Condition()
self._fail_fast = False
self._results_queue = []
self._current = 0
self._exec_count = 0
self._exec_depth = 0
def execute(self, concurrency, fail_fast):
self._fail_fast = fail_fast
self._results_queue = []
self._current = 0
self._exec_count = 0
with self._condition:
for n in xrange(concurrency):
if not self._execute_next():
break
return self._results()
def _execute_next(self):
# lock must be held
try:
(idx, (statement, params)) = next(self._enum_statements)
self._exec_count += 1
self._execute(idx, statement, params)
return True
except StopIteration:
pass
def _execute(self, idx, statement, params):
self._exec_depth += 1
try:
future = self.session.execute_async(statement, params, timeout=None)
args = (future, idx)
future.add_callbacks(
callback=self._on_success, callback_args=args,
errback=self._on_error, errback_args=args)
except Exception as exc:
# exc_info with fail_fast to preserve stack trace info when raising on the client thread
# (matches previous behavior -- not sure why we wouldn't want stack trace in the other case)
e = sys.exc_info() if self._fail_fast and six.PY2 else exc
# If we're not failing fast and all executions are raising, there is a chance of recursing
# here as subsequent requests are attempted. If we hit this threshold, schedule this result/retry
# and let the event loop thread return.
if self._exec_depth < self.max_error_recursion:
self._put_result(e, idx, False)
else:
self.session.submit(self._put_result, e, idx, False)
self._exec_depth -= 1
def _on_success(self, result, future, idx):
future.clear_callbacks()
self._put_result(ResultSet(future, result), idx, True)
def _on_error(self, result, future, idx):
self._put_result(result, idx, False)
@staticmethod
def _raise(exc):
if six.PY2 and isinstance(exc, tuple):
(exc_type, value, traceback) = exc
six.reraise(exc_type, value, traceback)
else:
raise exc
class ConcurrentExecutorGenResults(_ConcurrentExecutor):
def _put_result(self, result, idx, success):
with self._condition:
heappush(self._results_queue, (idx, ExecutionResult(success, result)))
self._execute_next()
self._condition.notify()
def _results(self):
with self._condition:
while self._current < self._exec_count:
while not self._results_queue or self._results_queue[0][0] != self._current:
self._condition.wait()
while self._results_queue and self._results_queue[0][0] == self._current:
_, res = heappop(self._results_queue)
try:
self._condition.release()
if self._fail_fast and not res[0]:
self._raise(res[1])
yield res
finally:
self._condition.acquire()
self._current += 1
class ConcurrentExecutorListResults(_ConcurrentExecutor):
_exception = None
def execute(self, concurrency, fail_fast):
self._exception = None
return super(ConcurrentExecutorListResults, self).execute(concurrency, fail_fast)
def _put_result(self, result, idx, success):
self._results_queue.append((idx, ExecutionResult(success, result)))
with self._condition:
self._current += 1
if not success and self._fail_fast:
if not self._exception:
self._exception = result
self._condition.notify()
elif not self._execute_next() and self._current == self._exec_count:
self._condition.notify()
def _results(self):
with self._condition:
while self._current < self._exec_count:
self._condition.wait()
if self._exception and self._fail_fast:
self._raise(self._exception)
if self._exception and self._fail_fast: # raise the exception even if there was no wait
self._raise(self._exception)
return [r[1] for r in sorted(self._results_queue)]
def execute_concurrent_with_args(session, statement, parameters, *args, **kwargs):
"""
Like :meth:`~cassandra.concurrent.execute_concurrent()`, but takes a single
statement and a sequence of parameters. Each item in ``parameters``
should be a sequence or :const:`None`.
Example usage::
statement = session.prepare("INSERT INTO mytable (a, b) VALUES (1, ?)")
parameters = [(x,) for x in range(1000)]
execute_concurrent_with_args(session, statement, parameters, concurrency=50)
"""
return execute_concurrent(session, zip(cycle((statement,)), parameters), *args, **kwargs)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,465 |
Qualified Agent for EMS, DHL, COPA COURIER and IB EXPRESS!
IB EXPRESS: Advantage of easy, fast and secured in Latin America Countries.
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DHL: One of the world's largest courier companies, offering outstanding express service by its wide logistics network and powerful transportation sources. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,332 |
est un film japonais réalisé par Kōzaburō Yoshimura et sorti en 1947.
Synopsis
La veille de la vente de sa propriété, une grande famille aristocratique organise un bal dans lequel maîtres et serviteurs, désormais égaux, se trouvent tous réunis. Le châtelain, ruiné, essaie de se suicider mais il est sauvé par sa fille qui lui redonne le goût de la vie...
Fiche technique
Titre du film : Le Bal de la famille Anjo
Titre original :
Réalisation : Kōzaburō Yoshimura
Scénario : Kaneto Shindo
Photographie : Toshio Ubukata
Musique : Chūji Kinoshita
Production : Takeshi Ogura
Société de production : Shōchiku
Pays d'origine :
Langue originale : japonais
Format : noir et blanc - 1,37:1 - 35 mm - son mono
Genre : drame
Durée : 89 minutes (métrage : 10 bobines - )
Dates de sortie :
:
:
Distribution
Osamu Takizawa : Tadahiko, le châtelain
Masayuki Mori : Masahiko, son premier fils
Setsuko Hara : sa deuxième fille, Atsuko
Yumeko Aizome : Akiko, la première fille
Masao Shimizu : Ryūzaburō Shinkawa
Keiko Tsushima : Yōko Shinkawa
Récompenses
Prix Kinema Junpō du meilleur film en 1948
Prix Mainichi du meilleur acteur Masayuki Mori en 1948
Commentaire
, dit Tadao Satō, puis il poursuit : .
Le Bal de la famille Anjo est, sur ce thème, l'un des films les plus tragiques. Une famille aristocratique est, ici, dépossédée de ses titres par l'abolition de la noblesse et ruinée par la réforme fiscale. Le bal est .
L'œuvre de Kōzaburō Yoshimura, aux accents tchekhoviens, ne manifeste aucune sympathie particulière à l'égard des deux classes. .
Setsuko Hara, la star japonaise qui a le mieux symbolisé l'esprit d'une époque, incarne, à nouveau, affectée par les bouleversements de l'après-guerre. Tout en sauvant son père du suicide, elle partage pleinement les évolutions en cours et va jusqu'à aider son père à se marier avec son ancienne maîtresse, une geisha.
Notes et références
Liens externes
Film japonais sorti en 1947
Film en japonais
Film dramatique japonais
Film de danse
Film de Shōchiku
Film réalisé par Kōzaburō Yoshimura | {
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Andrew McMahon to Delay Upcoming Solo Plans
Courtesy of blog.thephoenix.com
Indie-pop singer and former lead singer of Jack's Mannequin and Something Corporate, Andrew McMahon, has delayed his plans of becoming a solo artist. His four-song EP, The Pop Underground, in 2013 received mostly favorable reviews. As the lone single, "Synesthesia," though much different from past work, has been praised for being genuinely personal and one of the most reflective songs that he has ever released.
In July 2014, the lead singer shared on his website that Andrew McMahon in the Wilderness would be his latest musical project, and can expect an album release and tour to follow shortly thereafter. Within a week, McMahon, James Flannigan, and his longtime partner, Anders Grahn, released the first album single entitled, "Cecilia and the Satellite," in honor of his daughter who born just two weeks later.
On July 11, 2014, McMahon posted on his website that the next progression in his musical career would be to tour and release music under the moniker of. A few days later, the first single from the album was released; McMahon described his new work as a more refined version of what he was doing during The Pop Underground. Andrew McMahon in the Wilderness was officially released October 14, 2014.
McMahon was recently asked by Lehigh Valley Music about his decision to postpone his solo career so soon, "… I feel like it was a combination of the two things. The Wilderness for me is as much a project title as it is a reference to any band. I think sort of the stage I was working my way through when all of this came to light – including the EP – was sort of this wilderness moment, and whether or not I hang onto it for many record to come or if it's just this one."
He also went on to say that after his time working on the NBC television show Smash that he has considered working on a musical, although he would not share any further information on its development. The Somebody Told Me: A Night of Songs and Stories tour with McMahon, Brett Dennen, Glen Phillips, Jonathan Kingham, will be playing in Ann Arbor (Michigan) at the Ark, before leaving for the UK at the end of February.
Tags: Anders Grahn, Andrew McMahon, Andrew McMahon in the Wilderness, Brett Dennen, Cecilia and the Satellite, Glen Phillips, Jack's Mannequin, James Flannigan, Jonathan Kingham, Smash, solo artist, Somebody Told Me: A Night of Songs and Stories, Something Corporate, Synesthesia, The Pop Underground
Gene Simmons Wants to Coach Musical Gladiators
Andrew McMahon's Exclusive New Album Premiere
Records Still Worth Listening to a Decade Later
Andrew McMahon in the Wilderness: 'Cecilia and the Satellite' Music Video Review | {
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} | 8,552 |
Bangladesh Film Development Corporation or BFDC, is a government owned and operated corporation in Tejgaon, Dhaka, Bangladesh. Nuzhat Yeasmin is the managing director of the corporation.
History
The organization was founded in 1959 as the East Pakistan Film Development Corporation which was changed to Bangladesh Film Development Corporation after Bangladesh achieved independence in 1971. It signed an agreement with National Film Development Corporation of India in 2016 to jointly produce a documentary on Bangladesh Liberation war. It has faced criticism on mismanagement and waste of public funds. On 3 April of evey year the National Film day of Bangladesh is observed, the day is organized and celebrated by the corporation. The day marks the occasion when Sheikh Mujibur Rahman then Minister of Industries and Commerce of East Pakistan introduced the bill to formulate the East Pakistan Film development corporation.
References
1959 establishments in East Pakistan
Government agencies of Bangladesh
Organisations based in Dhaka
Film organisations in Bangladesh
Government-owned companies of Bangladesh
Ministry of Information and Broadcasting (Bangladesh)
Sheikh Mujibur Rahman | {
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// Copyright 2022 The Chromium Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
/**
* @fileoverview
* Declarations of prperaties added to global `window` for checking only (not in
* production).
*/
// Global `window` context.
declare global {
interface Window {
onAuthScriptLoaded: any;
supersize: any;
}
}
export {};
| {
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} | 5,226 |
{"url":"https:\/\/www.physicsforums.com\/threads\/help-with-inverse-of-derivative-function.792802\/","text":"# Help with inverse of derivative function\n\n1. Jan 17, 2015\n\n### Hazy001\n\n\u2022 Member warned about not using the homework template\nf(x) =\n3x^3 + 3x^2+ 2x + 1\n,a = 3\n\nformal is\n\nHomework is due tonight and this is the only problem i cant solve\n\n3=\n3x^3 + 3x^2 + 2x + 1\n, solve for xThe find the derivative of y=\n3x^3 + 3x^2 + 2x + 1\n, then plug x into that and put it under 1.\n\n2. Jan 17, 2015\n\n### Staff: Mentor\n\nWhat does a represent? I infer from your work below that you are taking it to mean the point (x0, a) on the graph of f.\n$3 = \\sqrt{3x^3 + 3x^2 + 2x + 1}$, and then square both sides. The resulting equation is not the easiest to solve, but it does have one real solution.\n\nLast edited: Jan 17, 2015\n3. Jan 17, 2015\n\n### Hazy001\n\nYes that is it but sadly i cannot solve it\n\n4. Jan 17, 2015\n\n### Ray Vickson\n\nYou want to solve $p(x) = 0$, where $p(x) = 3 x^3 + 3 x^2 + 2x - 8$.\n(i)One thing to try is the \"rational root theorem\"; see, eg.,\nhttp:\/\/en.wikipedia.org\/wiki\/Rational_root_theorem .\n(ii) Alternatively, plot the graph of $y = p(x)$ over some $x$-range, to see roughly where a root lies; that will suggest a factor of $p(x)$, which you can then verify exactly. (iii) If you are still desperate you can always submit the problem to an on-line solver, such as Wolfram Alpha.","date":"2017-11-19 07:06:29","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\": 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\": 0.6720426082611084, \"perplexity\": 1489.6632718957408}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934805417.47\/warc\/CC-MAIN-20171119061756-20171119081756-00167.warc.gz\"}"} | null | null |
Villaralbo est une commune espagnole de la province de Zamora dans la communauté autonome de Castille-et-León.
Géographie
Histoire
Démographie
Administration
Économie
Personnalités liées à la commune
César Alonso de las Heras (1913-2004), religieux et écrivain paraguayen, est né à Villaralbo.
Notes et références
Lien interne
Liste des communes de la province de Zamora
Lien externe
Site de la mairie de Villaralbo
Commune dans la province de Zamora | {
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Anhui Huanrui Heating Manufacturing Co.,Ltd is one of the most famous Chinese manufacturers and national high-tech enterprises, which takes the job of research and development, production, sale and service of self regulating heating cable, constant wattage heating cable, electric under floor heating system, water pipe freeze protection system, process temperature maintenance system, roof & gutter de-icing system, snow melting system etc. Provide the full solution for different projects, all products widely used in residential, commercial and industrial applications. Company passed ISO 9001 and ISO14001 system. Products approved by CE, CSA, ATEX, EAC, TUV, ROHS. Many certificates doing now.
Anhui Huanrui Heating Manufacturing Co.,Ltd located in city of National Science and Technology Innovation Pilot --- Hefei. It was found in 2004, the registered fund is 20.8 million RMB, have more than 100 employees. Company owns the first building area of 8476 square meters, and invest 120 million RMB to the second-stage factory, cover an area 16500 square meters. Company have more than 31 patented technology, more than 90 industrial certificates and honors from government.
Anhui Huanrui Heating Manufacturing Co.,Ltd has the strong R&D ability and professional team. Work with China top two founders of self regulating heating cable, who has more than 30 years experience research material chemical and work for the heat tracing project of China government. Company is one of few members of China self regulating heating cable national standard GB/T 19835-2005,GB/T 19835-2015 and GB/T 32348.1-2015/IEC 62395-1:2013. According to the international standard IEEE515, CSA C22.2-130 and IEC60800,62395,60079, Anhui Huanrui Heating Manufacturing Co.,Ltd build the most advanced laboratory,design and bring in high technology research and test equipment. So we can provide the OEM and ODM service for every customer from all over the world. What's more, all of the the exported product are 100% inspected.
Anhui Huanrui Heating Manufacturing Co.,Ltd has 19 patented technology, more than 90 industrial certificates and honors from government. In 2006, Anhui Huanrui Heating Manufacturing Co.,Ltd became the member of floor heating association of Hebei Province. In 2007, Anhui Huanrui Heating Manufacturing Co.,Ltd registered the brand "Jiuzi", which shows high quality and good faith of our enterprise.
In 2008, all the products of Anhui Huanrui Heating Manufacturing Co.,Ltd got the national explosion-proof certificates.
In 2011, Anhui Huanrui Heating Manufacturing Co.,Ltd got the certificate of ISO 9001:2008 quality management system.
In 2013, Anhui Huanrui Heating Manufacturing Co.,Ltd became the main compile organization for national standards.
In 2014, Anhui Huanrui Heating Manufacturing Co.,Ltd got the certificates of "National High and New Technology Enterprise", "Famous Brand of Anhui".
With the advanced production equipment and test devices of the world, and the continuous efforts of research and development team, Anhui Huanrui Heating Manufacturing Co.,Ltd works with the world top 500 enterprises, serves more than 20000 customers from all over the word. Our products widely used in oilfield, electric power, steel, chemical, coal and gas, refrigeration, food, construction, fire fighting, solar energy, electrothermal heating, geothermal cultivation, carriages, offshore oil platform industries and etc.
In 2007, Anhui Huanrui Heating Manufacturing Co.,Ltd became the supplier for the world top 500 ABB group.
In 2008, Anhui Huanrui Heating Manufacturing Co.,Ltd provided the self regulating heating cable for 2008 Olympic games.
Since 2009, Anhui Huanrui Heating Manufacturing Co.,Ltd became the long-term and fixed supplier for Dazing oilfield, Hubei oilfield, Songhua oilfield.
Since 2011, Anhui Huanrui Heating Manufacturing Co.,Ltd exported to Russia, UK, Australia, Hungary and etc.
In 2014, under the guidance of top two experts, Anhui Huanrui Heating Manufacturing Co.,Ltd produce a exclusive self regulating heating cable, using the latest advanced machine and highest technology, create a exclusive PTC heat matrix. Meanwhile obtain a patent of low resistance transition layer, PTC semi-conductive, Modified inner insulation layer process in one step, protect the PTC. No spacing between layer, so no impurity will hurt the PTC. It ensures that the cable will have a long work time, working at low starting current.
Until now everyone in company pay great effort, try to bring the high technology, energy saving and comfortable products to everyone. We believe that our company Anhui Huanrui Heating Manufacturing Co.,Ltd will be the great manufacturer in the future. Welcome customers from all over the world come to us and be familiar with products of Huanrui. | {
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} | 4,304 |
19 November, 2020 19 September, 2018 by Brent Brian
Maryborough is the birthplace of PL Travers, the creator and writer of the well-known story Mary Poppins. A bronze statue of Pamela Lyndon Travers is outside the building, with a brown sign for Cherry Tree Lane.
PL Travers was born in 1899 as Helen Lyndon Goff, sent to boarding school in Sydney. It was when she emigrated to England in 1933, she took the name Pamela Lyndon Travers having used it as a stage name in Australia.
P. L. Travers was used as her pen name while writing the first of the eight Mary Poppins books, published in 1934. The last sequel was published in 1988.
The story of Mary Poppins has her blown by the East wind to Number 17 Cherry Tree Lane, London. Cherry Tree Lane is also in the title of the 7th book, Mary Poppins in Cherry Tree Lane.
The life-size statue of PL Travers is on the corner of Kent Street and Richmond St, on the Richmond St side. The statue is holding a partly opened umbrella, symbolic of the scene where the wind blows away all of the waiting nannies and Mary Poppins gracefully flies in with her umbrella.
The red and green crossing lights at the intersection have been changed to look like Mary Poppins standing with her umbrella down and walking with her umbrella up, a nice touch.
A plaque in front of the statue reads;
Maryborough is the birthplace of P L Travers Creator of Mary Poppins.
This statue was unveiled on 9 August 2005, the 106th anniversary of P L Travers' birth by her friend Patricia Feltham.
It is a tribute to the author and the important place she holds in children's literature.
Sculptor: Dr Ryl Hinwood Hon D Phil (QLD)
The building has been purchased with intent to turn it into a museum about Mary Poppins and PL Travers. It had been cleared out when we were there and preparing for an open day to have a look inside.
An annual Mary Poppins Festival is held the first week in July, celebrating Mary Poppins and PL Travers and their connection to Maryborough.
Head south on Bruce Hwy, take the exit towards Maryborough and Hervey Bay onto Walker St. Follow Walker St for 3.5km and turn right into Ferry St. Follow Ferry St for 450m and turn left into Kent St. Follow Kent St for 950m, and the Mary Poppins Statue is on the left at Richmond St with the brown sign for Cherry Tree Lane.
Heading north on Bruce Hwy, on crossing Mary River continue on the highway for 900m passed the first exit for Maryborough and turning right at the next into Walker St. Follow Walker St for 3.5km and turn right into Ferry St. Follow Ferry St for 450m and turn left into Kent St. Follow Kent St for 950m, and the Mary Poppins Statue is on the left at Richmond St with the brown sign for Cherry Tree Lane.
Coming from Hervey Bay, head towards Maryborough on Saltwater Creek Rd. In Maryborough, Saltwater Creek Rd bends to the right becoming John St. Continue on John St for 1.1km to the end and turn left onto Kent St. Follow Kent St for 550m, and the Mary Poppins Statue is on the left at Richmond St with the brown sign for Cherry Tree Lane.
Cost:Adult $10.00, Child (5-17) $5.00, Concession $8.00, Family (2 Adults, 2+ Children) $25.00
Hours:Mon-Sun 10am-2pm
Toilets:Flushing Toilets
Tables/Seating:None
Wheelchair accessible:Yes
Categories Historical Tags Fraser Coast, QLD Post navigation
OLDS Engine House
Maryborough Heritage Gateway | {
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\section{Introduction and results}
\subsection{The model and the main result}
Let us consider the configuration model ${\mathrm{CM}}_n(\boldsymbol{d})$ on $n$ vertices, where the degrees $D_v, v\in \{1,2,\dots, n\}:=[n]$ are i.i.d.\ with a power-law tail distribution. That is, given the number of vertices $n$, to each vertex we assign a random number of half-edges drawn independently from a distribution $F$ and the half-edges are then paired randomly to form
edges. In case the total number of half-edges $\mathcal {L}_n:=\sum_{v\in[n]} D_v$ is not even, then we add one half-edge to $D_n$ (see below for more details).
We assume that
\begin{equation}\label{eq::F} \frac{c_1}{x^{\tau-1}}\le 1- F(x)= \mathbb{P}(D>x) \le \frac{C_1}{x^{\tau-1}},\end{equation}
with $\tau \in (2,3)$, and all edges have weight $1$. We assume that $\mathbb{P}(D\ge 2)=1$ guaranteeing that the graph has almost surely a unique connected component of size $n(1-o_{\mathbb{P}}(1))$ see e.g.\ \cite[Vol II., Theorem 4.1]{H10} or \cite{MolRee95, MolRee98}.\\
We further denote the mass function of (the \emph{size-biased version} of $D)-1$ by
\begin{equation}\label{def::size-biased1}
f^\star_{j}:=\frac{(j+1)\mathbb{P}(D=j+1)}{\mathbb{E}[D]}\text{, }\quad j\geq 0.
\end{equation}
We write $F^\star(x)$ for the distribution function $F^\star(x)=\sum_{j=0}^{\llcorner x\lrcorner}f^\star_{j}$.
It is not hard to see
that there exist $0<c_1^\star\le C_1^\star<\infty$, such that
\begin{equation} \label{eq::size-biased2} \frac{c_1^\star}{x^{\tau-2}}\le 1- F^\star(x)\le \frac{C_1^\star}{x^{\tau-2}}. \end{equation}
Pick two vertices $\mathcal R_0$ (the red source) and $\mathcal B_0$ (the blue source) uniformly at random in $[n]$,
and consider these as two sources of spreading infections.
Each infection spreads \emph{deterministically} on the graph: in general,
for color blue it takes $\lambda$ time units to pass through an edge, while color red needs $1$ unit of time for that.
Without loss of generality we can assume that $\lambda\ge 1$. The $\lambda>1$ case was treated in \cite{BarHofKom14}, while in this paper we study the case where $\lambda=1$. Each vertex is painted the color of the infection that reaches it
first, keeps its color forever, and starts coloring the outgoing edges at the speed of its color. When the two colors reach a vertex at the \emph{same} time, the vertex gets color red
or blue with probability $1/2$ each, independently of everything else. In general, this rule could be modified to an \emph{arbitrary adapted rule}, i.e., a rule that does not depend on the future. Throughout this paper, we handle the case when the rule is such that at all times, there is a strictly positive probability for both colors to get the vertex under consideration, and this decision is independent of the decision about other vertices. However, we emphasise that other rules can be handled analogously with possibly different outcomes for coexistence.
Let $\mathcal R_t:=\mathcal R_t(n)$ and $\mathcal B_t:=\mathcal B_t(n)$
denote the number of red and blue vertices occupied up to time $t$, respectively. We denote by $\mathcal {B}_{\infty}:=\mathcal {B}_{\infty}(n)$ the number of vertices eventually occupied by blue.
Roughly speaking, the first result of this paper, Theorems \ref{thm::main} and \ref{thm::main2} below, tell us that with high probability (whp), i.e., with probability tending to $1$ as the size of the graph tends to infinity, one out of two things can happen:
1.) When the local neighbourhoods of the source vertices are \emph{dissimilar enough}, i.e., if one of them grows at a significantly faster speed than the other, then there is no coexistence, and the color with faster-growing local neighbourhood gets $n-o_{\mathbb{P}}(n)$ vertices.
The number of vertices the other color paints is a polynomial of $n$ with a random exponent. More precisely, blue paints whp $n^{H_n(Y_r, Y_b)}$ many vertices, where the coefficient $H_n(Y_r, Y_b)<1$ is a random function that depends on $n, \tau$, and two random variables $Y_r$ and $Y_b$, that can intuitively be interpreted as some measure of `how fast' the neighbourhoods of the source vertices grow: the faster the local neighbourhoods grow, the larger these variables are. Moreover, $H_n(Y_r, Y_b)$ does not converge in distribution: it has an oscillatory part that exhibits `$\log\log$-periodicity'.
2.) If the local neighbourhoods of the source vertices are \emph{similar enough}, i.e., the random variables $Y_r, Y_b$, describing the speed of growth of the neighbourhoods, are within a factor $\tau-2$ of each other, then both colors get a linear proportion of vertices, i.e., there is asymptotic coexistence in the model.
More precisely, both $\mathcal {B}_\infty/n, \mathcal {R}_\infty/n$ stay strictly between $0$ and $1$ as $n\to \infty$. We also show that as the ratio of $Y_r, Y_b$ approaches $\tau-2$, the proportion of vertices painted by the color with the smaller $Y$ value tends to zero, and in this sense, the transition from coexistence to non-coexistence is smooth. These results are established by the analysis of a \emph{branching process random coloring scheme}, which we find interesting in its own right. In this coloring scheme, the BP is run until the maximal degree reaches some value $Q$, and then, the vertices in the last generation of the stopped BP are colored red, blue or stay uncolored according to their degrees. The colored vertices spread their color to earlier generation vertices on the BP tree, following a rule that is similar to the spreading dynamics for red and blue in the graph. We show that the root of the BP can get both colors with positive probability uniformly as $Q\to \infty$, which in turn implies coexistence in the configuration model. For more on this problem see Section \ref{sc::BP-color} below.
The heuristic interpretation of these results and the main result in \cite{BarHofKom14} in marketing terminology is as follows: if a company gains customers via word-of-mouth recommendations, then the one with faster spreading speed takes most of the network. On the other hand, if both of the companies gain customers at the same speed, then the one with better starting location can gain most of the network, at least if this location is significantly better than the starting location of the other company (companies).
This result reinforces the common sense about importance of location in (online) location-based advertisement: there is even a slogan called `Location, location, location', see e.g. \cite{BruKum07, Das05, LLL, XuOhTe09}.
According to our knowledge this is the first random graph model that can both produce asymptotic coexistence and non-coexistence when the spreading speeds are equal (i.e., both outcomes happen with positive probability). Further, which one of the two outcome happens depends on the starting location of the spreading colors, and moreover, the asymptotic probability of these events is explicitly computable.
The other main result, Theorem \ref{thm::distances}, describes the distribution of the fluctuations of typical distances in the graph. More precisely, it was shown in \cite{HHZ07} that the graph distance between two uniformly chosen vertices is concentrated around $2 \log \log n / |\log (\tau-2)|$, with bounded and non-converging fluctuations around this value.
Here, we provide a different, in some sense more natural, proof of this fact, and provide a different representation of the fluctuation: we describe it as a simple function of $n$ and two independent random variables that describe the growth rate of the local neighborhood of the source vertices. This function contains integer parts, hence, the same $\log \log$-periodicity phenomenon is present as the one in Theorems \ref{thm::main} and Theorem \ref{thm::main2}, coming from the fact that the edge weights are concentrated on a lattice. Although it is not apparent from their final forms, we emphasize that the result of \cite{HHZ07} and Theorem \ref{thm::distances} are \emph{the same}: we show this fact in Appendix \ref{appendix::distances}.
To be able to state the main theorem precisely, let us define the following random variables:
\begin{definition}[Galton-Watson limits]\label{def::limit-variables}
Let $Z_k^{\scriptscriptstyle{(r)}}, Z_k^{\scriptscriptstyle{(b)}}$ denote the number
of individuals in the $k$th generation of two independent copies of a Galton-Watson process described as follows: the size of the first generation has distribution $F$ satisfying \eqref{eq::F},
and all the further generations have offspring distribution $F^\star$ from \eqref{def::size-biased1}.
Then, for a fixed but small $\varrho>0$ let us define
\begin{equation}\label{def::yrn-ybn}Y_r^{\scriptscriptstyle{(n)}}:=(\tau-2)^{t(n^\varrho)} \log (Z^{\scriptscriptstyle{(r)}}_{t(n^{\varrho})}), \quad Y_b^{\scriptscriptstyle{(n)}}:=(\tau-2)^{ t(n^{\varrho})} \log (Z^{\scriptscriptstyle{(b)}}_{ t(n^{\varrho})}),\end{equation}
where $t(n^\varrho)=\inf_k\{\max\{ Z_k^{\scriptscriptstyle{(r)}}, Z_k^{\scriptscriptstyle{(b)}} \} \ge n^\varrho\}$.
Let us further introduce
\begin{equation}\label{def::Y} Y_r:= \lim_{k\to\infty} (\tau-2)^k \log (Z_k^{\scriptscriptstyle{(r)}}), \quad Y_b:=\lim_{k\to\infty} (\tau-2)^k \log (Z_k^{\scriptscriptstyle{(b)}}).\end{equation}
\end{definition}
We will see in Section \ref{sc::BP} below that these quantities are well-defined and that $(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})\buildrel {d}\over{\longrightarrow} (Y_r,Y_b)$ from \eqref{def::Y} as $n\to \infty$. To be able to state the results shortly, let us further define, for $j=r,b$,
\begin{equation}\label{def::ti-bi} T_j:=\left\lfloor\frac{\log\log n -\log((\tau-1)Y_j^{\scriptscriptstyle{(n)}})}{|\log (\tau-2)|}-1\right\rfloor, \quad
b_n^{(j)}:= \left\{\frac{\log\log n -\log((\tau-1)Y_j^{\scriptscriptstyle{(n)}})}{|\log (\tau-2)|}\right\},\end{equation}
where $\lfloor x\rfloor $ denotes the largest integer that is at most $x$ and $\{x\}= x-\lfloor x\rfloor$ denotes the fractional part of $x$.
Let us also introduce four events $E_<, E_>, O_<, O_>$ where $E,O$ stands for the events that $T_r+T_b-1$ is even or odd, respectively, and the subscript $<$ is added when $\tau-1<(\tau-2)^{b_n^{\sss{(r)}}}+(\tau-2)^{b_n^{\sss{(b)}}}$ and the subscript $>$ is added when $\tau-1>(\tau-2)^{b_n^{\sss{(r)}}}+(\tau-2)^{b_n^{\sss{(b)}}}$. We write $D_n^{\max}(t)$ for the degree of the maximal degree vertex occupied by the losing color at time $t$.
For sequences of random or deterministic variables $X_n, Y_n$ we write $X_n = o_{\mathbb{P}}(Y_n)$ and $X_n = O_{\mathbb{P}}(Y_n)$ if the sequence $X_n/Y_n \buildrel {\Pv}\over{\longrightarrow} 0 $ and is tight, respectively.
Recall that $\mathcal {B}_\infty, \mathcal {R}_\infty$ denotes the number of vertices eventually occupied by the blue and red colors, respectively. With these notations in mind, we can state our main results:
\begin{theorem}[Total number of vertices painted by the losing color]\label{thm::main} Let us assume wlog that $Y_r^{\scriptscriptstyle{(n)}}>Y_b^{\scriptscriptstyle{(n)}}$ in Definition \ref{def::limit-variables}, that is, the `losing' color is blue.
When $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} \le \tau-2$, then $\mathcal {R}_\infty/n\buildrel {\Pv}\over{\longrightarrow} 1$ whp, and
\[ \frac{\log (\mathcal {B}_\infty)}{\log n\cdot (\tau-1)^{-1} f_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) } \buildrel {d}\over{\longrightarrow} \sqrt{\frac{Y_b}{Y_r}}, \]
where $f_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ is an oscillating random variable given by
\begin{equation}\label{eq::main-f} f_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) \!=\! (\tau\!-\!2)^{(b_n^{\sss{(r)}}-b_n^{\sss{(b)}} - 1 - \mathbbm{1}_{O})/2}
\!\!\left(\! (\tau-2)^{b_n^{\sss{(b)}}\! + \mathbbm{1}_{O_<}}\! +\! (\tau\!-\!1\!-\!(\tau\!-\!2)^{b_n^{\sss{(r)}}})(\tau\!-\!2)^{ \mathbbm{1}_{O_>}}\! \right)\! , \end{equation}
where $O=O_< \cup O_>$.
\end{theorem}
\begin{remark}\normalfont
Note that in Theorem \ref{thm::main}, the function $f_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ filters out the oscillations coming from $\log\log$-periodicity, and hence, it depends on $Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}$. We emphasise that in general it is not true that $f_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) (\tau-1)^{-1}<1$. However, it is true that
\[ \sqrt{Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}} f_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) (\tau-1)^{-1} < 1, \]
hence we get the statement that $\mathcal {B}_\infty=o_{\mathbb{P}}(n)$. See Lemma \ref{lem::no-coexistence} below for the proof of this fact.
\end{remark}
The next theorem holds under the following (technical) assumption:
\begin{assumption}\label{assume::abs-cont}
The limiting random variable $Y=\lim_{n\to \infty} (\tau-2)^k\log Z_k$ of the BP described in Definition \ref{def::limit-variables} has an absolutely continuous distribution function with support containing an interval of the form $(0, K), \ K\in \mathbb{R}^+ \cup\{\infty\}$.
\end{assumption}
The criteria on $F$ required for this assumption to hold are not obvious: according to our knowledge, no necessary and sufficient condition for absolute continuity can be found in the literature. We provide some necessary criterion based on the work \cite{Sene73, Sene74} below in Assumption \ref{assume::convex}. This assumption is not tight, though, milder criteria on the slowly varying function hidden in \eqref{eq::F} can also guarantee the statement. We improve the already existing criteria in an upcoming short note to be published elsewhere \cite{HofKom15}.
\begin{theorem}[Asymptotic coexistence when $Y_b/Y_r>\tau-2$]\label{thm::main2}
Let us assume wlog that $Y_r^{\scriptscriptstyle{(n)}}>Y_b^{\scriptscriptstyle{(n)}}$ in Definition \ref{def::limit-variables}, that is, the `losing' color is blue, and that Assumption \ref{assume::abs-cont} holds. When $q:=Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} > \tau-2$, then both colors can paint a linear proportion of the vertices. More precisely, there exists deterministic constants $0<c(q)<C(q)<1$ such that whp as $n\to \infty$
\[ c(q)\le \frac{\mathcal {B}_\infty}{n}\le C(q). \]
Further, we have $c(q), C(q)\to 0$ as $q \searrow \tau-2$.
\end{theorem}
\begin{remark} For the specific values of $c(q)$ and $C(q)$ see the proof of the theorem on page 49.
\end{remark}
For an event $A$, we denote $\mathbb{P}_n(A):=\mathbb{P}(A| D_1, D_2, \dots, D_n)$. We say that the two competing spreading processes asymptotically coexist on a sequence of finite graphs indexed by $n$, if the limiting ratios $\mathcal {R}_\infty(n)/n$ and $\mathcal {B}_\infty(n)/n$ are both strictly positive with strictly positive probability, as $n\to \infty$.
An immediate consequence of Theorem \ref{thm::main2} is the following result:
\begin{corollary}[Probability of coexistence] In the above competition model with equal speeds, under Assumption \ref{assume::abs-cont}, whp
\[ \lim_{n\to \infty }\mathbb{P}_n( \text{ coexistence occurs } ) = \mathbb{P}\,(\, Y_b/ Y_r \in ( \tau-2, (\tau-2)^{-1}) ). \]
\end{corollary}
\begin{remark}\normalfont The core of the proof of Theorem \ref{thm::main2} is Proposition \ref{prop::BP-color} below. This proposition establishes that the probability that the root of a BP described in Definition \ref{def::limit-variables} gets both colors with strictly positive probability in a random coloring scheme, see Section \ref{sc::BP-color} below.
\end{remark}
\begin{remark}\label{rem::coexistence-precise}\normalfont
We conjecture that the statement of Theorem \ref{thm::main2} can be further sharpened, namely, we suspect that there exists a random variable $\widehat h_n (Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) \in (0,1)$ so that on the event $\{Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} \in (\tau-2,1)\}$, we have
$\mathcal {B}_\infty/(n\!\cdot\! \widehat h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})) \buildrel {\Pv}\over{\longrightarrow} 1$ as $n\to \infty$, where $\widehat h(n, Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ does not converge, but oscillates between two constants $ c_2(q)< C_2(q)$ with $n$, $c_2(q), C_2(q)$ satisfying $c(q)\le c_2(q)< C_2(q)\le C(q)$, where $q=Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}$, and both $c_2(q), C_2(q) \to 0$ as $q \searrow \tau-2$.
(Here, $c(q),C(q)$ are from Theorem \ref{thm::main2}.)
\end{remark}
\begin{remark} \normalfont The statements of Theorems \ref{thm::main} and \ref{thm::main2} remain valid also if the red and blue processes are started from either ends of a uniformly chosen edge. In this case, the laws of $Y_r, Y_b$ are limits of branching processes as in Definition \ref{def::limit-variables}, where also the root has offspring distribution $F^\star$.
\end{remark}
All the consecutive results hold again \emph{without} Assumption \ref{assume::abs-cont}.
Let us define $D_n^{\max}(t)$ as the degree of the maximal degree vertex that blue has colored before or at time $t$.
The next theorem is about the degree of the maximal degree vertex that each color can eventually paint:
\begin{theorem}[Maximal degree of the `losing' color]\label{thm::maxdegree} Let us assume wlog that $Y_b^{\scriptscriptstyle{(n)}}<Y_r^{\scriptscriptstyle{(n)}}$ in Definition \ref{def::limit-variables}, i.e., the losing color is blue. Then, with high probability,
the degree $D_n^{\max, \mathrm{red}}(\infty)$ of the maximal degree vertex that red can eventually paint always tends to the maximal degree in the graph, that is,
\[ D_{n}^{\max, \mathrm{red}}(\infty) = n^{1/(\tau-1)(1+o_{\mathbb{P}}(1))}. \]
The maximal degree vertex that blue can eventually paint satisfies the following:
(i) When $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}\le \tau-2$,
\[ \frac{\log D_n^{\max}(\infty)}{\log n \cdot (\tau-1)^{-1} h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})} \buildrel {d}\over{\longrightarrow} \sqrt{\frac{Y_b}{Y_r}}. \]
where $h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})\le 1$ is an oscillating random variable given by
\begin{equation}\begin{aligned}\label{eq::h1} h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) =& \mathbbm{1}_{ E_< \cup O_>} (\tau-2)^{(b_n^{\sss{(b)}}+b_n^{\sss{(r)}}-1 - \mathbbm{1}_{O_>})/2} +\\
&+\mathbbm{1}_{ E_> \cup O_<} (\tau-2)^{(b_n^{\sss{(r)}}-b_n^{\sss{(r)}}-1 - \mathbbm{1}_{O_<})/2} ((\tau-1)-(\tau-2)^{b_n^{\sss{(r)}}}). \end{aligned} \end{equation}
(ii) When $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} > \tau-2$, the maximal degree of blue sensitively depends on the fractional parts $b_n^{\sss{(r)}}, b_n^{\sss{(b)}}$. With $T_r, T_b$ as in \eqref{def::ti-bi},
\begin{enumerate}[(a)]
\item if $T_b-T_r=0$, then whp
\[ D_{n}^{\max}(\infty) = n^{1/(\tau-1)(1+o_{\mathbb{P}}(1))},\]
\item if $T_b-T_r=1$ and $\tau-1<(\tau-2)^{b_n^{\sss{(r)}}}+(\tau-2)^{b_n^{\sss{(b)}}}$, then whp
\[ D_{n}^{\max}(\infty) = n^{(\tau-2)^{b_n^{\sss{(b)}}}/(\tau-1)(1+o_{\mathbb{P}}(1))},\]
\item if $T_b-T_r=1$ and $\tau-1>(\tau-2)^{b_n^{\sss{(r)}}}+(\tau-2)^{b_n^{\sss{(b)}}}$, then whp
\[ D_{n}^{\max}(\infty) = n^{ ((\tau-1)-(\tau-2)^{b_n^{\sss{(r)}}})/(\tau-1)(1+o_{\mathbb{P}}(1))}.\]
\end{enumerate}
\end{theorem}
\begin{remark}\normalfont Compare Theorem \ref{thm::maxdegree} to Theorem \ref{thm::main} to see that Case (ii) corresponds to coexistence.
Also, we emphasise that there is a coupling of the graphs ${\mathrm{CM}}_n(\boldsymbol{d})$ for $n\ge 1$ so that even the stronger statement $(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) \buildrel {\Pv}\over{\longrightarrow} (Y_r, Y_b)$ is valid. In this coupling construction, the event $\{ Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}> \tau-2\}$ converges to the event $\{Y_b/Y_r > \tau-2\}$. Case (ii) in Theorem \ref{thm::maxdegree} heuristically says that even under this limiting event, the maximal degree of blue shows some oscillation with $n$. \end{remark}
As a side result of the proof of Theorem \ref{thm::maxdegree}, we get a new description of typical distances in the graph:
\begin{theorem}[Fluctuations of typical distances]\label{thm::distances}
In the configuration model with i.i.d. degrees from distribution $D$ satisfying \eqref{eq::F} with power-law exponent $\tau\in(2,3)$, the typical distance $\mathcal {D}_n(u,v)$ between two uniformly picked vertices $u:=\mathcal {R}_0,v:=\mathcal {B}_0$ can be described using $Y_r^{\scriptscriptstyle{(n)}}$ and $Y_b^{\scriptscriptstyle{(n)}}$ in \eqref{def::yrn-ybn} as
\begin{equation}\label{eq::duv-1} \begin{aligned} \mathcal {D}_n(u,v)=&\left\lfloor\frac{\log\log n -\log((\tau-1)Y_b^{\scriptscriptstyle{(n)}})}{|\log (\tau-2)|}\right\rfloor + \left\lfloor\frac{\log\log n -\log((\tau-1)Y_r^{\scriptscriptstyle{(n)}})}{|\log (\tau-2)|}\right\rfloor \\
&\, -1 + \mathbbm{1}\{(\tau-2)^{b_n^{(r)}}+(\tau-2)^{b_n^{(b)}}<\tau-1 \} + o_\mathbb{P}(1), \end{aligned}\end{equation}
whp, where $b_n^{(j)}=\left\{ \frac{\log\log n -\log((\tau-1)Y_j^{\scriptscriptstyle{(n)}})}{|\log (\tau-2)|}\right\}$ for $j=r,b$. Equivalently,
\[\mathcal {D}_n(u,v)- \frac{2\log\log n}{|\log (\tau-2)|}+1 +b_n^{(r)}+b_n^{(b)} - \mathbbm{1}\{\tau-1>(\tau-2)^{b_n^{(r)}}+(\tau-2)^{b_n^{(b)}}\} \buildrel {d}\over{\longrightarrow} \frac{-\log((\tau-1)^2 Y_rY_b)}{|\log (\tau-2)|}. \]
\end{theorem}
\begin{remark}\normalfont Note that Theorem \ref{thm::distances} implies that the typical distances in the graph are concentrated around $2 \log\log n / |\log (\tau-2)|$ with bounded fluctuations, a result that already appeared in \cite{HHZ07} under weaker assumptions on $F$. Our proof is considerably simpler than that in \cite{HHZ07}. The second statement of the theorem `filters out' the oscillations arising from fractional part issues: it is not hard to see that
\[ 1 +b_n^{(r)}+b_n^{(b)} - \mathbbm{1}\{\tau-1>(\tau-2)^{b_n^{(r)}}+(\tau-2)^{b_n^{(b)}}\}\in \Big[\frac{2 \log\tfrac{2}{\tau-1} } { |\log (\tau-2)|}, 2\Big)\]
oscillating with $n$. We emphasise here that the essential statement of Theorem \ref{thm::distances} and \cite[Theorem 1.2]{HHZ07} are the same, however, they provide a different description of typical distances.
\end{remark}
\subsection{Random coloring schemes for branching process trees}\label{sc::BP-color}
The crucial ingredient in the proof of Theorem \ref{thm::main2} boils down to the analysis of the following problem, that we find interesting in its own right. The specific version of the problem, which we solve in this paper, can be described as follows:
Suppose we have an infinite-mean Galton-Watson BP with offspring distribution given in \eqref{eq::size-biased2}. We let this BP grow until a vertex with degree at least $Q$ appears in the process.
Then, there is a \emph{starting rule}: We fix a parameter $\gamma \in (1, 1/(\tau-2))$. In the last generation of the stopped BP we paint every vertex with degree in the interval $[Q, Q^\gamma)$ red and vertices with degree in the interval $[Q^\gamma, Q^{1/(\tau-2)})$ red or blue with equal probability (if any).
After this, we sequentially color earlier generations, using a \emph{flow rule}: if a vertex has both red and blue children, it gets painted red or blue with equal probability; if it has children of only one color, then it takes that color; if it has no colored children, it stays uncolored, independently for each vertex in the same generation.
\begin{proposition}\label{prop::BP-color}
Fix a $\gamma\in (1,1/(\tau-2))$ and consider the above described coloring scheme of a branching process described in Definition \ref{def::limit-variables}. Assume further that Assumption \ref{assume::abs-cont} holds for the limiting random variable $Y$ of this BP.
Then there exist constants $0<c(\gamma)\le C(\gamma)<1$ such that
\[ c(\gamma)\le\liminf_{Q \to \infty} \mathbb{P}( \emph{root is painted blue}) \le \limsup_{Q \to \infty} \mathbb{P}( \emph{root is painted blue}) \le C(\gamma).\]
Further, $C(\gamma) \searrow 0$ as $\gamma\nearrow 1/(\tau-2)$.
\end{proposition}
This result is the core of the proof of the coexistence in Theorem \ref{thm::main2}. Note that this proposition is itself non-trivial since the proportion of blue vertices among all colored vertices in the last generation tends to zero as $Q \to \infty$, and further, the generation where the process is stopped also tends to infinity as $Q\to \infty$. As a result of these two effects, a smaller and smaller proportion of blue vertices have to `make their way' down to the root that is further and further away. Heuristically speaking, the rule that a vertex flips a coin that \emph{does not depend on
the number of its red and blue children} saves the blue color: this effect `exaggerates' the proportion of blue vertices as the generation number decreases towards the root.
To gain a more precise result on the proportion of blue vertices in the graph in Theorem \ref{thm::main2} (in particular, to prove the conjecture in Remark \ref{rem::coexistence-precise}), one has to gain a deeper understanding of the probability that the root is painted blue in this coloring scheme. In particular, the dependence of $\mathbb{P}(\text{root is blue})$ on the parameter $\gamma$ directly translates to the dependence of $\mathcal {B}_{\infty}/n$ on the ratio $q=Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} \in (\tau-2, 1)$. However, for a more detailed analysis of $\mathbb{P}(\text{root is blue})$, one has to know more about the shape of the density function of the limiting variable $Y=\lim_{k\to \infty} (\tau-2)^k \log Z_k$ similar as in \eqref{def::Y}. Without more specific assumptions on the offspring distribution than the one in \eqref{eq::F}, this is beyond our reach.
The random coloring scheme above might be generalised as follows: Suppose we have a branching process (a simple discrete time Galton-Watson BP in the case of this paper, but not necessarily in general) that we let grow until the (random) time when an individual with degree at least $Q$ appears, for some number $Q\gg 1$. Then, there is a \emph{starting rule} that colors some individuals in the BP, where the color depends on the degree of the individual. Further, this dependence is so that one color (say red) paints significantly more vertices than the other color (say blue), and the difference is exaggerated as $Q\to \infty$. Plus, the starting rule is so that only high-enough degrees get colored, the rest of the vertices in the BP remains unpainted.
Then, we prescribe a \emph{flow rule}: any vertex in the BP takes the color of one of its children according to some rule that might depend on the number of children of each color, but is independent for different vertices.
The question is: under what circumstances can the root be painted by both colors with strictly positive probability, as $Q \to \infty$? If this is possible, then how does this probability depend on the parameter of the starting rule?
We suspect that the property that the offspring distribution has infinite mean is crucial, as well as the strictly positive probability of taking each color in the presence of children of both colors. For instance, we conjecture that coexistence might not occur with other very natural degree dependent coloring rules, e.g., when a vertex takes the color of one of its neighbors proportional to the number of neighbors of that color.
\subsection{Discussion and open problems}
We gave an overview of related literature on first passage percolation, competitive spreading processes on lattices, applications such as word-of-mouth recommendations in online and offline marketing and epidemiology references in \cite[Section 1.2]{BarHofKom14}. Hence, we omit repetition and refer the reader for references about related mathematical and applied models there.
Here we review only results on competition on random graphs, and state our conjectures about possible generalizations.
Antunovic, Dekel, Mossel and Peres \cite{ADMP11}
give a detailed analysis of competition on random regular graphs (degree at least $3$) on $n$ vertices with i.i.d. exponential edge weights. They analyse scenarios where the speed of the two colors $\lambda_{1},\lambda_{2}$ might differ, and also the initial number of colored vertices might grow with $n$. They show that whp the color with higher rate occupies $n-o_{\mathbb{P}}(n)$ vertices and the slower color paints approximately $n^\beta$ vertices for some deterministic function $\beta(\lambda_1, \lambda_2)$. When the speeds are equal, they show coexistence starting from single sources. We conjecture that their result can be generalised for the configuration model with i.i.d.\ continuous edge weights as long as the second moment of the degree distribution is finite, i.e., $\tau >3$ holds in \eqref{eq::F}.
Next, van der Hofstad and Deijfen \cite{DH13} studied competition of two colors from uniformly picked single source vertices with i.i.d.\ exponential edge weights, on the configuration model with i.i.d.\ degrees satisfying \eqref{eq::F} with $\tau\in (2,3)$. They prove that even if the speeds are not equal, the `winner' color is \emph{random}, and the winning color paints all but a finite number of vertices. The randomness of the `winner' color comes from the fact that the underlying Markov branching process explodes in finite time, and the slower color has a positive chance to explode earlier than the faster color.
Then, in \cite{BarHofKom14} we treated \emph{fixed speed} competition on the configuration model with i.i.d. degrees satisfying \eqref{eq::F}, when the speeds of the two colors are not equal. Suppose it takes $1$ and $\lambda>1$ unit of time for red and blue to spread across an edge, respectively. We have shown that in case the two colors start to spread from uniformly chosen single source vertices, the red color paints $n-o_{\mathbb{P}}(n)$ vertices, while blue paints
\[ \exp \left\{ (\log n)^{2/(\lambda +1)} g_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})\right\}\]
many vertices, where $g_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ is a random variable that shows $\log\log$-periodicity, and can be rewritten in a similar ($\lambda$-dependent) form as the one in the statement of Theorem \ref{thm::main}.
Recently Cooper \emph{et al.} \cite{CooElsOgiRad15} have analysed a similar fixed speed competition model, also on the configuration model with power-law degrees with exponent $\tau\in(2,3)$. In what they call Model 2, one of the colors (say red) is called a `malicious information' (i.e., it might model the spread of a virus), while the other color (say blue) `immunizes' vertices. The main difference from the spreading rules of this paper is that in their model the infection does not spread to all the neighbors of a vertex, only a fixed subset of the nodes. Further, the immunization process starts from the not-yet infected neighbors of infected vertices (with a delay), and spreads then in a similar manner as the blue color in this paper. This implies a dependence between the two processes beyond the obvious `blocking' effect: if the red color paints a high-degree vertex (a hub), the blue color automatically reaches these hubs as well.
The authors show that in this competing scheme, the immunization process can block the spread of the infection, i.e., the infection can only spread to $o(n)$ many vertices.
Finally, this paper finishes the description of fixed speed competition when the speeds are equal.
Let us here compare the results to those in \cite{BarHofKom14}.
Theorem \ref{thm::main} and the first part of Theorem \ref{thm::maxdegree} correspond to \cite[Theorem 1.2]{BarHofKom14} and \cite[Theorem 1.4]{BarHofKom14}, respectively, and they could be interpreted as follows: if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}< \tau-2$, then the local neighbourhoods of the source vertices differ enough to build up a significant difference between the spread of the two colors. Hence, both the maximal degree and the total number of vertices occupied by blue can be obtained by substituting $\lambda=1$ in the formulas in \cite[Theorem 1.4]{BarHofKom14} and \cite[Theorem 1.2]{BarHofKom14}. However, if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}> \tau-2$, then the growth of the two colors does not differ enough, hence, both colors can paint a positive proportion of the hubs. Once the highest degree vertices are coloured, their coloring `rolls down' to smaller and smaller degree vertices to eventually occupy the whole graph in a manner that maintains the fact that `significantly many' of the vertices are coloured both blue and red. Eventually, there is asymptotic coexistence in the model.
\subsubsection*{Open problems}
The analysis of competition on the configuration model is far from complete. One can for instance ask about different spreading dynamics (edge lengths) and different power-law exponents. Further, one can ask what happens if the colors have entirely different passage time distributions (e.g.\ one is explosive and the other is not), or what happens if one of the colours have a main advantage by starting from one or many initial vertices of very high degree. These can correspond to e.g.\ competition advantage of different product on the network or to different marketing strategies.
Here we list some conjectures for competition on ${\mathrm{CM}}_n(\boldsymbol{d})$ with i.i.d. power law degrees of distribution $D$ with exponent $\tau$.
We further assume that the time to passage times can be represented as i.i.d. random variables on edges, from distribution $I_r$, $I_b$ for red and blue, respectively.
\emph{Uniformly chosen source vertices, $\tau\in (2,3)$:}
A. If the spreading dynamics are so that the underlying
branching processes defined by $D^\star,I_r$ and $D^\star, I_b$ are both explosive, then we conjecture that there is \emph{never} coexistence and either of the two colors can win. We conjecture that the number of vertices the losing color can paint depends on the behavior of density of the explosion time around $0$. A step towards proving this is to understand typical distances in ${\mathrm{CM}}_n(\boldsymbol{d})$ for arbitrary i.i.d. edge weights: this is done in an upcoming paper \cite{BarHofKomdist}.
B. If the underlying BP for one color is explosive while the other one is not, than we suspect that the explosive one always wins and the number of vertices the other color paints is tight.
C. If the edge weights are \emph{separated away from $0$}, in the sense that they can be written in the form $c+X$ for some random variable $X\ge 0$ and some constant $c>0$, then we conjecture that the different speed case ($\lambda >1$) will be similar to the results in \cite{BarHofKom14}: the faster color wins. We conjecture that even the number of vertices that the slower color paints should be the same as the result given in \cite{BarHofKom14} as long as the fluctuation of typical distances are \emph{tight} around $c \cdot 2 \log \log n/ |\log (\tau-2)|$. We investigate typical distances in this setting in the upcoming paper \cite{BarHofKomdist} and tightness in a subsequent paper. Tightness depends sensitively on the precise behavior of the random variable $X$ around the origin.
D. If the edge weights are so that the support of the distribution contains an interval $[0,\varepsilon]$ for some $\varepsilon>0$, and the underlying BP is not explosive, then even typical distances in the graph are not understood. This is mainly due to a lack of literature about conservative infinite mean BPs: a precise understanding of the time it takes in the BP to reach $m$ individuals would be necessary.
\emph{Special source vertices, $\tau \in (2,3)$:}
It would be interesting to study scenarios where at least one of the sources has a `big head start' in the sense that it is started from a vertex with degree that grows with $n$. Since the number of initial half-edges is the important parameter here, this problem is essentially equivalent to starting from multiple source vertices, where the number of source vertices grows with $n$. The first question that we might ask: how many half-edges are needed for a process to start from to guarantee the winning of that color? If it does not have the necessary amount of initial half-edges for winning, what is the number of painted vertices, and how does it depend on the initial size of the source set and on the speeds of the two colors?
Based on the results of this paper, it is reasonable to conjecture that at least in the fixed speed setting, if the slower color can occupy all the highest degree vertices (degrees larger than $n^{(\tau-2)/(\tau-1)}$), then it blocks the way of the other color from spreading, and as a result, it might flip the outcome and paint almost all vertices. If this is not the case, but the slower color has a head start, then similar estimates such as the maximal degree it can paint and the number of half-edges with this degree should determine the size it can eventually paint.
\emph{Uniformly chosen sources, $\tau>3$:}
As mentioned above, our conjecture for $\tau>3$ is mostly based on the result in \cite{ADMP11}.
We suspect that if the transmission times $I_r, I_b$ both have continuous distribution, and the branching process approximations of them have different Malthusian parameters, then there is no coexistence, and the number of vertices painted by the slower color is $n^{\beta}$ for some $\beta\in(0,1)$.
When the Malthusian parameters agree, we suspect that there is asymptotic co-existence.
\emph{Uniformly chosen sources, $\tau=3$:}
In this case $\mathbb{P}(D>x) = L(x)/ x^2$, with $L(x)$ a slowly varying function at infinity. We suspect that $L(x)$ and the transmission distributions $I_r, I_b$ jointly determine what happens:
In the case when one or more of the BPs might be explosive, we suspect that the outcome is similar to cases A and B above. The non-explosive cases might be harder, at least for the case when $L(x)$ is so that the $\mathbb{E}[D^\star] = \infty$.
\subsection{Overview of the proof and structure of the paper}
The heuristic idea of the initial parts of proof is the same as for the $\lambda>1$ case. The main idea is to extensively use the fact that in the configuration model, half-edges can be paired in an arbitrarily chosen order. This allows for a joint construction of the graph with the growing of the two colored clusters. The growth has six phases, out of which the first one (Section \ref{sc::climbup}) is essentially the same as for $\lambda>1$. In Section \ref{sc::peak}, the two cases, i.e., $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<\tau-2$ or $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$, separate and the proof of coexistence in the latter case is entirely new. The methodology of the proof for Theorem \ref{thm::main} for the $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<\tau-2$ remains in essence the same as the proof of \cite[Theorem 1.2]{BarHofKom14}, with some adjustments needed to handle larger error terms due to $\lambda=1$ instead of $\lambda>1$.
Wherever we can, we try to keep the overlap with \cite{BarHofKom14} minimal, and for a more detailed overview of the methodology for $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<\tau-2$ we refer the reader to \cite[Section 1.3]{BarHofKom14}.
We use the shorthand notation $X_n \buildrel {\mathbb{P}} \over{\sim} n^{a}$ if there exists a constant $b\ge 0$ such that $\mathbb{P}( X_n \in ( (\log n)^{-b} n^a, (\log n)^b n^a) ) \to 1$. We call vertices with degree at least $\buildrel {\mathbb{P}} \over{\sim} \!n^{(\tau-2)/(\tau-1)}$ \emph{hubs}.
\begin{enumeratei}
\item \label{ph::bp} \emph{Branching process phase.}\\ We couple the initial stages of the growth to two independent branching processes. The coupling fails when one of the colors (wlog we assume it is red)
reaches size $n^\varrho$ for some $\varrho>0$ sufficiently small.
\item \label{ph::montain_up}\emph{Mountain climbing phase.}\\
After the coupling fails, we build a path through higher and higher degree vertices to a vertex with degree at least $\buildrel {\mathbb{P}} \over{\sim} n^{(\tau-2)/(\tau-1)}$. The length of this path is of constant order. We denote the total time to reach such a vertex by red by $T_r$, and the time it would take for blue (if red would not be present at all) by $T_b$. While doing so, we arrange the vertices in layers of nested sets that can be though of as level sets of a (imaginary) mountain where the height function is linear in the $\log \log$( degree), hence the name of the phase.
\item \label{ph::peak}\emph{Crossing the peak of the mountain.} \\
The degree of the maximal degree vertex in the graph is $\buildrel {\mathbb{P}} \over{\sim} \!n^{1/(\tau-1)}$, and vertices of approximately this degree form a complete graph whp. Hence, when red occupies one of these hubs, in the next step it occupies all of them. We very carefully handle how red crosses the peak. If blue is at much lower degrees at time $T_r$ (meaning, $T_b\ge T_r+2$), then the proof follows the same method as for the $\lambda>1$ case, and red will occupy all vertices with degree higher than $\buildrel {\mathbb{P}} \over{\sim} \!n^{(\tau-2)^{\delta_n^{\sss{(r)}}}/(\tau-1)}$ for some $\delta_n^{\sss{(r)}} \in (0,1)$.
On the other hand, if $T_b=T_r$, then both colors arrive to the hubs at the same time, hence, in the next step, they arrive to almost all hubs at the same time, and hence, these vertices are painted with equal probability red and blue, respectively. Hence, we get Theorem \ref{thm::maxdegree} part 2(a).
Further, if $T_b=T_r+1$, then at time $T_r+1$ red can cross the peak, and occupies almost all vertices with degree higher than $\buildrel {\mathbb{P}} \over{\sim} \! n^{(\tau-2)^{\delta_n^{\sss{(r)}}}/(\tau-1)}$ for some $\delta_n^{\sss{(r)}} \in (0,1)$, while blue occupies a few vertices of degree higher than $\buildrel {\mathbb{P}} \over{\sim} \!n^{(\tau-2)/(\tau-1)}$, leading to Theorem \ref{thm::maxdegree} part 2(b), 2(c).
The question at this point becomes what happens at time $T_r+2$. We say that at time $T_r+2$, blue can also `cross over' the already red peak, if there is a $\delta_n^{\sss{(b)}}>\delta_n^{\sss{(r)}}$ so that blue occupies approximately half of the vertices between $\buildrel {\mathbb{P}} \over{\sim} \! n^{(\tau-2)^{\delta_n^{\sss{(b)}}}/(\tau-1)}$ and $\buildrel {\mathbb{P}} \over{\sim} \! n^{(\tau-2)^{\delta_n^{\sss{(r)}}}/(\tau-1)})$. We show that this happens if and only if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}> \tau-2$.
From here, the proofs separate for $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$ and $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<\tau-2$.
As a side-result, the analysis of the crossing of the mountain phase leads to the proof of Theorem \ref{thm::distances}.
Note that for typical distances, it is not necessary to let red and blue grow simultaneously. Hence, we let red grow $T_r$ steps, blue $T_b$ steps, show that they whp do not meet until this time, but they reach the top of the mountain, and then the same analysis as the one for crossing the peak shows that the typical distance is either $T_r+T_b+1$ or $T_r+T_b+2$.
\end{enumeratei}
The proof for $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<\tau-2$:
\begin{enumerate}
\item[(iv)(1)] \label{ph::mountain_down}\emph{Red avalanche from the peak.}\\
After crossing the mountain, red starts occupying all vertices of less and less degree. We call this the \emph{avalanche-phase of red}. This is similar to the $\lambda>1$ case.
\item[(v)(1)] \label{ph::meeting}\emph{At the collision time.}\\
Meanwhile, blue did its mountain climbing phase as well, and at time $T_r+1$, it is exactly $T_b-T_r-1$ many steps away from reaching the top of the mountain.
Then, approximately at time $T_r + 1 + (T_b- T_r-1)/2$ the red avalanche has sloped down to the same degree vertices as blue has climbed up to, hence they meet by arriving to vertices at the same time. Since this expression is not necessarily an integer, we have to investigate their meeting time\footnote{Note that this method provides and alternative proof for typical distances in Theorem \ref{thm::distances}. Of course, the two methods yield the exact same result, but the proof presented in Section \ref{sc::meetingtime} is much shorter.} and the maximal degree of blue more carefully. \item[(vi)(1)] \label{ph::after_meeting}\emph{Competing with the avalanche.}\\
After the meeting time, blue cannot occupy higher degree vertices anymore, since those are already all red.
Note that at this time most of the graph is still not reached by any color.
We estimate how many vertices blue can still paint in two steps: first we calculate the size of the `optional cluster of blue', i.e.\ we calculate the size of the $k$-neighborhood of blue half-edges via path-counting methods, yielding an upper bound.
Vertices in the optional cluster of blue are `close' to a blue half edge, hence, they will be blue unless they are occupied by red simply because they are `accidentally' also `close' to some red half-edge. In the second step we estimate the size of the intersection between the optional cluster of blue and the red cluster. The two steps together provide matching upper and lower bounds for the number of vertices that blue occupies after the intersection. \end{enumerate}
The proof for $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$:
\begin{enumeratei}
\item[(iv)(2)]\label{ph::mountain_down-mixed}\emph{Mixed avalanche from the peak.}
While most vertices close to the mountain-top are all red, there is an interval $(\delta_n^{\sss{(r)}}, \delta_n^{\sss{(b)}})$ in the $\log\log$( degrees) that has `mixed' coloring (see the interval in (iii)). We show that this pattern `rolls down' the mountain, i.e., for each $m\le \nu \log\log n/|\log(\tau-2)|$ for some $\nu<1$, each vertex with degree in between $\buildrel {\mathbb{P}} \over{\sim} \! n^{(\tau-2)^{m+\delta_n^{\sss{(b)}}}/(\tau-1)}$ and $\buildrel {\mathbb{P}} \over{\sim} \!n^{(\tau-2)^{m+\delta_n^{\sss{(r)}}}/(\tau-1)}$ is again painted red and blue with equal probability. We stop this process after approximately $\nu \log\log n/|\log(\tau-2)|$ steps and denote the degree of vertices `at the bottom of' the last colored interval by $Q$.
\item[(v)(2)] \label{ph::meeting}\emph{Coloring the neighbourhood of a random vertex.}\\
To determine the proportion of blue vertices, we look at a uniform random vertex $w$. We couple its local neighborhood in the graph to a branching process that is a copy of the BPs described in the branching process phase.
We introduce the random stopping time that is the number of generations needed for this BP to reach at least one colored vertex with degree at least $Q$. We look at the vertices in the last generation of the stopped BP with degree higher than $Q$. We color such a vertex red or blue with equal probability if its degree falls into a mixed interval, and color it red if it falls into an all-red interval, providing a partial coloring of the last generation of the stopped BP.
\item[(vi)(2)] \label{ph::random bootstrap}\emph{Random bootstrap percolation to the root.}\\
After the last generation of the BP tree with root $w$ has been partially colored, we do the following recursive procedure to (partially) color the earlier generations of the BP:
if a vertex has children of both colors, then we paint it blue and red with equal probability. If it has children of only one color, then we paint it in that color deterministically. If it has no colored children, then it stays uncolored. It is not hard to see that this is exactly the procedure how the two colors reach the local neighbourhood of a vertex $w$.
We show that in this random bootstrap procedure, the root gets color blue or red each with strictly positive probability (summing up to $1$) as $n\to \infty$. A second moment method - investigating two vertices instead of only one - finishes the proof of coexistence.
\end{enumeratei}
\subsubsection*{Notation}
We write $[n]$ for the set of integers $\{1,2,\dots, n\}$. We denote by the same name and add a superscript $(r), (b)$ to random variables, sets or other quantities belonging to the red and blue processes, respectively. We write $E({\mathrm{CM}}_n(\boldsymbol{d}))$ for the set of edges in ${\mathrm{CM}}_n(\boldsymbol{d})$. Note that multiple edges might also occur. For any set of vertices $S\subset [n]$, we write $N(S)$ for the set of their neighbors, i.e.,
\begin{equation} \label{def::ns}N(S)=\{y\in [n]: \exists x\in S, (x,y) \in E({\mathrm{CM}}_n(\boldsymbol{d}))\},\end{equation}
where $(x,y)$ might not be unique if there are multiple edges between a vertex $x\in S$ and $y\in N(S)$.
For any event $A$, $\mathbb{P}_n(A):=\mathbb{P}(A| D_1, D_2, \dots, D_n)$. As usual, we write i.i.d.\ for independent and identically distributed, lhs and rhs for left-hand side and right-hand side. We write $\lfloor x\rfloor, \lceil x \rceil$ for the lower and upper integer part of $x\in \mathbb{R}$, and $\{x\}$ for the fractional part of $x\in \mathbb{R}$. Slightly abusing the notation, we use curly brackets around set elements, events and long exponents as well.
We use $\buildrel {d}\over{\longrightarrow}, \buildrel {\Pv}\over{\longrightarrow}, \buildrel {a.s.}\over{\longrightarrow}$ for convergence in distribution, in probability and almost surely, respectively. We use the Landau symbols $o(\cdot), O(\cdot), o_{\mathbb{P}}(\cdot), O_{\mathbb{P}}(\cdot)$ in the usual way.
We say that a sequence of events $\mathcal E_n$ occurs with high probability (whp) when $\lim_{n\to \infty}\mathbb{P}(\mathcal E_n) = 1.$ In this paper, constants are typically denoted by $c$ in lower and $C$ in upper bounds (possible with indices to indicate which constant is coming from which bound), and their precise values might change from line to line. Typically, all the whp-events hold whp under the event $\{\mathcal {L}_n\in [ \mathbb{E}[D]n /2, 2 \mathbb{E}[D] n] \}.$
\section{The branching process phase}\label{sc::BP}
In this section we briefly summarise the key idea of this phase. In the construction of the configuration model, we start pairing the half-edges in an arbitrary order, and each time we pair a half-edge, we can pick the next one to pair as we wish. Hence, we can do the pairing in an order that corresponds to the spread of the two colors. More precisely, first we pair all the outgoing half-edges from the source vertices (time $t=1$), then we pair the outgoing half-edges from the neighbors of the source vertices (time $t=2$), and so on, in a breadth-first search manner. Whenever we finish pairing all the half-edges attached to vertices at a given graph distance from the source vertices, we increase the spreading time in the competitive coloring process by $1$. This process of joint construction of the competition and graph building is often called the \emph{exploration process} in the literature.
The key idea is that cycles in the exploration process are improbably as long as the total number of half-edges attached to colored vertices is small. Hence, the initial stage of the exploration can be coupled to a random tree, i.e., a branching process. For more details on the exploration process specific to this particular model, see \cite[Section 2]{BarHofKom14}.
Let $B_{i}$ stand for the \emph{forward-degree} of the $i$-th colored vertex $v_{i}$ for $i\ge 2$ in this exploration process, i.e., the number of half-edges incident to $v_i$ that are not paired yet when $v_i$ is reached in the exploration process. Since the probability of picking a half-edge that belongs to a vertex with degree $j+1$ is approximately equal to $(j+1)\mathbb{P}(D=j+1)/\mathbb{E}[D]$, we get the size-biased distribution \eqref{def::size-biased1}
as a natural candidate for the forward degrees of the vertices $v_i$ in the exploration process.
More precisely, \cite[Lemma 2.2]{BarHofKom14} based on \cite[Proposition 4.7]{BHH10} states that the number of vertices and their forward degrees in the exploration process can be coupled to i.i.d.\ degrees having distribution function $F^\star$ from \eqref{def::size-biased1}, as long as the total number of vertices of the colored clusters does not exceed $n^{\varrho'}$ for some small $\varrho'>0$.
An immediate consequence of this lemma is that locally we can consider the growth
of $\mathcal{R}_{t}$ and $\mathcal{B}_t $ as independent branching processes $(Z_{k})_{k>0}$ with offspring distribution $F^\star$ for the second and further generations, and with offspring distribution given by $F$ for the first generation.
The following theorem by Davies \cite{D78} describes the growth rate of a similar branching process:
\begin{theorem}[Branching process with infinite mean \cite{D78}]\label{thm::davies} Let $\widetilde Z_k$ denote the $k$-th generation of a branching process with offspring distribution given by the distribution function $F^\star$. Suppose there exists an $x_{0}>0$ and a function $x\mapsto\kappa(x)$ on $\mathbb{R}^+$ that satisfies the following conditions:
\begin{enumeratei}
\item $\kappa(x)$ is non-negative and non-increasing,
\item $x^{\kappa(x)}$ is non decreasing,
\item $\int\limits_0^\infty\kappa\left(\mathrm e^{\mathrm e^x}\right)\mathrm d x<\infty$.
\end{enumeratei}
Let us assume that the tail of the offspring distribution satisfies that for some $\tau\in(2,3)$ and for all $x\ge x_0$,
\begin{equation}\label{eq::davies_cond}
x^{-(\tau-2)-\kappa(x)}\leq 1-F^\star(x)\leq x^{-(\tau-2)+\kappa(x)}.
\end{equation}
Then $(\tau-2)^{k}\log(\widetilde{Z}_{k}\vee 1)$ converges almost surely to a random variable $\widetilde Y$. Further, the variable $\widetilde Y$ has exponential tails: if $J(x):=\mathbb{P}(\widetilde Y\le x)$, then
\begin{equation}\label{eq::exp-tails-Y} \lim_{x\to \infty} \frac{- \log (1-J(x))}{x} =1.\end{equation}
\end{theorem}
It is an elementary calculation to check that $F^\star$ in \eqref{def::size-biased1} satisfies the criterions of this theorem, and a simple modification yields that the same convergence holds for the branching process where the size of the first generation has distribution $F$ instead of $F^\star$, see \cite[Lemma 2.4]{BarHofKom14}.
Hence, we get that there exists a random variable $Y$ with exponentially decaying tail, such that this BP satisfies
\begin{equation}\label{eq::limitY} \lim_{k\to \infty} (\tau-2)^k \log Z_k \buildrel {a.s.}\over{\longrightarrow} Y,\end{equation}
since $Z_k\ge 1$ a.s., so $Z_k\vee 1 = Z_k$.
The random variables defined in Definition \ref{def::limit-variables} are finite time approximations of two independent copies of $Y$, denoted by $Y_r$ and $Y_b$ (standing for red and blue), respectively, at the time when the number of vertices colored by one of the colors reaches size $n^{\varrho}$ for some small $\varrho>0$. From now on, and without loss of generality, we assume that this color is red.
\section{Mountain-climbing phase}\label{sc::climbup}
In this section we briefly summarise the key ideas in the mountain-climbing phase and recall notation that will be needed later on. For a more detailed description we refer the reader to \cite[Section 3]{BarHofKom14}. Again, we assume that red is the color that reaches size $n^\varrho$ first.
Recall that the coupling of the number of vertices in the BP and in the growing cluster of the colors fails when one of the colors reaches $n^{\varrho'}$ many vertices. For our purposes later on, we rather want to guarantee that the number of vertices in the last generation of the BP is \emph{at least} some power of $n$. We can assure this by stopping the BP a bit earlier: let us first set some $\varrho<\varrho' (\tau-2)$ and define
\[ t(n^{\varrho})=\inf \{k: Z_k^{\scriptscriptstyle{(r)}} \ge n^{\varrho} \}.\]
Recall Definition \ref{def::limit-variables}, i.e.,
\begin{equation} \label{def::Y_r*}Y_r^{\scriptscriptstyle{(n)}}:=(\tau-2)^{t(n^{\varrho})}\log Z_{t(n^{\varrho})}. \end{equation}
An elementary rearrangement yields that, taking $Y_r^{\scriptscriptstyle{(n)}}$ as given and with $\{x\}=x-\lfloor x \rfloor$,
\begin{equation}\label{eq::an2} t(n^{\varrho}) = \frac{\log(\varrho/Y_r^{\scriptscriptstyle{(n)}}) + \log\log n}{|\log(\tau-2)|}+ 1-a_n^{\scriptscriptstyle{(r)}},\end{equation}
where
\begin{equation}\label{eq::an} a_n^{\scriptscriptstyle{(r)}}= \left\{ \frac{\log(\varrho/Y_r^{\scriptscriptstyle{(n)}}) + \log\log n}{|\log(\tau-2)|}\right\}. \end{equation}
Note that $1-a_n^{\scriptscriptstyle{(r)}}$ is there to make the expression on the rhs of $t(n^{\varrho})$ equal to its upper integer part. Due to this effect, the last generation has a bit more vertices than $n^{\varrho}$, so let us introduce the notation $\varrho^{\scriptscriptstyle{(r)}}$ for the random exponent of the overshoot
\begin{equation} \label{eq::rho_0} Z_{t(n^{\varrho})}^{\scriptscriptstyle{(r)}}= n^{ \varrho (\tau-2)^{a_n^{\scriptscriptstyle{(r)}}-1} }:= n^{\varrho^{\scriptscriptstyle{(r)}}},\end{equation}
We get this expression by rearranging \eqref{def::Y_r*} and using the value $t(n^{\varrho})$ from \eqref{eq::an2}.
The property $\varrho< \varrho'(\tau-2)$ and $a_n^{\scriptscriptstyle{(r)}} \in [0,1)$ implies that $\varrho^{\scriptscriptstyle{(r)}}< \varrho'$, which in turn guarantees that the coupling is still valid, i.e., we can also couple the \emph{degrees of vertices} in the $t(n^{\varrho})$th generation of the branching process to i.i.d.\ size-biased degrees.
The second step in this section is to decompose the high-degree vertices in the graph into the following sets, that we call \emph{layers}:
\begin{equation}\label{def::Gamma_i} \Gamma_i^{\scriptscriptstyle{(r)}}:=\{ v: D_v>u_i^{\scriptscriptstyle{(r)}} \}, \end{equation}
where $u_i^{\scriptscriptstyle{(r)}}$ is defined recursively by
\begin{equation}\label{eq::ui_recursion} u_{i+1}^{\scriptscriptstyle{(r)}} =\left(\frac{u_{i}^{\scriptscriptstyle{(r)}}}{C\log n}\right)^{1/(\tau-2)}, \ \ \qquad u_0^{\scriptscriptstyle{(r)}}:= \bigg(\frac{n^{\varrho^{\scriptscriptstyle{(r)}}}}{C\log n}\bigg)^{1/(\tau-2)} \end{equation}
for a large enough constant $C>0$ (e.g. $C= 8/c_1$ is sufficient, where $c_1$ is introduced in \eqref{eq::F}). It is not hard to see that
\begin{equation}\label{def::ui} u_{i}^{\scriptscriptstyle{(r)}}= n^{\varrho^{\scriptscriptstyle{(r)}} (\tau-2)^{-(i+1)}} (C\log n)^{-e_i}\quad \mbox{with} \quad
e_i = \frac{1}{3-\tau}\bigg( \Big( \frac{1}{\tau-2}\Big)^{i+1}-1\bigg). \end{equation}
Note that since $(\tau-2)^{-1}>1$, $u_i^{\scriptscriptstyle{(r)}}$ is growing, hence $\Gamma_0 \supset \Gamma_1 \supset \Gamma_2 \supset \dots$.
The third step is to show that $Z_{t(n^{\varrho})}$ has a nonempty intersection with the initial layer $\Gamma_0$.
The proof of this is based on the following lemma, that we cite here, since we repeatedly use it later on.
\begin{lemma}\label{lem::maxdegree}
Let $X_i, \ i=1, \dots, m$ be i.i.d.\ random variables with power-law exponent $\alpha$, i.e., the distribution function of $X_i$ satisfies \eqref{eq::F} with $\tau-1$ replaced by any $\alpha>0$.
Then for any $K>0$
\begin{equation}\label{eq::logwhp} \mathbb{P}\bigg(\max_{i=1,\dots, m} X_i < \Big(\frac{ m}{K\log m}\Big)^{1/\alpha} \bigg) \le \frac{1}{m^{c_1K}}, \end{equation}
where $c_1$ is introduced in \eqref{eq::F}.
\end{lemma}
Now we can apply this lemma since the last generation in the BP is an i.i.d.\ collection of $n^{\varrho^{\scriptscriptstyle{(b)}}}$ many power-law random variables, and the maximum of these behaves approximately as $n^{\varrho^{\scriptscriptstyle{(r)}}/(\tau-2)}$. The definition of $u_0^{\scriptscriptstyle{(r)}}$ adds a logarithmic correction term to this, hence, whp there is a vertex with degree at least $u_0^{\scriptscriptstyle{(r)}}$ in the last generation of the BP. For more details, see \cite[Lemma 3.1]{BarHofKom14}.
The fourth step is to show the existence of a red path from $\Gamma_0\cap \mathcal {R}_{t(n^{\varrho})}$ to a vertex that has degree at least $\buildrel {\mathbb{P}} \over{\sim} \!n^{(\tau-2)/(\tau-1)}$.
The existence of such a path is guaranteed by \cite[Lemma 3.4]{BarHofKom14}, based on simple concentration of binomial random variables, stating that whp
\begin{equation}\label{eq::gamma-i-connectivity} \forall i\ \ \Gamma_i \subset N(\Gamma_{i+1}), \end{equation}
where recall that $N(S)$ stands for the set of neighbors of $S$.
\begin{figure}\label{fig::mountain}
\includegraphics[width=0.5\textwidth]{Mountain-9p.pdf}
\caption{An illustration of the layers and the mountain climbing phase at time $t(n^{\varrho})+3$. Disclaimer: the degrees on the picture are only an illustration.}
\end{figure}
With \eqref{eq::gamma-i-connectivity} in hand, we can determine how long it takes to climb up through the layers $\Gamma_i^{\scriptscriptstyle{(r)}}$ to the highest-degree vertices in the graph. Lemma \ref{lem::maxdegree} with $X_i=D_i\sim F$, $\alpha=\tau-1$ shows that the maximal degree in ${\mathrm{CM}}_n(\boldsymbol{d})$ is of order $n^{1/(\tau-1)}$. We write $i_{\star\scriptscriptstyle{(r)}}$ for the last index\footnote{We put the brackets $(r)$ now in the subscript only because we want to avoid that quantities with both subscripts and superscripts look messy: compare $u_{ i_{\star\scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}}$ to $ u_{ i_{\star}^{\scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}}$.} when $\Gamma_{i}^{\scriptscriptstyle{(r)}}$ is whp nonempty, i.e.,
\begin{equation}\label{def::i*}i_{\star\scriptscriptstyle{(r)}}:=\inf \{ i: u_i^{\scriptscriptstyle{(r)}} \le n^{1/(\tau-1)} < u_{i+1}^{\scriptscriptstyle{(r)}}\}. \end{equation}
An easy calculation using \eqref{def::ui} shows that
\begin{equation}\label{eq::value_i*} i_{\star \scriptscriptstyle{(r)}}= -1+ \frac{-\log ((\tau-1)\varrho^{\scriptscriptstyle{(r)}})}{|\log(\tau-2)|}-b_n^{\sss{(r)}},
\quad \mbox{ with }\quad
b_n^{\sss{(r)}}=\left\{ \frac{-\log ((\tau-1)\varrho^{\scriptscriptstyle{(r)}})}{|\log(\tau-2)|}\right\}. \end{equation}
Using the value of the overshoot exponent $\varrho^{\scriptscriptstyle{(r)}}$ in \eqref{eq::rho_0} and then the value $a_n^{\scriptscriptstyle{(r)}}$ in \eqref{eq::an}, plus the fact that $\{x - 1+\{y\}\}=\{x+y\}$, we get that
\begin{equation}\label{eq::bn} b_n^{\sss{(r)}} = \left\{ \frac{-\log ((\tau-1)\varrho)}{|\log(\tau-2)|}+ a_n^{\scriptscriptstyle{(r)}}-1\right\} = \left\{ \frac{-\log ((\tau-1)Y_r^{\scriptscriptstyle{(n)}})+\log \log n}{|\log(\tau-2)|}\right\}.\end{equation}
From \eqref{def::ui} one can easily calculate that
\begin{equation}\label{eq::ui*} \begin{aligned} u_{i_{\star\scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}}&=n^{\frac{(\tau-2)^{b_n^{\sss{(r)}}}}{\tau-1}} (C \log n)^{-e_{i_{\star\scriptscriptstyle{(r)}}}}, \quad \mbox{ with} \\
e_{i_{\star\scriptscriptstyle{(r)}}}&=\frac{1}{3-\tau}\left(\frac{(\tau-2)^{b_n^{\sss{(r)}}}}{(\tau-1)\varrho^{\scriptscriptstyle{(r)}}} -1\right)\le\frac{1}{(3-\tau)}\left(\frac{1}{(\tau-1)\varrho^{\scriptscriptstyle{(r)}}}-1\right). \end{aligned} \end{equation}
In what follows, we will often use the total time to reach the top, so let us introduce the notation
\begin{equation}\label{eq::k*+i*} T_r:=t(n^{\varrho})+i_{\star\scriptscriptstyle{(r)}}=\frac{\log\log n-\log \left((\tau-1) Y_r^{\scriptscriptstyle{(n)}}\right)}{|\log(\tau-2)|} -1-b_n^{\sss{(r)}},\end{equation}
which only depends on $\varrho$ through the approximating $Y_r^{\scriptscriptstyle{(n)}}$, and $b_n^{\sss{(r)}}$ is exactly the fractional part of the expression on the rhs of $T_r$. Since also $Y_r^{\scriptscriptstyle{(n)}}\buildrel {d}\over{\longrightarrow} Y_r$ irrespective of the choice of $\varrho$, this establishes that the choice of $\varrho$ is not relevant in the proof.
Note that if color red would not be present in the graph, we could repeat the same procedure for blue, yielding the definitions \eqref{def::ti-bi}. It is not hard to see using \eqref{eq::an2} that
\begin{equation}\label{eq::BPblue} Z_{t(n^{\varrho})}^{\scriptscriptstyle{(b)}} = n^{ \varrho (\tau-2)^{a_n^{\scriptscriptstyle{(r)}}-1} Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} },\end{equation}
hence, using $\varrho^{\scriptscriptstyle{(b)}}:=\varrho (\tau-2)^{a_n^{\scriptscriptstyle{(r)}}-1} Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}$ instead of $\varrho^{\scriptscriptstyle{(r)}}$, we can define
\begin{equation}\label{eq::b-definitions}\begin{aligned} u_0^{\scriptscriptstyle{(b)}}&:= Z_{t(n^{\varrho})}^{\scriptscriptstyle{(b)}} = n^{ \varrho^{\scriptscriptstyle{(b)}} }, \qquad \ \ \
u_{i+1}^{\scriptscriptstyle{(b)}}:=\left(\frac{u_{i}^{\scriptscriptstyle{(b)}}}{C\log n}\right)^{1/(\tau-2)}, \\
\Gamma_i^{\scriptscriptstyle(b)}&:= \{ v: D_v>u_i^{\scriptscriptstyle{(b)}} \}, \qquad i_{\star \scriptscriptstyle{(b)}} := -1+ \frac{-\log ((\tau-1)\varrho^{\scriptscriptstyle{(b)}})}{|\log(\tau-2)|}-b_n^{\sss{(b)}}\end{aligned} \end{equation}
where $ b_n^{\sss{(b)}}=\left\{ \frac{-\log ((\tau-1)Y_b^{\scriptscriptstyle{(n)}})+\log \log n}{|\log(\tau-2)|}\right\}$ similarly as in \eqref{eq::bn}.
Note also that establishing the existence of a path to the top of the degree mountain only provides an \emph{upper bound} on how long it takes for each color to reach the top. However, with a bit of work we can turn these bounds into a \emph{matching lower bound} as well. This will be relevant later, since it also determines \emph{an upper bound on the maximal degree} of the colors in their climbing phase as well as the number of vertices in each layer they occupy.
Similarly as in \eqref{def::Gamma_i} and \eqref{eq::ui_recursion}, let us define, for $j=r,b$,
\begin{equation}\begin{aligned}\label{eq::uibar}
\widehat u_0^{\scriptscriptstyle(j)}&:= (Z^{\scriptscriptstyle{(j)}}_{t(n^{\varrho})}\cdot C \log n)^{1/(\tau-2)}, \\
\widehat u_{i+1}^{\scriptscriptstyle{(j)}}&:= (\widehat u_i^{\scriptscriptstyle{(j)}}\cdot C \log n)^{1/(\tau-2)}, \\
\widehat\Gamma_i^{\scriptscriptstyle{(j)}}&:=\{v \in {\mathrm{CM}}_n(\boldsymbol{d}): d_v \ge \widehat u_i^{\scriptscriptstyle{(j)}}\},
\end{aligned}\end{equation}
Note that $\widehat\Gamma_i^{\scriptscriptstyle{(j)}}$ grows faster than $\Gamma_i^{\scriptscriptstyle{(j)}}$ since there is always an extra $(C\log n)^2$ factor causing an initial `gap' of order $(\log n)^2$ between $u_0^{\scriptscriptstyle{(r)}}, \widehat u_0^{\scriptscriptstyle{(r)}}$ and `opening up' as $i$ gets larger.
The next lemma handles the upper bound on the maximal degree of red or blue at any time $t(n^{\varrho})+i$, but first some definitions. We say that a sequence of vertices and half-edges $(\pi_0, s_0, t_1, \pi_1, s_1, t_2, \dots, t_k, \pi_k)$ forms a \emph{path} in ${\mathrm{CM}}_n(\boldsymbol{d})$, if for all $0< i\le k$, the half edges $s_i, t_i$ are incident to the vertex $\pi_i$ and $(s_{i-1}, t_i)$ forms an edge between $\pi_{i-1},\pi_i$.
Let us denote the vertices in a path starting from a half-edge in $Z^{\scriptscriptstyle{(j)}}_{ t(n^{\varrho})}$ by $\pi_0, \pi_1, \dots $. We say that a path is \emph{good} if $\deg(\pi_i)\le\widehat u_i^{\scriptscriptstyle{(j)}}$ holds for every $i$. Otherwise we call it \emph{bad}. We decompose the set of red and blue bad paths in terms of where they turn bad, i.e.\ we say that a bad path of color $j$ is belonging to $\mathcal {B} ad \mathcal {P}_k^{\scriptscriptstyle{(j)}}$ if it turns bad at the $k$th step (for $j=r,b$):
\begin{equation}\label{def::badpaths} \begin{aligned} \mathcal {B} ad\mathcal {P}_k^{\scriptscriptstyle{(j)}} := &\{ (\pi_0, s_0, t_1, \pi_1, s_1 \dots, t_k, \pi_k) \text{ is a path, } \\
&\quad \pi_0\!\in\!\mathcal{C}^{\scriptscriptstyle{(j)}}_{ t(n^{\varrho}) },\ \deg(\pi_i)\!\le\! \widehat u_i^{\scriptscriptstyle{(j)}} \ \forall i\le k-1,\ \deg(\pi_k)\!>\!\widehat u_k^{\scriptscriptstyle{(j)}} \},\end{aligned}\end{equation} where $\mathcal{C}^{\scriptscriptstyle{(r)}}=\mathcal {R}, \mathcal{C}^{\scriptscriptstyle{(b)}}=\mathcal {B}$.
The following lemma tells us that the probability of having a bad path is tending to zero:
Note that $u_{i}^{\scriptscriptstyle{(j)}}, \widehat u_{i}^{\scriptscriptstyle{(j)}}$ and hence $\Gamma_i^{\scriptscriptstyle{(r)}}, \widehat \Gamma_{i}^{\scriptscriptstyle{(j)}}, i_{\star \scriptscriptstyle{(j)}}$ are random variables that depend (only) on $n, Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}$. For a short notation for conditional probabilities and expectation let us introduce
\begin{equation}\label{def::py-ey} \mathbb{P}_{Y,n}(\,\cdot\,):=\mathbb{P}(\,\cdot\, | Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}, n) \quad \mathbb{E}_{Y,n}[\,\cdot\,]:= \mathbb{E}[ \,\cdot\, | Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}, n].\end{equation}
\begin{lemma}\label{lem::badpaths}
The following bound on the probability of having any bad paths holds for color $j=r,b$:
\begin{equation} \mathbb{P}_{Y,n}( \exists k\le i_{\star\scriptscriptstyle{(j)}}: \mathcal {B} ad\mathcal {P}_k^{\scriptscriptstyle{(j)}} \neq \varnothing) \le \frac{2}{C\log n}.\end{equation}
\end{lemma}
\begin{proof}
The statement is a direct consequence of the proof of \cite[Lemma 5.2]{BarHofKom14} in the Appendix of that paper. In that proof, the estimate gets worse as the total number of layers grows, (which is of order $\log\log n$ there): here the total number of layers is bounded for color red and whp less then any increasing function for color blue. More precisely, for red, $\varrho\le \varrho^{\scriptscriptstyle{(r)}}\le \varrho(\tau-2)^{-1}$, $b_n^{\sss{(r)}}\in [0,1)$, hence $i_{\star \scriptscriptstyle{(r)}}$ is bounded (see \ref{eq::value_i*}). For $i_{\star \scriptscriptstyle{(b)}}$, pick any function $\omega(n)$ such that $\log \omega(n)=o(\log\log n)$ that tends to infinity (e.g. $\omega(n) = \log\log n$ will do). Then,
\[ \mathbb{P}( \varrho^{\scriptscriptstyle{(b)}}=\varrho' (\tau-2)^{a_n^{\scriptscriptstyle{(r)}}-1} Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} < \omega(n)^{-1}) \to 0 \]
as $n\to \infty$, since $a_n^{\scriptscriptstyle{(r)}}\in [0,1)$. Then, this implies that $i_{\star \scriptscriptstyle{(b)}} = -1+ \frac{-\log ((\tau-1)\varrho^{\scriptscriptstyle{(b)}})}{|\log(\tau-2)|}-b_n^{\sss{(b)}}$ is whp $o(\log\log n)$ and hence the proof of \cite[Lemma 5.2]{BarHofKom14} is also valid for the blue process. \end{proof}
An immediate consequence is the following:
\begin{corollary}\label{core::time-to-top-who}
For color red whp it takes time $T_r$ to reach a vertex with degree at least $n^{(\tau-2)/(\tau-1)}$, and for color blue it would take time $T_b$ whp to do so if red would not be present.
\end{corollary}
\begin{proof} We have seen that $T_j$, $j=r, b$, is a whp an upper bound to reach the top (due to \eqref{eq::gamma-i-connectivity} yielding the existence of a path of length $T_j$). Now we argue that $T_j$ is also whp a lower bound to reach the top, that is, there is whp no path to the top shorter than $T_j$.
On the event $\{ \mathcal {B} ad\mathcal {P}_k^{\scriptscriptstyle{(r)}} = \varnothing, \mathcal {B} ad\mathcal {P}_k^{\scriptscriptstyle{(b)}} \neq \varnothing \}$, which occurs whp, we can use the upper bound $\widehat u_i^{\scriptscriptstyle{(j)}}$ on the degrees at time $t(n^{\varrho})+i$ for all $i\ge 0$, hence
we get that the time it takes to reach a vertex of degree at least $n^{(\tau-2)/(\tau-1)}$ is at least
\[ \widehat i_{\star, \scriptscriptstyle{(j)}} :=\inf \{ i: \widehat u_i^{\scriptscriptstyle{(j)}} \le n^{1/(\tau-1)} \le \widehat u_{i+1}^{\scriptscriptstyle(j)} \}. \]
Combine this with the fact that
\begin{equation}\label{eq::wideui-ui} \widehat u_i^{\scriptscriptstyle{(j)}} = u_i^{\scriptscriptstyle{(j)}} (C\log n)^{\tfrac{2}{3-\tau}\Big( \big( \tfrac{1}{\tau-2}\big)^{i+1}-1\Big)},\end{equation} and we get that $\widehat i_{\star, \scriptscriptstyle{(j)}}=i_{\star, \scriptscriptstyle{(j)}} $ whp.
Hence, $T_b$ and $T_r$ are both upper and lower bounds for this quantity.
\end{proof}
Later, we will need to estimate the number of vertices in each layer $\Gamma_i^{\scriptscriptstyle{(j)}}$ at the time of occupation (without the presence of the other color).
Let us denote the set and number of color $j$ vertices in the $i$th layer $\Gamma_i^{\scriptscriptstyle{(j)}}$ right at the time when color $j$ (denoted by $\mathcal{C}^{\scriptscriptstyle{(j)}}$ below) reaches it, by
\begin{equation}\label{def::Ai} \mathcal {A}_i^{\scriptscriptstyle{(j)}}:=\mathcal{C}_{t(n^{\varrho})+ i}^{\scriptscriptstyle{(j)}}\cap\Gamma_i^{\scriptscriptstyle{(j)}}, \qquad A_i^{\scriptscriptstyle{(j)}}:=|\mathcal {A}_i^{\scriptscriptstyle{(j)}}|. \end{equation}
\begin{lemma}\label{lem::numberofverticesinGamma} For $j=r,b$, on the event $\{\mathcal {B} ad\mathcal {P}_k^{\scriptscriptstyle{(j)}}=\varnothing \ \forall k\le i_{\star \scriptscriptstyle{(j)}} \}$, whp for all $ i\le i_{\star \scriptscriptstyle{(j)}}$,
\begin{equation} \label{eq::aifinal} \ A_i^{\scriptscriptstyle{(j)}}\le\exp\left\{ \log (C\log n) \cdot \frac{2 (\tau-2)^{-i}}{(3-\tau)^2} \right\}. \end{equation}
\end{lemma}
\begin{proof} The proof is based on concentration of binomial random variables, applied in two settings: (1) In the last generation of the BP approximation the degrees are i.i.d., hence the number of vertices with degree at least $u_0^{\scriptscriptstyle{(j)}}$ is binomial\footnote{In the graph ${\mathrm{CM}}_n(\boldsymbol{d})$ is hypergeometric, but in the coupling to i.i.d degrees in the BP we simply have $Z_{t(n^{\varrho})}^{\scriptscriptstyle{(j)}}$ many i.i.d. variables with distribution $D^\star$, so the number of vertices with degree at least $u_0^{\scriptscriptstyle{(j)}}$ is binomial in the BP}, and so whp there are only $A_0^{\scriptscriptstyle{(j)}}:=C\log n$ many vertices with degree at least $u_0^{\scriptscriptstyle{(j)}}$. (2) Given that there are $A_i^{\scriptscriptstyle{(j)}}$ many color $j$ vertices in layer $\Gamma_i^{\scriptscriptstyle{(j)}}$ with degree at most $\widehat u_i^{\scriptscriptstyle{(j)}}$, the number of vertices that they connect to in $\Gamma_{i+1}^{\scriptscriptstyle{(j)}}$ is stochastically dominated by a \[ {\sf Bin}\left(A_i^{\scriptscriptstyle{(j)}} \widehat u_{i}^{\scriptscriptstyle{(j)}}, \frac{{\mathcal{E}}_{\ge u_{i+1}^{\scriptscriptstyle{(j)}}}}{\mathcal {L}_n(1+o(1))}\right)\]
random variable, where ${\mathcal{E}}_{\ge y}$ denotes the total number of half-edges incident to vertices with degree at least $y$. More details are worked out in the proof of \cite[Lemma 5.4]{BarHofKom14}.
\end{proof}
\section{Crossing the peak of the mountain}\label{sc::peak}
Next we investigate what happens when the path through the layers reaches the highest degree vertices. This is also the part where the two proofs for $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} < \tau-2$ or $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} >\tau-2$ separate. To be able to handle all the cases, we add the superscript $(r)$ and $(b)$ to quantities in the previous section, meaning that they belong to the description of the growth of the red and the blue cluster, respectively.
For red, we have just seen that $u_{i_{\star \scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}} \buildrel {\mathbb{P}} \over{\sim} n^{\frac{(\tau-2)^{b_n^{\sss{(r)}}}}{\tau-1}}$, where the exponent is $ \in \left( \frac{\tau-2}{\tau-1}, \frac{1}{\tau-1}\right)$. Since the maximum degree in the graph is $\buildrel {\mathbb{P}} \over{\sim} n^{\frac{1}{\tau-1}}$ whp, we have $\Gamma_{1+i_{\star \scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}}=\varnothing$.
Following the lines of \cite{BarHofKom14}, we cite the lemma from \cite[Volume II., Chapter 5]{H10}:
\begin{lemma}\label{lem::direct_connect} Consider two sets of vertices $A$ and $B$. If the number of half-edges $\mathcal S_A=o(n)$ and $\mathcal S_B$ satisfy
\[ \frac{\mathcal S_A \mathcal S_B}{n}> h(n),\]
for some function $h(n)$, then conditioned on the degree sequence with $\mathcal {L}_n\le 2\mathbb{E}[D] n$, the probability that the two sets are not directly connected can be bounded from above by
\[ \mathbb{P}_n(A\nleftrightarrow B)< {\mathrm e}^{-\tfrac{h(n)}{4\mathbb{E}[D]}}.\]
\end{lemma}
\begin{proof} See the proof of \cite[Lemma 4.1]{BarHofKom14}.
\end{proof}
Let us introduce
\begin{equation}\label{def::alpha} \alpha_n^{\scriptscriptstyle{(r)}}:=1-\frac{(\tau-2)^{b_n^{\scriptscriptstyle{(r)}}}}{\tau-1}, \quad
\beta_n^{\scriptscriptstyle{(r)}}:=1+\frac{1}{(3-\tau)}\left(\frac{1}{(\tau-1)\varrho^{\scriptscriptstyle{(r)}}}-1\right), \end{equation}
\begin{equation}\label{eq::wideu1} \widetilde u^{\sss{(r)}}_1:= (C \log n) n/u_{i_{\star\scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}}=n^{ \alpha_n^{\sss{(r)}}} (C\log n)^{\beta_n^{\scriptscriptstyle{(r)}}},\end{equation}
and the following layer:
\begin{equation}\label{def::witgamma}\widetilde \Gamma^{\sss{(r)}}_1:=\{v\in {\mathrm{CM}}_n(\boldsymbol{d}), D_v > \widetilde u^{\sss{(r)}}_1\}.\end{equation}
Then, we can apply Lemma \ref{lem::direct_connect} by picking $A$ to be a single red vertex with degree higher than $u_{i_{\star\scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}}$, and $B$ any vertex in $\widetilde \Gamma^{\sss{(r)}}_1$. As a result, \emph{as long as blue does not interfere}, \cite[Lemma 4.2]{BarHofKom14} stays valid, stating that all the vertices in $\widetilde \Gamma^{\sss{(r)}}_1$ are occupied by red at time $T_r+1$, i.e.,
\begin{equation}\label{eq::slopestart} \widetilde \Gamma^{\sss{(r)}}_1\subset \mathcal R_{T_r+1} \quad \mbox{whp.}\end{equation}
Now we investigate what happens if blue \emph{does} interfere with this stage.
To be able to do so, we define the similar quantities for blue as around \eqref{eq::BPblue}, and define
\begin{equation}\label{eq::uistar-b} u_{i_{\star \scriptscriptstyle{(b)}}}^{\scriptscriptstyle{(b)}}:= n^{(\tau-2)^{b_n^{\sss{(b)}}}/(\tau-1)} (C\log n)^{-e_{i_{\star\scriptscriptstyle{(b)}}}}, \end{equation}
with $e_{i_{\star \scriptscriptstyle{(b)}}}$ as in \eqref{eq::ui*} but $\varrho^{\scriptscriptstyle{(r)}}$ replaced by $\varrho^{\scriptscriptstyle{(b)}}$, and finally
\begin{equation} \label{eq::wideu1-b} \widetilde u^{\sss{(b)}}_1:= (C\log n) n / u_{i_{\star\scriptscriptstyle{(b)}}}^{\scriptscriptstyle{(b)}} =: n^{\alpha_n^{\sss{(b)}}} (C\log n)^{\beta_n^{\sss{(b)}}},\end{equation}
where $\alpha_n^{\sss{(b)}}, \beta_n^{\sss{(b)}}$ are as in \eqref{def::alpha}, but the superscript $(r)$ and $\varrho^{\scriptscriptstyle{(r)}}$ replaced by $(b)$ and $\varrho^{\scriptscriptstyle{(b)}}$, respectively.
First of all, note that blue would occupy some vertices with degree at least $n^{(\tau-2)^{b_n^{\sss{(b)}}}/(\tau-1)(1+o_{\mathbb{P}}(1))}$ at time $T_b$, and it is working its way towards this degree by increasing its maximal degree by a factor of $1/(\tau-2)$ in the exponent with each step. Hence, at time $T_r$, blue occupies some vertices with $\log$(degree)/$\log n$ that is
\[ \frac{(\tau-2)^{b_n^{\sss{(b)}}}}{\tau-1}(\tau-2)^{T_b-T_r}(1+o_{\mathbb{P}}(1)).\]
Recall that $D_n^{\max}(t)$ stands for the degree of the maximal degree vertex that blue paints at or before time $t$. We have to distinguish three cases.
\begin{description}
\item[\namedlabel{case::tb-tr>2}{$(1)$}] $T_b-T_r\ge2$. In this case, at time $T_r+1$, blue occupies some vertices with $\log$(degree)/$\log n$
that is \[ \frac{\log (D_n^{\max}(T_r+1))}{\log n} = \frac{(\tau-2)^{b_n^{\scriptscriptstyle{(b)}}}}{\tau-1}(\tau-2)^{T_b-T_r-1}(1+o_{\mathbb{P}}(1)),\]
while red crosses the peak and tries to occupy every vertex in $\widetilde \Gamma^{\sss{(r)}}_1$ by \eqref{eq::slopestart}. It is not hard to see in \eqref{def::alpha} that $\alpha_n^{\sss{(r)}}>(\tau-2)/(\tau-1)$, hence
\[ \frac{\log \widetilde u^{\sss{(r)}}_1}{\log n} = \alpha_n^{\sss{(r)}} (1+ o_{\mathbb{P}}(1)) > \frac{\log (D_n^{\max}(T_r+1))}{\log n}. \]
In other words, if $T_b\ge T_r+2$ then blue does not reach any red vertex at time $T_r+1$. Red can cross the peak, and \eqref{eq::slopestart} is valid.
\item[\namedlabel{case::tb=tr}{$(2)$}] $T_b-T_r=0$. In this case, red and blue arrive to the top of the mountain at the same time. By the assumption that $Y_r^{\scriptscriptstyle{(n)}}>Y_b^{\scriptscriptstyle{(n)}}$, we have the easy statement
\begin{equation}\label{eq::frac-compare} \frac{\log\log n - \log ((\tau-1)Y_r^{\scriptscriptstyle{(n)}})}{|\log (\tau-2)|}-1 < \frac{\log\log n - \log ((\tau-1)Y_b^{\scriptscriptstyle{(n)}})}{|\log (\tau-2)|}-1 \end{equation}
Note that $T_r$ and $T_b$ are the integer parts of these expression. Hence, we conclude that $T_b=T_r$ is only possible if the two expressions in \eqref{eq::frac-compare} differ by at most $1$. This translates exactly to $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$. Secondly, also note that \eqref{eq::frac-compare} can be rewritten as
\begin{equation}\label{eq::key-1} T_r + b_n^{\sss{(r)}} < T_b + b_n^{\sss{(b)}}. \end{equation}
Note that on the event $T_r=T_b$, at time $T_r+1$, both colors try to cross the mountain at the same time, and red tries to occupy every vertex with degree at least $n^{\alpha_n^{\sss{(r)}}(1+o_{\mathbb{P}}(1))}$, while blue tries to occupy every vertex with degree at least $n^{\alpha_n^{\sss{(b)}}(1+o_{\mathbb{P}}(1))}$.
Now, \eqref{eq::key-1} implies that if $T_r=T_b$, we must have $b_n^{\sss{(r)}}<b_n^{\sss{(b)}}$, which in turn implies that $\alpha_n^{\sss{(r)}}<\alpha_n^{\sss{(b)}}$.
Let us introduce $\delta_n^{\scriptscriptstyle{(j)}}$ for $j=r,b$ implicitly by
\begin{equation}\label{def::delta} \alpha_n^{\scriptscriptstyle{(j)}} =: \frac{(\tau-2)^{\delta_n^{\scriptscriptstyle{(j)}}}}{\tau-1}. \end{equation}
It is not hard to see from \eqref{def::alpha} that $\delta_n^{\scriptscriptstyle{(j)}}\in(0,1]$, and $\delta_n^{\sss{(r)}}>\delta_n^{\sss{(b)}}$ when \eqref{eq::key-1} holds together with $T_r=T_b$.
Recall the rule that if red and blue arrive at a vertex at the same time, then the vertex gets each color with equal probability, independently of everything else. This rule, combined with the previous observation, implies that at time $T_r+1$, approximately half of the vertices with degree at least $\buildrel {\mathbb{P}} \over{\sim} \! n^{\alpha_n^{\sss{(b)}}}$ are red and blue, respectively, while almost all\footnote{In fact, a few vertices might be blue even in this interval, but the proportion of these vertices is negligible. An estimate on how many vertices might be blue is given in Lemma \ref{lem::red-intervals} below. For the purpose of understanding, at this point it is enough that `almost all' vertices are red.} vertices between degree
$(\buildrel {\mathbb{P}} \over{\sim} \!n^{\alpha_n^{\sss{(r)}}}, \buildrel {\mathbb{P}} \over{\sim} \!n^{\alpha_n^{\sss{(b)}}})$ are red.
We introduce the shorthand notation for this as
\begin{equation}\label{eq::shorthand}(0, \delta_n^{\sss{(b)}}] \in \text{Mix}, \quad (\delta_n^{\sss{(b)}}, \delta_n^{\sss{(r)}}]\in \mathcal {R}. \end{equation} For an illustration see Fig.\ \ref{fig::cross-1}. \begin{figure}
\includegraphics[width=0.6\textwidth]{Mountain-cross-1.pdf}
\caption{Case (2): An illustration of the crossing of the mountain peak when $T_b=T_r$ at time $T_r+1$. The crosshatched (darker) and dotted (lighter) areas indicate the part of the mountain where vertices are red and blue with equal probability, and almost all red, respectively.}\label{fig::cross-1}
\end{figure}
\item[\namedlabel{case::tb-tr=1}{$(3)$}]\label{case::tb-tr=1}$T_b=T_r+1$. In this case, red arrives to the top of the mountain just one step before blue. At time $T_r+1$, blue occupies some vertices of $\log$(degree)/$\log n$ approximately $(\tau-2)^{b_n^{\sss{(b)}}}/(\tau-1)(1+o_{\mathbb{P}}(1))$. Note that by Lemma \ref{lem::numberofverticesinGamma} blue occupies only a few vertices of exactly this degree, but quite a few vertices slightly below this value, i.e., we can add a slightly larger logarithmic correction term at this step and have plenty of vertices that blue tries to color at time $T_r+1$. So, even if the degree of these vertices is higher than $\widetilde u^{\sss{(r)}}_1$ (by the equal probability upon same arrival rule), blue will whp occury some vertices around this value. Meanwhile, red occupies every other vertex with $\log$(degree)/$\log n$ at least $\alpha_n^{\sss{(r)}}$.
The question is what happens at time $T_r+2$ in this case.
Apply Lemma \ref{lem::direct_connect} for the blue color at time $T_r+1$: one can see that the blue vertices with $\log$(degree)/$\log n$ at least $(\tau-2)^{b_n^{\sss{(b)}}}/(\tau-1)(1+o_{\mathbb{P}}(1))$ are whp connected to \emph{every vertex} with degree higher than $n^{\alpha_n^{\sss{(b)}}}$ (the proof is identical to that of \eqref{eq::slopestart}). Here, there are two cases: either $\alpha_n^{\sss{(b)}}>\alpha_n^{\sss{(r)}}$, and then these vertices are already all red, hence blue cannot occupy more vertices. Or, $\alpha_n^{\sss{(b)}}<\alpha_n^{\sss{(r)}}$, and in this case blue and red arrives to vertices with $\log$(degree)/$\log n$ in the interval $[\alpha_n^{\sss{(b)}}, \alpha_n^{\sss{(r)}})$ at the same time. Meanwhile, red occupies every vertex in the interval $[(\tau-2)\alpha_n^{\sss{(r)}}, \alpha_n^{\sss{(b)}})$.\footnote{This fact is verified in \eqref{eq::widetildegamm} below.}
To check which case of the two scenarios can happen we turn to \eqref{eq::frac-compare} and \eqref{eq::key-1}.
Since $T_r, T_b$ are the integer parts of the two sides of \eqref{eq::frac-compare}, $T_b=T_r+1$ can only happen if
$Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}> (\tau-2)^2$. Let us write shortly $q:=Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}$. There are two cases:
\begin{description}
\item[\namedlabel{casetb-tr=1,nc}{$(3)(a)$}] If $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} \in ((\tau-2)^2, \tau-2]$, then $-\log q/|\log(\tau-2)| \in [1,2).$ Hence, using $Y_b^{\scriptscriptstyle{(n)}}=qY_r^{\scriptscriptstyle{(n)}}$, the right hand side of \eqref{eq::frac-compare} can be rewritten in two ways as
\begin{equation}\label{eq::key-2} T_b+ b_n^{\sss{(b)}} = T_r + b_n^{\sss{(r)}} -\log q/|\log(\tau-2)|.\end{equation}
Since the last term is between $1$ and $2$, $T_b=T_r+1$ is only possible if $b_n^{\sss{(b)}}=(-\log q/|\log(\tau-2)|-1) + b_n^{\sss{(r)}}$, more importantly, if $b_n^{\sss{(b)}}>b_n^{\sss{(r)}}$. This in turn implies $\alpha_n^{\sss{(b)}}>\alpha_n^{\sss{(r)}}$. This means that at time $T_r+2$ blue tries to occupy vertices that are already colored red. Hence, blue cannot increase its degree anymore and is left with the few maximal degree vertices with $\log$(degree)/$\log n$ approximately $(\tau-2)^{b_n^{\sss{(b)}}}/(\tau-1)$. This means that similarly to Case \ref{case::tb-tr>2}, \emph{only red} can cross the peak and \eqref{eq::slopestart} is valid.
\item[\namedlabel{casetb-tr=1,coex}{$(3)(b)$}] If $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} \in ((\tau-2), 1]$ then $-\log q/|\log(\tau-2)| \in [0,1).$ Using the rewrite \eqref{eq::key-2} again, we see that $T_b=T_r+1$ in this case is only possible if $-\log q/|\log(\tau-2)| + b_n^{\sss{(r)}} >1$, and in this case $b_n^{\sss{(b)}}<b_n^{\sss{(r)}}$. (Otherwise, $T_r=T_b$ and we are back to Case (2) above.) This in turn implies $\alpha_n^{\sss{(b)}}<\alpha_n^{\sss{(r)}}$, hence, at time $T_r+2$,
blue and red both try to occupy vertices with $\log$(degree)/$\log n$ in the interval $[\alpha_n^{\sss{(b)}},\alpha_n^{\sss{(r)}})$: here, each vertex has an equal chance to be colored red or blue. On the other hand, at the same time red also occupies almost all vertices in the interval $[(\tau-2)\alpha_n^{\sss{(r)}}, \alpha_n^{\sss{(b)}})$.\footnote{Again, by \eqref{eq::widetildegamm} below. A few vertices might be blue here as well, see Lemma \ref{lem::red-intervals} for error bounds.} Recall $\delta_n^{\sss{(b)}}, \delta_n^{\sss{(r)}}$ from \eqref{def::delta}. In the shorthand notation introduced in \eqref{eq::shorthand}, we can write in this case
\begin{equation}\label{eq::shorthand-2} (\delta_n^{\sss{(r)}}, \delta_n^{\sss{(b)}}] \in \text{Mix}, \quad (\delta_n^{\sss{(b)}}, \delta_n^{\sss{(r)}}+1] \in \mathcal {R}. \end{equation} For an illustration see Fig. \ref{fig::cross-2}.
\end{description}
\end{description}
\begin{figure}
\includegraphics[width=0.6\textwidth]{Mountain-cross-2.pdf}
\caption{Case (3)(b): An illustration of the crossing of the mountain peak when $T_b=T_r+1$ at time $T_r+2$, and $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$. The crosshatched (darker) and dotted (lighter) areas indicate the parts of the mountain where vertices are red and blue with equal probability, and almost all red, respectively.}\label{fig::cross-2}
\end{figure}
This completes the \emph{crossing the peak of the mountain} phase.
\section{Red or mixed avalanche from the peak}\label{sc::slopedown}
In this section we describe how the (potentially) two colors roll down the mountain. As much as possible, we try to keep together the proofs for Cases (1) -- (3) above. From now on, we write $\mathbb{P}_Y(A):=\mathbb{P}(A| Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}, n)$. Note that all the quantities below are actually random variables that depend on $Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}$. We omit the dependence on $n$ in the notation.
Recall $\widetilde \Gamma^{\sss{(r)}}_1$ and $\widetilde u^{\sss{(r)}}_1, \widetilde u^{\sss{(b)}}_1$ from \eqref{def::witgamma} and \eqref{eq::wideu1} and \eqref{eq::wideu1-b}, as well as define for $j=r,b$
\begin{equation} \label{eq::wideui_recursion} \widetilde u_{\ell+1}^{\scriptscriptstyle{(j)}}=C \log n\!\cdot\!(\widetilde u_{\ell}^{\scriptscriptstyle{(j)}})^{\tau-2}. \end{equation}
and also the increasing sequence of sets
\begin{equation}\label{eq::wgar} \widetilde \Gamma_\ell^{\scriptscriptstyle{(j)}}:=\{ v: D_v > \widetilde u_\ell^{\scriptscriptstyle{(j)}}\},\end{equation}
i.e., now $\widetilde \Gamma^{\sss{(r)}}_1 \subset \widetilde \Gamma^{\sss{(r)}}_2 \subset \dots$ holds, and in Case (2) $\widetilde \Gamma^{\sss{(b)}}_1 \subset \widetilde \Gamma^{\sss{(r)}}_1 \subset \widetilde \Gamma^{\sss{(b)}}_2 \subset \widetilde \Gamma^{\sss{(r)}}_2 \dots$, (see Fig.\ \ref{fig::avalanche-2}), while in Case (3)(b) we have $\widetilde \Gamma^{\sss{(b)}}_1 \subset \widetilde \Gamma^{\sss{(r)}}_2 \subset \widetilde \Gamma^{\sss{(b)}}_2 \subset \widetilde \Gamma^{\sss{(r)}}_3 \dots$ (see Fig. \ref{fig::avalanche-3}).
Since \eqref{eq::wideui_recursion} is the very same as the recursion in \eqref{eq::ui_recursion}
with indices exchanged, we can apply equation \eqref{eq::gamma-i-connectivity} to $\big(\widetilde \Gamma^{\sss{(r)}}_\ell\big)_{\ell\ge 1}$, and to $\big(\widetilde \Gamma^{\sss{(b)}}_\ell\big)_{\ell\ge 1}$ now yielding that, for some $\nu<1$, for all $\ell< \nu \log\log n / |\log(\tau-2)|+ O_{\mathbb{P}}(1)$\footnote{This inequality on $\ell$ guaranties that the logarithmic correction terms in $\widetilde u_\ell$ are of less order than the main factor.},
\begin{equation}\label{eq::widetildegamm} \widetilde \Gamma^{\sss{(r)}}_{\ell+1}\subset N(\widetilde \Gamma^{\sss{(r)}}_\ell)\quad\mbox{ and }\quad \widetilde \Gamma^{\sss{(b)}}_{\ell+1} \subset N(\widetilde \Gamma^{\sss{(b)}}_{\ell}) \quad \mbox{whp.} \end{equation}
This means that
\begin{enumerate}
\item[(1)] if $T_b-T_r\ge 2$, whp red occupies whp \emph{all} vertices in $\widetilde\Gamma_\ell$ at time $T_r+\ell$. More precisely, as long as $\widetilde u^{\sss{(r)}}_\ell>D_{\max}(T_r+\ell)$, every vertex in $\widetilde \Gamma^{\sss{(r)}}_\ell$ is colored red. If this inequality is not true anymore, at time $T_r+\ell$, red still occupies every vertex in $\widetilde \Gamma^{\sss{(r)}}_\ell$ that is not in the $\ell$-neighborhood of a vertex that is already blue at time $T_r$. We work out the number of such vertices in Section \ref{sc::path-counting}.
\item[(2)] if $T_b-T_r=0$, we have seen in Section \ref{sc::peak} that at time $T_r+1$, every vertex in $\widetilde \Gamma^{\sss{(b)}}_1$ is red or blue with equal probability, and every vertex in $\widetilde \Gamma^{\sss{(r)}}_1\setminus \widetilde \Gamma^{\sss{(b)}}_1$ is red, see Fig. \ref{fig::cross-1}. We show in Lemma \ref{lem::independence} below that whp every vertex in $\widetilde \Gamma^{\sss{(b)}}_2\setminus \widetilde \Gamma^{\sss{(r)}}_1$ has plenty of neighbors in $\widetilde \Gamma^{\sss{(b)}}_1$, hence, the probability that none of them is blue or none of them is red is negligible. As a result, these vertices are again painted red and blue with equal probability at time $T_r+2$. On the other hand, almost all vertices in $\widetilde \Gamma^{\sss{(r)}}_2\setminus \widetilde \Gamma^{\sss{(b)}}_2$ whp only have neighbors\footnote{For an error bound on the number of blue vertices, see Lemma \ref{lem::red-intervals} below.} in $\widetilde \Gamma^{\sss{(r)}}_1\setminus \widetilde \Gamma^{\sss{(b)}}_2$, hence, most vertices in $\widetilde \Gamma^{\sss{(r)}}_2\setminus \widetilde \Gamma^{\sss{(b)}}_2$ are painted red whp. In the shorthand notation, we have
\[ (\delta_n^{\sss{(r)}}, \delta_n^{\sss{(b)}}+1] \in \mathcal{M}ix, \quad (\delta_n^{\sss{(b)}}+1, \delta_n^{\sss{(r)}}+1] \in \mathcal {R} ed\]
One can continue inductively to see that at time $T_r+\ell$, every $m\le \ell$
\begin{equation}\label{shorthand-case2-ell} (\delta_n^{\sss{(r)}}+m-1, \delta_n^{\sss{(b)}}+m)\in \mathcal {M} ix, \quad (\delta_n^{\sss{(b)}}+m, \delta_n^{\sss{(r)}}+m] \in \mathcal {R} ed,\end{equation}
that is, the `mixed-and-red' interval pattern `rolls down' the mountain. For the precise statement, see Lemmas \ref{lem::independence} and \ref{lem::red-intervals} below.
\item[(3)(a)] If $T_b-T_r=1$ and $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<\tau-2$, only a small number of vertices are blue. At time $T_r+\ell$, red occupies every vertex in $\widetilde \Gamma^{\sss{(r)}}_\ell$ that is not in the $\ell$-neighborhood of a vertex that is already blue at time $T_r$. This case is exactly the same as Case (1) after red and blue have already met.
\item[(3)(b)] If $T_b-T_r=1$ and $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$, we have seen that at time $T_r+1$, red paints almost all vertices in $\widetilde \Gamma^{\sss{(r)}}_1$ and blue reaches some vertices with degree at least $u_{i^\star\scriptscriptstyle{(b)}}^{\scriptscriptstyle{(b)}}$. Then, at time $T_r+2$, the previous section's Case (3)(b) analysis implies that $\widetilde \Gamma^{\sss{(r)}}_1\subset\widetilde \Gamma^{\sss{(b)}}_1$. Combining this with the fact that $\widetilde \Gamma^{\sss{(b)}}_1\subset \widetilde \Gamma^{\sss{(r)}}_2$, we see that all vertices in $\widetilde \Gamma^{\sss{(b)}}_1\setminus \widetilde \Gamma^{\sss{(r)}}_1$ are coloured red or blue with equal probability, while most vertices in $\widetilde \Gamma^{\sss{(r)}}_2\setminus \widetilde \Gamma^{\sss{(b)}}_1$ are whp red. This verifies \eqref{eq::shorthand-2}, and Fig. \ref{fig::cross-2}.
From here, analogously as the reasoning for Case (2), we have that at time $T_r+\ell$, each vertex in the set $\widetilde \Gamma^{\sss{(b)}}_{\ell-1}\setminus \widetilde \Gamma^{\sss{(r)}}_{\ell-1}$ is coloured red and blue with equal probability whp, while most vertices in the set $\widetilde \Gamma^{\sss{(r)}}_{\ell}\setminus \widetilde \Gamma^{\sss{(b)}}_{\ell-1}$ are red whp.
In shorthand notation, at time $T_r+\ell$, for every positive integer $m\le \ell$,
\begin{equation}\label{eq::shorthand-case3b-ell} (\delta_n^{\sss{(r)}}+m-1, \delta_n^{\sss{(b)}}+m-1] \in \mathcal {M} ix, \quad (\delta_n^{\sss{(b)}}+m-1, \delta_n^{\sss{(r)}}+m] \in \mathcal {R} ed.\end{equation}
Note that both in Case (2) and (3)(b), at any particular time, the coloring always `ends' with a red layer, that is, the layer with smallest degree that is colored is red.
\end{enumerate}
\begin{figure}[ht]
\centering
\subfigure[Case (2): $T_b=T_r$]{\label{fig::avalanche-2}
\includegraphics[keepaspectratio,width=6cm]{Avalanche-mixed-2}}
\subfigure[Case (3)(b): $T_b=T_r+1$, $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$]{\label{fig::avalanche-3}
\includegraphics[keepaspectratio,width=6cm]{Avalanche-mixed-3}}
\caption{On these pictures, the mixed avalanche is illustrated when $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>(\tau-2)$, and it shows the state of the avalanche at time $T_r+4$. Dotted (lighter) areas indicate vertices that are almost all coloured red, while cross-hatched (darker) areas indicate vertices with equal probability to be blue or red. On both pictures, $\mathcal {M}_k$ and $\mathcal {R} ed_k$ are occupied at time $T_b+k$, $\mathcal {M}_k$ is mixed while $\mathcal {R} ed_k$ is mostly red.}\label{fig::avalanches}
\end{figure}
Solving the recursion \eqref{eq::wideui_recursion} yields that
\begin{equation}\label{eq::ul} \widetilde u^{\sss{(r)}}_\ell= n^{\alpha_n^{\sss{(r)}}(\tau-2)^{\ell-1}} (C\log n)^{b_n^{\sss{(r)}} (\tau-2)^{\ell -1} + \frac{1}{3-\tau}\left(1-(\tau-2)^{\ell-1}\right)},\end{equation}
where $\alpha_n^{\sss{(r)}}$ and $b_n^{\sss{(r)}}$ were defined in \eqref{def::alpha}.
To shorten the too complicated notation for the next two lemmas, let us introduce the following notation for the all-red and equal probability (mixed) layers in the avalanche:
For $T_b=T_r$, for $\ell\ge 1$ let
\[ \mathcal {R}{ed}_\ell:= \widetilde \Gamma^{\sss{(r)}}_\ell \setminus \widetilde \Gamma^{\sss{(b)}}_{\ell}, \quad \mathcal {M}_\ell:= \widetilde \Gamma^{\sss{(b)}}_\ell \setminus \widetilde \Gamma^{\sss{(r)}}_{\ell-1}, \]
where we mean $\widetilde \Gamma^{\sss{(r)}}_0:=\varnothing$.
For $T_b=T_r+1$, for $\ell \ge 0$ we let
\[ \mathcal {R}{ed}_\ell:= \widetilde \Gamma^{\sss{(r)}}_{\ell+1} \setminus \widetilde \Gamma^{\sss{(b)}}_{\ell}, \quad \mathcal {M}_\ell:= \widetilde \Gamma^{\sss{(b)}}_\ell \setminus \widetilde \Gamma^{\sss{(r)}}_{\ell}, \]
with $\widetilde \Gamma^{\sss{(b)}}_0:=\varnothing$. This means that for $T_b=T_r+1$ the red layer at the very top of the avalanche is denoted by $\mathcal {R} ed_0$, while $\mathcal {M}_0$ can be ignored.
See Figure \ref{fig::avalanches} for this notation.
More importantly, note that in both settings, $\mathcal {M}_\ell$ and $\mathcal {R} ed_\ell$ are colored at the same time, at time $T_b+\ell$. In what follows, we write $d_w$ for the degree of vertex $w$.
\begin{lemma}[Independence of mixed coloring]\label{lem::independence}
Suppose $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$, and let us condition on the values $Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}$, and write $\widetilde\mathbbm{1}=\mathbbm{1}_{\{T_b=T_r\}}$. Define $\widetilde u^{\sss{(r)}}_0:=n^{1/(\tau-1)}C \log n$. Then, the two colors red and blue whp reach the vertices in $\widetilde \Gamma^{\sss{(b)}}_\ell \setminus \widetilde \Gamma^{\sss{(r)}}_{\ell-\widetilde\mathbbm{1}}$ at the same time (at $T_b+\ell$), for any $1\le \ell< \nu \log\log n / {|\log (\tau-2)|} + O_{\mathbb{P}}(1)$, for some $\nu<1$.
Further, the color of vertices with degree in the interval $[ \widetilde u^{\sss{(b)}}_\ell)^{1+\varepsilon}, \widetilde u^{\sss{(r)}}_{\ell-\widetilde\mathbbm{1}}] $ for some $\varepsilon>0$, can be described as i.i.d. random variables taking value red and blue with equal probability.
\end{lemma}
\begin{proof}
We proceed by induction. Recall that if a not-yet-colored vertex is reached by the red and blue clusters at the same time, then its color is decided by a fair coin flip independently of everything else. Also recall the arguments in Section \ref{sc::peak}, that the vertices in $\mathcal {M}_1:=\widetilde \Gamma^{\sss{(b)}}_1 \setminus \widetilde \Gamma^{\sss{(r)}}_{1-\widetilde \mathbbm{1}}$ are reached at the same time by the two colors (see \eqref{eq::shorthand} and \eqref{eq::shorthand-2}). Hence, independent coin flips will decide the color of these vertices, yielding that the statement holds for $\ell=1$.
Next we advance the induction.
Suppose the statement holds for all indices at most $\ell-1$, then we show that it also holds for $\ell$. For this, we show that whp any vertex $w$ satisfying the criterion of the lemma is connected to at least one red and at least one blue vertex that is in $\mathcal {M}_{\ell-1}$. Hence, a coin flip will decide its color, so its coloring is red and blue with equal probability, and moreover, this coloring is independent of the coloring of other vertices in the same interval, that is, in $\mathcal {M}_\ell$.
First we show that whp $w$ is connected to many vertices in $\mathcal {M}_{\ell-1}$.
Using the fact that $d_w$ is at least a power $\varepsilon$ away from the boundary of $\mathcal {M}_\ell$, let us fix an $\varepsilon_\ell <\varepsilon$ and define
\begin{equation}\label{eq::i-ell} I_{\ell-1}:=[(d_w)^{(1-\varepsilon_{\ell})/(\tau-2)},(d_w)^{1/(\tau-2)}] \subset \mathcal {M}_{\ell-1}.\end{equation}
By a concentration of binomial random variables (see e.g. \cite[Lemma 3.2]{BarHofKom14}) and the fact that the degrees are i.i.d. in ${\mathrm{CM}}_n(\boldsymbol{d})$, using \eqref{eq::F}, the number of vertices in $I_{\ell-1}$
is whp within the interval
\begin{equation}\label{eq::interval-vertices} [ n (d_w)^{\frac{1-\varepsilon_{\ell}}{\tau-2}(1-\tau)} c_1/2 , 2 C_1 n (d_w)^{\frac{1-\varepsilon_{\ell}}{\tau-2}(1-\tau)}],\end{equation}
as long as $(d_w)^{\varepsilon_\ell (1-\tau)} \to 0$.
Hence, the number of half-edges incident to these vertices is at least $n (d_w)^{\varepsilon_{\ell}-1} c_1/2$ whp. Now, on the event $\{\mathcal {L}_n/n \in (\mathbb{E}[D]/2, 2\mathbb{E}[D])\}$, the expected number of half-edges connecting $w$ to vertices in $I_{\ell-1}$ is at least
\[ d_w \frac{n (d_w)^{\varepsilon_{\ell}-1} c_1/2 }{\mathbb{E}[D]n/2}= \widetilde C(d_w)^{\varepsilon_{\ell}}, \] for some constant $\widetilde C$. Next we show that most of these half-edges connect to disjoint vertices:
the probability that there are $2$ half-edges of $w$ that connect to the same vertex is at most
\[\begin{aligned} {d_w \choose 2} \sum_{v\in I_{\ell-1} } \frac{d_v^2}{(n\mathbb{E}[D]/2)^2} &\le \widetilde C (d_w)^{2} \frac{(d_w)^{\frac{2}{\tau-2} }}{n^2} n (d_w)^{\frac{1-\varepsilon_{\ell}}{\tau-2}(1-\tau)}\\
&\le \widetilde C (d_w)^{\frac{\tau-1}{\tau-2}(1+\varepsilon_{\ell})}/n, \end{aligned}\]
which is $o(1)$ as long as $d_w< n^{(\tau-2)(1-\varepsilon')/(\tau-1)}$ for some $\varepsilon'>0$. In particular, this holds for every $\ell\ge 3$, since then $d_w< \widetilde u^{\sss{(b)}}_2=n^{(\tau-2)^{\delta_n^{\sss{(b)}}+1}/(\tau-1)(1+o_{\mathbb{P}}(1))}$, and also holds for $\ell=2, \widetilde\mathbbm{1}=0$, since in this case $d_w< \widetilde u^{\sss{(r)}}_2 = n^{(\tau-2)^{\delta_n^{\sss{(r)}}+1}/(\tau-1)(1+o_{\mathbb{P}}(1))}$. If $\ell=2, \widetilde\mathbbm{1}=1$, and $n^{(\tau-2)/(\tau-1)}<d_w< \widetilde u^{\sss{(r)}}_1$, then $w$ is whp connected to \emph{all} the vertices in $\widetilde \Gamma^{\sss{(b)}}_1$:
by Lemma \ref{lem::direct_connect}, since the sum of the exponents of $n$ of the degree of $w$ and any vertex in $\widetilde \Gamma^{\sss{(b)}}_1$ is
\[ \frac{(\tau-2)^{\delta_n^{\sss{(r)}}} + (\tau-2)}{ \tau-1} > 1, \]
since $\delta_n^{\sss{(r)}} \in [0,1)$. The number of vertices in $\widetilde \Gamma^{\sss{(b)}}_1$ is at least $n^{1-(\tau-2)^{b_n^{\sss{(b)}}} (1+ o_{\mathbb{P}}(1))}$, which is still plenty.
Summarizing, we get that $w$ is connected to at least $\widetilde C(d_w)^{\varepsilon_{\ell}}$ many vertices in $\mathcal {M}_{\ell-1}$.
By the induction hypothesis, these vertices are colored red and blue with equal probability. Hence, the probability that all of them are blue or all of them are red is at most $2\cdot 2^{-\widetilde C(d_w)^{\varepsilon_{\ell}}}$, which tends to zero as long as $d_w> (C\log n)^\sigma$ for some power $\sigma>0$.
Hence, $w$ has whp at least one red and at least one blue neighbor that is colored a time-step earlier, so an independent coin flip will decide the color of $w$.
Note that for the induction to hold true, we need to use a decreasing sequence of $\varepsilon'>\varepsilon_\ell> \varepsilon_{\ell-1}>\varepsilon_{\ell-2} \dots$ to reach higher and higher intervals. First, \eqref{eq::interval-vertices} needs to hold true, and also $(\widetilde u^{\sss{(b)}}_i)^{\varepsilon_i} \to \infty$ for all $i$. These are all guaranteed if all $\varepsilon_i>\varepsilon'/2$, for instance.
\end{proof}
In the proof of the next lemma, we repeatedly use the following claim:
\begin{claim}\label{claim::Sbound} Let ${\mathcal{E}}_{\ge y_n}$ denote the total number of half-edges incident to vertices with degree at least $y_n$, and $V_{\ge y_n}$ the total number of vertices of degree at least $y_n$, for a sequence $y=y_n$. Then, for any $\omega(n)\to \infty$, and a large enough constant $C<\infty$, a small enough constant $0<c$ and for some constant $0<c_2<\infty$,
\begin{equation}\label{eq::Sbound} \begin{aligned} \mathbb{P} ( {\mathcal{E}}_{\ge y_n} \ge C \omega(n) \!\cdot\! n\!\cdot\! y_n^{2-\tau} ) &\le c_2/ \omega(n), \\
\mathbb{P} ( V_{\ge y_n} \ge C\!\cdot\! n\!\cdot\! y_n^{1-\tau}) &\le \exp\{- c_2\cdot n y_n^{1-\tau}\},\\
\mathbb{P} ( V_{\ge y_n} \le c\!\cdot\! n\!\cdot\! y_n^{1-\tau}) &\le \exp\{- c_2\cdot n y_n^{1-\tau}\}
\end{aligned}
\end{equation}
\end{claim}
\begin{proof} Note that
\[ {\mathcal{E}}_{\ge y_n}= \sum_{i=1}^n D_i \mathbbm{1}_{\{ D_i \ge y_n\}} \le \sum_{i=1}^n \sum_{k=1}^\infty y_n 2^{k} \mathbbm{1}_{\{ 2^{k-1} y_n< D_i \le 2^k y_n\}}. \]
Exchanging sums we can write
\[ {\mathcal{E}}_{\ge y_n} \le \sum_{k=1}^{\infty} y_n 2^k X_k^{(n)}, \]
where the marginals of the random variables on the rhs are $X_k^{(n)}\ {\buildrel d \over \le }\ \mathrm{Bin}(n, C_1 (y_n 2^{k-1})^{1-\tau} )$, where $C_1$ is from \eqref{eq::F}. Calculating the expected value and using Markov's inequality yields \eqref{eq::Sbound}. The proof for $V_{\ge y_n}$ is easier and directly follows from the fact that $V_{\ge y_n} \sim \mathrm{Bin}(n, 1-F(y_n))$ and usual concentration of Binomial random variables (see e.g. \cite[Lemma 3.2]{BarHofKom14}).
\end{proof}
Recall that $\buildrel {\mathbb{P}} \over{\sim} , \overset{\mathbb{P}}{\lesssim}$ means whp equality/ inequality up to a multiplicative factor of finite powers of $C \log n$, and that
we assume that the event $\{ \mathcal {L}_n/n \in [ \mathbb{E}[D]/2, \mathbb{E}[D] ] \}$ holds.
\begin{lemma}[Red coloring in red intervals]\label{lem::red-intervals}
Recall $\widetilde u^{\sss{(b)}}_0:=n^{1/(\tau-1)}C \log n$, and set some $0<\nu<1$.
Then, for any $\le \ell< \nu \log\log n / {|\log (\tau-2)|} + O_\mathbb{P}(1)$, `almost every' vertex $w$ that satisfies
$\widetilde u^{\sss{(r)}}_{\ell+\widetilde\mathbbm{1}}<d_w < (\widetilde u^{\sss{(b)}}_{\ell})^{1-\varepsilon}$ for some $\varepsilon>0$, is whp painted red, where again $\widetilde\mathbbm{1}=\mathbbm{1}_{\{T_b=T_r\}}$. More precisely, for a uniformly picked vertex $v$ in $\mathcal {R} ed_{\ell}$
\[ \mathbb{P}_{Y,n}( v \in \mathcal {R} ed_\ell \text{\ is blue\ }) \le (C \log n)^{x+\ell}\, \widetilde u^{\sss{(r)}}_{\ell+\widetilde \mathbbm{1} }/\widetilde u^{\sss{(b)}}_\ell ,\]
for some finite $x\in \mathbb{N}$ that depends only on $Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}, n$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem::red-intervals} when $T_b=T_r+1$.]
Due to the shift of indices that will become visible later, we handle the cases $T_b=T_r$ and $T_b=T_r+1$ separately. We start with $T_b=T_r+1$.
We have seen in Section \ref{sc::peak}, that if $T_b=T_r+1$ then all vertices in $\mathcal {R} ed_0$ are red at time $T_r+1$, except maybe those that are maximal degree blue vertices. (In case $\widehat u_{i_{\star \scriptscriptstyle{(b)}} }^{\scriptscriptstyle{(b)}} > \widetilde u^{\sss{(r)}}_1$.) Let us call the set of blue vertices (if any) in $\mathcal {R} ed_0$ by $\mathcal {B} ad_0$. By Lemma \ref{lem::numberofverticesinGamma}, there are at most $(C\log n)^{x_0}$ many such vertices (where $x_0$ is a rv that depends only on $Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}, n$), which is much smaller than the total size of $\mathcal {R} ed_0$ ($\buildrel {\mathbb{P}} \over{\sim} \! n (\widetilde u^{\sss{(r)}}_1)^{1-\tau}$).
To some (minor) extent, $\ell=1$ is different from $\ell\ge 2$, so let us calculate one more step. We want to give an estimate on the total number of blue vertices. Any vertex in this interval or in $\mathcal {M}_1$ is colored at time $T_r+2$: for each vertex in $\mathcal {R} ed_1$, there is at least a neighbor in $\mathcal {R} ed_0$ (since $\widetilde \Gamma^{\sss{(r)}}_2 \subset N(\widetilde \Gamma^{\sss{(r)}}_1)$), but some vertices in $\mathcal {R} ed_1$ might be connected to maximal degree blue vertices as well (i.e., vertices in $\mathcal {B} ad_0$.)
Hence, a vertex in $\mathcal {R} ed_1$ might be blue if it is a neighbor of $\mathcal {B} ad_0$.
Thus we define as the potentially blue set
\[ \mathcal {B} ad_1:= N( \mathcal {B} ad_0) \cap \mathcal {R} ed_1.\]
Since $\mathcal {R} ed_1$ has minimal degree $\widetilde u^{\sss{(r)}}_2$, by Claim \ref{claim::Sbound}, with $\omega(n):=C\log n$ the total number of half-edges incident to vertices in $\mathcal {R} ed _1$ is at most $ n (\widetilde u^{\sss{(r)}}_2)^{2-\tau} C\log n$.
Then, the expected number of half-edges from $\mathcal {B} ad_0$ paired to $\mathcal {R} ed_1$ is bounded by
\begin{equation}\label{eq::b1-r2} \mathbb{E}_Y[ \#\{ \mathcal {B} ad_0 \leftrightarrow \mathcal {R} ed_1\} ] \le (C \log n)^{x_0+1} \widehat u_{i_{\star \scriptscriptstyle{(b)}} }^{\scriptscriptstyle{(b)}} \frac{n (\widetilde u^{\sss{(r)}}_2)^{2-\tau}}{n \mathbb{E}[D]/2} \le (C \log n)^{x_0+x_1+1} \frac{n (\widetilde u^{\sss{(r)}}_2)^{2-\tau}}{\widetilde u^{\sss{(b)}}_1}, \end{equation}
where we used that $\widehat u_{i_{\star \scriptscriptstyle{(b)}} }^{\scriptscriptstyle{(b)}}= (C \log n)^{x_1} u_{i_{\star \scriptscriptstyle{(b)}} }^{\scriptscriptstyle{(b)}} = n (C\log n)^{x_1+1}/ \widetilde u^{\sss{(b)}}_1$, for some $x_1$ depending on $Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}, n$, see \eqref{eq::wideu1}.
We need to show that the right hand side of \eqref{eq::b1-r2} is of smaller order than the total number of vertices in $\mathcal {R} ed_1$, which is $\buildrel {\mathbb{P}} \over{\sim} n (\widetilde u^{\sss{(r)}}_2)^{1-\tau}$ by Claim \ref{claim::Sbound}. Then we can write
\[ \frac{n (\widetilde u^{\sss{(r)}}_2)^{2-\tau}}{\widetilde u^{\sss{(b)}}_1} = \frac{\widetilde u^{\sss{(r)}}_2}{\widetilde u^{\sss{(b)}}_1} n (\widetilde u^{\sss{(r)}}_2)^{1-\tau}\buildrel {\mathbb{P}} \over{\sim} \frac{\widetilde u^{\sss{(r)}}_2}{\widetilde u^{\sss{(b)}}_1} |\mathcal {R} ed_1|,\]
Note that $\widetilde u^{\sss{(r)}}_2 =o(\widetilde u^{\sss{(b)}}_1)$, (see Figure \ref{fig::avalanche-3}), since if $T_b=T_r+1$ then $\widetilde u^{\sss{(r)}}_2\le n^{(\tau-2)/(\tau-1)}<\widetilde u^{\sss{(b)}}_1$. The statement of the lemma follows with $x=x_0+x_1+2$ for $\ell=1$, where we added the extra $C \log n$ factor for a Markov's inequality.
In what follows we show that for $\mathcal {B} ad_\ell$, the blue vertices in $\mathcal {R} ed_\ell$, for every $\ell<\nu \log\log n /|\log (\tau-2)|+O_{\mathbb{P}}(1)$,
\begin{equation} \label{eq::induction-step-canceled}|\mathcal {B} ad_\ell| \le n (C \log n)^{x+2\ell}(\widetilde u^{\sss{(r)}}_{\ell+1})^{2-\tau}/\widetilde u^{\sss{(b)}}_{\ell},
\end{equation}
where again, $x= x(n)=x_0+x_1+1$ as for $\ell=1$, is a rv that depends only on $Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}, n$, and forms a tight sequence of rv-s.
By the arguments above, the statement holds for $\ell=1$. Next we show it for general $\ell$. Let us write $N_k(S)$ for the set of vertices $k$ edges away from the set $S$ ($N(S)=N_1(S)$ in this notation).
For this, we decompose $\mathcal {B} ad_\ell$, $\ell\ge k$ into subsets, according to how many `generations' a vertex has to go back to an $\mathcal {M}_i$, $i\le \ell$ as follows:
\begin{equation} \mathcal {B} ad_\ell= \bigcup_{k=1}^{\ell} \mathcal {B} ad_{\ell}(k), \end{equation}
where
\begin{equation} \begin{aligned} \mathcal {B} ad_\ell(1)&=N(\mathcal {M}_{\ell-1}) \cap \mathcal {R} ed_\ell, \\
\mathcal {B} ad_\ell(k)&= N (\mathcal {B} ad_{\ell-1} (k-1)) \cap \mathcal {R} ed_\ell, \quad k\ge 1. \end{aligned} \end{equation}
This means that we divide bad vertices according to `how many generations' are they already bad: for instance, a bad vertex $v$ in $\mathcal {B} ad_3$ might be `first generation' bad, being a neighbor of a vertex in $\mathcal {M}_1$. Or, second generation bad, being a neighbor of a first generation bad vertex in $\mathcal {B} ad_2$, when a path $\mathcal {M}_1 \to \mathcal {B} ad_2 (1) \to v$ is present in the graph. Or `third generation' bad being a neighbor of a second generation bad vertex in $\mathcal {B} ad_2$, when a path $\mathcal {B} ad_0 \to \mathcal {B} ad_1 \to \mathcal {B} ad_2(2) \to v$ is present in the graph\footnote{Note that it is also possible to have a blue vertex in $\mathcal {B} ad_3$ that is the first or second neighbor of $\mathcal {B} ad_0$, or the first neighbor of $\mathcal {M}_1$, but, using the same methods as in \eqref{eq::badellk-bound} below it is easy to show that these, even summed up, are negligible compared to the main terms, represented by $\mathcal {B} ad_\ell(k)$.}.
\begin{figure}\label{fig::badsets}
\includegraphics[width=0.7\textwidth]{Badsets.pdf}
\caption{An illustration of the structure of bad sets up to $\ell=3$.}
\end{figure}
Next we calculate the expected size of each $\mathcal {B} ad_\ell(k).$ For a vertex in $\mathcal {B} ad_\ell(k)$, there must be a path of length $k$
\begin{equation} \mathcal {M}_{\ell-k} \to \mathcal {B} ad_{\ell-k+1}(1) \to \mathcal {B} ad_{\ell-k+2}(2) \to \dots \mathcal {B} ad_{\ell-1}(k-1) \to \mathcal {B} ad_\ell(k)\end{equation} present in ${\mathrm{CM}}_n(\boldsymbol{d})$.
Since the total number of half edges $\mathbb{E}_Y[H(\mathcal {M}_{\ell-k})]\le C n (\widetilde u^{\sss{(b)}}_{\ell-k})^{2-\tau} $ for some constant $C$ by Claim \ref{claim::Sbound}, the expected number of such paths and hence the size of $\mathcal {B} ad_{\ell}(k)$ can be estimated , for all $1\le k < \ell$ as
\begin{equation}\label{eq::badellk} \mathbb{E}_{Y,n}[|\mathcal {B} ad_\ell(k)| ] \le n (\widetilde u^{\sss{(b)}}_{\ell-k})^{2-\tau} \prod_{i=1}^{k-1} \left(\sum_{v \in \mathcal {R} ed_{\ell-k+i}} \frac{d_v (d_v-1)}{n}\right) \cdot \sum_{v \in \mathcal {R} ed_{\ell}} \frac{ d_v}{n},\end{equation}
while for $k=\ell$ we have an extra factor $(C \log n)^{x_0+x_1}$ on the right hand side.
Now, it is not hard to see that using \eqref{eq::size-biased2}, the sums inside the product can be approximated by
\begin{equation}(2\mathbb{E}[D])^{-1} \mathbb{E}[D(D-1) \mathbbm{1}_{\{\widetilde u^{\sss{(r)}}_
{\ell-k+i+1}<D< \widetilde u^{\sss{(b)}}_{\ell-k+i}\}}] \le C (\widetilde u^{\sss{(b)}}_{\ell-k+i})^{3-\tau}, \end{equation}
and hence, multiplying the rhs here with $C \log n$ implies
\begin{equation}\label{eq::sum-to-expected} \mathbb{P}_{Y,n}\left( \sum_{v \in \mathcal {R} ed_{\ell-k+i}} \frac{d_v (d_v-1)}{n} \ge C \log n (\widetilde u^{\sss{(b)}}_{\ell-k+i})^{3-\tau} \right) \le \frac{1}{C \log n}.\end{equation}
The last sum in \eqref{eq::badellk} is at most $C \log n (\widetilde u^{\sss{(r)}}_{\ell+1})^{2-\tau}$ whp by Claim \ref{claim::Sbound}.
Thus, the rhs of \eqref{eq::badellk} is at most, as $n\to \infty$,
\begin{equation} \label{eq::badellk-2} \mathbb{E}_{Y,n}[|\mathcal {B} ad_\ell(k)| ] \le (C \log n)^{1+\mathbbm{1}_{k=\ell }(x_0+ x_1)} n (\widetilde u^{\sss{(b)}}_{\ell-k})^{2-\tau} \prod_{i=1}^{k-1} \left(C\log n(\widetilde u^{\sss{(b)}}_{\ell-k+i})^{3-\tau}\right) \cdot (\widetilde u^{\sss{(r)}}_{\ell+1})^{2-\tau}. \end{equation}
Now we can repeatedly apply \eqref{eq::wideui_recursion} in the form $(\widetilde u^{\sss{(b)}}_s)^{2-\tau}= C \log n / \widetilde u^{\sss{(b)}}_{s+1}$,
and then \eqref{eq::badellk-2} simplifies to
\begin{equation}\label{eq::badellk-bound} \mathbb{E}_{Y,n}[|\mathcal {B} ad_\ell(k)| ] \le n (C\log n)^{2k+1+\mathbbm{1}_{k=\ell}(x_0+ x_1) } \frac{(\widetilde u^{\sss{(r)}}_{\ell+1})^{2-\tau}}{\widetilde u^{\sss{(b)}}_{\ell}}. \end{equation}
Hence, we get that for some $\hat C$ constant,
\begin{equation} \mathbb{E}_{Y,n}[|\mathcal {B} ad_\ell|] \le \sum_{k=1}^{\ell} \mathbb{E}[\mathcal {B} ad_{\ell}(k)] \le \hat C (C \log n)^{x-1+2\ell} n \frac{(\widetilde u^{\sss{(r)}}_{\ell+1})^{2-\tau}}{\widetilde u^{\sss{(b)}}_{\ell}}, \end{equation}
with $x=x_0+x_1+2$.
Adding an extra factor of $C \log n$ to the rhs, the inequality holds whp without the expected value on the lhs as well.
Finally, by the concentration of binomial random variables, $|\mathcal {R} ed_\ell| \ge c n (\widetilde u^{\sss{(r)}}_{\ell+1})^{1-\tau}$ whp, for some constant $c$.
As a result, the probability that a vertex in $\mathcal {R} ed_\ell$ is in $\mathcal {B} ad_\ell$, conditioned on $Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}$
\begin{equation}\label{eq::prob-of-bad}\mathbb{P}_{Y,n}( v \in \mathcal {B} ad_\ell | v\in \mathcal {R} ed_\ell ) \le \hat C (C \log n)^{2\ell+x} \frac{\widetilde u^{\sss{(r)}}_{\ell+1}}{\widetilde u^{\sss{(b)}}_\ell}, \end{equation}
whp.
If $\ell\le \nu \log \log n/ |\log (\tau-2)|$, for some $\nu<1$, then using \eqref{eq::ul}
\begin{equation}\label{eq::prob-of-bad-bound} \widetilde u^{\sss{(r)}}_{\ell+1}/\widetilde u^{\sss{(b)}}_\ell = \exp\left\{ - (\log n)^{1-\nu} (\alpha_n^{\sss{(b)}}-(\tau-2)\alpha_n^{\sss{(r)}}) \right\} (C \log n)^{1/(3-\tau)} (1+ o(1)), \end{equation}
where recall that $\alpha_n^{\sss{(b)}}>(\tau-2)\alpha_n^{\sss{(r)}}$ when $T_b=T_r+1$. On the other hand, $(C \log n)^{2\ell}\le \exp\{ c (\log \log n)^2\}$, hence the contribution of this term is negligible if $n$ large enough.
As a result, the rhs of \eqref{eq::prob-of-bad} tends to zero for all $\ell< \nu \log \log n /|\log (\tau-2)|$, as $n\to \infty$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem::red-intervals} when $T_b=T_r$.]
For $T_b=T_r$, the induction starts slightly differently. Namely, in this case $\widetilde u^{\sss{(b)}}_1>\widetilde u^{\sss{(r)}}_1$ and the highest degree vertices belong to $\mathcal {M}_1$, see Figure \ref{fig::avalanche-2}. $\mathcal {M}_1$ and $\mathcal {R} ed_1$ are occupied at the same time: all vertices in $\mathcal {R} ed_1$ are connected to maximal degree red vertices, but some of them might be also connected to maximal degree blue vertices. These maximal degree blue vertices might or might not be inside the set $\mathcal {R} ed_1$: let us call them $\mathcal {B} ad_0$. Further, let us write $\mathcal {B} ad_1$ for the set of blue vertices in $\mathcal {R} ed_1$, then the size of this set can be bounded by the expected number of half-edges from maximal degree blue vertices to $\mathcal {R} ed_1$, similarly as it was done in \eqref{eq::b1-r2}:
\begin{equation}\label{eq::b0-r1-2} \mathbb{E}_{Y,n}[ \#\{ \mathcal {B} ad_0 \leftrightarrow \mathcal {R} ed_1\} ] \le (C \log n)^{x_0} \widehat u_{i_{\star \scriptscriptstyle{(b)}} }^{\scriptscriptstyle{(b)}} \frac{n (\widetilde u^{\sss{(r)}}_1)^{2-\tau}}{n \mathbb{E}[D]/2} \le (C \log n)^{x_0+x_1+1}\frac{n (\widetilde u^{\sss{(r)}}_1)^{2-\tau}}{\widetilde u^{\sss{(b)}}_1}.
\end{equation}
On the other hand, the total size of $\mathcal {R} ed_1$ is $\buildrel {\mathbb{P}} \over{\sim} n (\widetilde u^{\sss{(r)}}_1)^{1-\tau}$ by Claim \ref{claim::Sbound}, and since $\widetilde u^{\sss{(r)}}_1=o(\widetilde u^{\sss{(b)}}_1)$ if $T_r=T_b$ (see Figure \ref{fig::avalanche-3}), the rhs of \eqref{eq::b0-r1-2} is of less order than that.
From here, the exact same proof as for $T_b=T_r+1$ can be repeated, with the indices of $\widetilde u^{\sss{(r)}}_i$-s shifted by $-1$. In this case $\mathcal {R} ed_\ell$ has total size of $n (\widetilde u^{\sss{(r)}}_\ell)^{1-\tau}$, and also $\widetilde u^{\sss{(r)}}_\ell=o(\widetilde u^{\sss{(b)}}_\ell)$. This implies that the rhs of equation \eqref{eq::prob-of-bad-bound} contains $\alpha_n^{\sss{(b)}}-\alpha_n^{\sss{(r)}}>0$ instead of $\alpha_n^{\sss{(b)}}-(\tau-2) \alpha_n^{\sss{(r)}}$. As a result, the corresponding version of \eqref{eq::prob-of-bad} tends to zero also in this case, meaning that the statement of the lemma implies that the proportion of blue vertices in $\mathcal {R} ed_\ell$ tends to zero also when $T_b=T_r$.
\end{proof}
\section{Typical distances and the maximal degree of blue}\label{sc::meetingtime}
In this section we describe how the two colors meet and prove Theorem \ref{thm::distances}. As a result of the analysis, we also prove Theorem \ref{thm::maxdegree}.
Recall that the time to reach the top of the mountain for the two colors for $j=r,b$ is denoted by
\begin{equation}\label{eq::Tj} T_j = \left\lfloor\frac{\log\log n-\log \left((\tau-1) Y_j^{\scriptscriptstyle{(n)}}\right)}{|\log(\tau-2)|}-1\right\rfloor = \frac{\log\log n-\log \left((\tau-1) Y_j^{\scriptscriptstyle{(n)}}\right)}{|\log(\tau-2)|} -1-b_n^{\scriptscriptstyle{(j)}}. \end{equation}
\begin{proof}[Proof of Theorem \ref{thm::distances}]
For the upper bound, we show that there is whp a path that connects red and blue in at most
\begin{equation}\label{eq::distance-try} \mathcal {D}(\mathcal {R}_0, \mathcal {B}_0) = T_r + T_b + 1 + \mathbbm{1}\{ \tau-1>(\tau-2)^{b_n^{\sss{(r)}}} + (\tau-2)^{b_n^{\sss{(b)}}}\} \end{equation}
many steps.
We have seen in Section \ref{sc::climbup} that whp there exists a path of length at most $T_r$ that connects the red source to the top of the mountain, i.e., to some vertex with $\log$(degree)/$\log n$ that is $(\tau-2)^{b_n^{\sss{(r)}}}/(\tau-1)(1+o_{\mathbb{P}}(1))$.
The crucial observation is, that in terms of the distance of the sources, the timing of the colors does not matter, that is, imaginarily, we can stop the spread of red at this moment. Now, the same method for blue shows that there exists a path of length at most $T_b$ steps, and blue occupies some vertices with $\log$(degree)/$\log n$
that is $(\tau-2)^{b_n^{\sss{(r)}}}/(\tau-1)(1+o_{\mathbb{P}}(1))$. For an upper bound on typical distances, we can simply assume that these `climbing clusters' are disjoint, that is, we assume $\mathcal {R}_{T_r} \cap \mathcal {B}_{T_b} = \varnothing$.
Now we let the two colors jump, and we reduced the problem to the analysis of Case (2): If we let them do one jump, then whp the maximal degree red vertex connects to every vertex in layer $\widetilde \Gamma^{\sss{(r)}}_1$, while the maximal degree blue vertex connects to every vertex in $\widetilde \Gamma^{\sss{(b)}}_1$ (see Fig. \ref{fig::cross-1}). Note that the distance is then $T_r+T_b + 1$ if and only if the maximal degree blue vertex is in $\widetilde \Gamma^{\sss{(r)}}_1$ or the maximal degree red vertex is in $\widetilde \Gamma^{\sss{(b)}}_1$, that is, if
\[ u_{i_{\star \scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}} \ge \widetilde u_1^{\scriptscriptstyle{(b)}} \quad \mbox{or} \quad u_{i_{\star \scriptscriptstyle{(b)}}}^{\scriptscriptstyle{(b)}} \ge \widetilde u_1^{\scriptscriptstyle{(r)}}. \]
Otherwise, we can pick an arbitrary vertex in $\mathcal {M}_1=\widetilde \Gamma^{\sss{(b)}}_1\cap \widetilde \Gamma^{\sss{(r)}}_1$, and both the maximal degree blue and the maximal degree red vertex connect to that vertex whp, hence, the distance is $T_r+T_b+2$.
Since for large enough $n$, the logarithmic factors in $u_{i_{\star \scriptscriptstyle{(j)}}}^{\scriptscriptstyle{(j)}}$ and $\widetilde u_1^{\scriptscriptstyle{(j)}}$ are negligible, to have distance $T_r+T_b+2$ we need for large enough $n$
\[ \frac{(\tau-2)^{b_n^{\sss{(b)}}} }{\tau-1}< \alpha_n^{\sss{(r)}} \quad \mbox{and}\quad \frac{(\tau-2)^{b_n^{\sss{(r)}}}}{\tau-1} < \alpha_n^{\sss{(b)}}. \]
Using \eqref{def::alpha}, we see that both criteria are equivalent to
\begin{equation}\label{eq::cond-larger}\tau-1>(\tau-2)^{b_n^{\sss{(r)}}}+(\tau-2)^{b_n^{\sss{(b)}}}.\end{equation} The region at which this is satisfied is above the red curve on Fig~\ref{fig::contourplot}. Hence, the distance between the two sources is at most \eqref{eq::distance-try}.
Note that the form at which the theorem is presented is a simple rewrite of this equation.
With this, we have finished the upper bound of the proof of Theorem \ref{thm::distances}.
\begin{figure}[ht]
\includegraphics[height=6cm]{Contourplot}
\caption{The contour plot of the function $0.2^x+0.2^y$ with a red line indicating $0.2^x+0.2^y=1.2$, that is $\tau=2.2$. Darker colors represent smaller values, that is, $b_n^{\sss{(b)}} >\delta_n^{\sss{(r)}}$ is satisfied above the red curve. }\label{fig::contourplot}
\end{figure}
For the lower bound, we argue as follows: we have seen in Lemma \ref{lem::badpaths} that $T_r, T_b$ are also lower bounds for the time to reach the top of the mountain. We need to show that these `climbing clusters' are disjoint whp, i.e., any vertex that is distance at most $T_b$ away from the blue source is different from the vertices that are distance at most $T_r$ away from the red source:
\begin{equation} \label{eq::no-early-meeting} \mathcal {R}_{T_r}\cap \mathcal {B}_{T_b} = \varnothing. \end{equation}
Let us postpone the proof of this statement till Claim \ref{cl::no-early-meeting} and assume that it holds, implying that \begin{equation}\label{eq::distance-lowerbound} \mathcal {D}(\mathcal {R}_0, \mathcal {B}_0 ) \ge T_b+T_r+1. \end{equation}
So, we only have to show that if
\begin{equation}\label{eq::equivalent-assumption} \tau-1> (\tau-2)^{b_n^{\sss{(r)}}} + (\tau-2)^{b_n^{\sss{(b)}}},\end{equation}
then we need one extra edge to connect the two clusters. Recalling the definitions \eqref{eq::ui*}, \eqref{eq::uistar-b}, it is not hard to see that \eqref{eq::equivalent-assumption} for large $n$ is equivalent to
\begin{equation}\label{eq::ui-wideu1} u_{i_{\star \scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}} \le \widetilde u_1^{\scriptscriptstyle{(b)}}= \frac{C\log n\cdot n}{u_{i_{\star \scriptscriptstyle{(b)}}}^{\scriptscriptstyle{(b)}} } \quad \mbox{and } \quad u_{i_{\star \scriptscriptstyle{(b)}}}^{\scriptscriptstyle{(b)}} \le \widetilde u_1^{\scriptscriptstyle{(r)}}=\frac{C\log n \cdot n}{u_{i_{\star \scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}} }.\end{equation}
Now, recall Lemma \ref{lem::badpaths}: on the whp event that $\{ j=r,b, \forall i\le i_{\star \scriptscriptstyle{(j)}}\ \mathcal {B} ad \mathcal {P}^{\scriptscriptstyle{(j)}} =\varnothing\}$, $\widehat u_{i}^{\scriptscriptstyle{(j)}} $ as in \eqref{eq::uibar} serves as an upper bound on the maximal degree of red and blue at time $t(n^{\varrho})+ i_{\star \scriptscriptstyle{(j)}}, j=r,b$.
Note also that by Lemma \ref{lem::numberofverticesinGamma}, there are only $\exp\{ \log (C \log n) \hat C (\tau-2)^{-i_{\star, \scriptscriptstyle{(j)}}}\}$ many vertices with degree at most $\widehat u^{\scriptscriptstyle{(j)}}_{i_{\star \scriptscriptstyle{(j)}}}$ for $j=r,b$ with $\hat C=2/(3-\tau)^2$.
Note that we need only one edge to connect $\mathcal {R}_{T_r}$ to $\mathcal {B}_{T_b}$ if some maximal degree red vertex is connected to some maximal degree blue vertex. Hence, on the whp event that the total number of half-edges satisfies $\{ \mathcal {L}_n/n \in (\mathbb{E}[D]/2, \mathbb{E}[D] )\}$, by a simple union bound, the probability that any of the maximal degree red vertices is connected to any of the maximal degree vertices by an edge is at most
\begin{equation}\label{eq::top-connect-bound} \mathbb{P}_{Y,n}(\mathcal {R}_{T_r} \leftrightarrow \mathcal {B}_{T_b}) \le \frac{2}{\mathbb{E}[D] n} (C \log n)^{ \hat C (\tau-2)^{-i_{\star \scriptscriptstyle{(b)}}}} \widehat u^{\scriptscriptstyle{(j)}}_{i_{\star \scriptscriptstyle{(b)}}} \cdot
(C \log n)^ {\hat C (\tau-2)^{-i_{\star \scriptscriptstyle{(r)}}}} \widehat u^{\scriptscriptstyle{(j)}}_{i_{\star \scriptscriptstyle{(r)}}},\end{equation}
with $\hat C=2/(3-\tau)^2$.
Since $i_{\star \scriptscriptstyle{(j)}}$, $j=r,b$ are tight random variables that do not grow with $n$, by \eqref{eq::wideui-ui},
\begin{equation}\label{eq::ui-ui-estimate}\widehat u_{i_{\star \scriptscriptstyle{(j)}}}^{\scriptscriptstyle{(j)}}\buildrel {\mathbb{P}} \over{\sim} u_{i_{\star \scriptscriptstyle{(j)}}}^{\scriptscriptstyle{(j)}} \buildrel {\mathbb{P}} \over{\sim} n^{(\tau-2)^{b_n^{\scriptscriptstyle{(j)}}}/(\tau-1)} \end{equation}
Note that the extra powers of $C \log n$ in \eqref{eq::top-connect-bound}, only depend on $i_{\star,\scriptscriptstyle{(j)}}$ and hence do not grow with $n$. So, picking $n$ large enough and using \eqref{eq::ui-ui-estimate}, the expression on the rhs of \eqref{eq::top-connect-bound} is $\buildrel {\mathbb{P}} \over{\sim} (u_{i_{\star \scriptscriptstyle{(b)}}}^{\scriptscriptstyle{(b)}} u_{i_{\star \scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}})/n$. This is $o(1)$ when \eqref{eq::equivalent-assumption} holds.
Hence, under the condition \eqref{eq::equivalent-assumption} we need at least $T_r+T_b+2$ edges to connect, while we need at least $T_r+T_b+1$ edges in either case by \eqref{eq::no-early-meeting}.
To finish the lower bound of the proof of Theorem \ref{thm::distances}, it is left to show \eqref{eq::no-early-meeting}, that we handle in the following lemma.
\end{proof}
\begin{lemma}\label{cl::no-early-meeting} On the event $\{ \mathcal {B} ad \mathcal {P}^{\scriptscriptstyle{(j)}} =\varnothing \ \forall i\le i_{\star \scriptscriptstyle{(j)}} \mbox{ for } j=r,b\} $, the event $\{\mathcal {R}_{T_r} \cap \mathcal {B}_{T_b} = \varnothing\}$ holds whp.
\end{lemma}
In the proof of this lemma, and also later on in Section \ref{sc::coexistence}, we use the following technical, rather easy claim:
\begin{claim}\label{cl::stoch-dom-alpha-stable} Recall that $\widehat u_0^{\scriptscriptstyle{(r)}} = (C \log n \cdot Z_{t(n^{\varrho})}^{\scriptscriptstyle{(r)}} )^{1/(\tau-2)} $. Then there exist a $0<c<\infty$, so that
\[ \mathbb{P}\left( \sum_{i=1}^{Z_{t(n^{\varrho})}} D_i^\star \ge \widehat u_0^{\scriptscriptstyle{(r)}} \right) \le \frac{c}{(\log n)^{\tau -2}}. \]
\end{claim}
\begin{proof}
We prove the claim conditioned on the value $Z_{t(n^{\varrho})}^{\scriptscriptstyle{(r)}}:=m$. Note that $m=n^{\varrho^{\scriptscriptstyle{(r)}}}$ is a polynomial of $n$.
First, notice that we can pick a $b>0$ so that $D^\star\ {\buildrel {d}\over{\le}} \ b+X$, where $X$ is a continuous random variable with distribution function $\mathbb{P}(X\le x)= 1- C_1^\star\!\cdot\! x^{2-\tau}$ on $[0, \infty)$, where $C_1^\star$ is from \eqref{eq::size-biased2}. Then, $b+X$ is a totally asymmetric stable random variable with skewness $\kappa=1$, shift $b$ and some scale parameter $c$.
As a result, the moment generating function of $X$ is of the form
\[ \mathbb{E}[{\mathrm e}^{-\theta (b+X)}] = \exp\{ - \theta b + c' \theta^{\tau-2} \}. \]
Then, calculating the moment generating function of $\sum_{i=1}^m X_i$ for i.i.d.\ $X_i\sim X$ gives that
\begin{equation}\label{eq::sum-domination} \sum_{i=1}^m D_i^\star \ {\buildrel {d}\over{\le}} \ \sum_{i=1}^m b+X_i \ {\buildrel {d}\over{=}}\ b m + m^{1/(\tau-2)}X' \ {\buildrel {d}\over{\le}\ }m^{1/(\tau-2)} (b + X'),\end{equation}
where $X'\sim X$.
Then, by the stochastic domination and the tail distribution of $X'$, for an arbitrary $0<C'\le \infty$ there is a $c'<\infty$ so that
\[ \mathbb{P}\left( \sum_{i=1}^m D_i^\star \ge m^{\frac{1}{\tau-2}} C' \log m\right) \le \mathbb{P}( b + X' \ge C' \log m ) \le \frac{c'}{(\log m)^{\tau-2}}.\]
The statement of the claim follows by noticing that $\log m=\varrho^{\scriptscriptstyle{(r)}} \log n$, with $\varrho'(\tau-2)<\varrho^{\scriptscriptstyle{(r)}}<\varrho'$ which modifies the constant $C'$, $c'$ to $C$ and $c$, respectively.
\end{proof}
\begin{proof}[Proof of Lemma \ref{cl::no-early-meeting}]
Recall that we write $\mathbb{P}_{Y,n}(\cdot), \mathbb{E}_{Y,n}[\cdot]$ for probabilities of events and expectations of random variables conditioned on the values $Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}$.
To prove the lemma we first calculate the total number of free (unpaired) half-edges going out of the set $\mathcal {R}_{T_r}$, which we denote by $H(\mathcal {R}_{T_r})$. We do this via counting the number of paths \emph{with free ends}, that is, now we say that a sequence of vertices and half-edges $(\pi_0, s_0, t_1, \pi_1, s_1, t_2, \dots, t_k, \pi_k, s_k)$ forms a \emph{path with free end} in ${\mathrm{CM}}_n(\boldsymbol{d})$, if for all $0< i\le k$, the half edges $s_i, t_i$ are incident to the vertex $\pi_i$ and $(s_{i-1}, t_i)$ forms an edge between vertices $\pi_{i-1},\pi_i$. Clearly, since the same vertex might be approached on several paths, the total number of free half-edges in $\mathcal {R}_{T_r}$ can be bounded from above by the number of paths with free end of length $T_r$, starting from the red source vertex $\mathcal {R}_0$. Now, on the event $\{ \mathcal {B} ad \mathcal {P}^{\scriptscriptstyle{(j)}} =\varnothing, \ \forall i\le i_{\star \scriptscriptstyle{(j)}}\
\mbox{ for } j=r,b,\} $, at time $t(n^{\varrho}) + i$ $\widehat u_{i}^{\scriptscriptstyle{(j)}}$ (see definition in \ref{eq::uibar}) is an upper bound on the degrees of color $j$ vertices.
Let us note that
\begin{equation}\label{eq::d-star-indicator} \mathbb{E}_{Y,n}[D^\star \mathbbm{1}_{\{D^\star < \widehat u_{i}^{\scriptscriptstyle{(j)}}\}} ] \le C_1^\star (\widehat u_{i}^{\scriptscriptstyle{(j)}})^{3-\tau}\end{equation}
by \eqref{eq::size-biased2}. We write $N_k(\mathcal {A}, \text{free})$ for the total number of $k$-length paths with an unpaired half-edge starting from set $\mathcal {A}$. Then, since $T_r=t(n^{\varrho})+i_{\star \scriptscriptstyle{(r)}}$ (see \eqref{eq::k*+i*}),
\begin{equation}\label{eq::half-edge-to-path}H(\mathcal {R}_{T_r}) \le N_{i_{\star \scriptscriptstyle{(r)}}}(\mathcal {R}_{t(n^{\varrho})}, \text{free}), \end{equation}
and recall $\mathcal {R}_{t(n^{\varrho})}$ is coupled to the branching process described in Section \ref{sc::BP}. Hence, the degrees in the last generation of the BP phase are i.i.d.\ having distribution $D^\star$ satisfying \eqref{eq::size-biased2}.
By Claim \ref{cl::stoch-dom-alpha-stable}, the total number of half-edges in this last generation is whp
\[ \sum_{i=1}^{Z_{t(n^{\varrho})}} D_i^\star \le \widehat u_0^{\scriptscriptstyle{(r)}}. \]
The path counting method described in \cite[Appendix]{BarHofKom14}
gives that the expected number of paths with free ends of length $i_{\star \scriptscriptstyle{(r)}}$ under the assumption of the claim satisfies
\begin{equation}\label{eq::nkab} \mathbb{E}_{Y,n}[ N_{i_{\star \scriptscriptstyle{(r)}}}(\mathcal {R}_{t(n^{\varrho})}, \text{free})] \le \widehat u_0^{\scriptscriptstyle{(r)}} \left(\prod_{i=1}^{i_{\star \scriptscriptstyle{(r)}}}\frac{\mathcal {L}_n}{\mathcal {L}_n-2i+1}\right)\cdot\sideset{}{^\star}\sum_{\substack{ \pi_1,\dots,\pi_{ i_{\star \scriptscriptstyle{(r)}}} \\ \forall i\ \pi_i \in \Lambda_i}} \left(\prod_{i=1}^{i_{\star \scriptscriptstyle{(r)}} }\frac{d_{\pi_i} (d_{\pi_i}-1)}{\mathcal {L}_n}\right) \end{equation}
where $\sideset{}{^\star}\sum$ means that we sum over distinct vertices, $d_{\pi}$ denotes the degree of vertex $\pi$, and $\Lambda_i=\{v \in [n]: D_v \le \widehat u_i^{\scriptscriptstyle{(r)}} \}$.
Now, we can apply \eqref{eq::sum-to-expected} $i_{\star\scriptscriptstyle{(r)}}$ many times (with a union bound) and get, that on the event $\mathcal {L}_n/n \in (\mathbb{E}[D]/2, \mathbb{E}[D])$, whp,
\begin{equation} \mathbb{E}_{Y,n}[ N_{i_{\star \scriptscriptstyle{(r)}}}(\mathcal {R}_{t(n^{\varrho})}, \text{free})] \le (C \log n)^{i_{\star\scriptscriptstyle{(r)}}}\cdot \widehat u_0^{\scriptscriptstyle{(r)}} \cdot \left(\prod_{i=1}^{i_{\star \scriptscriptstyle{(r)}} } (\widehat u_i^{\scriptscriptstyle{(r)}})^{3-\tau} \right) {\mathrm e}^{2 i_{\star \scriptscriptstyle{(r)}}^2/ (\mathbb{E}[D] n) }. \end{equation}
Note that $i_{\star \scriptscriptstyle{(r)}}$ is a tight sequence of rv-s, see \eqref{eq::value_i*}. The value of $\widehat u_i^{\scriptscriptstyle{(r)}}$ can be calculated from \eqref{eq::uibar} and is the same as the rhs of \eqref{def::ui} with $+e_i$ in the exponent of $C \log n$. Hence, the product of the first three factors on rhs of the previous formula, after some longish but elementary calculations, equals
\[ n^{\varrho^{\scriptscriptstyle{(r)}} (\tau-2)^{ - (i_{\star \scriptscriptstyle{(r)}} +1) } }(C\log n)^{ \frac{1}{3-\tau} ( (\tau-2)^{ - (i_{\star \scriptscriptstyle{(r)}}^{\scriptscriptstyle{(r)}} +1) } -1 ) } = \widehat u_{i_{\star \scriptscriptstyle{(r)}}}. \]
Recall that the value of $\widehat u_{i_{\star \scriptscriptstyle{(r)}}}^{\scriptscriptstyle{(r)}}$ is the same as the rhs of \eqref{eq::ui*} with $+e_{i_{\star \scriptscriptstyle{(r)}}}$ in the exponent of $C \log n$, so by \eqref{eq::half-edge-to-path}, whp,
\[ \mathbb{E}_{Y,n}[ H(\mathcal {R}_{T_r})] \le n^{(\tau-2)^{b_n^{\sss{(r)}}} / (\tau-1) (1+o_{\mathbb{P}}(1)) }. \]
Multiplying the rhs with a constant $C \log n$ factor to allow for an application of Markov's inequality, we also have
\begin{equation}\label{eq::half-edge-red} \mathbb{P}_{Y,n}\left(H(\mathcal {R}_{T_r}) \ge C \log n \cdot n^{(\tau-2)^{b_n^{\sss{(r)}}} / (\tau-1) (1+o_{\mathbb{P}}(1)) } \right) \le 1/ (C\log n). \end{equation}
Now, let us apply the exact same technique for $H(\mathcal {B}_{T_b-i})$, the half edges from the blue cluster at $T_b-i$, for all $i=1, 2, \dots, i_{\star \scriptscriptstyle{(b)}}$. Since now we stop the paths at $t(n^\varrho)+i_{\star \scriptscriptstyle{(b)}}-1, t(n^\varrho)+i_{\star \scriptscriptstyle{(b)}}-2, \dots, t(n^{\varrho})$, we get that
\[ \mathbb{P}_{Y,n}\left(H(\mathcal {B}_{T_b-i}) \ge (C \log n)^{i} n^{(\tau-2)^{b_n^{\sss{(b)}}+i}/(\tau-1) (1+o_{\mathbb{P}}(1)) } \right) \le 1/ (C\log n)^{i}. \]
Summing the error terms up, we get
\begin{equation}\label{eq::half-edge-blue} \mathbb{P}_{Y,n}\left(H(\mathcal {B}_{T_b-i}) \le (C \log n)^{i} n^{(\tau-2)^{b_n^{\sss{(b)}}+i}/(\tau-1) (1+o_{\mathbb{P}}(1)) } \ \forall i=1, \dots, i_{\star \scriptscriptstyle{(b)}}, \right) \ge 1-1/ (C\log n). \end{equation}
Now, to see that $\mathcal {R}_{T_r}$ and $\mathcal {B}_{T_b}$ are disjoint, we apply the following procedure:
It is easy to see that $H(\mathcal {R}_{T_r-i})$ is maximised at $i=0$. Hence, we grow the red cluster first till time $T_r$, and then stop it. Then, we grow the blue cluster step by step, looking at $\mathcal {B}_1, \mathcal {B}_2, \dots, \mathcal {B}_{T_b}$, and at each step we check if any of the half edges paired are actually paired to a red half-edge. If this happens for any time before or at $T_b-1$, then an early connection happens and the distance is at most $T_b+T_r$. (Note that the distance is $T_r+i$ if we pair a blue half-edge that is at the end of a path with free end of length $i-1$, to a red half-edge. Hence, to have a connection of $T_r+T_b$, we need to find this connection at the latest when pairing the half-edges in $H(\mathcal {B}_{T_r-1})$. )
The probability that there is a connection before or at $t(n^{\varrho})$ is of the same order as the probability that there is a connection at time $t(n^{\varrho})$, since the total degree in the whole BP is the same order of magnitude as the total degree in the last generation.
The probability that $H(\mathcal {B}_{T_b-i})$ connects to $H(\mathcal {R}_{T_r})$ is then at most, by a union bound,
\[ \mathbb{P}_{Y,n}( H(\mathcal {B}_{T_b-i}) \leftrightarrow H(\mathcal {R}_{T_r}) ) \le \frac{H(\mathcal {B}_{T_b-i}) H(\mathcal {R}_{T_r})}{\mathcal {L}_n}. \]
On the event that $\{\mathcal {L}_n \in (\mathbb{E}[D]/2, \mathbb{E}[D)]\}$, using \eqref{eq::half-edge-blue} and \eqref{eq::half-edge-red}, we sum up the error terms for $i=1, \dots, i_{\star \scriptscriptstyle{(b)}}$,
\begin{equation} \label{eq::early-error} \mathbb{P}_{Y,n}( \mathcal {R}_{T_r} \cap \mathcal {B}_{T_b}\neq \varnothing ) \le \frac{2}{\mathbb{E}[D]n} n^{(\tau-2)^{b_n^{\sss{(r)}}}(\tau-1)(1+o_{\mathbb{P}}(1)) } \sum_{i=1}^{i_{\star \scriptscriptstyle{(b)}}}n^{(\tau-2)^{b_n^{\sss{(b)}}+i} /(\tau-1)(1+o_{\mathbb{P}}(1)) }. \end{equation}
Note that the exponent of $n$ in the dominant term in the numerator is
\[ \frac{(\tau-2)^{b_n^{\sss{(r)}}}+ (\tau-2)^{b_n^{\sss{(b)}}+1}}{\tau-1} \le 1, \]
since $b_n^{\sss{(b)}}, b_n^{\sss{(r)}} \in [0,1)$. Note that $b_n^{\sss{(b)}}=b_n^{\sss{(r)}}=0$ happens with probability tending to zero.
If $n$ is so enough that that the logarithmic factors (hidden in the $(1+o_\mathbb{P}(1))$ factor in the exponent) are negligible, the rhs of \eqref{eq::early-error} tends to zero with $n$.
This finishes the proof of the claim, and hence, the proof of the lower bound of Theorem \ref{thm::distances}.
\end{proof}
We continue to prove Theorem \ref{thm::maxdegree}, for which we need to investigate the maximal degree of a vertex occupied by blue.
\begin{proof}[Proof of Theorem \ref{thm::maxdegree}]
Before we start, recall the various notations of `$u$'-s: $u_{i}^{\scriptscriptstyle{(j)}}$ denotes the climbing degree of color $j$ at time $t(n^\varrho) +i$, while $\widetilde u_\ell^{\scriptscriptstyle{(j)}}$ denotes the avalanche degree of color $j$ at time $T_j+\ell$ (we need $j=r$ in this section).
First we handle Case (1), i.e., we assume $T_b-T_r\ge2$. Recall the mountain climbing phase, i.e., the fact that $u_{i+1}^{\scriptscriptstyle{(b)}} = (u_{i}^{\scriptscriptstyle{(b)}})^{1/(\tau-2)}(1+o_{\mathbb{P}}(1))$. At time $T_r+k$, since blue would reach the top of the mountain at time $T_b$, blue is $T_b-T_r-k$ steps away from the top of the mountain, i.e., it is at vertices with $\log$(degree)/$\log n$ equal to
\begin{equation}\label{eq::bluedeg-k}\frac{(\tau-2)^{b_n^{\sss{(b)}}}}{\tau-1}(\tau-2)^{T_b-T_r-k}(1+o_{\mathbb{P}}(1))\end{equation}
while red at this moment is at $\log$(degree)/$\log n$ that is $\alpha_n^{\sss{(r)}}(\tau-2)^k(1+o_{\mathbb{P}}(1))$. Recall that $\alpha_n^{\sss{(r)}}=(\tau-2)^{\delta_n^{\sss{(r)}}}$ and that $\delta_n^{\sss{(r)}}\in[0,1)$. From here, the mountain climbing phase for blue and the avalanche phase for red can be continued, as long as the lowest degree where red occupies all the vertices is still higher than the maximal degree of blue. Let us define the \emph{real} time $t_c$ when this happens as the solution of the equation
\begin{equation}\label{eq::intersect}(\tau-2)^{{\delta_n^{\sss{(r)}}}}(\tau-2)^{t_c-1}=(\tau-2)^{b_n^{\sss{(b)}}}(\tau-2)^{T_b-T_r-t_c}\end{equation}
which yields\footnote{We remark that formula \eqref{eq::tc2} is the same as applying \cite[Equation 6.5]{BarHofKom14} with $\lambda=1$ and using the identity $\log (\alpha_n^{\sss{(r)}} (\tau-1))/|\log (\tau-2)|={\delta_n^{\sss{(r)}}}$.}
\begin{equation}\label{eq::tc2} t_c=\frac{T_b-T_r+1}{2} + \frac{b_n^{\sss{(b)}}-{\delta_n^{\sss{(r)}}}}{2}=\frac{1}{2}\frac{\log(Y_1^{\scriptscriptstyle{(n)}}/Y_2^{\scriptscriptstyle{(n)}})}{|\log (\tau-2)|} + \frac{b_n^{\sss{(r)}} + 1-{\delta_n^{\sss{(r)}}}}{2}. \end{equation}
Due to the shifts $\delta_n^{\sss{(r)}},b_n^{\sss{(b)}}$, we note that $t_c$ is typically not an integer. Note that the definition of $t_c$ is so that at time $\lfloor t_c\rfloor$, the maximal degree of blue, given by $u_{\lfloor t_c\rfloor}^{\scriptscriptstyle{(b)}}$ is just `slightly less' than the location of the red avalanche at degrees $\widetilde u^{\sss{(r)}}_{\lfloor t_c\rfloor}$.
In what follows, we determine what happens at the next jump, at time $\lfloor t_c\rfloor +1$.
Note that
writing $\lfloor t_c\rfloor= t_c-\{t_c\}$ and using \eqref{eq::tc2} combined with \eqref{eq::bluedeg-k}, the maximal degree of blue at $\lfloor t_c\rfloor$ and $\widetilde u^{\sss{(r)}}_{\lfloor t_c\rfloor}$ satisfy
\begin{align} \log (D^{\max}_n(\lfloor t_c\rfloor))/\log n &=\frac{ (\tau-2)^{\frac{b_n^{\sss{(b)}}+\delta_n^{\sss{(r)}}}{2}}}{\tau-1} (\tau-2)^{\frac{T_b-T_r-1}{2}} (\tau-2)^{\{t_c\}}(1+o_{\mathbb{P}}(1)), \label{eq::dmax-tc} \\
\log (\widetilde u^{\sss{(r)}}_{\lfloor t_c\rfloor}) / \log n &= \frac{(\tau-2)^{\frac{b_n^{\sss{(b)}}+\delta_n^{\sss{(r)}}}{2}}}{\tau-1} (\tau-2)^{\frac{T_b-T_r-1}{2}} (\tau-2)^{-\{t_c\}}(1+o_{\mathbb{P}}(1)). \label{eq::wur-tc}
\end{align}
Neglecting the $(1+o_{\mathbb{P}}(1))$ terms, we get that the logarithm of \eqref{eq::dmax-tc} divided by \eqref{eq::wur-tc}, divided by $|\log (\tau-2)|$, i.e., the distance between the two exponents is given by
\begin{equation}\label{eq::dtc2} 2\{t_c\} = 2\left\{\frac{T_b-T_r+1}{2} + \frac{b_n^{\sss{(b)}}-\delta_n^{\sss{(r)}}}{2}\right\}. \end{equation}
Clearly, $2\{t_c\} \in [0,2)$. %
At time $T_r + \lfloor t_c \rfloor+1$, when $2\{t_c\}>1$, the maximal degree of blue is increased by a factor $1/(\tau-2)$ in the exponent, while, when $2\{t_c\}<1$, the vertices `slightly below'\footnote{`Slightly below' here means $\widetilde u^{\sss{(r)}}_{\lfloor t_c \rfloor}(C\log n)^{-2} $: the number of vertices between degrees $\widetilde u^{\sss{(r)}}_{\lfloor t_c \rfloor}(C\log n)^{-2} $ and $\widetilde u^{\sss{(r)}}_{\lfloor t_c \rfloor}$ is then large enough so that whp at least one of them will get color blue at time $\lfloor t_c \rfloor+1$.} $\widetilde u^{\sss{(r)}}_{\lfloor t_c \rfloor}$ are reached both by color red and blue at the same time (at time $\lfloor t_c \rfloor+1$), and hence each of these vertices their color is red and blue with equal probability. Hence, the maximal degree of blue can be described as
\begin{equation}\label{eq::max-degree-final} \frac{\log D_n^{\max}(\infty)}{(\tau-1)^{-1}\log n}=(\tau-2)^{\frac{b_n^{\sss{(b)}}+\delta_n^{\sss{(r)}}}{2}} (\tau-2)^{
\frac{T_b-T_r-1}{2}} (\tau-2)^{(\{t_c\}-1)\mathbbm{1}\{2\{t_c\}>1\} - \{t_c\}\mathbbm{1}\{2\{t_c\}<1\} }(1+o_{\mathbb{P}}(1)).\end{equation}
Next we analyse $\{t_c\}$, and then substitute the results into this formula.
Since $T_r,T_b$ are integers, the first term in \eqref{eq::dtc2} contributes either $0$ or $1/2$ to the fractional part, depending on the parity, while the second term is between $(-1/2,1/2)$. Hence we can distinguish four cases depending on the parity of $T_r+T_b$ and $b_n^{\sss{(b)}}-{\delta_n^{\sss{(r)}}} <0$ or $b_n^{\sss{(b)}}-{\delta_n^{\sss{(r)}}} >0$.
From \eqref{def::alpha} and \eqref{def::delta} it follows that the condition $b_n^{\sss{(b)}}-{\delta_n^{\sss{(r)}}}>0$ is equivalent to
\begin{equation}\label{cond::alpha<bn2} \tau-1>(\tau-2)^{b_n^{\sss{(b)}}}+(\tau-2)^{b_n^{\sss{(r)}}},\end{equation} corresponding to Cases $(E_>)$ and $(O_>)$ below. The region at which this is satisfied is above the red curve in Fig~\ref{fig::contourplot}.
Here we list what happens with the maximal degree of blue in the four cases:
\begin{description}
\item[\namedlabel{caseodd1}{$(O_{>})$}]$T_b-(T_r+1)=2k+1$ and $b_n^{\sss{(b)}}>\delta_n^{\sss{(r)}}$. In this case both fractional parts in \eqref{eq::dtc2} are at least $1/2$, thus $\{t_c\}=(1+b_n^{\sss{(b)}}-\delta_n^{\sss{(r)}})/2\ge 1/2$, and $\lfloor t_c \rfloor=(T_b-T_r)/2$, $2\{t_c\}\ge 1$.
The right hand side of \eqref{eq::max-degree-final} equals $(\tau-2)^{b_n^{\sss{(b)}}} (\tau-2)^{(T_b-T_r-2)/2}$. Note that $(T_b-T_r-2)/2\ge 0$ if $T_b-T_r\ge 2$.
See Fig~\ref{fig::odd1}.
\item[\namedlabel{caseeven1}{$(E_{>})$}]$T_b-(T_r+1)=2k$ and $b_n^{\sss{(b)}}> \delta_n^{\sss{(r)}}$. In this case $2\{t_c\}=b_n^{\sss{(b)}}-{\delta_n^{\sss{(r)}}}<1$, and $\lfloor t_c \rfloor=(T_b-T_r+1)/2$. The right hand side of \eqref{eq::max-degree-final} becomes $ (\tau-2)^{\delta_n^{\sss{(r)}}} (\tau-2)^{(T_b-T_r-1)/2}$. Note that $(T_b-T_r-1)/2$ is integer-valued and $\ge 0$ if $T_b-T_r\ge 3$. See Fig~\ref{fig::even1}.
\item[\namedlabel{caseodd2}{$(O_<)$}]$T_b-(T_r+1)=2k+1$ and $b_n^{\sss{(b)}}<\delta_n^{\sss{(r)}}$. In this case $\{t_c\}=(1+b_n^{\sss{(b)}}-\delta_n^{\sss{(r)}})/2$ and $\lfloor t_c \rfloor=(T_b-T_r)/2$, $2\{t_c\}<1$.
The right hand side of \eqref{eq::max-degree-final} is $(\tau-2)^{\delta_n^{\sss{(r)}}} (\tau-2)^{(T_b-T_r-2)/2}$. Note that $(T_b-T_r-2)/2\ge 0$ if $T_b-T_r\ge 2$.
See Fig~\ref{fig::odd2}.
\item[\namedlabel{caseeven2}{$(E_<)$}] $T_b-(T_r+1)=2k$ and $b_n^{\sss{(b)}}<\delta_n^{\sss{(r)}}$. In this case $\{t_c\}=1+(b_n^{\sss{(b)}}-\delta_n^{\sss{(r)}})/2$ and $\lfloor t_c \rfloor=(T_b-T_r-1)/2$, and $2\{t_c\}>1$.
The right hand side of \eqref{eq::max-degree-final} becomes $(\tau-2)^{b_n^{\sss{(b)}}} (\tau-2)^{(T_b-T_r-1)/2}$. Note that $(T_b-T_r-1)/2\ge0$ if $T_b-T_r\ge 3$.
See Fig~\ref{fig::even2}.
\end{description}
\begin{figure}[t]
\includegraphics[width=12cm]{cases1b}
\caption{Case $(O_>)$: $b_n^{\sss{(b)}}>\delta_n^{\sss{(r)}}$ and $T_b-(T_r+1)$ is odd ($=3$ in the picture). The picture shows the exponent of $\tau-2$ in the formula for $\log$(degrees)/$\log n$. At time $T_r+1$, blue is at $b_n^{\sss{(b)}}+3$ and red is at $\delta_n^{\sss{(r)}}$ in the picture; the full and dashed arrows indicate their jump at time $T_r+2$ and $T_r+3$, respectively. Since $b_n^{\sss{(b)}}>\delta_n^{\sss{(r)}}$ and $T_b-T_r-1=3$ is odd, the distance between the two colors just before merging (at time $T_r+2$) is $b_n^{\sss{(b)}} +2-(\delta_n^{\sss{(r)}}+1)>1$. So, at time $T_r+3$, there is a region between them where they jump to vertices at the same time (indicated by dashed horizontal lines). Hence, the exponent of $\tau-2$ in the maximal degree of blue becomes $b_n^{\sss{(b)}}+(T_b-T_r-2)/2$, which is $b_n^{\sss{(b)}}+1$ in the figure.}\label{fig::odd1}
\end{figure}
\begin{figure}[t]
\includegraphics[width=14cm]{cases1a}
\caption{Case $(E_>)$: $b_n^{\sss{(b)}}>\delta_n^{\sss{(r)}}$ and $T_b-(T_r+1)$ is even ($=4$ in the picture).
The picture shows the exponent of $\tau-2$ in the formula for $\log$(degrees)/$\log n$. At time $T_r+1$, blue is at $b_n^{\sss{(b)}}+4$ and red is at $\delta_n^{\sss{(r)}}$ in the picture; the full arrows indicate their jumps at time $T_r+2$ and $T_r+3$. Since $b_n^{\sss{(b)}}>\delta_n^{\sss{(r)}}$ and $T_b-T_r-1=4$ is even, the distance between the two colors just before merging (at time $T_r+3$) is less than $1$. So, at time $T_r+4$, there is a region where they jump to vertices at the same time (indicated by dashed horizontal lines), i.e., the exponent of $\tau-2$ in the maximal degree of blue is $\delta_n^{\sss{(r)}}+(T_b-T_r-1)/2$, which is $\delta_n^{\sss{(r)}}+2$ in the figure.}
\label{fig::even1}
\end{figure}
\begin{figure}[ht]
\includegraphics[width=12cm]{cases2b}
\caption{Case $(O_<)$: $b_n^{\sss{(b)}}<\delta_n^{\sss{(r)}}$ and $T_b-(T_r+1)$ is odd ($=3$ on the picture). The picture shows the exponent of $\tau-2$ in the formula for $\log$(degrees)/$\log n$. At time $T_r+1$, blue is at $b_n^{\sss{(b)}}+3$ and red is at $\delta_n^{\sss{(r)}}$ in the picture; the full arrows indicate their jumps at time $T_r+2$. Since $b_n^{\sss{(b)}}<\delta_n^{\sss{(r)}}$ and $T_b-T_r-1=3$ is odd, the distance between the two colors just before merging (at time $T_r+2$) is less than $1$. So, at time $T_r+3$, there is a region where they jump to vertices at the same time (indicated by dashed horizontal lines), i.e., the exponent of $\tau-2$ in the maximal degree of blue is $\delta_n^{\sss{(r)}}+(T_b-T_r-2)/2$, which is $\delta_n^{\sss{(r)}}+1$ in the figure.}\label{fig::odd2}
\end{figure}
\begin{figure}[t]
\includegraphics[width=14cm]{cases2a}
\caption{Case $(E_<)$: $b_n^{\sss{(b)}}<\delta_n^{\sss{(r)}}$ and $T_b-(T_r+1)$ is even ($=4$ on the picture). The picture shows the exponent of $\tau-2$ in the formula for $\log$(degrees)/$\log n$. At time $T_r+1$, blue is at $b_n^{\sss{(b)}}+4$ and red is at $\delta_n^{\sss{(r)}}$ in the picture; the full and dashed arrows indicate their jumps at time $T_r+2$ and $T_r+3$. Since $b_n^{\sss{(b)}}<\delta_n^{\sss{(r)}}$ and $T_b-T_r-1=4$ is even, the distance between the two colors just before merging (at time $T_r+2$) is larger than $1$. So, at time $T_r+3$, there is a region where they jump to vertices at the same time (indicated by dashed horizontal lines), i.e., the exponent of $\tau-2$ in the maximal degree of blue is $b_n^{\sss{(b)}}+(T_b-T_r-1)/2$, which is $b_n^{\sss{(b)}}+2$ in the figure.}\label{fig::even2}
\end{figure}
We are left to calculate the expressions for the maximal degree using \eqref{eq::Tj}:
\begin{equation}\label{eq::t2-t1exp}(\tau-2)^{(T_b-T_r)/2} = \sqrt{Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}} (\tau-2)^{\frac{b_n^{\sss{(r)}}-b_n^{\sss{(b)}}}{2}}.\end{equation}
Combining this with the maximal degree listed in the four cases above, it is not hard to see that
\begin{equation}\label{eq::max-deg-2} \frac{\log \left(D_n^{\max}(\infty)\right)}{\log n} = \sqrt{Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}} \frac{1}{\tau-1} h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})(1+o_{\mathbb{P}}(1)),\end{equation}
with
\[\begin{aligned} h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) =& \mathbbm{1}_{ E_< \cup O_>} (\tau-2)^{(b_n^{\sss{(r)}}+b_n^{\sss{(b)}}-1 - \mathbbm{1}_{O_>})/2} +\\
&+\mathbbm{1}_{ E_> \cup O_<} (\tau-2)^{(b_n^{\sss{(r)}}-b_n^{\sss{(b)}}-1 - \mathbbm{1}_{O_<})/2} ((\tau-1)-(\tau-2)^{b_n^{\sss{(r)}}}), \end{aligned} \]
exactly as in \eqref{eq::h1} in the statement of Theorem \ref{thm::maxdegree}. Note that in all cases, using the representations listed in the four cases, the maximal degree is either $(\tau-2)^{b_n^{\sss{(b)}}+\ell}/(\tau-1)\le 1$ or $(\tau-2)^{\delta_n^{\sss{(r)}}+\ell}/(\tau-1)\le 1$ for some integer $\ell\ge 1$ when $T_b-(T_r+1)\ge 2$. Hence, the maximal degree in all cases is a random power of $n$ that is at most $(\tau-2)/(\tau-1)$.
It remains to investigate the cases when $T_b-T_r\in \{0,1\}$.
\begin{enumerate}
\item[(2)] If $T_b-T_r=0$, then in Section \ref{sc::peak} we have established that red and blue paint every vertex in the set $\widetilde \Gamma^{\sss{(b)}}_1$ with equal probability.
As a result, the maximal degree of blue whp tends to the maximal degree in ${\mathrm{CM}}_n(\boldsymbol{d})$, that is, it is of order $n^{1/(\tau-1)(1+o_{\mathbb{P}}(1))}$ (see Fig. \ref{fig::cross-1}).
\item[(3)(a)] If $T_b-T_r=1$ but $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<\tau-2$, then we have seen in Section \ref{sc::peak} that at time $T_r+1$, blue arrives to a few vertices with $\log$(degree)/$\log n$ that is $(\tau-2)^{b_n^{\sss{(b)}}}/(\tau-1)(1+o_{\mathbb{P}}(1))$, while red occupies all vertices with $\log$(degree)/$\log n$ that is $\alpha_n^{\sss{(r)}}=(\tau-2)^{\delta_n^{\sss{(r)}}}/(\tau-1)$. Two scenarios are possible:
\begin{enumerate}
\item[(3)(a.1)] If $T_b-T_r=1$ and $b_n^{\sss{(b)}}>\delta_n^{\sss{(r)}}$ then at time $T_b=T_r+1$ red is still at higher degree vertices than blue. At $T_r+2$, blue can increase its exponent to some vertices with $\log$(degree)/$\log n$ at most $(\tau-2)^{\delta_n^{\sss{(r)}}}/(\tau-1).$ Since $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$ implies that $\delta_n^{\sss{(r)}}>\delta_n^{\sss{(b)}}$ in this case, the vertices blue tries to occupy via crossing the mountain are already all painted red. Note that the maximal degree is a power of $n$ that is less then $1/(\tau-1)$. This case can be merged into Case (1), $(E_>)$ above.
\item[(3)(a.2)] If $T_b-T_r=1$ and $b_n^{\sss{(b)}}<\delta_n^{\sss{(r)}}$ then red has already occupied lower degree vertices, so the maximal degree of the blue remains at exponent $(\tau-2)^{b_n^{\sss{(b)}}}/(\tau-1)$. Further, again, since $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$ implies that $\delta_n^{\sss{(r)}}>\delta_n^{\sss{(b)}}$ in this case, the vertices blue tries to occupy via crossing the mountain are already all painted red. Note again that this is a power that is less than $1/(\tau-1)$. This case can be merged into Case (1), $(E_<)$ above.
\end{enumerate}
\item[(3)(b)] If $T_b-T_r=1$ and $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$. Note that $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$ implies $\delta_n^{\sss{(b)}}>\delta_n^{\sss{(r)}}$, and hence the maximal degree vertex blue can paint depends on the fact if $b_n^{\sss{(b)}}>\delta_n^{\sss{(r)}}$ or $b_n^{\sss{(b)}}<\delta_n^{\sss{(r)}}$:
The same argument works here as for Cases (3)(a.1) and (3)(a.2), i.e., if $b_n^{\sss{(b)}}>\delta_n^{\sss{(r)}}$, then the maximal degree of blue increases its exponent to $(\tau-2)^{\delta_n^{\sss{(r)}}}/(\tau-1)$, while if $b_n^{\sss{(b)}}<\delta_n^{\sss{(r)}}$, then the maximal degree of blue remains at the exponent $(\tau-2)^{b_n^{\sss{(r)}}}/(\tau-1)$. Geometrically, this means that the line where blue jumped from to the mixed area on Figure \ref{fig::cross-2} can be both above or below the bottom of the all-red area on the top of the mountain.
\end{enumerate}
Summarizing, if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} <\tau-2$, then the maximal degree of blue is always less than $1/(\tau-1)$, and is described by the cases $E_<, E_>, O_<, O_>$ above. Dividing by $h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ we see that
\[ \frac{\log (D_n^{\max}(\infty))}{(\tau-1)^{-1}h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})\log n} = \sqrt{Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}} (1+o_{\mathbb{P}}(1)),
\] and the right hand side tends to $\sqrt{Y_b/Y_r}$ in distribution.
Note that we did not investigate one issue, namely, what part of the proof was an upper and what part was a lower bound. First, Section \ref{sc::climbup} establishes the existence a blue vertex at time $t(n^{\varrho}) +i$ of degree at least $u_i^{\scriptscriptstyle{(b)}}$. On the other hand, Lemma \ref{lem::badpaths} states that whp blue does not paint any vertex with degree higher than $\widetilde u_i^{\scriptscriptstyle{(b)}}$ at the same time. Hence, on the blue part, the estimates we used were both upper and lower bounds, up to the $(1+o_{\mathbb{P}}(1))$ factors that exactly describe the ratio $\log (\widehat u_i^{\scriptscriptstyle{(b)}}/u_i^{\scriptscriptstyle{(b)}})$, and they tend to zero when divided by $\log n$ in the statement of the theorem.
One might imagine that the avalanche might actually roll down `faster' than described by the layers $\widetilde \Gamma^{\sss{(r)}}_\ell$s. To eliminate this problem we argue as follows: even though it might happen that the red avalanche occupies at time $T_r+\ell$ some lower degree vertices than $\widetilde u^{\sss{(r)}}_\ell$, but if so, only a small number of them, not all of them.\footnote{A similar argument could be used as in Lemma \ref{lem::red-intervals} to show that the number of these vertices is small, at least proportionally to the number of vertices in any given layer. However, the argument given here is easier.}
If it would meet blue earlier than it should (i.e., at time $\lfloor t_c \rfloor$ or $\lfloor t_c \rfloor+1$ as described by the four cases $E_<, E_>, O_<, O_>$ above), then the path formed by vertices in the red fast avalanche and the blue climbing path together would establish a path to the top of the mountain for blue that violates Lemma \ref{lem::badpaths}. Hence, this will not happen whp. As a result, our bounds on the maximal degree of blue are both upper and lower bounds, i.e., they hold whp.
\end{proof}
\section{Coexistence when $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$.}\label{sc::coexistence}
\subsection{Excursion to absolute continuity and some general estimates}
Before we move on to the proof of coexistence, we need some preliminaries. In Proposition \ref{prop::no-gap} below, we collect some results on the generating function of $D, D^\star$ and on the limiting random variable $Y$ of the branching process described in in Definition \ref{def::limit-variables}.
To show coexistence in the competition model, we will later need that $Y$ has an absolutely continuous distribution with support including some interval $(0, K)$, for some constant $K>0$. Hence the following assumption:
\begin{assumption}\label{assume::convex}
Let us write $h^{\star}(s):=\sum_{k=1}^\infty \mathbb{P}(D^{\star}\!=k) s^k$ where $D^\star$ follows the distribution \eqref{def::size-biased1}.
We assume that $f(t):=-\log( 1- h^\star(1-{\mathrm e}^{-t}) )$ is convex or concave on $\mathbb{R}^+$.
\end{assumption}
\begin{proposition}\label{prop::no-gap} Recall the degree distribution $D$ from \eqref{eq::F} and its size-biased version $D^\star$ from \eqref{def::size-biased1}.
Then
(i) the generating functions $h(s):=\sum_{k=2}^\infty \mathbb{P}(D=k) s^k$ and $h^{\star}(s):=\sum_{k=1}^\infty \mathbb{P}(D^{\star}=k) s^k$ satisfy
\begin{equation}\label{eq::gen-func} 1-h(s)=(1-s)^{\tau-1} L(1/(1-s)), \quad 1-h^{\star}(s)=(1-s)^{\tau-2} L^\star(1/(1-s)), \end{equation}
where $0<L(x), L^\star(x)<\infty$ are bounded slowly varying functions.
(ii) Consider $\widetilde Z_k$, the size of the $k$th generation in a Galton-Watson branching process with offspring distribution $D^\star$. Under Assumption \ref{assume::convex}, $\widetilde Y=\lim_{k\to \infty} (\tau-2)^k \log \widetilde Z_k$ has an absolutely continuous distribution function $J(x)$ with full support on $(0, \infty)$, that can be written as
\begin{equation}\label{eq::jx} J(x)=1-\exp\{ - \Delta(x) \},\end{equation}
where $\Delta(x)$ is continuous, strictly increasing, and can be written as
\[ \Delta(x) = \int_0^x p(t) \mathrm dt,\]
where $p(t)$ is strictly positive; increasing if $f$ is convex, and decreasing if $f$ is concave. Further, $\lim_{x\to \infty} \Delta(x)/x=1$, in agreement with \eqref{eq::exp-tails-Y}.
\end{proposition}
\begin{proof}The proof of (i) can be found in general in \cite{Feller71}, and the fact that the slowly varying functions are bounded is a consequence of the condition in \eqref{eq::F}, i.e., the slowly varying function hidden in the distribution function is bounded.
Part (ii) is a rewrite of \cite[Theorem C, Theorem 4]{Sene74}.
\end{proof}
The rest of this section holds under the milder condition that Assumption \ref{assume::abs-cont} holds, but we will use the notations $J(x)$ and $\Delta(x)$ as in Proposition \ref{prop::no-gap}. Note that already Assumption \ref{assume::abs-cont} implies that $J(x)$ is strictly increasing, hence it can also be written in a form as in \eqref{eq::jx}, with a strictly increasing, though not necessarily continuous, $\Delta(x)$.
Note that if we would like to approximate the exploration of the graph from a uniformly chosen vertex $w$ in ${\mathrm{CM}}_n(\boldsymbol{d})$, then the root of the approximating BP has offspring distribution $D$, and all the further individuals have offspring from distribution $D^\star$. We denote the number of individuals in this BP by $Z_k^w$. To identify the limit random variable $Y^w:=\lim_{k\to \infty} (\tau-2)^k \log Z_k^w$, we use \cite[Lemma 2.4]{BarHofKom14}, stating
$Y^w=(\tau-2) \max_{i=1, \dots D_w} \widetilde Y^{(i)}$. From this representation and Proposition \ref{prop::no-gap} it is obvious that under Assumption \ref{assume::abs-cont} $Y^w$ also has full support on $(0, K)$, with strictly increasing distribution function $J^w(x)$, since
\[ J^w(x):=\mathbb{P}(Y^w \le x) = \sum_{k=2}^\infty \mathbb{P}(D=k) J(x)^k,\]
by dominated convergence and the fact that each term is strictly increasing by Assumption \ref{assume::abs-cont}. Hence,
$\Delta^w(x) := - \log (1- J^w(x))$ is strictly increasing.
Further, using \eqref{eq::jx} and \eqref{eq::gen-func},
\[ J^w(x) = \mathbb{E}[ (1- {\mathrm e}^{-\Delta(x/(\tau-2))})^D ]= 1- {\mathrm e}^{ -(\tau-1) \Delta(x/(\tau-2)) } L({\mathrm e}^{\Delta(x/(\tau-2))}),\]
where $L$ is defined in \eqref{eq::gen-func}.
Since $L(x)$ is a bounded positive function, $Y^w$ has also exponential decay with exponent
\begin{equation}\label{eq::jw-tail} \lim_{x\to \infty} \Delta^w(x)/x=(\tau-1)/(\tau-2). \end{equation}
Next, we investigate the relationship between the sum of the degrees and the maximum of the degrees in each generation in the branching process with root $w$. We write $\mathcal {G}_k$ for the set of individuals in the $k$-th generation. Clearly $|\mathcal {G}_k|=Z_k^w$.
\begin{claim}\label{cl::max-sum-relation}
Let $M_k:=\max_{i\in \mathcal {G}_k} D_i^\star$. On the event $\lim_{k\to \infty}(\tau-2)^k \log Z_k^w = Y^w$, the limit $\lim_{k\to \infty}(\tau-2)^k \log M_k = Y^w/(\tau-2)$ holds.
\end{claim}
\begin{proof} Intuitively, the statement of the lemma should hold since $M_k\approx Z_{k+1}^w$. In a bit more detail,
similarly as in the proof of Claim \ref{cl::stoch-dom-alpha-stable}, we can pick $0\le b_1\le b_2$ so that $b_1+ X^{\scriptscriptstyle{(1)}}\ {\buildrel {d}\over \le } \ D^\star\ {\buildrel {d}\over \le } \ b_2+ X^{\scriptscriptstyle{(2)}}$, where the random variable $X^{\scriptscriptstyle{(1)}}$ has distribution function $1-c_1^\star/ x^{\tau-2}$ on $[0, \infty)$, and $X^{\scriptscriptstyle{(2)}}$ has distribution function $1-C_1^\star/x^{\tau-2}$ on $[0, \infty)$.
Both of these random variables are totally asymmetric stable distributions with skewness $\kappa=1$, shifts $b_1$ and $b_2$, and some scale parameters $c^{\scriptscriptstyle{(1)}}$ and $c^{\scriptscriptstyle{(2)}}$, respectively.
Then for any $m\in \mathbb{N}$, $x\in \mathbb{R}_+$
\begin{equation}\label{eq::stoch-dom-above} \mathbb{P}\left( \max_{i=1}^m X_i^{\scriptscriptstyle{(1)}} \le m^{1/(\tau-2)} x \right) = \left(1- \frac{c_1^\star} {x^{1/(\tau-2)} m}\right)^m \le \exp\{ -c_1^\star/ x^{1/(\tau-2)} \}, \end{equation}
while
\begin{equation}\label{eq::stoch-dom-below} \mathbb{P}\left( \max_{i=1}^m X_i^{\scriptscriptstyle{(2)}} \le m^{1/(\tau-2)} x \right) = \left(1- \frac{C_1^\star} {x^{1/(\tau-2)} m}\right)^m \ge \exp\{ -\tfrac12 C_1^\star x^{1/(\tau-2)} \}, \end{equation}
for large enough $x\in \mathbb{R}$.
Let us denote random variables with distribution function given by the right hand side of \eqref{eq::stoch-dom-above} and \eqref{eq::stoch-dom-below} by $M^{\scriptscriptstyle{(1)}}$ and $M^{\scriptscriptstyle{(2)}}$, respectively. Then
\[ \frac{b_1}{m^{1/(\tau-2)}} + M^{\scriptscriptstyle{(1)}}\ {\buildrel {d}\over \le }\ \frac{\max_{i=1}^m D_i^\star}{m^{1/(\tau-2)}}\ {\buildrel {d}\over \le } \ \frac{b_2}{m^{1/(\tau-2)}}+ M^{\scriptscriptstyle{(2)}}. \]
On the other hand, using the method in the proof of Claim \ref{cl::stoch-dom-alpha-stable}, the stochastic domination in \eqref{eq::sum-domination} can be used to estimate the moment generating function of
$\sum_{i=1}^m D_i^\star/m^{1/(\tau-2)}$ which yields
\[ b_1+ X^{\scriptscriptstyle{(1)}}\ {\buildrel {d}\over \le }\ \frac{\sum_{i=1}^m D_i^\star}{m^{1/(\tau-2)}}\ {\buildrel {d}\over \le }\ b_2+ X^{\scriptscriptstyle{(2)}}\ \]
Combining the last two estimates yields
\begin{equation}\label{eq::ratio-domination} \frac{M^{\scriptscriptstyle{(1)}}}{b_2+ X^{\scriptscriptstyle{(2)}}}\ \ {\buildrel {d}\over \le }\ \frac{M_k}{Z_{k+1}}\ {\buildrel {d}\over \le }\ \frac{M^{\scriptscriptstyle{(2)}} + b_2/m^{1/(\tau-2)}}{b_1+X^{\scriptscriptstyle{(1)}}}, \end{equation}
where all random variables are positive a.s. Hence, writing
\[ (\tau-2)^k\log M_k = (\tau-2)^k \log \left(\frac{M_k}{Z_{k+1}} \right) + \frac{1}{\tau-2}(\tau-2)^{k+1} \log Z_{k+1}, \]
we conclude that the first term tends to zero by the stochastic dominations in \eqref{eq::ratio-domination}, and the second term tends to $\widetilde Y^w / (\tau-2)$ by Theorem \ref{thm::davies}.
\end{proof}
\subsection{Coexistence}
Next we turn to the proof of coexistence when $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2.$
To understand the proportion of vertices that blue eventually paints, we use the usual trick that
\begin{equation}\label{eq::empirical-blue} \frac{\mathcal {B}_\infty}{n}= \frac{1}{n}\sum_{v\in [n]} \mathbbm{1}_{\{ v \text { is eventually blue} \}},\end{equation}
which can be interpreted as the empirical measure of blue. To show coexistence it is thus enough to show that this expression is strictly positive with positive probability. We do this via the first and second moment method. The first moment gives the probability (conditional on $n, Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}$) that a uniformly chosen vertex is eventually painted blue, while for the second moment we need to investigate the probability that two uniformly chosen vertices are both eventually blue.
We have seen in Section \ref{sc::slopedown} that when $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}>\tau-2$ there is a `mixed' avalanche occupying lower and lower degree vertices. In both cases (i.e., $T_b=T_r$ or $T_b=T_r+1$), with each additional time unit, there is a new interval on the log-log scale that gets colored, namely $\mathcal {M}_\ell \cup \mathcal {R} ed_\ell$ at time $T_b+\ell$. In both cases vertices in the `top' part of this interval (in $\mathcal {M}_\ell$) are red and blue with equal probability (Lemma \ref{lem::independence}), while almost all vertices in the `bottom' part of the interval (in $\mathcal {R} ed_\ell$) are painted red (Lemma \ref{lem::red-intervals}), see Fig. \ref{fig::avalanches}.
The idea to prove coexistence is as follows: we have to avoid the problem of the degrees getting too small (and as a result, our estimates getting too noisy). Hence, we only `run' this mixed avalanche as long as it stays in relatively high vertices, the `core' of the graph, that we call $\mathrm{Core}_n$. We choose this $\mathrm{Core_n}$ in a way that `coincides' with the layers of the avalanche. After the avalanche reaches the boundary of the core, we stop it. As a result, every vertex in the core has been painted whp.
Then, we investigate how the neighbourhood of a random vertex $w$ `enters' the painted core of the graph: we approximate the neighborhood of the vertex by a branching process that is described in Definition \ref{def::limit-variables} and we let this BP grow until the random generation when it first hits the core of the graph. Depending on what the degrees are, the vertices in this last generation might be red, red or blue with equal probability, or uncolored. This gives a partial coloring of the last generation of the BP.
These colors then `percolate' through the BP tree towards the root, following the rules of the coloring scheme, that is, if an uncolored vertex at any time has neighbors of only one color, then it takes that color in the next time step, while if it has neighbors of more than one color, then it picks a color with equal probability. We will show that in this random coloring scheme, the root of the BP gets both colors with strictly positive probability that depends on the ratio $q=Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}$, and the probability that the root is blue tends to zero as $q \searrow (\tau-2)$. After this have been shown, a second moment method finishes the proof.
We start to investigate the first moment. In this section we condition on $Y_r^{\scriptscriptstyle{(n)}}$ and $Y_b^{\scriptscriptstyle{(n)}}$. Recall the definition of $\widetilde u^{\sss{(r)}}_\ell, \widetilde u^{\sss{(b)}}_\ell$ from \eqref{eq::wideui_recursion} and $\widetilde \Gamma^{\sss{(r)}}_\ell, \widetilde \Gamma^{\sss{(b)}}_\ell$ from \eqref{eq::wgar}. Throughout, we write $\widetilde\mathbbm{1}=\mathbbm{1}_{\{T_b=T_r\}}$ and $\buildrel {\mathbb{P}} \over{\sim} $ for whp equality up to factor of finite powers of $\log n$.
Note that if $\ell=\nu\log\log n / |\log(\tau-2)| +1 + x_n$, for some $\nu<1$, and $x_n$ is chosen so that the expression is an integer, then \eqref{eq::wideui_recursion} gives, for $j=r,b$,
\begin{equation} \label{eq::core-ell} \widetilde u_\ell^{\scriptscriptstyle{(j)}} = {\mathrm e}^{(\log n)^{1-\nu}\alpha_n^{\scriptscriptstyle{(j)}} (\tau-2)^{x_n}} (C \log n)^{1/(3-\tau)} (1+o_{\mathbb{P}}(1)).\end{equation}
Also, by Lemma \ref{lem::red-intervals}, the probability that a vertex in a red interval is not red is of order $\buildrel {\mathbb{P}} \over{\sim} \widetilde u^{\sss{(r)}}_{\ell+\widetilde\mathbbm{1}}/ \widetilde u^{\sss{(b)}}_{\ell}$. We would like to keep this probability small, and we also would like that $\widetilde u^{\sss{(b)}}_{\ell}\buildrel {\mathbb{P}} \over{\sim} (\widetilde u^{\sss{(r)}}_{\ell+\widetilde\mathbbm{1}})^{\alpha_n^{\sss{(b)}} (\tau-2)^{\widetilde\mathbbm{1}-1} / \alpha_n^{\sss{(r)}}}$ holds.
This holds as long as $0<\nu<1$. Note that for any fixed positive $\nu<1$, $\widetilde u^{\sss{(r)}}_\ell$ and $\widetilde u^{\sss{(b)}}_\ell$ are sub-polynomial in $n$.
So, let us fix a $0\!<\!\nu\!<\!1$ and then set $\ell_{\max}:=\lfloor \nu \log\log n / |\log (\tau-2)|\rfloor$, and define
\begin{equation}\label{def::core-n} \mathrm{Core}_n=\{ v\in {\mathrm{CM}}_n(\boldsymbol{d}): D_v> \widetilde u^{\sss{(r)}}_{\ell_{\max}} \}.\end{equation}
From now on we simply write $Q:=\widetilde u^{\sss{(r)}}_{\ell_{\max}}.$ Note that the bottom of the last mixed interval is at degree $\widetilde u^{\sss{(b)}}_{\ell_{\max}-\widetilde\mathbbm{1}}\buildrel {\mathbb{P}} \over{\sim} Q^{\alpha_n^{\sss{(b)}}(\tau-2)^{\widetilde\mathbbm{1}-1}/\alpha_n^{\sss{(r)}}}$, which implies that almost all vertices with degree in the interval
$[Q, \buildrel {\mathbb{P}} \over{\sim} \!Q^{\alpha_n^{\sss{(b)}}(\tau-2)^{\widetilde\mathbbm{1}-1}/\alpha_n^{\sss{(r)}}} )$ are painted red, while vertices with degree in the interval $[\buildrel {\mathbb{P}} \over{\sim} Q^{\alpha_n^{\sss{(b)}}(\tau-2)^{\widetilde \mathbbm{1}-1}/\alpha_n^{\sss{(r)}}}, \buildrel {\mathbb{P}} \over{\sim} \!Q^{1/(\tau-2)})$ are colored red and blue with equal probability.
We will simply write
\begin{equation}\label{eq::color-rule} [Q, Q^{\gamma}) \in \mathcal {R} ed, \quad [Q^{\gamma}, Q^{1/(\tau-2)}) \in \mathcal Mix, \end{equation}
with \begin{equation}\label{def::gamma}\gamma:=\alpha_n^{\sss{(b)}}(\tau-2)^{\widetilde \mathbbm{1}-1}/\alpha_n^{\sss{(r)}}.\end{equation}
Note that $\gamma \in (1,(\tau-2)^{-1}).$
By Lemma \ref{lem::red-intervals}, the proportion of potentially blue vertices in $\mathcal {R} ed_{\ell_{\max}}=[Q, Q^{\gamma}) $ is
\begin{equation}\label{eq::p-error} p_e:= \frac {\widetilde u^{\sss{(r)}}_{\ell_{\max}}}{\widetilde u^{\sss{(b)}}_{\ell_{\max} - \widetilde \mathbbm{1}}} (1+o_{\mathbb{P}}(1))\buildrel {\mathbb{P}} \over{\sim} Q^{1-\gamma}.\end{equation}
In this section, we write $\mathbb{P}_\gamma(.):= \mathbb{P}(.| \gamma, Q, Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}), \mathbb{E}_\gamma[.]:= \mathbb{E}[.| \gamma, Q, Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}]$.
\begin{remark}\label{rem::q-gamma-relation}\normalfont
It is intuitively clear that $q\to Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}\approx(\tau-2)$ if and only if $\gamma\to\tfrac{1}{\tau-2}$, while $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} \approx 1$ implies $\gamma \approx 1$.
To see this, let $\varepsilon, \varepsilon', \varepsilon'', \varepsilon'''$ be small positive numbers.
Recall that $\alpha_n^{\sss{(b)}}, \alpha_n^{\sss{(r)}} \in (\tfrac{\tau-2}{\tau-1}, \tfrac{1}{\tau-1}]$ (see \ref{def::alpha}).
When $T_r=T_b$, then $\gamma =\alpha_n^{\sss{(b)}}/\alpha_n^{\sss{(r)}}= \tfrac{1-\varepsilon}{\tau-2}$ is only possible if $\alpha_n^{\sss{(b)}}= \tfrac{1-\varepsilon'}{\tau-1}$, $\alpha_n^{\sss{(r)}}\approx \tfrac{\tau-2+\varepsilon''}{\tau-1}$ and this in turn implies $b_n^{\sss{(b)}}=1-\varepsilon'$ and $b_n^{\sss{(r)}} = \varepsilon''$. By \eqref{eq::key-2}, this is only possible if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} = (\tau-2)^{1-\varepsilon' -\varepsilon'' }$ and $n$ is so that $\tfrac{\log\log n - \log Y_j^{\scriptscriptstyle{n}}}{ |\log (\tau-2)}$ is close to an integer for both $j=r,b$.
When $T_b=T_r+1$, then since $\gamma=\alpha_n^{\sss{(b)}}/(\alpha_n^{\sss{(r)}}(\tau-2)) $ in this case, and $\alpha_n^{\sss{(b)}}<\alpha_n^{\sss{(r)}}$, $\gamma = \tfrac{1-\varepsilon}{\tau-2}$ is only possible if $\alpha_n^{\sss{(b)}} = \alpha_n^{\sss{(r)}}-\varepsilon'$, that is, if $b_n^{\sss{(b)}} = b_n^{\sss{(r)}}- \varepsilon''$, which, combined with $T_r=T_b+1$ again implies that this is only possible if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} \approx (\tau-2)^{1-\varepsilon'''}$.
Summarizing, we see that if $\gamma\approx 1/(\tau-2)$ then $ q=Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}\approx \tau-2$. The other direction can be treated similarly, as well as the equivalence between $\gamma \approx 1$ and $q=Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}\approx 1$.
\end{remark}
To investigate the color of a uniform vertex $w$, we couple its local neighborhood to an independent copy of a branching process described in Section \ref{sc::BP}, independent of the blue and red BPs. We will denote this BP by $\mathrm{BP}_w$ and run it until the stopping time when its maximum degree reaches $Q$.
This coupling can be achieved by extending \cite[Lemma 2.2]{BarHofKom14} to three vertices, where first we couple the degrees of the local neighbourhoods of the source vertices to the red and blue BP, determining $Y_b^{\scriptscriptstyle{(n)}}, Y_r^{\scriptscriptstyle{(n)}}$, then couple the degrees of the third vertex to a BP, till it reaches maximum degree $Q$, and then we can continue with everything else. Note that since $Q=o(n^\varepsilon)$ for every $\varepsilon>0$, this can easily be done, since the maximal degree is the same order of magnitude as the total number of vertices in the next generation by Claim \ref{cl::max-sum-relation}, and hence the size of the $\mathrm{BP}_w$ at stopping will be still sub-polynomial in $n$. Hence, the coupling can be done with coupling error that tends to zero with $n$.
Recall that every vertex except the root has degree from distribution $F^\star$ from \eqref{def::size-biased1}. We write $D^\star_x$ for the degree of vertex $x$ in $\mathrm{BP}_w$. Denote the set of vertices in generation $i$ of $\mathrm{BP}_w$ by $\mathcal {G}_i$ and define the stopping time $\kappa$ as
\[\kappa:= \inf\{j: \max_{x \in \mathcal {G}_j} D_x^\star \ge Q\}.\]
This definition of $\kappa$ ensures that $\kappa$ is the first generation where the local neighbourhood of vertex $w$ meets colored vertices, i.e., the shortest path from $w$ to $\mathrm{Core}_n$ and hence \emph{to any of the colored vertices} is of length $\kappa$:
\begin{equation}\label{eq::shortest-kappa} \mathcal {D}(w, \mathrm{Core}_n)=\kappa. \end{equation}
As a consequence, vertex $w$ will be coloured \emph{exactly} $\kappa$ time unit later than the coloring of the last interval in $\mathrm{Core}_n$ (that is, $w$ is colored at time $T_r+\ell_{\max}+ \kappa$).
We color vertices in $\mathcal {G}_{\kappa}$ so that their coloring in $\mathrm{BP}_w$ corresponds to the coloring of $\mathrm{Core}_n$. To provide an upper and a lower bound on the number of blue vertices in $\mathcal {G}_\kappa$, we describe two colorings that we couple together:
By Lemmas \ref{lem::independence} and \ref{lem::red-intervals} we color vertices of $\mathcal {G}_\kappa$ \emph{independently} of each other according to following rules, where $p_e$ is from \eqref{eq::p-error}:
\begin{framed}
\emph{Starting Rule 1:} a vertex $x\in \mathcal {G}_\kappa$ gets color
\begin{enumerate}[(i)]
\item red when $D_x \in [Q, Q^{\gamma})$,
\item red or blue with equal probability when $D_x\ge Q^{\gamma}$,
\item neutral when $D_x<Q$.
\end{enumerate}
\end{framed}
\begin{framed}
\emph{Starting Rule 2:}
a vertex $x\in \mathcal {G}_\kappa$ gets color
\begin{enumerate}[(i)]
\item red with probability $1-p_e$ and blue with probability $p_e$ when $D_x \in [Q, Q^{\gamma})$,
\item red or blue with equal probability when $D_x\ge Q^{\gamma}$,
\item neutral when $D_x<Q$.
\end{enumerate}
\end{framed}
Neutral color means no coloring, that is, the vertex is not in the core of the graph.
The two rules can be naturally coupled to the coloring of the core and also to each other: we can use a $p_e$-coin flip in Rule 2 to decide what happens when $D_x\in [Q, Q^{\gamma})$. Under this coupling, the number of blue vertices in Rule 1 and in Rule 2 is a stochastic lower and upper bound on the blue vertices in the intersection of the last generation of $\mathrm{BP}_w$ and $\mathrm{Core}_n$, respectively, due to Lemmas \ref{lem::independence} and \ref{lem::red-intervals}.
After we colored $\mathcal {G}_\kappa$, we color the vertices independently of each other in subsequent generations $\kappa-1, \kappa-2, ...., 1, 0$ as follows, (for both starting rules):
\begin{framed}
\emph{Flow rule:}
If a vertex $x\in \mathcal {G}_i$ has children in $\mathcal {G}_{i+1}$ that
\begin{enumerate}[(i)]
\item are all neutral, vertex $x$ gets color neutral,
\item are all either neutral or red, then it gets color red,
\item are all either neutral or blue, then it gets color blue,
\item have both colors red and blue, then it gets color red and blue with equal probability.
\end{enumerate}
\end{framed}
Note that this rule exactly corresponds to the rule that we set at the beginning of the paper for the spread of each color: if a vertex gets color $\mathcal{C}$ at time $t$, it colors its not yet colored neighbors to color $\mathcal{C}$ at time $t+1$, and succeeds to do so for each not yet colored neighbor $z$ unless $z$ has a neighbor of the other color as well, in which case $z$ gets color red and blue with equal probability. We can rewrite this rule from the point of view of $z$: $z$ stays neutral as long as it has no colored neighbors; and if some neighbors become colored at time $t$, $z$ takes the same color at time $t+1$ if it is a unique color (only red or only blue) while picks a color with equal probability otherwise.
Let us introduce some notation: we write $Z_i:= |\mathcal {G}_i|$, $M_i:=\max_{x \in \mathcal {G}_i} D_x^\star$,
so that the definition of $\kappa$ is equivalent to $\kappa=\inf\{ i: M_i\ge Q\}$.
To start with, let us introduce \begin{equation}\label{eq::Y_k^w} Y_k^w:=(\tau-2)^{k+1} \log M_{k},\end{equation}
and then we have $Y_k^w\buildrel {a.s.}\over{\longrightarrow} Y^w$ by Claim \ref{cl::max-sum-relation}. Rewrite this formula for $k=\kappa$, and compare it to the value $Q$:
\begin{equation}\label{eq::mkappa} M_{\kappa}=\exp\{ Y_{\kappa}^w (\tau-2)^{- (\kappa+1)}\}\ge Q. \end{equation} The solution to this inequality, conditioned on $Y_\kappa^w$, is given by
\begin{equation}\label{eq::kappa}\kappa=\left\lceil\frac{\log\log Q-\log Y_\kappa^w}{|\log (\tau-2)|}-1\right\rceil := \frac{\log\log Q-\log Y_\kappa^w}{|\log (\tau-2)|}-c_Q, \end{equation}
with $c_Q:=\left\{\frac{\log\log Q-\log Y_\kappa^w}{|\log (\tau-2)|}\right\}\in[0,1).$
In what follows, we analyse the structure of the BP under the conditioning that we stopped it at $\kappa$.
This conditioning implies that
\begin{enumeratei}
\item the maximal degree vertex in $\mathcal {G}_\kappa$ has degree $M_{\kappa}\ge Q$;
\item all other vertices in $\mathcal {G}_\kappa$ have degree $\le M_{\kappa}$;
\item all vertices in $\mathcal {G}_i, \ i<\kappa$ have degree $<Q$.
\end{enumeratei}
Without loss of generality we can assume that the maximal degree vertex $v^\star$ is unique in $\mathcal {G}_{\kappa}$ (if not, we pick the `leftmost' one).
Let us call the unique path to the root from $v^\star$ the \emph{ray}, and let us number its vertices backwards, that is, $v^\star:=v_0$, $v_1$ is the parent of $v^\star$, $v_2$ is the parent of $v_1$, etc, finally, $v_{\kappa}:=w$.
\begin{figure}[ht]
\includegraphics[width=\textwidth]{maxdegree-ray-2}
\end{figure}
We would like to show that $\mathbb{P}(w \text{ is red})$ and $\mathbb{P}(w \text{ is blue})$ are both strictly positive.
For the first, it is enough to show that the probability that there are no blue vertices at all in $\mathcal {G}_\kappa$ is strictly positive. For the second, it is enough to show that the probability $r_{\kappa}$ that there is a blue sibling of $v_{\kappa-1}$ is strictly positive, since then, $w$ gets color blue with probability at least $r_{\kappa}/2$.
As a preliminary result, we show that under both starting rules, both $\mathbb{P}(v^\star \text{ is blue})$ and $\mathbb{P}(v^\star \text{ is red,\ } D_{v^\star} <Q^\gamma)$ are strictly positive (and hence both strictly less than $1$). Note that under Starting Rule 1, the latter event implies that \emph{all} vertices in $\mathcal {G}_\kappa$ are colored red, hence, $\mathbb{P}(v^\star \text{ is red}, D_{v^\star} < Q^\gamma)$ gives a lower bound on $\mathbb{P}(w \text{\ is red})$.
\begin{lemma} \label{lem::q-delta} Let $\gamma\in (1,1/(\tau-2))$ and apply Starting Rule 1 or 2 for coloring the generation $\kappa$ of the stopped BP_w. Then there exist strictly positive numbers $0<q_{\gamma}, q_\gamma^{\text{red}}< 1$ such that
\begin{equation}\label{eq::q-delta-lem} \mathbb{P}_{\gamma}(v^\star \text{ is blue\,}) \ge q_{\gamma}, \quad \mathbb{P}_{\gamma}(v^\star \text{ is red,\ } D_{v^\star} <Q^\gamma) \ge q_{\gamma}^{\text{red}}.\end{equation}
Further, $q_\gamma \searrow 0$ and $q_\gamma^{\text{red}} \nearrow 1$ as $\gamma\nearrow 1/(\tau-2)$.
\end{lemma}
\begin{proof}
Under Starting Rule 1, $ \{ D_{v^\star}<Q^\gamma\} \subset \{v^\star \text{is red}\}$, hence,
\[ \mathbb{P}(D_{v^\star} < Q^\gamma) = \mathbb{P}_{\gamma}(M_{\kappa} < Q^{\gamma}), \quad \mathbb{P}(v^\star \text{ is blue}) \ge \mathbb{P}_{\gamma}(M_{\kappa} \ge Q^{\gamma})/2. \]
Using \eqref{eq::mkappa} and \eqref{eq::kappa} we get that $M_{\kappa}=Q^{(\tau-2)^{c_Q-1}}(1+o_\mathbb{P}(1))$, hence $M_{\kappa} < Q^{\gamma}$ if
\[ (\tau-2)^{c_Q-1}<\gamma.\]
This holds if and only if \begin{equation}\label{eq::crit} c_Q > 1-\frac{\log \gamma}{|\log (\tau-2)|}:=\widetilde\gamma. \end{equation}
Note also that $c_Q\in [0,1)$ by definition, and since $\gamma\in [1, 1/(\tau-2))$ whp, $\widetilde \gamma \in [0,1)$ whp.
By the definition of the fractional part,
\begin{equation}\label{eq::only-red}\begin{aligned} \mathbb{P}_{\gamma}(M_\kappa <Q^\gamma) &= \mathbb{P}_{\gamma}( c_Q \in (\widetilde\gamma, 1) )\\
&= \mathbb{P}_{\gamma}\left( Y_\kappa^w \in \bigcup_{m\in \mathbb{N}}\left((\tau-2)^{m+1} \log n , (\tau-2)^{m+\widetilde \gamma} \log n\right) \right). \end{aligned}\end{equation}
Similarly,
\begin{equation}\label{eq::not-only-red} \begin{aligned} \mathbb{P}_{\gamma}(M_\kappa \ge Q^\gamma ) &= \mathbb{P}_{\gamma}( c_Q \in [0, \widetilde\gamma) )\\
&= \mathbb{P}_{\gamma}\!\left( \! Y_\kappa^w \in\! \bigcup_{m\in \mathbb{N}}\!\left((\tau-2)^{m+\widetilde \gamma} \log n , (\tau-2)^{m} \log n\right) \!\right)\\
&= \mathbb{P}_{\gamma}(Y_\kappa^w \in \bigcup_{m\in \mathbb{N}}\!\! A_{n,\widetilde\gamma}(m)). \end{aligned}\end{equation}
We need to show that both \eqref{eq::only-red} and \eqref{eq::not-only-red} are strictly positive for all $n$ and $\gamma\in(1, (\tau-2)^{-1})$.
We only deal with $ \eqref{eq::not-only-red}$, but \eqref{eq::only-red} can be handled analogously. Note that since $Y_\kappa^w=Y_{\kappa(n)}^w \buildrel {a.s.}\over{\longrightarrow} Y^w$ as $n\to \infty$, \eqref{eq::not-only-red} can be rewritten in the form
$\mathbb{P}(Y_n \in A_{n,\widetilde\gamma})$, where $Y_n=Y_{\kappa(n)}^w$, and $A_{n,\widetilde\gamma}:=\cup_{m\in \mathbb{N}} A_{n,\widetilde\gamma}(m)$. The problem is that $A_{n,\widetilde\gamma}\subset \mathbb{R}^+$ does not converge with $n$.
We can overcome this issue as follows: first of all, recall that one of our assumptions was that $(0,K) \subset \mathrm{supp} (Y^w)$ for some $K>0$. Let us assume wlog that $K>1$: if this does not hold, replace $1$ in the following proof by any number that is smaller then $K$. Let us restrict our interest to the interval $[0,1]$, i.e., we investigate only those $A_{n,\widetilde\gamma}(m)$ for which $(\tau-2)^m\log n<1$.
We can then write
\begin{equation}\label{eq::yn-decompose} \mathbb{P}_{\gamma}(Y_n \in A_{n, \widetilde\gamma} \mathbbm{1}_{[0,1]}) \ge \mathbb{P}_{\gamma}(Y^w \in A_{n, \widetilde \gamma} \mathbbm{1}_{[0,1]} ) - \mathbb{P}_{\gamma}(Y^w \in A_{n, \widetilde \gamma}\mathbbm{1}_{[0,1]}, Y_n \notin A_{n, \widetilde \gamma}\mathbbm{1}_{[0,1]}). \end{equation}
Our goal is to show that the second term is sufficiently small for large enough $n$. Below in \eqref{eq::q-delta} we aim to show $\mathbb{P}_{\gamma}(Y^w \in A_{n, \widetilde \gamma} \mathbbm{1}_{[0,1]} )\ge r_{\widetilde\gamma}>0$. Given this number $r_{\widetilde\gamma}$, we show
\begin{equation} \label{eq::second-error} \mathbb{P}_{\gamma}(Y^w \in A_{n, \widetilde \gamma}\mathbbm{1}_{[0,1]}, Y_n \notin A_{n, \widetilde \gamma}\mathbbm{1}_{[0,1]}) \le r_{\widetilde \gamma}/2. \end{equation}
For this, let us introduce for $\varepsilon>0$
\begin{equation}\label{eq::and-epsilon}\begin{aligned} A_{n, \widetilde\gamma}^{-\varepsilon}(m):&= ((\tau-2)^{m+\widetilde\gamma-\varepsilon} \log n, (\tau-2)^{m+\varepsilon} \log n), \\
A_{n, \widetilde\gamma}^{-\varepsilon}&:= \bigcup_{m\in \mathbb{N}}A_{n, \widetilde\gamma}^{-\varepsilon}(m).\end{aligned}\end{equation}
Clearly, $A_{n, \widetilde\gamma}^{-\varepsilon}(m)\subset A_{n, \widetilde\gamma}(m).$
Using this notation, we have the upper bound
\begin{equation}\label{eq::error-n-d} \begin{aligned} \mathbb{P}_{\gamma}( Y^w \in A_{n, \widetilde \gamma}\mathbbm{1}_{[0,1]},& Y_n \notin A_{n, \widetilde \gamma}\mathbbm{1}_{[0,1]}) \le \mathbb{P}_{\gamma}( Y^w \in \mathbbm{1}_{[0,1]} A_{n, \widetilde\gamma}\setminus A_{n, \widetilde\gamma}^{-\varepsilon} ) \\
&+\!\!\!\!\!\!\!\!\!\! \sum_{\substack{m\in \mathbb{N}\\ (\tau-2)^m\log n\le1 } }\!\!\!\!\!\!\mathbb{P}_{\gamma}( Y_n \notin A_{n, \widetilde \gamma}(m)| Y^w \in A_{n, \widetilde \gamma}^{-\varepsilon}(m) ) \mathbb{P}_{\gamma}( Y^w \in A_{n, \widetilde\gamma}^{-\varepsilon}(m)). \end{aligned}\end{equation}
Now pick $\varepsilon$ small enough so that $\mathbb{P}_{\gamma}( Y^w \in \mathbbm{1}_{[0,1]} A_{n, \widetilde\gamma}\setminus A_{n, \widetilde\gamma}^{-\varepsilon} )\le r_{\widetilde\gamma}/4$. This can be done uniformly in $n$ by the absolute continuity of the measure of $Y^w$ and the compactness of the set $[0,1]$: the set $A_{n, \widetilde \gamma}\mathbbm{1}_{[0,1]}$ can be generalised to the form
\[ B_{\widetilde\gamma,x}:= \bigcup_{m\in \mathbb{N}} (x(\tau-2)^{m+\widetilde\gamma}, x(\tau-2)^m), \quad x \in [0,1].\] Note that the set $B_{\widetilde\gamma,x}$ only depends on the initial point $x$ and on the `width' given by the factor $(\tau-2)^{\widetilde \gamma}$. Now, we can define $B_{\widetilde\gamma,x}^{-\varepsilon}(m)$ similarly as for $A_{n, \widetilde\gamma}(m)$ for all $m\in \mathbb{N}$, and pick
an $\varepsilon=\varepsilon(\widetilde\gamma)$ so small that the total length of the union of intervals $B_{\widetilde\gamma, x}(m)\setminus B_{\widetilde\gamma,x}^{-\varepsilon}(m)$ is small enough even for the worst starting point $x$. Then by the absolute continuity of the measure (see \cite[Theorem 3.5]{Foll84}) it follows that $\mathbb{P}_{\gamma}(Y^w \in B_{\widetilde\gamma,x}\setminus B_{\widetilde\gamma,x}^{-\varepsilon}) ) \le r_{\widetilde\gamma}/4$ for all possible initial point $x$.
For the second term on the right hand side of \eqref{eq::error-n-d}, a simple calculation shows that $ \mathbb{P}_{\gamma}( Y_n \notin A_{n, \widetilde \gamma}(m)| Y^w \in A_{n, \widetilde \gamma}^{-\varepsilon}(m) ) $ implies either $Y_n/Y^w > (\tau-2)^{-\varepsilon}$ or $Y_n/Y^w\le (\tau-2)^\varepsilon$, hence,
\[ \mathbb{P}_{\gamma}\Big( Y_n \notin A_{n, \widetilde \gamma}(m)\Big| Y^w \in A_{n, \widetilde \gamma}^{-\varepsilon}(m) \Big) \le \mathbb{P}_{\gamma}\Big( \log Y_n - \log Y^w \notin ( -\widetilde \varepsilon, \widetilde \varepsilon ) \Big), \]
where $\widetilde\varepsilon:= \varepsilon | \log (\tau-2)|$. Now since $Y_n=Y_{\kappa(n)} \buildrel {a.s.}\over{\longrightarrow} Y^w$, clearly, $\log Y_n \buildrel {a.s.}\over{\longrightarrow} \log Y^w$ and hence the latter probability tends to zero for each fixed $\varepsilon$ as $n\to \infty$. Pick $n_0=n_0(\varepsilon)$ large enough so that this probability is at most $r_{\widetilde\gamma}/4$ for all $n\ge n_0(\varepsilon)$. Then, combining the estimates for the two terms and the fact that $A_{n,\widetilde\gamma}^{-\varepsilon} \subset A_{n, \widetilde\gamma}$, we get
\[ \mathbb{P}_{\gamma}( Y^w \in A_{n, \widetilde \gamma}\mathbbm{1}_{[0,1]}, Y_n \notin A_{n, \widetilde \gamma}\mathbbm{1}_{[0,1]}) \le r_{\widetilde\gamma}/4 + \mathbb{P}_{\gamma}(Y^w \in A_{n, \widetilde \gamma}\mathbbm{1}_{[0,1]} ) r_{\widetilde\gamma}/4 \le r_{\widetilde\gamma}/2.\]
To finish the proof that \eqref{eq::yn-decompose} is at most $r_{\widetilde\gamma}/2$, we are left to give the uniform lower bound $r_{\widetilde\gamma}$ on $\mathbb{P}(Y^w \in A_{n, \widetilde\gamma}\mathbbm{1}_{[0,1]})$.
Clearly,
\begin{equation}\label{eq::indicator-Y-in-01} \begin{aligned} \mathbb{P}_{\gamma}(Y^w \in A_{n, \widetilde\gamma}\mathbbm{1}_{[0,1]}) &=\!\!\!\!\!\!\!\!\! \sum_{\substack{m\in \mathbb{N} \\ (\tau-2)^m\log n< 1}} \!\!\!\!\!\!\!\!\exp\{- \Delta^w( (\tau-2)^{m+\widetilde\gamma} \log n)\} - \exp\{- \Delta^w( (\tau-2)^{m} \log n )\}\\
&\ge \exp\Big\{- \Delta^w( (\tau-2)^{1-c_n+\widetilde\gamma})\Big\} - \exp\Big\{- \Delta^w( (\tau-2)^{1-c_n+\widetilde\gamma})\Big\}
\end{aligned} \end{equation}
where we took the first $m$ that satisfies the criterion, i.e., $m=\lfloor\log\log n / | \log (\tau-2)|\rfloor + 1$, and introduced $c_n:=\{\log\log n / | \log (\tau-2)|\}\in [0,1)$.
Using that $(\tau-2)^{1-c_n} \in (\tau-2, 1)$ and $\Delta^w$ is strictly increasing (i.e., the difference is strictly positive), the right hand side is at least
\begin{equation}\label{eq::q-delta} r_{\widetilde\gamma}:=\min_{x \in [\tau-2,1]} \left\{ \exp\big\{\!-\! \Delta^w( x(\tau-2)^{\widetilde\gamma})\big\}
-\exp \big\{ \!-\!\Delta^w( x)\big\} \right\}.
\end{equation}
Note that this minimum exists and is strictly positive as well, since if it would be zero, that would imply the presence of an interval of size at least $(\tau-2)(1-(\tau-2)^{\widetilde\gamma})$ where $\Delta^w$ is a constant, which contradicts the fact that it is strictly increasing.\footnote{Without further assumption on the form of the generating function of $D^\star$ it is not possible to determine where this minimum is taken. E.g. it is not hard to determine using Proposition \ref{prop::no-gap} Part (2) that the minimum is taken at $(\tau-2)$ if $f$ is convex and at $1$ if $f$ is concave in Proposition \ref{prop::no-gap}.}
Further and more importantly, $r_{\widetilde\gamma}$ only depends on $\widetilde\gamma$, but not on $n$, hence, it provides a uniform lower bound on the probability that $v^\star$ is painted blue.
Combining everything, we get that
\[ \mathbb{P}_{\gamma}(v^\star \text { is blue}) = \mathbb{P}_{\gamma}( M_\kappa > Q^{\gamma})/2 = \mathbb{P}_{\gamma}( c_Q \in (0, \widetilde\gamma))/2 \ge r_{\widetilde\gamma}/4.\]
Setting $q_\gamma:= r_{\widetilde\gamma}/2$ finishes the proof for $\mathbb{P}(v^\star \text{\ is blue\,})$.
Further, since $\widetilde\gamma=1- \log \gamma/ |\log (\tau-2)|$, $\widetilde\gamma\searrow 0$ if $\gamma\nearrow 1/(\tau-2)$. So, the interval $[0, \widetilde\gamma]$ vanishes, and hence the lower bounds for \eqref{eq::not-only-red} vanish
by the continuity of measure of $Y^w$. As a result, $q_\gamma=r_{\widetilde\gamma}/4\to 0$.
A lower bound on \eqref{eq::only-red} can be given in a similar manner, defining $q_\gamma^{\text{red}}$.
To see that $q_\gamma^{\text{red}} \to 1$ uniformly in $n$ as $\gamma\to \tfrac{1}{\tau-2}$, we also need to show that \eqref{eq::not-only-red} tends to $0$ when $\widetilde\gamma\to 0$, uniformly in $n$.
We fix a constant $L>0$ to be chosen later and write
\begin{equation}\label{eq::upper-bound-on-and} \mathbb{P}_{\gamma}(Y_n \in A_{n, \widetilde\gamma}) \le \mathbb{P}_{\gamma}(Y^w > L) + \mathbb{P}(Y^w\in A_{n, \widetilde\gamma}\mathbbm{1}_{[0,L]}) + \mathbb{P}_{\gamma}(Y^w \notin A_{n, \widetilde \gamma}, Y^w \le L, Y_n \in A_{n, \widetilde \gamma}). \end{equation}
In the second term, the total Lebesque measure of the union intervals $A_{n, \widetilde \gamma}(m)\mathbbm{1}_{[0,L]}$ is at most
\[ (1-(\tau-2)^{\widetilde\gamma}) L \sum_{m\ge 1} (\tau-2)^m =(1-(\tau-2)^{\widetilde\gamma}) L \frac{\tau-2}{3-\tau}. \]
Now, if $L^2(1-(\tau-2)^{\widetilde\gamma})<1$, then the last term is at most $\frac{\tau-2}{3-\tau}/L$. Hence, we can pick $L=L(\widetilde\gamma)$ satisfying this, and use the continuity of the measure \cite[Theorem 3.5]{Foll84} of $Y^w$ on $[0,L]$ to get $\mathbb{P}( Y^w \in A_{n, \widetilde \gamma} \mathbbm{1}_{[0,L]})\le C/L.$ The first term is at most $\exp\{ - \Delta(L)\}$, hence it is sufficiently small when $L$ is large enough. This can be guaranteed even when $L^2(1-(\tau-2)^{\widetilde\gamma})<1$, when $\widetilde\gamma$ is sufficiently close to $0$.
The problem with the third term in \eqref{eq::upper-bound-on-and} is that $Y^w\notin A_{n, \widetilde \gamma}$ is a very likely event. But, we can apply a similar trick as in \eqref{eq::and-epsilon} and define for an $\varepsilon>0$
\begin{equation}\label{eq::and+epsilon}\begin{aligned} A_{n, \widetilde\gamma}^{+\varepsilon}(m):&=
((\tau-2)^{m+\widetilde\gamma+\varepsilon} \log n, (\tau-2)^{m-\varepsilon} \log n), \\
A_{n, \widetilde\gamma}^{+\varepsilon}&:= \bigcup_{m\in \mathbb{N}}A_{n, \widetilde\gamma}^{+\varepsilon}(m).\end{aligned}\end{equation}
Clearly, $A_{n, \widetilde\gamma}^{+\varepsilon}(m) \supset A_{n, \widetilde\gamma}(m)$.
We can then estimate the third term in \eqref{eq::upper-bound-on-and} as
\[\begin{aligned} \mathbb{P}_{\gamma}(Y^w \notin A_{n, \widetilde \gamma}, Y^w \le L, Y_n \in A_{n, \widetilde \gamma}) &\le \mathbb{P}(Y^w \in A_{n, \widetilde\gamma}^{+\varepsilon}\setminus A_{n, \widetilde\gamma}, Y\le L ) \\
&\ \ +\mathbb{P}_{\gamma}(Y^w \notin A_{n, \widetilde \gamma}^{+\varepsilon},Y_n \in A_{n, \widetilde \gamma}, Y^w\le L). \end{aligned}\]
Notice that the first term is small when $\varepsilon$ is small enough by the continuity of the measure of $Y^w$ and the fact that $[0,L]$ is a bounded interval. The second term is small if $n\ge n_0(\varepsilon)$ since it implies that $Y_n/Y^w \notin ((\tau-2)^\varepsilon, (\tau-2)^{-\varepsilon})$, and $Y_n\buildrel {a.s.}\over{\longrightarrow} Y^w$.
This shows that $\mathbb{P}_\gamma( w \text{ is red}) \to 1$ uniformly in $n$ as $\gamma \to 1/(\tau-2)$.
To finish up the proof of Lemma \ref{lem::q-delta}, we are left to show what error bound we have if we had applied Starting Rule 2 instead of Starting Rule 1. First of all note that the estimates on $\mathbb{P}(v^\star \text{ is blue})$ and
$\mathbb{P}(M_\kappa< Q^\delta)$ do not change. However, it might happen that even though $M_\kappa < Q^\delta$, there are some blue vertices in $G_\kappa$. This fact only might ruin $\mathbb{P}( v^\star \text{ is red, } D_{v^\star}<Q^\gamma)$.
However, note that the probability that any vertex is mis-colored, $p_e\to 0$ as $Q\to \infty$, which is true due to the choice of $Q$.
\end{proof}
As an immediate corollary of Lemma \ref{lem::q-delta}, we get the following.
\begin{corollary}\label{cor::red-first-moment} For both starting rules, the probability that the root is red is strictly positive for all $\gamma\in (1, 1/(\tau-2))$. Further, this probability tends to $1$ when $\gamma\nearrow 1/(\tau-2)$.
\end{corollary}
\begin{proof}
Under Starting Rule 1, $\{ w \text{ is red} \} = \{ D_{v^\star} < Q^\delta \}$, since on this event \emph{all} vertices get color red in $\mathcal {G}_\kappa$. As a result, $\mathbb{P}(w \text{ is red}) \ge q_\gamma$, and the result follows from Lemma \ref{lem::q-delta}.
It is left to investigate the effect of Starting Rule 2. Recall that in the interval $[Q^\gamma, Q^{1/(\tau-2)}]$ all vertices are painted red and blue with equal probability, and in the interval $[Q, Q^\gamma)$ they are painted red with probability $1-p_e$ and blue with $p_e$, where $p_e\to 0$ as $n \to \infty$. When $\gamma=1$, then $\mathbb{P}(w \text{ is red}) =1/2$ by symmetry. Then, a simple coupling argument and monotonicity of the Flow Rule implies that the event $\{w \text{ is red}\}$ is an \emph{increasing event}\footnote{If we write $\omega_i=0,1$ if the $i$th individual in $\mathcal {G}_\kappa$ is blue or red, respectively, and two colorings $\underline\omega\le \underline\omega'$ iff $\omega_i\le \omega_i'$ for all $i\in \mathcal {G}_\kappa$, then $\mathbb{P}(w \text{ is red } | \mathcal {G}_\kappa, \underline\omega ) \le \mathbb{P}(w \text{ is red } |\mathcal {G}_\kappa, \underline\omega')$.} in the number of red vertices in $\mathcal {G}_\kappa$.
This implies that $\mathbb{P}(w \text{ is red}) \ge 1/2$ for all $\gamma \in [1,1/(\tau-2)).$
Thus, the statement that $\mathbb{P}(w \text{ is red})$ strictly positive also holds under Starting Rule 2. It is left to investigate what happens when $\gamma\nearrow 1/(\tau-2)$. Let us denote the number of differently colored vertices in the two colourings by $\mathrm{Blue}_\gamma$, that is, the number of blue vertices in $\mathcal {G}_\kappa$ with degree in $ [Q, Q^\gamma)$.
We would like to show that
\[ \mathbb{P}(\mathrm{Blue}_\gamma\ge 1) \to 0 \]
as $\gamma \nearrow (\tau-2)^{-1}$.
Note that $\mathrm{Blue}_\gamma\ {\buildrel d \over \le }\ \mathrm{Bin}(\#\{x\in G_\kappa, D_x\ge Q\} , p_e)$, hence, first we aim to give an estimate on $|\mathcal {G}_\kappa|=Z_\kappa$, then on the proportion of vertices with degree $D_x\ge Q^\gamma$.
Using the stochastic domination argument in the proof of Claim \ref{cl::max-sum-relation}, we get
\[ X^{\scriptscriptstyle{(1)}}\ {\buildrel {d} \over \le }\ \frac{Z_{i+1}}{Z_i^{1/(\tau-2)}} \ {\buildrel {d} \over \le }\ b_2+X^{\scriptscriptstyle{(2)}}, \]
for some positive random variables $X^{\scriptscriptstyle{(1)}}, X^{\scriptscriptstyle{(2)}}$ and shift $b_2>0$. This fact, combined with Claim \ref{cl::max-sum-relation} and the \emph{proof} of Claim \ref{cl::stoch-dom-alpha-stable} gives that for some logarithmic correction term, if $Q$ is large enough, whp
\[ Z_\kappa \le (\log Q) (M_\kappa)^{(\tau-2)} = Q^{(\tau-2)^{c_Q}} \log Q. \]
Recall that in the $\kappa$th generation of the stopped BP, all vertices have i.i.d.\ degrees with distribution $D^\star$ conditioned on being $\le M_\kappa$.
Hence, the probability that the degree of a vertex $v\neq v^\star$ falls in the interval $[Q, Q^{\gamma})$ is at most
\[ \mathbb{P}(D^\star \ge Q | D^\star \le M_\kappa, M_\kappa) \le \frac{1-F^\star(Q)}{F^\star(M_\kappa)}\le 2 C_1^\star Q^{2-\tau}.\]
According to Starting Rule 2, each vertex in this interval gets color blue with probability $\buildrel {\mathbb{P}} \over{\sim} Q^{1-\gamma}$ independently of each other,
hence, $B_\gamma$ can be stochastically dominated by a binomial random variable, and so
\[ \mathbb{E}[\mathrm{Blue}_\gamma] \le C Q^{(\tau-2)^{c_Q} + 2-\tau+ 1-\gamma} \log Q.\]
Writing $\gamma:=1/(\tau-2) - x$, and recalling that $c_Q\in [0,1)$,
the exponent of $Q$ is at most $-(3-\tau)^2 +x$, with equality if $c_Q=0$.
Hence, if $\gamma\nearrow 1/(\tau-2)$, $x\searrow 0$ and so we can pick a small enough $x$ so that the exponent is negative.
By Markov's inequality, $\mathbb{P}(\mathrm{Blue}_\gamma\ge 1) \to 0$ in this case, and so the coloring in Starting Rule 2 is whp the same as the coloring in Starting Rule 1. For the latter, $\mathbb{P}_\gamma(w \text{ is red\,}) \to 1$ has been already shown in Lemma \ref{lem::q-delta}.
\end{proof}
The next lemma is the crucial ingredient for the proof of coexistence, and it shows that the probability that the blue color reaches the root is uniformly positive in $n$. It also implies Proposition \ref{prop::BP-color}.
\begin{lemma}\label{lem::blue-first-moment} Let $\gamma\in (1, 1/(\tau-2))$.
For both starting rules, the probability that the root $w$ of the branching process is painted blue is at least
\[ \mathbb{P}(w \text{ is blue}) \ge \frac12 {\mathrm e}^{-\gamma} q_{\gamma(1+\varepsilon)}, \]
where $\varepsilon>0$ is such that $\gamma(1+\varepsilon)<1/(\tau-2)$ holds, and $q_{\gamma}$ is from Lemma \ref{lem::q-delta}.
\end{lemma}
\begin{proof}[Proof of Proposition \ref{prop::BP-color}].
The lower bound in statement of the proposition directly follows from Lemma \ref{lem::blue-first-moment} with $c(\gamma):=\frac12 {\mathrm e}^{-\gamma} q_{\gamma(1+\varepsilon)}$. Further, recall that $q_\gamma\searrow 0$ as $\gamma \nearrow 1/(\tau-2)$ holds, see the statement of Lemma \ref{lem::q-delta}. The upper bound in the proposition follows from Lemma \ref{lem::q-delta} and Corollary \ref{cor::red-first-moment}, with $C(\gamma):=1-q_\gamma^{\text{red}}$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem::blue-first-moment}]
Throughout the proof, we analyse the worst starting scenario for blue, that is, Starting Rule 1. Recall that the maximal degree vertex in generation $\mathcal {G}_\kappa$ is denoted by $v^\star$, and $v_{\kappa-1}$ is the child of the root $w$ so that the subtree of $v_{\kappa-1}$ contains $v^\star$.
Lemma \ref{lem::q-delta} implies that there is a positive chance that there are blue vertices in generation $\mathcal {G}_\kappa$ of the BP. Note that by the same argument, for any small $\varepsilon$ such that $\gamma(1+\varepsilon) < 1/(\tau-2)$, we also have $\mathbb{P}(M_\kappa> Q^{ \gamma(1+\varepsilon)})\ge r_{\widetilde\gamma-\log(1+\varepsilon)/|\log(\tau-2)|}/4 >0$, with $\widetilde\gamma$ defined in \eqref{eq::crit}.
Hence, let us fix an $\varepsilon$ for which $\widetilde\gamma-\log(1+\varepsilon)/|\log(\tau-2)| \ge \widetilde\gamma/2$, and we aim to show that \emph{conditioned on} $M_\kappa> Q^{\gamma(1+\varepsilon)}$, the probability that a sibling of $v_{\kappa-1}$ is blue is strictly positive, i.e., $r_\kappa>0$.
Let us write $\mathcal {T}^{(i)}$ for the subtree of the $i$th sister of $v_{\kappa-1}$, and let
\[ \mathcal {T}_{-r}:= \bigcup_{i=1}^{D_w-1} \mathcal {T}^{(i)}\]
The reason to restrict our attention to the subtrees of the siblings of $v_{\kappa-1}$ is that the degrees in these subtrees are conditionally independent of each other and also of $D_w$.\footnote{On the other hand, $D_w$ is not independent of $\kappa$ and $M_\kappa$, and also the degrees of vertices on the ray, starting from $v_{\kappa-i}$ for $1\le i\le \kappa-1$ are not conditionally independent: the conditioning that $v_{\kappa-i}$ leads to a maximal degree vertex influences the degree of $v_{\kappa-i}$.} By the definition of $\kappa$, conditioned on $M_\kappa$ and the position of the ray, the degrees of vertices in every generation in $\mathcal {T}_{-r}$ are \emph{independent}, and the degrees in $\mathcal {G}_\kappa \cap \mathcal {T}_{-r}$ have distribution $D^\star| D^\star\le M_\kappa$, while in every earlier generations the degrees have distribution $D^\star| D^\star<Q$.
From now on, we work under the assumption that $M_\kappa\ge Q^{\gamma(1+\varepsilon)}$, (otherwise, we get that $s_0$ below is zero). Then, for any vertex in $\mathcal {G}_\kappa\cap \mathcal {T}_{-r}$, using \eqref{eq::size-biased2},
\begin{equation}\label{eq::s0} \begin{aligned} \mathbb{P}_{\gamma}(x \in \mathcal {G}_{\kappa} \cap \mathcal {T}_{-r} \text{ is blue}\, | M_\kappa ) &= \mathbb{P}_{\gamma}(D^\star> Q^{\gamma}| D^\star \le M_\kappa, M_\kappa)/2\\
&\ge \frac{c_1^\star}{2} Q^{-\gamma(\tau-2)}\frac{1- Q^{-\gamma\varepsilon(\tau-2)}}{1-c_1^\star Q^{-\gamma(1+\varepsilon)(\tau-2)}} \\
&\ge \frac{c_1^\star}{4} Q^{-\gamma(\tau-2)}=:s_0,\end{aligned} \end{equation}
where we have used that the last ratio is at least $1/2$ if $Q$ is large enough (which clearly holds since $Q\ge \exp\{ (\log n)^{1-\nu-c} \}$ for some small $c>0$). Recall that the Flow rule ensures that a vertex in $\mathcal {G}_{\kappa-1}$ gets color blue with probability at least $1/2$ if it has at least one blue child. Also,
recall that the degree of any vertex in $\mathcal {G}_{\kappa-1}\cup \mathcal {T}_{-r}$ has distribution $D^\star| D^\star < Q$.
Then, the probability that any vertex in $\mathcal {G}_{\kappa-1}$ has a blue child is at least
\begin{equation}\label{eq::before-s1} \mathbb{P}_{\gamma}(x\in \mathcal {G}_{\kappa-1} \cap \mathcal {T}_{-r} \text{ is blue} ) \ge \frac12 \mathbb{E}_{\gamma}[ 1- (1-s_0)^{D^\star} | D^\star<Q]=\frac12(1-\widehat h(1-s_0)),\end{equation}
where $\widehat h$ is the generating function of $D^\star | D^\star<Q$. We calculate using \eqref{eq::gen-func} from Proposition \ref{prop::no-gap} that
\[ \widehat h(s)=\frac{\sum_{k=1}^\infty \mathbb{P}(D^\star=k)s^k - \sum_{k=Q}^\infty \mathbb{P}(D^\star=k)s^k}{\mathbb{P}(D^\star< Q)} \
\le \frac{1-(1-s)^{\tau-2} L^\star(\tfrac{1}{1-s}) }{1-C_1^\star Q^{-(\tau-2)} }.\]
Hence, \eqref{eq::before-s1} becomes
\begin{equation}\label{eq::at-s1} \mathbb{P}_{\gamma}(x\in \mathcal {G}_{\kappa-1} \cap \mathcal {T}_{-r} \text{ is blue} )\ge \frac{1}{2} \frac{s_0^{\tau-2}L^\star(\frac{1}{s_0})- C_1^\star Q^{-(\tau-2)}}{1-C_1^\star Q^{-(\tau-2)}} \ge \frac{c_2^\star}{4} s_0^{\tau-2}=:s_1,
\end{equation}
where the last inequality is true since $L^\star(\cdot)>c_2^\star$ is a strictly positive bounded function by Proposition \ref{prop::no-gap}, further, $\gamma\in(1, 1/(\tau-2))$ implies that $s_0^{\tau-2}\gg Q^{-(\tau-2)}$ (see \eqref{eq::s0}).
For any $i\ge 1$, any vertex in generation $\mathcal {G}_{\kappa-(i+1)}$ is blue with at least a probability $1/2$ if it has at least one blue child in generation $\mathcal {G}_{\kappa-i}$. Hence, writing
\[ \mathbb{P}_{\gamma}(x\in \mathcal {G}_{\kappa-i} \cap \mathcal {T}_{-r} \text{ is blue} )\ge s_i,
\] we can repeat \eqref{eq::before-s1} and \eqref{eq::at-s1} using $s_i$ instead of $s_0$. Note that the condition $s_i^{\tau-2}\gg Q^{-(\tau-2)}$ is also satisfied for all $i$. This yields that
that $s_i$ satisfies the recursion $s_{i+1}=\frac{c_2^\star}{4} s_{i}^{\tau-2}$, hence
\begin{equation}\label{eq::s-kappa-1}s_{\kappa-1}= s_0^{(\tau-2)^{\kappa-1}} \left(\frac{c_2^\star}{4}\right)^{(1-(\tau-2)^{\kappa-1})/(3-\tau)}.\end{equation}
By definition, $s_{\kappa-1}=\mathbb{P}( x \in \mathcal {G}_1 \cap \mathcal {T}_{-r} \text{ is blue})$. Also, since every vertex has degree at least $2$, $D_w-1\ge1$ and hence the root has at least one child that is not on the ray. As a result,
\[ \mathbb{P}(w \text{ is blue}) \ge s_{\kappa-1} /2.\]
Using the definition of $\kappa$ in \eqref{eq::kappa}, we calculate
\[ (\tau-2)^{\kappa-1}=\frac{Y_\kappa^w}{\log Q} (\tau-2)^{-c_Q-1}.\]
Combining this, \eqref{eq::s-kappa-1} and the value of $s_0$ from \eqref{eq::s0}, conditioned on the value of $\kappa$,
\begin{equation}\label{eq::s-kappa} \begin{aligned} s_{\kappa-1}&\ge \left(\frac{c_2^\star}{4}\right)^{\frac{1}{3-\tau}} \left(\frac{c_1^\star 4^{\frac{\tau-2}{3-\tau}} }{(c_2^\star)^{\frac{1}{3-\tau}}}\right)^{ (\tau-2)^{\kappa-1}} \exp\{-(\log Q) \gamma(\tau-2) \frac{Y_\kappa^w}{\log Q} (\tau-2)^{-c_Q-1}\} \\
&\ge C \exp\{ -\gamma(\tau-2)^{-c_Q} Y_\kappa^w\} \ge C \exp\{ -\gamma Y_\kappa^w\}, \end{aligned}\end{equation}
where we used that $c_Q \in (0,1).$
Note that this expression is conditioned on the value of $Y_\kappa^w$, hence, it is left to evaluate its expectation (conditioned on $\gamma$), but recall, that we also have assumed that $M_\kappa \ge Q^{\gamma(1+\varepsilon)}$, which event is the same as $c_Q \in (0, \widetilde\gamma- \log(1+\varepsilon)/|\log (\tau-2)| )$ by \eqref{eq::crit}.
Hence, combining \eqref{eq::s-kappa} with this conditioning, we get the probability that the root $w$ is blue can be bounded from below by
\[ \mathbb{E}_{\gamma}\left[ \exp\{ -\gamma Y_\kappa^w\} \mathbbm{1}_{ \left\{ \left\{\frac{\log\log n - \log Y_\kappa^w}{|\log (\tau-2)|}\right\} \in (0, \widetilde\gamma- \frac{\log(1+\varepsilon)}{|\log (\tau-2)|}) \right\}}\right]\]
We can get a lower bound on this expression by restricting $Y_\kappa^w$ to $[0,1]$, in which case, the factor before the indicator is at least $\exp\{ - \gamma \}$, while the expectation of the indicator is treated in the \emph{proof of} Lemma \ref{lem::q-delta}, and is at least $r_{\widetilde\gamma - \frac{\log(1+\varepsilon)}{|\log (\tau-2)|} }$ (where $r_{\widetilde\gamma}$ is defined in \eqref{eq::q-delta}, see also \eqref{eq::indicator-Y-in-01}).
Combining these we arrive at
\[ \mathbb{P}_{\gamma}(w \text{ is blue }) \ge \frac12 {\mathrm e}^{-\gamma} r_{\widetilde\gamma - \frac{\log(1+\varepsilon)}{|\log (\tau-2)|} }, \]
and since $\varepsilon$ satisfies that $\widetilde\gamma-\frac{\log(1+\varepsilon)}{|\log (\tau-2)|}\ge 0$, this finishes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm::main2}]
In the proof of this theorem, we work conditionally on $q=Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} \in (\tau-2, 1)$. Of course, if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} \in (1, 1/(\tau-2))$, then the same statements are true, with the role of red and blue exchanged. Recall that $\mathbb{E}_\gamma[\cdot] = \mathbb{E}_[ \cdot | \gamma, Q, Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}]$, where $\gamma, Q$ were defined in \eqref{def::gamma} and just after \eqref{def::core-n}, respectively.
We aim to show that there exist constants $c(q), c(q)^{\text{red}}>0$, so that $\mathcal {B}_\infty/n\ge c(q)$ and $\mathcal {R}_\infty/n\ge c(q)^{\text{red}}$ hold whp. The upper bound in the statement of the theorem is then obvious with $C(q):=1- c(q)^{\text{red}}$ since $\mathcal {B}_\infty + \mathcal {R}_\infty =n$.
We only show this statement for $\mathcal {B}_\infty/n$, but $\mathcal {R}_\infty/n\ge q_{\gamma}^{\text{red}}/2:=c(q)^{\text{red}}$ can be treated analogously, using Lemma \ref{lem::q-delta} and Corollary \ref{cor::red-first-moment} by setting $\gamma$ in these to be equal to the value in \eqref{def::gamma}.
Let us write $2p_{\gamma}:= \exp\{ - \gamma \} r_{\widetilde \gamma- \frac{\log(1+\varepsilon)}{|\log (\tau-2)|}}/2$.
Lemma \ref{lem::blue-first-moment} ensures that, with $w$ being a uniformly chosen vertex,
\[ \mathbb{E}_{\gamma}\left[\mathcal {B}_\infty/n\right] = \sum_{v\in [n]} \mathbb{E}_{\gamma}[ \mathbbm{1}_{\{ v \text{ is blue}\}}]/n = \mathbb{P}_{\gamma}(w \text{ is blue} ) \ge 2 p_{\gamma}. \]
Then we use Chebishev's inequality to get
\[ \mathbb{P}_{\gamma}( \mathcal {B}_\infty/ n \le p_{\gamma} ) \le \mathbb{P}_{\gamma}( |\mathcal {B}_\infty/ n - \mathbb{E}_{\gamma}\left[\mathcal {B}_\infty/n\right]| \ge p_{\gamma} ) \le \frac{{\rm Var}_{\gamma}[ \mathcal {B}_\infty/n ]}{ p_{\gamma}^2 }. \]
To prove the statement of the theorem, we need to show that the rhs tends to zero with $n$.
We write
\[\begin{aligned} {\rm Var}_{\gamma}[\mathcal {B}_\infty /n] &\le \frac{1}{n^2} \sum_{w \in [n]} \mathbb{P}_{\gamma}(w \text{ is blue}) \\&+ \frac{1}{n^2} \sum_{w\neq z \in [n]} \mathbb{P}_{\gamma}(w,z \text{ is blue}) - \mathbb{P}_{\gamma}(w \text{ is blue})\mathbb{P}_{\gamma}(z \text{ is blue}). \end{aligned}\]
Clearly, the first sum is at most $n$, hence, the first term is at most $1/n$.
For the second term we need to show that for a uniformly chosen pair of vertices $w,z$
\[ \mathbb{P}_{\gamma}(w,z \text{ is blue}) \to \mathbb{P}_{\gamma}(w \text{ is blue})\mathbb{P}_{\gamma}(z \text{ is blue}).\]
This statement is a consequence of the coupling to independent branching processes. Indeed, if we have two uniformly chosen red and blue source vertices $u,v$, and two other uniformly chosen pair of vertices $w,z$, then one can generalise
\cite[Lemma 2.2]{BarHofKom14}: the local neighbourhoods of these four vertices can be coupled to four independent branching processes, where in each of them the degree of the root is an i.i.d copy of $D$, and all other forward degrees are distributed as $D^\star$. This coupling can be achieved by extending \cite[Lemma 2.2]{BarHofKom14} to four vertices in the following way: first we finish coupling the degrees of the local neighbourhoods of the source vertices $u,v$ to the red and blue BP up to total size $n^{\varrho^{\scriptscriptstyle{(r)}}}$, determining $Y_b^{\scriptscriptstyle{(n)}}, Y_r^{\scriptscriptstyle{(n)}}, Q, \gamma$. Then, we imaginarily stop the spreading of these two colors at this point, and we couple the degrees in the exploration process of the local neighbourhood of $w,z$ to a collection of independent random variables, forming another two independent BPs, till their maximum degree reaches $Q$, and then we can continue with everything else.
If we pick $\varrho$ in Section \ref{sc::climbup} sufficiently small, the total number of vertices explored in the four processes together is still at most of order of magnitude $n^{\varrho/\tau-2}< n^{\varrho'}$ (where $\varrho'$ is from \cite[Lemma 2.2]{BarHofKom14}), hence, the coupling of the forward degrees is still valid by \cite[Lemma 2.2]{BarHofKom14}.
Clearly, conditioned on the value $Q$ and $\gamma$, (determined by the red and blue BPs), the probability that the roots of the two independent BP-s are blue under this independent coupling is $\mathbb{P}_{\gamma}(w\text{ is blue}\,)\mathbb{P}_{\gamma}(z\text{ is blue}\,)$.
Hence, the probability that $\mathbb{P}_{\gamma}(w,z \text{\ is blue\,}) \neq \mathbb{P}_{\gamma}(w\text{ is blue\,})\mathbb{P}_{\gamma}(z\text{ is blue\,})$ is exactly the probability that the coupling fails, which tends to zero as $n\to \infty$ (an estimate on order of magnitude of the coupling error can be found in \cite[Appendix A.2]{BHH10}, where it is shown to be $O(n^{-\varepsilon})$ for some small but positive $\varepsilon$.)
This finishes the proof of Theorem \ref{thm::main2} with $c(q):= p_\gamma$ and $C(q):=1- q_\gamma^{\text{red}}/2.$
Note that $ p_\gamma \to 0, q_\gamma^{\text{red}} \to 1$ as $\gamma \nearrow 1/(\tau-2)$ exactly implies $c(q), C(q) \to 0$ as $q\searrow (\tau-2)$, see Remark \ref{rem::q-gamma-relation}.
\end{proof}
\section{Number of maximum degree vertices}\label{sc::mbn}
In what follows, we aim for the proof of Theorem \ref{thm::main} if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<\tau-2$, but we still need plenty of preparation for that.
Recall from Section \ref{sc::slopedown}, page 19 that Case (1) stands for $T_b-T_r\ge 2$, while Case (3)(a) means $T_b-T_r=1$ with $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} <\tau-2$.
We have seen in the proof of Theorem \ref{thm::maxdegree} (see page 30-33), that $D^{\max}_n(\infty)$, the degree of the highest degree vertex that blue can paint, can be expressed by distinguishing four cases $O_>, E_>, O_<, E_<$ (see page 31) representing $T_b-(T_r+1)$ being odd or even and $\tau-1>(\tau-2)^{b_n^{\sss{(r)}}} + (\tau-2)^{b_n^{\sss{(b)}}}$ holds or not. Recall that $h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ captures the oscillating part that depends on these 4 cases for the normalising oscillating random variable in Theorem \ref{thm::maxdegree}.
In this section we investigate \emph{how many} maximum degree vertices are reached by blue. Here, by `maximum degree' vertex we mean any vertex $w$ that satisfies $\log (\deg_w) = \log (D^{\max}_n(\infty)) (1+ o_{\mathbb{P}}(1))$. We show that in Cases $E_>,O_<$, the number of these vertices is in fact so large that it corresponds to an additional factor for the total number of half-edges in maximum degree vertices of blue.
More precisely, let us denote the set of outgoing half edges from the maximal degree vertices by $\mathcal M^{\sss{(b)}}_n$, and its size by $M^{\sss{(b)}}_n$.
\begin{lemma}\label{lem::verticeswithmaxdegree}
The number of outgoing half-edges from the set of maximal degree vertices, i.e. the sum of the forward degrees of vertices for which \eqref{eq::max-deg-2} holds, satisfies
\[ \frac{\logM^{\sss{(b)}}_n}{(\log n)\!\cdot\! (\tau-1)^{-1} h_n^{\text{half-edge}}( Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) } \buildrel {d}\over{\longrightarrow} \sqrt{\frac{Y_b}{Y_r}}, \]
where $h_n^{\text{half-edge}}( Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ stochastically dominates $h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$, and is a bounded random variable given below in formula \eqref{def::gfunction}.
\end{lemma}
\begin{proof}
The proof is analogous to that of \cite[Lemma 6.3]{BarHofKom14}, but some details differ. Hence, we work out the proof here.
To start with, recall from the proof of Theorem \ref{thm::maxdegree} that in Cases $E_<, O_>$, blue finishes its last jump at a certain layer $\Gamma_{i}^{\scriptscriptstyle{(b)}}$. See the argument after \eqref{eq::dtc2} for this, and the four cases on page 31.
Thus, in Case $E_<, O_>$ the statement is a direct consequence of Lemma \ref{lem::numberofverticesinGamma}, since blue is stuck with its maximal degree at layer $\Gamma_{T_r+\lfloor t_c \rfloor-t(n^{\varrho})+1}^{\scriptscriptstyle{(b)}}$, and hence $M^{\sss{(b)}}_n= A_{T_r+\lfloor t_c \rfloor-t(n^{\varrho})+1}^{\scriptscriptstyle{(b)}} D_n^{\max}(\infty)$. Let us write from now on in this proof $i_{\max}:=T_r+\lfloor t_c \rfloor-t(n^{\varrho})+\mathbbm{1}_{E_< \cup O_>}$. Recall that $i_{\star \scriptscriptstyle{(b)}}$ denotes the total number of layers blue can go through if red would not be present (see \eqref{eq::b-definitions}), so we clearly have $i_{\max}\le i_{\star\scriptscriptstyle{(b)}}$.
Then, for Cases $E_<, O_>$, we get the bound
\[\log M^{\sss{(b)}}_n \le \log D_n^{\max}(\infty)+ \log A_{i_{\max}}^{\scriptscriptstyle{(b)}},\] and since $i_{\max} < i_{\star \scriptscriptstyle{(b)}}$ is whp a bounded random variable, the last term disappears when we divide by $\log n$.
We are left with handling the cases were the last jump of blue is not a full layer, i.e.,\ Cases $E_>, O_<$. In these cases, after reaching layer $\Gamma_{i_{\max}}^{\scriptscriptstyle{(b)}}$, blue still jumps up, but not a full layer:
due to the presence of red color the forward degrees are truncated at $\widetilde u_{\lfloor t_c \rfloor}^{\scriptscriptstyle{(r)}}$.
First, we apply Lemma \ref{lem::numberofverticesinGamma} to see that $\log A_{i_{\max}}^{\scriptscriptstyle{(b)}}$ in the last `full' layer $\Gamma_{i_{\max}}^{\scriptscriptstyle{(b)}}$ is small.
Let us recall the notation $u_{i_{\max}}^{\scriptscriptstyle{(b)}}=D_n^{\max}(T_r+\lfloor t_c \rfloor)$, and the fact that the `size' of the last jump of blue in the exponent of $(\tau-2)$ is $2\{t_c\}<1$ at time $T_r+\lfloor t_c \rfloor +1$ for Case $E_<, O_>$ (see \eqref{eq::dmax-tc} and \eqref{eq::wur-tc} to get \eqref{eq::dtc2}). Let us introduce as the extra factor of the $\log$(degrees)/$\log n$ reached at time $T_r+\lfloor t_c \rfloor+1$ by blue
\begin{equation}\label{def::xi} \xi:=(\tau-2)^{-2\{t_c\} }, \end{equation} and then we introduce a new layer
\[ \Gamma^\diamond:=\left\{ v\in {\mathrm{CM}}_n(\boldsymbol{d}): d_v \ge \frac{(u^{\scriptscriptstyle{(b)}}_{i_{\max}})^\xi}{(C\log n)^{1/(\tau-2)} } \right\},\]
and we denote the number of half-edges in this set by ${\mathcal{E}}_{\xi}$.
By Lemma \ref{lem::badpaths}, whp blue does not reach higher degree vertices than $\widehat u_{i_{\max}}^{\scriptscriptstyle{(b)}}$ at time $T_r+\lfloor t_c \rfloor$. So,
conditioned that there are $A_{i_{\max}}^{\scriptscriptstyle{(b)}}$ many vertices in layer $\Gamma_{i_{\max}}^{\scriptscriptstyle{(b)}}$, the number of vertices in $\Gamma^\diamond$ to which blue is connected to is dominated by
\begin{equation}\label{eq::bindom3}\mathcal B \cap \Gamma^\diamond \ {\buildrel {d}\over{ \le }}\ {\sf Bin} \left( A_{i_{\max}}^{\scriptscriptstyle{(b)}} \widehat u_{i_{\max}}^{\scriptscriptstyle{(b)}}, \frac{{\mathcal{E}}_{\xi}}{\mathcal {L}_n (1+o(1))} \right), \end{equation}
where we recall that $\mathcal {L}_n$ is the total number of half-edges in ${\mathrm{CM}}_n(\boldsymbol{d})$. We can bound
\[ {\mathcal{E}}_\xi\le n (u^{\scriptscriptstyle{(b)}}_{i_{\max}})^{-\xi(\tau-2)} C\log n\] by using Claim \ref{claim::Sbound}.
Thus, conditioned on $A_{i_{\max}}^{\scriptscriptstyle{(b)}}$, on the event $\mathcal {L}_n/n\in \{\mathbb{E}[D]/2, \mathbb{E}[D]\}$ the expected value of the binomial variable in \eqref{eq::bindom3} is whp bounded from above by
\[ \frac{2 C_1^2}{c_1} (C \log n) A_{i_{\max}}^{\scriptscriptstyle{(b)}} \widehat u_{i_{\max}}^{\scriptscriptstyle{(b)}} (u_{i_{\max}}^{\scriptscriptstyle{(b)}})^{-\xi(\tau-2)}
= \frac{2 C_1^2}{c_1} (C \log n) A_{i_{\max}}^{\scriptscriptstyle{(b)}}\frac{\widehat u_{i_{\max}}^{\scriptscriptstyle{(b)}}}{u_{i_{\max}}^{\scriptscriptstyle{(b)}}} (u_{i_{\max}}^{\scriptscriptstyle{(b)}})^{1-\xi(\tau-2)}. \]
This gives an upper bound on the number of \emph{vertices} with degree at least $(u^{\scriptscriptstyle{(b)}}_{i_{\max}})^\xi/ (C \log n)^{1/(\tau-2)}$, thus the total number of \emph{half-edges} going out from maximal degree vertices can be bounded from above by
\begin{equation}\label{eq::addgamma} \frac{2 C_1^2 (C \log n)^{\frac{\tau-3}{\tau-2}}}{c_1} A_{i_{\max}}^{\scriptscriptstyle{(b)}}\frac{\widehat u_{i_{\max}}^{\scriptscriptstyle{(b)}}}{u_{i_{\max}}^{\scriptscriptstyle{(b)}}} (u_{i_{\max}}^{\scriptscriptstyle{(b)}})^{1+\xi(3-\tau)}.\end{equation}
Since $i_{\max}\le i_{\star\scriptscriptstyle{(b)}}$ is bounded, we can use \eqref{eq::aifinal}, \eqref{eq::uibar} with \eqref{def::ui} to see that
\[ A_{i_{\max}}^{\scriptscriptstyle{(b)}}\frac{\widehat u_{i_{\max}}^{\scriptscriptstyle{(b)}}}{u_{i_{\max}}^{\scriptscriptstyle{(b)}}}\le (A_{i_{\max}}^{\scriptscriptstyle{(b)}})^2 \le (A_{i_{\star \scriptscriptstyle{(b)}}}^{\scriptscriptstyle{(b)}})^2\] is some bounded power of $C \log n$, and thus it disappears when taking logarithm and dividing by $\log n$.
Hence, we get that in Cases $E_<, O_>$ with $\xi=(\tau-2)^{-2\{t_c\}}$, whp
\begin{equation}\label{eq::xi-estimate-1}\log M^{\sss{(b)}}_n\le(1+\xi(3-\tau))\log (u_{i_{\max}}^{\scriptscriptstyle{(b)}} ).\end{equation}
The \emph{lower bound} on the number of blue vertices in $\Gamma^\diamond$ is easier: we can establish a similar binomial domination argument (now from below), using that there is at least one blue vertex in $\Gamma_{i_{\max}}^{\scriptscriptstyle{(b)}}$ with degree at least $u_i^{\scriptscriptstyle{(b)}}$, and that the number of half edges ${\mathcal{E}}_\xi\ {\buildrel d \over \ge y_n} \mathrm{Bin}(n, c_1 y_n^{1-\tau})$, with $y_n= (u^{\scriptscriptstyle{(b)}}_{i_{\max}})^\xi/(C\log n)^{1/(\tau-2)} $, and using the concentration of the binomial random variables. The estimate gives the same order of magnitude as the rhs of \eqref{eq::xi-estimate-1}, with small error probabilities: the details are left for the reader.
Recall the definition of $\xi$ from \eqref{def::xi} and that in Cases $E_<, O_>$
\[ \log(D_n^{\max}(\infty)) = \xi \log u_{i_{\max}}^{\scriptscriptstyle{(b)}} (1+o_{\mathbb{P}}(1)) \]
holds. Comparing this with \eqref{eq::xi-estimate-1} we arrive at
\begin{equation}\label{eq::logmbn-to-logdmax} \log M^{\sss{(b)}}_n = (\xi^{-1} + (3-\tau) ) \log D_{\max}^{\scriptscriptstyle{(b,n)}}(\infty) (1+o_{\mathbb{P}}(1)).\end{equation}
To get the final expression for total number of half-edges at the last up-jump, we need to multiply the function $h_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ in \eqref{eq::h1} by $(\xi^{-1} + (3-\tau))$ in Cases $O_<, E_>$, using that $2\{t_c\}=b_n^{\sss{(b)}}-\delta_n^{\sss{(r)}}$ for Case $E_>$, while $2\{t_c\}=1+b_n^{\sss{(b)}}-\delta_n^{\sss{(r)}}$ for Case $O_<$. As a result, we obtain
\[ \frac{ \log M^{\sss{(b)}}_n} {(\tau-1)^{-1}\log n} = \sqrt{\frac{Y_b^{\scriptscriptstyle{(n)}}}{Y_r^{\scriptscriptstyle{(n)}}}} h^{\text{half-edge}}_n( Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) (1+o_{\mathbb{P}}(1)),\]
with
\begin{equation}\label{def::gfunction} \begin{aligned} &h^{\text{half-edge}}_n( Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}):=\mathbbm{1}_{ E_< \cup O_>} (\tau-2)^{(b_n^{\sss{(r)}}+b_n^{\sss{(b)}}-1 - \mathbbm{1}_{O_>})/2} +\\
&+\mathbbm{1}_{ E_> \cup O_<} (\tau-2)^{(b_n^{\sss{(r)}}-b_n^{\sss{(b)}}-1 - \mathbbm{1}_{O_<})/2} ((\tau-1)-(\tau-2)^{b_n^{\sss{(r)}}})(3-\tau)\\
&+\mathbbm{1}_{ E_> \cup O_<} (\tau-2)^{(b_n^{\sss{(r)}}+b_n^{\sss{(b)}}- \mathbbm{1}_{E_>})/2}. \end{aligned}\end{equation} Dividing by $h^{\text{half-edge}}_n( Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ and using that $(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})\buildrel {d}\over{\longrightarrow} (Y_r, Y_b)$ finishes the proof of Lemma \ref{lem::verticeswithmaxdegree}.
\end{proof}
Before moving on to the next section, let us introduce the time when the maximal degree is reached, which is nothing else but the time of the last possible up-jump of blue, i.e., for all four cases $O_<, O_>, E_<, E_>$ it is
\begin{equation}\label{def::t_b}\begin{aligned} t_b&:=T_r+\lfloor t_c \rfloor+1.
\end{aligned} \end{equation}
\section{Path counting methods for blue}\label{sc::path-counting} We have seen in Section \ref{sc::mbn} that if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<(\tau-2)$, blue has $M^{\sss{(b)}}_n$ many half-edges of highest degree at time $t_b$. At this time, only $o(n)$ vertices are reached by red and blue together\footnote{This statement needs verification, but it follows simply from the fact that typical distances in ${\mathrm{CM}}_n(\boldsymbol{d})$ are $2 \log \log n/|\log (\tau-2)|+O_{\mathbb{P}}(1)$ while $t_b= \log \log n / |\log (\tau-2)| + O_{\mathbb{P}}(1)$, hence most vertices have not been reached by any color at time $t_b$ yet.}
-- most of the vertices are still not colored. Thus, it still remains to determine how many vertices blue can reach \emph{after} time $t_b$. We do this via giving a matching upper and a lower bound on how many vertices blue occupies in this last phase. This part is a direct application of the methods described in \cite[Section 7]{BarHofKom14}, thus, we only describe the idea and check that each condition in the lemmas there is satisfied.
For the upper bound, the idea is that we count the close neighborhood of the half-edges that are just occupied at time $t_b$. Since the red avalanche continues to be in its avalanche phase and occupies all vertices at lower and lower degrees as time passes, the spreading of blue is more and more restricted, so this local neighborhood is quite small. We call this the \emph{optional cluster} of blue. We give a concentration result on its size. (That is, a concentrated upper bound on what blue can get.)
For the lower bound, we estimate how many vertices red might `bite out' of this optional cluster. This can happen since even a constant degree vertex might be close to both colors. We show that this intersection of the clusters is negligible compared to the size of the optional cluster.
We start describing the upper bound -- the optional cluster of blue -- in more detail. At time $t_b$, the half-edges in the set $\mathcal M^{\sss{(b)}}_n$ start their own \emph{exploration clusters}, i.e., an exploration process from the half-edge to not-yet
occupied vertices. At time $t_b+ j$, we color every vertex $v$ blue, whose distance is exactly $j$ from some half-edge $h$ in $\mathcal M^{\sss{(b)}}_n$, and the degrees of vertices on the path from $h$ to $v$ are less than what red occupied by that time. That is, the degree of the $j+1$st vertex on the path must be less than $\widetilde u_{ t_b-T_r+ j-1}^{\scriptscriptstyle{(r)}}$.
We do this via estimating the number of paths with degree restrictions from $\mathcal M^{\sss{(b)}}_n$ and call this the \emph{optional cluster of blue}, and denote the set by $\mathcal{O}_{\max}$ and its size by $\mathrm{O}_{\max}$. Corollary \ref{cor::chebisev} below determines its asymptotic behavior.
On the other hand, not just half edges in $\mathcal M^{\sss{(b)}}_n$ can gain extra blue vertices: from half edges in $\mathcal {A}_{i_{\max}-z}^{\scriptscriptstyle{(b)}},\ z=0,1,2\dots$ the explorations start earlier (at time $t_b-z$) towards small degree vertices. Let us denote the vertices reached via half-edges from layer $\mathcal {A}_{i_{\max}-z}^{\scriptscriptstyle{(b)}}\setminus \mathcal {A}_{i_{\max}-z+1}^{\scriptscriptstyle{(b)}}$ by $\mathcal{O}_{-z}$ and its size by $\mathrm{O}_{-z},\, z\ge 0$. At time $t_b- z+ j$, we color every vertex $v$ blue, whose distance is exactly $j$ from a half-edge $h$ in $\mathcal {A}_{i_{\max}-z}^{\scriptscriptstyle{(b)}}$, and the degrees of vertices on the path from $h$ to $v$ are less than $u_{i_{\max}-z+j}^{\scriptscriptstyle{(b)}}$, and also what red has already occupied at that moment, i.e. the degree of the $j$-th vertex on the path must be less than $\min\{u_{i_{\max}-z+j}^{\scriptscriptstyle{(b)}}, \widetilde u_{ t_b-z+ j-1-T_r}^{\scriptscriptstyle{(r)}}$\}. This extra truncation is needed since we do not want to count vertices explored from $\mathcal {A}_{i_{\max}-z}^{\scriptscriptstyle{(b)}}$ towards $\mathcal {A}_{i_{\max}-z+1}^{\scriptscriptstyle{(b)}}$, hence the additional restriction. We show that the total number of optional blue vertices in lower layers, $\sum_{z\ge 0}\mathrm{O}_{-z}$ with these additional explorations is at most the same order as $\mathrm{O}_{\max}$ in Lemma \ref{lem::opt-z-lemma}.
For the lower bound of what blue can occupy after $t_b$, note that not every vertex in $\mathcal{O}_{\max}$ will be occupied by blue: red can still bite out some parts of these vertices by simply randomly being close to some parts of the blue cluster. We estimate the number of vertices in the intersection of $\mathcal{O}_{\max}$ and red, and then subtracting the gained estimate from the lower bound on $\mathrm{O}_{\max}$ gives a lower bound on what blue occupies from the graph after $t_b$, see Lemma \ref{lem::red-blue-intersection} below. Now we turn to the calculations.
As before, we use $\mathbb{P}_{Y,n}(\cdot), \mathbb{E}_{Y,n}[\cdot]$ defined in \eqref{def::py-ey}.
We introduce the expected truncated degree of a vertex that is distance $j$ away from the set $\mathcal M^{\sss{(b)}}_n$ by
\begin{equation}\label{def::nuj} \nu_j:=\mathbb{E}_{Y,n}\left[D^\star \mathbbm{1}_{\left\{D^\star< \widetilde u_{t_b+j-1-T_r}^{\scriptscriptstyle{(r)}}\right\}} \right], \end{equation}
Then \eqref{eq::d-star-indicator} yields an upper bound on $\nu_j$, and the same expression with $C_1^\star$ replaced by $c_1^\star$ serves as a lower bound.
Let us also define
\[ \kappa_j:=\frac{1}{\mathbb{E}[D]}\mathbb{E}\left[D(D-1)(D-2)\mathbbm{1}_{\left\{D< \widetilde u_{t_b+ j-1-T_r}^{\scriptscriptstyle{(r)}}\right\}}\right] \]
Then, again by \eqref{eq::F},
\[ \frac{c_1}{\mathbb{E}[D]} \left(\widetilde u_{t_b+ j-1-T_r}^{\scriptscriptstyle{(r)}}\right)^{4-\tau}\le \kappa_j \le \frac{C_1}{\mathbb{E}[D]} \left(\widetilde u_{ t_b+ j-1-T_r}^{\scriptscriptstyle{(r)}}\right)^{4-\tau}.\]
Recall the definition of a path from page 13. This time, let us call a path of length $k$ from $\mathcal M^{\sss{(b)}}_n$ with vertices $\left(\pi_j\right)_{j\le k}$ \emph{good} if $\pi_j\le \widetilde u_{ t_b+ j-1-T_r}^{\scriptscriptstyle{(r)}}$, and \emph{good-directed} if $\widetilde u_{ t_b+ j-T_r}^{\scriptscriptstyle{(r)}}\le \pi_j\le \widetilde u_{ t_b+ j-1-T_r}^{\scriptscriptstyle{(r)}}$.
\begin{lemma}\label{lem::wihZk} For $k\ge 0$, denote by $\mathrm{O}_{\max}(k), \mathrm{O}_{\max}^{\mathrm{d}}(k)$ the number of vertices that are on good and good-directed paths of distance $k$ away from $\mathcal M^{\sss{(b)}}_n$, respectively. Then,
\begin{equation} \label{eq::wihZk_expected} M^{\sss{(b)}}_n \cdot \prod_{j=1}^{k} \nu_j \le \mathbb{E}[\mathrm{O}_{\max}(k)\mid M^{\sss{(b)}}_n ]\le M^{\sss{(b)}}_n \cdot \prod_{j=1}^{k} \nu_j \cdot \left(1+O\left(\frac{k^2}{n}\right)\right)\end{equation}
and
\begin{equation} \label{eq::wihZk_expected2} M^{\sss{(b)}}_n \cdot \prod_{j=1}^{k} (\nu_j-\nu_{j+1}) \le \mathbb{E}[\mathrm{O}_{\max}^{\mathrm{d}}(k)\mid M^{\sss{(b)}}_n ]\le M^{\sss{(b)}}_n \cdot \prod_{j=1}^{k} (\nu_j-\nu_{j+1}) \cdot \left(1+O\left(\frac{k^2}{n}\right)\right)\end{equation}
while for the variance of the latter:
\begin{equation} \begin{aligned} \label{eq::wihZk_variance}&{\text{\bf Var}}[\mathrm{O}_{\max}^{\mathrm{d}}(k)|M^{\sss{(b)}}_n] \le \mathbb{E}[\mathrm{O}_{\max}^{\mathrm{d}}(k)|M^{\sss{(b)}}_n] \\
&\ + \overline{\mathbb{E}[\mathrm{O}_{\max}^{\mathrm{d}}(k)|M^{\sss{(b)}}_n]}^2 \cdot \left(\frac{\nu_{k-1}}{(\nu_{k-1}-1)}\frac{\kappa_1}{\nu_{1}^2}\left(\frac{1}{M^{\sss{(b)}}_n} + \frac{2}{\mathcal {L}_n}\right)+ \frac{\nu_{k-1}^2}{(\nu_{k-1}^2-1)^2}\frac{\kappa_1^2}{\nu_{1}^4} \frac{2}{M^{\sss{(b)}}_n \mathcal {L}_n}+e_{k,n}\right), \end{aligned}\end{equation}
where $\overline{\mathbb{E}[\mathrm{O}_{\max}^{\mathrm{d}}(k)|M^{\sss{(b)}}_n]}$ means the upper bound on $\mathbb{E}[\mathrm{O}_{\max}^{\mathrm{d}}(k)|M^{\sss{(b)}}_n]$ in \eqref{eq::wihZk_expected2}, and the error term $e_{k,n}$ is given by
\begin{equation}\begin{aligned}\label{def::ekn} e_{k,n}&=\left( \prod_{i=1}^{k} \frac{\mathcal {L}_n-2i+1}{\mathcal {L}_n-2i-2k+1}-1\right)\\
&\ \ + \left(1+\frac{\kappa_1\nu_{k-1}}{\nu_1^2}\frac{1}{M^{\sss{(b)}}_n} \right)\left(1+\frac{\kappa_1 \nu_{k-1}}{\nu_1^2 }\frac{1}{c\mathcal {L}_n}\right)\frac{k}{\nu_{k-1}-1}\left(e^{k^2 \kappa_1^2 \nu_{k-1}/(\nu_{1}^4\mathcal {L}_n)}-1\right).\end{aligned}\end{equation}
\end{lemma}
The proof of this lemma uses path counting methods and is similar to that of \cite[Lemma 5.1]{Janson10}. Similar techniques can also be found in \cite[Volume II.]{H10}. The detailed proof can be found in \cite[Appendix]{BarHofKom14}, where $\lambda=1$ can be set.
Now we state the immediate corollary of Lemma \ref{lem::wihZk}. Recall the definition of $t_b$ from \eqref{def::t_b}.
\begin{corollary}[Chebyshev's inequality for blue vertices]\label{cor::chebisev} Take $c_3\le (1-\varepsilon) |\log(\tau-2)|^{-1}$ and any $k\le c_3\log\log n.$
Then, conditioned on the number of blue half-edges $M^{\sss{(b)}}_n$ at time $t_b$, the number of vertices optionally occupied by blue up to time $t_b+ k$ satisfies that, conditionally on $M^{\sss{(b)}}_n$,
\[\frac{\log (\mathrm{O}_{\max}(k))}{\log M^{\sss{(b)}}_n + \sum_{i=1}^{k-1} \log \nu_i} \buildrel {\Pv}\over{\longrightarrow} 1. \]
\end{corollary}
\begin{proof}
In this proof below, all expectations and probabilities are conditional wrt.\ $M^{\sss{(b)}}_n$: since $M^{\sss{(b)}}_n$ is a deterministic function $n,Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}},$ we express this conditioning by using the $\mathbb{P}_{Y,n}(\cdot), \mathbb{E}_{Y,n}[\cdot]$ notation. Let us write
\[ J:=\mathbb{P}_{Y,n}\left( \big|\mathrm{O}_{\max}(k) -\mathbb{E}_{Y,n}[\mathrm{O}_{\max}(k) ]\big|\ge \frac12 \mathbb{E}_{Y,n}[\mathrm{O}_{\max}(k)] \right).\]
Then by a simple triangle inequality,
\begin{equation}\label{eq::triangle} \begin{aligned} J &\le \mathbb{P}_{Y,n}\left(|\mathrm{O}_{\max}(k) - \mathrm{O}_{\max}^{\text{d}}(k)| \ge \mathbb{E}_{Y,n}[\mathrm{O}_{\max}(k)]/6 \right) \\
&+\mathbb{P}_{Y,n}\left( \big|\mathrm{O}_{\max}^{\text{d}}(k) -\mathbb{E}_{Y,n}[\mathrm{O}_{\max}^{\mathrm{d}}(k) ]\big|\ge \mathbb{E}_{Y,n}[\mathrm{O}_{\max}(k)]/6 \right)\\
&+\mathbb{P}_{Y,n}\left( |\mathbb{E}_{Y,n}[\mathrm{O}_{\max}^{\text{d}}(k) - \mathbb{E}_{Y,n}[ \mathrm{O}_{\max}(k)] |\ge \mathbb{E}[\mathrm{O}_{\max}(k)]/6 \right)\\
&:= J_1+J_2+J_3. \end{aligned}\end{equation}
We can apply Chebyshev's inequality on the second term, using the lower bound in \eqref{eq::wihZk_expected}, the upper bound in \eqref{eq::wihZk_expected} as an upper bound on $\overline{\mathbb{E}[\mathrm{O}_{\max}^{\mathrm{d}}(k)|M^{\sss{(b)}}_n]}$, and the variance formula in Lemma \ref{lem::wihZk}:
\begin{equation}\label{eq::J}\begin{aligned} J_2&\le \frac{9 {\text{\bf Var}}[\mathrm{O}_{\max}^{\text{d}}(k)| M^{\sss{(b)}}_n]}{\mathbb{E}[\mathrm{O}_{\max}(k)|M^{\sss{(b)}}_n]^2} \\
&\le \left( \frac{1}{M^{\sss{(b)}}_n } \frac{\kappa_1}{\nu_1^2}\frac{\nu_{k-1}}{(\nu_{k-1}-1)} +\frac{ \kappa_1^2}{\nu_{1}^4 \mathcal {L}_n} \frac{2k^4 \nu_{k-1}}{\nu_{k-1}-1} \Big(1+ O\big(\frac{1}{M^{\sss{(b)}}_n } \frac{\kappa_1}{\nu_1^2}\big)\Big) \right) \left(1+ O(\tfrac{k^2}{n})\right). \end{aligned}\end{equation}
The term containing $\kappa_1^2/\nu_{1}^4 \mathcal {L}_n$ comes from the Taylor expansion of the exponential factor in the formula for $e_{k,n}$.
We have to verify that the rhs to $0$. For this we need $\kappa_1/(\nu_1^2 M^{\sss{(b)}}_n) \to 0$ and also $\kappa_1^2/(\nu_1^4 \mathcal {L}_n) \to 0$.
For $\kappa_1/(\nu_1^2 M^{\sss{(b)}}_n)$ note that $M^{\sss{(b)}}_n\ge D_n^{\max}(\infty)$, since it counts the number of half-edges with maximal degree $D_n^{\max}(\infty)$. Further, $\kappa_1/\nu_1^2= (\widetilde u_{ t_b -T_r}^{\scriptscriptstyle{(r)}})^{\tau-2}=(\widetilde u_{\lfloor t_c \rfloor+1}^{\scriptscriptstyle{(r)}})^{\tau-2}=o( D_n^{\max}(\infty))$, since it is not hard to see that at time $t_b$, the degree above which red occupies everything (i.e., $\widetilde u_{\lfloor t_c\rfloor+1}^{\scriptscriptstyle{(r)}}$) is already less than $D_n^{\max}(\infty)$,
otherwise blue could have still increased its maximal degree at $t_b+1$ by an extra jump.
(Alternatively, compare the exact values of $D_n^{\max}(\infty)$ in \eqref{eq::max-deg-2}, and compare it to that
of $(\widetilde u_{ t_c \rfloor+1}^{\scriptscriptstyle{(r)}})^{(\tau-2)^2}$, which can be derived by multiplying the rhs of \eqref{eq::wur-tc} by $(\tau-2)^2$.)
Similarly, the second term, $\kappa_1^2/(\nu_1^4 \mathcal {L}_n)=(\widetilde u_{\lfloor t_c\rfloor+1}^{\scriptscriptstyle{(r)}})^{2(\tau-2)}/\mathcal {L}_n$ is small as long as $(\widetilde u^{\sss{(r)}}_{ \lfloor t_c\rfloor+1})^{\tau-2}=o(\sqrt{n})$. This is always true, since already $(\widetilde u^{\sss{(r)}}_1)^{\tau-2}=o(\sqrt{n})$.
To handle $J_1$ in \eqref{eq::triangle}, we Markov's inequality, conditioned on $M^{\sss{(b)}}_n$:
\[ J_1 \le \frac{\mathbb{E}_{Y,n}[\mathrm{O}_{\max}(k)|M^{\sss{(b)}}_n]-\mathbb{E}_{Y,n}[\mathrm{O}_{\max}^{\text{d}}(k)|M^{\sss{(b)}}_n]}{6^{-1}\mathbb{E}_{Y,n}[\mathrm{O}_{\max}(k)|M^{\sss{(b)}}_n]} \le 6(1+O(\tfrac{k^2}{n}))\Big(1- \prod_{j=1}^k \big(1-\frac{\nu_{j+1}}{\nu_j}\big)\Big). \]
We use $\nu_i/(\widetilde u^{\sss{(r)}}_{t_b+ i-1 -T_r})^{3-\tau}\in [c_1, C_1]$ hence the last factor on the rhs is at most
\[ \sum_{j=1}^k \frac{\nu_{j+1}}{\nu_{j}} \le \sum_{\ell=t_b -T_r}^{t_b+ k-1 -T_r }\frac{C_1}{c_1} \left(\frac{\widetilde u_{\ell+1}}{\widetilde u_\ell}\right)^{3-\tau} \le \sum_{\ell=[t_b -T_r]}^{t_b+k-1 -T_r} \frac{C_1(C\log n)^{3-\tau}}{c_1(\widetilde u^{\sss{(r)}}_\ell)^{(3-\tau)^2}},\]
where we have used the recursion $\widetilde u^{\sss{(r)}}_{\ell+1}=C\log n\,(\widetilde u^{\sss{(r)}}_{\ell})^{\tau-2}$ in \eqref{eq::wideui_recursion}.
Again, by the same recursion, for some large enough constant $C'$, the sum on the rhs is at most
\[ \frac{C_1(C\log n)^{3-\tau}}{c_1}\frac{C'}{(\widetilde u^{\sss{(r)}}_{t_b+k -1-T_r} )^{(3-\tau)^2}},\]
which is small as long as $ (\widetilde u^{\sss{(r)}}_{t_b+k -1-T_r})^{3-\tau}$ is of larger order than $C\log n$. Note that this holds for an appropriate choice of $k=k(n)$ by \eqref{eq::core-ell}.
Finally, notice that $J_3$ in \eqref{eq::triangle} is deterministic under the measure $\mathbb{P}_{Y,n}(\cdot)$. It is not hard to see that this inequality always holds whenever
\[ \frac{\mathbb{E}_{Y,n}[\mathrm{O}_{\max}^{\text{d}}]}{\mathbb{E}_{Y,n}[\mathrm{O}_{\max}^{\text{d}}]}\ge \prod_{i=1}^{k} \big(1- \frac{\nu_{j-1}}{\nu_j}\big) \big(1+ O(\tfrac{k^2}{n})\big)^{-1}\ge 5/6. \]
By the same argument as the one used for the term $J_1$, this holds for large enough $n$.
\end{proof}
We cite the next lemma without its proof from \cite[Lemma 7.3]{BarHofKom14}, showing that $\sum_{z\ge 0}\mathrm{O}_{-z}(k)$, the vertices reached via half-edges from vertices in the sets $\mathcal {A}_{i_{\max}-z}\setminus \mathcal {A}_{i_{\max}-z+1}$, (and not going through a higher layer), are at most the same order of magnitude as $\mathrm{O}_{\max}(k)$.
\begin{lemma}[\cite{BarHofKom14}]\label{lem::opt-z-lemma} With the notation introduced before,
\[ \log \Big(\sum_{z\ge 0}\mathrm{O}_{-z}(k) \Big)\le \log (\mathrm{O}_{\max}(k)) (1+o_{\mathbb{P}}(1)). \]
\end{lemma}
Having analysed the size of the optional cluster of blue, we are ready to finish the upper bound of Theorem \ref{thm::main} by combining the previous results.
\begin{proof}[Proof of the upper bound in Theorem \ref{thm::main}.]
First, fix $k=k(n)\to \infty$ so that $k(n)=o(\log\log n)$.
Then, Lemma \ref{lem::opt-z-lemma} implies that the logarithm of the total number of vertices that blue paints in the last phase is at most $\log \mathrm{O}_{\max}(k) (1+o_{\mathbb{P}}(1))$. Corollary \ref{cor::chebisev} says that the order of magnitude of $\log (\mathrm{O}_{\max}(k))=\log M^{\sss{(b)}}_n + \sum_{j=1}^{k-1}\log \nu_j + o_{\mathbb{P}}(1)$, where $M^{\sss{(b)}}_n$ is the number of blue half-edges in the highest layer that blue can reach.
Further, Lemma \ref{lem::verticeswithmaxdegree} determines the order of magnitude of $\log M^{\sss{(b)}}_n$, which equals
\begin{equation}\label{eq::order-mbn}\log M^{\sss{(b)}}_n=\sqrt{Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}} \log n \cdot \frac{h_n^{\text{half-edge}}(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})}{\tau-1} (1+o_{\mathbb{P}}(1)),\end{equation} and hence converges in distribution to $(Y_b/Y_r)^{1/2}$ when divided by the second two factors.
Thus, to get the asymptotic behavior of $\log (\mathrm{O}_{\max}(k))$, it remains to calculate $\sum_{j=1}^k \log \nu_j$ and compare it to the order of $\log M^{\sss{(b)}}_n$. For this recall the definitions of $\nu_j$ in \eqref{def::nuj}, $t_b$ in \eqref{def::t_b}, $t_c$ in \eqref{eq::tc2}, $\widetilde u^{\sss{(r)}}_\ell$ in \eqref{eq::ul}, and the upper bound on $\nu_j$ in \eqref{eq::nujbound}.
\begin{align} \label{eq::prodnuj1}\sum_{j=1}^{k} \log \nu_j \le& \sum_{j=1}^k \log\left( C_1 \left(\widetilde u_{ t_b+ j-1-T_r\rfloor}\right)^{3-\tau}\right)\\
\le& \sum_{j=0}^{k-1} \left\{\left(\alpha_n^{\sss{(r)}}\log n \!+\! b_n^{\sss{(r)}} \log (C\log n)\right)(\tau-2)^{\lfloor t_c \rfloor+j} (3\!-\!\tau)\right\} \!+\!k \log (C_1 C\log n).\nonumber \end{align}
Note that this is simply a geometric sum. Using \eqref{def::delta} we obtain
\begin{equation}\label{eq::orderoflognuj} \sum_{j=1}^{k} \log \nu_j\le\ (1+o_{\mathbb{P}}(1))\cdot \log n
\frac{(\tau-2)^{\delta_n^{\sss{(r)}}}}{\tau-1}(\tau-2)^{\lfloor t_c \rfloor}(1- (\tau-2)^{k}).
\end{equation}
Recall that $\log (\widetilde u^{\sss{(r)}}_\ell)/\log n=(\tau-2)^{\delta_n^{\sss{(r)}}+\ell-1}/(\tau-1)(1+o_{\mathbb{P}}(1))$, (by \eqref{eq::ul} and \eqref{def::delta}), and hence the expression in \eqref{eq::orderoflognuj} is essentially $\log \widetilde u^{\sss{(r)}}_{\lfloor t_c \rfloor+1}$, multiplied by by a factor of $(1-(\tau-2)^k)$ in the exponent. Using the rhs of \eqref{eq::wur-tc}, \eqref{eq::t2-t1exp} and the value of $\{t_c\}$ in the four cases $(E_<, O_>, E_>, O_<)$,
\begin{equation}\label{eq::log-nuj} \sum_{j=1}^{k} \log \nu_j\le\ (1+o_{\mathbb{P}}(1))\cdot \log n \left(\frac{Y_b^{\scriptscriptstyle{(n)}}}{Y_r^{\scriptscriptstyle{(n)}}}\right)^{1/2}
\frac{(\tau-2)^{\delta_n^{\sss{(r)}} + \frac{b_n^{\sss{(r)}}-b_n^{\sss{(b)}}}{2} }}{\tau-1}(\tau-2)^{(\mathbbm{1}_{E_>}-\mathbbm{1}_{E_<})/2}(1- (\tau-2)^{k}). \end{equation}
Recall that $(\tau-2)^{\delta_n^{\sss{(r)}}}=((\tau-1)-(\tau-2)^{b_n^{\sss{(r)}}})/(\tau-1)$, hence, let us introduce
\begin{equation} \label{def::dnY} h^{\text{paths,k}}_n( Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}):=(\tau-1-(\tau-2)^{b_n^{(r)}}) (\tau-2)^{(b_n^{\sss{(r)}}-b_n^{\sss{(b)}}+\mathbbm{1}_{E_>} - \mathbbm{1}_{E_<})/2 }(1-(\tau-2)^k).\end{equation}
Let \begin{equation} \label{def::Cpath}h^{\text{paths}}_n( Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}):=\lim_{k\to \infty} h^{\text{paths,k}}_n( Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}).\end{equation}
Now, recall again that $\log \mathrm{O}_{\max}(k)=\log M^{\sss{(b)}}_n+ \sum_{j=1}^k \nu_j+ o_{\mathbb{P}}(1)$ by Corollary \ref{cor::chebisev}, and combine \eqref{eq::log-nuj} with \eqref{eq::order-mbn}
\[ \frac{\log \mathrm{O}_{\max}(k) }{
\log n \!\cdot\! (\tau-1)^{-1}\left( h_n^{\text{half-edge}}(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) + h^{\text{paths}}_n( Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})\right)} \le \sqrt{ \frac{Y_b^{\scriptscriptstyle{(n)}}}{Y_r^{\scriptscriptstyle{(n)}}}}(1+ o_{\mathbb{P}}(1)). \]
Now we can finally use $Y_r^{\scriptscriptstyle{(n)}}\buildrel {d}\over{\longrightarrow} Y_r, \ Y_b^{\scriptscriptstyle{(n)}}\buildrel {d}\over{\longrightarrow} Y_b$ (by Theorem \ref{thm::davies}
). Hence, the right hand side converges to $\sqrt{Y_b/Y_r}$. Note that $f_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}):=h_n^{\text{half-edge}}(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) + h^{\text{paths}}_n( Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ after some elementary rearrangements and simplifications exactly gives \eqref{eq::main-f}, finishing the proof of the upper bound.
\begin{remark}\normalfont It is not entirely obvious from the formulas that the factor multiplying $\log n$, $\sqrt{Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}} f_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}}) (\tau-1)^{-1}$, is always strictly less than $1$. We investigate this issue below in Lemma \ref{lem::no-coexistence}. \end{remark}
\end{proof}
Recall that the vertices reached via half-edges from layer $\mathcal {A}_{i_{\max}-z}^{\scriptscriptstyle{(b)}}\setminus \mathcal {A}_{i_{\max}-z+1}^{\scriptscriptstyle{(b)}}$ are denoted by $\mathcal{O}_{-z}, z\ge 0.$
For the lower bound of the proof of the first part of Theorem \ref{thm::main}, let us introduce the notation $\mathcal{O}(k):=\bigcup_{z\ge0} \mathcal{O}_{-z}(k) \cup \mathcal{O}_{\max}(k)$, and set $\mathrm{Opt}(k):=|\mathcal{O}(k)|$,
where $k$ stands for the length of the paths we are counting.
The next lemma shows that essentially all the vertices in $\mathcal{O}(k)$ for some $k=k_n=o(\log\log n)$ will indeed be painted blue, i.e., red cannot accidentally bite out too much from this set:
\begin{lemma}\label{lem::red-blue-intersection}
Set $k=k(n)=o(\log\log n)$.
The expected number of vertices in the intersection of $\mathcal {R}_{t_b+ k}$ and $\mathcal{O}(k)$ is small:
\[ \mathbb{E}_{Y,n}\left[|\mathcal{O}(k)\cap \mathcal {R}_{t_b+ k}|\right]= o_{\mathbb{P}}\left(\mathrm{O}_{\max}(k)\right),\]
hence $|\mathcal{O}(k)\setminus(\mathcal{O}(k)\cap \mathcal {R}_{t_b+ k})|=\mathrm{O}_{\max}(k)(1+o_{\mathbb{P}}(1))$.
\end{lemma}
\begin{proof}
The proof of this lemma is identical to that of \cite[Lemma 7.4]{BarHofKom14}, with setting $\lambda=1$, hence, we refer the reader there.
\end{proof}
\begin{proof}[Proof of the lower bound of Theorem \ref{thm::main}]
The proof is identical for the proof of the lower bound of \cite[Theorem 1.2]{BarHofKom14}, hence we again refer the reader there.
\end{proof}
\begin{lemma}\label{lem::no-coexistence}
If $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<\tau-2$, then there is no coexistence, i.e. $\mathcal {B}_\infty=o_{\mathbb{P}}(n)$.
\end{lemma}
\begin{proof}
To show the statement, we have to recall from the proof of Lemma \ref{lem::verticeswithmaxdegree} that
\[ \frac{\log M^{\sss{(b)}}_n}{(\tau-1)^{-1}\log n} =\left( \mathbbm{1}_{ O_< \cup E_>} ((\tau-2)^{2 \{t_c\} } + (3-\tau)) + \mathbbm{1}_{O_> \cup E_< } \right)\frac{\log (D_n^{\max} (\infty))}{(\tau-1)^{-1}\log n}, \]
and now use the representation of the maximal degree that is listed in the four cases at the bottom of page 31 to get that in Cases $O_<, E_>$
\begin{equation}\label{eq::mbn-ole-ege} \frac{\log M^{\sss{(b)}}_n}{(\tau-1)^{-1}\log n} = (\tau-2)^{(T_b - T_r-1-\mathbbm{1}_{O_< })/2} (\tau-2)^{\delta_n^{\sss{(r)}}} \left( (\tau-2)^{b_n^{\sss{(b)}} - \delta_n^{\sss{(r)}} + \mathbbm{1}_{O_<}} + (3-\tau) \right), \end{equation}
while in Cases $O_>, E_<$
\begin{equation}\label{eq::mbn-oge-ele} \frac{\log M^{\sss{(b)}}_n}{(\tau-1)^{-1}\log n} = (\tau-2)^{(T_b - T_r-1-\mathbbm{1}_{O_> })/2} (\tau-2)^{b_n^{\sss{(b)}}}. \end{equation}
We also have to recall that the path counting method gives \eqref{eq::orderoflognuj}, combined with the observation below \eqref{eq::orderoflognuj}, namely that the rhs of \eqref{eq::orderoflognuj} equals the right hand side of \eqref{eq::wur-tc} multiplied by $\tau-2$. Hence,
\[ \frac{\sum_{j=1}^\infty \log \nu_j }{(\tau-1)^{-1} \log n} = (\tau-2)^{(T_b-T_r + 1)/2} (\tau-2)^{(b_n^{\sss{(b)}}+\delta_n^{\sss{(r)}})/2 - \{t_c\}}. \]
Combining this again with the value of $\{t_c\}$ in the four cases listed at the bottom of page 31, we get
\begin{equation}\label{eq::lognuj-rewrite} \frac{\sum_{j=1}^\infty \log \nu_j }{(\tau-1)^{-1} \log n} = (\tau-2)^{(T_b-T_r + \mathbbm{1}_{E_>} - \mathbbm{1}_{E_<})/2} (\tau-2)^{\delta_n^{\sss{(r)}}}. \end{equation}
To show that $\mathcal {B}_\infty = o_{\mathbb{P}}(n)$ we need to show that the sum of the rhs of \eqref{eq::mbn-ole-ege} or \eqref{eq::mbn-oge-ele} plus the rhs of \eqref{eq::lognuj-rewrite} is less than $(\tau-1)$.
We analyse the four cases separately.
\emph{Case $O_>$}. Note that by the definition of the event $O_>$, $T_b-T_r-1$ is odd, hence $T_b-T_r\ge 2$ is even. The sum of the right hand sides of \eqref{eq::mbn-oge-ele} and \eqref{eq::lognuj-rewrite} in this case equals
\[ (\tau-2)^{(T_b-T_r-2)/2} \left( (\tau-2)^{b_n^{\sss{(b)}}} + (\tau-2)^{\delta_n^{\sss{(r)}}+1} \right) < (\tau-2)^0 + (\tau-2)^1 = \tau-1, \]
where we have used that $T_b-T_r-2\ge 0$, hence the first factor is at most $1$, and also $b_n^{\sss{(b)}}\ge 0, \delta_n^{\sss{(r)}}>0$ by \eqref{def::delta}.
\emph{Case $O_<$}. Note that by the definition of the event $O_>$, $T_b-T_r-1\ge 1$ is odd. The sum of the right hand sides of \eqref{eq::mbn-ole-ege} and \eqref{eq::lognuj-rewrite} in this case equals
\[ (\tau-2)^{(T_b-T_r-1)/2} \left( (\tau-2)^{b_n^{\sss{(b)}}+1} + (\tau-2)^{\delta_n^{\sss{(r)}}} \right) < (\tau-2)^1 + (\tau-2)^0 = \tau-1,\]
since $T_b-T_r-2\ge 0$, and again, $b_n^{\sss{(b)}}\ge 0, \delta_n^{\sss{(r)}}>0$ by \eqref{def::delta}.
\emph{Cases $E_<$ and $E_>$}. Note that by the definition of the event $E_<, E_>$, in these cases $T_b-T_r- 1\ge 0$ is even. The sum of the right hand sides of \eqref{eq::mbn-oge-ele} or \eqref{eq::mbn-ole-ege} and \eqref{eq::lognuj-rewrite} in both cases equals
\[ (\tau-2)^{(T_b-T_r-1)/2} \left( (\tau-2)^{\delta_n^{\sss{(r)}}} + (\tau-2)^{b_n^{\sss{(b)}}} \right). \]
Clearly, if $T_b\ge T_r + 3$, then this expression is at most $2(\tau-2) < \tau-1$ since $\tau<3$.
Now, if $T_b= T_r+1$, then the first factor is $1$ and the second factor equals $\tau-1- (\tau-2)^{b_n^{\sss{(r)}}} +(\tau-2)^{b_n^{\sss{(b)}}}$, which is at most $\tau-1$ if and only if $b_n^{\sss{(b)}}>b_n^{\sss{(r)}}$. Recall the analysis of the crossing the peak in Section \ref{sc::peak}, Case (3): we see that if $T_b=T_r+1$, then $b_n^{\sss{(b)}}>b_n^{\sss{(r)}}$ if and only if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}<\tau-2$, which is exactly what we assume here throughout. Hence, the expression is less than $\tau-1$ in these cases.
\begin{remark}\normalfont
We now comment on why we did not prove the statement of the lemma directly using the formula for $f_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$. Note that by \eqref{eq::t2-t1exp}, $(\tau-2)^{(T_b-T_r)/2} = \sqrt{Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}} (\tau-2)^{(b_n^{\sss{(r)}}-b_n^{\sss{(b)}})/2}$. Hence, for instance in Cases $E_<, E_>$,
\[ \frac{\log \mathcal {B}_\infty}{\log n} = \sqrt{Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}} (\tau-2)^{(b_n^{\sss{(r)}}-b_n^{\sss{(b)}}-1)/2} \left( (\tau-2)^{\delta_n^{\sss{(r)}}} + (\tau-2)^{b_n^{\sss{(b)}}} \right).\]
Notice that the first factor on the right hand side is at most $\sqrt{\tau-2}$, hence we need that the other factors are at most $(\tau-1) / \sqrt{\tau-2}$ in these cases. This is however only true under the extra information that if $\sqrt{Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}}$ is close to $\tau-2$, so that this formula holds (not the ones for $O_<, O_>$), then necessarily $b_n^{\sss{(b)}}\ge b_n^{\sss{(r)}}$, and $b_n^{\sss{(b)}}\searrow b_n^{\sss{(r)}}$ as $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}} \nearrow \tau-2$. On the other hand, if $Y_b^{\scriptscriptstyle{(n)}}/Y_r^{\scriptscriptstyle{(n)}}\ll\tau-2$, then the relationship between $b_n^{\sss{(r)}}, b_n^{\sss{(b)}}$ is not necessarily the same, and hence, e.g. on the event $b_n^{\sss{(b)}} \nearrow 1$ the maximum of $f_n(Y_r^{\scriptscriptstyle{(n)}}, Y_b^{\scriptscriptstyle{(n)}})$ is $(\tfrac{2\tau-3}{3})^{3/2} 2/(\tau-2)$ which is in fact strictly larger than $(\tau-1)/\sqrt{\tau-2}$.
\end{remark}
\end{proof}
\section{Acknowledgement}
The work of RvdH and JK was supported in part by the Netherlands Organisation for Scientific Research (NWO) through VICI grant 639.033.806.
The work of JK was supported in part by NWO through the STAR cluster, the VENI grant and the work of RvdH was supported in part by NWO through Gravitation grant 024.002.003. JK thanks the Probability Group at The University of British Columbia for their hospitality while working on this project.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,333 |
**FELINE INTUITION**
While I scanned the tiny handwriting of the registrar of the 1840s, I heard snatches of conversation. I paid them scant attention, focusing on my task. But when I heard the words _murder_ and _Priest_ , I started listening.
I found the incident oddly unsettling, though I couldn't say why. I supposed the conversation was about Godfrey Priest, since he was _the_ hot topic in Athena at the moment. And hearing the word _murder_ in conjunction with his name wasn't that odd. The man did write murder mysteries. Then I heard the sound of a throat clearing on the other side of my desk. My eyes widened in surprise as I recognized the man. It was Godfrey Priest. What the heck was he doing here?
"Good morning, Godfrey," I said, extending a hand in greeting. Diesel padded right behind me. "It's been a long time."
"What is that? A cat?" Godfrey asked, watching as Diesel made a slow circle around him. Evidently unimpressed, Diesel walked back to the window and jumped up to his bed. Yawning, he turned his back on both of us and settled down for a nap.
"He's a Maine coon," I said. "They're larger than most cats."
"That's the first time I've ever been snubbed by a cat." Godfrey laughed, but his expression revealed annoyance. "They always love me because they can tell I'm a cat person."
I tried not to laugh. "Diesel doesn't take to everybody."
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MURDER PAST DUE
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**ACKNOWLEDGMENTS**
My first thanks go to Michelle Vega and Natalee Rosenstein, for keeping me in the family. Their support means more than they will ever know. Nancy Yost, my agent, handles the really fun part of the business, so I don't have to.
The Tuesday night crew gave me valuable input on the early stages of the manuscript. Thanks to Amy, Bob, Joe, Kay, Laura, Leann, and Millie for their insights and advice. A special thanks goes to Enzo, Pumpkin, Curry, and their two-legged staff, Susie, Isabella, and Charlie, for allowing us to invade their home each week to critique in such a warm and inviting venue.
I owe a very special thanks to Terry Farmer, Ph.D., proud mom of three Maine coons, Figo, Anya, and Katie, for serving as my technical advisor in all matters having to do with Maine coon cats. Any mistakes in my portrayal of Diesel and his behavior are mine and not hers. (I do have two cats of my own—but neither of them is a Maine coon.) Finally, my love and gratitude to two very dear friends who never fail to encourage me, Patricia Orr and Julie Herman.
Table of Contents
Title Page
Copyright Page
Acknowledgements
ONE
TWO
THREE
FOUR
FIVE
SIX
SEVEN
EIGHT
NINE
TEN
ELEVEN
TWELVE
THIRTEEN
FOURTEEN
FIFTEEN
SIXTEEN
SEVENTEEN
EIGHTEEN
NINETEEN
TWENTY
TWENTY-ONE
TWENTY-TWO
TWENTY-THREE
TWENTY-FOUR
TWENTY-FIVE
TWENTY-SIX
TWENTY-SEVEN
TWENTY-EIGHT
TWENTY-NINE
THIRTY
THIRTY-ONE
THIRTY-TWO
THIRTY-THREE
_Excerpt from_ Claws for Concern
**ONE**
A hurricane slammed through my kitchen this morning, and his name was Justin.
I sighed, surveying the aftermath of my boarder's breakfast. What had gotten into the boy?
An open milk carton sat on the table, accompanied by a bowl, a spoon, and a box of cereal. Justin hadn't closed the box, and he'd left a sprinkle of cereal on the table. Splatters of milk surrounded the bowl and an abandoned plate with a half-eaten piece of toast.
I glanced toward the counter at an open loaf of bread and an uncovered butter dish, sitting in a beam of sunlight. Two pieces of bread occupied slots in the toaster, but from what I could see, Justin had forgotten to press the lever down. I strode over and picked up my newspaper from beside the sink. Justin had somehow managed to dribble water over the paper. I was glad I'd read it earlier, because now it was stuck together.
I stared out the kitchen window into the backyard for a few seconds, calming myself. I turned back. Okay, maybe it wasn't a hurricane. Just a minor tropical disturbance. I was not one of those neat freaks who hyperventilated at the first sign of a mess.
Like most men, I can be messy—but I'm happier when things are clean and well kept. I really shouldn't let myself feel so annoyed over something so trivial.
Maybe Justin was in a hurry to make his first class, but the Athena College campus was only three blocks away. He could sprint there in five minutes tops.
He was acting out of character, and that's what was really bothering me. The eighteen-year-old had been boarding with me for a couple of months and usually was careful to pick up after himself. The past few days, however, he had become increasingly careless about leaving his things lying around the house and not cleaning up in the kitchen after his meals.
Perhaps I should have expected something like this when I relaxed my rule about accepting only older students, preferably those in graduate school, as boarders. They were generally much too focused on their work to cause any disturbances, and I valued the quiet, orderly life I had created for myself these past three years.
But I had accepted Justin as a favor to an old friend. His mother, Julia Wardlaw, and I had known each other since high school and all the way through college. Justin, an only child, wasn't ready for the rough and tumble of dorm life, she said. She wanted him to have a quieter, more homelike atmosphere for his first year in college. After the grilling Julia had put me through, I felt almost honored that she was entrusting her precious chick to my care.
A large paw pushed against my leg. Diesel, my two-year-old Maine coon cat, chirped in sympathy when I looked down at him. He withdrew his paw and stared up at me.
"I know, Diesel." I shook my head. "Justin has a problem, or he wouldn't be acting this way."
Diesel responded with another chirp—many Maine coons don't meow like other cats—and I reached down to rub his head. He still had his lighter summer coat, soft as down. His neck ruff and tail were less bushy than they would be during the colder months ahead. The tufts on the tips of his ears stood out as he stared up at me, a patient expression on his face. He was a gray tabby with dark markings, and at the age of two hadn't reached full maturity yet, weighing in at twenty-five to thirty pounds. With their broad chests and muscular bodies, Maine coons are the defensive tackles of the cat world.
"We'll have to have a talk with our boarder," I said. Diesel liked Justin and often visited him in his third-floor bedroom. "Just think what Azalea would do if she came in some morning and found a mess like this. She'd skin both Justin _and_ me." Diesel returned my rueful glance with a solemn gaze.
Azalea Berry, the housekeeper I inherited along with the house when my beloved Aunt Dottie died, had strict notions about keeping a clean home. She also had strong opinions about large cats as house pets, but she and Diesel somehow managed to reach detente when I brought him home with me a couple of years ago. Even when he was a kitten, Diesel had been smart enough to pick up on Azalea's basic antipathy to cats.
Azalea had more tolerance for college-age boys, but that didn't mean she would allow Justin to get away with leaving the kitchen a mess, even a minor one. Maybe I could help him with whatever his problem was before he did it again and Azalea got after him.
I couldn't blame Azalea for her devotion to the house. Aunt Dottie had lavished her money—and her decorating abilities—on what she considered the center of any home. The kitchen occupied the southeast corner of the house, and the morning sun poured in through the large windows on both outside walls. Light suffused the room, helped by the pale yellow paint on the walls and the white ceramic tile on the floor. The cabinets shone a delicate blue and blended well with the darker hue of the table and chairs.
I could almost smell the scent of the ginger cookies Aunt Dottie used to make when I was a boy. There were only happy memories in this room, but for a moment I ached with the loss of my dear aunt and of my beloved wife, Jackie. They both died within a few weeks of each other three years ago. I pictured them at the table together, laughing and chatting.
Coming out of my reverie, I glanced at Diesel again, and I could swear he had a sympathetic look on his face. "Enough of that," I told him. He twitched his tail, turned, and padded off in the direction of the utility room and his litter box.
I cleared up Justin's mess, and as I was putting the cereal box away in the cupboard, Justin popped into the kitchen.
"Mr. Charlie," Justin said, stopping in the doorway. "I was planning to clean up." One hand clutched a worn backpack, and the other smoothed dark hair out of his eyes. The boy needed a haircut, or else he needed a ponytail.
Diesel reappeared and rubbed against his friend's jean-clad leg. Justin squatted for a moment, scratching the cat's head but watching me through his bangs.
"I thought you already left for your class," I said. "If this was one of Azalea's days, she might have a few things to say about finding the kitchen the way you left it."
I kept my tone mild, but Justin flushed anyway. Down went his head, and his hair swung forward, shielding his face. He mumbled something as he stood. Diesel sat beside him, staring up at his face.
"What did you say?"
Justin shrugged. "Sorry," he said, more clearly this time, avoiding my gaze for the moment. "I really meant to clean up, but I just lost track of time." He shot me a quick glance, then stared down at his feet again.
"No real harm done, Justin. But it seems to me you've been a little careless the last few days. That's not like you."
He shrugged. "Well, I'm gonna be late. Bye, Diesel." He turned and disappeared down the hall. In a moment I heard the front door open. I was relieved not to hear it slam shut.
We were definitely due for a chat, Justin and I. Something was bothering him, and he was bordering on rudeness. In the two months that he'd lived here, he hadn't been the most outgoing young man, but he had been civil until recently.
As the father of two former teenagers, I knew that a change in behavior could signal any one of several problems. I hoped this wasn't a substance abuse problem. His father, a conservative Evangelical preacher, would probably yank him out of college and take him home if that was the case. Julia wouldn't be too happy either, and might even blame me for letting him get in trouble.
The last thing I wanted was to get involved in the life of one of my boarders. If Justin's problem turned out to be serious, he would have to go home to his parents. I wasn't ready to cope with anything big.
Diesel padded beside me to the wall rack by the back door, and I lifted his harness and leash off the hook. He purred as I got him street-ready, emitting the rumbling sound that inspired his name. He loved going to work with me.
"Let me get my coat and my satchel," I said. I checked my tie for coffee and food stains and examined my pants for cat hair. Why did dark colors attract pet hair like magnets? I did a quick removal job with a lint brush, and then Diesel and I were ready to go.
In the past two years, since I'd first found a shivering kitten in the parking lot of the public library, most people in my hometown of Athena, Mississippi, had grown used to seeing me walking my cat on a leash. As Diesel grew bigger, some of them wondered if he wasn't part bobcat, but that's only because no one in town—including me—had ever seen a Maine coon. What they'd think when he was fully grown in another year, I had no idea.
Strangers sometimes stopped us on the street to ask if he was a weird-looking dog—and I'd swear Diesel looked offended when they did. He was a sociable critter, but he didn't tolerate fools lightly—a trait _I_ found endearing.
I detected a hint of wood smoke in the crisp autumn air. It seemed early to be lighting a fire in the fireplace, but evidently one of my neighbors disagreed. The odor reminded me of times by the fire in my parents' house on cold winter days.
The homes on my street were over a century old, many occupied by the same families for generations. The graceful architecture, the classic landscaping, and the feeling of a real neighborhood gave me a sense of security after I lost my wife.
Putting thoughts of Jackie aside, I started walking, Diesel preceding me by a few paces. The campus of Athena College—our destination this morning—lay three blocks to the east. A walk that should have taken five minutes usually took fifteen or twenty, because Diesel and I stopped several times so his many admirers could say hello. He took it all in stride, chirping and purring, putting smiles on faces, including my own. One or two even remembered to say, "Morning, Charlie," so I wasn't completely ignored.
Jordan Thompson, owner of the Athenaeum, our local independent bookstore, was out for her morning run. She waved as she zoomed past. I had to admire her dedication to exercise, and wished I could emulate it.
Diesel and I arrived at the college library just as Rick Tackett, the operations manager for the two library buildings, was unlocking the front door. Eight o'clock on the dot, and not a second sooner. We stood on the veranda of the antebellum Greek Revival mansion that housed the library's administrative offices, archives, and rare book collections. The main building, known as Hawksworth Library, stood next door.
Rick nodded in response to my "Good morning" and stepped aside to let me and Diesel enter. About a decade my senior, Rick was pleasant though difficult to engage in conversation. Since he spent much of his time in the main building next door, I saw him only infrequently.
Diesel led the way up the central stairs to the second floor. On the landing, he turned left and paused in front of a door with RARE BOOK ROOM emblazoned in gold leaf on the glass.
When I unlocked the door and pushed it open, I dropped Diesel's leash, then I went around the room turning on lights. By the time I finished, Diesel was settled on his favorite perch—a bed placed on the wide windowsill behind my desk. I unhooked his leash, coiled it, and stuck it beside his bed.
Diesel purred while I readied myself for the workday. My jacket and satchel stowed, I sat at my desk, turned on my computer, and began to organize my day mentally.
I worked two days a week here as a cataloger and archivist, although I enjoyed what I did so much that I never really thought of it as labor. During my career as a librarian in Houston, I had spent much of my time as an administrator. Being able to catalog again was almost heaven after dealing for so long with budgets and personnel. I was content to be here with Diesel and the rare books.
I had barely begun to read my e-mail when I heard a tap at the door.
"Morning, Charlie." Melba Gilley advanced into the room. She was as sleek as she had been in high school. Spectacular figure back then, and she still did. I liked Melba a lot, and she liked me, but so far we had both been content with simply reestablishing our friendship. I wasn't ready to date yet, and now that I was approaching fifty, I wasn't sure when or if I would be. I didn't want the emotional complications a new relationship would bring.
"Morning, Melba," I said. "How are you?"
Melba plopped into the chair by my desk, picked a nonexistent piece of fluff from her immaculate aubergine pantsuit, and said, "Excited. Aren't you?" She looked past me at the window. "Morning, Diesel, honey."
Diesel warbled a response but didn't leave his bed.
"About what?" I frowned. What had I forgotten?
"The big reception tonight. What else would I be excited about?" Melba smiled. "It's not every day Athena welcomes home a golden boy."
"Oh, that," I said. "Big deal."
Melba shook her head at me. "Charlie Harris, you can't tell me you're not curious to see what Godfrey Priest is like after all these years. I know y'all didn't get along in high school, but surely you want to see a famous author in the flesh." She laughed.
I shook my head at her. "He was a jerkwad thirty-two years ago, and he's probably an even bigger jerkwad now, just a rich one."
"He's had four wives," Melba said. "Or so I hear tell. But I guess he can afford the alimony, as much as he makes off his books."
"He hasn't made much off me," I said. "At least not since his first few books came out."
"So you _have_ read his books." Melba almost crowed in triumph.
"I'll admit it," I said. "I was curious, like everybody else in Athena. And I even liked the first few. They were entertaining. But then he started writing those violent thrillers, and the plots got more and more unbelievable." My mouth twisted in distaste. "Not to mention the violence against women. Surely you don't still read him?"
Melba shook her head. "No, I quit a few books back. Same reasons as you."
"Then why are you so excited?"
"He's a bestseller, a celebrity," Melba said. "How often does a celebrity come to Athena? We could use some excitement."
I rolled my eyes. "Surely you're not forgetting the time Roberta Hill spray-painted her husband pink when she caught him dead drunk and naked in Liz Graham's trailer? I hear tell that was exciting, especially when he woke up and chased her with an ax down Main Street."
Melba hooted with laughter. "Oh honey, I wish you'd been here. I've never seen anything so funny in my life. Delbert all pink, and his personal bits flopping all over the place. I was coming out of the bank when he and his wife went flying by. He's lucky Roberta didn't do something worse, like take that ax to his weenie."
I laughed too, trying not to think about an axed weenie. From behind me I heard Diesel purring, like he was laughing along with us.
Melba's cell phone rang. Grimacing, she pulled it from the holster at her waist. She handled it like a gunslinger with years of practice, twirling it in her hand before holding it still to read the display. "His Majesty wants me." She answered the call to assure our boss she'd be with him right away. Then she ended the conversation and replaced her phone after another twirl.
"Duty calls," she said as she stood. "I swear, that man couldn't find his tiny rear end if I wasn't there to show him where it was." She snickered.
"That's why you're such a valuable administrative assistant," I said. "You know where everything is."
Melba grinned. "See you later, hon."
I chuckled. Peter Vanderkeller, the director of the library, resembled nothing more than a garden rake. His size thirteen feet seemed out of proportion to his emaciated six-foot-four frame. Melba swears she's never seen him eat anything, and most of the time I believe her. I'd never seen him put anything in his mouth other than a pen or a pencil. He invariably chewed on one during meetings.
The silence after Melba's departure felt good. Quiet is the last word anyone would use to describe her.
I turned back to my e-mail, wincing my way through Peter's weekly letter to the troops, as we library employees not-so-fondly called it. Last year at Halloween several of us dressed up in uniform for a staff meeting. Peter didn't get the joke. He never did. I felt sorry for him sometimes.
The subject of this week's homily was recycling. Peter exhorted everyone to stop bringing bottled water to work and to use instead the filtered tap water in the staff lounge. I glanced at my satchel. It usually contained at least two bottles of water. I resolved to use those bottles for refills from the tap once they were empty. Perhaps that would satisfy the boss.
The last e-mail I read reminded me of the gala reception tonight in honor of our celebrity at the president's house. I'd been telling myself I wouldn't go, but I knew curiosity would get the better of me. As much as I detested Godfrey Priest, I wanted to get a look at him after all this time.
Back in high school, I'd let Godfrey intimidate me. He was taller and better looking and was always flaunting his success with girls. I resented him in high school and in college—we were both alumni of Athena College—but that was long ago. Surely I'd left all that behind me?
Perhaps I hadn't, but if living well truly was the best revenge, I wanted to show the jerkwad I was doing fine.
Shaking my head over my foolishness, I turned away from the computer to examine some papers on my desk. Where had I put that letter? I lifted one or two of the piles until I had located what I wanted.
Besides cataloging, I also handled certain kinds of reference questions, those that related to some historical aspect of the college or the library's archives and rare books. Yesterday I had received a request from an elderly woman in Vicksburg who was trying to track down a stray twig on her family tree. Said twig was supposed to have attended Athena College back in the 1840s, not long after the school was founded.
Glancing through the letter, I found the name I needed. Laying the letter aside, I left my desk and approached a shelf of reference materials to the left of the door. What I sought was an old book of attendance records that should answer the question. One of these days I hoped to get a grant to have the records computerized, but until that happened, the old-fashioned way would have to do.
I pulled the book off the shelf and gently turned the pages until I found the years I wanted. Sounds from other parts of the library drifted up. The acoustics often behaved oddly, the grand stairway and the high-ceilinged foyer serving to bounce voices around.
While I scanned the precisely formed but tiny handwriting of the registrar of the 1840s, searching for one Bushrod Kennington, I heard snatches of conversation. I paid them scant attention, focusing on my task. But when I heard the words _murder_ and _Priest_ , I started listening.
**TWO**
I kept listening but could discern no other words. The voices faded.
I found the incident oddly unsettling, though I couldn't say why. I supposed the conversation was about Godfrey Priest, since he was _the_ hot topic in Athena at the moment. And hearing the word _murder_ in conjunction with his name wasn't that odd. The man did write murder mysteries.
I stopped listening and resumed my search until I located old Bushrod.
Back at my desk I made a few notes, planning to respond to the letter after lunch. This morning I intended to spend my time cataloging. I retrieved the truck of books I'd been working on and pulled the next book to catalog. After logging in to the cataloging module of our integrated library system (or ILS, in library parlance), I began to examine the book.
Part of a collection of nineteenth-century medical books, this particular volume was an 1807 treatise on midwifery by Thomas Denman. The binding was in excellent condition, but I opened the book with great care, as always. By now I was accustomed to handling books two centuries old and even older, but I still felt a sense of wonder when I touched them. So sturdy, able to survive two hundred years with proper care, but at the same time so fragile, so easily destroyed. A faint mustiness tickled my nose, and my fingers caressed the cool softness of the pages.
The particular fun of cataloging something this old was noting anything about the copy in hand—inscriptions, stamps, notations—that would set it apart from another copy. In the book I held, the front free endpaper bore, in faded ink, a previous owner's name and date: "Dr. Francis Henshall, March 18, 1809." As I delved further into the book, I found notations in ink in the same handwriting. Dr. Henshall had added comments to the text, based on his own patients.
I turned to the computer and called up the record I had previously downloaded into our system from a bibliographic utility. All the basics were there—title, publisher, date, and so on—and I added the notes to identify the copy in hand.
Engrossed in my work, I started when I heard the sound of a throat clearing on the other side of my desk.
I suppressed my irritation at the interruption as I turned to face the newcomer. Then my eyes widened in surprise as I recognized the man.
Hastily saving my work, I mumbled, "Just a moment."
"Take your time, Charlie," Godfrey Priest said, his voice booming in the quiet of the rare book room.
Beside me, Diesel stretched and yawned. He enjoyed visitors, and he hopped down from his perch to welcome Godfrey.
What the heck was he doing here? We hadn't been that close in high school or college, so why seek me out?
"Good morning, Godfrey," I said, standing. I came around the desk and extended a hand in greeting. Diesel padded right behind me. "It's been a long time."
"It sure has," Godfrey said, his tones still hearty. He clasped my hand in his bigger one and gave it a firm squeeze and a shake. "You're looking good."
"You, too," I said, trying not to wince. I flexed my fingers slightly when Godfrey released my hand.
He was even taller than I remembered. I glanced down at his feet and I could see why. He was wearing an expensive pair of cowboy boots with heels that made him about two inches taller than his normal six-four.
"What is that? A cat?" Godfrey asked, watching as Diesel made a slow circle around him. Evidently unimpressed, Diesel walked back to the window and jumped up to his bed. Yawning, he turned his back on both of us and settled down for a nap. I'd give him a treat later.
"He's a Maine coon," I said. "They're larger than most cats."
"That's the first time I've ever been snubbed by a cat." Godfrey laughed, but his expression revealed annoyance. "They always love me because they can tell I'm a cat person."
I tried not to laugh. "Diesel doesn't take to everybody. Don't pay any attention."
I continued to take in my visitor. Though we were the same age, he looked ten years older. His skin resembled leather, and years of exposure to the sun had added lines to the skin around his eyes. His hair, now a bleached straw mop, had suffered, too. His clothes screamed designer labels, and the Rolex watch he consulted ostentatiously, along with a chunky gold bracelet, made the point that he had plenty of money.
"What can I do for you, Godfrey?" I went back to my desk to sit down. With a wave I indicated he should sit, too. "Did you drop by to talk about the good old days?"
"I have it on good authority that you are the archivist here," he said, patently ignoring my little dig. He settled his long frame into the chair and crossed his arms.
"I am," I said. _Pompous as ever_. I waited.
Godfrey glanced past me toward the sleeping Diesel. "They let you bring that cat to work?" His fingers tensed on his arms, and his eyes searched the room. He seemed nervous, but I had no idea why.
"Obviously."
Godfrey's cheeks reddened as he faced me. I remembered that he had never cared for sarcasm, particularly when it had been directed at him.
"When did you return to Athena?" Godfrey asked. "I don't get here often myself. My schedule is so demanding—book tours, interviews, talking to guys in Hollywood." Again his gaze roved around the room. Was he ever going to get to the point of this visit? How much self-aggrandizement would I have to endure?
"I moved back three years ago," I replied, trying not to sound impatient. Did he think I'd be impressed by his busy life? "Not long after my wife died, my aunt left me her house here."
"Your Aunt Dottie?" Godfrey asked, frowning. "So your aunt died, too?"
"Shortly after my wife."
"Sorry to hear that," Godfrey said. "That's too bad, their dying so close together."
"It was rough." Then a memory surfaced. "You lived with Aunt Dottie for a couple of semesters, didn't you?"
Godfrey nodded. "That would have been my senior year. My parents sold up and moved to Alabama, to Fairhope, and I didn't want to live in the dorm anymore. I was lucky Miss Dottie had a room available. She was a wonderful lady." His face softened with a reminiscent smile.
"She certainly was." This was a side of Godfrey I didn't remember seeing. He had obviously been fond of my aunt. "You're doing well these days. Bestseller list with every new book. That's pretty exciting."
"Thanks. My last seven books have debuted at number one," Godfrey said, the smile giving way to a smug look. "And that's kind of why I'm here."
"I heard you're getting an award for being a distinguished alumnus," I said.
Godfrey shook his head. "That's not what I meant, although that's the ostensible reason I'm back in town. No, I meant the reason I was here talking to you."
_Finally_. "And that would be . . . ?" I asked, my voice trailing off.
"The archive," Godfrey replied. "I am giving my papers to the university archive. I plan to make the announcement tonight at the dinner." He stared at me. "How do we do this?"
The university administration would be delighted by such a gift, and I thought it was an excellent idea. On one condition. "I know the university would love to have your papers," I said. "But giving them is one thing. Are you willing to donate money to help with the preparation, cataloging, and maintenance?"
"Sure," Godfrey said. "What do you have to do, other than put them on the shelves?" He waved a hand in the direction of the bookshelves. "And how much money? I'm sure I can afford it."
"The papers have to be organized and cataloged," I said, ignoring that last sentence. "That could take some time, depending on the extent of the collection. I'm the archivist, but I work only part-time. It could take years to get your papers done, considering all the other books and collections waiting to be processed here."
"If I give enough money, could you hire someone to catalog my papers and get them done sooner?" He frowned. "I don't want them sitting in boxes, gathering dust."
"Yes," I replied. "We have a tiny budget, and we rely on donations."
"How much?"
"How many papers are we talking about?" I pointed to a nearby box, roughly the size of a box of computer paper. "How many boxes of stuff?"
Godfrey stared at the box. After a moment, he answered. "There are manuscripts of all my novels, and I've published twenty-three. Then there's correspondence, plus copies of my books, in English and other languages." He paused. "Say fifty-four boxes."
That was oddly precise, I thought. Had he already boxed everything? He would never imagine the university would turn down his gift.
"And you would continue to add to it," I said, doing some mental calculations.
"Sure," Godfrey said. "I'll be writing for a long time to come, knock on wood." He rapped my desk with his knuckles.
I found a pad and pencil and made some rough calculations. I named a figure, and Godfrey didn't blink.
"Sounds good," he replied. "I'll double it, just to be safe. That should take care of things for a few years, right?"
"Yes," I said. Hearing the voice of my boss in my head, I added something, though I didn't like doing it. "And of course you might want to put a bequest in your will, too. It never hurts."
Godfrey laughed. "You have to say that, don't you?"
"Yes," I said, trying to suppress a sour expression.
"Don't worry, I'm used to it. People are holding their hands out for money all the time." He grinned. "I'll call my lawyer this afternoon and take care of it."
"You'll need to talk to some of the administrative people tonight after you make your announcement," I said.
He nodded.
I thought our business was done, but Godfrey didn't move from the chair.
I waited a moment.
"You're living in Miss Dottie's house, huh?" Godfrey said.
"Yes."
"Are you taking in student boarders like she did?" He stared past me at the window where Diesel still slept.
"Yes," I said. "It's what she wanted, and it's not so bad having someone in the house, now that my own two children are grown and out of the nest."
"You have two kids?" Godfrey glanced at me, an odd look on his face.
"A son and a daughter," I replied. "Sean is twenty-seven, and Laura is twenty-three."
"That's nice," Godfrey said, his voice soft. "Having kids, I mean."
Maybe I had accomplished something Godfrey hadn't. As far as I knew, he didn't have any children. I was lucky, even if I wasn't a rich writer.
Godfrey shifted in his chair. "What are the boarders like, the ones living there now?"
"Both nice young men," I said, puzzled by the conversation. Why was he asking about my current boarders?
"One of them is named Justin, right?" Godfrey examined his hands with care.
"Yes, there is a Justin boarding with me. The other one, Matt, is actually spending a semester in Madrid, doing research for his dissertation." I was getting more and more uneasy. "Look, Godfrey, what's going on here? Why these questions? Do you know Justin?"
"No, I don't," Godfrey replied. "But I'd like to." He paused for a deep breath. Then he faced me. "He's my son, Charlie, but he doesn't know it."
**THREE**
"You're Justin's father?" I stared at Godfrey, feeling as if this was a bizarre joke. Back in high school he had a reputation for outlandish pranks.
Godfrey nodded, and I was sure he was serious.
But why the heck was he telling me this? Simply because Justin boarded with me?
"This is incredible." A fatuous reaction, but I had to say something.
"Yeah, it is," Godfrey said. Looking down at his hands, he continued. "I had no idea until about six months ago that I had a son. I can't believe Julia never told me." His voice had an odd note in it.
"Julia Wardlaw?" I sounded like a not-very-bright parrot, I decided.
Godfrey glanced up at me. "Yes, surely you remember her from high school. Julia Peterson. God, she was beautiful." He smiled.
Julia _had_ been a knockout thirty years ago. I saw her on a weekly basis now, when she came on Fridays to pick up Justin and take him home for the weekend. Sadly, the years had not been kind. "Have you seen her lately?" I said.
"No, but I've talked to her," Godfrey said. "She wrote to me through my website. Told me about Justin, and I about fell through the floor."
"I can imagine." Knowing this helped me put a few things together. When Julia brought Justin to my house, helping him move in his things, she told me more about her family. Obviously as reluctant to tell me as I was to hear it, she seemed to feel it her duty anyway. Justin and his father, Ezra, argued over Justin's choice of schools. Ezra Wardlaw wanted Justin to attend a small Bible college and follow him into the ministry. Justin rebelled, supported by Julia. He was their only child, and the betrayal—that was the very word Julia had used, quoting her husband—had hit Ezra hard.
"This is really none of my business," I said, "but are you sure Justin is your son?"
"Absolutely." Godfrey looked at me like I was an idiot. "You don't think I'd take someone's word for it? But I knew it was a possibility. In my position, I have to be sure, so I insisted on a DNA test."
"Naturally," I said, my tone wry. "It's still none of my business, but what do you plan to do?"
"I want to meet Justin," Godfrey said. "Talk to him, explain the situation. Now that I know, I want to be part of his life."
Perhaps Justin already knew about his famous biological father, I thought. Julia could have told him recently, knowing that Godfrey was coming to Athena. It had been announced in the local paper a couple of weeks ago.
If Justin knew, that might explain his behavior the past few days. News that his father wasn't Ezra Wardlaw, but Godfrey Priest, would have come as a huge shock. Poor kid, I thought.
"What is it?" Godfrey stared at me.
"Thinking about Justin, that's all." I was not going to share my thoughts on this with Godfrey. Besides, I was only speculating.
"You like him? Think he's a good kid?" Godfrey sounded so eager, I felt sorry for him. But I was more concerned with Justin's reactions to all this. Would he be able to cope with another father in his life?
"Yes, I do. He's a nice, intelligent young man." Behind me, Diesel added his opinion, emitting a few trills and chirps. He knew whom we were discussing. "He's a son you can be proud to acknowledge."
"Thank you, Charlie," Godfrey said. "You have no idea what that means to me." He sounded pathetically grateful, and I sympathized.
"When will you talk to him?"
"I'm supposed to meet Julia for lunch," Godfrey said. He checked his watch. "If she can get away from Misery, that is. I still can't believe she married that guy."
"Misery" was an old nickname for Ezra Wardlaw. He was several years older than Godfrey, Julia, and I, and by the time we were in high school, Ezra already had a reputation as a fire-and-brimstone Evangelical preacher.
"I was in Texas by the time they got married," I said. "The last time we had heard from her, she was dating Rick Tackett and it sounded pretty serious." I had forgotten about that until now. Funny how things popped back into the memory sometimes.
"Rick Tackett?" Godfrey's tone was sharp. "How do you know that?"
"We came to spend Christmas with Aunt Dottie around then, and we ran into Julia and Rick somewhere. At the grocery store, I think." I paused. "You know him?"
"Yeah, I know him," Godfrey said, but it didn't sound like he cared much for him.
"Anyway, that's why I was so surprised when I heard she married Ezra."
Godfrey stared down at his hands. "Guess that's my fault. I spent a few months here, about nineteen years ago, doing research for one of my early books. We spent some time together. I'd divorced my first wife, and Julia seemed to be on her own."
"Seemed to be?"
"She was attending Ezra's church, and they had been dating. But while I was here she didn't see much of him." He squirmed in the chair, his eyes still cast downward. "I had no idea I'd gotten her pregnant. I finished my research and went back to California."
I'd bet there was more to the story than Godfrey told me. I got the feeling he lied about not knowing about Julia's pregnancy.
"You left, and she married Ezra Wardlaw." I watched him, wondering if he would look at me. "And she never got in touch with you to let you know she about Justin?"
Godfrey shifted in his chair again. "No, she didn't."
This behavior convinced me he was lying, but I wouldn't call him on it. What would be the point?
"Why are you telling me all this?" I said.
Examining his hands again, Godfrey said, "You're the only old friend I have in Athena—besides Julia, of course—and my son is living in your house. I thought you should know, since I'll be in town for a while, hoping I can get to know my son."
His only "old friend"? I almost laughed at that. Considering our history in high school and college, Godfrey had to be pretty desperate to call me his friend. After the way he treated me, I shouldn't even be talking to him.
Even though he was still a jerk, I couldn't, in good conscience, turn my back on him. As a father, I could sympathize with his situation.
"I'll do what I can," I told him.
His face brightened.
"But you need to keep in mind that Justin is in his first semester of college. He's facing a lot of stress as it is, and you need to be careful about adding more."
"I understand," Godfrey said. "I just want to be a part of his life, if he'll let me." He leaned back in his chair. "But I'd really like to take him to California with me for a while. We could get to know each other, he could have a little fun—which we both know he hasn't had with Misery in the picture. Maybe take off for Europe, if he'd like that."
Protesting would do no good, I realized. Julia would be horrified if she heard this, because I knew she wouldn't want Justin that far away from her. But they would have to work this out themselves. It was none of my business.
Diesel jumped down from the window and walked over to Godfrey's chair. He sat up on his hind legs and stretched out one of his front legs, resting his paw on Godfrey's knee. Godfrey stared down at Diesel in surprise.
"He does that sometimes," I said. "He seems to sense people's feelings, and he tries to comfort them."
"Thank you, buddy," Godfrey said, his voice soft as he touched Diesel's paw lightly with his hand.
Diesel muttered, withdrew his paw, and went back to his window.
Godfrey shook his head. "I've got to put him in a book. That was freaky."
"Diesel is a special cat." I smiled and reached for a piece of scrap paper. Picking up a pen, I jotted down my telephone number. "Call me and let me know when you'll be coming by the house, okay?"
Godfrey accepted the paper. "Thanks, Charlie. I've got another appointment before I meet Julia for lunch." He stood.
I stood too and accepted the hand he proffered. "Good luck. I hope things work out for you with Justin."
Godfrey thanked me again, and I watched as he strode out of the room.
"What a mess," I said.
I didn't realize I had spoken the words until I felt Diesel's paw on my shoulder. When I turned to face him, Diesel gazed at me, his head cocked a bit to the right. It was uncanny, the things this cat seemed to understand. "Yes, it's a messy situation. Poor Justin." I shook my head. "Poor Godfrey, poor Julia, and even poor Ezra."
Diesel trilled a couple of times.
"Somehow I think we'll be in the middle of it, too. With Justin in our house, there's no way to avoid it."
Diesel chirped.
"We'll support Justin any way we can." I spoke with more assurance than I felt. "If he wants our help, that is." I couldn't believe I was saying such things. What had happened to my resolve to steer away from emotional complications?
I stared at my cat as if he could answer that for me. Diesel blinked slowly before settling back down to nap.
For a few minutes, I thought about what had just occurred in my office. Godfrey started out as I remembered him—cocksure, swaggering, self-involved. His manner changed, though, his self-assurance seemingly gone when he told me about Justin. Perhaps having a son humbled him.
But there was more to it. The way he squirmed when talking about Julia. He lied to me about that. He had been lying to himself for years. He knew all along Julia was pregnant, but for whatever reason he hadn't been willing to acknowledge it. Until now. Why? I wondered. Maybe hitting fifty had done it.
I turned to my computer. All this speculation gave me a headache. I needed to focus on work and forget about distractions.
I managed to keep myself busy until lunchtime, continuing with my cataloging. Around eleven-thirty I put down my pencil, set my computer to standby, and stood.
"Come on, boy." I rubbed Diesel's head. "Let's go home for lunch."
A few minutes later, Diesel and I were heading for home. The temperature had risen a few degrees, but the weather remained pleasantly cool. Being back in northern Mississippi, where there were actually four seasons a year, made a welcome change from Houston with its two seasons. Summer and not-summer, as I liked to call them.
As I inserted my key in the front door, I heard voices inside. Loud voices, full of anger.
I opened the door, Diesel on my heels.
Ezra Wardlaw stood in the living room, shaking a finger at Justin. The boy sat, head bowed, on the sofa. They were so involved in their argument, they paid no attention to Diesel and me.
". . . get your things right now. You're coming home." Ezra's face was so red I thought he might stroke out.
"I'm not leaving." Justin looked up at his father and yelled back at him. "And you're not my father!"
"Don't you dare speak to me like that." Ezra's arm drew back.
I winced as Ezra's hand connected with Justin's face. Justin's head rocked back, and Ezra stepped closer.
I stepped forward, determined to prevent any more violence in my house.
"Stop that." I dropped Diesel's leash, and I heard, rather than saw, Diesel run out of the room. He hated loud voices. "Don't strike that boy again."
Ezra whirled to face me. "This is none of your business. Keep out of it."
Justin rubbed his face gingerly. He looked straight at me. He mouthed the word _please_.
"It _is_ my business. You're in _my_ house." I kept my voice and tone firm. "You will not strike anyone in this house, or I'll call the police. Understood?" I took a step closer to him. I was taller, by about three inches, and I outweighed him by a few pounds, too. If I had to, I'd knock some sense into him.
Ezra glared at me, but his hands stayed by his sides as he turned back to his son. "Get your things. Now."
"I'm eighteen. I don't have to go anywhere with you." Justin stared up at Ezra, resolve in his eyes.
Ezra's chest heaved. He seemed to be struggling for breath.
"You should leave now." I waited, ready to intervene if necessary.
Backing away from his son, but never losing eye contact, Ezra said, "This isn't over. That bastard will rot in hell before he takes you away from me."
**FOUR**
After that statement, Ezra stomped out of the room. Moments later, the door slammed so hard the windows in the living room rattled.
"I hate him." Justin's voice bore such loathing. What had Ezra done to this boy to make Justin despise him so?
"Come with me, son. We need to put some ice on your cheek. It's swelling already."
Justin blinked at me. I think he'd forgotten I was in the room. "Yes, sir." He stood but didn't move forward.
I took him gently by the arm and led him into the kitchen. After his last outburst, he appeared listless, watching me with dulled eyes.
He leaned against the sink while I got ice cubes from the dispenser and wrapped them in a dish towel.
"Here," I said. "Hold this to your cheek. It will feel better."
His cheek was still an angry red. He was going to have a terrific bruise there.
Justin accepted the towel and put it against his face. He winced, but he held the towel in place.
As I watched, concerned, wondering what else I could do for him, he started crying. Quietly, at first. Then harder and louder, the sobs beginning to wrack his body.
Poor kid. This was more than he should have to bear. I put an arm around his shoulder, and he hugged himself to me with his free arm.
I spoke to him, keeping my voice low and soothing, and the sobs diminished.
Feeling a cat rubbing against my leg, I looked down. Diesel had come out of hiding, and now he watched me, wanting to help.
"Justin, look. Diesel's here. He wants to talk to you."
Sniffling, Justin pulled away from me, gazing down at the anxious feline face. He sat down on the floor, towel still against his cheek. Diesel rubbed his head on Justin's chin.
The cat climbed into the boy's lap, his rumbling purr loud in the room. Head bent, Justin let Diesel lick his uncovered cheek.
Smiling, I left the kitchen, knowing that Justin was in good paws. Diesel could bring him comfort, and Justin needed it.
I used the downstairs bathroom, taking my time washing my hands. I stared at my reflection. For all my talk of minding my own business, I had walked right into a messy situation. How would Julia react when she found out what Ezra had done? She had a fiery temper as a young woman. She might light into Ezra the way he had lit into Justin. What a mess.
Finished washing my hands, I judged it okay to go back to the kitchen.
Justin now sat in a chair, Diesel in his lap. Boy and cat glanced at me. Justin seemed calmer, and Diesel no longer looked anxious. A bruise was forming on the boy's cheek.
"How about some lunch, guys?" I went to the refrigerator. "Diesel has his crunchies if he wants them, but I need something else."
I stared into the fridge, waiting for Justin to respond. He was probably embarrassed, poor kid. He might be eighteen in years, but he was still a boy in so many ways.
"There's still plenty of that ham Azalea baked. I think I'll make some sandwiches." I turned to face the table. "How about you, Justin? I make a pretty good ham sandwich."
Justin's head dipped down for a moment. He rubbed Diesel's head. "That sounds good. But I can make my own."
"Tell you what," I said. "I'll slice the ham, and you can get everything else together. Okay?"
"Yes, sir," Justin said. Diesel jumped down from his lap and padded off in search of his own lunch.
Justin came to the sink and washed his hands, still avoiding looking directly at me.
I set the plated ham on the counter, found a knife, and started carving thick slices. Azalea cooked a mighty fine ham, and my mouth was already watering.
Justin retrieved mustard and mayo from the fridge, along with a jar of Azalea's homemade pickles. He set it all on the table, along with the bread and a big bag of potato chips. Next he found plates and knives, along with napkins, and arranged them.
"Would you get me a can of Diet Coke?" I asked.
Justin went back to the refrigerator, pulling out my Diet Coke and a can of regular for him.
He sat down at the table, waiting for me to finish. I had sliced enough ham for four or five sandwiches, I figured. That should do.
I brought the ham to the table and sat down, cater-corner from Justin. He held out the loaf of bread to me, and I took four slices. "I don't know about you, but I'm betting I can eat at least two sandwiches."
"I'm kinda hungry too." He seemed surprised that he had an appetite. He waited while I helped myself to the mayo and mustard before making his first sandwich.
I poured some chips onto my plate, watching as Justin carefully spread a thick layer of mayo on two slices of wheat bread.
He still wouldn't look at me.
"I want you to know, son," I said, "that you can talk to me, if you want to. I'll help you any way I can, and Diesel will, too."
Justin smiled at that and looked me in the face finally. "Thank you, Mr. Charlie. I appreciate that." He took a bite of his sandwich and winced. When he finished chewing—slowly—he spoke again.
"I'm glad you came home when you did." He paused for a moment. His gaze shifted away. "He would've beat the crap out of me if you hadn't."
My stomach clenched in anger. "Has he beaten you before?"
Justin nodded. "Yes, sir. He doesn't like it when I defy him."
He said it so matter-of-factly that my heart ached for him. "You don't have to put up with that anymore. Don't let him in the house when I'm not here."
"No, sir, I won't." Justin ate some more of his sandwich. He touched his bruised cheek a couple of times. I was sure it was pretty sore.
Trying to appear calm, I was stewing inside. I'm not normally a violent man—far from it—but violence against children makes me furious. My father had been, like Ezra Wardlaw, a devout Evangelical. Stern, demanding, but he never once raised his hand against me. I tried his patience often enough, but his firm and loving discipline taught me what I needed to know. I felt the back of my mother's hairbrush on my bottom a few times, but she never struck hard enough even to bruise me.
Justin cleared his throat. "Um, guess I should explain why I said he isn't my father." He pushed some potato chips around on his plate. "Not my biological father anyway. But Mom is really my mother." He watched my face carefully for a reaction.
Feigning surprise at this point would be ridiculous. Justin deserved the truth.
"I know," I said. "Your biological father came to see me this morning."
"You know him? I suppose you would, you and him both being from Athena." Justin tried to appear nonchalant, but his curiosity was obvious.
"We grew up together. Same class in school and at the college, too."
"That's cool." Justin ate in silence for a couple of minutes.
I could have volunteered information, but I thought it was better to let Justin ask me what he wanted to know. I'd have to be diplomatic, though. I didn't want to tell the boy his biological dad was a jerk, in my opinion.
Finished with my first sandwich, I started on the second one after a sip of my drink. By this time Diesel had come back. He crawled into the chair opposite mine and sat, looking back and forth between Justin and me.
"It's so funny how he does that." Justin laughed. "Do you ever let him eat at the table?"
"No, because he doesn't get people food very often." I arched an eyebrow at my boarder. "Remember?"
Justin nodded, a guilty expression flashing across his face. "Yes, sir, I promise I won't do it again unless you say I can."
"Thank you."
Diesel trilled a few times.
"Yes, we're talking about you," I said. "And don't think you can con any ham or potato chips out of me or Justin."
If cats could frown, I'd swear Diesel frowned at me then.
Justin snickered. After drinking some of his Coke, he set the can down and looked at me. "What's he like? Godfrey Priest, I mean. I've, like, seen him on TV, and I even read some of his books. But I don't know much about him."
Definitely the time for tact. "We always knew Godfrey would do something big." I sat back in my chair and regarded Justin. "Even as a boy, he made plans. Talked about traveling all over the world. At first he was going to be a reporter, and by the time he was a teenager, he decided he was going to be a famous writer."
"That's pretty amazing," Justin said. His eyes glowed with the beginnings of hero worship. Godfrey might have a lot to live up to with Justin.
"When Godfrey set his mind to do something, he did it." _No matter what it cost anyone else_ , I added silently. "He always had the drive and the ambition. I don't think anyone who knew him doubted he'd succeed."
"Were you friends?"
"Not really. I was pretty competitive too, and we were always vying for the same honors in school." With a rueful laugh I admitted, "Godfrey usually won. The only thing I ever beat him in was math."
"Yeah, I know what that's like." Justin shook his head. "This girl in my class was always beating me for things. I hated coming in second."
"I did, too," I said. Odd how the memories of those many defeats still rankled on occasion. "But I had plenty of other accomplishments to be proud of. You will, too."
Justin nodded his thanks. I could see he was burning to ask another question but was probably afraid to.
I wanted to set his mind at ease. "He's looking forward to meeting you. I know he wanted to talk to your mother first, but I'm sure he'll come to see you as soon as he can."
"Yeah, I guess," Justin said. "But he's this rich, famous writer, and I'm a hick from a little town in Mississippi."
I suppressed a smile. "He's from this same little town. He knows he has a son now, and that's the only thing that matters. You could be purple with seven eyes, and he wouldn't care."
Justin laughed at that, and Diesel joined in, chirping. The sound of a ringing phone interrupted their merriment.
"Excuse me," Justin said. He stood and pulled a cell phone from the pocket of his jeans. "It's my mom," he said after glancing at the display. "Be right back."
Justin walked out of the kitchen as he answered the call. "Hi, Mom."
That was the last Diesel and I heard. Diesel stared hopefully at the potato chips left on Justin's plate.
"No, siree," I said. I picked up the plate and took it over to the sink. "That's not Diesel food."
I walked back to the table where Diesel sat. As I scratched his head, his rumbling purr started.
"Mr. Charlie."
Justin stood in the kitchen doorway, a stricken look on his face.
"What's wrong?"
"My father—Ezra, I mean—is in the hospital. He got in a fight, and now he's in bad shape." He paused, his body trembling. "Can you take me to the hospital?"
**FIVE**
I hate hospitals. I have spent far too much time in them, first with my parents and then with my wife. As I parked my car in a visitors' lot at Athena Regional Medical Center, I remembered the last time I was here—when Aunt Dottie succumbed to pancreatic cancer. I was at her side, trying to see not her ravaged face and body, but the happy, healthy woman I adored.
Beside me, Justin unbuckled his seat belt, the sound breaking into my reverie. "I don't like hospitals," he said. "But I guess I have to go in." He made no move to open his door, but he touched his bruised cheek a couple of times.
"I don't like them either," I said. "But your mother wants you to be here. She needs your support." I opened my door. "Come on, let's go in."
Justin sighed heavily, but he did as I instructed.
He followed me, lagging a little behind, to the emergency room entrance. I couldn't blame him for not wanting to go inside, not after what Ezra had done to him earlier.
I had no idea how serious Ezra's condition might be. "Bad shape" could mean any number of things. Julia hadn't given her son any details, but I doubted Ezra was in critical condition.
And if Ezra had been fighting, who was his opponent? The logical answer was Godfrey Priest. Was he injured as well?
Inside the ER, we paused at the desk to inquire about Ezra, but before I finished speaking, Julia appeared beside us.
She was better dressed than I remembered seeing her for a long time. Her usual shapeless cotton or polyester frock was gone, replaced by a serviceable black dress. Probably the one she wore to funerals, I decided. It gave her a certain dignity, offering a sharp contrast to her gray hair, pulled into a severe bun at the nape of her neck.
"Thank you for bringing him, Charlie," Julia said. She touched Justin's arm. Then she gasped as she saw the bruise on his cheek. She touched it gingerly, and Justin shied away. "Sweetie, what happened?"
"He hit me." Justin glared at his mother.
Julia whirled to face me. "What on earth do you mean, striking him like that?" By the fire in her eyes I could see she was about ready to strike out at me.
"Not me," I said, holding up a hand. "Calm down, Julia."
"Then who?" Julia asked, turning back to Justin.
"Ezra." He said the word with such loathing, even Julia flinched. "A little while ago. He came to try to make me go back home with him. But I told him I wouldn't go, Mama. I said he wasn't my father, and he couldn't make me. That's when he hit me."
Julia threw her arms around him and hugged him close. "My poor little lamb. I don't know what's gotten into the man, I swear to the Lord. He was very upset this morning, honey. It's my fault. I should have handled him better."
Justin pulled away from his mother. "I don't want to see him ever again."
"Honey, that's foolish," Julia said. "He _is_ your father, in all the ways that count. Even if he struck you like that. You have to give him the chance to apologize to you. By now he must be very upset with himself for doing it."
Justin had a mulish expression on his face. "I don't have anything to say to him."
"Just do what I tell you." Julia's sharp tone surprised both Justin and me.
"Yes, ma'am."
"Come with me." Julia turned and walked away. Justin, after a quick glance at me, trailed after her. I wasn't sure forcing Justin to talk to Ezra right now was a good idea, but Julia would no doubt have brushed aside any objections I could raise.
I moved to the small waiting room and sat down. Resigning myself to an indeterminate period of twiddling my thumbs, I wished I had brought a book with me. Or Diesel.
Diesel was confused when I told him he had to stay home. He went almost everywhere with me, except to church, and he knew today wasn't Sunday. He sat in the kitchen, watching as Justin and I went out the back door to the garage. I knew he'd still be sitting there when I came home again.
I glanced around me. There were only a couple of other people in the waiting room, an anxious-looking elderly woman and a man who had to be her son. He had the same nose, the same angles to his face. He kept patting his mother's hand, speaking in low tones, but she didn't seem to be hearing him. Who were they here for? I wondered. Her husband, his father? I hoped whoever it was would be all right.
Julia appeared in front of me, blocking out the harsh fluorescent lighting for a moment. I looked up into her face, not surprised to see the weariness and anger there.
"How is Ezra?" I asked as I stood. I motioned Julia to the seat next to me, and she sank into it like a woman twice her age.
"Are you okay?" The stiffness of her movements worried me.
Julia grimaced. "Just getting old, Charlie. And tired."
"You're the same age I am. You're not old." I tried to keep my tone light, but Julia heard the concern in my voice.
"It's not the years, it's the mileage. Isn't that what they say?" The specter of a smile passed across Julia's face. "I'm okay. Tired is all. The past couple of months have been pretty rough."
"Ever since Justin left home."
Julia nodded. "Ezra has been beside himself for months. He loves that boy with all his heart, and Justin defying him the way he has, well, it's about broken his heart." She paused. "But I'm about ready to wring his neck over what he did. He should never have struck Justin like that. He almost cried, though, when he saw the bruise he made."
I forbore commenting on Ezra's behavior at the moment. "Justin has the right to live his own life." I probably should have kept my mouth shut, let Julia talk.
"I know that as well as you do. I had to make a choice when Justin told me he didn't want to be a preacher, and I made it." Julia's angry tone didn't offend me. I was treading on ground where I had no business stepping.
"Do you regret it?" I was prying, but instinct told me Julia needed to talk about all this.
"No, I don't." Julia closed her eyes and leaned back in the hard plastic chair. "You're a parent. Would _you_?"
"No." I waited a moment, but she didn't reply. "How did Ezra wind up in the hospital? Justin mentioned a fight."
Julia turned her head and looked me in the eye. "Godfrey told me he came to see you this morning. Told you everything."
"Yes, he did. I'm sure you'd rather he kept this private."
"He's bound and determined to make this some kind of public spectacle. But if he thinks he's going to take Justin away from me, just because of all his money and his fame, he's going to get a rude awakening." Julia sat up. The loathing in her tone didn't surprise me. Godfrey had that effect on people, at least in my experience.
"If I can do anything to help, you know I will." I wasn't thrilled about being dragged into this mess, but for Justin's sake, I wanted to do what I could.
"Thank you. You always were a good man, even when you were a boy. Did the right thing, and stood by your friends." Julia smiled, some of the tension and anger draining out of her. She sounded fatigued as she continued. "It's such a nightmare. I was sitting in a restaurant having lunch with Godfrey so we could talk about Justin. And then Ezra walked in. He and Godfrey started arguing. I tried to get between them, but I couldn't."
"I doubt you could have done anything to stop Ezra," I said, recalling the scene at my house. "When a man's that angry . . ."
"Ezra has a terrible temper. He has prayed for so long, asking the Lord to help him overcome it. But he never can."
Did he take that terrible temper out on his wife as well as his son? After what I witnessed earlier, I was afraid he did. Could that explain the stiffness in Julia's movements?
Julia perhaps sensed my concern. She looked at me again. "Ezra has never struck me, if that's what you're thinking."
"I'm relieved to hear it. But he struck Justin this morning. Very hard."
Julia's hands clenched, and her breathing grew labored. From the glint in her eye, I figured it was just as well Ezra was already in the hospital or Julia might have put him there herself.
"How badly is Ezra hurt?"
"Not as bad as he's going to be if he strikes Justin again." Julia made an effort to regain control of herself. For a moment I thought she might go flying into Ezra's room. "Sorry, Charlie, this is all so sordid."
"You don't need to worry about that with me. We've known each other too long." I took her right hand and patted it. "Now, tell me what happened to Ezra."
"It was ridiculous, a man his size laying into Godfrey like that. Godfrey hit him twice, once in the nose and once in the eye, and it was all over. I don't think Godfrey did anything except bruise his knuckles a little."
"Did he break Ezra's nose? Or injure the eye badly?" I could picture the scene all too easily. I fought Godfrey a couple of times myself in the folly of adolescence. Godfrey won both times, but thankfully my face didn't suffer lasting harm.
"Ezra's nose is pretty swollen. So is his eye. I don't think there'll be permanent damage, except to his pride. There were several of our church members in the restaurant. Ezra shamed himself in front of them." The grim satisfaction in Julia's voice didn't bode well for Ezra. Any sympathy I could have felt for him evaporated the moment he struck Justin. "What can I do for you?" I said.
"I'll be fine," Julia replied. "I'm sorry that you got dragged into this, but I know you'll help Justin if you can. Just be there if he needs someone to talk to, if you don't mind."
"Of course I don't." I paused for a moment. There was a question I felt I had to ask, but Julia might not want to answer it. "What made you decide to get in touch with Godfrey after all this time and tell him about Justin?"
Julia threw me an odd look, but any answer she might have given me was forestalled by Justin's abrupt entrance into the waiting room.
His stormy face said it all. The time spent with his father had not gone well.
Julia stood and held out her arms. Justin walked into them, and they embraced. I turned away to give them some privacy. The elderly woman and her son were gone now. I got up and moved to the other end of the room. Julia and Justin conversed in low voices.
After a few minutes, while I stared out a window into the parking lot, Julia called my name. I strolled back to her and Justin.
"Thank you for coming," Julia said, one arm still around her son. "I think Justin's ready to go. I really appreciate your bringing him here."
Mother and son both appeared worn to the bone now. The best thing I could do was to get Justin home and let him have some privacy or maybe some time with Diesel. My cat had a tonic effect on people, and Justin needed that now.
"Glad to do it," I said. I took her free hand and held it between both of mine for a moment. Julia smiled, and I released her hand. "If there's anything else I can do, let me know."
Julia nodded. "Put some ice on that bruise if it hurts very much." Justin gave her a quick kiss on the cheek, and I walked out of the waiting room with him a couple of steps behind.
In the car, Justin didn't say anything. After he buckled his seat belt, he leaned back and closed his eyes. I kept silent. If he wanted to talk, I'd listen.
Justin stirred, opened his eyes, and looked out the window. "He apologized for hitting me." He touched his cheek briefly and then let his hand slide back down to the seat.
I left the car in park. "I'm glad to hear that," I said.
"He said he'd never hit me again. He was crying." Justin turned to look at me. "You think he means that?"
"I sure hope so." Faced with alienating his son completely, perhaps Ezra was trying to change his behavior.
"He kept telling me he was my father, that he had taken care of me for eighteen years. Like he wanted me to be grateful, I guess."
"In his way I think he's trying to tell you that he loves you and considers you his son," I said, choosing my words with care. "I don't think he's really looking for gratitude. He doesn't want to lose you, but he might not be able to find the right words to tell you that."
Justin frowned. "He won't listen when I try to tell him things. He just preaches at me and tells me what I ought to do, instead of trying to understand me. I'm not him."
"No, you aren't. But sometimes fathers have a hard time letting their sons be their own men. I think some fathers feel their sons have to be just like them in order to justify their own choices in life. Does that make sense?"
Justin's eyes had grown big. "I never thought about it like that. That's why he wants me to be a preacher too, huh?" He turned to gaze out the window again, his head against the glass.
I had given him enough to think about. I backed the car out of the parking space and headed for home. Justin stayed silent the whole way.
When I turned onto my street, I glanced ahead and swore under my breath. A strange car, a late model Jaguar, sat on the street in front of my house. It could only be Godfrey.
**SIX**
I was tempted to drive right by. Justin needed some time to himself, I thought. But this meeting with Godfrey was inevitable. Maybe it was better to get it over with.
As I passed the car I looked inside. Sure enough, Godfrey waved as I turned into the driveway. I clicked the garage door opener. Justin stirred as I drove inside.
I turned off the car and clicked the opener again. The door came down behind us.
In the dim light provided by two windows high in the wall in front of us, I examined Justin's face. He still bore signs of strain from his time with Ezra.
"That's him in the car out there, isn't it?" Justin unbuckled his seat belt.
"If you're not ready to talk to him, you don't have to."
Justin blinked a couple of times. "No, I want to talk to him." He paused. "But what do I call him?"
"Only what you feel comfortable with. He'll understand if you call him Mr. Priest. You both need to know each other better before you decide anything else." I smiled at him.
Justin nodded. He opened his door and got out.
I followed him into the house, and sure enough, Diesel was waiting near the kitchen door. Justin knelt on the floor beside the cat and rubbed Diesel's head.
"You talk to Diesel for a few minutes," I said. "I'll let Godfrey in. I want to have a word with him first if you don't mind."
"Yes, sir," Justin said. Diesel climbed into his lap and was butting the boy's chin with his head.
For a moment Justin looked much younger than eighteen, and I worried about the burdens piling up on those boyish shoulders.
Godfrey was waiting on the doorstep. I motioned him inside.
"Hi, Charlie. Where have y'all been?" As he stepped past me into the hallway, he showed no signs of his fight with Ezra.
"At the hospital," I said, closing the door behind him. "Julia called and asked Justin to come."
"The hospital?" Godfrey shook his head. "Man, I didn't hit Ezra that hard, did I?"
"They wanted to make sure his nose isn't broken," I said. I led the way into the living room and motioned for Godfrey to sit down in one of the two overstuffed armchairs. I sat in the other, and we regarded each other for a moment.
"Ezra will probably be fine," I said. "Though I don't think Julia's very happy with him at the moment. Or with you."
"Julia." Godfrey leaned back in his chair. "I wouldn't have recognized her, she's changed so much since the last time I saw her." He was frowning.
"We're all fifty years old," I said, my tone deliberately harsh. "You don't look like you did thirty years ago either, you know."
Godfrey scowled at me. "You think I don't know that? I wasn't criticizing Julia, anyway. It was just a bit of a shock."
"Forget about Julia and Ezra for the moment. Let's talk about Justin."
"Where is he? I really want to see him." He turned in his chair, half rising, and looked toward the door.
"He's in the kitchen with Diesel. He'll be here in a minute. I wanted to talk to you first, though." I held up a hand, and Godfrey sat back.
"So talk." Godfrey folded his arms across his chest. "What are you going to lecture me about now?"
"I'm not going to lecture you," I said, wanting to add an epithet or two but restraining myself. "Julia has entrusted Justin to my care, and I simply wanted to tell you to move slowly with him. He's had a rough day so far, and he doesn't need you charging into his life like a bull in a china shop. You need to focus on what Justin needs, and not so much on what you want."
"Yes, Mr. Harris. Thank you for telling me what to do." Godfrey's tone mocked me, but I ignored that.
"I have no reason to expect that you've changed much in thirty years, Mr. Priest," I said just as mockingly. "You never did think much about anyone but yourself. But you have a son now, and that has to change."
Godfrey stared at me. "Lord, I had no idea you despised me so much. What did I ever do to you?"
I almost laughed in his face. The man had a colossal ego. "We don't have enough time to go into that. Just stop and think for a moment about what you did to Julia nineteen years ago. Walking away and leaving her pregnant, knowing she would probably marry Ezra. You have a lot to answer for."
Godfrey's face whitened, and I knew I was right. He had lied about not knowing Julia was pregnant when he left her. To his credit, he didn't try to deny it now.
"I'll go get Justin," I said, rising from my chair. "And you take it easy with him."
Godfrey didn't answer. I left him gazing at the wall.
In the kitchen, Justin and Diesel were still on the floor. Justin's face was buried in Diesel's neck, and Diesel was muttering away. "Are you okay?" I stopped a few paces away from the pair.
Justin looked up at me, his face slightly tearstained. "Yes, sir."
"Why don't you wash your face and hands?" I said. "Do you still want to see Mr. Priest?"
Nodding as he got his feet, Justin went to the sink and splashed his face with water. After patting himself dry with a towel, he washed his hands.
"I'm ready," he said as he turned to me.
I put a hand on his shoulder and kept it there as he preceded me out of the kitchen and to the living room. His steps were slow but steady.
We paused in the doorway of the living room. Godfrey stood, facing us as we came into the room.
Justin stopped several feet away from his biological father, and Godfrey drank in the sight of his son like a man who hasn't had water for weeks.
"Justin, this is Godfrey Priest. Godfrey, this is Justin Wardlaw."
"It's nice to meet you, sir," Justin said. He took a step forward, his hand out, but Godfrey didn't move. Justin faltered.
Godfrey started to speak. He stopped to clear his throat. "It's nice to meet you, too." He finally held out his hand, and Justin stepped forward to take it.
Godfrey shook his son's hand, his eyes still fixed on the boy's face. Now that I saw the two of them together, I spotted certain features they shared. Justin had Julia's coloring and her eyes, but his nose and cheekbones were just like Godfrey's.
Godfrey drew Justin toward the couch, and they both sat down, neither one of them speaking, each simply staring at the other.
"What happened to your face?" Godfrey asked.
I turned and stole away, leaving father and son alone together. I would let Justin explain the bruise.
Back in the kitchen, I picked up the phone and punched in Melba Gilley's number. I had called her earlier, before I took Justin to the hospital, to tell her I might not be back this afternoon. As Diesel rubbed against my legs, I glanced at the clock. It was now almost two-thirty.
"Hey, Melba, it's Charlie." I listened for a moment as I leaned back against the kitchen counter. "I'm not sure. I might be back a little later. Oh, so you've already heard about that?"
I shouldn't have been surprised that news of Ezra's set-to with Godfrey had already hit the Athena grape-vine. And trust Melba to be one of the early grapes on the vine.
"Yes, I do know what it's all about. I'm surprised your informant didn't tell you that, too."
Melba squawked a bit in my ear.
"You'll find out soon enough." I hated the fact that this scandal would be all over town, and all over the college, before long. Justin and Julia deserved some privacy, but thanks to Ezra and Godfrey, they had lost all chance of that.
"I'll tell you more about it when I see you," I said. She might as well have the real story from me instead of who-knew-what wild rumors were flying around.
Diesel had his paw on my thigh now. He chirped at me.
"Gotta go now. I'll talk to you later." I listened a moment longer and then hung up the phone.
"What is it, boy?" Diesel was talking away.
Then I heard the front door close.
Diesel followed me from the kitchen into the hallway. The living room was empty.
"Justin? Where are you?"
There was no answer. I went to the window and looked out in time to see Justin getting in the car with Godfrey.
"Well, they're gone," I said to Diesel. "That's what you were trying to tell me, weren't you?"
Diesel looked up at me as if to say, _Of course_.
"I really wish they hadn't," I said, heading back to the kitchen. "But nothing I can do about it now. Guess we'll go back to work, okay, boy?"
About fifteen minutes later, back in my office, Diesel and I were settled in for the remainder of the afternoon. I planned to work till around six, then we would head home. I needed to change for the big dinner tonight, an occasion I did not anticipate with much joy.
I had hardly sat down in my chair before Melba popped in, eager to get the scoop from me. I gave her a bare outline of the facts, and her jaw dropped a couple of times.
"Poor Julia," she said when I finished. "That Godfrey is a rat bastard, if I do say so myself. Running off and leaving her pregnant like that."
I hadn't needed to spell it out for Melba. Anyone who knew Godfrey in our high school days wouldn't be a bit surprised.
Melba left after a few more comments on Godfrey and his behavior, and I was able to work for a while with no interruptions.
Around four o'clock I realized I was thirsty. I rummaged in my bag, but I had forgotten to bring any bottled water with me. Taking a large plastic mug with me, I headed downstairs to the staff lounge for the filtered water cooler there. Diesel yawned at me, declining to come with me.
The walk down and up the stairs would do me good. I spent so much time hunched over the computer that my back generally ached by the time I got home at night. I hardly ever remembered to get up and stretch the way I should.
I rounded the bottom of the staircase and walked down the short hallway to the back of the house. The room that had once been the study-cum-office of the master of the house had been converted into a congenial space where library employees could eat lunch, have some coffee, and relax.
I hadn't expected to find anyone in the lounge at this time of the afternoon, but Willie Clark sat at one of the tables, frowning down at the legal pad in front of him. He put down his pen as he heard me enter and scowled at me.
Since this was Willie's general greeting to everyone, I took no offense. He, too, had been one of my classmates in high school. He had never been friendly, but that probably wasn't his fault. He was the kid who was always the butt of the joke, the one the football team—Godfrey was captain our senior year—never failed to harass. Even those who, like me, tried to be nice to him didn't get very far. He hadn't changed much as an adult, sad to say.
"How are you, Willie?" I regarded him with a smile as I filled my mug from the cooler.
"Fine," he snapped back at me. For someone who served as the head of the library's reference department, Willie was lacking in people skills. "Trying to work, if people will let me."
As long as I had known him, Willie had been scribbling words on pieces of paper. I presumed he wanted to be a writer, but I never heard that he managed to publish anything.
"Sorry, didn't mean to bother you," I said. I turned to leave, but Willie spoke again, and I turned back.
"Godfrey Priest came to see you," Willie said. "Heard he got into a fight, too." He smirked.
"Yes, he did," I said. "I guess the whole town has heard about it now."
"Too bad Ezra didn't put Godfrey in the hospital," Willie said, his face dark with hatred. "Or in the grave, where he belongs."
**SEVEN**
Willie was so often the target of Godfrey's bizarre practical jokes in high school, it didn't surprise me that he harbored intense feelings against his old nemesis.
But wishing Godfrey dead?
"That's a bit strong," I said, trying to keep a mild tone.
Willie sucked at his prominent front teeth—an irritating habit—as he glared at me. I remembered that Godfrey started calling Willie "Bugs" because of those teeth. The nickname stuck, unfortunately for Willie.
"Godfrey's a colossal jackass, and you know it." Willie slapped a hand down on his legal pad. "He made you look like a fool more than once."
"Yes, he did," I said. "I don't like him either, but that doesn't mean I wish he was dead."
"More fool you, then." The contempt in Willie's voice surprised me. "You don't know everything he's done. No one does. But I do." He stood, pushing his chair back with a violent gesture, grabbed his pad and pen, and stalked out of the room.
The nickname "Bugs" was cruelly apt, because physically Willie was a rabbit-like specimen. Godfrey and I both towered over him, and I knew Willie resented us for that. Godfrey hadn't been content with physical domination, however. He enjoyed tormenting Willie because Willie always reacted. That simply egged Godfrey on.
I wasn't the only one who tried to make Godfrey leave Willie alone, but Godfrey wouldn't—or couldn't—stop.
Having Godfrey in Athena was bringing back too many unpleasant memories from the past, and I had an uneasy feeling more unpleasantness lay ahead, as long as Godfrey stayed around. I wondered briefly what Willie had been talking about when he said "everything he's done." Probably his own list of grievances against Godfrey, and I had no doubt they were legion.
I left the staff lounge and was about to mount the stairs when a voice hailed me. I turned to see Peter Vanderkeller, the library's director, standing in the doorway to his office suite.
"Afternoon, Peter," I said. "Did you want to see me?"
"Yes, please," he said before he turned and disappeared.
I suppressed a sigh of irritation and followed him. Conversations with Peter on occasion lasted an hour or more. Melba rolled her eyes at me as I passed her desk—her signal when our boss was in one of his odd moods.
"Please shut the door behind you," Peter said when I entered his office.
I did as he asked and then advanced toward his desk. Peter stood behind it, hands on hips, so thin he made me think of the old TV character Gumby. If Peter were green, he'd give a fair imitation. I dismissed the foolish notion as Peter gestured to one of the comfortable chairs in front of his desk.
This was my favorite room in the house. Originally, Peter's office and Melba's had been one larger room, the front parlor. The high ceilings with their ornate moldings bore witness to the era in which the house was built. A magnificent mahogany dining table served as Peter's desk, though he used a contemporary office chair with it. I envied Peter that table. The machines of modern technology—computer, printer, and telephone—looked sadly out of place. If I closed my eyes for a moment, I could easily conjure up the figure of a woman in a hoop skirt, her beau paying court.
"What can I do for you?" I sipped at my water while I waited for a response.
Peter removed his horn-rimmed glasses and twirled them idly by one earpiece. He blinked at me. "It has come to my attention that our eminent alumnus and hometown boy wishes to endow our institution's archive with his papers, accompanied by a considerable sum of money. It has also come to my attention that he has discussed this matter with you."
"Yes, on both counts," I said. Listening to Peter made me want to be as terse in response as a character in a Dashiell Hammett novel. "I should have told you about it right after Godfrey spoke to me. But I guess I just got busy and didn't think about it."
"That is quite okay." Peter waved my apology away. "No doubt the man believes he has bestowed an honor of great magnitude on his alma mater." His mouth twisted in a grimace. "If it were in my power to do so, I would tell Mr. Priest we don't wish to house the work of a man who has prostituted himself to the bestseller lists."
I had no idea Peter held such a low opinion of Godfrey and his work. I had never considered Peter a literary snob, either. He read fiction widely and counted several Mississippi mystery writers, like Carolyn Haines and Charlaine Harris, among his favorites. They had both spoken at Athena College, and Peter had been beside himself with excitement during their visits.
Why did he have such disgust for Godfrey Priest, then?
"I don't think the president would be very happy if you did such a thing," I said.
"No, he wouldn't," Peter replied. "More's the pity. Athena College has always prided itself on its rich literary heritage." He smiled sadly. "And now, having to add the work of a hack to our archives is a sad comedown and a none-too-subtle comment on the priorities of our current administration."
"It's not so bad. We also have the complete works of that nutty doctor from the nineteen fifties who fancied himself the next Walt Whitman." One hundred and twenty-three privately bound, handwritten volumes of poetry so execrable it made rap songs sound like Shakespearean sonnets—but the man had left the college three quarters of a million dollars along with his so-called art.
Peter ignored that. "I should thank you, I suppose, for confirming the awful truth for me. And so I do. I know that I can leave the matter in your more-than-capable hands, Charles."
"You certainly may, Peter," I said. Peter never unbent so far as to call me Charlie. I stood. "If that's all, then?"
Peter nodded. "I suppose I shall see you tonight at this absurd fête the president has planned?"
"Yes, I'll be there."
Nodding again, he turned his attention to the papers on his desk.
I retreated to the door and let myself out, careful to close it softly behind me.
When I turned, I saw Diesel on top of Melba's desk. Woman and cat were enjoying a conversation.
"Diesel got lonely, I guess." Melba glanced at me over the cat's head.
"Diesel, get down off the desk," I said. "You know you're not supposed to be up there."
The cat muttered as he jumped to the floor. Padding to the doorway, he sat down and started to groom himself.
"He was looking for you," Melba said.
"I know. He doesn't like being left alone for long."
"Did Peter talk to you about Godfrey Priest?" Melba leaned back in her chair.
"Yes," I said. "I suppose I should have come and talked to him earlier, right after Godfrey dropped by this morning." Now I felt a bit guilty. Peter should have heard the news from me.
"You know how he hates to think he's always the last to find out something." Melba glanced toward Peter's door. "Like when he found out his wife was having an affair with Godfrey."
"What?" I stepped closer to the desk. "When was this?"
"About ten years ago," Melba said. "Not long before Peter came to Athena, in fact."
"He was at some small college in California before, wasn't he?"
Melba nodded. "Near Los Angeles. And guess who Mrs. Vanderkeller became friends with?"
"Godfrey. How did they meet?"
"Apparently she had these big plans to be a fancy Hollywood screenwriter." Melba's nose wrinkled in disgust. "The way I heard it, she was always dragging Peter to any party she could get herself invited to. She was supposed to be real attractive, and she met Godfrey at one of the parties."
"She left Peter for Godfrey?" This was beginning to sound like the story line of a soap opera.
"She did. She divorced Peter and married Godfrey. His second wife, I think." Melba thought a moment. "Yes, his second. His first wife was some C-list actress who actually made porn films, from what I heard." The scandalized look on her face was priceless.
"How did Peter end up here, of all places? Didn't he know this was Godfrey's hometown?"
"No, poor man, just his luck." Melba glanced toward Peter's door again. "I guess he wanted to get as far away from California as he could, but he had no idea until after he got here that Godfrey was from Athena."
The man was jinxed. I felt sorry for Peter. No wonder he had such a venomous attitude toward Godfrey.
"Is there anyone Godfrey _hasn't_ pissed off?" I gave Melba a rueful smile.
"I'm beginning to think not." A buzzer sounded. Melba looked cross. "I'd better find out what he wants. I'll see you tonight."
"We'll be there," I said, heading for the door. "Come on, Diesel. Let's finish upstairs and go home."
Diesel bounded up ahead of me. He knew the word _home_.
I finished cataloging a couple more items, and when I remembered to look at my watch, I was surprised to see that it read 5:37. "Definitely time to go," I said.
Diesel was ready, practically pulling me down the stairs once I locked the door to the archive behind us.
Back home again, I freed Diesel from his harness, and off he went to find crunchies and water. I headed to my bedroom on the second floor for a quick shower. I paused on the landing to listen for sounds of habitation on the third floor, where Justin's room was.
"Justin? Are you there?"
I waited a moment and called again. There was no answer, only silence. I supposed he could still be with Godfrey, but Godfrey was due at his reception at seven. I went up the stairs to Justin's door and tapped lightly. I called his name, but there was no response.
I listened for a moment longer and then tried his door. It was unlocked. Normally I wouldn't have done it, because my boarders deserved their privacy. I couldn't shake the feeling that Justin could be in trouble.
The room was empty, the bed unmade.
I shut the door and walked slowly back down the stairs to my room, telling myself not to worry. There was surely some innocuous reason for Justin's absence.
Coming out of the bathroom twenty minutes later, still toweling my hair, I spotted Diesel in the middle of my bed. His head was cocked to the side, as if he were asking me a question.
"Yes, I'm going out, and yes, you're going with me." I frowned. I talked to Diesel a lot, I realized. Some people might find that odd, but it didn't really matter. So I talked to my cat.
I was about to tie my tie when the phone rang. It might be Sean, my son. He sometimes called around this hour.
"Hello."
Melba's excited voice boomed out at me. "You won't believe this. The party's canceled."
**EIGHT**
"Canceled?" I was stunned. "But why?"
"I heard Godfrey called the president's office about half an hour ago and told them he was too ill to come. Peter just called me, and I called you right away."
"Thanks for letting me know. But Godfrey seemed perfectly fine earlier."
"It sure is strange," Melba said, and I agreed.
"Look, Melba, I've got to go. Diesel's demanding his dinner." I knew that was the easiest way to get her off the phone. "See you tomorrow. Bye."
I hung up and looked down at Diesel, lying half-asleep on my bed. He turned his head to look up at me, then blinked and yawned.
What had happened to Godfrey?
He reveled in attention, so there had to be a serious reason he'd canceled on a party in his honor.
Did it involve Justin in some way?
On that thought, I left my bedroom and proceeded back upstairs to the boy's room. I might as well check in case he had come home while I was in the shower. Diesel passed me and scooted up the stairs well ahead of me. He was sitting in front of Justin's door when I reached the third floor.
I knocked, but there was no answer. I opened the door, and there was still no sign of Justin.
"Come on, boy," I said to Diesel as I shut the door. "He's not here."
Diesel bounded down the stairs before me. I headed for the kitchen, thinking vaguely about having something to eat. But by the time I reached the kitchen, my uneasiness over Justin's absence had become more urgent.
I couldn't help feeling that something was wrong. Why had Godfrey skipped out on his own party? Where was Justin?
Even if it turned out to be a waste of time and there was some simple explanation for all this, I decided not to sit home and wait.
_So much for my not getting emotionally involved in this mess._ But my paternal instincts were kicking in, I guess.
Diesel followed me to the back door, but I told him he couldn't come with me this time. He assumed a long-suffering expression, as if I were always abandoning him.
"I won't be gone too long," I said, adding, "I hope," under my breath.
Farrington House, Athena's ritziest hotel, was my destination. Godfrey had to be staying there, probably in their best suite. The hotel occupied half a block on the town square, about ten minutes' drive from my house. It was dark outside now, and I switched on my headlights as I backed the car out of the garage.
I hoped Justin was safe with Godfrey, probably in the hotel still talking and getting to know his biological father.
There were no empty parking spaces in front of the hotel. I had to settle for one across the street, facing the square. As I turned off my lights, preparing to get out of the car, I observed someone sitting in the shadows of an old gazebo about ten yards in front of me.
He moved, and I recognized him as Justin. He watched me, obviously nervous, as I approached. The night was cool, and Justin was in short sleeves. He shivered a bit, his face turned away from me as I sat down next to him. The stone bench chilled me even through the wool of my pants.
"What are you doing sitting out here?" I asked in a mild tone. "Aren't you getting cold?"
"A little, I guess." Justin's teeth chattered. "I, um, I can't go back in there."
"The hotel? Why not?"
"I just can't."
The desperation in Justin's voice alarmed me. I put a hand on his shoulder to try to calm him a bit.
"What's wrong? You can tell me." I spoke in my most benign, fatherly manner.
"No, it's horrible." The boy shuddered, whether from the cold or something else, I wasn't sure.
"Is something wrong with Godfrey?" I tried to keep my voice even.
Justin nodded. He still wouldn't look at me.
"Does he need a doctor?" I stood. "We'd better go and check on him then."
"It won't do any good," Justin said, his voice barely above a whisper.
"Why not?"
"It just won't."
I was getting a really bad feeling about this. "Did you call nine-one-one?"
Justin shook his head. He still seemed too dazed to do anything.
I pulled out my phone and punched in the emergency number. I reported a possibly injured man.
"What's the room number?" I asked Justin.
He shrugged. "The Lee Room, I think."
I gave the operator the information and hung up before she tried to keep me on the line.
"Come on." I grasped the boy gently by the arm and pulled him up. "Maybe there's something we can do." I was afraid Godfrey was beyond help at this point, but I had to try.
I expected resistance, but for whatever reason Justin came docilely enough. Though I peppered him with questions on the way, he wasn't saying anything more.
In the bright light of the lobby, I could see that Justin was pale and in obvious distress. I felt an even stronger sense of urgency. Had Godfrey had a heart attack?
"He gave me a key," Justin said when I veered toward the front desk.
"Fine." I headed for the elevator, my hand still firmly on Justin's arm. Inside, Justin punched the button for the fourth floor.
There were five large suites on this, the top floor. All were named for Confederate generals. Justin headed for the grandest of the five, the Robert E. Lee (of course), and paused in front of the door.
I took the card from Justin and inserted it in the lock. I opened the door and stepped aside. If anything, Justin's face was paler now than it had been before. He leaned against the wall.
I caught a whiff of mingled scents through the open door, and my sense of unease grew. After I took two steps inside, I knew by some instinct that this was a crime scene. I had to move with care and not disturb anything. But I still had to check on Godfrey.
Overhead lights illuminated the room, the reception area of the suite. The smells were stronger now, and I approached one of the two couches with dread, certain of what I would find.
Three steps more and I could look down over the back of the sofa.
Godfrey sprawled prostrate on the floor. The back of his head was a bloody mess.
The coppery tang of blood had mingled with the unpleasant scent of Godfrey's bowels, loosened in death.
I heard a faint roaring in my head. I couldn't tear my gaze away from Godfrey's corpse. He had to be dead with a wound like that.
But in the faint hope that he was still alive, I steeled myself to walk around the couch and approach Godfrey's body. I bent down and grasped his left wrist, lifting the arm just enough to get my fingers in the right spot. I detected no pulse, though I held the wrist for what seemed an eternity.
As I put the arm gently back against the carpet, I caught a glimpse of something sticking out from under Godfrey's waist. My brain didn't register it for a moment. I was going to throw up if I didn't put some distance between me and that horror on the floor.
Out in the corridor again, I drew a deep breath of clear air. I closed my eyes for a moment, but all I could see was Godfrey, his head bashed in.
And Justin's cell phone by the body.
It had to be his. The phone in that room was purple, and Justin had a purple phone.
Justin had his arms wrapped around his body, and he was shivering. He looked at me, fear in his eyes.
What had happened in that room between father and son? Had they argued? Over what? They hadn't met before today.
But as I looked at Justin, I couldn't believe he was responsible. He wasn't a killer, not this miserable, frightened boy.
Even though I had already called 911, I took out my cell phone and punched in the number of the sheriff's department. When the dispatcher came on the line, I gave him my name and a quick report. "I'll be in the lobby, waiting for you."
Justin was crying now, quietly. I was torn. I wanted to comfort him, but I was also tempted to get back into Godfrey's room and retrieve the cell phone. I couldn't believe I was even considering doing something like that, but the last thing Justin needed right now was to be the chief suspect in Godfrey's murder.
The sheriff's department was only three blocks away. They'd be here in five minutes or less. The EMTs should be here any second too, though there wasn't anything they could do for poor Godfrey.
As I wavered, the decision was made for me. The elevator opened, and an elderly couple stepped out.
"Come on, son, let's go downstairs." I put an arm around Justin's shoulders.
The couple cast inquiring looks our way as we passed them, but I ignored them. I had to get Justin downstairs and find us both something hot and sweet to drink—my aunt's favorite cure for any kind of shock.
The elevator seemed to take forever, and the bland music playing in it stretched my already frayed nerves even further. Finally the door opened into the lobby, and I led Justin to the restaurant.
The hostess took one look at my face and the crying teenager with me and asked, "What do you need?"
"Hot coffee, two cups, with a lot of sugar."
I sat Justin down at the closest table, and the hostess returned right away with the coffee. "Here, son, drink this. You need it."
Justin stared at me for a moment, but with trembling hands he picked up the cup and began to drink. The hostess hovered, a worried look on her face.
"Is he going to be okay?" she said.
I nodded. "Just a bit of a shock." I took a drink of my own coffee, feeling the welcome warmth spread through me.
"Okay," she said. "If you need anything else, let me know."
I thanked her, watching Justin as the color slowly came back into his face. I pulled a handkerchief from my pocket and handed it to him. He scrubbed at his face with it, drying away the tears, and then he blew his nose.
"Thank you," he said. He wiped his nose again. "I guess I kind of freaked out when I found him."
"I understand," I said. "I don't blame you. I'm kind of freaked out myself." I took another sip of my coffee. "How long were you sitting out there on the square?"
"I don't know," Justin said. "I'm not really sure." He sipped at his coffee. "Who would kill him? It's crazy."
"I know," I said. "It doesn't make any sense right now, probably never will." I debated whether to tell him about the cell phone. Would it be better for him to know now?
If I told him, though, his reaction later might seem more suspicious. Probably best not to say anything so whatever he told the sheriff's department would be unrehearsed.
Lord, what a mess.
"Do you think my dad, I mean Ezra, killed him? He was so angry earlier." The expression on Justin's face was heartbreaking.
_Lord_ , I prayed, _please don't let it be Ezra. Or Julia_. _I don't think Justin could take it_.
What could I say to the boy now that could possibly comfort him? I had no assurances for him. This would force him to grow up brutally fast.
"I just don't know," I finally said.
Before I could say anything else I spotted an arrival in the lobby. "You don't move from this spot, and ask for more coffee if you need it," I said. "The EMTs and the sheriff's deputies are here, and I need to talk to them."
"Yes, sir," Justin said. "I won't go anywhere."
I had an inspiration. I gave Justin my cell phone. "Call your mother. Ask her to come here as quickly as she can. I don't know how long I'll be, and I don't want you to be by yourself."
Justin nodded. He picked up my phone, examined it for a moment, and then punched in a number.
I left him at the table and braced myself for the coming interview. I had been so concerned for Justin that I hadn't taken time to alert the hotel staff. The manager on duty, I could see now, was reacting badly to the news of a dead body in her hotel. She handed one of the EMTs a key, and they headed for the elevators with a couple of deputies.
As I neared the front desk, a tall, thin black woman in uniform turned to face me. Her expression was enigmatic, to say the least. She had her hair scraped back into a bun, and she regarded me with cold brown eyes.
"Mr. Harris," she said, her voice neutral. "You reported this incident."
"Yes, Deputy Berry, I did." I stopped a couple of feet away from her.
Kanesha Berry and I had a difficult relationship that stemmed from the fact her mother was my housekeeper. Kanesha had tried, once Aunt Dottie passed away, to get her mother to retire. Azalea paid no attention to her daughter. She wasn't ready to stop working, and she told me the day I moved into the house that she was going to look after me and she wasn't about to listen to any arguments.
Since it would take a braver man than I—or a more foolhardy one—to argue with Azalea, I simply smiled and said, "Thank you."
Kanesha couldn't argue with her mother, so she chose to blame me. Every time I encountered her, I felt like I'd run up against a buzz saw.
After glaring at me for a moment, Kanesha summoned another officer. "Deputy Bates," she said, her voice taut, "Mr. Harris here called it in. Go with him and take a preliminary statement."
"Yes, ma'am." Bates gazed into the distance, not at Kanesha, when he spoke. His tone verged on insubordinate.
Kanesha's eyes narrowed for a moment. She did not reply. She turned and walked in the direction of the elevator.
"Come with me, sir," Bates said. "We'll be using the manager's office." I had seen Bates around town, but I didn't know him. He appeared to be about Kanesha's age, mid-thirties.
Praying that Julia could get here quickly, I followed Bates around the desk. The manager came with us into her office and flapped about for a moment, still obviously unsettled. Bates calmed her down and asked her to step outside.
When we were alone, Bates sat behind the desk and motioned for me to sit across from him.
I sank into the chair, my stomach churning. Images of Godfrey, dead on the floor, flashed through my mind. Lord, I needed something to settle me down. A nice shot of brandy would do the trick, but I doubted Bates would let me ask for one.
Bates asked me my name, address, and so on. Then he got down to the meat of the interview.
After a couple of false starts, I was able to give an organized account of finding the body. I carefully omitted for the moment that I hadn't been alone.
"How'd you get in the room?" Bates asked when I finished.
"I had a key."
"And how'd you come by that?" Bates eyed me with suspicion.
"Godfrey's son gave it to me. Godfrey had given it to him earlier." That much was true.
Bates jotted something into his notebook, then he asked me to go through it all again, starting with why I came to check on Godfrey in the first place.
I explained again about the party having been canceled and my worries that Godfrey was ill, because he would have never, as far as I knew, skipped an opportunity for a lot of attention.
"I tried calling, but there was no answer."
"And you didn't ask somebody here to check on him?" Bates watched me, his face blank.
"No, I didn't think of that," I said. "My house isn't that far away, so I just hopped in the car and came here."
Bates nodded, and I continued telling my story for the second time.
During all this, I continued to worry about Justin. Surely Julia had arrived by now. She'd be upset at the news too, but the most important thing right now was looking after her son.
Bates sat examining his notes, and I ventured a question.
"Is Deputy Berry in charge of this investigation?"
Bates nodded, his expression unreadable. "Acting Chief Deputy Berry," he said. "Chief Deputy Dan Stout is out on medical leave right now."
"I see." And I did. This case could be a big break for Kanesha if she managed to solve it quickly. She was the only African-American woman deputy in the department, and I knew her well enough to understand how ambitious she was.
Justin and I both were probably in for a rough time. Kanesha wouldn't put on kid gloves for us, even though I'd bet her mother would have a few things to say if she pushed us too hard.
The door opened, and Bates rose to his feet. I turned in my chair.
Kanesha stood a couple of paces inside the room. She held up a plastic bag with a cell phone inside. A purple cell phone.
"Mr. Harris, did you lose this?"
**NINE**
My eyes fixated on the cell phone in the plastic bag. "No, that's not mine." The land mines lay ahead, and I had to avoid them.
"Do you know whose it is?" Kanesha lowered the bag but her gaze did not waver from my face.
"I do not." That much was true. I thought it was Justin's, but I didn't know for sure.
"Do you know someone named Justin?"
I did not reply for a moment. Kanesha ought to know very well that I had a boarder named Justin. She lived with her mother, and I couldn't imagine that Azalea hadn't mentioned Justin to her.
"I do," I finally said. "Justin Wardlaw. He boards with me."
"Where is he now?" she asked.
"I don't know." That was the truth. Justin might still be waiting in the hotel restaurant, but it was likely that Julia had arrived and taken him away. "If you want to come by my house tonight or tomorrow, you'll probably find him there."
Kanesha nodded. She moved closer to the desk and set the bag down. Deputy Bates vacated his chair, and Kanesha took his place. She held out a hand, and Bates gave her his notebook. She read through his notes, and I saw her frown a couple of times.
When she finished, she pushed the notebook aside, and Bates retrieved it. Kanesha settled back in the chair and regarded me, her eyelids slightly hooded.
I tried not to squirm in the chair.
"I believe that's all we need for the moment," Kanesha said.
_What the heck?_ I thought. _She can't be serious_.
"I'll have more questions for you and young Mr. Wardlaw later, but I know where you live." The ghost of a smile played across her lips.
It wasn't a benevolent ghost.
"Fine," I said. "I'll be at home tonight." As I rose, I nodded at both deputies. I knew I'd eventually catch hell from Kanesha for not telling her I wasn't alone when I discovered Godfrey's corpse. But I don't think I would have done anything differently. Justin needed time to recover from the shock of Godfrey's violent death before he had to deal with the law. I had bought him some time, though it could cause me trouble.
I made a beeline for the hotel restaurant, but the table where I left Justin was unoccupied. I hurried to my car. When I reached home a few minutes later, an aging Honda was parked near my mailbox.
In the kitchen, Julia and Justin sat at the table. Diesel was ensconced in Justin's lap, and the boy cuddled the cat to his chest. Teapot, cups, spoons, and cream pitcher were neatly arranged on the table between mother and son.
Julia looked up at me, her face troubled. "Oh Charlie, what an unholy mess this is. I'm so sorry you and Justin had to see something like that."
"I'm okay," I said, giving her shoulder a reassuring squeeze. "Justin, how are you?"
"Okay," he said, his voice muffled. He was rubbing his head against Diesel's neck. Diesel's purr rumbled through the room.
I fetched a cup and saucer from the cupboard, and when I sat down Julia poured me some tea. I added cream and sugar and sipped. The warmth of the tea soothed and comforted, as always.
After a few sips, I set the cup down. Neither Julia nor Justin had spoken, perhaps waiting for me to break the silence. I could feel the tension in the room emanating from both mother and son.
"I spoke with the investigators from the sheriff's department," I said. "The person in charge of the investigation is Kanesha Berry."
"Azalea's daughter?" Julia frowned. "I hadn't heard she was promoted, but that's good for her, I guess."
"It's apparently temporary." I shrugged. "The chief deputy is out on medical leave, and Kanesha is the acting chief deputy."
"Why aren't the police in charge?" Justin asked.
"In a big city, they would be," I said. "But here it's the sheriff's department that investigates homicides. Our police department isn't equipped for a major crime."
"Thank the Lord things like this don't happen very often in Athena." Julia cradled her teacup in both hands and gazed down into it.
"I didn't tell Deputy Berry that Justin was with me at the hotel," I said. There was no gentle way of doing this.
"Thank you." Julia smiled, but the lines across her forehead deepened.
"But they're going to know, of course, that Justin spent time with Godfrey in that room today," I said.
"Yes, he did. But other people could have visited him. Obviously, someone else did." Julia's tone was as sharp as her gaze at me.
"Obviously." I looked at Justin. "When was the last time you used your cell phone?"
Justin didn't meet my gaze. Instead, he leaned back in his chair and stuck his hand in his jeans pocket. He pulled out my cell phone and returned it to me without a word.
"Thank you." I put the phone on the table. "Now how about answering my question?" My tone was as cutting as Julia's had been, moments ago. Justin's lack of response annoyed me.
"Don't speak to him that way." Julia glared at me. "He's done nothing wrong."
"I didn't say he had." I returned Julia's fierce look. "But he needs to answer my question. It's important."
Julia leaned back in her chair, arms crossed over her chest. I thought she was going to speak again, but after an intake of breath, she remained silent.
I repeated my question to Justin.
Diesel rubbed his head against Justin's chin, and the boy finally looked at me. "I dunno, sometime this afternoon, I guess."
"Do you have your cell phone now?"
Julia watched me intently.
"No, sir." Justin pushed his dark bangs back from his face. "I lost it."
"Why is this so important?" Julia leaned her elbows on the table, her hands clasped together. "People lose cell phones all the time."
"Because they found Justin's cell phone with Godfrey's body." I watched them both carefully to gauge their reactions.
Julia blinked rapidly, and Justin remained mute.
"Justin, was Godfrey alive when you left him this afternoon?" I made my tone neutral, unthreatening.
"Yes, sir," the boy said.
"But you went back to his room and found him dead. How long was it between the time you left him and came back?"
Justin thought for a moment. "About an hour, I guess." He glanced at his mother, clearly concerned about something. She avoided his gaze.
"Why did you go back?"
"I wanted to talk to him some more." Justin focused on me again. "He didn't answer when I knocked, so I used the key and went in." His voice caught. "I thought maybe he'd gone somewhere, and I was just going to write him a note. But . . ." His voice trailed off, and I thought he might start to cry.
"When he found Godfrey's body," Julia said, her expression bleak, "he got out his phone to call someone. But he panicked."
"And he dropped the phone and didn't retrieve it." I could picture it very easily.
"Yes, sir," Justin said. He had his voice under control. "I was really scared. I got out of the hotel, but I didn't know what to do. So I went across the street and sat there until you came."
"What are we going to do, Charlie?" Julia sounded angry. "Kanesha will probably think Justin killed Godfrey. But he didn't. The idea is totally ridiculous. I'm not going to let anyone treat my son like a vicious killer."
"No, I didn't kill anybody." Justin hugged Diesel to him. He eyed his mother warily, it seemed to me.
"I believe you," I said, putting as much conviction into my voice as I could. "I mean, what motive would you have?"
Justin shifted in his chair, and Diesel protested. He rubbed the cat's head, but then he lifted Diesel and set him on the floor. Diesel chirped before stalking off to the utility room.
"Sorry," Justin looked straight at me. "We did have an argument, and I guess I kind of yelled at him." He paused and turned to his mother.
Julia sighed. "Godfrey wanted Justin to go back to California with him, stay a few months. But he didn't want to go. Godfrey . . . well, you know what he was like when he didn't get his way. As if I'd let him take Justin to California anyway."
"I remember only too well." I remembered Godfrey's rages when he was thwarted. "He could be extremely unpleasant."
"He was," Justin said, his eyes sad. "He kinda calmed down after I yelled back. But I could tell he was still mad. So I left."
"And when Justin went back, Godfrey was dead." The horror in Julia's voice brought the nasty scene back to mind.
I squirmed in my chair. It would be a long time before I could get that vision out of my head.
I sipped at my tea, hoping to settle my stomach a bit. "You'll have to tell all this to Deputy Berry."
Justin looked mutinous, and Julia was none too happy, either.
"I know you don't want to," I said. "I'd like to keep Justin out of this too, but I don't think we can."
"If you don't say anything, they're not going to know Justin was there." Julia looked ready to throw the teapot at me. She intended to protect Justin at all costs, it seemed. She wasn't being rational.
"Show some sense." I had just about had it with both of them. I had tried to buy Justin a little time to recover, but there was no way I was going to lie any further about his presence at the hotel. "They have his cell phone, and they'll probably find his fingerprints all over the place. How are you going to explain that away?"
"I suppose you're right," Julia said after a moment's silence. "Honey, we have to be truthful. They need to find out who really did this, and if we lie to them, it only makes things more difficult."
"Is it okay if I go up to my room now?" Justin stood up. "I'm really tired."
"Of course, honey," Julia said. "Get some rest. But you haven't had any dinner. Are you hungry?"
"I've got something to eat in my room," Justin said. "Can I just go, Mama?"
"Yes, you can." Julia stood and held out her arms. Justin approached her for a hug, but he didn't let it last long. He pulled free from his mother and walked swiftly out of the kitchen. Moments later we heard him clumping up the stairs.
Julia and I sat at the table and watched each other for a moment. She was still unhappy with me, I could tell, but I wasn't particularly happy with her, either. Protecting her son was one thing, but trying to pretend he was never in the hotel was completely absurd.
"How about you?" I said. "Have you had anything to eat? I haven't." As an olive branch, it might do.
"No, I was still at the hospital when Justin called. I got over to the hotel as quickly as I could to get him away from there." Julia relaxed enough to lean back in her chair.
"Is Ezra still in the hospital?" I had forgotten about him until now.
"Yes." Julia glanced away from me.
"Was he that seriously hurt?" I frowned. Something didn't add up here. "Surely Godfrey didn't injure him that badly."
"He didn't," Julia said, her tone flat. "But being in a fight didn't help anything."
"What's wrong with him?"
"He's dying." Julia burst into tears.
**TEN**
"I'm so sorry," I said, knowing how inadequate that was. "I had no idea."
I got up and found a box of tissues for Julia. She pulled a couple from the box and dabbed at her eyes as I sat down again.
"No one knows except Ezra and me and his doctors." Julia's voice was husky from the tears.
"You haven't told Justin yet?"
"No," Julia said. "But I have to, I know. I was putting it off until after he met Godfrey, but now . . ." Her voice trailed away.
"Is it cancer?" I asked. I had noticed, when Ezra was in my house assaulting Justin, that he looked thinner and older than I remembered.
Julia nodded. "Pancreatic cancer."
"I'm so sorry," I said. "My wife had it too."
"I know," Julia said softly.
"Where has he been going for treatment?"
"Memphis," she said. "They wanted to send him to Houston, to that big cancer hospital, but Ezra doesn't want to go."
That big cancer hospital in Houston, M.D. Anderson, had done its best for Jackie, but the cancer had won.
"The survival rate is so small," I said.
"Yes, it is." Julia rubbed her temples as if her head ached.
"There's not much you can do, then," I said.
"No, there isn't." Julia smiled so sadly that I wished I could do something to comfort her. "And miracles seem to be in short supply at the moment."
Before I could respond, I heard the doorbell.
Startled, Julia glanced at me.
"Probably Kanesha," I said, rising.
Julia paled. "I wish I didn't have to talk to her tonight."
"It's best to get it over with. Maybe I can stay with you while you talk to her." I smiled at her before I left the room.
I peered through the peephole in the front door. Kanesha Berry, along with Deputy Bates, stood on the front porch. I couldn't postpone this, no matter how much I wanted to. I opened the door.
"Hello again, Mr. Harris." Kanesha nodded at me. "I have more questions for you, like why you forgot to mention the fact that you weren't alone at the hotel."
"I'll be happy to explain that. Come in, please," I said, standing aside.
"I also want to speak to Justin Wardlaw. Is he here?" Kanesha remained on the doorstep.
"He is, and so is his mother. I think she would like to speak to you first." I motioned for her to enter, and this time she did.
"How long has Mrs. Wardlaw been here?" Kanesha turned to face me after I closed the door.
I thought for a moment. "Perhaps half an hour."
Kanesha grimaced, and I could tell she was not happy about this. She probably thought Julia and I had been cooking up alibis together.
"This way, please," I said. "Mrs. Wardlaw is in the kitchen, if you don't mind talking to her in there."
"Wherever you like." Kanesha and Bates followed me.
Julia was standing by the table when we entered the kitchen. Diesel had disappeared, probably upstairs with Justin. Julia appeared composed, but I knew how anxious she must be.
"Good evening, Mrs. Wardlaw," Kanesha said, halting on the other side of the table from Julia. "This is Deputy Bates."
"Ma'am," Bates said, removing his hat and sticking it under his arm.
"Good evening." Julia nodded at them each in turn. "Is it okay if Mr. Harris stays with me?"
"He might as well." Kanesha's tone was sharp enough to cut through stone. "I'm sure he's already heard everything you have to say."
Julia frowned at that, and I shrugged. The damage was done. Any investigator with a shred of intelligence wouldn't take anything we said at face value anyway.
"Why don't we sit down?" I indicated empty chairs. "Can I get you something to drink?"
Both deputies declined my offer. Julia and I sat first, then the two deputies took seats. Bates pulled his notebook and a pencil from a pocket and prepared to take notes.
"Charlie told me you're in charge of the investigation," Julia said. Her complete attention seemed to be focused on Kanesha.
"That's correct," Kanesha said. "I'm sure you're aware by now that Godfrey Priest was found dead under suspicious circumstances. We are investigating his death, and I have some questions for you and also for your son." She nodded in my direction. "And for Mr. Harris, too."
"I'll be happy to answer your questions," Julia said.
Kanesha regarded Julia with a bland expression. "Mrs. Wardlaw, what was your relationship to the deceased?"
"I've known him most of my life," Julia said. "We were not particularly close, at least in recent years." Her face colored slightly. "But I suppose you could say we were friends."
"I see," Kanesha said. "And your son? What was his relationship to Mr. Priest?"
Did Kanesha already know? The way gossip traveled in Athena, I figured she must.
But why didn't she ask directly?
"Godfrey was Justin's biological father." Julia's cheeks stayed red. "They met for the first time today."
"Is your husband aware of this?" Kanesha was an excellent poker player, I was willing to bet.
"Yes, he is," Julia said.
"How does he feel about it?"
"He's not happy," Julia said, in a tone that indicated she thought it a stupid question. "He has always considered Justin his own son."
"He knew he's not the boy's biological father?" Kanesha was poking at every possible sore spot—and none too gently.
"Yes, he knew. He has always known." Julia's color heightened.
"Where is Mr. Wardlaw?" Kanesha asked.
"In the hospital," Julia said. "Where he has been since about one o'clock this afternoon. I was with him until about thirty minutes ago."
Julia had stated her own alibi and Ezra's very clearly, but Kanesha did not appear interested.
"There was an altercation between your husband and Mr. Priest today." Kanesha appeared to be well informed about the day's events.
"They had words." Julia frowned and crossed her arms over her chest. "My husband struck Godfrey, and Godfrey hit him back. In the face. My husband was bleeding and in pain, so I took him to the emergency room."
"Did you see Mr. Priest again after that?"
Julia hesitated. "No, I did not."
That was the first question Julia hadn't answered right away. Was she lying?
"You're sure about that?" Kanesha had noticed the hesitation, too. She looked ready to pounce.
"I am." This time Julia didn't falter.
"So you have no further knowledge of Mr. Priest's movements after you saw him at lunchtime?" Kanesha leaned back in her chair, relaxing her rigid posture for the first time since the interview began.
"Only what Justin and Charlie have told me." Julia smiled briefly. "But I'm sure you'd rather hear it from them."
"Yes," Kanesha said, "and I'll also need to talk to your husband. How long will he be in the hospital?"
"It's possible he'll be released tomorrow." Julia relaxed her arms, letting them slide into her lap. "But he's ill. You cannot upset him."
Kanesha pulled a business card from one of her shirt pockets. "Please call me at this number tomorrow, and let me know when I can speak with him."
Julia accepted the card and placed it on the table in front of her. "Certainly."
"Thank you. I may have more questions for you later." Kanesha said. "Now I'd like to speak to your son."
"I'll get him, if you like," I said to Julia.
"Thank you, Charlie," she said. "If you don't mind."
"Back in a minute," I said, rising from my chair.
As I left the kitchen I heard no further conversation. Would Kanesha continue to question Julia while I was out of the room? I was concerned about Julia and that one hesitation in answering. I doubted Kanesha would let that go for long.
Nothing I could do about it now, I thought as I climbed the stairs to the third floor.
I knocked on Justin's door and waited for a response. When none was forthcoming, I opened the door and looked inside. Justin, still dressed and with his shoes on, appeared to be sound asleep on the bed. Diesel, stretched out beside him, raised his head and blinked at me.
I entered the room, moving quietly. I found three candy wrappers on the floor by Justin's bed. Some dinner, the poor kid. He'd had a terrible day, and it wasn't finished yet.
Laying a hand on his shoulder, I shook him gently and called his name.
Justin's eyes popped open, and he stared up at me in confusion. "Mr. Charlie? What . . . ?" The memories of the day evidently came back, and he sat up, rubbing his face.
"Sorry to wake you," I said. "But the deputies are here. You need to come downstairs now."
"Yes, sir," Justin said, his voice dull.
Diesel jumped to the floor and rubbed against my legs as I turned to leave.
"Please wait. I need to ask you something." I turned to face him as Justin stood.
Diesel began chirping at me, and I reached down to rub his head. The cat pushed his head against my hand, and I rubbed a little harder. He really loved head rubs.
"What is it, son?" I asked.
"I don't know what to do." Justin ran a hand through his hair.
"About what?"
Diesel walked over to Justin and rubbed against his legs.
"What if I found something in the hotel room?" Justin asked. "Something that could get somebody in trouble?"
**ELEVEN**
"In the hotel room, you mean?" I examined Justin's face. He was clearly worried.
Justin nodded.
"You should tell the deputies, like we discussed earlier." My tone was firm. "It's best to be truthful. What you found could help them solve the case."
"I'm afraid to." Justin looked miserable. "And you're the only person I can talk to about it."
"What did you find?" I asked. Who was he afraid of incriminating? I had an uneasy feeling I knew.
Justin stuck a hand into his pocket and withdrew something. He held his palm out to me, and lying across it was a gold pen.
"It's my dad's."
"Your dad's? You mean Ezra's." I was so shocked by what Justin had, I didn't know quite how to respond.
"Yes, sir," Justin said. "I gave it to him for his birthday last year. I had it engraved." He held the pen closer, pointing to the letters on the shaft with his free hand.
I suddenly recalled Justin's question when we were sitting in the hotel restaurant. He'd asked me if I thought Ezra had killed Godfrey.
Now I knew why. The evidence lay in Justin's hand.
But how could Ezra have left the hospital and made it over to the hotel without anyone knowing he was gone? Julia said she had been with him all day, and surely the hospital staff would have noticed if he had been gone for very long.
"What should I do?" Justin asked as we both stared at the pen.
"You have to show it to the deputies," I said.
"I can't," Justin said. "Even after what he did to me."
I stared at the pen in his hand, torn. Justin's fingerprints were all over the pen now, and had probably obliterated any others. Deputy Berry already knew he was in the room.
I made a snap decision—with my heart, not my head. I wasn't keen on suppressing evidence, but until I knew how that pen ended up in Godfrey's room, I was going to continue to protect Justin—and whichever of his parents left that pen behind.
"Put it back in your pocket, and don't say anything about it to them yet. Let's see how things go with their questions."
Obviously relieved, Justin nodded. He shoved the pen back into his pocket. "Guess I'd better go downstairs."
"Yes." I strode to the door and held it open. "Come on, boys."
Diesel scampered out, and Justin and I followed. The cat had already disappeared by the time Justin and I reached the stairs.
Back in the kitchen we found Deputy Bates regarding Diesel with awe. "That's really a cat?"
"Yes," I said. "He's a Maine coon, and they get to be as big as thirty or thirty-five pounds."
"Dang, he's bigger'n my dog." Bates shook his head.
Kanesha frowned at her subordinate while I exchanged an amused glance with Julia. If nothing else, Bates's reaction to Diesel had relieved some of the tension.
"Deputy Berry, this is my son, Justin Wardlaw." Julia stepped forward, stretching an arm around the boy's waist and pulling him close.
Kanesha introduced herself and Bates to Justin, then motioned for him to have a seat. "I'd like to speak to Justin alone," Kanesha said.
"No." Julia shook her head. "No, I need to be with him."
Kanesha frowned. "How old are you, Justin?"
"Eighteen," he said.
Kanesha nodded. "In that case, Mrs. Wardlaw, I have to insist I talk to Justin without you. Justin is an adult."
Julia bridled at that. She looked like a lioness about to attack. "That's ridiculous. If you're not going to let me stay with him, then I don't think he should talk to you."
Kanesha opened her mouth, but I spoke first.
"Julia, Deputy Berry is right. Justin is an adult now, and I think it's best to let her talk to him now. Otherwise, I imagine he might end up doing it down at the sheriff's department. Wouldn't you rather avoid that?"
Kanesha shot me a severe look. She wasn't happy with my interference, but I thought I'd better do something before Julia got really riled up.
"Very well. You're right. I don't want that." Julia let go of Justin after a moment. "I'll be close by if you need me." She smiled at her son.
Justin nodded. He might be eighteen, and therefore an adult in the eyes of the law, but when I looked at him, I saw a tired, frightened boy. I hoped Kanesha would handle him gently. He'd had a very difficult day.
I offered Julia my arm and escorted her across the front hall into the living room. She sank into an armchair and covered her face with her hands.
At the moment I wasn't sure how to comfort her, other than by patting her shoulder a few times.
I pulled a chair close to hers and sat down. "Julia, it's going to be okay." I hoped I sounded convincing.
Julia let her hands fall to her lap. Her face was wet with tears. "Oh Lord, what are we going to do?"
"They won't arrest Justin," I said, to myself adding, _at least not tonight, I hope_.
"Please, Lord, no," Julia said, her voice soft. She leaned back in the chair, looking suddenly older than her fifty years. "Ezra will be so upset when he finds out about all this."
"He'll have to know," I said, and Julia nodded. "And while we're talking about Ezra, I have a question."
Julia regarded me warily.
"You were with him at the hospital all day? Until you left to get Justin from the hotel?" I watched her carefully.
Her gaze dropped away for a moment. "Yes, I was."
"Then there was no way Ezra could have left the hospital?"
Startled, Julia sat upright. "Of course not. Whatever made you ask something like that?" She frowned. "Ezra did not kill Godfrey. You can get that notion out of your head right this minute."
Julia was getting angry with me again, but I wasn't going to back down.
"Okay," I said. "I'll get it out of my head for a moment. But let me ask you another question." I hated doing this, but Julia was obviously—at least to me—lying about her time at the hospital.
"Well, go ahead." Julia glared at me.
"If I asked Ezra whether you were with him all day at the hospital, what would he say?"
Julia's eyes narrowed. "What is that supposed to mean? Of course Ezra would say I was with him."
"Then one of you would be lying." I said it as gently as I could. "It's no use. I know one of you was in Godfrey's room today." I paused for a moment to let the words sink in.
Julia paled, and I knew I was right. I felt no satisfaction from it.
"How?"
"Justin found something belonging to Ezra in the room," I said. "When he went back and found Godfrey dead."
"Oh dear." The fight seemed to have gone out of her—for the moment. "Is Justin going to give it, whatever it is, to the deputies?"
"No, I told him not to, for now," I said. "He's very worried about it, but I wanted to find out who left it there. You haven't told him yet that you went there, have you?"
"No, I haven't." Julia shook her head. "I see now I should have told him."
"You have to tell _me_ the truth too, if I'm going to be any use to Justin," I said.
"I will." Julia's tone was firm.
"The question is," I said, "well, two questions, actually. First, which one of you went to the hotel? And second, when?"
She leaned back in the chair again. "It was me. Ezra never left the hospital. What did I leave there?"
"Ezra's pen, one Justin gave him."
"Of course. How stupid of me." Julia shook her head. "Godfrey wanted to give me a check, but he didn't have a pen. I had Ezra's things in my purse, so I must have pulled out that pen and then left it behind."
"Godfrey was alive when you left?" I hated to ask it, but I had to.
"Yes." Julia's eyes flashed. "Alive and mad as a hornet."
"Why?"
"Because I read him the riot act," Julia said. "I went there expecting to find Justin with him. And he wasn't. When I asked Godfrey where Justin was, he told me they'd argued and why."
"So you lit into him?" I had to suppress a smile. I recalled that Julia had a rather fiery temper as a girl.
"I did," Julia said with a trace of satisfaction in her voice. "I told him he had to think more about what was good for Justin, and not what he wanted. He got the point."
"I'm sure he did," I said. "How long were you there?"
Julia thought for a moment. "No more than fifteen minutes, maybe only ten. I had to get back to the hospital."
"What time was that?"
"I got back a little after three. They were changing shifts."
"So you left Godfrey alive before three?" I was trying to get the chronology straight in my head.
"Yes."
"You didn't try to find Justin before you went back to the hospital?" I asked.
"No. I was concerned about him," Julia said. "But I had to check on Ezra, and I wanted to stop by the bank to deposit Godfrey's check. Justin would have called if he needed me." She turned away for a moment. "Or so I thought."
"You'll have to tell the deputies about this," I said.
"I know. They'll think Justin or I killed him. Or maybe that we did it together." Julia rubbed a hand across her eyes as she faced me again. "Lord, I wish I could hear what's going on in your kitchen right now."
"I know. I wish I could, too," I said. "But there's nothing we can do at the moment except wait."
Julia nodded.
"It's possible they might think you or Justin killed him," I said. "Depends on what they think your motive is. Revenge, maybe."
"Why would I suddenly decide I wanted revenge now?" Julia snorted. "If I'd wanted to kill Godfrey because he got me pregnant and ran out on me, I would have done it years ago."
"Possibly," I said. "But now your husband is terminally ill, Godfrey appears and wants to take your son back to California, and maybe you're so stressed you lose control and strike him down."
Julia blanched. "I hadn't thought of that. It does sound plausible when you put it that way. The Lord knows my stress level is through the roof."
"No wonder," I said in sympathy. "Anybody's would be, with what you're going through with Ezra."
Julia smiled her thanks. "But I didn't kill Godfrey, and neither did my husband nor my son."
"Then we have to look elsewhere." I paused. "How often did Godfrey come back to Athena over the years?"
Julia thought for a moment. "Every few years, probably. A few times he came on a book tour. Other times for research of some kind."
"Once his parents left Athena, did he have that many ties here, other than college?"
Julia didn't appear to have heard me.
"What is it? Have you remembered something?" I leaned forward in my chair.
"Talking about book tours made me think of it," Julia finally said, focusing on me again. "When I was leaving the hotel earlier today, I saw somebody at the front desk with a box of books." She shrugged. "At least, that's what I thought it must be, because I saw the name of Godfrey's latest book on the side of the box."
"Who was it?" A potential new suspect, I hoped. All the better for Justin and Julia.
"That woman who owns the bookstore on the square, Jordan Thompson," Julia said. "And I know for a fact she hated Godfrey with a passion."
**TWELVE**
"I didn't think ministers' wives listened to gossip." I said it teasingly, but Julia didn't take it that way.
"I don't run around gossiping with anyone." Julia's tone was frosty enough to make me wish I was wearing a sweater. "But people tell me things, even when I don't ask them to. Besides, Melba Gilley's niece Patty works there. Has since she got out of high school five years ago. She used to babysit Justin, and whenever I run into her, she always wants to talk."
I nodded. I knew Melba's niece, Patty Simpson. Plus, I knew Melba. If Patty was at all like her aunt, she knew what was going on around her within a ten-mile radius.
"Okay, let's say something happened between Godfrey and Jordan Thompson." I regarded Julia warily. "Something that pissed off Jordan so much she wanted Godfrey dead. How the heck are we supposed to find out what that was? Other than calling up Patty Simpson and asking her, since she seems to know everything."
"I'm not suggesting that." Julia scowled. "Although I have no doubt Patty would be happy to tell you that, and a dozen other things besides." She paused. "I know you go into the bookstore. I've seen you there myself, several times."
"Yes, I do. I go in there at least every couple of weeks." I have always loved bookstores, and though I have plenty of access to books through the two libraries where I work and volunteer, I can't resist the lure of the bookstore.
"Then go by there tomorrow and talk to Jordan," Julia said. "She's fond of older men, from what I've seen. You can probably get her to talk to you."
"Julia, I can't believe you're suggesting such a thing." I pretended to be shocked, but I was more amused than anything. I couldn't see myself in the role of _homme fatal_ , persuading attractive young women to spill their secrets.
She didn't respond. Instead, she turned in her chair and peered in the direction of the kitchen. "What's taking so long? Shouldn't they be done by now?" She started to rise.
"No, I'll go." I motioned for her to stay where she was, and she sat down again. "Kanesha won't like it, I'm sure, but she's probably already so annoyed at me it won't make much difference."
A few feet from the kitchen I could hear the low murmur of voices. Then one rose above the rest—Justin's.
"Yes, I went back, but he was dead. I keep telling you that. Why do you keep asking me?"
The note of near-hysteria in the boy's voice worried me. When I stepped into the kitchen, I could see Diesel in Justin's lap, peering angrily at Kanesha. He looked like he was ready to launch himself over the table at her.
"Diesel, no."
At the sound of my voice, the cat warbled, and I could tell he was upset. But some of the tension left his body, and he sat back against Justin. The signs of exhaustion in Justin's face emboldened me.
Kanesha stood up and faced me. "I'd appreciate it if you'd remove that cat from the room."
I didn't care for the way she said _that cat_. "It's his house too, and if he wants to be in this room, he can. What are you doing that's upsetting him?"
The surprise in the deputy's face pleased me. Obviously she hadn't expected me to talk back to her. I pressed my advantage without allowing her to answer.
"I think you've had enough time now to ask Justin your questions," I said. "He's had a long and very upsetting day. Unless you're going to charge him with something, I think this interview should be over."
Over Kanesha's shoulder, I caught sight of a smirking Bates. That wasn't good. Kanesha might take it out on Justin because she knew she had to prove herself in front of her good ol' boy of a subordinate.
"I am conducting an investigation into what looks like homicide, Mr. Harris." Kanesha enunciated each word so carefully, I could tell she was furious. "I will conduct the investigation as I see fit, and that means questioning anyone with any connection to the victim." The intensity of her gaze made me want to take a step back. "Do you understand that?"
"I do." A smarter man would have tucked tail and run. She was one pissed-off deputy, but another look at Justin's face was all I needed to make me stand my ground. "My point is, you've questioned Justin and his mother. You've pushed your luck far enough as it is, since Justin hasn't had time to talk to a lawyer. They're both very upset about what happened, and if you have a humane bone in your body, you'll give them time to recover. They haven't even had their dinner yet, and neither have I. You can continue this tomorrow."
Bates stood and moved close to Kanesha. He appeared ready to step between Kanesha and me.
I must have looked more threatening than I realized, because now both deputies were glaring at me. I took a step back, my hands up to show that I meant no harm.
Kanesha jerked her head once, and Bates moved away.
"I have more questions for you, Mr. Harris." Kanesha folded her arms over her chest. "But they can wait until tomorrow. I'm sure I'll have more questions for Justin and Mrs. Wardlaw as well. Have a good evening."
She stalked past me, Bates behind her. He gave me a cocky grin as he went.
Moments later I heard the front door open and close, and Julia appeared in the kitchen right after. She took one look at Justin, then hurried to his side. Diesel jumped down from the boy's lap and came to rub against my legs.
"Honey, how are you? Did they mistreat you?" Julia examined Justin, her fingers trembling as she touched his face.
"No, I'm okay, Mama." Justin leaned against her, his head at her waist. Julia stroked his hair. "It was pretty intense. She kept asking me the same questions over and over."
I went to the refrigerator and retrieved a can of Coke for Justin.
"Thank you, sir," he said as he accepted it. "You should've heard Mr. Charlie, Mama. He came in here and told the deputy that she should stop. And she did. But boy was she mad." He popped the top on the can and took a long swig of Coke.
"Thank you." Julia threw me a glance full of gratitude.
"You're welcome," I said. "Now, anyone hungry? How about I order us a pizza?"
Both Julia and Justin shook their heads. "Not for me, thanks," Justin said.
When a teenager turned down pizza, he was obviously worn out.
"Okay," I said. "Why don't you go on up to bed? And if you get hungry in the night, there's plenty of food in the fridge."
"Yes, sir." Justin stood, his shoulders tensed. "Mama, will you come upstairs with me for a few minutes? I need to talk to you about something." He glanced at me, and I nodded. He relaxed.
I knew he wanted to talk to Julia about the pen he found in Godfrey's room. Julia could explain, and then they could decide what to do about it.
"If you need anything, let me know." I watched as they left the room, Julia's arm around her son.
Diesel started after them, but I called him back. "Not now, boy. Justin and his mother need to be alone. You stay with me."
The cat looked at me for a moment, then sat down and started cleaning his left front paw.
I've never had a pet that seemed to understand what I said so well. Sometimes it freaked me out a little. I watched Diesel a little longer, until hunger pangs claimed my attention.
I decided against pizza and settled for scrambled eggs with cheddar cheese and a couple of pieces of toast instead. When my meal was ready, I poured a glass of water and carried it with my plate to the table.
Diesel could smell the eggs and cheese, and approached my chair, chirping. He loved scrambled eggs, and I usually gave him a few bites. If I didn't, I got a heavy paw on my leg as a reminder.
I took my time eating, and when I was done I cleaned up the kitchen. Azalea would be back in the morning, and I didn't want her to find a mess. She would have enough to do with dusting, vacuuming, and laundry without having to clean up in here.
Julia still hadn't come back down by the time I finished. I was tired and ready to climb into bed with a book, but I didn't want to go upstairs without seeing Julia to her car. Aunt Dottie would haunt me if I neglected my duties as a host.
I went up for the book I was reading, planning to take it back downstairs while I waited for Julia. Diesel trotted along with me. I retrieved the book from my bedside table, and Diesel followed as I left my room and walked back to the stairs.
From above me I heard Julia telling Justin good night, and moments later she was coming down the stairs. I moved forward to intercept her as she reached the second-floor landing.
"You must be about ready for bed." Julia paused, her hand on the banister. "What a day this has been."
"Yes, it has." Somehow it seemed three days long, but it was only this morning that Godfrey Priest had appeared in the archive. "I wanted to see you out and ask if there's anything else I can do."
Julia placed a hand on my arm as we walked down the stairs together. Diesel had zipped ahead and disappeared before we were halfway down.
"You're a good friend," Julia said. "And I'm so sorry if I was rude to you earlier. I'm just terrified of what's going to come of this." Her grip tightened on my arm. "I have to keep Justin safe."
"How is he?"
"Very tired and confused, poor lamb." Julia sighed. "Like both of us, I expect. We talked, and I explained about the pen."
"Good." I wanted to ask whether they decided to tell Kanesha about it, but Julia looked exhausted.
We reached the bottom of the stairs, and I turned to face her as her hand dropped from my arm. "I'll keep my eye on him, and I'll keep my ears open, too. Someone else had a powerful motive, and I'm sure the truth will come out. It's just going to take some digging."
"You're a good man, Charlie Harris." Julia surprised me with a peck on the cheek, and I could feel my face redden a bit. "I'll get my purse and be right back."
I waited, hoping my face had lost any vestige of red by the time she returned.
When Julia reappeared, purse clutched in her hand, I moved to open the door for her. I started to follow her down the walk, but she insisted that I not.
"It's not that far to the car, and I'll be fine. I'm going by the hospital to check on Ezra, and then I'll head home and collapse." She smiled before she turned and moved down the walk to the street.
"Good night, then," I called after her. I waited until she pulled her car away from the curb before shutting the door.
I turned off the lights downstairs, watching for Diesel, but there was no sign of him. I found him sprawled across my bed when I got back upstairs.
After putting my book back on the nightstand, I undressed and got ready for bed myself. I was tired, but my brain was buzzing with all kinds of thoughts about the events of the day.
I read for a while, trying hard to concentrate on my book, and eventually I put it aside and turned out the light. Diesel snuggled close to my legs.
Praying that I wouldn't have nightmares about dead bodies all night long, I did my best to fall asleep.
**THIRTEEN**
If I dreamed about corpses, I didn't remember it when I woke the next morning. I came out of a sound sleep to feel a paw gently prodding my nose and then a head butting lightly against my chin.
With Diesel around I had no need of an alarm clock. He got me up most mornings by six-thirty, and today was no exception.
After I came out of the bathroom, wearing my robe over my pajamas, I went down to the kitchen, where I knew Diesel would be waiting. I filled his bowl with fresh water and replenished his food. He began eating his breakfast with enthusiasm.
I hadn't remembered to fill the coffeepot last night and set it so that I would have coffee when I got up. And no wonder. I felt dazed as I recalled the events of the day before.
While I waited for the coffee, I went to the front door to retrieve the paper. Standing on the doorstep, breathing in the fresh, cool air, I began to feel more awake. I scanned the front page, but there was no mention of Godfrey's death. Tomorrow's paper would be full of it, I was sure. And there would probably be national news crews all over the place. The mysterious death of a bestselling writer would attract attention across the country.
I was working the crossword and sipping coffee when the back door opened. I looked up to greet Azalea Berry. Today, Wednesday, was one of her three weekdays at my house. She had other clients on Tuesdays and Thursdays.
At nearly six feet tall, Azalea was an imposing figure. She had a regal bearing and she rarely smiled, but she was kind, with a warmth that belied her reserve. She was only about three or four years older than I, but she possessed the poise of a grande dame in her eighties.
"Good morning," I said.
"Good morning, Mr. Charlie," Azalea replied. She closed the door behind her and set her purse and keys on the counter nearby. "It sure is some beautiful morning."
"Yes, it is." I wondered whether she had heard about Godfrey Priest's death. Surely Kanesha had mentioned it to her mother.
"Terrible thing about that poor man." Azalea retrieved her apron from a hook by the back door and put it on.
"It sure was. It seems like a nightmare instead of something real."
"And you finding him that way." Azalea shook her head. "It's a wonder you wasn't up all night."
"It was pretty grisly." I took a sip of my coffee.
"How is Justin this morning?" She shook her head. "That poor child."
"I haven't seen him this morning. He was completely worn out last night."
"Then he's going to need a good breakfast. Build up his strength. You, too." She went to the refrigerator and began pulling out eggs, sausage, and milk. Next she retrieved the flour canister, and I knew she was going to make hotcakes.
My mouth began watering. Azalea made wonderful hotcakes.
Diesel wandered into the kitchen and sat down a few feet away from Azalea.
She regarded him with a gimlet eye, and he stared back unfazed. "I don't need no help from you," Azalea said.
Diesel warbled at her, and Azalea turned her back on him, busying herself with preparing breakfast.
"Diesel, let's go see if Justin is up." I put my coffee cup aside and stood. "Come on, boy."
Diesel was off like a streak. I followed at a much more leisurely pace.
When I reached Justin's room, I found the door open and Justin sitting at his computer with Diesel climbing into his lap. I tapped lightly on the door, and Justin looked up at me.
The worn, frightened look had left his face, and this morning he appeared more his usual self, I was glad to see.
"Good morning." I smiled. "Azalea's downstairs making hotcakes for breakfast."
Justin's face lit up. "I sure am hungry." His head ducked down for a moment. "Uh, about yesterday . . ."
"Yes," I said when he paused.
"Thank you," Justin said, raising his head to look at me. "I'm glad you were there, sir."
"You're welcome." He seemed younger than eighteen right then. He'd had more than one deep shock yesterday, and the Lord only knew how it would all affect him in the long run. "Come on downstairs when you're ready. Breakfast will be on the table soon."
"Yes, sir. I will." Justin rubbed Diesel's head, and the cat chirped happily.
I reached the kitchen in time to answer the phone. The appetizing smells emanating from the stove made my stomach rumble. Justin wasn't the only hungry one.
"Hello."
"Good morning, Mr. Harris. This is Ray Appleby from the _Athena Daily Register_. I'd like to talk to you about the murder of Godfrey Priest."
I glanced at the clock. It was only seven-fifteen.
"You're calling pretty early, Mr. Appleby. I haven't had my breakfast yet." My tone was sharp, but I didn't care.
"I apologize if I woke you," Appleby said. He didn't sound apologetic. "But I really need to talk to you as soon as possible. According to my sources you found the body."
"If you want to call back at a more civilized hour, I _might_ be willing to talk to you. Until then, I have nothing more to say." I hung up the phone.
I turned to find Azalea regarding me, her expression inquisitive.
"Somebody from the paper, wanting to talk to me about yesterday." I sat down at the table.
"That's mighty rude, calling somebody this early." She turned back to the stove. "People just ain't raised right these days."
"It's only going to get worse," I said. I picked up my coffee cup and, seeing that it was empty, got up to refill it.
"I guess he was pretty big news." Azalea expertly flipped a couple of hotcakes as I poured the coffee.
"He was, and there'll probably be news crews from all over the country here." I stirred some sugar substitute into the coffee. "And it looks like your daughter may be center stage, since she's in charge of the investigation."
Azalea made a noise that sounded like _hmmph_.
"It's a big chance for her." I sat down at the table again and drank some coffee.
"That girl wanna be on TV, she should've been an actress." Azalea set a plate with three hotcakes and three sausages on the table in front of me.
"Thank you," I said, reaching for the syrup she had placed on the table, along with a napkin and cutlery.
Justin appeared a few minutes later when Diesel was begging for another bite of hotcake. Justin saw it and grinned.
"Good morning, child." Azalea treated the young man to one of her rare smiles. "You set on down here and eat you some breakfast. You need your strength."
"Yes, ma'am," Justin said, eyeing the plate of hotcakes and sausage avidly. "Thank you, Miss Azalea. I'm starving."
Azalea stood, arms folded, watching Justin eat for a moment. Then she inspected my plate. "How about some more?"
I groaned and pushed my plate away. "No, thank you. That was delicious, but if I eat any more I'll have to go run around the track for two hours."
The housekeeper cocked an eyebrow at that. She knew I was not a runner. "Just go up and down them stairs a few times. That'll do it."
The doorbell rang, and I started to get up from the table.
"You set still." Azalea motioned me back into my chair. "I'll take care of whatever heathen that is, ringing the bell this time of the morning."
"Thank you," I said. I knew better than to argue with her.
As I watched Justin shovel the food into his mouth with Diesel sitting hopefully by his chair, I heard raised voices come from the hallway. I recognized one of them and sighed.
The voices neared.
"I done told you, girl, you ain't going in that kitchen. You go and set yourself down in the living room. Mr. Charlie'll come in there when he's done finished with his breakfast."
"Mama, this is ridiculous." Kanesha Berry sounded angry.
"Git on in there like I told you. Ain't gonna hurt you to wait five minutes."
"Oh good Lord. If this don't beat all."
Justin stared at me, round-eyed, and I tried hard not to laugh. The stern, commanding deputy of the night before was starting to sound like a petulant teenager.
Azalea entered the kitchen alone, and I hastily drank some coffee to hide my smile. Justin dipped his head down and stuck another forkful of hotcake and sausage in his mouth.
"Miss High and Mighty Deputy is waiting to talk to you, Mr. Charlie, when you be done with your breakfast." Azalea proceeded to the stove as if Justin and I had heard nothing of the argument between her and Kanesha.
"Thank you, Azalea," I said. "I'm not exactly dressed for an interview, but I don't think she'll want to wait while I shower and dress."
Something like _hmmph_ sounded from the direction of the stove, and I shrugged at Justin.
"Are you going to your classes this morning?"
Justin regarded his plate for a moment. "I guess so. Do you think I should? Or maybe I should go over to the hospital?"
"Why don't you call your mother and talk to her about it? My guess is she'll say you should go to your classes and keep busy."
"Yes, sir." Justin appeared relieved.
The last thing he needed was to be hanging around the house all day. I was hoping the media didn't know who he was yet, so they'd leave him alone.
"But if people start pestering you with questions," I said as I thought about it, "you come on back here and don't worry about your classes, okay?"
"You mean like newspeople?"
I nodded.
Justin made a face. "I'm not talking to them. I don't want to be in the news."
"Then you don't have to talk to them. Remember that."
"Okay."
I stood. "Now I'd better go talk to Deputy Berry."
As I was leaving the room, I glanced back to see Azalea serving Justin more hotcakes and sausage. Diesel remained with him, ever hopeful.
If only I had a metabolism like that, I thought wistfully.
I tightened the belt of my robe before I entered the living room. I should have washed my hands, I realized too late. Oh well.
"Good morning, Deputy. You wanted to see me?"
Kanesha turned from studying one of the bookcases against the far wall. Lines of tiredness had etched her face overnight. I wondered whether she'd had any sleep at all.
"Good morning, Mr. Harris." Kanesha frowned. Whether she was still riled up from the argument with her mother, I couldn't tell. "Yes, I do."
"Why don't we sit down?" I gestured toward the sofa and chairs.
Kanesha chose one of the chairs, and I sat in the other, bracing myself for an onslaught of questions.
"I came by to tell you not to speak to any newspeople." Kanesha glared at me. "I don't want this investigation compromised by someone letting details get loose."
"I'm in no particular hurry to talk to any reporters," I said, somewhat stung by the sharpness of her tone. "One of them called me already this morning, but I hung up on him."
"And who was that?"
"Ray Appleby, from the local paper."
Kanesha's eyes narrowed. "I've already given him a statement. If he bothers you, let me know. Same thing goes for any other reporters."
"Thank you, I will. I have no desire to see myself or anyone in this house on national television." I crossed my arms and gazed blandly back at her.
"So far none of them know that Justin Wardlaw was with you last night." Kanesha shifted position in the chair. "I'd like to keep it that way as long as possible. They'll find out eventually, though."
"They won't find out from me," I said. "He wants to go to his classes today. Do you think that's a good idea?"
Kanesha considered that for a moment. "I don't see why not. I need to question him again, but I have other things to do this morning." She stood.
"That's all?" I shrugged. "I thought for sure you had more questions for me."
"I do, but they can wait. You'll hear from me."
I stood, ready to show her to the front door.
"I'll see myself out."
I nodded as she walked past me toward the hallway. Moments later I heard the door open and close behind her.
I headed back to the kitchen. Justin was gone, along with Diesel. Azalea was clearing the table, putting things in the dishwasher.
She probably wouldn't ask me what Kanesha had said to me, so I told her.
"Any of them come sniffing around the house, I'll just turn on the water hose."
I laughed. I could see her doing it. "Go right ahead."
As I picked up my coffee cup to get a refill, I saw Azalea regarding me with a frown.
"Is something wrong?" I asked.
"She gone need some help." Azalea, for the first time in my acquaintance, looked worried.
"Kanesha?"
Azalea nodded.
"She seems pretty capable to me," I said. "She seems to know what she's doing."
"She's a smart girl, I know. Always worked real hard. Ambitious, too." Azalea smoothed her apron, and I waited for her to continue.
"But people ain't gonna talk to her. You know what they're like." Azalea looked at me expectantly.
"You mean because she's black." There was no other way to say it, and I knew what Azalea meant. Old attitudes die hard, and many people in Athena weren't used to the idea of a young black woman in a position of such authority. That could cause Kanesha some problems.
"I sure do," Azalea said. Her eyes bored into mine. "That's why you got to help her, only don't let on like you're doing it."
**FOURTEEN**
Showered, shaved, and dressed, I contemplated the day ahead. Wednesday is my day for errands. I worked at the college library on Mondays, Tuesdays, and Thursdays, and on Fridays I volunteered at the public library.
Justin had gone off to his classes, and Azalea would be here most of the day working and keeping an eye on things. She had already taken another phone call from the local reporter, Ray Appleby, and I doubted he would call back anytime soon.
Before I went upstairs, Azalea extracted a promise from me to help Kanesha as discreetly as possible. I knew there was some truth to what Azalea said, and I couldn't help being curious about who had killed Godfrey and why. Everyone else in town would be talking about it, so there was no reason I couldn't, too. And slip in the occasional question.
Was this how the Hardy Boys got started? I laughed at myself in the mirror. I didn't have a famous detective for a father, but I had read hundreds of mystery novels. I would poke around, but I wasn't planning to investigate houses on cliffs, old mills, or secret caves anytime soon.
Diesel followed me to the room next to my bedroom, another bedroom that Aunt Dottie had converted into a sitting room for herself. With a few small changes I had turned it into an office of sorts, mainly by adding my computer and printer.
The cat jumped up onto the desk by the computer—his usual spot—and watched as I turned the computer on and got comfortable in my chair. I had a little time to kill—an unfortunate phrase, I realized—before shops would open, so I might as well check my e-mail.
The first message I opened was from my daughter Laura, who had moved to Los Angeles two years ago to pursue a career as an actress.
The news about Godfrey had apparently hit the media in California last night, because Laura's message was full of questions. She had no idea, of course, how closely involved I was in the case. I glanced at the time stamp on her message. She had sent it around two A.M. Pacific time.
I replied to her message at some length, explaining what I knew about Godfrey's death and my own involvement. I knew there would be many questions to come, because Laura loved mysteries as much as I did. As a ten-year-old she wrote her own plays based on the Nancy Drew books, and naturally she starred as Nancy. If I wasn't careful, she'd hop on the first plane home, determined to help me.
Then I remembered she was in a successful play at the moment, so I was safe from her enthusiastic assistance. Smiling, I clicked the SEND button.
There was no message from my son, Sean, but that wasn't unusual. Much more taciturn than his younger sister, Sean wrote me an e-mail every week or so and called about as often. He and his mother had been very close, as Laura and I are, and I knew he was still struggling to come to terms with Jackie's death.
Finished with e-mail, I shut down the computer. Diesel yawned at me, and I reached out to scratch his head.
"Are you ready to go, boy? It's almost ten."
The cat hopped to the floor and rubbed against my legs. He knew the word _go_.
Downstairs I heard Azalea running the vacuum in the living room. I fastened Diesel into his harness, and soon we were on our way in the car. I had decided not to walk this morning, despite fine weather, in case I needed to get somewhere quickly.
My first destination this morning was the independent bookstore, the Athenaeum. Some locals and visitors might scratch their heads over the name, but I thought it was clever. Its present location was on the town square, across from Farrington House, but it had started life about twenty years ago in a house on a street near downtown. The present owner, Jordan Thompson, had inherited it from her father, and when I moved back to Athena, I was delighted to find it thriving.
It was a few minutes past ten when I pulled my car into a spot directly in front of the store. The neon OPEN sign was on. Diesel hopped down from the car, eager to go inside. Jordan always made a fuss over him and gave him a kitty treat or two. Or five. Diesel sometimes went into starving-cat mode around her, and I pretended not to notice.
I paused at the front window. There was a large pile of Godfrey's latest book, a hardcover with a garish cover, on display. It would probably sell even more copies now that he was dead.
With that morbid thought, I entered the bookstore, Diesel stepping ahead of me. The bell hung from the door handle jangled and, as usual, Diesel swatted at it until I pulled him away.
"Good morning." I called out the words because I didn't see any staff members in evidence.
The head of one of Jordan's assistants popped up from behind the counter. "Let me know if you need help with anything." The head disappeared.
"Thanks." The head belonged to Jordan's younger brother, Jack, who was about the same age as Justin. He was always in a hurry, it seemed, and I took no offense at his abrupt manner.
The Athenaeum occupied about four thousand square feet, and there were thousands and thousands of books lining the shelves. I could easily spend two hours here—and often had—such was the wealth of the printed word available. I headed for the mystery section, where I usually started. I had been here the previous Saturday, so there might not be anything new in. It never hurt to look, though.
I was checking the _H_ s for three of my favorites—Haines, Harris, and Hart—when I heard a voice behind me.
"Good morning, Diesel. What a beautiful boy you are."
Diesel tugged at the leash, and I let him go as I turned to greet Jordan.
"Good morning, Charlie," she said, bending down to give my cat an affectionate greeting.
Her long red hair pulled back in a ponytail, Jordan looked younger than thirty-one. Tall and willowy, she was a striking woman, with flawless skin and flashing green eyes. "Good morning," I said. "How are you?"
"I'm fine," she said, standing up. "Can Diesel have a treat or two?"
"Sure," I said. "You spoil him, you know."
Jordan laughed. "He's a big guy. Needs to keep up his strength."
I passed over the leash, and Diesel padded happily after Jordan.
I hoped that by the time Jordan brought Diesel back I could figure out an approach. I scanned the bookshelves in front of me, as if I could find inspiration there.
I heard the bell on the door, and then a voice called out, "Morning, everybody. I brought doughnuts."
Recognizing the voice of Patty Simpson, I smiled. With Patty here it shouldn't be too hard to steer the conversation around to the death of the town's famous writer.
I left the mystery section for the front counter. Her back to me, Patty was setting down a box of doughnuts along with a purse and a bag of books. Jack Thompson had disappeared from behind the counter.
"I finished the galley of that new thriller you gave me," Patty said without turning her head. "It was pretty awful, so I don't think you should order more than one."
Then she turned and saw me. "Oh, sorry, I thought it was Jordan. How are you, Mr. Harris? Would you like a doughnut?"
I would very much have liked a doughnut, but after the breakfast I had consumed, I knew I shouldn't. "No, thank you," I said, surprised that the words actually came out of my mouth.
"There's plenty," Patty said.
"No, really, I'm okay. But thanks for the offer."
"I'll be right back." Patty grabbed her purse and scurried off in the direction of the back room.
While I waited, I turned my back on the doughnuts, lest I be tempted further. Instead I focused on a nearby display of diet cookbooks. I ought to buy one, but I knew I'd never cook anything out of it.
When Patty returned, she eyed the box of doughnuts. She helped herself to one, stuffing half of it in her mouth. Judging from the plumpness of her figure, she wasn't interested in diet cookbooks any more than I was.
"Have you heard the big news?" She popped the rest of the doughnut into her mouth while she waited for my reply.
There was no point in playing coy. Sooner or later everyone would know I found the body.
"Yes, I have. Poor Godfrey."
Patty swallowed. Her expression turned sour, and I didn't think it was from the doughnut.
"He was a colossal jerk, that's what he was." Patty reached for a second doughnut before pulling her hand back.
"I went to school with him," I said. "He wasn't always a nice person. Did you know him?"
"Only through the bookstore. And from things my Aunt Melba has told me about him." She shot me an arch look. "I know you know my Aunt Melba. Don't you think she looks good for someone her age?"
I suppressed a laugh. Patty was anything but subtle. "She sure does."
Patty grinned, and I knew the minute I left she'd be on the phone with Melba, reporting my comment.
"So Godfrey came to the store to sign books, I guess." If I didn't steer the subject back to Godfrey, no telling what Patty, trying to get a response from me, would say about Melba next.
"Not as often as he should have." Patty frowned. "You'd think Mr. High-and-Mighty Bestselling Author would have the decency to help out his hometown bookstore. But not him. He was too good for us."
"You mean he wouldn't sign here?" That was rather ugly of Godfrey, if it was true.
"Well, he did sign a couple of times," Patty said. "But the last time he was going to come, he canceled at the last minute and went over to that big chain bookstore out on the highway instead. The jerk."
"What are you talking about?" Jordan and Diesel walked up to us, and I could tell that Jordan wasn't happy as she gave Diesel's leash back to me. Diesel stared back and forth between us, sensing the sudden tension in the room.
"About Godfrey Priest," Patty said, not the least fazed by Jordan's forbidding expression. "And about the dirty trick he played on us the last time he was supposed to sign here."
"We all have better things to do with our time than talk about that jackass," Jordan said. "You need to finish checking those backlist orders." She turned and stalked off.
Patty waited until Jordan was safely out of earshot before moving a step closer to me and Diesel. "She used to be in love with him, you know."
"Really?" I felt awkward. This was the kind of thing I had come for, but it suddenly seemed a bit embarrassing.
Patty was not embarrassed. "Oh yeah, she would go off to those mystery conventions, when he was still showing up at them, and I think they had a big ol' fling. But then he must've dumped her."
"That's too bad. He did have a terrible reputation with women, though." I kept an eye on the back room. Jordan might reappear at any moment, and I didn't want her to catch us.
"And that was when he stopped coming to the store." Patty sounded triumphant, as if she'd just solved a puzzle.
That was interesting. Hell hath no fury, etc. Not to mention a bookstore owner whose business could be hurt by the defection of a big-selling writer.
Jordan stuck her head out of the back room. "Patty, have you started on that backlist order yet?"
"Just starting it now." Patty's tone was cheerful in reply. She winked at me. "If you need help with anything, you just let me know." She turned to look around the counter for something. She picked up a printout and brandished it at me. "I have to go through the romance section and decide what we need to reorder. I'm the expert for that section."
As she turned, her foot caught on something, and she stumbled toward me. I put out a hand to steady her, and Diesel scooted out of the way.
"Thank you," she said. "Now what's this doing here?"
She stooped down and picked up a box of books that had been sitting on the floor behind the counter. She set the box, labeled with Godfrey's name and the title of his new book, on the counter next to the doughnuts.
Julia's words from last night flashed into my mind. "Could I have a look at one of those? I haven't read it yet."
Shrugging, Patty pulled one out of the box and handed it to me. "It's pretty awful. I gave up after fifty pages."
I heard her only dimly as I opened the book to the title page. There, below the printed name, was Godfrey's signature.
And yesterday's date.
**FIFTEEN**
"This is pretty interesting." I held the book out to Patty.
She took it from me and glanced down at the title page.
"Whoa. This is going to be worth something, let me tell you." She snapped the book shut and stuck it back in the box.
"I suppose so." I was annoyed she hadn't given the book back to me, but perhaps she was so surprised she didn't realize her rudeness.
"So that's where she was." Patty muttered the words under her breath, but I was close enough to make them out.
"What _are_ you doing?"
Neither of us had heard Jordan approach. Patty stared at her boss like a fox caught in the proverbial henhouse, while I mustered as innocent-looking a smile as I could.
"Just looking at this box of books," I said. "I was thinking about buying one. I haven't read it yet."
Jordan's eyes narrowed with suspicion as she looked from Patty to me. "These aren't for sale."
"Then what are you going to do with them?" I thought that was a reasonable question. She could surely sell them for a lot more than list price—books signed by a famous mystery writer the day he was murdered. Talk about collectible.
"I meant they're all spoken for," Jordan said in a more conciliatory tone. "They're all special orders." She turned and reached for the box.
"You know, I saw Godfrey yesterday morning," I said. "I know he had plans for lunch. Was that when he signed them?"
Jordan stepped back from the box and glared at me.
Patty watched avidly, her eyes going back and forth between her boss and me.
"No, it wasn't," Jordan said, her face flushing. "If you must know, he signed them yesterday afternoon. I went by his hotel room."
"Gosh, then maybe you were the last person to see him alive." Patty could hardly contain her glee. "I bet the police will want to talk to you."
Jordan, in the act of reaching for the box again, stumbled against the counter. When she turned, her face was dead white. For a moment I thought she was going to faint, but she rallied. She pulled a high-seated stool over and sat down on it. "What happened?"
"You haven't heard?" I was surprised. She was probably the one person in Athena who hadn't. "Godfrey was found dead in his hotel room last night. The sheriff's department is treating it as a suspicious death."
"Oh dear Lord." Jordan muttered the words over and over.
"Can I get you some coffee or something?" Patty, suddenly contrite, appeared anxious.
Jordan waved her away. "No, just go do your job for once."
Patty's sulky expression didn't bode well for her dedication to the task, but she went away quietly.
"Are you okay?" I asked, concerned by how shaken Jordan still seemed to be.
Diesel, sensing her distress, stood up on his hind legs and stretched his right paw out, touching her thigh. Jordan gave him a shaky smile and a rub on the head.
"If she ever does penance for anything, it'll be for that double-jointed tongue of hers." Jordan paused and breathed deeply. "Yeah, I'll be okay. It's a shock, hearing news like that. So completely unexpected." She continued rubbing Diesel's head.
"You really had no idea?" Was she a consummate actress, only pretending to be stunned?
Jordan shook her head. "No, why should I? I never make it to the ten o'clock news. I'm always too tired. And nobody called me, either." She snorted. "Though I'm surprised Patty didn't."
"Was Godfrey a particular friend of yours?" I wasn't sure how she would react. This might be my last visit to her bookstore if I wasn't careful, and I certainly wouldn't like that. "Sorry, but you seem pretty shaken up."
"More than a bookstore owner should be for a writer who hadn't deigned to enter her premises in five years?" Jordan laughed, a bitter sound. Diesel sat back on his haunches and stared up at her.
"I suppose so, yes. If you put it that way." Perhaps I should have excused myself and gotten the heck out of there, but curiosity kept me.
"I'll tell you one thing: I'm not sorry the bastard's dead." Jordan stood up, and Diesel scooted back beside me. "He embarrassed the hell out of me by not showing up here—twice—for advertized events. Not to mention the money I lost on returning hundreds of copies of his books—books I could easily have sold. But he didn't have to balls to show his face in here."
"That's too bad. No wonder you were pissed at him." I didn't know what else to say. The passion in her voice startled me. Right now, she sounded angry enough to have killed him.
But anger this intense because of business?
Or was there something more personal behind it, as Patty claimed?
I couldn't ask her that outright, or I really would be banned from the bookstore. At the moment I couldn't think of a subtle way of getting at the information either.
"Now, is there something I can help you with?" Jordan became very businesslike.
"I would still like a copy of Godfrey's latest book." I nodded at the box of signed copies. "If those aren't available, an unsigned copy will do."
Jordan stared at the box for a moment before reaching into it and pulling out a book. "It's okay. You can buy one."
"Thank you." I went around to the front of the counter, Diesel at my heels. As Jordan rang up my purchase and bagged it, I pulled out my debit card.
The transaction finished, Jordan returned my card and handed me the bag. "Thank you very much." She didn't smile the way she usually did, but she also didn't look like she never wanted to see me in her store again. That was a relief.
"Come on, Diesel. Got to finish our errands." I flashed Jordan a smile as the cat and I headed for the door, but the bookstore owner had already turned away.
Outside the store, I paused. Diesel sat down and looked up at me. I gazed back at him, lost in thought.
Why had Jordan changed her mind and let me buy one of the signed copies?
Should I take it as some sort of bribe? Because the book would probably soon be worth a lot more than the $26.95 plus tax I paid for it.
Or was it Jordan's way of telling me she had nothing to do with Godfrey's death?
Short of asking her point-blank, I didn't see any way to answer those questions for now.
Diesel warbled at me, bringing me out of my wool-gathering. "Time to move on. I know."
I put the book in the car, and Diesel and I walked down the block to the bakery.
Helen Louise Brady, another of my Athena High School classmates, had opened a patisserie and café a few years before I moved back. It quickly thrived, patronized by many of the college faculty and students, and plenty of townspeople as well. Helen Louise's pastries and cakes were sinfully delicious, and I never could resist popping in for something to take home.
Another point in the bakery's favor was that Helen Louise didn't mind having Diesel come in with me. The first few times I took him in some of her regulars raised their eyebrows, but Helen Louise had been known to ban customers who annoyed her. If she said it was okay for Diesel to be there, no one was going to argue with her.
Rake-thin and nearly six feet tall, her hair jet black, Helen Louise beamed with joy when she spotted Diesel. "Ah, _mon chat très beau_." Helen Louise often lapsed into French. She had lived in Paris for nearly ten years before coming back to Athena and to open the patisserie. "Let me find something for you."
I sometimes marveled that Diesel didn't weigh fifty pounds, so many people wanted to feed him. I kept an eye, though, on his little treats, and at home we had play sessions designed to help him burn off the extra calories.
Helen Louise came around the counter with some creamy frosting on her fingers and bent to let Diesel lick it off. He purred, and Helen Louise smiled again.
"Thank you." I smiled back. "I know Diesel thanks you, too. He's going to have to run an extra lap or two on the stairs at home, but I'm sure it's worth it."
"I should hope so." Helen Louise laughed. She went behind the counter to a sink and washed her hands. As she dried them, she asked, "And what can I get for you today, Charlie?"
She made a wicked chocolate gateau, and I pointed to one in the glass case. "That will do quite nicely. And I'll have to run up and down the stairs a few times myself." I grinned.
" _Quel dommage_. But every mouthful a little heaven on the tongue." Helen Louise expertly boxed my selection and rang it up at the register.
" _Oui, certainement_." I knew some French too, and Helen Louise laughed.
"Come again soon," she said. "You too, Charlie."
I grinned as I led Diesel to the door. Helen Louise was charming, and her personality was one ingredient in her success.
I put the gateau carefully on the backseat of the car, while Diesel sprang into the front. My two most important errands of the morning accomplished, I thought Diesel and I might drop by the public library for a few minutes. It was only a few blocks away, on the route home.
I was about to back out of the parking space when my cell phone rang. I shifted back to park and pulled the phone out of my shirt pocket. Glancing at the number on the display, I frowned. Someone from the college was calling, but I didn't recognize the number.
"Hello, this is Charlie Harris."
"Hey, Charlie, it's Rick. How you doing?" Rick Tackett was operations manager for the college library.
"Doing fine, and you?"
"Pretty busy," Rick said. "Got a big delivery for you, and I wondered if you wanted it up in your office maybe? Or somewhere else?"
"How big?" I asked, puzzled. I wasn't expecting anything.
"Fifty-four boxes," Rick said. "Pretty heavy. Maybe somebody's papers or something."
Papers?
For a moment I couldn't remember any recent agreement to take someone's papers for the archive.
Then it hit me.
Could these be Godfrey Priest's papers?
**SIXTEEN**
Who else could the papers have belonged to? Godfrey had estimated he had fifty or sixty boxes of papers and books to give to the college archive.
But when had he shipped them?
"Charlie, you still there?"
Rick's voice brought me back to the conversation. "Yeah, I'm still here. Just a bit stunned, that's all."
Rick chuckled into my ear. "Yeah, it's a huge shipment. And pretty heavy, too. Probably cost a coupla thousand bucks, I bet."
"If they belonged to whom I think they did, he had plenty of money." Yeah, the papers were Godfrey's. He must have called someone and had them shipped right after our conversation yesterday.
"Must be nice." Rick laughed again. "Anyway, they're here on the loading dock. Oh, and there's a letter, too." There was silence for a moment. When Rick spoke again his tone was somber. "Return address says it's from Godfrey Priest. I heard he died last night."
"Yes, he did." What should I do with Godfrey's boxes? The sheriff's department would probably impound them if they knew about them, though I couldn't imagine what use they would be to Kanesha Berry. Technically they were now the property of Athena College, although I didn't think Godfrey had signed anything to that effect yet.
Maybe there was something in his letter that stated his intentions.
"I'd better come over there. I'll meet you on the loading dock in a few minutes."
"Sure," Rick said. "I'll be here."
I ended the call and stuck the phone back in my pocket. Diesel butted my elbow with his head.
"No, I didn't forget about you," I told him. "But we've got to take a detour. Sit."
Diesel sat in the passenger seat. I'd been meaning to get him one of those pet car seats, but since I mostly just drive around town, and pretty slowly at that, I kept putting it off.
About six minutes later I pulled into the loading dock of Hawksworth Library. Built in the 1920s and added to several times over the past eight decades, it was named for an illustrious president of the college who had served right after the Civil War. Altogether it occupied half a block of the street on the north side of the antebellum mansion that housed the archive and some administrative offices.
Rick Tackett, a friendly, stocky fireplug of a man about ten years my senior, stood on the loading dock beside a pallet of boxes.
I rolled the front windows down a little before shutting off the car. "You stay in the car, boy. I won't be long."
Diesel yawned at me and curled up on the seat. Sometimes, like now, he was remarkably obedient. Other times he was as headstrong as a Brahma bull. I never knew how he'd react to a command.
Or a suggestion, from a feline point of view.
I climbed up onto the loading dock and shook Rick's extended hand.
"Morning, Charlie," he said. He nodded at the neatly stacked and shrink-wrapped boxes. "Here's the letter." He pulled it from his back pocket.
The envelope, made of heavyweight paper, screamed _expensive_ , as did the gold-embossed return address bearing Godfrey Priest's name—or rather, "Godfrey Priest Enterprises Inc." I guess being a big bestseller was something like running a business.
"Thanks," I said. "I'll just open this and have a quick look, if you don't mind."
"Sure thing," Rick said. "Here." He handed me a penknife.
I took it and slit open the envelope with the blade and returned the knife to Rick. I extracted the contents, two pages of heavy bond paper.
The top sheet, bearing last Wednesday's date, was a letter from one Gail Enderby, apparently Godfrey's administrative assistant. Ms. Enderby explained that she had prepared the boxes for shipping per her boss's instructions. Each box, she said, contained an inventory of its contents, and box number one—I glanced over at the pallet, and the boxes I could see did bear numbers—contained a master inventory.
With the amount of time all this would have taken to organize, Godfrey had evidently been planning this donation for several months.
The second sheet was a letter from Godfrey himself, dated the day before his assistant's note. He proclaimed his intent to donate his papers to the Athena College archive. He didn't mention giving any money along with the papers to cover the costs of processing and housing the collection, but at least this letter ought to give the college clear ownership.
"Good news?" Rick asked when I looked up from the letters.
"Yes. Now I feel like I can answer your question about what to do with these boxes."
"Great. Where do you want them? Over in your building?"
I eyed the pallet, trying to estimate how much space I had in one of the storage rooms allotted to the archive. "Could you have the boxes numbered one through ten put in the office? I think the rest of them will fit in the archive storage room."
"No problem." Rick glanced at his watch. "My guys'll be having lunch soon. How about they get 'em up there by two? That do?"
"That's fine." I was itching to get into box number one and take a gander at Ms. Enderby's master inventory, but that could wait. "Thanks. I really appreciate it."
Rick smiled, and I climbed down from the loading dock and rejoined Diesel in the car.
We sat there for a moment as I stared at the boxes above us. How bizarre this was. And yet, how typical of the man.
Godfrey, with his irrepressible ego, was so sure the college would want his papers, he had them boxed and ready to go. What would he have done yesterday, I wondered, if someone had told him the college wasn't interested?
He would have found a home for them somewhere, but in reality, Athena College, like most private schools these days, couldn't afford to turn down a gift from a prominent alumnus like Godfrey. Athena would accept anything in the hope that more money would follow.
Diesel rubbed against my arm and chirped loudly, interrupting my train of thought.
"Let's go home for lunch," I said, scratching his head. "Then we're going to come back this afternoon and take a look at those boxes."
I drove us home and, once we were in the kitchen and I'd taken off his harness, Diesel headed straight for his litter box in the utility room. I went to the refrigerator, feeling a bit peckish. After the big breakfast I'd had, I didn't want much for lunch.
Azalea had anticipated that, for I discovered a bowl of salad with mixed greens, chopped egg, and cheese sitting on the top shelf. Add some of Azalea's homemade thousand island dressing to that, and it would be just fine.
I prepared my salad and filled a glass from the pitcher of fresh tea on the counter and took them to the table. Diesel came back and settled on the floor near my chair. The house was quiet, and I figured Azalea had probably gone to the grocery store, one of her usual Wednesday activities.
About fifteen minutes later, finished with lunch, I put my dirty dishes in the sink. Upstairs, I brushed my teeth while Diesel lolled on my freshly made bed. When I first moved back I had made it myself on the days Azalea was due, vaguely embarrassed to have her doing it instead. She quickly informed me that if she wanted me to do her job she'd let me know, and after that I left the bed-making to her. She did it even better than I did anyway.
I glanced at the clock—a few minutes before twelve-thirty. There was no sense rushing back to the archive, because the boxes wouldn't be delivered for at least another hour or so. Spotting Godfrey's latest book on the bedside table, I decided I might as well read a bit of it to pass the time.
I picked up my reading glasses and the book and settled into a comfortable armchair near the window. Diesel appeared to be sound asleep, for which I was grateful. Sometimes he insisted on sitting in my lap while I read, and that could get uncomfortable because of his weight.
Godfrey's book was titled _Moon of the Hunter_. I skipped reading the jacket blurb because sometimes it gave away too much of the plot. I turned past the title page and started reading his acknowledgments. I always found them interesting. Occasionally an author gushed, thanking everyone he knew. Others made poignant remarks about loved ones. Sometimes they were just plain funny.
Godfrey was pompous. He thanked his various agents—in New York, Hollywood, and London—along with members of his staff in California, including Gail Enderby, for ensuring that his life ran smoothly. He mentioned a couple of technical experts he had consulted, and that was it.
The last time I'd read one of his books was probably six or seven years ago, and as I read the first page, I remembered why I stopped. The graphic violence in the opening paragraphs was shocking in its intensity, but somehow compelling. I didn't like the fact that I found it compelling and wanted to read further. But I ignored that and kept turning the pages. Godfrey knew how to pace a story.
A hundred pages later I remembered to check my watch. It was now almost quarter to two. By the time I reached the archive, the boxes of Godfrey's books and papers should be waiting for me. I stuck a bookmark in the book and laid it aside, albeit a bit reluctantly. _Moon of the Hunter_ was the story of a serial killer who lured young women to his isolated cabin in the mountains of east Tennessee and the determined sister of one of his victims who was intent on tracking him down and killing him.
I could easily have sat in the chair and finished the book in another couple of hours, but my curiosity over Godfrey's boxes won out. I got up from the chair, stretched, and approached the bed.
"Come on, boy, let's go."
Diesel yawned and rolled over on his back. I reached down and rubbed his stomach. He purred loudly in appreciation.
"I'm not going to stand here and do this for the next two hours." I gave him a final rub and withdrew my hand. "Come on."
A few minutes before two, I unlocked the door to the archive storeroom. Rick's assistants had delivered the boxes, and there was little open space left in the room now. I did a quick count while Diesel sniffed around the boxes. There were forty-four of them, all numbered. Boxes one through ten should be in the office.
I pulled Diesel away from his perusal and headed down the hall to the archive. Inside, the lights on, I dropped Diesel's leash, and he began inspecting the boxes stacked on the floor in three piles in front of my desk. Diesel hopped on top of the first pile of three, and then I realized there were eleven boxes, not ten. The other two piles had four boxes each.
Then I noticed that ten of the boxes were numbered, one through ten, but the eleventh box didn't have a number.
That was interesting. It had to be part of the shipment, because Rick didn't mention any other delivery for the archive today.
I moved forward to pull the eleventh box from the bottom of the center stack but my cell phone rang. I pulled the phone out of my pocket and glanced at the number display. It was the sheriff's department. I answered and identified myself.
"Good afternoon, Mr. Harris." Kanesha Berry's voice was cool and professional. "I'd like to talk to you right away. Can you come down to the sheriff's department please?"
Dang. I really wanted to delve into the boxes, especially the oddly unnumbered one. But I didn't think putting the deputy off would be a good move. I might as well get it over with.
"Okay, I'll be there in a few minutes." I ended the call, stuck my phone in my pocket, and got Diesel down from atop the boxes.
"Come on, boy. Off to jail we go."
**SEVENTEEN**
I parked a few spaces down from the front door of the Athena County Sheriff's Department. If I had ever been inside the building, I didn't remember it. The building dated from before the Second World War, but there was a new jail behind it, built about five years ago.
"This will be a new experience for both of us," I told Diesel as we approached the door. Diesel's nose twitched in anticipation. He was always curious about strange places.
Inside, the chilled air and fluorescent lighting reminded me of a hospital. Diesel strained against his leash several paces ahead of me. He had spotted the reception desk and a uniformed man sitting behind it. He wanted to go say hello.
"Good afternoon," I said as I approached. "I'm Charles Harris. I'm here to see Deputy Berry. She's expecting me."
The officer behind the desk was too busy staring at Diesel to acknowledge me at first. I cleared my throat a couple of times, and he finally looked up at me.
"Sorry, sir, what did you say?" Before I could respond, he continued. "What kind of cat is that?"
"He's a Maine coon. They get to be pretty big." I smiled at his reaction to my cat. I repeated my name and the purpose for my visit.
"Sure," the deputy said. "She's got someone with her right now. Why don't y'all have a seat over there, and soon as she's done, I'll take you back to her."
"Okay," I said, disconcerted. I led Diesel to the chairs the deputy indicated and sat down. Diesel climbed onto the chair next to me and looked around.
If Kanesha wanted to see me right away, why was I being made to wait? Was this some little power trip on her part? Or had someone turned up to talk to her before I arrived?
I kept checking my watch as I waited. Five minutes passed. Then ten. Fifteen.
Finally, twenty-one minutes after I sat down, I looked up to see Julia Wardlaw coming out of door behind the reception area.
The deputy let her through the security gate, and she came straight to me. I stood to greet her.
"Hello, Charlie." Dark circles under her eyes told me she'd had little sleep since last night. She reached down to stroke Diesel's head.
"Are you okay? You look exhausted." Not the most gallant thing to say, but it was the truth.
"I am," Julia said. "I was up most of the night at the hospital with Ezra. They moved him to a room yesterday, and he's not doing very well at the moment."
"I'm so sorry." Such inadequate words.
"Thank you." Julia gave me a weak smile. "I'm going home for a bit now to try to get some sleep."
"Good idea," I said. "Did Kanesha call you in? She did me."
Julia nodded. "She had more questions. And I told her about going to the hotel to see Godfrey yesterday. She wasn't happy, but it's done."
"Did you tell her about seeing Jordan Thompson there?"
"I did."
"I'm sorry," I said. "I shouldn't hold you up. You need some rest."
Julia gave me a quick peck on the cheek and another tired smile. "I'll talk to you later. I want to come by to see Justin."
"Of course," I said. "Anytime."
As she turned to leave, the deputy on duty called me. I approached the desk, Diesel in tow.
"Come on back. I'll show you the way." The deputy let us through the security gate before escorting us part of the way down a corridor. "Deputy Berry's in the last room on the left, sir."
I thanked him, and Diesel and I moved on toward the room he indicated.
I paused at the open door and knocked. Kanesha Berry looked up from a computer and frowned when she spotted Diesel with me. She stood. "Come in, Mr. Harris. Please have a seat." She indicated a chair in front of her desk.
The office, about ten by ten, held two desks, bookshelves, a few chairs, and stacks of paper. Kanesha's desk appeared orderly, in contrast to the haphazard piles on her office mate's desk.
I pulled another chair next to mine for Diesel, and the cat and I sat. On the way over from the library I speculated why Kanesha had waited so long to question me when she'd had several opportunities already. This morning, of course, she had been effectively routed by her mother, because I doubted she had come to my house simply to tell me to avoid the news media.
"What can I do for you, Deputy?" I squirmed a bit in the hard chair, probably chosen for its discomfort factor. Beside me Diesel sat up and watched Kanesha, his head almost level with my own.
Kanesha seemed not to be able to take her eyes off the cat for a moment. Then she shook her head and focused on me.
"When we questioned you at the hotel, why didn't you tell us Justin Wardlaw was with you?"
I had to choose my words with care, because I didn't want to give her the impression I suspected Justin of killing Godfrey. "Justin had a pretty rough time of it yesterday. I don't know how much he might have told you about the events of the day, but I was concerned. I thought he needed a little time to get himself together before talking to anyone."
"That was very thoughtful of you." Kanesha's jaws clenched for a moment. "You were obstructing an investigation. You do realize that?"
She was definitely pissed.
"Yes, I suppose so," I said. "But I did what I did for the boy's sake. If you have to charge me with something, go right ahead."
"Believe me, I'm very tempted." She paused. "I'm not happy about it, but I've got to live with it. You, Justin, and Mrs. Wardlaw had time enough to collude on your stories by the time Bates and I got to your house last night. I'm not happy about that either, but if I find out one of you lied to me about anything— _any thing_ —I will come down hard on you."
"Understood." She was angry already, so I might as well ask for an answer to something that had puzzled me since last night. "Why didn't we do this last night?"
"Because I chose not to."
_In other words_ , I thought, _you goofed and won't admit it_. The murder rate in Athena County was very low, and Kanesha probably had little real experience investigating homicides. The last murder here—that I knew about—occurred seven or eight years ago when an outraged husband killed the man who'd been sleeping with his wife. Since he did it in plain sight of several people, there wasn't much to investigate.
Kanesha picked up a pen from the desk and put a notepad in front of her. She scribbled something. "Take me through your day yesterday, starting with Godfrey Priest's arrival in your office."
Suppressing a sigh, I complied with her request. Diesel curled up and went to sleep, and I talked for what seemed like half an hour.
Kanesha interrupted me only twice before I reached the point when I decided to go to the hotel to check on Godfrey and Justin.
"Why were you so concerned about a man you barely knew anymore? A man you said you didn't really like. And, according to your statement, one you hadn't seen in nearly thirty years. I'm not sure I understand."
I thought for a moment. "I suppose I was really more concerned about Justin and the fact that he was still gone. He was under a considerable emotional strain yesterday, even before he found Godfrey dead. Plus, Godfrey probably never missed any opportunity for people to pay lots of attention to him. It just seemed odd somehow."
"Why are you taking such an interest in Justin? He's not your son." Kanesha leaned back in her chair, her gaze cool, as if she were eying a specimen of some kind.
"No, he's not. But he is in my care, in a way. He boards with me, and naturally I take an interest in the welfare of someone who lives under my roof. He's also the son of an old friend."
"I see" was all she said in response.
I decided to venture a question of my own. "Are you aware of how much Godfrey was disliked by people who knew him?"
A faint smile played on the deputy's lips. "I've picked up on that, yes."
"Then you must realize there were probably people who had far stronger motives to kill him than either Julia or Justin. Or me." Mindful of Azalea's plea to me this morning, I decided I had better share the gossip I had gleaned. I didn't like having to implicate someone possibly innocent of Godfrey's murder, but I had little choice if I was to help Justin.
"Such as?" She put her pen down on the desk and leaned back in her chair.
"Jordan Thompson for one. I spoke to Julia just now, and she said she told you about seeing Jordan at the hotel yesterday when she was leaving."
"She did," Kanesha said. "But I have no proof as yet that Ms. Thompson saw the victim yesterday."
"Well, I have it," I said, trying not to sound triumphant. "A signed copy of Godfrey Priest's new book. It's dated, too. Yesterday's date."
Kanesha blinked. That interested her. She picked up her pen and jotted something down. "How did you get this signed and dated copy?"
I told her about my visit to the bookstore this morning, including the gossip from Patty Simpson about Jordan's affair with Godfrey. Kanesha scribbled more notes as I talked.
"Since Jordan saw Godfrey _after_ Julia did, it seems to me she's a better suspect. And one with a pretty strong motive, perhaps."
"Possibly." Kanesha laid the pen down again. "I'll check it out, of course, but that doesn't mean anyone else is off the hook now."
"Of course," I said, refusing to be nettled by her dismissive tone.
"Any other little tidbits you want to share?" Kanesha's lip curled. "Especially since you seem to be so up on the latest dirt."
She wasn't making this easy.
With a quick mental apology to my boss I told her what I knew about Peter Vanderkeller's intense dislike of Godfrey and the cause of it.
Once again she took a few notes, but this didn't appear to impress her any more than the information about Jordan Thompson.
"I think that's all, Mr. Harris. If I have further questions, I'll be in touch."
That was a bit abrupt, I thought. "Good day, then." I stood, and Diesel jumped to the floor. Kanesha turned to her computer and started typing.
Azalea would be appalled at her daughter's lack of manners, I thought. Kanesha could have at least thanked me for coming in after her peremptory summons.
Diesel and I left her office and headed back up the corridor to the reception area. We paused at the desk for the deputy to open the security gate. I could tell Diesel wanted to explore around the desk and visit with the deputy, but I couldn't wait to get away from here.
"Come on, boy," I said, tugging lightly at the leash. "Time to go back to work."
"Bye, kitty," the deputy said. Diesel rewarded him with a few trills as we moved toward the door.
Outside I blinked a few times, adjusting to the afternoon sunshine. Kanesha's manner still rankled, but I supposed I shouldn't have expected anything different. At least I had given her two new potential suspects to consider.
Back in the car, I drove to the college library and parked in the lot behind it. Diesel and I entered the house through the back door, near the staff lounge. I was thirsty, and I figured Diesel might be also. I led him into the lounge, unoccupied at the moment. I found an oversized mug in the cupboard and filled it from the cooler. I drank it down quickly and then refilled it and set it on the floor. Diesel lapped at the water. When he was finished I would wash out the mug in the sink.
"Hello, boys. What are you two doing here this afternoon?"
I looked up to see Melba Gilley in the doorway of the lounge. She advanced with a smile, a mug in her hand.
"There's something I want to check on upstairs," I said.
After she exchanged further greetings with Diesel, Melba filled her mug with coffee and took a sip. She made a face. "This has been sitting here awhile. But it'll have to do." She sipped again. "You talking about all those boxes? What the heck are they anyway?"
"They're full of Godfrey's papers," I said. "He had them shipped last week."
"Without even waiting to see if we'd take them." Melba laughed. "Typical." She shook her head. "I never dreamed when I called you last night that he was dead. Bizarre."
"Yes, it is." Diesel was finished drinking. I took the mug to the sink and turned on the hot water. Raising my voice over the sound of the water, I continued. "The whole thing is really bizarre. Godfrey probably ticked off a lot of people, but who hated him enough to kill him?"
"The Lord only knows." Melba moved closer to the sink. "Maybe one of his ex-wives sneaked into town and did it."
I squirted a little dish soap in the mug and scrubbed it with a brush. I gave it quick rinse and set it upside down on the draining board.
As I dried my hands on a towel, I said, "That's possible, I guess, but why would one of them have waited until now to do it? I think it's somebody right here in Athena."
"You're probably right." Melba poured the remains of her coffee out and set the mug in the sink. "You think you'll find anything interesting in Godfrey's papers?"
"I might. I'm sure they'll be interesting," I said.
"Maybe there's a clue to his murder."
Before I replied, we both heard a floorboard squeak out in the hall.
Melba and I exchanged glances.
I waited a moment to see if whoever was in the hallway entered the room. No one did.
I took a step toward the door. "Who's there?"
There was no answer.
**EIGHTEEN**
The floorboard creaked again, and then we heard the sound of footsteps in rapid retreat.
I strode over to the door, about six feet away, but whoever was listening to our conversation had disappeared. I walked down the hall and around by the stairs, but I still didn't see anyone. Nor did I hear anything other than the muted sound of street traffic.
Melba and Diesel had followed me out of the staff lounge.
"That was peculiar." Melba frowned. "And kind of creepy."
"It was definitely odd."
"I'm going back to my office and keep an eye on the door." Melba stepped past me, smiling uneasily. "Don't turn your back on anyone."
I picked up Diesel's leash. "Don't worry. I won't."
I waited until Melba disappeared into the director's office suite. "Come on, boy. Let's go upstairs."
Before I unlocked the door of the archive office, I checked inside the storeroom. Nothing seemed to have been disturbed. I shut the door and examined the lock. It looked sturdy enough, like the one on the office door.
The eleven boxes in the office hadn't been touched, as far as I could tell. Diesel started sniffing around them again, and I had to push him gently away in order to uncover the unnumbered box. When I pulled it free, I restacked the three cartons that had been on top of it before picking it up and setting it on my desk.
Diesel hopped on top of the middle tier of boxes and watched while I cut open the box. After I pulled out the wads of paper used for packing material, I found several smaller boxes and trays of computer disks and even a couple of thumb drives. The disks probably contained the texts of Godfrey's books and perhaps some of his correspondence.
I wondered why the box hadn't been numbered. Perhaps this box hadn't been intended for inclusion in Godfrey's archive.
The master inventory in box number one ought to answer that question. I moved around my desk to check. The box I wanted was underneath the one Diesel was sitting on. I moved him aside to the sound of annoyed chirping.
I extracted box one and set in on the floor. I retrieved my scissors from the desk and cut open the box. Right on top, under more packing material, lay a small report folder labeled "Inventory."
Back at my desk, folder in hand, I sat down and began skimming through it while Diesel played with the discarded packing material on the floor.
Calling these few sheets of paper a master inventory was a gross overstatement. Each box was listed, but there was little detail of the contents. Godfrey's assistant had merely listed categories, like fan letters, business letters, reviews, awards, newspaper clippings, contracts, review copies, books in English, books in other languages, convention programs, and speeches. Nowhere in the inventory did the words _disk_ or _diskette_ appear.
It seemed fairly clear to me the box of disks had been shipped by mistake. Otherwise it would have been numbered and included on the inventory. The number of boxes in the inventory matched the quantity of numbered boxes received.
What should I do with it? Send it back to Ms. Enderby in California?
I found the two letters on my desk and scanned the one from Gail Enderby. There was a phone number included. I might as well call her and ask.
I used my cell phone, rather than the office phone, because I could never remember the long distance dialing code I was supposed to enter to authorize a call.
The call went to voice mail after five rings. A perky, young-sounding voice informed me that Gail Enderby was on vacation, and her stated return date was a couple of weeks away. She gave no alternate contact information. I wondered if she had seen the news yet about her boss's death. I left a message, asking her to call.
That was that. The disks were in my custody for now. I replaced the packing material and re-taped the box. Instead of putting the box back with the others, I put it behind some shelves a few feet away from my desk. Perhaps the mysterious eavesdropper had spooked me, but the disks might be valuable. As long as I was the only one who knew they were here, I might as well keep it that way.
I picked up box one and placed it on my desk. Consulting the inventory list, I saw that this box contained fan mail. Curious, I pulled out one of the folders, dated twenty years ago, and began leafing through it.
The first couple of letters were full of praise for Godfrey.
" _Trapped_ kept me up until three in the morning," one fan wrote.
Another one said, "I had to get up and check all the locks in the house when I finished _Midnight Killer_."
On most of the letters I examined there were notes that indicated when Godfrey responded, though copies of Godfrey's answering letters were not in the folder.
The most interesting letter of those I read was one that took Godfrey to task for abandoning the gentler, more traditional mysteries he wrote at the beginning of his career in favor of "bloodthirsty, needlessly violent trash." Godfrey's note on this one was a terse "no response."
I laid the folder aside and was about to pick up another one when my office phone rang.
"Good, you're still here," Melba said when I answered. "Peter wants to see you right away. I told him about the boxes."
"I'll be right down." Sighing, I hung up. I wasn't in the mood for a talk with Peter, but then I realized it was a good opportunity to do a bit of sleuthing.
I picked up the letters that came with the boxes and called to Diesel. "Come on, boy. Let's go."
I paused long enough to lock the office door behind me before following Diesel down the stairs. I found him in Melba's office on top of her desk.
"It's okay," Melba said, flashing me a guilty look. "I let him get up there."
"I guess there's no point in arguing. You'll keep an eye on him while I talk to Peter?"
"Of course." Melba rubbed the cat's head. "You go right on in."
I knocked on Peter's door and then opened it.
"Ah, Charles," he said, rising from his chair. "Do come in."
I took a seat, and Peter resumed his.
"Melba tells me that you have received a shipment of the late Mr. Priest's archival material." Peter tented his fingers together and regarded me owlishly.
"Yes, the boxes arrived today." I leaned forward and handed him the two letters. "It's all very well organized, so he must have been planning this for some time."
Peter read through the letters quickly. He laid them on his desk. "No doubt. Given the colossal ego that man possessed, he would have assumed the college would accept his papers without demur." He sniffed.
"I agree," I said. "But he certainly had no idea he was going to die so soon, and in such a brutal fashion."
"One cannot pretend to feel sorrow for such an unmitigated bastard, despite the distasteful manner of his death. The drivel he wrote will sell even better now, though he won't be able to reap the benefits." Peter smiled with grim satisfaction.
I never suspected our library director possessed such a deep streak of vindictiveness. He really had hated Godfrey.
"His sales will jump, for a while at least," I said. "You're probably right about that. But I wonder who _will_ benefit." Oddly enough, this was the first time I had stopped to think about the matter. Who would inherit Godfrey's wealth? Justin?
"One can only hope he made suitable provision in his will to enable the college to house and process his collection of papers. Otherwise they will have to remain as they are." Peter lifted his chin in a determined manner as he regarded me. "I trust we are in agreement on that point."
"Certainly," I said. I had more than enough to do as a part-time employee. I would far rather catalog rare books than process Godfrey's papers, despite my curiosity.
"Excellent." Peter beamed at me.
"Barring some provision in Godfrey's will, do you think that letter is sufficient for the college's ownership of the collection?"
"I should think so," Peter said. He picked up the letter and read it again. "He states his intentions perfectly clearly, though it is a great pity he did not mention any pecuniary bequest to accompany it."
"All this is going to generate a lot of publicity for the college and for the town," I said.
"Sadly, I fear you are correct." Peter frowned, his distaste evident. "Why the man had to come here to get himself murdered, I simply do not understand."
Peter colored faintly, perhaps having realized the fatuousness of that remark. I decided to ignore it.
"The whole thing is very odd," I said. "There are a lot of things I'm curious about. For one thing, that call Godfrey made to say he was too ill to attend the dinner in his honor last night. It seems a little too pat."
Peter didn't respond. He just stared at me.
"I wonder if it was Godfrey who really called?"
"Why shouldn't it be?" Peter said, his fingers tapping on his desk.
I shrugged. "Just a thought. When Melba called me, she said Godfrey had called the president's office to inform him. Then I guess someone from his office must have called you."
Peter's fingers ceased their rhythmless tattoo on his desk. "Actually, that is not quite accurate."
"Why not?"
"Melba, I'm afraid, somehow misunderstood." Peter paused for moment. "She quite often does because she fails to listen properly, and I have spoken to her severely on the subject several times."
I waited, and after a moment he continued.
"You see, I was the one who spoke to Godfrey and who in turn informed the president's office, at his request."
**NINETEEN**
That was definitely odd. Why would Godfrey call someone in the library, rather than the president's office?
"When I spoke to him," Peter continued, "he complained of a rather nasty stomach virus. He regretted the inconvenience—or used words to that effect—and asked me to pass along the word. As I did." His fingers resumed their tattoo upon the desk.
"Out of curiosity," I said in a diffident tone, "do you remember what time that was?"
"Around five-thirty, I suppose," Peter said after a moment's thought.
"Has anyone from the sheriff's department spoken with you yet?"
"Whatever for?" Peter paled slightly. "One would not wish to be involved in something so sordid as a murder investigation."
"No, one wouldn't," I said, a wry twist to my voice. "But unfortunately one already is." I was beginning to lose patience with the man. He was being overly fastidious, in my opinion. "You might have been the last person—barring the killer, of course—to speak to Godfrey. The deputy in charge of the investigation needs to know that."
"I see." Peter reached for a glass of water on the credenza behind his desk and took a long swallow. He set the glass down with a hand that trembled. "Then one must do one's duty."
He was still pale, obviously unsettled, but apparently willing to follow through. I dictated the number of the sheriff's department and told him to ask for Deputy Berry. He laid the pen aside and said he would call.
"Very well," I said. "Shall I leave these letters with you?" I pointed to his desk as I stood.
"Yes, for now. I shall have Melba make copies of them for you. One imagines that the college's legal counsel will want to keep the originals."
"Of course. Well, if that's all, I'll get back to work," I said.
Peter nodded, and I turned for the door.
"Oh dear, I almost forgot."
I turned back. "Yes, Peter?"
He made a moue of distaste. "I received a call from the president's office, shortly before you came, informing me that there is to be a memorial service for Godfrey this Saturday afternoon at two in the college chapel. I suppose I shall have to attend, though one could easily think of far more pleasant things to do on a Saturday." He sighed.
"It would be the proper thing to do," I said. "I'll have to attend, too."
Peter didn't reply. I don't think he heard me, because he had turned to look out the window behind his desk.
I left his office, shutting the door gently behind me. He was an odd duck, no two ways about it.
Diesel still sat on Melba's desk, watching her as she worked at her computer. The keys clicked at a rapid pace, and the cat appeared mesmerized by Melba's flying fingers.
"Sorry to interrupt," I said. "Come on, Diesel, back upstairs."
Melba ceased typing and turned to smile at me. "See you later, then, boys." She gave the cat an affectionate scratch on his head. Diesel purred his thanks.
"Come on now," I said, and Diesel leaped gracefully to the floor. He followed me to the stairs and dashed up them as soon as I placed my foot on the first step.
Back in the office, Diesel began to play with the loose packing material, batting it around and then leaping on top of it. I watched him for a moment. He was still very kittenish, despite his size.
As I sat down at my desk, I noticed the message light blinking on the phone. I listened to a message from circulation at Hawksworth Library next door informing me that a book I'd requested was available.
I checked my watch—it was nearly five o'clock now. Time to head home. I could delve more into Godfrey's papers tomorrow. Before we left, though, I repacked the open box on my desk, taking away Diesel's toy. "You can play with it again tomorrow."
He turned and sat with his back to me until I headed for the door. I attached the leash to his harness, locked the door behind us, and set off down the stairs and out the back door. I wanted to pick up the book, but first I had to put Diesel in the car. Hawksworth was one of the few places I couldn't take him. A couple of staff members had complained that his presence was too disruptive, because invariably students clustered around him, wanting to pet him. They made too much noise, according to the complainants.
So, into the car Diesel went. The day was cool, and I cracked the front windows enough to allow air to circulate—but not enough for a large and enterprising cat to squeeze through.
"I'll be back in five minutes," I told him, but I could tell he wasn't happy at being left behind. He never was.
Inside the library, I went straight to the circulation desk. While I waited for the student worker to find my book, a recent study of the late antiquity and the early Middle Ages, I listened idly to a conversation at the nearby reference desk. Willie Clark was on duty and being his usual charming self while helping a female student.
"No, we haven't received that issue yet. Can't you read the screen? Do you see any mention of volume thirty-three, issue ten?"
I watched as Willie tapped the computer screen in front of him while the student, red-faced, mumbled something.
"Then you'd better go back and check your citation again. You probably wrote it down wrong." The disgust in his voice was obvious.
Head down, the student scurried away. She was probably a freshman. Older female students learned to avoid the reference desk when Willie sat behind it. He could be gruff with male students as well, but his voice had a particular edge to it whenever he talked to a woman.
Not surprising, then, that he had never married. He wasn't gay either, as far as I knew. Too crabby, in my experience, for a partner of either sex to put up with long enough to establish a relationship.
Willie caught me looking at him, my expression no doubt critical. He scowled at me and turned away.
Book in hand, I left the library and went back to my car. Diesel complained nonstop to me on the short drive home, and I scratched his head a couple of times in apology for having abandoned him in the car.
The moment I opened the kitchen door appetizing smells tickled my nostrils. Diesel sniffed appreciatively too, though he was bound to be disappointed. I tried not to feed him from the table, though he often sat nearby and stared hard, as if hoping to bend me to his will.
I glanced at the clock after I released Diesel from his harness. It was a little after five, and Azalea had left for the day. There was a pot of green beans on the stove, and when I peeked in the still-warm oven I found a chicken, mushroom, and brown rice casserole. There was a tossed salad in the fridge as well and, as usual, Azalea had prepared enough food for at least four people.
I checked Diesel's bowls, and Azalea had taken care of them already. She might fuss at him sometimes, but she wasn't about to let anyone in the house go hungry. Diesel examined them before loping off to the utility room.
The doorbell rang. I hoped it wasn't Kanesha Berry, dropping by with more questions.
Julia Wardlaw stood on my doorstep, looking wan and tired.
"I apologize for dropping by like this without calling first," she said as I stepped aside for her to enter. "But I wanted to see Justin before I went home."
"You're always welcome here, Julia," I said. "You have an open invitation to visit whenever you like." I shut the door and examined her with concern.
"Thank you," she said.
"How are you? And how is Ezra?"
"I'm tired, but Ezra's doing better, thank the Lord. They're keeping him one more night, and he should be able to come home tomorrow."
"That's good," I said. "Why don't you come on in the kitchen and sit down. Let me get you something to drink, and I'll go get Justin for you, if he's here. I just got home myself, and I haven't seen him yet."
"I'd appreciate that," Julia said as she followed me. "Right now I don't feel up to climbing those stairs, I have to say."
Diesel came to greet our visitor, and Julia petted and talked to him while I poured her a glass of the sweet tea Azalea had made.
As I climbed the stairs I thought, not for the first time, about having an intercom system installed. But then I reflected that I could always use the exercise.
Puffing slightly by the time I reached Justin's door, I knocked.
"Come in."
I opened the door and took a step inside. Justin sat at his desk, working at his computer. He tapped the keys a moment longer before he turned to greet me. "Hello, sir."
"Hello," I said. "Your mother is downstairs. She'd like to talk to you."
"Thank you," he said. "I'll be right down. I need to do one more thing to this"—he indicated the computer with a quick nod—"but that won't take two minutes."
"Fine," I said. "I'll tell her." I backed out and shut the door. Justin seemed a bit more animated today. All day yesterday he had appeared depressed, occasionally almost catatonic in his lack of response. A good night's rest had helped, I supposed, along with a little distance from the events of yesterday.
Julia had finished her tea by the time I got back to the kitchen, and I offered her more after I relayed Justin's message. She declined.
"You're welcome to visit with Justin in here," I said, "but you might be more comfortable in the living room."
"This is fine," Julia said. "As long as you don't mind. This is such a lovely, comforting room."
I glanced around it with affection. Yes, it was comforting. When Aunt Dottie was alive, it was usually the center of the house, the room where she spent so much of her time. I liked to think her warmth and generosity lingered here.
"It is that," I said. "Why don't you stay and have dinner with me, you and Justin both? Azalea left more than enough for the three of us, and I can guarantee it will be delicious. That woman is a wonderful cook."
Julia smiled. "I really shouldn't impose on you after all you've done already. But I can't face the thought of going home to cook for myself. Thank you. I'd love to have dinner with you."
"Hi, Mama." Justin came clattering into the kitchen. Yes, he was definitely more animated tonight. He bent to kiss his mother on the cheek. She touched his head as he did so, and he didn't move for a moment.
"If you'll excuse me, I'll just run upstairs for a few minutes," I said. "Then if you're both ready to eat, we'll have dinner."
Julia smiled her thanks, and as I headed for the stairs I heard her relaying my invitation to her son.
I dawdled in my bedroom, wanting to give Julia and Justin enough time to talk. I wondered whether Julia was going to tell her son about Ezra's health problems. She ought to do it soon. Postponing it wouldn't be doing Justin any favors in the long run.
Diesel did not appear, and I figured he was downstairs with Justin. He was really fond of the boy, and Justin certainly seemed attached to the cat. Diesel always seemed to have the ability to sense when someone needed comfort, and right now Justin did. If Diesel could help Justin through the difficult times ahead, I was delighted and very thankful that such a special four-legged friend had come into my life.
Almost half an hour passed by the time I went back downstairs. Julia and Justin were quiet when I entered the kitchen. It looked as though Justin had been crying, but now he appeared calm. Diesel jumped down from the boy's lap and came to greet me.
"I told Justin about his father," Julia said simply.
I nodded. "I can't tell you both how sorry I am." I reached down to rub the cat's head.
"Thank you," mother and son said in unison.
Julia stood. "If you'll excuse me a moment, I'd like to freshen up a bit. Justin, why don't you help Charlie set the table?"
"Yes, Mama," Justin said. He got up from the table and went to the cabinet. Diesel padded after him.
I started to point Julia toward the downstairs bathroom, but she waved me away with a smile. "No need for directions."
Justin brought three plates out and set them on the table, Diesel matching him step for step. "Thank you for inviting my mother to dinner."
"You're both very welcome," I said. "If you'll finish setting the table, I'll get the food there."
Justin nodded and worked in silence for a moment. As I was putting on oven mitts, he spoke again.
"Um, sir, I guess there's something I need to tell you." He stood, utensils in hand, his head slightly down. He appeared embarrassed. Diesel rubbed himself against the boy's legs, but Justin didn't seem to notice.
"What's that?" I asked as I reached into the oven for the casserole dish. I thought it might be easier for him to talk if I wasn't looking at him.
"It's about what I told you yesterday," Justin said. "About my dad—Ezra—hitting me."
I set the casserole dish on top of the stove, realizing I needed to put a trivet on the table first.
"Go on," I said, my voice neutral.
"I guess I kind of lied about it," Justin said. His face colored. "Yesterday was the only time he ever hit me like that."
"Why did you lie about it, then?"
Justin shrugged. "He was being so weird about the whole thing, about Godfrey Priest being my dad, too. He kind of freaked out, maybe, and I guess I wanted to get back at him by making him sound bad."
"I can understand that," I said. "What he did yesterday is inexcusable. He never should have struck you like that."
"No, sir." Justin began to lay the utensils at each place.
"I can't blame you for being angry with him. No one could. But I'm glad to know that yesterday was the only time something like that happened."
"Yes, sir." Justin smiled briefly. "And he promised me at the hospital that he'd never ever hit me again, no matter what." His face crumpled. "And now he's going to die, too." Diesel rubbed against his legs again.
Julia came back in time to hear that last sentence, and she gathered her son into her arms. Diesel moved away from them but sat nearby, watching. Justin wept for a moment, and Julia regarded me with a question in her eyes.
"Justin told me he lied to me about Ezra beating him," I said, my voice soft.
"Good," Julia said. "I told him he had to."
Justin pulled away from his mother. "I'm sorry, Mama."
"I know, sweetie." Julia patted his cheek. "Why don't you go wash your face and blow your nose?"
Justin nodded and headed for the bathroom in the hall. Diesel went with him.
"He really is a good boy most of the time," Julia said when Justin was out of earshot.
"I know," I said with a smile. "Diesel wouldn't be so fond of him if he weren't."
Julia laughed. "That cat is such a little character."
I politely refused Julia's offer of help, and by the time Justin returned to the kitchen everything was ready.
We all sat, Julia to one side and Justin across from me. I asked Justin to say grace.
He bent his head over his plate. "Bless this food, oh Lord, to the nourishment of our bodies. We thank you for our many gifts, and we pray that you will watch over us and over the loved ones who are not with us. Amen."
Julia and I echoed his _amen_. I held my hand out for Julia's plate and filled it with casserole and green beans while Julia filled her bowl with salad.
For a few minutes we were busy preparing our plates and bowls of salad, passing things back and forth. Diesel sat near my chair, watching every movement of my hands with great interest. When no tidbits were forthcoming, he moved to the other end of the table to try his luck with Justin.
By unspoken agreement, it seemed, we spoke of things other than the events of the day before. Julia asked Justin about his classes, and he expressed enthusiasm for his freshman English and history courses. He was not so fond of the science and math classes, however.
I talked a bit about my work cataloging rare books, and Julia listened to each of us in turn. Occasionally she prompted with a question, but for the most part she appeared content to let the males at the table carry the burden of conversation. I turned a blind eye to the occasional morsel of chicken or green bean that Justin so casually slipped from his plate.
An hour passed pleasantly, and I realized how much I missed having dinner with other people. I wished Sean and Laura, my children, weren't so far away. But most of all, of course, I wished Jackie and Aunt Dottie could sit at the table with us, too.
Even Azalea's chocolate turtle cheesecake couldn't tempt Julia to stay for dessert. She looked much better now than when she had first arrived, but she was still tired and ready to go home for some rest.
I waved away any offers to help clear the table and set to work while Justin saw his mother out.
He stepped into the kitchen long enough to thank me again, and Diesel followed him upstairs when he said he had to get back to studying.
I took my time in the kitchen, doing my best to keep my mind off Godfrey's death and Ezra's terminal illness. It all seemed too much somehow, and I needed a mental break.
Finished at last, I turned off the lights downstairs and headed up to my bedroom.
After brushing my teeth and changing into my pajamas, I climbed into bed. Diesel was absent, no doubt still with Justin. He would appear eventually to claim his share—and more—of my bed.
I reached for Godfrey's book and got comfortable. I read twenty pages or so before putting the book aside. The heroine wasn't a particularly likable person, and I remembered that was another aspect of Godfrey's books that had always bothered me. There was a strain of misogyny in the books that made me uncomfortable. For all the women Godfrey had apparently married and romanced, he didn't seem to like women very much.
Still not ready to turn off the light and go to sleep, I retrieved my library book. Reading nonfiction would be a good way to cleanse my palate, I decided.
At some point I must have nodded off, book on my chest, because when Diesel jumped on the bed, I came to with a jerk. The book slid off me, and I yawned. While Diesel made himself comfortable, I put the book on the nightstand, turned off the light, and settled down to sleep.
**TWENTY**
The next morning, as I unlocked the door to the archive office a little after eight, I thought about Godfrey Priest. Only two days ago he walked in here, very much the Godfrey I knew in my youth, self-involved and full of life, and less than twelve hours later he was dead. I never liked him, but he didn't deserve to be murdered.
Diesel couldn't wait to investigate those intriguing boxes, and he sniffed around them while I got comfortable in my chair and turned on the computer.
I heard a small sound, as if something had fallen onto the floor, and looked up. Diesel, from the top of one tier of boxes, chirped at me and began grooming himself. What had he knocked over?
On the floor in front of the boxes I found the folder containing the inventory of Godfrey's papers. As I bent down to pick it up, I frowned. What was it doing on top of the boxes? I was sure I had left it on my desk yesterday.
If it had been on my desk, Diesel couldn't have knocked it off onto the floor in front of the boxes, I reasoned. He was a clever feline, but even he could not have picked it up off my desk and then dropped it onto the floor.
The more I thought about it, the more convinced I was that the inventory had been on my desk when I left the office yesterday.
That meant someone had been in here meddling last night or early this morning.
I went back to my chair and placed the inventory on the desk. I examined it, and it was intact, no pages missing.
Who could it have been?
Had the intruder tampered with the boxes? That was an unpleasant thought. If the intruder had taken anything, I would never know, because the inventory wasn't detailed enough.
I got up and examined the boxes. They appeared to be intact. I went down the hall to check the boxes in the storeroom. They also seemed to be undisturbed. I also checked the locks on both doors, and from what I could see, no one had forced them.
Back at my desk, I considered the problem. As far as I knew there were only three sets of keys to the office and the storeroom. I had one set, Melba had one, and the operations staff had the third.
I couldn't imagine Melba or any operations staff member coming in here after I was gone to poke around. I should check with Melba, though, to make sure she still had her set of keys.
And what about the extra box?
I was relieved to find it where I had left it, untouched as far as I could tell. It was a good thing I put it out of the way, I thought. I went back to my desk.
How many people even knew that Godfrey's papers had arrived here? Rick Tackett or some of his staff could have mentioned it to someone, but I couldn't see that it would be that exciting a bit of conversation. Peter knew, naturally, and so did Melba.
Then I remembered the odd incident yesterday, when someone had eavesdropped on my conversation with Melba—a conversation about Godfrey's papers.
Was that who had done it?
If so, that put a different spin on the incident.
Someone was interested in Godfrey's papers but didn't want anyone else to know.
Why?
Was it a deranged fan seeking mementoes of a dead idol?
Or was it simply someone sly and secretive who liked to poke around in things?
Even if it turned out to be someone harmless, I didn't want anyone entering the archive without supervision.
"Come on, Diesel. Let's go see Melba." Time to check on the status of her keys.
I locked the office door behind us. In the past I hadn't done it while I was in the building, but perhaps I needed to change that.
Melba was on the phone when Diesel and I walked into her office. She smiled and held up a finger, by which I understood the call was almost done.
"Sure thing, hon," Melba said. "I'll let Peter know." She hung up the phone. "Geneva Watterson. Sick again, poor thing."
Geneva was one of the reference librarians, and she seemed to have a lot of health problems. I had pinch-hit for her a couple of times when the reference department was short-staffed. I made commiserating noises, but at the moment I had other things on my mind.
"What's up with you two?" Melba smiled as Diesel hopped up on her desk.
I shook my head at the cat, but he ignored me. I sat down in the chair by Melba's desk. "I think someone was poking around in the archive without my permission."
"What?" Melba's eyes nearly bugged out of her head. "That's outrageous. Did they make a mess?"
I explained my reasoning, and Melba nodded. "I know how you like things a certain way, and if you say you left that inventory on your desk, then that's where you left it."
"Thanks," I said with a grin. "Whoever it was got in with a key. There's no sign of forced entry. I have my keys. Where is your set?"
"Right here in this drawer," Melba said promptly. She pulled the drawer open. "I keep them in this tray."
I leaned forward to look, and Melba hissed in annoyance. "If that don't beat all. Someone got in my desk and took those keys."
"Do you lock your desk at night?"
"I sure do." Melba's tone dared me to argue with her.
"What about during the day, if you leave your desk for a few minutes? Or for lunch, say?"
"When I go to lunch, I lock it," Melba said. "But if I'm just going to run to the bathroom or to the lounge for coffee, I don't usually take the time."
"What about the keys to get into your office after hours? Could someone do that?" I was trying to think of all the possibilities.
Melba nodded. "All the department heads have a key to this office, in case of emergency. So one of them could have got into my office last night, I guess."
"That's one possibility," I said. "But it's also possible the intruder saw you were away from your desk, found the keys, and took them. The other question is, who would know you had the keys and where they were?"
Melba thought about that for a moment. "People are always dropping by to chat," she said slowly. "This drawer gets opened a lot, because it's where I keep aspirin and antacids and stuff like that. People come by all the time asking for things because they know I keep them on hand."
"Then anyone could have seen the keys in the drawer," I said. "But how would they know what they're for?"
"Because I had a tag attached to them that said 'Archive, '" Melba said, sighing. "Labeled on both sides, of course."
"You _always_ lock your desk at night?" I wanted to be sure.
"Yes, of course I do."
"And the lock hasn't been tampered with?"
"No, it hasn't, or I would have noticed this morning." Melba was getting a bit testy with this drawn-out interrogation.
"Sorry, just trying to get the facts straight." I smiled, and she relaxed. "Various people have access to this office after hours, but your desk is kept locked. That rules out someone coming in after hours to get the keys."
"Sounds reasonable to me," Melba said.
Diesel, apparently annoyed by the lack of attention, reached out and prodded Melba's arm with his paw. She smiled as she rubbed his head.
"Therefore the intruder must have swiped the keys while you were away from your desk late yesterday afternoon.
"The boxes weren't in there until around two, and at that point very few people even knew they were there." I frowned. "Either the intruder saw the boxes being moved over here from the loading dock at Hawksworth and asked Rick and his guys about them, or else it was probably the eavesdropper. Remember?"
Melba shivered. "That was creepy."
"There could be some other explanation, but that's all I can think of at the moment."
"You need to tell Peter about this," Melba said.
"I know. I think we also need to get the locks changed right away. You think he'll go for that?" With Peter, one never knew.
"I don't see why not," Melba said. "He's due at a meeting at nine-thirty, but he should have time to talk to you now. He got here a few minutes before you and Diesel." She buzzed his office.
When Peter answered, Melba told him I needed to talk to him about something urgent. She listened for a moment. "Go on in," she said, hanging up the phone. "I'll keep an eye on Diesel."
"Thanks," I said. I got up, a bit reluctantly. I did not relish repeating all this to Peter, because he could be amazingly obsessive sometimes about the tiniest details. I might be in for an extended inquisition.
I opened the door and stepped into Peter's office. "Good morning, Peter," I said.
Peter looked up from his desk. "Good morning, Charles. I am most pleased to see you. There is something I feel I should discuss with you. I value your judgment, and I know you will offer sage advice."
He seemed to have forgotten that I wanted to talk to him, but I knew there was no point in trying to divert his attention.
Suppressing a sigh, I sat down. "Tell me about it, and I'll do my best."
Peter stared at me, as if suddenly mute. As I watched him, beginning to grow concerned, his face reddened. Was he having some kind of attack?
"Peter, what's the matter? Do you need a doctor?" I rose from my chair, ready to yell for Melba.
He waved a hand, indicating that I should sit down. "There is no need," he said, his voice low. "I am simply embarrassed by what I have to tell you in order to solicit your counsel."
"There's no need to feel embarrassed," I said. "I won't betray your confidence, I assure you."
"Thank you," Peter said. "I know you are a man of honor." He sighed. "And that is the crux of the matter. I fear that I have acted in a dishonorable manner."
"How so?" I did my best to maintain a patient tone, but Peter could be maddeningly slow getting to the point.
"I refer to the matter of the phone call which we discussed yesterday," Peter said.
"You mean the call from Godfrey? About his feeling too ill to attend the dinner in his honor?"
"Yes, that is correct." Peter drew a deep breath. "I lied to you, Charles. There was no phone call."
**TWENTY-ONE**
"No phone call?" I stared at Peter in disbelief. "Were you playing some kind of joke on Godfrey?"
"Would that were all it was," Peter said, his face reflecting his pompous tone. "The message from Godfrey Priest was real enough. The method of delivery was quite different, I regret to say."
Peter's conversational style was giving me a headache. I yearned to grab him and shake him a few times, maybe knock loose some of the extra words so he would get to the point sooner.
"Then tell me how and when you talked to Godfrey." That came out far calmer than I expected.
Peter grimaced. "I yielded to a base impulse. I have just cause to despise the man, though perhaps you are unaware of said cause." He paused for a moment.
I didn't want to tell him I already knew about his wife, in case he asked me for the source of my information. Melba wouldn't thank me for snitching on her.
When I failed to respond, Peter continued. "Before I came to Athena College to assume the position as director of the library, I lived for many years in California. In Los Angeles, to be exact. I was also married then, and my former wife had personal ambitions centered upon writing for the cinema."
I decided to speed things up a bit. "And somehow she must have met Godfrey and hoped to use his connections to break into screenwriting."
"Yes, that is more or less what happened," Peter said with a pained look. "Not content, however, simply to approach that detestable man for assistance and befriend him, my former spouse decided the only way to achieve her goal was to marry the man. After divorcing me first, naturally."
"I see. One couldn't blame you for not liking Godfrey," I said, "though your ex-wife certainly bears a lot of the blame."
"Oh, most definitely, the balance of the opprobrium lies with her," Peter said. "But one cannot exculpate Priest. He encouraged her and, after all, he did marry her once our divorce was final." He snickered. "The marriage lasted little more than two years, I believe."
Time to steer the conversation back onto the right track. "We've established your reasons for despising Godfrey," I said. "Now what about your talking to him and getting the message that he was too ill to attend the dinner?"
"Ah yes," Peter said with a frown. "As I have mentioned already, I yielded to impulse and decided to confront the man. Having no wish to make an ass of myself in front of anyone else, I decided that the best place for such an affray was his hotel. There, one assumed, one could be assured of some privacy."
"You went to Farrington House to talk to Godfrey," I said, holding on to my patience because he was actually approaching the point. "What time was that?"
Peter considered for a moment, his head cocked to one side. "Around five-thirty," he said. "Yes, I waited until I was through with my day here, which is generally around five or five-fifteen. Then I proceeded directly to the hotel."
"How did you know where he was staying?"
"I was privy to the arrangements," Peter said. "The president's office consulted me, perhaps because the man was a writer. Had he been an alumnus of the athletic type, they no doubt would have consulted the athletic director."
"And you knew what room he was in?" Things couldn't have been more convenient for Peter, I thought. And surely Peter realized that his having gone to the hotel put him high on the list of suspects.
But with Peter one never knew what tortuous path his thought processes might take.
"Yes, I did," Peter said. "That was indeed fortunate, because had I stopped at the front desk and asked them to ring and announce me, the man might well have refused to see me. Thus I decided the better plan of attack would be to knock on his door and gain admittance before he realized who I was."
I had a sudden vision of Kanesha Berry questioning Peter. How would she handle his inability to get right to the point? She might arrest him out of sheer irritation.
"So you went up to his suite and knocked on the door?"
"Yes, I did." Peter frowned. "But things did not proceed thenceforth in the way I had postulated. Godfrey did not open the door. I was forced to raise my voice slightly and entreat him to let me in. He refused."
"Did he give a reason?" If Godfrey hadn't wanted to talk to anyone, why did he bother answering?
But at least I knew he had been alive at five-thirty if he had talked to Peter.
"He explained that he felt quite ill and that he feared it was some kind of stomach virus. He had no desire to inflict it upon anyone in case it was infectious. He intended to stay in his room until he recovered."
"That was kind of him," I said. But odd, I thought. The bug must have had a quick onset, because he had appeared perfectly fine when I had last seen him at my house.
"Perhaps. I inquired of him whether he had informed the president's office of his illness, and he said he had not. He then asked if I would be so good as to do it for him. Then he excused himself, saying he had to rush back to the bathroom. Seeing no point in remaining there any longer, I went back to my car and used my cell phone to call the president's office. I also called Melba and asked her to inform others on the library staff who were planning to attend, as one presumes she informed you."
"She did," I said. "You need to talk to the deputy in charge of the investigation even more urgently now. This will help narrow down the time of death."
"I suppose so," Peter said, obviously unhappy about the idea. "One has so little desire to embroil oneself in such a sordid happening."
"I quite understand," I said. "But still, one must do one's duty." I stood.
"Thank you for your forbearance, Charles," Peter said. "I appreciate you hearing me out."
"Not a problem," I said. "I'll talk to you later."
I opened the door, ready to leave, when an unsettling thought struck me. Why hadn't it occurred to me sooner? I turned and walked back to the chair and sat down again.
"What is it?" Peter asked.
"During the time when your ex-wife was pursuing Godfrey," I said, "did you ever meet him?"
"Yes, a few times at parties," Peter said. "Though I must say I quite often tried to avoid the man, finding him hideously conceited, with only one subject of conversation—himself."
"How long ago was that?"
"Seven years ago," Peter said. "Why do you ask?"
"When Godfrey spoke to you through the door at the hotel, did you recognize his voice?"
"What a peculiar question," Peter said, clearly taken aback. "One simply assumed that one was talking to him because it was his room." He paused. "I cannot be absolutely certain that it was indeed Godfrey I conversed with, given the circumstances. There is the additional fact that the man claimed to be ill, and I did detect what I thought was a note of strain in his voice."
"But you can't swear that it was actually Godfrey on the other side of the door?"
Justin hadn't said anything about Godfrey's feeling ill, and surely he would have noticed. It wouldn't have been easy for Godfrey to conceal a stomach bug of some kind from his son if he had to rush off to the bathroom periodically.
"No, I cannot," Peter said. "But if it was not Godfrey with whom I spoke, then who was it?"
"It might have been the murderer," I said.
Peter turned so white I thought he was going to faint. I started to get up to attend to him, but he rallied enough to say, "No, thank you, I'm all right. Just a bit of a shock, you know."
"Yes, I know," I said. "It's a shock to me, too. But the more I think about it, the more inclined I am to believe that Godfrey was already dead when you went to the hotel." I kept my eye on him. If he was the murderer, he was putting on quite an act to convince me otherwise. I couldn't see him as a killer unless he were talking someone to death.
The problem was, I couldn't see anyone—at the moment—as a killer, but someone had murdered Godfrey.
"I must say, that is quite an unsettling notion." Peter was slowly regaining some color—not that he had much to begin with, poor man. "To have been that close to the perpetrator of such a vicious act—well, the mind frankly boggles, as I am certain you can understand."
"I can," I said. "Now you have to tell the deputy about what you did. She'll probably draw the same conclusion." Or at least, she should, I amended silently. Kanesha might be a pain in the neck sometimes, but she was bright.
"Yes, I will," Peter said.
"Good. I'll leave you then," I said, and once again I made it to the door. But the memory of why I had come down to see Peter surfaced, and I turned back.
"I forgot something," I said as I walked back toward the desk. "We need to get the locks on the archive office and the storeroom changed right away."
"What?" Peter looked alarmed. "What has happened?"
I explained tersely. Peter shook his head. "I shall certainly speak to Rick Tackett immediately," he said. "This is a serious breach of our security. I wonder whether I should discuss this with the head of the campus police."
"I don't think that's necessary just yet," I said. "Getting the locks changed today, if at all possible, is the most important thing."
"I shall see to it." Peter sighed. "So many phone calls to make." He brightened. "I shall have Melba make the necessary contact with Rick, however."
"That's probably not a bad idea," I said. I knew Melba could probably get results from Rick faster than Peter could. "Good luck with Deputy Berry." I thought a reminder couldn't hurt.
Peter was picking up the phone as I left.
In the outer office Diesel was stretched out on the credenza behind Melba's desk while Melba worked at her computer. She looked up when I shut Peter's door behind me.
"You sure were in there a long time," she said. "It couldn't have taken that long, even with Peter, to talk about what happened."
"It didn't," I said. "There was something else Peter wanted to discuss." I threw up a hand. "And before you ask me, I can't tell you. If Peter chooses to tell you, fine, but don't ask me, please."
Melba pouted for a moment, but she never could stay annoyed or angry with anyone for long. "All right, Charlie. Spoilsport." She grinned at me. "I'll get it out of Peter somehow."
I smiled at her, not doubting for a moment that she could. "Come on, Diesel. Let's go."
Diesel sat up and yawned. Then he stretched for a moment before jumping down. He came up to me and rubbed against my legs.
"We'll see you later." I waved at Melba as I followed my cat out of the office and toward the stairs.
I had plenty to think about when Diesel and I were once again installed in our accustomed places in the archive office. While the cat settled down for a nap, I stared at the computer screen. I should have been checking e-mail, but instead I kept running a list of suspects through my mind:
Julia Wardlaw
Justin Wardlaw
Jordan Thompson
Peter Vanderkeller
Any or all of them could be lying.
For example, Julia could have seen Jordan Thompson _leaving_ as she herself arrived, rather than the other way around. Peter could be lying about speaking to someone through the door, or it could have been any one of the others in the room when Peter came to speak to Godfrey. Justin could have killed Godfrey, run out of the room terrified by what he had done, and then sat on the bench in the square until I found him.
Then there was the unknown factor: Mr. X or Ms. X.
Godfrey seemed to have angered enough people in his life that there were probably others in Athena who might have wanted to kill him.
But how to find out who they were, that was the question.
I glanced at the inventory of Godfrey's papers lying on my desk. I knew one place to start.
Sighing, I picked up the inventory and began jotting down the box numbers that contained correspondence. It was going to be a long day.
**TWENTY-TWO**
I worked my way steadily through Godfrey's correspondence, stopping only for lunch and the occasional insistent demand for attention from Diesel. At some point Rick Tackett appeared to change the locks on the office door and the storeroom, but until he came to offer me the new keys and take the old ones away, I hardly noticed him.
He stood in front of my desk for a moment, surveying the boxes. "Lotta stuff here. What are you gonna do with it?"
"Keep it in storage until I have a chance to go through it all and catalog it. But that's going to be a while. I have a lot of other things to see to first."
"Seems like a lotta work for just a bunch of paper," he said.
I shrugged. "Someone may be interested in them at some point, want to do a dissertation perhaps. You never know what kinds of interesting stuff you'll find in a collection like this."
"Is it valuable?"
"Possibly," I said. "Like anything, it depends on how much someone would be willing to pay for it. I doubt the college would want to sell the collection, though."
Rick nodded and turned away. I watched him go, somewhat surprised by the conversation. This was the first time I had heard him express any curiosity about anything archival in nature. In the past when he'd delivered packages to the office he had never asked even one question.
It was probably because of Godfrey's murder, I reasoned. I went back to my work.
Godfrey had accumulated several boxes full of fan mail, not to mention other kinds of correspondence. I scanned each letter as quickly as I could, looking for evidence of some kind of threat to—or ill feeling toward—Godfrey. There were indeed some of the latter but none of the former. If he ever received a threatening letter, Godfrey hadn't kept it, apparently. I also skimmed any notations that Godfrey made on the letters, but I gleaned nothing worthwhile.
By five o'clock I had achieved nothing more than a bad headache and a case of eyestrain. There was still the other correspondence to go through, chiefly business stuff, but that would have to wait. I needed a break, and Diesel was more than ready to go home. I usually spent only half a day in the archive on Thursday anyway.
The walk home helped my headache. Being out in the cool late-afternoon air, plus getting some exercise, made a difference. By the time Diesel and I reached the house I was feeling quite a bit better.
After filling Diesel's food and water bowls and cleaning out the litter box—something I had neglected to do this morning—I contemplated preparing dinner. I found a fresh package of ground beef in the fridge and decided that hamburgers were just the thing. A check of the pantry turned up a large can of baked beans. Add a salad to that, and I'd have a pretty tasty and filling meal for both my boarder and me.
Justin, with Diesel right on his heels, appeared in the kitchen as I was finishing up the burgers. "Good timing," I said. "I'll let you fix your hamburgers for yourself." I pointed with the spatula toward the table. "There's salad there and baked beans in the pot."
"Thank you, sir," Justin said with a shy smile. "I'm really hungry."
"There's plenty." I returned his smile.
Justin never had much dinner-table conversation, and tonight was no exception. I waited until he had dispatched one burger, a large helping of salad, and two helpings of beans before I ventured a question.
"How are you doing?" Still hungry, I reached for the salad bowl, thinking more salad would be better for me than another round of beans.
Justin shrugged. "I'm okay, I guess. It all seems like a really bad dream, you know?"
"I do," I said. "I know it might be difficult to talk about, but I was hoping you wouldn't mind telling me a few things." I had been thinking about the time Justin spent with Godfrey, wondering whether Justin had heard or seen anything that might be a clue to the murder.
"I don't mind," Justin said. He got up from the table to fix himself another burger.
"I'm sure Deputy Berry asked you the same questions I'll probably ask," I said. "The reason I'm doing this, I want you to understand, is because I'm concerned for you and your mother."
"Yes, sir, I know," Justin said. "I know Mama really appreciates it, and I do, too." Finished at the counter, he returned to the table. He smiled at me again, not so shyly this time.
Good, I thought, he's beginning to get some of his old spark back.
"You spent several hours with Godfrey yesterday," I said. "Did anything unusual happen?"
Justin chewed for a moment, and after he swallowed he replied, "Most of the time we just talked. We argued, like I told you, but nothing weird happened."
"Were you in his hotel room most of the time?"
"Yes, sir," Justin said. "He said he didn't want anyone bothering us, so it was better to be somewhere private." He frowned. "That didn't stop people from calling him, though."
"How many phone calls did he get?" These might be slim leads, but they were better than nothing.
"Just two." Justin ate a forkful of beans. "The first one was from his agent, he said. They only talked a few minutes, and he went into the bedroom to do that."
"What about the second call?" I asked.
"Somebody else," Justin said. "He went into the bedroom again, but he wasn't there long."
"Did he say who it was?"
"Not exactly," Justin said. He thought for a moment. "When he came out of the bedroom he was muttering to himself, so I asked him if everything was okay. He said it was just some guy he knew bugging him about reading a book."
That didn't sound like a clue to anything. "That was all?"
Justin frowned. "Now that I think about it, he didn't say _book_ , he said _manuscript_. That's different, I guess." He paused. "I asked him if he read other people's manuscripts and why, and he said he did sometimes because they wanted him to give them some kind of quote to use on the book when it was published. Then he said a lot of times people wanted him to read their stuff because they thought he would help them get it published. But he said this guy was a pest with no talent, and he wasn't going to do it." His face reddened a bit. "Actually the way he said it was a lot ruder, but I'm not going to repeat his actual words."
"I think I can guess the gist of it," I said. Justin was very different from my son, Sean, at that age. Sean had delighted in trying to shock his mother and me with rude language. "So that was it? Just those two phone calls?"
"Yes, sir," Justin said. "Oh, I almost forgot. I did ask him about the guy and how he knew he had no talent if he wasn't going to read the guy's manuscript."
"What did he say to that?" As big a bestseller as Godfrey was, aspiring writers who wanted his help probably approached him all the time. Knowing Godfrey, he probably wasn't that gracious about it, either.
"He said he'd known this guy a long time, but he wouldn't take no for an answer." Justin pushed a couple of beans around on his plate with his fork. "You don't think somebody like that would get mad enough to kill him, do you?"
"I don't know," I said. "It depends on how desperate the man was. And how stable. Someone with mental health problems might respond violently to being thwarted."
"That's pretty freaky," Justin said. He set his fork aside.
"Yes," I said. There was something odd about that second conversation. "Which phone did Godfrey talk on? The hotel phone or his cell phone?"
"His cell phone," Justin said.
"For both calls?"
"Yes, sir."
"Why would Godfrey give his cell phone number to someone he described as a pest?" That was what was bothering me about the second call. "That doesn't make much sense."
"You're right," Justin said. "It doesn't. He talked to me about his writing and stuff, and he said once his books started selling really well, he had people coming out of the woodwork all the time. He had his own security guards at his house in California to keep the crazies away from him." Justin turned a bit pink again, and I figured Godfrey had used a much coarser term than _crazies_.
"I'm not surprised. I've seen it happen before when really big-name authors have signed at the bookstore here in town. I remember one woman who held up the signing line to tell the author in detail about the book she had written. It was sure to be a bestseller, if only she could get someone to read it—according to her. The author politely—and tactfully—declined, but the store staff had to intercede to get the woman out of the line. Even then, she hung around waiting to accost the author again. The staff finally had to eject her. It was embarrassing for everybody."
"Sure sounds like it," Justin said. His mouth twisted in obvious distaste. "But how would they find the person he was talking to?"
"They probably could get a record of his calls and trace the number that way," I said. "Of course, we have no idea where the person was calling from. There's no reason to think he was here in Athena."
"That's true," Justin said.
"You told all of this—everything you told me just now—to Deputy Berry?" I wanted to be sure.
"Yes, sir," Justin said. "I told her, but she didn't say much, just kept asking questions."
"As long as she received the information," I said. "That's the key thing." I stood up, ready to clear the table.
Justin forestalled me. "I'll clean up, Mr. Charlie. Why don't you go relax?"
"Thanks, I think I will." I smiled and looked down at Diesel, who had been napping on the floor near my chair during dinner. "What about you, boy? You want to come up with me or stay here and help Justin?"
Diesel sat up and warbled at me. He stretched a moment before getting to his feet and walking over to Justin's chair. "There's my answer," I said. "See you later, then."
I left the two of them and climbed the stairs to my bedroom. I wanted to change out of my clothes and relax with a book—the history book, not Godfrey's novel. I wasn't in the mood for it right now.
But when I was comfortably in my pajamas, slouched into my chair, book in hand, I found I couldn't concentrate on late antiquity. My mind kept returning to the murder.
Was there someone else who might have a motive for wanting Godfrey dead? The mysterious Mr. or Ms. X?
I needed to know more about Godfrey's past. I needed dirt, if there was any. And I knew the right person to call. Putting my book aside, I retrieved my cell phone and settled in for a long chat.
**TWENTY-THREE**
The obvious person to call was Melba Gilley. With her healthy interest in the doings of her fellow Athenians and her long-term involvement in a variety of community activities, she was a prime source of information.
Calling her, however, meant that I would have to tell her why I was so involved in Godfrey's murder. If she had somehow heard that I—really, Justin and I—discovered the body, she hadn't let on, and such behavior would be totally out of character.
I found her number in my cell phone's address book and initiated the call. She picked up after two rings.
"Good evening," I said. "It's Charlie. How are you?"
"Hey, Charlie, I'm doing fine. How about you?" She sounded as chipper as ever.
"I'm okay," I said. "Is this a good time? Am I interrupting anything?"
"Only some lame show on TV," she said, laughing. "Sometimes I don't know why I even turn the dang thing on, except it's company. What's on your mind?"
"I need to talk to you about Godfrey," I said. "I need to find out some things, and I figured you were the person to ask."
She gave a hearty chuckle. "You mean you called the biggest gossip you know."
I had to laugh. "Well, if you want to put it like that."
"I'm nosy. I admit it," Melba said. "So what do you want to know? But maybe I should ask _why_ first. Godfrey wasn't exactly a buddy of yours."
"No, he wasn't," I said. "And if it weren't for certain circumstances I'd be happy to keep my nose out of it."
"And they would be?"
"First off, you know Justin Wardlaw is boarding with me," I replied. "And I'm sure by now you've heard about his relationship to Godfrey."
"Yes, I have," Melba said. "Can't say I'm surprised. I remember how hard Godfrey was running after Julia back then. And frankly, honey, if I had to pick between Godfrey and Ezra, I'd pick Godfrey. Even knowing he was a class A jerk most of the time."
I started to speak, but Melba went on. "And her already engaged to Ezra. That's what got me. Julia never seemed like the type, but I guess you never can tell, can you? When the baby was born, people started counting up, but it was close enough that no one knew for sure."
Poor Julia, I thought. Having to be the cynosure of all those suspicious people in town, many of them gleefully assuming the worst.
"It was certainly a surprise to me," I said. "I feel a certain amount of responsibility for Justin because he's boarding with me, and I can't help feeling concerned for him and Julia, naturally."
"Of course," Melba said. "They need support right now, because I'm sure the sheriff's department is looking pretty closely at them."
"They are," I said. "But the other reason I'm concerned about this is . . . well, I found the body, basically." I didn't see any point in giving Melba the full details of the situation. This would be more than enough to make her eyes pop.
"You dog," Melba said. "You never said a word to me." She chuckled. "But I won't hold it against you."
"Thanks," I said. "Trust me, I'd rather not have been the one. It wasn't pleasant."
"No, I'm sure it wasn't," Melba said, her tone serious. "It's one thing to read about it in a book, like one of Godfrey's, but it's something else to experience it for real."
"It sure is," I said, doing my best to suppress that ugly image from reappearing in my head. "I guess you can see now why I'm curious about it all."
"Sure," Melba said. "What was it you wanted to ask me?"
"I know Godfrey came back to Athena on a regular basis," I said. "He had quite a track record with women, like the episodes with Julia and Peter Vanderkeller's wife. Are there any other outraged husbands or boyfriends or spurned women in the area?"
Melba was silent for a moment. "The first one that comes to mind is the woman who owns that bookstore on the square. Can't remember her name at the moment."
"Jordan Thompson," I said. "I heard about her. Can you think of anyone else?"
"The other one that comes to mind is Frank Ledbetter." Melba sighed into the phone.
"Frank Ledbetter?" Why did that name sound familiar?
"My ex-husband," Melba said.
"Oh," I said, too stunned for the moment to say anything else.
"I know," Melba said, sounding sheepish. "It's not something I'm proud of, let me tell you. But I had a brief fling with Godfrey about ten years ago, and it cost me my marriage."
"I'm so sorry," I said. "I had no idea." Poor Melba. I knew she was divorced, but that was pretty much the extent of my knowledge.
"Frank and I were going through a bad patch," Melba said. "Old, old story. And here comes Godfrey on a book tour. I went to hear him talk, there were some sparks, and I went out to dinner with him afterward. And you can guess the rest."
I could, but I was curious about one thing. "I do have to ask you one question."
"Shoot," Melba said.
"Did Godfrey hang around town while you had this, er, relationship?"
"I wasn't stupid enough to run off to California with him, thank the Lord," Melba said. "Yeah, Athena was the last stop on his tour, and he was planning to stay here for a couple of months, doing research for a book."
"And when he finished, he went back to California?"
"And I stayed here," Melba said. "By then I'd realized what a fool I'd made of myself, and poor Frank, too. He filed for divorce right away."
"Another question," I said, "and forgive me for asking it, but I have to. Did either you or Frank hate Godfrey enough to kill him?"
"Ten years ago Frank was ready to skin him alive—and he loved to hunt," Melba said. "But by the time Godfrey came back to town a couple years later, Frank was remarried."
"And you?" I prompted her gently.
"I hated him, too," Melba said. "But I hated myself more, let me tell you. I learned my lesson from that." She laughed, a little wildly, I thought. "I got back at Godfrey in my own way, though."
"How so?" I was almost afraid to ask, uncertain of what I would hear.
"I took every one of his books I owned and sat down in front of the fireplace. I ripped out every page, one by one, and burned them. It felt pretty good, though of course I don't normally hold with burning books."
"That's a good thing," I said, trying to ease the tension a little. "Especially since you work in a library."
She laughed, and I felt relieved. I couldn't completely cross her off the suspect list, but it sounded to me like she had worked through her feelings.
I wondered if things might be awkward between us at work after her confession, and I hoped they wouldn't be. I liked Melba. Her sunny, upbeat disposition made her fun to be around, and I would hate for her to feel embarrassed with me.
"I appreciate you telling me all this," I said.
"No big deal," Melba said, though her tone belied the words. "I figured somebody was bound to bring it up sooner or later, and I'd rather you heard it from me."
It was time to move on. "Can you think of anybody else?" I didn't want to say it, but Godfrey, I was sure, probably found a different, but willing, woman whenever he popped back into town for a visit.
"Besides Julia, the bookstore lady, and me?" Melba snorted. "The notches that man must have had on his bed. Well, there is one more that I know of. Janette Turnipseed."
"That's an odd name," I said. "I don't recall anybody by that name."
"No reason you should," Melba said. "She was a professor at Athena, a post-doc in the English department. She was here about six years ago. Godfrey spent three months here one fall in that writer-in-residence program they have, and apparently they had quite an affair."
"She left after her post-doc?" I said.
"Before it was finished," Melba replied. "Poor woman. She left at the end of the fall semester. I think I heard she went to some school out in Nebraska or Oklahoma."
"Why did women keep falling for him? Surely they knew about his past?" I realized my questions could be insulting, but Melba seemed not to take them that way.
"He could be incredibly charming when he wanted to," Melba said. "He'd focus those eyes on you, and suddenly you felt like the most desirable woman in the world." She laughed. "Listen to me. I sound like a teenager. But he made me feel that way."
"I'll have to take your word for it," I said wryly. "I've been reading his new book, and I have to say, the way he writes about women makes him seem like a misogynist."
"That's the weird thing," Melba said. "I got that from his books, too, but in person he wasn't like that. I think he truly liked women, and that was his problem. He liked them so much he couldn't limit himself to one, or even one at a time."
"Then I wonder why he wrote about them with such disdain?"
"Only his shrink knows for sure," Melba said.
Diesel appeared in the doorway and ambled over to my chair. He leaped into my lap, and I tried not to wince from the impact. I grunted into the phone anyway.
"Are you okay?" Melba asked.
"I'm fine," I said. "Just a bit winded from being the landing spot for a large cat."
Melba laughed. "I can see it now. He's pretty big for a lap kitty."
"Try to tell _him_ that," I said as I shifted in the chair to redistribute some of Diesel's weight. "Okay, now I can breathe again."
Diesel chirped at me, and I rubbed his head with my free hand. He would be happy to sit like this for an hour or two—or until my hand cramped and my legs went to sleep, that is.
He laid his head against my chest, and when he did that, I forgave him everything. He was such a loving companion, and if it hadn't been for him, I don't know what I would have done the past couple of years.
"Thanks for all the information," I said. "So far, though, I don't think any of these people—especially you—sounds like a good suspect, though. Can you think of anyone else? I believe Godfrey's parents have passed away, but did he have any other family?"
"Some distant cousins, I think," Melba said. "On his father's side. But they live in south Alabama. I don't think Godfrey's parents had much contact with them, though."
I was about to ask another question when Melba continued.
"The only other one I can think of is his half brother," she said.
"Half brother?" That was news to me. I didn't remember ever hearing that Godfrey had any siblings.
"Yeah, he's about ten years older. Godfrey's mother was married to someone else before she married Mr. Priest."
"I never knew that," I said. "And I sure didn't know about a half brother."
"You know him. You just don't know you know him," Melba said, an impish tone in her voice.
"Okay, I give. Who is it?" I said. I really had no idea.
"Rick Tackett, the operations manager."
**TWENTY-FOUR**
"So Rick is Godfrey's half brother," I said.
Was that why he was so interested in the value of Godfrey's papers?
"Yeah, I guess a lot of people probably don't know that. I don't think they ever had much to do with each other," Melba said. "What I heard was that the boys' mama left her first husband, Mr. Tackett, and Rick for Godfrey's daddy. This was back in the fifties, and I reckon it was a real scandal for a while."
"She left both her first husband _and_ her son?" I said. "That's really sad for the son."
"I know," Melba said. "I've always felt a bit sorry for Rick on account of it. His daddy wouldn't let him have anything to do with his mama. But apparently Godfrey took after _his_ daddy in a lot of ways."
"So old man Priest was a ladies' man, too?"
"From what my mama told me, he was," Melba said. "But they stayed married, even if he did run around on her. She got paid back for running out on her son like that."
"They sound like such lovely people," I said, my tone sour. I had no respect for men who behaved like that. Or for women who ran off and left their children for some man. I didn't know all the details, so I could be misjudging Rick and Godfrey's mama, but still.
"Mrs. Tackett, as she was then, was the organist at her church, and her sister was the preacher's wife. _That_ sure caused some talk."
"I guess Peyton Place had nothing on Athena," I said.
"Not then, or now," Melba said. "People don't change that much. They're always gonna get themselves into all kinds of messes."
"I guess so," I said. "But what I have to wonder is, could one of those messes be related to Godfrey's death? What about Rick, for example?"
"Like I said, I don't think he and Godfrey ever had much to do with each other. I never heard that they did, anyway. Rick, though, has had a pretty hard life."
"I don't really know anything about him," I said, knowing that Melba would fill in the details.
"For one thing, he was in Vietnam, right at the tail end of the war, and the Lord only knows how that affected him," Melba said, the pity obvious in her voice. "Old Mr. Tackett was a hard man, they say. He was a farmer, and you know that's not an easy life. Rick worked on the farm until he was old enough to enlist in the army."
"Couldn't wait to get away, probably," I said. "That's what it sounds like."
"Probably," Melba said. "Rick's been married a couple of times too, and has three kids, I think." She paused for a moment. "Yeah, three. Two boys and a girl. He had another daughter, though, but she died when she was only nine or ten."
"That's awful," I said. I couldn't imagine anything worse than the death of one's child.
"She had some kind of cancer," Melba said. "They took her to St. Jude in Memphis, and I reckon they did everything they could for her. But they couldn't save her."
"Poor Rick," I said.
"Yeah, he's had a hard row to hoe all his life," Melba said. "And there's his half brother, rich as all get out from his books, and Rick struggling to raise his family and get them through school."
"Godfrey probably never did a thing to help them," I said. I couldn't imagine Godfrey being that charitable. Of course, Rick might not have wanted anything from his brother.
"Not that I ever heard," Melba said. "Those kids are smart, though, and one of them's really talented. I heard her sing in the church choir. She has a beautiful voice, and the last I heard she was off to one of those high-toned music schools back east. I think she wants to be an opera singer."
That wouldn't be cheap, I thought. Godfrey's money could make all the difference, if he were inclined to help.
I chatted a few minutes longer with Melba, but she had no further skeletons to reveal. I finally ended the call and put my phone aside.
Diesel was sound asleep in my lap, and my legs were beginning to ache a bit from his weight. I woke him gently and shifted him off my lap to the floor. He blinked up at me, yawned, and hopped on the bed where he went back to sleep.
I got up and stretched my legs before going into the bathroom for a drink of water. My throat was dry after the long conversation with Melba.
After I brushed my teeth I climbed into bed beside Diesel, thankful that for once he had left me plenty of room, so I didn't have to move him.
I lay there in the dark for a while, thinking about all Melba had told me. Godfrey had caused a lot of heartache—as had his mother and father, apparently. But out of all those sad little stories, was there one relevant to Godfrey's death?
The one that seemed like the only real possibility was that Godfrey had a half brother. I could see where Rick might resent his brother, especially since Godfrey had become rich and famous while he had to struggle just to get by.
Even if Rick resented Godfrey, though, was that a strong enough motive for murder? From what Melba told me, it didn't sound likely to me that Godfrey would have put Rick and his family in his will. There was probably no monetary gain, then, from Godfrey's death.
Or could it have been the product of sheer envy, turned deadly by years of disappointment and resentment?
Troubled by these questions, I had a hard time going to sleep. Eventually I drifted off, Diesel by my side.
When the alarm sounded the next morning, I felt groggy and inclined to stay in bed. My sleep hadn't been restful, and I knew I'd be logy all day. Diesel poked his nose close to mine and warbled at me. When I didn't move, he put a paw on my arm and warbled again.
I opened my eyes and glared at him. "Oh, all right. I'll get up. I'm sure you'll faint from hunger if I don't get up and feed you right this minute."
Diesel walked across me and jumped down onto the floor, ignoring my grunt of pain as he stepped on my stomach. That put me in my place, so I got up.
Downstairs I put the coffee on, filled up Diesel's bowls, and cleaned the litter box. While the cat munched happily on his food, I retrieved the paper from the front lawn and sat down to read it. _That coffeepot better hurry up_ , I thought. I needed the caffeine to kick-start my brain into gear.
Godfrey's death was still front-page news, though I was pleased to see the article mentioned no names of suspects. I'd have to thank Kanesha Berry later for keeping my name, as well as those of Julia and Justin, out of the paper. I realized then that the local reporter had not been pestering me any further either, so I owed Kanesha for that, too.
Azalea came bustling in while I was finishing my second cup of coffee.
"Good morning," she said, and I returned her greeting.
"How's Kanesha doing?" I asked.
"Sore as a long-tailed cat in a roomful of rocking chairs," Azalea said. "She better get this thing figured out soon, else I don't know what I'm going to do with her."
"She's under a lot of pressure," I said. "I can understand why she might be feeling stressed out."
"That's why I been hoping _somebody_ might find out something to help her with," Azalea said with a pointed look at me.
"I'm doing my best," I said. "I've been digging, but so far I haven't come up with anything solid. Godfrey managed to make a lot of people angry with him over the years, and it seems like his parents did, too."
Azalea sniffed. "With that trashy mama of his, and that hound dog of a daddy, it ain't no wonder."
"Did you know them?" From the disdain in her voice, I thought Azalea must have had personal experience of the Priests.
"I sure did," Azalea said with a pained expression. "Worked for that woman six weeks or so when I was sixteen. You ain't never heard so much yellin' and cussin' in your life as the two of them, and that poor child having to hear it all. No wonder he turned out like his daddy."
"Sounds pretty awful," I said.
"It was," Azalea replied. "Wasn't no amount of money worth working for them folks, let me tell you. I got me another job fast as I could. That's when I come to work for Miss Dottie." Her face softened with a smile. "She was a true lady, and I loved every minute of working for her."
I knew that Aunt Dottie had treasured Azalea, but I didn't dare say so. This was about as sentimental as I had ever seen Azalea, and I didn't want to offend her by some well-meant but unwelcome comment.
Instead I said, "Yes, she was one of a kind."
Azalea turned back to the stove. "I'm going to be scrambling some eggs and frying up some bacon."
"Sounds good to me," I said. "I think I'll head upstairs for my shower. I'll be back down in about fifteen minutes."
Azalea nodded, and I left her in the kitchen. Diesel followed me up, but he kept on going when I stopped on the second floor. He would make sure Justin was up in time for breakfast.
I was almost finished eating by the time Justin appeared. He ate quickly, explaining that he needed to get to the library before class to meet a friend for a study date. The way he bolted his food down, I doubt he tasted much of it, but I remembered the hasty meals of my own student days and forbore commenting.
Diesel and I spend three Fridays a month at the public library where I volunteer. I fill in as needed, helping with reference, doing a bit of cataloging, and running one of the reading groups for retirees. Today, however, was not one of those Fridays, so I decided to go instead to the archive and poke around some more in Godfrey's papers.
By the time Diesel and I reached the campus, the building was open. I debated seeking Rick out and talking to him about Godfrey, but what pretext could I use? Nothing that wouldn't make me sound like a tabloid journalist on the hunt, I realized. I decided to wait and see if a good opportunity presented itself. Perhaps Rick would attend the memorial service tomorrow.
We made it upstairs without Melba spotting us. I would just as soon she didn't know—at least for a while—that Diesel and I were here today. I wanted to focus on Godfrey's papers, and Melba would only be a distraction.
I shut the door behind us and turned on the lights. The boxes of Godfrey's papers appeared undisturbed, and I hoped that the change of locks would keep them that way.
Diesel made himself comfortable in the window. I eyed the inventory as I sat down at my desk. Where to start?
I didn't want to read more letters this morning, so I decided against starting on Godfrey's business correspondence. While I sat there, I remembered the box of computer disks. I might as well see what was on them and start making an inventory of their contents.
I retrieved the box and set it on my desk. I pulled out one of the containers of disks from inside and opened it. They were the large floppy disks that hadn't been used for years. Under normal circumstances these disks would cause a problem, since few people these days had computers that could accommodate them.
The archive, however, was prepared for just such a contingency. I had a computer that could handle them, and it was loaded with various word-processing programs. I ought to be able to read the contents of the disks with one of them.
This computer was on a desk in a corner, behind a range of bookshelves. I took all the disks with me and turned on the computer. While I waited for it to boot up, I examined some of the disks. They were labeled, and I recognized the words as the titles of some of Godfrey's early books. There were also dates on them, so I could put them in chronological order.
When the computer was ready, I inserted the earliest disk of the group and executed a DOS command to see the directory of its contents. Judging from the file extensions, I didn't think I'd have any trouble opening them. I scanned the directory. There were only twelve files, and they all had numerical names. Chapters one through twelve no doubt.
I opened the file named "one" and scanned through it. I recognized the text of what I thought was Godfrey's first thriller, _Count the Cost_. The change in style from his early, more traditional mysteries, was clear. I closed the file and removed the disk. I didn't see much point in reading through the text of the books, because I wasn't interested in analyzing Godfrey's prose.
There were three disks labeled "Cost." I inserted the third one in the drive and executed the directory command. There were more files names with numbers, but there was one file called "letter." I opened it and began to read.
The letter was addressed simply to "G." I presumed that meant Godfrey. The writer started by thanking G for taking time to read the manuscript and expressed the hope that G would like it enough to help get it published. The letter referred to the title _Count the Cost_.
By the time I finished the letter—unsigned, unfortunately—I was convinced Godfrey had not written a book that bore his name.
**TWENTY-FIVE**
Stunned by the contents of the letter, I stared blankly at the computer screen, trying to get my mind back into working gear.
If this letter wasn't some kind of joke, then the implications were clear. Godfrey had stolen the work of another writer and published it as his own.
But how had he been able to get away with such a thing? Surely the writer, Mr. or Ms. X, would have figured it out. Godfrey even used the same title referenced in the letter.
I read through the letter again, more slowly this time, searching for any possible clues to the identity of the writer.
_Here's the manuscript I told you about when you were here a few months ago. Thanks for taking the time to read it. I hope you'll like it enough to want to help me get it published. It's different from your books—a lot darker and harder-edged—but you said you liked thrillers when you talked to the group. I call it_ Count the Cost _, but that might not be the best title. Any suggestions you have about that would be appreciated, too. I know a catchy title seems to be important, but you know more about the business than I do. At least for now, that is. I'm hoping to know a lot more about it one of these days. Thanks again. I'm looking forward to hearing what you have to say._
There were no real clues to the letter writer's identity, not even a hint of the gender. Two things might be helpful: "when you were here a few months ago" and "when you talked to the group." There was no date in the text of the letter, but then I got the bright idea of looking at the date stamp in the directory of files on the disk.
Before I did that, however, I printed a copy of the letter. Once that was done, I called up the directory and looked at the date: August 3, nineteen years ago. The last time the file had been altered was nineteen years ago.
Nineteen years ago. I thought for a moment.
Justin was eighteen.
Godfrey would have been in Athena roughly nineteen years ago.
Could this mean the letter writer lived in Athena?
He or she must. There had to be a local connection to Godfrey's murder. Otherwise, why was he killed here and not somewhere else?
_Slow down_ , I told myself. _You're jumping to conclusions pretty fast_.
I did a screen print of the directory and clipped it to the letter.
Before I examined any of the other disks, I wanted to check something. This computer was not connected to the Internet, so I went back to my desk. Diesel appeared sound asleep in the window when I glanced at him. I connected to the library's online catalog and searched for Godfrey's name. I wanted to check the publication dates for his books. The library should have all of them in the collection because he was a local writer.
No doubt Godfrey had a website that provided the information, and I would check that later. But I preferred the information from the catalog—it was probably more accurate than the website.
I performed the operations necessary to create a brief citation list of all of Godfrey's titles and printed it out. The citation included publisher and date of publication.
When the printer finished, I examined the sheets. I had sorted the citations in ascending publication order, so I could trace Godfrey's books as they were published, from the first one to the most recent.
His first five books were published within four years, and then there was a four-year gap before his sixth novel, _Count the Cost_. It was published seventeen years ago, and that meant a two-year gap, roughly, between the date stamp on the disk and the actual publication of the book.
Godfrey's first five books were different in style and tone from the later books. Light, amusing, fluffy, they featured a bickering duo of amateur detectives who fought their attraction to each other as they stumbled over dead bodies. _Count the Cost_ signaled an almost radical change, and if I had thought about it at all, I probably assumed Godfrey did it for commercial reasons. He wasn't a bestseller before, as far as I was aware, but _Count the Cost_ made the bestseller list. He had been a fixture there ever since.
In that letter was the reason for the abrupt shift in Godfrey's work.
It wasn't his, plain and simple.
But was that the only one? What about the other fifteen books published in the years since _Count the Cost_?
Diesel stood up from his perch, stretching and yawning. I reached over to rub his head. He rewarded me with a couple of happy chirps. He jumped down and accompanied me as I took the list of Godfrey's books back to the other computer. The book published the year after _Count the Cost_ was entitled _Abide with Me_. I checked the box of disks, and there were three labeled "Abide."
I inserted the first one into the drive and looked at the list of files. Only numbers. I checked the second and third disks as well. Same thing. No letter this time. Diesel rubbed against my legs a few times before he wandered off to prowl around the office. He was not usually destructive so I let him have free run in the room.
I checked the next book on the list: _Dead Men's Plans_. There were four disks for this one. This time I put the disk numbered four in and checked the contents.
Bingo. Another letter, again addressed simply to G. No signature. That was frustrating.
_The arrangement seems to be working out pretty well, though I thought you were at least going to mention me in the acknowledgments. I don't mind you getting all the attention from the media—I hate that kind of thing. But why couldn't you at least include my name somewhere? I expected a bit more gratitude, frankly. The money's good—I'm not complaining about that, but if it weren't for me, your name wouldn't be on the bestseller list, you know. I'm glad it's time for a new contract. I'm going to want some changes, but I'll let you know what they are after I've had a chance to think more about them._
The letter went on to describe, only in the briefest terms, the idea for the next book and asked for feedback on it.
This was pretty clear evidence that Godfrey was putting his name on another writer's work. It also sounded like X was getting a bit restive over the lack of attention.
Why had Godfrey kept these disks? If he'd had any sense, he would have destroyed them years ago. They clearly weren't intended to be part of his archive, since the box was not included on the inventory. Knowing the man's arrogance, however, I figured he never thought it possible that anyone would find out about his deception.
Some overzealous person must have seen the box and simply included it with the rest. Godfrey would no doubt have had a fit if he had known.
But what a stroke of luck. I silently thanked whoever had stuck the box of disks in with everything else.
This was something Kanesha Berry needed to know about, and I planned to tell her.
First, though, I wanted to dig further and see if there were other letters. Had Godfrey continued to put his name to X's work?
I realized I should also check the acknowledgments in all of the books to see if there were any clues in them.
_Did his agent know?_ I wondered. I would have to start keeping note of the questions that occurred to me.
Back to checking the disks. I went through them chronologically. There was no letter included on the disks for the next book on the list. I went on to the fifth book.
Pay dirt. My eyes widened as I read this letter.
_You bastard, I should have known better than to trust you. You haven't changed, still intent on screwing anybody to get what you want. I thought a contract would protect me, but you saw to that, didn't you? I can't believe how naive and stupid I was. I should have talked to a lawyer, but you said it wasn't worth the money. Did your agent really review the contract like you said? Frankly, I don't believe it. I have a good mind to call her and have a little talk with her. But I can't afford to pay back the money, you bastard, and you know that. I'm stuck, but this doesn't mean I won't try to find some way out._
_What kind of contract had X signed?_ I wondered. Godfrey had only too obviously taken advantage of X's naivete and lack of experience, but this sounded bad, even for Godfrey.
There was a mention of contracts in the inventory, but I suspected that there wouldn't be a copy of the contract with X. Even Godfrey wouldn't be arrogant enough—or stupid enough—to include something that potentially damaging with the rest of his papers.
That contract, however, could be the key to everything. Had X finally become so enraged over the unfairness and decided that Godfrey had to die? If only I could find it—or figure out some way of discovering who X was.
The answer might be in the box of disks. All it would take was time.
Stopping long enough only to walk home for a quick lunch, I spent most of the day at the computer, going through every disk in the box.
I found a few more letters, some of them filled with X's anger over Godfrey's behavior, others with a tone of resignation. Occasionally X mentioned the increasing profits the books brought. X apparently had no complaints there.
That was another point to consider. Godfrey had made millions from these books. Even I had read some of the articles in magazines about his lifestyle. X must have made some pretty significant money as well.
Was there someone in Athena living beyond his or her apparent means? Had X resisted the temptation to spend conspicuously? That would be something Kanesha could check better than I—once one of us could put a name to X, that is.
After all the time spent with the disks, I had no solid clue to the identity of X. X had obviously known Godfrey a long time, but there were plenty of people in Athena and elsewhere who had.
There was the mention of a group—probably a writers' group—but there were such groups all over the place.
I believed there was a local connection, though. It was at least a place to start. I could talk to one of the librarians who had been at the public library for nearly thirty years. If there was a writers' group in the area, she would know about it.
I finished printing copies of the letters on the disks and arranged them chronologically. I made sure the disks were organized in their containers as well.
_Time to quit stalling_ , I told myself. I couldn't put it off any longer, though I wasn't looking forward to the inevitable explosion.
I went back to my desk, noticing Diesel once again asleep on the windowsill. I picked up the phone and called the sheriff's department.
**TWENTY-SIX**
When Kanesha Berry walked into my office, I could see the thunderclouds forming. "What is so urgent? I don't have time for some amateur interfering in this investigation, Mr. Harris."
"I understand that, Deputy," I said. "If I didn't think this was significant, I wouldn't have called you away from what you were doing."
She did not appear mollified by my placatory tone. I gestured to the chair by my desk. "Please, have a seat, and let me tell you what I've found."
Behind me on the windowsill, Diesel stirred. He always reacted to a harsh tone of voice, and Kanesha had disturbed him.
The deputy took the proffered seat, but her glare did not diminish.
Before I sat down I handed her a folder of the letters I had printed out for her.
"What's this?" She accepted the folder but didn't open it.
"It could be evidence of a strong motive for Godfrey Priest's murder," I said. "Let me tell you about what I found, and I think you'll agree this is serious."
Kanesha nodded before glancing pointedly at her watch.
"When Godfrey showed up in my office three days ago, he told me he wanted to donate his papers to the college archive," I said. "What I didn't know at the time was that he had already made arrangements to ship his papers here. They arrived the day after his death."
"And you waited two days to tell me about this?" The intensity of Kanesha's glare sharpened.
"Yes, I did," I said. "Godfrey's papers basically belong to the college now. In a letter that came with the papers Godfrey pretty much assigns ownership to the college."
"That may be," Kanesha said. "But that doesn't mean you can suppress information that could be relevant to this case. I have a good mind to charge you with interfering with an official investigation."
"I'm not suppressing it, and I'm not trying to interfere," I said. "There was simply a delay in telling you about them. I realize that's not an excuse, but as the person who will have to process the collection at some point, I wanted a chance to see what it contained. A lot of the content won't be of any use to your investigation whatsoever."
"It's kind of you to make that judgment for me," Kanesha said, the sarcasm dripping from her words. "And how do I know you haven't already destroyed anything in these papers that might link you to the crime? Or link someone else, like Julia or Justin Wardlaw?"
"You don't," I said with what I hoped was a disarming smile. "And if you want to charge me with anything, go right ahead. But first, at least let me tell you what I did find. I think it might be the key to Godfrey's murder."
"Go ahead," Kanesha said. "I'll listen." _But not for long_ , her expression told me.
I picked up the inventory of the papers and handed it to her. "This is the inventory that came with all the boxes. It's very general, which is unfortunate. But the interesting thing is that there is an extra box."
"What do you mean?" Kanesha was scanning the inventory.
"All the boxes were numbered except one. And the numbers match the inventory. The unnumbered box contains computer disks."
"You think that box wasn't meant to be included?" Kanesha handed the inventory back to me.
"Judging by what I found on some of the disks, no, I don't think Godfrey wanted anyone else to see them. I don't know why he kept them, other than his unbelievable arrogance. He probably figured no one would ever see them and he would be safe."
"Safe from what?" Kanesha glanced at her watch again.
"From the letters I found on some of the disks." I pointed to the folder I gave her earlier. "They're all there in chronological order. Take a look at them, and I think you'll see very quickly."
She still thought I was wasting her time. I could see it in her face. She was also angry that I hadn't let her know about the boxes sooner. But after clenching and unclenching her jaw for a moment, she opened the folder and began to read.
I watched. She read quickly, and after the second letter she glanced up at me with a frown. I maintained a bland expression, and she went back to the letters. I believe I had finally piqued her curiosity.
Eight minutes later—I timed her—she was done. She closed the folder and looked at me, her expression thoughtful.
"He basically paid someone else to write the books for him," she said. "And whoever he paid wasn't happy over the terms of the contract."
"Exactly," I said. "I think X—that's what I've been calling the unknown writer—might finally have become so incensed over Godfrey's treatment that he—or she—killed him."
"Uh-huh," she said. She handed the folder back to me. "If X was so unhappy about the contract, why didn't he or she hire a lawyer and take Mr. Priest to court?"
"Not knowing what the contract stipulated," I said, "I don't have a solid answer for that. But reading between the lines, I figure that the contract between Godfrey and X must have bound X to complete secrecy, otherwise the deal was off."
"That's a possibility, I suppose," Kanesha said. "But what's at stake here? Mr. Priest's reputation, of course, but what about money? How much could he make from the books?"
"While I was waiting for you to arrive, I did some research on the Internet," I said. "I found an article published a little over a year ago that ranked the top-selling American writers by their projected annual incomes. Godfrey was in the top ten. According to the article, his annual earnings were in the neighborhood of twenty million dollars."
Kanesha wasn't expecting that. Her eyes popped wide open. "That's significant money," she said. "How could he make so much?"
"For one thing, the books are published in something like thirty languages, and they apparently sell really well all over Europe, and in Japan, too. Then there are the movie adaptations. If Godfrey had a cut of the profits, that could add up to a lot of money, too. Several of the films based on the books have been big hits, both domestically and in foreign countries."
"Twenty million dollars a year." Kanesha shook her head as if she still couldn't take it in.
"X had to know the books were generating huge income," I said. "And what if his cut was small compared to what Godfrey was raking in? Add to that the fact that he's not getting any credit for his work, and he might have become more and more frustrated every year."
"It's possible," Kanesha said. "I grant you that. And it makes as much sense as anything else I've been able to discover. But how the heck am I going to figure out who X is? I don't even know where X lives."
"I think X lives in Athena," I said. I explained my reasoning, and Kanesha picked up the folder again and glanced through the letters.
"It makes sense," she said when she finished. "Now all I have to do is track down some writers' group that X belonged to." She rolled her eyes. "Talk about looking for needles in haystacks."
"I can help with that," I said. "There's a librarian at the public library who's been there for about thirty years. If anyone would know about local writers' groups, she would. Her name is Teresa Farmer."
"I know her," Kanesha said. "She does the summer reading program for kids."
"That's her," I said. I tapped the inventory list on my desk. "We can also look in the boxes that contain contracts. There might be some information there. And you can always talk to Godfrey's agent. She's supposed to be here for the memorial service tomorrow."
Kanesha nodded a couple of times. "I already have an appointment with her. She gets into Memphis late tonight and is driving down tomorrow morning."
"Good," I said.
"Why are you doing all this?" Kanesha regarded me, her eyes narrowed.
"You mean poking my nose into your investigation?" I said, trying not to sound too flippant.
Kanesha nodded.
"Natural instincts, I suppose." I shrugged. "Librarians are trained to help find answers, and the identity of Godfrey's killer is an important question. Plus I find myself involved because of Justin and Julia Wardlaw. I can't believe that either of them killed Godfrey, and I want to do what I can for their sakes."
I wanted to add—but knew I didn't dare— _And because your mother asked me to, and I couldn't figure out a way to say no_.
"If anything you've done compromises my investigation in any way, you are going to be in a lot of trouble. Are you clear on that?"
I resisted the urge to salute. Kanesha sounded like a drill sergeant dealing with a bunch of raw recruits fresh out of the cotton patch. I hadn't expected her to turn suddenly effusive with gratitude, but I guess I shouldn't have been surprised she was taking a dim view of my "interference."
"Yes, I am," I said. "But let me ask you this: If I had called you the minute these boxes of papers arrived, what would you have done? Would you have rushed over and impounded them, or whatever the term is, for your investigation? And would you have gone through those disks the way I did? Do you even have a computer that can handle old floppy disks?"
"Whoa." She held up a hand. I had gotten a bit carried away. "You might have saved me some time on these disks, but I still can't be sure that you haven't destroyed other evidence. Frankly, how can I even be sure that you didn't create this stuff about X yourself as a smoke screen?"
Aghast, I stared at her. Never would I have thought she'd react this way. I didn't know what to say.
"By not letting me know the minute these papers arrived, you basically tainted the evidence—if any of this could be called evidence. If I could have been assured that the contents of these boxes were untouched once they arrived here, I'd feel a lot more comfortable with all this. But you took it on yourself to investigate, and now I'm left with a difficult situation."
"I'm sorry," I finally managed to say. I had never thought about any of this, and I realized I had goofed big time. "I really don't know what to say other than I'm sorry."
"Where are the disks?"
They were still in the box back by the other computer. I retrieved them without a word and handed the box to Kanesha. She looked inside.
"I see what you mean about old disks," she said.
If that was some kind of olive branch, I'd accept it.
"I'll take them," she said. "If you can make out a receipt, I'll sign it. But these need to go with me."
"I'll do that," I said. "But what about the rest of Godfrey's papers?" I realized suddenly that I should tell her about the unauthorized person who had been in my office Wednesday night. She was going to be even happier with me.
"Where are the rest of the boxes?" she asked.
"In a storeroom down the hall," I said.
"Is it secure?"
"It is now," I said. "I have something else I need to tell you."
"Go ahead." She had one of those pained _What now?_ looks on her face.
As briefly as I could, I related the facts about the unauthorized visitor.
"And you still didn't call me." She examined me like I was an exotic insect that she found distasteful.
"No," I said. "But I made sure the locks were changed that day. There are only three sets of keys. I have one, of course. Melba Gilley, the library director's assistant, has a set, and Rick Tackett, the head of operations for the library, has the other set."
Kanesha continued to regard me like a specimen.
"By the way," I said, feeling more uneasy by the second, "did you know that Godfrey had a half brother? It's Rick Tackett. Do you know him?"
"I don't know him," Kanesha said. "But I knew of the relationship. You might find this hard to believe, but I did make an effort to find out as much as I could about the victim and his family right off the bat."
"Sorry," I said, abashed.
She simply stared at me, and I tried not to squirm. Feeling like a delinquent ten-year-old was a new experience for me.
"For now we will move these boxes into the storeroom with the others, and I will seal the room until I can send someone here to pick them up and bring them down to the sheriff's department."
"Okay," I said. "Sounds like a good idea to me." I wasn't going to argue. If the college for some reason had a problem with this, I would deal with it later.
I had an inspiration. I found some of the cotton gloves I used to handle rare books and offered her a pair. She accepted them with a nod, and then we moved the boxes from my office down to the storeroom.
Diesel, very curious about where we were going with the boxes, followed us back and forth until we completed the move.
"Since we're here," I said, waving a hand to indicate the storeroom, "why don't you have a quick look at some of the contracts? That would answer one question pretty quickly."
"Meaning you want to know, so you want me to do it right now and tell you before these boxes are removed." Kanesha didn't smile, but I could have sworn she was the tiniest bit amused. That was a surprise.
"Well, yes," I said.
"Go get the inventory and the box of disks, and I'll stay here," Kanesha said. "I'll have a look at the contracts."
I nodded and went back to my office to retrieve the inventory. Diesel remained with Kanesha. He was prowling around the boxes as I left the room. I thought Kanesha might object. She ignored him, however.
I came back with the box of disks and the inventory. "The contracts are in box twelve," I said. "I looked already." I set the box down on the floor.
We both turned to hunt for box twelve. Kanesha spotted it first, at the bottom of a stack of four. I helped her move the three on top, and she pulled box twelve out of the way.
"It's pretty light," she said, frowning up at me.
She squatted down and removed the lid.
It was empty.
**TWENTY-SEVEN**
If I could have crawled into that empty box and pulled the lid over me, I would have. Kanesha's expression was implacable enough to stop a charging rhino in its tracks.
She stood after replacing the lid on the box. She walked out into the hall, and I fancied I could see the anger in every step she took. She turned and waited for Diesel and me to join her in the hall.
"Please lock the door," she said.
When I had done so she held out her hand for the keys. "I'm sorry," I said as I complied.
She didn't respond. Instead she walked down the hall to my office, unlocked the door, and stood pointedly by the door. Diesel followed me as I went down the hall and into the office.
"I'll be back in a few minutes," she said. "I'll return your keys then." She headed for the stairs.
I went to my desk and sat down. I had really screwed things up, and Kanesha had every right to be furious with me. I had let myself get too caught up in the situation and hadn't thought things through clearly enough. I wasn't one of the Hardy Boys, happily assisting my famous detective father.
As a basically law-abiding citizen, however, I had interpreted what I saw as my civic duty to assist the deputy in her inquiry a little too broadly. I did think I had helped her in some ways. How long would it have taken her to find the letters on those disks, for example? But did that outweigh my blunder in allowing someone to steal a whole boxful—if not more—of Godfrey's papers?
To distract myself from spinning mental wheels to no effect, I turned to my computer to check my e-mail. Diesel, seeming to sense my inner turmoil, kept rubbing against my legs and purring. I scratched his head and, as always, that made me feel better. Seeing his pleasure from the attention was an effective calming agent.
After a couple of minutes of scratching, Diesel pulled his head away and climbed into his window seat. Still purring he settled down for a siesta while I tried to focus on work.
As I read through my e-mail, I heard Kanesha return, but I didn't look up from my task. Some minutes later I was aware that she entered my office, and I swiveled in my chair to face her.
"Here are your keys," she said as she placed them on my desk. "I've put an official seal on the room, but I'll be sending someone here within the hour to remove those boxes to the sheriff's department. If you will have a receipt ready, I'd appreciate it."
I glanced at her face. Her expression had lost some of the rigidity it had earlier, and I relaxed a bit myself. Maybe she wasn't going to bless me out after all.
"I'll be glad to do that," I said.
"Fine." She glared at me a moment. "I realize that you had good intentions, Mr. Harris, and generally we appreciate cooperation from the public. But you stepped too far over the line. You realize that, don't you?"
"Yes, I do," I said. "I can't tell you how much I regret not notifying you right away about Godfrey's papers. I can only hope this won't cause serious problems for your investigation."
She listened, but when I finished, she simply nodded and walked out.
After that I tried to focus on my e-mail, but it was no use. I was still too unsettled by what had transpired between Kanesha and me. I glanced at my watch. It was almost four-thirty. _Might as well get out of here._
"Come on, boy, time to go," I told Diesel as I shut down my computer.
Yawning, he sat up and stretched. He stood patiently, as always, for me to put him in his harness, and a few minutes later we were ready to go. The afternoon was cool but sunny when we left the building. During the brief walk home, I thought about what I might do this evening.
A quiet night at home would be just the thing. That's what I told myself, but a little niggling voice kept insisting that there was something else I could do.
Teresa Farmer, the librarian I mentioned to Kanesha Berry, was usually at the public library until six on Friday evenings. I had time to go over there and have a quiet chat with her and find out what she might know about local writers' groups.
This would mean treading on Kanesha's toes again, but I knew I could trust Teresa's discretion. If I told her why I was asking, she would not talk about it to anyone until the deputy asked for her assistance.
When you want something, you can generally come up with the reasoning to justify it, I have discovered over the years. Even when you know you shouldn't.
At home all was quiet. I let Diesel have time to use the litter box and eat something before heading to the public library in the car. I contemplated leaving him home, but if I walked through the front door of the library without him, I would have to look at any number of disappointed faces. Diesel was very popular there.
I pulled into the parking lot at the library a few minutes after five. Diesel walked ahead of me, pulling a bit on the leash, eager to go inside. He enjoyed the public library because of the attention he always received.
The first few minutes inside we spent accepting greetings from some of the children who were there, not to mention the adults on the library staff. Teresa was not at the reference desk, and I was afraid for a moment that she wasn't at work today.
But a few minutes later she appeared from the office area behind the reference desk, alerted no doubt by the increase in noise. A petite dynamo a few years my senior, Teresa smiled broadly when she discovered the reason for the noise.
As soon as I could I extracted Diesel from his cadre of young admirers and led him behind the reference desk where Teresa waited. She had three cats of her own, and she was as fond of Diesel as anyone here.
"Charlie, what are you doing here? This is an unexpected pleasure," Teresa said. "And Diesel, how are you?" She squatted in front of the cat in order to give him some attention, rubbing his head affectionately.
Diesel purred and warbled while I explained.
"I came to see you," I said. "I need your help with something."
Teresa stood. "Sure, come on back to my office."
Diesel and I followed her. Teresa was the head of reference for the library as well as the assistant director. She also supervised the library's few volunteers, and I had worked closely with her for almost three years now.
She sat down behind her desk and motioned for me to take a seat across from her. I did so and unhooked Diesel's leash from his harness. He padded around the desk and climbed up into Teresa's lap. When he sat up his head was actually a bit higher than hers, and I had to smile at the sight.
"What can I do for you, Charlie?" Teresa said as she rubbed Diesel under the chin.
"It has to do with Godfrey Priest," I said.
Startled, Teresa looked at me. "That's odd," she said.
"How so?"
"I had a call just a few minutes ago from Detective Berry," she said. "She's coming in tomorrow morning to talk to me about something to do with Godfrey. She didn't say what, exactly, just that she needed some information and someone had suggested me to her. Was that you?"
Kanesha had acted more quickly than I expected. At least she had accepted my suggestion, I thought.
"Yes, it was," I said. "I'm being really naughty in coming to talk to you before she does, but I'm letting my curiosity get the better of me, I'm afraid."
Teresa laughed. "I promise not to rat on you. What is it you and Deputy Berry want to know?"
"Information on local writers' groups," I said. "If there are any, I figured you'd be bound to know."
"Thanks," Teresa said. "We do try to keep track of any community activities to be prepared for the inevitable questions."
"I know," I said, grinning. "I'll never forget the time I got a call from a woman—this was in Houston—who was looking for information on an organization for cats." I had to laugh, just thinking about it.
"What's so funny about that?" Teresa asked.
"She had heard about a group that knitted socks for cats, she said, and she wanted to join them," I said. I chuckled again.
Teresa joined in my laughter. "I can't imagine one of my cats allowing me to put socks on her or him. They'd have a fit."
"I thought it was pretty funny," I said. "But of course I couldn't tell her that. So I found her the name of a contact person for a local cat fanciers' group. I never heard whether she found what she was looking for."
"At least you gave her something," Teresa said, still smiling. "Now, about writers' groups. Yes, I can think of several. There's one group that's been meeting here at the library for about twenty years. They're all poets, though, and somehow I don't think that's what you're looking for. Not if it has something to do with Godfrey Priest."
"Right," I said. "I want to know if there was a group he was ever a part of, or maybe whether he spoke to local groups when he came back to Athena."
"And you can't tell me exactly why you want this information?"
"No, I can't," I said with regret. "You don't mind, I hope."
"I can live with it," Teresa said wryly. "Okay. Godfrey Priest and writers' groups." She frowned as she thought. By now Diesel had settled down in her lap, his head against her chest as he purred in deep contentment. Teresa stroked his head gently.
I kept quiet while she dredged through her memory banks. She had amazing recall—one reason she was such a terrific reference librarian. If there were something to find, she'd find it.
"It has to be at least twenty years ago now," Teresa said. "Godfrey Priest hasn't spoken at this library in at least that long. He did participate in a fund-raiser we had about seven years ago, spoke at a Friends of the Library dinner, but that was it."
"What about twenty years ago?" I said, prompting her gently when she fell silent again.
"There was a group that met here occasionally back then," Teresa said. "Seven or eight people, I think. They weren't together that long, or at least they didn't ask to use our meeting room for long. They could have continued meeting somewhere else."
"Do you recall who was in the group?" I kept my fingers crossed.
"I can do better than that," Teresa said with a smile. "I can show you a picture of them." She scratched Diesel's head. "But you're going to have to let me up." Diesel sat up, butted his head against her chin, and jumped to the floor at her gentle urging.
"A picture would be great," I said as Diesel came around the desk to sit by my chair.
Teresa got up from her desk. "I'll be back in a minute. What I want is in one of the filing cabinets behind the reference desk."
Diesel and I waited quietly for her return. She was gone less than five minutes.
When she returned she handed me a folder. I examined the label: "Library Annual Reports."
"I put the relevant one on top," Teresa said as she resumed her seat behind the desk.
I extracted it from the folder and laid the rest aside on top of her desk. The report consisted of only a few pages, and it was on page four that I found the photograph Teresa wanted me to see. It was rather small, and the caption only said, "Writers' Group Meets with Local Novelist."
In the center of a group of six people was Godfrey Priest—looking much younger and much less successful than he did when I saw him a few days ago. That was only natural. This picture was taken before he hit it big.
I examined the faces of the others in the group. I recognized two of them right away, and I was stunned as I put the names to the faces.
Julia Wardlaw and Rick Tackett stood on either side of Godfrey, both smiling into the camera.
**TWENTY-EIGHT**
"You look shocked," Teresa said. "Is anything wrong?"
"I'm just really surprised," I said. "I see two people in this group I never expected to see. Two people I had no idea were interested in writing."
I examined the other faces in the group. Two of them looked vaguely familiar, but I couldn't quite place them. If only the caption to the picture had included their names.
I was about to hand the report back to Teresa to ask whether she knew who they all were when I spotted something odd in the picture. I held it closer and squinted. The resolution wasn't great, but I thought I saw the top of another head peeking out from behind Julia's shoulder, the one next to Godfrey.
"Looks like there's another person here in the background," I said. I held the report across the desk to Teresa. "See what you think. Also, do you know who all the people are?"
Teresa examined the picture for a moment before laying the report aside. She opened one of her desk drawers and rummaged through it. "Ah, here it is," she said. She brandished a magnifying glass. She picked up the report again and examined the picture with the aid of the glass.
"I think you're right," she said after a moment. "That does look like someone's head. It's odd, though. Why wouldn't whoever it was want to be visible in the picture?"
"Beats me," I said. My heartbeat picked up though, because I wondered if the mystery person behind Julia was X. Based on the letters I had read, X shunned the spotlight, and it could be that he or she avoided having photos taken.
Teresa laid the glass aside and looked at the picture again. "I recognize all of them," she said. She named them, and in addition to Julia and Rick, I recognized the names of a couple of professors at Athena, one from the history department and the other from English.
"Would you mind writing those down for me?" I said. "And do all of them still live in the area?"
"One of them passed away a few years go," she said. "I'll put an _X_ next to her name. But the others—except Mr. Priest, of course—are still around."
"Thanks," I said. "I really appreciate your help with this. I can't tell you how, or why, but this may be the key to Godfrey's murder."
"That's definitely intriguing," Teresa said. She finished writing and handed a piece of paper across the desk to me. "I'm sure it would help if we knew who the other person was lurking in the background. I've been mulling it over, and I seem to remember that there were a couple of people who met with the group a few times, but the six you see here were the core. They met together for four or five years, I think."
"I know one of the people in the group pretty well," I said. "And I work with another one."
"That's right, Rick Tackett works at the college library," Teresa said. "He's a nice man, pretty quiet. Reads a lot. I hope he's not involved in this."
"I hope so, too," I said. "I agree he's a nice guy. But I think one of the people here may very well be the one Deputy Berry is looking for."
"That's unsettling, to say the least," Teresa said with a frown. "I hope she manages to figure it out soon. If one of them comes in the library before she does, I think I'll be a bit nervous."
"No need to be, I'm sure," I said. "There's no reason for anyone to think you're involved in any way."
"Other than assisting the official inquiry, you mean." Teresa's smile was impish. "And the unofficial one."
"Yes," I said, hoping that my face wasn't turning pink. "I appreciate your help, but I think Diesel and I ought to head home now." I stood, and Teresa came around the desk to shake my hand. "I'll see you next Friday, of course."
"We always look forward to it," Teresa said as she escorted Diesel and me out of the office. "Our volunteers are a huge help, and we definitely appreciate what you do for us." She bent to rub Diesel's head. "And you too, big guy."
Diesel chirped at her, and I smiled as I led him away. We managed to make it out the door after only a few minutes' delay for more attention to Diesel. He loved every second of it, the ham.
Back in the car I examined the list of names for a moment while Diesel settled down on the passenger seat beside me. I might as well start with Julia, I reckoned. Seeing her in the picture had really knocked me for a loop. Her connections to this case were so strong, and though I was sure she had to be innocent of Godfrey's murder, I knew her having been part of the writers' group might make Kanesha Berry focus more intently on her.
Rick Tackett seemed like a stronger possibility in many ways. He was Godfrey's half brother, for one thing, and as the library's custodian, he had easy access to my office and to the archive storeroom. No one would have thought twice about it if he had been spotted upstairs near the storeroom on the evening when someone had obviously entered my office and examined the boxes.
I had to hope that whoever it was hadn't destroyed the contracts. If Kanesha could find those in his—or her, I added, to be completely fair—possession, that would be an important link to the murder.
Surely, however, there were other copies of the contracts. Godfrey's agent had to have copies. I brightened at that thought. The agent would be at Godfrey's memorial service tomorrow. By then she would already have talked with Kanesha, and perhaps I could slip in a few questions without objection.
On the short drive home I pondered the questions I wanted to pose to the agent. How should I start? What preface to my questions could I use to disarm her enough to talk to me?
One big question occurred to me right away, and I knew I would have to be very careful in asking it. Did the agent know that Godfrey wasn't writing the books himself?
Then I remembered that Kanesha would probably be asking her that question, not to mention countless others, tonight. I would try and see how far I could get.
Julia's car was parked in front of the house, and Diesel and I found her in the kitchen. We exchanged greetings as I released Diesel from his harness. He went to greet Julia before trotting off to his litter box.
"I came to pick up Justin," Julia said. "He's coming home to have dinner with us, and he'll probably spend the night."
Justin often spent Friday nights with his parents, and that meant I had the house to myself one night a week. At least until the spring semester, I reminded myself. My other boarder would be back from his semester abroad then.
"How is Ezra doing?" I asked. "Would you like something to drink while you wait?" I went to the fridge for a diet drink.
"No, thank you, Justin should be down any minute," Julia said. "Ezra is doing okay. Very happy to be home from the hospital, naturally."
"Good," I said, popping the top on the can and having a swallow. I came to the table and sat down. "I've come across something interesting, and I'd like to talk to you about it, if you have a moment."
Julia frowned. "This isn't a good time, I'm afraid. I really need to get back to Ezra. Justin needs to get a move on."
"I understand," I said. "But when you do have a moment, it's important."
"Okay," Julia said. "Perhaps when I bring Justin back tomorrow morning. It depends on how Ezra's doing, though."
"Of course," I said. I had to be content with that. Julia obviously wasn't in the mood to talk.
Justin came clattering down the stairs then, his backpack slung over his shoulder and his hair over his eyes. "I'm ready, Mama." He spotted me then and said, "Evening, sir."
"Good evening, Justin. I'll miss your company at dinner tonight," I said, and I realized I meant it. I had grown accustomed to having someone at the table with me—besides Diesel, that is.
"Thank you, sir," Justin said, coloring slightly. He bent to pet Diesel, who had reappeared in the kitchen the moment he heard Justin coming down the stairs.
Julia stood. "We'd better be going. I'll see you tomorrow morning, Charlie."
"I look forward to it," I said.
Julia flashed me a questioning look, but she didn't linger. I hoped she would have time to talk in the morning. For now, however, I would have to curb my impatience.
Diesel followed them to the door, and I heard Justin say good-bye to him before the door shut. The cat came back into the kitchen as I was checking to see what Azalea had left for tonight's dinner.
There was a roast in the oven, along with a baked potato wrapped in foil. A pot of green beans on the stove rounded out the meal. I sighed in contentment. Azalea's roasts were tender and delicious, and I looked forward to my dinner.
On the way upstairs to change clothes, I considered calling Rick Tackett, but I quickly rejected the idea. I could think of no reasonable pretext for calling him out of the blue—for he was probably home by now. Plus I knew such a call could get me into deeper trouble with Kanesha Berry. I discounted my chat with Teresa Farmer and the forthcoming talk with Julia. Concerning the latter, I figured it possible that Kanesha might talk to Julia before I did. She often didn't bring Justin back to my house until lunchtime or after on Saturdays. If Kanesha talked to Teresa early enough tomorrow, she would probably get on to Julia right away.
My son Sean called as I was ready to go downstairs for my dinner. Instead, I sat on the bed and chatted with him for nearly half an hour. That was a long call for Sean. Our conversations usually lasted no more than ten minutes, but tonight I sensed that Sean needed to talk, and I wasn't going to hurry him.
By now he had heard about Godfrey Priest's murder, and I told him of my involvement. Sean, in his second year out of law school, worked for a large firm in Houston that specialized in civil law. He expressed concern for me, and I assured him I was fine.
He kept talking about innocuous things, but all the time he spoke I sensed an undercurrent. Finally, I decided to ask him outright what was wrong.
Sean sighed into the phone. "I don't know, Dad. It's a number of things. The job, for one. It's not really what I thought it would be, and the hours are crazy. I work all the time."
"It's hard, I know. Those big firms work junior lawyers really hard for the first few years." Now that he was talking openly, I could sense a certain amount of relief.
"Yeah, that's part of the problem. It's going to be years before it gets any better, and I'm not sure I'm cut out for this."
That concerned me, because Sean knew he wanted to be a lawyer from the age of twelve when he first read _To Kill a Mockingbird_.
"You wanted to be Atticus Finch," I said.
"I did," Sean said. "I was pretty naive, wasn't I?"
"Idealistic," I said. "There's a difference."
"Well, it's hard to hold on to your ideals when you're working on cases involving millions of dollars and representing big companies who are trying to sidestep the law any way they can."
"What are you going to do about it?" I asked.
Sean didn't answer for a moment. "I'm not sure. I'm still thinking about it. I thought I might spend a couple weeks at Christmas with you, if that's okay. Do you think Laura's coming home then?"
"She hasn't said yet, but I certainly hope she will. And you know, son, you can come and stay as long as you like. There's plenty of room." I didn't dare be too effusive. Sean turned prickly over displays of paternal emotion. He had always been closer to his mother. "Thanks, Dad," he said, the relief obvious in his voice. "I'll let you know when I can get away."
"Good. I'm looking forward to seeing you," I said. I actually hadn't seen him since his law school graduation over two years ago. He was always too busy to visit me, and whenever I suggested coming to Houston, he put me off.
We chatted a few more minutes, and when I put down the phone I was thoughtful. Sean was in distress, and I wanted to help him. I would have to wait until the holidays, though. I tried not to dwell too much on the possibility that he might leave Houston permanently for Athena. I didn't want to be disappointed. By December he might change his mind about even coming here for the holidays.
Dinner was every bit as delectable as I expected, and when I finished I thought ruefully about that third helping of roast. I felt discomfort in my stomach, and I scolded myself for overeating. I put it down to my concern for Sean. I had always been a stress eater.
I had a restless night as well, partly because I'd overeaten, but in large part due to worries about my son. When I rose the next morning, bleary-eyed from not getting enough quality sleep, Diesel hopped out of bed, perky as ever. On mornings like this he reminded me of one of my college roommates, who invariably rose from bed chipper and happy. There were times when I could cheerfully have whacked him over the head and stuffed his body in the closet.
Diesel was safe, however. He was much too fast for me.
On Saturday mornings I pottered about the house once I had read the newspaper and eaten my breakfast. Sometimes I worked in the yard, and I knew a couple of the flowerbeds in the backyard needed attention. I was not the world's most enthusiastic gardener, but I knew it would do me good to be out in the clear, cool air, engaged in a useful activity.
Besides, Diesel loved exploring the backyard. The lot was large, and there were plenty of spots for an enterprising feline to delve into in hopes of finding something fun to play with. As I weeded the flowerbeds, Diesel popped into and out of them, batting fallen leaves about and cheering me up to no end.
Near noon I decided to break for lunch. There had been no sign of Julia and Justin, and I hoped they would appear soon. I was eager to talk to Julia about the writers' group.
As I was washing my hands in the kitchen sink, I heard the front door opening. Justin had a key, so I assumed it was he and Julia. Diesel scampered off. He would accompany Justin upstairs, I was sure.
"Good afternoon," Julia said moments later, as she paused in the doorway. "You look like you've been working out in the yard today."
I glanced down and saw the streaks of dirt on my old khakis. "Weeding flowerbeds while Diesel stalked the jungle in search of dangerous leaves."
Julia laughed at that.
"Come in and have a seat," I said. "Would you like something to drink?"
"I'm fine," Julia said as she came to the table. "We finished lunch a short time ago. Justin was anxious to get back. He has a paper due for his English class on Monday."
I filled a glass of water from the tap and sat down at the table. "How are things?"
"Okay," Julia said. "Though we had a visit this morning from Kanesha Berry."
"I see," I said. "I have an idea what you might have talked about."
"How would you know?" Julia asked. "Is she taking you into her confidence?"
"Not exactly," I said wryly. "But I did manage to find out a few things that she didn't know."
"Something to do with a writers' group that I used to belong to." Julia said it flatly. She looked annoyed, whether with me or Kanesha, I wasn't sure.
"Yes," I said.
"Why all the interest in something that happened twenty years ago?" Julia frowned. "I can't see what my belonging to that group for a couple of years has to do with anything." She paused for a moment, a faraway look in her eyes. "Though that _is_ when I had my fling with Godfrey, the Lord forgive me, and got pregnant with Justin."
"I can't really say why Kanesha is interested, or why I am either," I said. "But I do think it's important. I never knew you were interested in writing."
Julia shrugged. "I tried my hand at several things back then, trying to figure out what I could do besides being a preacher's wife. I'd always made good grades in English, so I assumed—wrongly, as it turned out—that I had potential as a writer." She laughed suddenly, a bitter sound. "I had visions of becoming the new Phyllis Whitney or Victoria Holt. Not only were books like that not being published anymore—unless you were Phyllis Whitney or Victoria Holt—but I wasn't very good at writing them. Godfrey might have been a jerk in many ways, but at least he convinced me to stop wasting my time."
"You weren't interested in writing thrillers?" She had sounded sincere when talking about her writing, but I needed to be sure she wasn't X and trying to put me off the scent.
"Heavens, no." She laughed again, this time sounding amused. "I almost never read them. I never had a desire to write them, I promise you."
"Good," I said. "What about the other members of the group? Were any of them interested in writing thrillers?"
"Not that I recall," Julia said. She thought for a moment. "Rick Tackett was writing a book about Vietnam. I think it was therapy for him, more than anything else. The other two women in the group were writing romance novels, and one of them was working on a western. The history professor—I think he's actually teaching Justin this semester—was writing this horrendously awful historical novel about an oversexed druid in ancient Britain."
"That's six of you," I said. "Were there others in the group?"
"Occasionally," Julia said. "We had three people join for a brief time, if I remember correctly, but they never lasted."
"Do you remember who they were?" I was thinking of the person lurking behind Julia in the photograph. "Someone who might have been part of the group when Godfrey spoke to you twenty years ago?"
"That's what Kanesha Berry wanted to know," Julia said, her head tilted to one side.
"Oh," I said. "And did you have an answer for her?"
Julia looked at me for a moment. "There was this strange little man who came a few times, but he never showed us any of his writing. Shortly after Godfrey talked to us, he stopped coming."
"Who was he?" I said. I had the feeling Julia was deliberately dragging this out.
"He was one of our classmates in high school," Julia said. She paused for a moment, and I thought I would have to prompt her again. Then she spoke. "It was Willie Clark. He always was an oddball, you know."
**TWENTY-NINE**
"Willie." Of course, I thought. Who I had seen the day Godfrey died, scribbling away at something in the staff lounge.
Then I put another piece of the puzzle together. The misogyny of the books. Who had a reputation for it? Willie did. I remembered the conversation I had overheard the other day in Hawksworth Library. Willie didn't like women, while Godfrey did.
Perhaps I was jumping to conclusions, but for me, that clinched it. Willie was the X who wrote the books.
And who had a powerful motive for killing Godfrey.
"Charlie." Julia's voice brought me back to earth. "What is it? Why are you so excited about Willie?"
I tried to restrain myself. I didn't want to give away anything to Julia, not without talking to Kanesha Berry first.
"I can't really say," I told her. "But knowing that Willie was part of your group, even briefly, helps fill in some missing pieces of the puzzle."
Julia scrutinized me for a moment, as if she were trying to read my mind. "It's the oddest thing," she finally said.
"What is?" I asked when she fell silent again.
"About Willie," Julia replied. "Now that I'm thinking about it, I could have sworn I saw him at Farrington House on Tuesday."
"You did?" This was even better—a witness to place him near the scene of the crime.
Julia nodded. "I think it was him. You know how it is, when you're in a hurry and you catch sight of someone in the corner of your eye. I don't think it really registered at the time who he was." She paused and closed her eyes for a moment, as if trying to visualize the scene. "As I was leaving, I was aware of someone in the revolving door, entering the hotel. But I was in a hurry to get to the bank and then back to the hospital, so I didn't think much about it at the time."
"And it was Willie?" This put both Willie and Jordan Thompson in the hotel. I knew Jordan had seen Godfrey. The signed and dated copies of his new book were evidence of that.
"Yes, I'm pretty sure, the more I think about it," Julia said.
If Willie was the killer, he saw Godfrey after Jordan did. According to her, she didn't stay that long with Godfrey. Then in comes Willie with a strong grievance over Godfrey's treatment of him. Perhaps he had wanted more money for his part of the deal, or maybe he simply was tired of the anonymity of his position and wanted recognition.
Whatever the motive, he might have become enraged by Godfrey's attitude and struck Godfrey down on impulse.
Yes, that sounded like a believable scenario.
"When you talked to Kanesha about the writing group," I said, "did you happen to mention that Willie was a member for a while?"
"Yes," Julia said. "She had a picture with her. Actually an annual report from the library. I had forgotten all about that picture. Willie was there that day, I remembered, but he hid behind me. At the time I thought it was peculiar, but you know how he was in high school. Always scurrying from one place to another, trying not to be noticed."
"So the football team wouldn't pick on him, as I recall." Willie's life in high school had to have been pretty miserable. "And Godfrey was one of the worst." How ironic that was, if I was right about Willie being X.
"Yes, he was." Julia sighed. "He really was an out-and-out bastard a lot of the time."
"You need to tell Kanesha that you saw Willie at the hotel that day."
"Of course. As soon as I get the chance." Julia glanced at her watch. "Perhaps I'd better go hurry Justin along."
"Are you going somewhere this afternoon?" I asked.
Julia nodded. "Godfrey's memorial service. I promised Justin I would go with him." She gestured at my clothes. "Doesn't look like you were planning to go."
The moment Julia mentioned it, I realized I had forgotten all about it. I checked my watch. It was 12:32. If I hurried, I could clean up and get dressed and still make it to the service just about on time.
"I can't believe I forgot about it," I said, rising. "If you'll excuse me, I'll run upstairs and get ready. I'll see you and Justin there." _So much for lunch._ But there would be food after the memorial service, I remembered.
"Good. We'll save a spot for you, if we can. I expect a lot of people will turn out, just out of curiosity."
"Probably so," I said. "See you soon." I hurried up the stairs.
I met Justin on the second floor landing. He was wearing a dark suit and looking pale but composed.
"Hello, sir," he said. "Are you coming to the service?" He eyed my clothes with doubt.
"Yes, just running a little late," I said. "I'll see you there. Was Diesel with you?"
"He was," Justin said, pausing on his way down the stairs. "But he disappeared while I was in the shower." He hesitated, as if he was about to add something, but then he turned and continued down.
Diesel was napping on my bed, his head on one of the pillows. He opened one eye when I came in the room, regarded me for a moment, then shut it again. His tail twitched a couple of times while I took off my clothes, but after that he appeared to be sound asleep.
Just as well, I thought. The memorial service was one place I shouldn't really take him. I hoped he would stay asleep while I got ready.
I took a very quick shower, and as I was toweling off, I reconsidered my decision not to take Diesel with me. I remembered Justin's hesitation before he went on down the stairs. This memorial service was bound to be difficult for him, and I guessed he might have been planning to ask me to bring Diesel. Cat and young man seemed to have a special bond, and Justin needed support right now.
Diesel could come with me after all. For Justin's sake.
I dressed quickly into one of my own dark suits. Diesel woke up when I sat on the bed to tie my shoes. "Come on, boy," I said. "Let's go."
Diesel hopped off the bed and was at the door in a flash. He knew those words too well.
I glanced at my watch as I hurried down the stairs, Diesel ahead of me. It was 12:52. I would just about make it.
I had Diesel in his harness in record time, and then we headed out the door. It would be just as fast to walk to the college chapel as to drive and try to find a place to park, I reasoned.
We set off at a brisk pace, and the carillon on campus was chiming one as we approached the chapel, which was down the street from the library buildings.
Campus police were in evidence, as well as members of the sheriff's department and the city police force. I spotted all three uniforms moving among the crowd of reporters and photographers on the lawn outside the chapel. I should have realized that Godfrey's memorial service would attract the media. As far as I knew, however, they were still unaware of my role in the case. I really owed Kanesha Berry for that.
Diesel and I weren't the only late arrivals, though I was the only one accompanied by a cat. Diesel's presence occasioned a few frowns, but I didn't care. Justin mattered more than what these people thought.
A couple of reporters tried to get my attention, probably because of Diesel. I knew cameras were busy snapping shots of us as we hurried up the walk toward the front door of the chapel. One reporter with a microphone and a cameraman tried to step around the cordon the police had placed, but a campus officer quickly stepped in and forced her back behind the barrier. Diesel and I scooted into the chapel. I hoped we could avoid them again after the service.
I paused at the entrance to the sanctuary, trying to find Julia and Justin in the crowd. There were very few open seats, and the sanctuary could easily hold three hundred people. I spotted Melba Gilley and Peter Vanderkeller near the front. Willie Clark was here too, in the back row to my left. Jordan Thompson sat nearby, two rows in front of Willie. Standing in the back to my right was Kanesha Berry, dressed in a black skirt and jacket instead of her usual uniform. She saw me and acknowledged me with a brief nod.
I scanned the crowd again and finally picked out Julia and Justin about halfway down on the right in the middle of a pew. There was an empty space next to Justin, and I led Diesel toward it.
I mumbled, "Excuse me," several times as Diesel and I made our way to the middle of the pew. One woman hissed, "Well, I never." A vaguely familiar man with her told her to hush. "That's the cat I told you about," I heard him tell her in an undertone.
I flashed him a quick smile, and then I reached the empty space. I sat, and Diesel moved between Justin's legs and stared up at him.
"Thank you," Justin whispered to me. He bent forward and began to rub Diesel's head. I just hoped the cat wouldn't purr too loudly and annoy the people sitting around us.
Julia glanced down and shook her head, but smiled. She had her arm around her son's shoulders.
The organist began playing. The service had started.
The choir sang two hymns, and the chaplain spoke briefly about Godfrey's accomplishments and lamented a life cut short by violence. The president also spoke and said a few words about Godfrey's generosity to the school over the years. Godfrey had always given money on condition of anonymity, and that surprised me. He always seemed to want to be the center of attention. Knowing this made me think slightly better of him.
The president introduced Godfrey's agent, a petite blonde named Andrea Ferris, who said a few words about the effect of his death on his millions of fans around the world. She herself didn't seem all that grief stricken, however. Perhaps she was simply putting up a brave front. The president stepped back in front of the microphone to invite everyone to move into the chapel meeting room for a reception in the dear departed's memory.
Then it was over. It was mercifully brief, but the whole time I had been aware of the tension coming from mother and son beside me. There had been no mention of Godfrey's recently discovered son during the service, and I imagined that both Julia and Justin were greatly relieved. The last thing they wanted right now was that kind of attention, especially with the media waiting right outside.
I remained seated with mother, son, and cat while the pews around us slowly emptied. Julia was making no move to leave, and I wondered if she planned to go home now and skip the reception.
"Are you leaving now?" I asked when most of the people around us were gone.
"No," Julia said. "We should put in an appearance at the reception. And I want to have a word with Godfrey's agent."
"So do I," I said, smiling briefly. "Shall we?" I stood.
I exited the pew, leading Diesel on his leash, and Justin and Julia followed me through the sanctuary to the meeting room behind.
Not everyone who attended the service stayed for the reception. There were only about a hundred or so people in the room, and I was thankful for that. I tended to be a bit claustrophobic when a large number of people occupied a small space, and this room wasn't really designed to hold as many people as the sanctuary.
Mindful of my lack of lunch today, I followed Julia and Justin as they joined the line of people at the buffet table. From my place in line I could see some of the food. It appeared to be mostly cocktail party-type snacks. Not ideal, but enough. I could easily fill up on cheese and crackers and fruit. There were also deviled eggs, a staple of this kind of gathering—at least in Mississippi. I would have to watch Diesel, though, in case he decided he wanted to investigate the food. When he stood on his hind legs, he was tall enough to reach out and scoop something from the table.
We made it through the line without incident, and along with Julia and Justin I found a place to stand against the wall. While the two of them nibbled at the few things on their plates, I had to restrain myself from gobbling it down. I was hungrier than I realized.
I was chewing my last bit of cheese and cracker when Kanesha Berry approached us.
"Good afternoon." Her voice was low, her demeanor wary.
I returned her greeting, echoed by Julia and Justin. Diesel chirped at her, and she glanced down for a moment. I could almost swear I spotted a brief smile, but when she looked up, her expression was blandly official.
"Julia has something she needs to tell you," I said, eager to the point of rudeness. Now that the solution to the murder was so close, I really wanted to see things happen. Once Willie was arrested—for at this point I had no doubt he, as X, had the best motive for murder, and according to Julia he also had opportunity—we would all rest much easier.
Kanesha turned to Julia with an expectant look.
Julia frowned slightly. "I'm not sure this is the place," she said.
Justin surprised us all by interrupting. "Mr. Charlie, would you mind if I took Diesel for a walk?" He had a slightly desperate look, and I wondered whether the occasion was proving too much for him.
I handed over the leash. "Sure, but why don't you just go into the sanctuary? It should be pretty quiet in there, and I don't think going outside right now is a good idea."
"Yes, sir," Justin said. "Come on, Diesel."
I watched boy and cat make their way through the crowd. Poor kid. So much had happened to him so quickly. No wonder he wanted to find a quiet place.
"You have something to tell me?" Kanesha spoke firmly to Julia.
"I suppose so," Julia replied with a sidelong glance at me. "During a chat with Charlie before the service, I recalled something that happened when I went to the hotel to see Godfrey."
"I see. What was that?" Kanesha shifted her weight from one foot to the other.
"It was talking about the writers' group that brought it back to mind," Julia said. "I remembered that, when I was leaving the hotel that day, I saw someone in the revolving door, entering as I was going out." She paused for a moment. "It was Willie Clark. Charlie seems to think that's significant for some reason."
"How so?" Kanesha could have been discussing today's weather, I thought. She didn't seem particularly interested in Julia's revelation.
I thought I could get her interested, however. I said, "Willie is X."
**THIRTY**
Kanesha flashed me a warning look, her head moving ever so slightly in Julia's direction.
"X? What does that mean?" Julia frowned at me. "Are you telling me that _Willie_ murdered Godfrey?"
I was relieved that she kept her voice down, otherwise the people nearby would have heard it all.
"I really cannot discuss that with you, Mrs. Wardlaw. Do not repeat this conversation to anyone."
Julia nodded. "Certainly I won't."
Kanesha was clearly annoyed with me for speaking in front of Julia. She took my arm and started leading me away. "I need to speak to Mr. Harris alone."
I went without protest. I should have restrained myself and waited until I could speak to her alone, but sometimes I got a bit carried away. I recalled an expression my grandmother used when I did something like this as a child: "His head knows better, but his feet can't stand it."
In other words, despite knowing better, I sometimes put my foot in it.
Kanesha led me back out into the sanctuary. I spotted Justin and Diesel in the choir loft, away from the few people sitting in pews, eating and talking. Kanesha found a spot a good ten feet away from anyone else and pointed to a pew.
I sat.
She sat down beside me, about a foot away on the pew. Her right hand gripped the back of the pew in front of us, and I saw her knuckles tighten. "You cannot blurt out things like that."
"I know," I said, feeling foolish. "I'm sorry. It's just that, now that I know who killed Godfrey, I want this to be over."
Kanesha closed her eyes for a moment, and I wondered whether she was praying for patience. Her grip on the pew didn't loosen.
"You know who killed Godfrey Priest?" Her eyes opened. "I suppose you think Willie Clark did it."
"Yes," I said, eager to atone for my goof. "Once I found out he was part of the writers' group, and knowing what we know about someone else writing Godfrey's books, it all fell into place."
"How so?" Kanesha let go of the pew and folded her arms across her chest.
"The attitude toward women in the books," I said. "Look, have you ever read one of the books?"
"Yes, a few of them," Kanesha said. "I like to read them and find all the mistakes in police procedure." She shook her head. "His books were pretty bad in that respect. But I know what you mean about the women in his books. He didn't like them."
"Well, that wasn't Godfrey. From what everyone says, Godfrey truly liked women. He just couldn't settle down with one. It's Willie who's the big-time misogynist. You should hear him talking to female staff and students sometimes. He can be a real jerk."
"Okay," Kanesha said. "He's a misogynist. I'd need more evidence than that, though. Even if he did write the books. I have to have something that links him to the actual murder."
"According to Julia, he was at the hotel that day. He had to have gone there to talk to Godfrey." I was feeling a bit deflated by her lack of excitement. I thought surely she would see the picture as clearly as I did.
But she was an officer of the law, and I was a librarian. This was her job, not mine.
"I will ask him about it," Kanesha said. "But unless he admits to being there, I'm going to need more than Mrs. Wardlaw's glimpse of him in a revolving door to go on."
"Of course," I said. "You need physical evidence for a stronger case." I had read enough mysteries to know that. "But at least now you have motive and opportunity."
"There are other suspects who have motives and who also had the opportunity," Kanesha said, her logic relentless.
"Okay, you win," I said. Here I thought I had come up with the answer, and she was refusing to accept it.
What if I was wrong? That was an unwelcome thought. There were things Kanesha knew that I didn't—if her investigation had turned up any kind of evidence from the scene of the crime. I didn't even know what the murder weapon was.
"You did help—a little. I found out some things faster because of your interference." Her tone was grudging, but I knew better than to expect outright gratitude.
I nodded.
"But you're done," she said. "Back off now, and leave me to finish this."
I saw the glint in her eye. "You know who did it, don't you?"
Kanesha regarded me for a moment. "I do. I have a few more things to check, however, and I don't want you getting in my way again."
"I won't, I promise," I said.
"Good." Kanesha stood and made her way out of the pew. She disappeared through the door into the meeting room.
I sat there, thinking about our conversation. Kanesha seemed awfully sure she knew who the murderer was. Was that because of what Julia and I had told her about Willie? Or had she known already?
Perhaps that meant Willie wasn't the killer.
Not knowing was going to annoy me to no end. I had a sudden suspicion that was why Kanesha had told me she knew the killer's identity. If so, I supposed it was adequate payback for the annoyance I had caused her.
It was time to head back to the reception. I would have to be careful about what I said, and to whom, though. I had pushed Kanesha far enough.
I stood in the doorway and looked around, searching for Julia. After a moment, I spotted her in the far corner to my right, talking to someone, but I couldn't see who it was. As I moved closer, I could peer through the crowd, and I recognized Godfrey's agent, Andrea Ferris.
At the same time I also spotted one of the campus blowhards, an elderly English professor named Pemberton Galsworthy. Many suspected the name was his own invention because it was so pompous sounding. But in that respect it was apt. He was a self-important windbag who never had an opinion he wasn't willing to share with anyone within hearing distance.
I almost turned away, knowing that I could be stuck there for an hour if I joined the group. Galsworthy never had conversations. He performed soliloquies.
But Julia caught sight of me, and I couldn't ignore the plea in her eyes. I didn't know why she thought I could do anything to stop the deluge of words. We had suffered through Galsworthy's sophomore literature course together, and she knew him as well as I did.
I moved forward and sidled up next to Julia.
Galsworthy noticed me—in itself noteworthy—and interrupted himself to acknowledge my presence. He peered at me. "Harris, isn't it? Librarian, aren't you?"
Without waiting for an answer, he resumed his peroration, peering now at Godfrey's agent. "Contemporary literature has obviously been bastardized to the point of utter banality. Crass commercialism, naturally. Publishing was once the profession of gentlemen—educated, sophisticated, cultured—who chose works for their literary merit and their ability to enlighten and transform. Not because they would sell in the millions and cater to the tastes of the lowest common denominator, so sadly low these days, one fears for the intellectual survival of the species."
He had more to say in that vein, but I tuned him out for a moment, though I faced him with a rapt expression. I had learned to do it in his class, and thankfully it was a skill I hadn't completely forgotten.
I sneaked a glance at Andrea Ferris, dressed smartly in a dark suit and spike heels that made her stand about five-two. She had that glazed look common to anyone in Galsworthy's presence for more than ten seconds.
Julia nudged me, and I looked at her. She frowned and bobbed her head in Galsworthy's direction. I knew what she wanted, but short of clapping my hand over the man's mouth and shoving him into a closet, I didn't know how to shut him up. We could simply have turned and walked away, but generations of Southern grandmothers would spin in their graves if we behaved so rudely. That was the curse of being raised to have good manners and to treat one's elders with respect—no matter how irritating they were.
I tuned back in at the sound of Godfrey's name.
". . . a sad example of a young man with a good mind—a good mind, you understand, not a fine one—but, yes, a young man with a good mind who could have accomplished something more lasting than such ephemera as he chose to create. Then there is his appalling portrayal of females in his work. One has little doubt that a psychiatrist could have helped the poor boy work through his obvious feelings of hatred toward women. Yet I have no doubt that his female readers little suspected his opinion of them."
I exchanged amused glances with Julia. Galsworthy had obviously read some of Godfrey's work, though one wondered why he had allowed his intellect to be sullied by entertainment of such dubious value to mankind.
Galsworthy blathered on, but I could see that Andrea Ferris was about ready to pop. She cut him off suddenly in mid-sentence.
"I'll be delighted to share your observations of contemporary publishing with my colleagues in New York," Andrea said, her tone deceptively sweet. "I have little doubt they will respond immediately by pulping anything that smacks of lowbrow entertainment and instead start printing—in huge quantities, of course—works that cannot fail to _enlighten_ and _transform_. This will revolutionize publishing around the world, and your name, professor, will be on everyone's lips."
After his initial shock at being interrupted, Galsworthy appeared delighted to have his opinions received so well. But Andrea's tone altered as she spoke, becoming more waspish by the syllable, until even Galsworthy had to recognize the sarcasm.
"Good day to you, young woman." Galsworthy glared at Andrea, and so upset was he that he failed to include Julia and me in his farewell.
Julia and I both sighed audibly as he stalked off.
"What a pretentious snot," Andrea said. She sniffed. "If I had a dollar for every one of his kind I've met, I could retire." She turned to me and stuck out her hand. "Andrea Ferris, the late Godfrey Priest's agent."
"Charlie Harris," I said. "Archivist here at the college. Like Mrs. Wardlaw, I went to school with Godfrey eons ago."
Andrea nodded, her eyes on Julia. "You're the mother of his son, aren't you?"
Startled, Julia nodded. "I asked Godfrey to keep it to himself for a while, but obviously he didn't."
"Oh, Godfrey told me everything," Andrea said. "He was my biggest client, you know."
Had Godfrey really told her everything, I wondered? Did Andrea know about the ghostwriter?
"I'm not surprised," I said. "Godfrey made millions."
"He sure did." Andrea's smile was smug. "No complaints there." She cocked her head to one side, thinking about something. "But you know, the old windbag did have a point about something."
"What was that?" I said, though I knew what she meant.
"The bit about Godfrey's treatment of women in the books," Andrea replied. "That always bothered me, because Godfrey liked women. No doubt about that. I never could figure out why the tone of the books was so antifemale."
"Did you ever ask him about it?" Julia seemed intrigued by the question, too.
"I did, early on," Andrea said. "I wasn't his agent for his first few books. I took him on on the strength of _Count the Cost_ , his first bestseller." She frowned. "I went back and read one of his earlier books, and the tone was very different."
"What was Godfrey's response when you asked him about it?" I said to get her back to the point.
"He just shrugged and said that was the way the book came out. He claimed most thrillers were like that anyway, so why should his be different?"
"And that made sense to you?" Julia didn't sound convinced.
"As much as anything else," Andrea said. "Frankly, he started making so much money for both of us, I didn't really care."
I decided to risk a question. "Did you ever think someone else might have written the books? I mean, because the tone was so different."
Andrea laughed. "Don't be ridiculous. Who else could have written them? Godfrey changed his style, that's all. He wanted to break out and make serious money, and he did."
She seemed sincere, and I thought Godfrey had kept his ghostwriter a secret from her, too. She was in for a rude shock, though.
Julia regarded me, obviously curious. She knew that I wouldn't have asked such a question without a reason.
Before either of us could respond to Andrea's last remark, she spoke again. "He made it after all." She waved at someone.
Julia and I turned our heads to look. "Who is it?" I asked.
"The tall man in the suit there, talking to the deputy. You know who I mean, don't you?"
"Yes," Julia and I said in unison.
About twenty feet away Kanesha Berry was deep in conversation with a distinguished-looking man about sixty years old.
"Who is he?" Julia asked.
"Miles Burton," Andrea replied. "Godfrey's attorney." She grinned at Julia. "And if Godfrey managed to get his will changed like he was planning to, your son is going to be really rich, Mrs. Wardlaw."
**THIRTY-ONE**
After her announcement, Andrea excused herself, saying she wanted to talk to Miles Burton.
"That was hardly discreet," I said as she walked away.
"No," Julia said. "But I already knew that. Godfrey told me he changed his will to include Justin and acknowledge him as his son." She smiled with what appeared to me to be grim satisfaction. "And he died before he could change it again."
"Why would he want to change it again?"
For a moment Julia looked uneasy. "Well, Justin did quarrel with him, and you know how nasty Godfrey could be when he didn't get his way."
That didn't make much sense to me. One disagreement on the day Godfrey met his son for the first time didn't mean he would disinherit Justin. Godfrey was too excited about having a son, I figured, to do something vindictive after one meeting.
I didn't express my doubts to Julia, though. She was watching Andrea Ferris speak with Kanesha Berry and Miles Burton. Her face betrayed her avid interest. I wondered why she didn't simply go up to them and introduce herself to the lawyer.
Kanesha saved her the trouble. She beckoned for Julia to join them, and I decided I was included in the invitation. Julia needed support, especially since Ezra wasn't here with her.
Kanesha frowned at me as she introduced Julia to Miles Burton.
"I regret that we are meeting under such tragic circumstances," Burton said, his voice a mellow baritone. "Where is your son? Did he attend the service?"
"Yes, he did," Julia said. "He's here somewhere."
"He was in the sanctuary, up in the choir loft the last time I saw him." I introduced myself. "Would you like to speak to him?"
"Yes, I would," Burton said with a grave smile. "I have matters to discuss with him and with Mrs. Wardlaw."
"I'll go find him," I said, and Burton nodded his thanks.
As I left them, Julia was asking Burton how long he had been Godfrey's attorney. I didn't hear the answer.
Out in the sanctuary, I turned to look up into the choir loft. Justin and Diesel weren't there. I scanned the sanctuary, but there was no sign of them. Perhaps Justin had gone to the restroom.
I went down the hall on the side of the chapel opposite the meeting room and checked inside the men's room. All was quiet, and I didn't see any legs, human or feline, in any of the stalls. Had Justin and Diesel gone home?
How had they made it past the media outside? I had visions of Justin being pinned to the front steps of the chapel while reporters bombarded him with questions. But I realized that, unless they knew who Justin was, they probably would have asked him only general questions. Like why did he have a cat with him?
On a hunch, I went further down the hall to the back of the chapel. There was another short hallway running across the rear of the building, which led to a back door. I opened the door and peeked outside. There were no reporters out here.
I stepped outside on the stoop and looked around. No sign of boy and cat here, but I realized that Justin and Diesel could easily have slipped away without attracting attention. They could have taken a roundabout way to the house without having to cross paths with the media.
I made my way back into the meeting room to report to Miles Burton and the others. As I approached them, Andrea Ferris was speaking.
". . . shame that after the new book is out next fall, there won't be any more. Thank goodness Godfrey finished it before he came here." She tittered. "It's the best thing he's done yet, and I predict it will outsell his last two."
"How tragic," Julia said. She turned to look at me.
I answered the unspoken question. "No sign of Justin, nor of Diesel. I think they slipped out the back and went home."
"Diesel? Who is that?" Miles Burton frowned.
"My cat," I said. "Justin is very attached to him, and I brought him along to the service to help comfort the boy. This has all been a severe shock to him."
"Naturally," Burton said, though he eyed me doubtfully. "I would like to speak with the young man sometime today, if possible. My plane leaves Memphis very early tomorrow morning. I have a case coming to trial on Tuesday in LA."
"I can bring him to your hotel," Julia said.
"Or you can come back to my house now." I made the offer with a smile. "There will probably be reporters at the hotel, and if we go out the back way to my house, you can avoid all that."
"Excellent idea," Burton said. He turned to Julia. "If that is okay with you, Mrs. Wardlaw." He looked in Kanesha's direction. "And you too, Deputy."
"It's fine," Julia said.
"Okay with me," Kanesha said. "I'm in no hurry to make a statement to the media, and I need to hear what Mr. Burton has to say."
"In that case," Burton said, pulling a small notebook from the jacket of suit. He opened it and flipped through a few pages. "If Mr. Harris wouldn't mind, I'd like to request that a few others be present as well. I might as well address all beneficiaries of Godfrey's will at one time."
"It's fine with me," I said. "You're welcome to use my living room."
Kanesha frowned. Would this interfere with her plans for arresting the murderer? To me it looked like she was figuring something out, and after a moment the frown relaxed.
"I think that's okay," Kanesha said. "Who else do you need to speak to?"
Burton consulted his list. "Richard Tackett and William Clark. And a representative of the college, if possible."
Beside me, Julia tensed. What was bothering her? I was surprised at hearing Willie's name, and no doubt she was, too. But perhaps it was the mention of Godfrey's half brother that concerned her. After all, she had dated Rick for a while before Godfrey and she had the fling that produced Justin. And she knew perfectly well, unlike me until recently, that the men were half brothers.
"I work for the college," I said. "And I'm the archivist. Godfrey spoke with me earlier in the week about donating his papers to us. I saw the president leave a few minutes ago."
"You should be sufficient as a representative for the moment," Burton said. "Official notice will come later, and that can be addressed to the president and board of trustees."
I scanned the crowd in the meeting room. I caught sight of Rick and pointed him out to the lawyer. Burton strode off to speak to him.
"Do you see Willie anywhere?" I asked Julia. "I'll go around the room. He's short enough that we might not be able to see him in a crowd."
"I'll help you look." Julia started off in one direction and I in the other to make a circuit of the room. Andrea Ferris chattered at Kanesha.
I found Willie behind a clump of people, hectoring a history professor about his students and their lack of library skills. I had heard this song many times before.
When I interrupted Willie, the history professor shot me a grateful look and disappeared quickly.
"What do _you_ want?" Willie was gracious as ever.
"Godfrey Priest's lawyer is here, and he wants to speak with you."
Willie looked taken aback at first, but then a smile spread on his homely face. "Maybe there will be justice at last." He started forward.
"What do you mean by that?" I asked, although I was certain I knew the answer.
"You'll find out soon enough," Willie said. "Where is this lawyer?"
"Over here," I said. Burton had rejoined the three women, and Rick Tackett was with him.
"This is William Clark," I said when we reached them.
Willie stuck out his hand. "I sure am pleased to meet you."
"And I you." Burton shook the hand. "You are all acquainted already with Mr. Tackett, of course. Why don't we proceed now to Mr. Harris's house?"
"Just follow me," I said. I offered my arm to Julia, and she clasped it with a trembling hand. I shot her a sideways glance and was surprised to see her pale. Was she excited or nervous? I couldn't tell. Perhaps it had something to do with Rick Tackett. I had seen him watching her intently when Willie and I joined the group.
I led the group on a slightly circuitous route to my house, but even with the detour we arrived in less than ten minutes. I unlocked the front door and ushered everyone into the living room.
Miles Burton set his document case on the coffee table as the others found seats. I offered refreshments, but everyone declined.
"I'll go check on Justin," I said. "I'm sure he's upstairs with Diesel."
_He'd better be_ , I thought as I climbed the stairs. I couldn't imagine where else he could be right now.
Sure enough, he and Diesel were in his room. Justin was lying on his back in his bed, still wearing his suit. Diesel was stretched out beside him, purring as the boy rubbed his head.
"How are you doing?" I asked.
"Okay."
He didn't look okay. He looked miserable, and no wonder. I wished he could have been spared all this. The months—and perhaps years—ahead were going to be hard for him. To have his biological father snatched away from him so cruelly just after they'd met for the first time—it was truly tragic.
"Godfrey's lawyer is here, along with your mother and a few other people," I said. "The lawyer needs to speak to all of us about Godfrey's will."
"I don't care about his will," Justin said. "Can't everyone just leave me alone?" He turned his face into the pillow, away from me.
I sat on the edge of the bed. "Son, I'm sorry you have to go through all this. But you need to come downstairs and hear the lawyer out. Godfrey obviously remembered you in his will, and for his sake, you need to listen."
Justin lay there, unresponsive for a moment. I waited, and then he sat up. He had been crying.
"Go wash your face," I said gently. "Then we'll go downstairs."
Justin nodded and got out of bed. Diesel stretched and came over to me on the bed. I scratched behind his ears. I suspected he was going to be spending a lot of time with Justin in the near future. I hoped Diesel could provide the comfort the boy would need.
Justin came back, and we set off down the stairs, Diesel running ahead of us.
All heads turned when the three of us entered the living room. Miles Burton came forward, hand extended. He introduced himself and shook Justin's hand. I could see the sympathy he felt for the boy.
Burton led Justin to a seat on the sofa next to Julia and near his own chair. Diesel climbed into Justin's lap. Andrea Ferris, who occupied the other spot on the sofa, stared at Diesel in fascination. Rick Tackett and Willie Clark had pulled chairs close in a semicircle.
Kanesha stood a few feet away, arms crossed. I found another chair and offered it to her, but she shook her head. I took it instead, sitting a little behind Rick Tackett. I had a clear view of Julia and Justin and the lawyer from this vantage point.
Miles Burton held a thick document in his hands. "I regret deeply the occasion that has brought all of us together. Godfrey Priest was my client for many years, and I wish he could have been with all of us for many more." He glanced down at the papers he held. "But it is now my duty to share with his beneficiaries the terms of his will. Godfrey changed his will recently because of the knowledge that he had a son.
"He was thrilled with the knowledge, and I also deeply regret that he had such a short time with this young man. I know how excited Godfrey was to meet him for the first time." He smiled at Justin, who ducked his head. Diesel rubbed his head against the boy's cheek.
"I will spare you all the unnecessary details of a testament such as this. You must realize that, in the case of such a large estate, there are many details that have to be considered. Those, however, are of little concern to you at the moment.
"There are a number of relatively small bequests to which I will return in a few minutes. The important point is that Godfrey stipulated that these small requests should be paid first, and the remainder of the estate would be divided as follows:
"'To my biological son, known as Justin Henry Wardlaw, I leave two-thirds of my estate;
"'To my half brother, Richard Horace Tackett Jr., I leave the remaining third of my estate.'"
Burton paused, as if to gauge the impact of his words. In front of me, I could see Rick Tackett's shoulders relax and his head go down. I thought I could hear him muttering a prayer of thanks.
Julia's eyes glittered with triumph, and her smile was wide. Justin stared at the lawyer, as if he found it difficult to understand what the man had said.
"What are we talking about, in real terms?" Julia surprised me by the obvious note of greed in her voice.
Miles Burton eyed her with what I presumed to be slight distaste. "A conservative estimate of your son's share, Mrs. Wardlaw, would be something in the range of seventy million dollars."
Justin's mouth dropped open, and even Julia appeared thunderstruck. She had obviously never realized how rich Godfrey was.
Burton turned to Rick Tackett. "And that means Mr. Tackett's share would approach thirty-five million."
"I can't believe it," Rick said. "After all these years of ignoring me, why now?" He kept shaking his head.
"Godfrey made no explanation," Burton said.
"It was his way of saying he was sorry, probably," Andrea Ferris said. "He was like that. He always thought money could excuse anything."
"What about me?" Willie Clark startled everyone. "What did he say about me?"
Burton frowned as he consulted Godfrey's will.
"To my high school friend, William Ebenezer Clark, I leave the sum of one million dollars and my grateful thanks for his friendship over the years."
I waited for the eruption, and it came almost immediately.
"That's all? That's all he had to say?" Willie was screaming. He leaped out of his chair and tried to snatch the will from Burton.
Kanesha stepped forward and put herself between Willie and the lawyer. "Sit down, Mr. Clark. Now."
Willie backed away, but he was still furious. His face was so red, he looked like he was going to have a stroke any moment now.
"That cheating bastard, I can't believe he did this to me. Even dead he's screwing me."
"What are you talking about?" Andrea Ferris glared at Willie. "I don't think a million bucks is anything to feel bad about."
"You stupid cow," Willie said. "I wrote the freaking books, not Godfrey."
**THIRTY-TWO**
There was dead silence for a moment. Andrea Ferris jumped up from the sofa, the outrage plain in her face.
"You are totally nuts, you little creep." For a moment I thought she was going to crawl over the coffee table to get to Willie. "I was Godfrey's agent, and I know damn well he wrote those books."
"Shows how much you know, you ignorant bitch," Willie said, not in the least cowed by Andrea's response. "I have proof that I wrote the books. Godfrey always said you knew, but I guess he didn't trust you enough to tell you the truth."
"What kind of proof?" Andrea sounded a little less certain now. "You're damn well going to have to prove it."
"Well, for one thing," Willie said in a smug tone, "I can give you a copy of the manuscript for the book that's coming out next September. How do you think I'd have a copy of it if I didn't write it?"
"Godfrey could have asked you to read it for some reason," Andrea said. "It's set here in Mississippi again, and he could have asked you to do some fact-checking for him."
During this exchange I had been trying to get Kanesha's attention, but she steadfastly ignored me. She was intent on the argument between Willie and Andrea.
Willie laughed. "You can argue all you like, woman. It's not going to change anything. I wrote those books, and I have proof. I have a contract with Godfrey."
"You can produce this contract?" Miles Burton frowned. "This is a serious allegation, you understand. I'm not certain what the ramifications will be, because Godfrey assigned his copyrights to his son."
Willie howled in rage and made a move toward the lawyer. Kanesha, who was still standing between Willie and Burton, held a hand up in front of Willie's face. "Calm down. Now. Or I will have you taken out of here. You understand?"
Faced with Kanesha's commanding tone and stance, Willie backed down. He resumed his seat, and I could feel the tension in the room drop a little.
"I can and will produce the contract," Willie said. "We will discuss it later, you can be damn sure."
Kanesha stepped to one side of the lawyer, but her gaze remained fixed on Willie.
For a moment there was silence, and in that brief interval I heard a car pull up in front of my house. I got up and went to the window. The curtains were open, but I had to pull aside the sheers in order to see clearly.
There were two sheriff's department cars in front of my house. I glanced toward Kanesha, and she was watching me. She inclined her head a fraction, and I went back to my chair, thinking hard. She was going to arrest someone in my house. My heart started beating faster. I wasn't sure I liked the idea.
Burton resumed announcing the contents of Godfrey's will. "There is a bequest of five million dollars to Athena College, and of that amount two hundred and fifty-thousand dollars is to be used for the processing and preservation of the papers he is donating to the school." Burton glanced at me.
"I'm sure our president and trustees will be delighted," I said. Godfrey had obviously made his plans for the archive before he ever consulted me. He came to me simply to talk about Justin, and I could understand that.
"There are bequests to a few charities," Burton said. "And that is it."
"How soon will my son actually be able to receive his bequest?" Julia leaned forward on the sofa, watching Burton like a proverbial hawk.
"The will must go through probate, naturally. There is also the investigation into Godfrey's death," Burton said. "Until that is concluded, nothing much can happen. And now there is an additional issue to consider, the true authorship of the novels that bear my client's name. I really cannot say how that will affect the disposition of Godfrey's estate."
"What does the investigation have to do with it?" Rick Tackett asked. "I'm not sure I understand."
"In Mississippi," Kanesha said, "murderers are not allowed to profit from their crimes. If one of the beneficiaries in the will is found guilty of Mr. Priest's murder, he or she will not inherit."
"Is this true?" Julia looked right at Miles Burton.
"I'm sure Deputy Berry knows this particular statute better than I," Burton said. "Since the crime occurred here, that law would obviously be in effect."
Julia now directed her gaze at Willie Clark. I had been watching her in fascination, seeing a side of her that I hadn't expected to see. She was far more avaricious than I would have guessed, judging by her behavior these last few minutes.
Julia pointed across the coffee table to Willie. "I think you have a pretty darn good motive. Plus, I know you were in the hotel that afternoon."
"Me? You're nuts, Julia." Willie's voice came out in a squeak.
"I saw you," Julia said. "In the revolving door as I was leaving." She sat back and crossed her arms across her chest. Her smile was grim.
Willie laughed, startling us all. "Yes, I was there. I went to see Godfrey to talk about the new book. Not the one that's coming out next year, the one after that."
"And you got into a fight and bashed him over the head." Julia nodded. "I can see it now."
I looked to Kanesha to intervene, but she didn't. She simply stood there and watched.
"Well, I saw you too, Julia." Willie did not appear in the least perturbed by Julia's accusation. "But you've got it backward. I was in the revolving door with you, but I was the one leaving, not you. I saw Godfrey around two-thirty, after waiting for him almost twenty minutes. He was upset about something when I finally did get in to see him, and he said we'd have to talk later. By then I couldn't really hang around any longer either. I was due back on the reference desk at three. One of my staff called in sick that morning, and I had to take his stint at the desk."
"The reference desk?" Julia had paled.
"Yep," Willie said. "At three, and in full view of plenty of people for two hours, because I manned the desk until five. Then I had a meeting with the chair of the history department, and I was with him until nearly six."
It appeared that Willie had a pretty good alibi for Godfrey's murder. Based on what Julia had told me, it was nearly three when she left Godfrey. That statement lent credence to Willie's assertion.
But if she had lied about when she saw Willie, had she lied about anything else?
Kanesha broke the tense silence that had fallen. "I have to ask you, Mrs. Wardlaw, if you would like to revise what you told me earlier. Is Mr. Clark correct? Did you see him as you were _entering_ the hotel?"
"Perhaps I got it wrong, and I did see Willie as I was entering," Julia said. "But he could have come back later and killed Godfrey."
"I most certainly did not," Willie said. "After I finished the meeting with the head of the history department, I walked over to the patisserie for something to eat, and then I went to the bookstore for a poetry reading. I didn't have time to go to the hotel and kill anybody."
All eyes appeared to be on Julia now. Except for Justin's. He had his head against Diesel, hugging the cat closely to him.
"Mrs. Wardlaw, refresh my memory. What was it you did after you left the hotel and your interview with Mr. Priest?" Kanesha took a step closer to the sofa.
Julia watched Kanesha, the unease evident in her face. "I went to the bank to deposit a check Godfrey had given me. Then I went to the hospital. I got there in time for the shift change, a little after three."
"Were you given a receipt for your deposit, Mrs. Wardlaw?"
What was going on here? From Kanesha's demeanor I began to wonder if she had decided Julia was the murderer. My stomach began to knot up in distress.
"Yes, I suppose so," Julia said, shrugging. "Don't they always give you one?"
"They're supposed to," Kanesha said. "And generally those receipts record the time of the deposit. Were you aware of that, Mrs. Wardlaw?"
The relentless use of _Mrs. Wardlaw_ was like a nail being slowly hammered into a coffin.
Julia stared at the deputy but didn't respond. It was clear that she had never given a second thought to the time stamp on her bank receipt.
"I believe also that the bank is open until six P.M. during the week," Kanesha said. "I can of course check with the bank, and I will, to determine at what time you made your deposit, Mrs. Wardlaw. I have already spoken with hospital personnel in order to verify your whereabouts."
Kanesha paused, but there was only the sound of hard breathing. Julia was afraid, and the fear was almost palpable in the room.
"Do you have anything you wish to say about the time you made your bank deposit, Mrs. Wardlaw? It's only a matter of time before I know the truth."
Julia took a deep breath. "It was a few minutes before six."
Justin raised his head and looked at his mother. "Mama, what's going on? Why did you lie about the stupid bank deposit?"
"I guess I was just mixed up," Julia said, but even Justin didn't believe her. The pain in his eyes as he looked at his mother was heartrending.
"Mr. Priest wanted to take Justin back to California, didn't he? You were afraid you might lose your son, weren't you? And you weren't going to let that happen."
"No, that's not right. Godfrey wasn't going to do that. I talked to him and he promised he wouldn't, at least not until Justin finished college." Julia sounded desperate, but at this point I didn't think anyone believed her.
"Can I ask a question?" Rick Tackett spoke, his voice low and hesitant.
"Yes, Mr. Tackett, what is it?" Kanesha appeared surprised at the interruption, but she nodded encouragement when Rick failed to speak right away.
"Justin, when is your birthday?" Rick watched Justin, his hands on his knees. I saw that his knuckles were white.
"August fourth," Justin said after clearing his throat. Then he added the year.
"Thank you," Rick said. "He wasn't premature, was he, Julia?"
Tears welled in Julia's eyes. "No, he wasn't." We could barely hear her.
Rick nodded. He took a deep breath as he looked straight at Justin.
"He's not Godfrey's son," he said. "He's mine."
**THIRTY-THREE**
I wasn't the only one in the room who was stunned. I sneaked a quick look at Kanesha's face, and I could have laughed at her expression. The English have a term for it: _gobsmacked_. Translated roughly, it means _utterly astounded_.
That's exactly how Kanesha looked.
Rick spoke again. "Son, I'm truly sorry you had to find out this way."
"Mama, is it true?" Justin put a trembling hand on Julia's arm.
Julia didn't answer.
"It has to be," Rick said, his voice steady. "I suspected it for a long time, and I just let it go, I guess. Julia had dumped me for Godfrey. And then she went and married Ezra. She made it clear she didn't want me, even though I'd asked her to marry me." He paused. "I didn't realize until today that she was claiming Godfrey was the boy's father. I couldn't let the lie go on any further."
"How can you be sure?" Kanesha asked.
Rick shrugged. "The last time I saw Julia back then"—and we all understood that _saw_ was a euphemism—"was in early December. Godfrey didn't blow into town until mid-January."
We could all do the math. If Rick was right, Godfrey couldn't have been Justin's father.
"Did Mr. Priest know about your relationship with Mr. Tackett?" Kanesha went back on the attack.
"No," Julia said. "He was only here for about two weeks that time, and I made sure he didn't hear about it. He never knew."
I had to speak up, though it hurt me to do so. "He found out about it on Tuesday," I said. "I told him. It just came up in the conversation. My family and I were here for Christmas that year, and we saw Julia and Rick together. I told Godfrey that, and he seemed surprised by it."
"She told Godfrey the boy was a preemie." Andrea Ferris got off the sofa and came to stand near me. "When Godfrey first told me about it, he said he was thankful the boy hadn't had any significant health problems despite being two months premature."
"Mr. Priest confronted you that afternoon, Mrs. Wardlaw. He had figured out that he might not be Justin's father. I imagine he was very angry with you." Kanesha glared at Julia.
Julia was sobbing now. All she could do was nod.
Rick got up from his chair and extended a hand across the coffee table to Justin. "Son, I think you should come with me." He glanced at Kanesha, and she nodded.
Justin, obviously torn, still clutching Diesel, looked first at Rick and then at his mother. Julia said, "Go. Please." She wouldn't look at her son.
Justin hesitated, then kissed her cheek. He gently pushed Diesel aside and got up from the couch. He moved from behind the coffee table, and Rick put an arm around the boy's shoulders. We all watched as he led Justin from the room. Diesel came to sit by my chair.
"Mr. Harris, would you go to the door and wave at the cars parked outside? They'll know what it means." Kanesha moved closer to Julia, and I got up from my chair to do what the deputy asked.
As I headed for the front door, Andrea, Willie, and Miles Burton all moved to the other side of the room.
I opened the door and waved. A moment later three deputies stepped out of the cars and headed up the walk. I moved aside to let them in. I kept an eye out for Diesel in case he decided to wander outside.
I saw him scampering up the stairs when I closed the door behind the deputies. I was tempted to follow him, because I didn't think I could bear to see Julia being arrested. I was appalled by what she had done, but I also hated the thought of her being so alone now.
I went back into the living room and sat down on the couch with her. Kanesha had begun the process of arresting her for murder.
On Monday morning when I was about to leave for the college library, Justin walked into the kitchen. With Julia in custody, he had gone home to Ezra Saturday afternoon. I went with him, to try to explain to a very bewildered man what had happened.
Ezra's illness was taking its toll and Justin stayed with him until the evening, when Rick Tackett arrived. The boy was too dazed to make any decisions for himself, and I encouraged him to go with Rick. He was going to need a father, and Rick had the quiet strength, I thought, to help his son.
All Julia had wanted to do was help her son, too, but she had gone about it the wrong way. Godfrey had treated her badly, driving her to choose Ezra instead of going back to Rick. I had no doubt now she bitterly regretted that choice. She seemed determined, however, to make Godfrey pay for what he had done, and even though Godfrey realized Justin wasn't his son, he must have felt guilty enough to give her money anyway. He probably thought he could simply buy her off, but by then Julia was, I suspected, so irrational that she simply acted without any consideration for the consequences. Otherwise she wouldn't have forgotten Justin's cell phone or have let something so simple as the time stamp on her deposit receipt trip her up.
The tragedy of it all was stunning, and I felt such pity for Julia. I could do something for her, though, by continuing to look after her son however I could.
"How are you?" I examined Justin with concern. He looked like he had slept very little the past two nights.
Justin shrugged. "I don't really know. It's all too freaky."
Diesel rubbed against his legs, and Justin squatted down to hug the cat.
"Yes, it is," I said. "I want you to know, though, if I can do anything to help you, I will."
"Thank you," Justin said, looking up at me. Besides the fatigue, I thought I could see the beginnings of a new maturity in his face. He stood.
"Actually, there is one thing you can do for me, if you will." Justin watched me calmly. "I'd like to stay here with you for now."
"Of course you can," I said. I had to speak around a lump in my throat. "Diesel would miss you terribly, you know."
Justin gave me the ghost of a smile. "I'd miss him, too. Rick wants me to move in with him and my brothers and sisters." He shook his head. "That sounds so weird. I have brothers and sisters now. Half, that is, but still."
"I'm glad. It's good to have family." I paused. "But it can be a bit confusing to try to get to know them all at once. Maybe you need a little time to get used to the idea."
"Yes, sir," Justin said. "Thank you, Mr. Charlie, and you too, Diesel."
He stood there for a moment, and my heart ached for him. But Diesel and I would do our best to help him.
"I think I'll go up to my room and take a nap," Justin said.
"Sounds like a good idea." I smiled at him. "And I'll bet you can talk Diesel into coming with you."
"Come on, boy," Justin said, waggling his fingers at the cat. "Let's go upstairs."
I sat down at the table, forgetting about work for the moment, as boy and cat left the kitchen. I heard Justin clumping up the stairs, and I realized what a reassuring sound that was.
In the years since my wife died, I had done my best to isolate myself from all but the necessary daily contacts with other people. With my son and daughter off living their own lives, I had only Diesel for any kind of emotional companionship.
That had been enough for a while. But the shock of the events of the past week had broken through that shell I had almost unknowingly put up around me.
For a moment I fancied I could see both Jackie and Aunt Dottie sitting at the table with me. "It's time," Jackie would say, and Aunt Dottie would nod in agreement.
I smiled as the images I conjured up faded away, leaving only the glow of happy memories.
Yes, it was time.
I gathered my things and headed for work.
Read an excerpt from Miranda James's new Cat in the Stacks series mystery, _Claws for Concern_. Coming in hardcover in February 2018
**Chapter One**
I couldn't stop checking the clock on the wall nearby. "Come on, three o'clock," I muttered under my breath. "Get here already."
The wretched clock refused to cooperate. It read two forty-seven, and the second hand seemed to be taking way too long to sweep around the clock's face. Thirteen minutes until I could pack up and head home.
Diesel, my Maine Coon cat and near-constant companion, warbled anxiously from the area next to my feet under the reference desk. He always picked up on my emotions, and I forced myself to calm down. There was no point in getting a nearly forty-pound cat all wound up. Nor myself, actually.
"It's all okay, boy," I told him in a low voice before I reached under the desk to scratch his head. "We'll be home soon." I think the cat knew what—or really, who—was waiting for us at home, and he was as eager as I to be there.
_Clock check_. Only eleven minutes to go. I could leave now if I really wanted to. I volunteered at the Athena Public Library. I did not earn a paycheck from the place. I knew, though, how much the director, Teresa Farmer, and the other staff appreciated my help on Fridays, and I wasn't going to cut my time short. I settled back into my chair for the remaining minutes and glanced around me.
On this late July afternoon, the only people I saw in the library were adults, mostly my own age or older. Some, no doubt, sought relief from the punishing heat. The soaring temperatures taxed air conditioners, and there were many elderly people in Athena who couldn't afford to cool their houses. I knew most of those who came into the library to get relief, at least by name.
One man was a definite stranger, however. I first noticed him a week ago. Tall, a bit stooped, with a shambling gait, he looked to be about ten years older than me, so that put him in his midsixties, though he might have been older. I'd not had any interaction with him last week, and he had not come near the reference desk today. He had glanced my way a couple of times, his expression a puzzled frown.
I wondered whether he knew me or thought that he might. I had never seen him before that I could recall, though there was an elusive familiarity about his face. Maybe I had run across him thirty years ago, I mused, before I left Athena to move to Texas for graduate school in library science. I couldn't place him, but I hadn't spent much energy trying. I had learned over the years to let such things resolve themselves on their own schedule. The answer to this particular puzzle, if I knew it, would occur to me in due course.
Earlier today I had thought about approaching him and simply asking him who he was, but I hesitated to follow through on that. He appeared reserved and perhaps shy, and I didn't want to intrude if he truly had no desire to talk to people. I glanced his way again, and he looked up for a moment. Then he dipped his head down, focused once more on the book in his lap, and I read that as a clear signal that he did not want to be interrupted.
Diesel chirped and laid a large paw on my knee, as if he were asking me the time, and I checked the clock. Two minutes to three. Bronwyn Forster, one of the full-time librarians, should be here to relieve me any moment now. Sure enough, when I looked toward the area where the offices were, I saw her emerge from the doorway and head toward us.
After we greeted her, and she and I exchanged places, Diesel stayed with Bronwyn while I went to gather my things. He had to be sure to get his full quota of rubs on the head and under the chin before we left. Bronwyn, like the other staff and many of the patrons, never hesitated to oblige him. No wonder he loved coming to work with me.
Back at the desk again, I spoke to Bronwyn. "Would you mind keeping him with you for a minute? It's so hot outside, I want to get the car started and cooling off before I put him in it."
Bronwyn gave me her habitual sweet smile. "Of course, Charlie. Diesel won't mind getting loved on for a couple more minutes."
I heard a happy warble from behind the desk and knew Diesel would be content until I was ready to take him out. "Back in a minute, then." I headed for the door.
The second I stepped outside the heat swarmed around me like a cloud of gnats. I could feel the sweat starting to form as I made my way through the parking lot to the far side where the staff usually parked. This morning I had found a spot beneath the largest tree that cast shade over the lot. That meant the inside of my car was a few degrees cooler than it might have been otherwise.
I backed the car out and drove it around the lot to the closest spot to the front door. I left the engine running and went into the building to retrieve my cat. The moment I called, he came running around the desk toward me.
I scooped him up in both arms and backed out the door, with one last farewell to Bronwyn. The temperature was too high today—hovering around the century mark—to let Diesel walk over the hot asphalt and concrete. In weather like this I carried Diesel to and from buildings where the sidewalks and parking areas were in the direct sunlight. I didn't want him blistering the pads of his feet.
The drive home took less than ten minutes, and once I had the car parked in my garage, I let Diesel out of the back seat. He preceded me into the kitchen where I knew we would find the object of our intense interest.
When I stepped into the room I saw my daughter, Laura, sitting at the table, feeding my grandson, and chatting with Azalea Berry, my housekeeper. Diesel approached Laura slowly. When he reached her side, he looked up at her and chirped twice.
"Hello, handsome boy," Laura said. "We're almost through here, and then I'll let you see him. How about that?"
Diesel warbled in happiness. He loved the baby and could sit near him and watch him for long periods of time. Until both he and the baby fell asleep, that is.
I greeted both women and put my things on the small table by the door. I moved closer to Laura and my grandson and watched for a moment. Then my vision blurred, and I had to slip my handkerchief out of my pocket to wipe my eyes.
"How is young Master Charles Franklin Salisbury?" I asked, my voice husky.
Laura laughed as she looked up at me. "Like his grandfather, always ready for a meal."
Azalea chuckled at that. "That baby sure is a chip off this old block." She slid a sly glance in my direction. "How about something for you, Mr. Granddaddy?"
I grinned at the silly nickname Azalea gave me not long after baby Charlie was born. We were all giddy with happiness over the arrival of this child. I hated that his grandmother and his great-great aunt weren't alive to see him, but I knew they were watching over him. I sometimes felt their presence here in the kitchen. Like now, when a whisper of air passed my right ear.
"I wouldn't mind a cold glass of water," I replied. "This horrible heat wave makes me thirsty."
"I'm glad to hear you ask for water instead of a diet soda." Laura smiled at me. "You were drinking way too much of it. It's good that you've cut back."
After a satisfying couple of sips from the glass Azalea handed me, I raised an eyebrow at Laura and sniffed. "You wouldn't say that if you hadn't had to give up the exact same beverage while you were breastfeeding my grandson."
"Ha-ha," Laura replied. "By the time this young man is off the breast, I will have completely forgotten what the ambrosia tasted like." She sighed. "No wine until then, either. _That_ I will definitely go back to, believe me."
"I didn't think you had to cut caffeine out completely, though," I said as I took a seat across the table from Laura and baby Charlie.
"No, I don't, but it needs to be limited," Laura said. "I still have a little, mostly coffee or tea, but nothing like what my intake used to be."
I recalled her teenage years—and probably the years spent in California while she pursued her acting career—when she seemed to live on diet drinks, salads, wine, and cheese, with the occasional hamburger and french fries on the side.
"How is his rash?" I nodded toward the baby.
"Almost completely cleared up," Laura said. "His doctor said that infant acne is fairly common and usually clears up on its own."
"I don't remember whether you or Sean had that," I said. "I don't think you did, though."
"It's all due to maternal hormones," Laura replied. "His little face will be completely smooth again in another day or two."
Diesel chirped as if to acknowledge gratification at this news.
Azalea placed a stack of mail in front of me. "You hit the jackpot today."
"Half of it at least will go in the recycling bin." I eyed the pile with a jaundiced glance. There were three catalogs, several circulars, and four letters. I pushed the catalogs and circulars aside and picked up the envelopes.
"When I was a little girl," Laura said, "you used to give me all the mail you didn't want so that I could pretend it was my mail."
"Yes, I did." I laughed. "You would sit and read through it so solemnly."
Laura rolled her eyes. "I must have been adorable, thinking I was important enough to have mail like my father."
"You were, and still are, adorable," I told her.
"Mushy," Laura said, but she smiled.
The first two letters were junk mail. The third was a legitimate bill. The fourth, however, seemed to be of a personal nature. The return address and my name and address were all handwritten.
I glanced at the return address. The name was Jack Pemberton, and the town was Tullahoma, a smaller town about eighty miles southwest of Athens. I didn't recognize the sender of the letter. I couldn't recall ever having met someone of that name.
Curious, I opened the letter by tearing a small strip off one end. I extracted the pages inside, along with what looked like a bookmark. Pemberton might be a writer, I decided. Was he trying a direct-mail approach to selling books?
I examined the bookmark first. One side showed two images of book covers, both with slightly lurid illustrations. I laid the bookmark aside and opened the letter. I scanned it quickly, impatient as usual to figure out what the import was. Then I went back and read it more carefully.
I frowned and laid the pages aside.
"What is it, Dad?" Laura said. "Bad news?"
"Not news at all." I laughed, suddenly struck by the seeming absurdity of the letter writer's intent. "A man from Tullahoma wants to write a book about me."
"About you? Why?" Azalea asked, obviously puzzled. Then suddenly her face cleared and she scowled. "About all your murders, you mean."
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1. Title Page
2. Copyright Page
3. Acknowledgements
4. ONE
5. TWO
6. THREE
7. FOUR
8. FIVE
9. SIX
10. SEVEN
11. EIGHT
12. NINE
13. TEN
14. ELEVEN
15. TWELVE
16. THIRTEEN
17. FOURTEEN
18. FIFTEEN
19. SIXTEEN
20. SEVENTEEN
21. EIGHTEEN
22. NINETEEN
23. TWENTY
24. TWENTY-ONE
25. TWENTY-TWO
26. TWENTY-THREE
27. TWENTY-FOUR
28. TWENTY-FIVE
29. TWENTY-SIX
30. TWENTY-SEVEN
31. TWENTY-EIGHT
32. TWENTY-NINE
33. THIRTY
34. THIRTY-ONE
35. THIRTY-TWO
36. THIRTY-THREE
37. Excerpt from CLAWS FOR CONCERN
1. Table of Contents
2. Start Here
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Warm Apple Cinnamon Galette with a scoop of vanilla ice cream is dessert perfection. So simple to make – wrap flaky pie dough around fresh cinnamon-sugar apples, then bake, and you're done.
I've said it before, and I'll say it again; put some fruit into a pastry crust, bake it up until it's golden brown and delicious, et voilà, my idea of the perfect dessert. Pies and tarts of all sorts are my favorite, but I have a special place in my heart for the simple galette, or free-form tart. When I make a galette it's always with seasonal fruit, minimal sugar and a great crust.
One of my all-time favorites is my Plum Galette with a walnut crust. The juicy plums pair perfectly with the nutty crust. My Pear-Ginger Galette is very similar to this Apple Cinnamon Galette. As you can see, the possibilities are endless. Simply swap in whichever fruit is seasonal and use a spice that highlights that fruit. You can also use whichever pastry dough you like to work with. I'll admit that some pastries can be kind of fussy to put together, but a free form tart is meant to be rustic so no special pastry skills are needed for this dessert.
My Wood-Fired Apple Cinnamon Galette!
The most fun part about this particular Apple Cinnamon Galette (well, the most fun besides eating it) was that I baked it in a wood-fired, brick-lined oven. The super hot oven with the hot stone floor was the perfect environment for a galette. Is that little bit of char on the crust a mistake? Not to me, that's my favorite part.
You can bake this galette on a parchment lined baking sheet or if you have a baking stone pre-heat it and bake the galette directly on the stone (I would put some parchment under it in case it leaks a little.
Gerry, the homeowner of the house with the fantastic wood fired oven. He's enjoying the aroma of fresh baked Apple Cinnamon Galette.
If using a baking stone place it in the oven to preheat at least an hour before baking.
Bake directly on the baking stone or on the sheet pan until the apples are tender and the crust is well browned. Enjoy warm or room temperature.
If you have a baking stone you can bake the galette directly on the stone. Build the galette on the parchment and slide it onto the stone using a peel or the back of a sheet pan. Otherwise bake it on a parchment-lined sheet pan.
Thanks, Dominique. The wood-fired oven was the perfect place to bake a galette, plus it's so much fun. Galettes are probably my favorite dessert of all.
I would be in heaven if I had access to a wood fired oven…. your gallette is gorgeous!
Thanks, Michele. It is so fun to bake in the wood fired oven. Especially for a baking geek like me.
Oooh, I am so jealous of that wood-fired oven! And what a lovely and delicious looking galette!
Thanks, Shannon. It was so much fun!
I am not a baker, but this recipe makes me want to try it!
Thanks, Anne. It's such a simple recipe.
This is so beautiful! Love rustic gallettes! Some day I will have to try baking in a wood fired oven!
Thanks, Maria. If you get a chance to bake in a wood fired oven, jump on it!
That looks amazing Eileen! I seriously want a wood fired oven!!!
I can just imagine the aroma of this very rewarding dessert! I prefer fruit desserts to have minimal sugar – and this is perfect! | {
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{"url":"https:\/\/www.physicsforums.com\/threads\/motion-in-magnetic-field-co-ordinates.42233\/","text":"# Motion in Magnetic Field : Co-Ordinates\n\n1. Sep 8, 2004\n\n### HIGHLYTOXIC\n\nIs there any way we can find the co-ordinates of a charged particle undergoing uniform circular motion in uniform magnetic field, in space ?\n\nI tried it using the angular velocity of the particle but it becomes quite complex..\n\nCan anyone help? How about co-ordinates in Helical Motion? Any Chance?\n\n2. Sep 8, 2004\n\n### Tide\n\nYes - it's pretty straightforward. For example, take the magnetic field to be in the $\\hat z$ direction and your equations of motion become\n$$\\frac {d v_x}{dt} = \\Omega v_y$$\n$$\\frac {d v_y}{dt} = - \\Omega v_x$$\n$$\\frac {d v_z}{dt} = 0$$\n\nwhere $\\Omega$ is all the magnetic field, charge and mass folded into a single parameter. You shouldn't have any difficulty solving them if you have any experience with differential equations.\n\n3. Sep 8, 2004\n\n### Tide\n\nOh, and once you have the velocity you can find the position by integrating\n$$\\frac {d \\vec x}{dt} = \\vec v$$\n\n4. Sep 8, 2004\n\n### HIGHLYTOXIC\n\nYeah, that does make it simple..Thanx for the help!\n\n5. Oct 23, 2004\n\n### SoberSteve2121\n\nHow would you prove using Newton's Second law that the trajectory of a charged particle with Velocity in the i direction and magnetic field in the -k direction must be circular?\n\n6. Oct 23, 2004\n\n### Tide\n\nThe preceding posts showed exactly how to do that.\n\n7. Oct 24, 2004\n\n### SoberSteve2121\n\nWould you be able to lay it out for me because I don't understand that?\n\n8. Oct 25, 2004\n\n### Tide\n\nSteve,\n\nThe equations I wrote in Post #2 in this thread ARE Newton's Law! Acceleration is force divided by mass and you see the left side of the equations represent the acceleration vector. The right side of the equations are the force vector $q \\vec \\times \\vec B$ divided by the mass (I rolled all the constants into the constant $\\Omega$.\n\nIf you can solve those equations then you have your answer. You should at least be able to convince yourself that $\\cos \\Omega t$ and $\\sin \\Omega t$ are solutions of the first two equations so all you would have to do is apply initial conditions to the general solution\n\n$$\\vec x = \\hat i x + \\hat j y = \\vec a \\cos \\Omega t + \\vec b \\sin \\Omega t$$\n\nto find the constants a and b. The result is a parametric representation of a circle!","date":"2017-11-25 04:45:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.6311912536621094, \"perplexity\": 397.45765455176746}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934809392.94\/warc\/CC-MAIN-20171125032456-20171125052456-00168.warc.gz\"}"} | null | null |
Hammaptera sabrosa är en fjärilsart som beskrevs av Paul Dognin 1893. Hammaptera sabrosa ingår i släktet Hammaptera och familjen mätare. Inga underarter finns listade i Catalogue of Life.
Källor
Mätare
sabrosa | {
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\section{Introduction}
\IEEEPARstart{E}{gomotion} and scene perception plays a vital role in autonomous systems. Self-driving and mobile robots are normally equipped with a combination of sensors, e.g. camera, inertial measurement unit (IMU) and LIDAR, for self-motion and scene perception, which further supports path planning and decision making.
A representative method is visual odometry (VO), which tracks and matches visual features between consecutive images, and establishes multi-view geometry model to calculate relative pose. However, pose estimates from monocular VO are scale-ambiguous. Stereo VO exploits stereo camera to recover the absolute scale of pose estimates, and is generally more robust because of information fusion from two cameras. Visual-inertial odometry (VIO) attracts attentions, as it combines camera with inertial sensor to estimate system states that can recover global-scale, and improve the robustness in complex environments.
Recently, deep learning based methods construct end-to-end deep neural network model to learn pose from images directly. They can be generally categorized into supervised learning based and unsupervised learning based approaches. Supervised learning based VO/VIOs require ground-truth pose as labels to train the neural network model. Though they show good performance in challenging scenes, e.g. in featureless areas or in complex light-conditions,
labelling data is costly and time-consuming. In addition, the trained model is hard to generalize to new environments with completely different scene geometry and appearance. Thus, there are a number of attempts on
unsupervised learning based VO/VIOs that can jointly learn to produce relative pose and scene depths, by constructing photometric loss between the real image and wrapped image from novel view synthesis. However, They still suffer from scale ambiguity problem.
To solve this problem, previous researchers use prior information, e.g. velocity or camera height to recover the scale metric. A recent work, i.e. VIOLearner \cite{VIOLearner}, calculates the interpolated depths from pointcloud data to replace depth learning, and use the depths with absolute scale to obtain pose with absolute scale. However, the performance of scale recovery highly depends on the quality of pointcloud data, i.e. sparsity.
Moreover, the interpolated depths do not conform to the multi-view geometry, and thus can be not used as supervised labels for depth learning, so that VIOLearner only produces pose estimates.
We instead propose a self-supervised learning framework to jointly estimate ego-motion and depths with global scale metric from image sequences, via a novel Bi-directional coarse-to-fine scale recovery process. Instead of interpolating depths directly, in our model, point clouds data are used to compare with a pretrained depth model to calculate coarse scale ratio and depths. A two-stages process of scale recovery is proposed to firstly produce coarse pose via a single-directional scale recovery conditioned on coarse depths, and then refine depths and poses together through a bi-directional coarse-to-fine scale recovery. Our model is capable of generating both depths and poses with global scale accurately and robustly, even in low-light conditions (i.e. driving at night). In addition, inertial data can also be introduced into the framework via an attention based fusion module that further improves the performance of rotation estimation.
Our contributions can be summarized as follows:
\begin{itemize}
\item We propose a novel self-supervised learning framework that can jointly estimate ego-motion and depths with global scale metric from image sequences.
\item We propose a novel Bi-directional coarse-to-fine scale recovery process that jointly refines coarse poses and depths. Depth scale is recovered via comparing point cloud data against a pretrained model that ensures the consistency of photometric loss.
\item Extensive experiments were conducted to validate the effectiveness of our proposed model. Our model produces pose and depth estimates robustly, even in low-light condition (i.e. a driving scene at night), a very challenging task for traditional VO/SLAM.
\end{itemize}
\section{Related Works}
\subsection{Traditional Visual Odometry}
\begin{figure*}[htp]
\centering
\includegraphics[width=18cm]{framework.pdf}
\caption{overview of our framework}
\label{fig1:overview of our framework}
\end{figure*}
Visual Odometry (egomotion estimation) is to calculate the relative pose of a moving camera through the matching feature points between different image frames via multi-view geometry model. Its main processes can be divided into: feature point extraction, feature point matching, outlier elimination,
and bundle adjustment. LIBVISO\cite{libviso} is a classic model-based visual odometry, that uses two different types of convolution kernels to detect the maximum response points in the image as feature points, and uses RANSAC\cite{FISCHLER1987726} algorithm to filter the matching feature points. PTAM\cite{PTAM} uses FAST \cite{FAST} algorithm to detect feature points on images with different resolutions, in order to improve the performance of odometry at different scales. ORB-SLAM \cite{ORB} adds stereo camera and depth camera to extend its application.
In the texture-missing environment, the direct-method shows better robustness in pose estimation than feature-point based methods. DSO\cite{DSO} and LSD-SLAM\cite{engel14eccv} use the corresponding pixel difference between adjacent images to construct photometric loss model to calculate camera pose. Under low-light environments, the performance of visual odometry decreases due to the failure of feature dection and matching.
Combined IMU with camera, visual-inertial odometry (VIO) obtains better robustness and higher accuracy in pose estimation. \cite{forster2016manifold} uses IMU pre-integration to integrate IMU raw data and optimizes the re-projection loss using matching feature points and point clouds. VINS\cite{VINS} adds automatic initialization, relocation, loop closure detection and other functions to form a more complete visual-inertial odometry.
\subsection{The supervised learning of ego-estimation estimation}
DeepVO\cite{deepVO} is a representative work of end-to-end learning based visual odometry (VO). It uses the convolution layers of FlowNet \cite{FlowNet} as the visual feature extractor, and a LSTM\cite{article} network to integrate the visual features of multiple images. Fully-connected layers are added after the LSTM network to transform features into pose predictions. Leveraging high-accuracy pose labels to train the model, DeepVO can learn to predict absolute pose predictions.
\cite{clark2017vinet} uses the LSTM network to process inertial data to obtain the motion features from a sequence of inertial data. It combines the visual features and inertial features to form the fused features. \cite{chen2019selective} adds an attention module into the visual-inertial odometry network to improve the robustness, and shows better performance in multiple complex environments. \cite{chen2022learning} introduces to fuse point clouds with inertial information to form a multi-sensor odometry estimation network.
\cite{xue2019beyond} introduces a memory and refining module that preseves important contextual information and based on that further improves poses.
DAVO \cite{KuoLLLCL20} improves the performance of supervised learning based VO by fusing the inputs from semantic segmentation, optical flow and RGB image via an attention module.
\subsection{The Unsupervised learning of ego-estimation estimation}
Unsupervised learning of egomotion methods have attracted attention, as they do not require high-precision pose labels. SfMLearner\cite{SfMLearner} designs a double network structure: PoseNet and DepthNet. It takes three consecutive images as the optimization sequence. PoseNet is responsible for estimating the relative pose between each two images. DepthNet is responsible for predicting the target frame depth. It combines relative pose estimates, target depth prediction and target/source images pixels to construct the photometric loss. In order to improve the accuracy of depth prediction, DepthNet adopts U-Net\cite{U-Net} like structure, which connects the multi-scale feature map of decoder and encoder. \cite{godard2019digging} uses ResNet18\cite{ResNET18} as the backbone of PoseNet and DepthNet. In order to obtain a more accurate pose and depth estimate, it upsamples different resolution depth predictions from DepthNet decoder to raw image resolution. And then it combines multi-scale depth predictions to form photometric loss. Besides, it adds the per-pixel minimum reprojection loss to eliminate the outliers in the photometric loss. \cite{9710474} predicts disparity map at all scales in DepthNet encoder and decoder, and it transforms disparity maps into pseudo-labels. It uses pseudo-labels as self-distillation signals in training process. Meanwhile, it introduces depth hint labels into the photometric loss to facilitate depth learning. To avoid overfitting, it also proposes a data augmentation method for depth estimation. In order to solve the problem of inconsistent depth prediction scales, \cite{bian2019unsupervised} proposes a depth scale consistency loss to constrain the depth and pose estimate to a unified scale. After multiple training iterations, the depth prediction in whole training set will keep a global consistent scale metric. \cite{Wang2021CanSM} designs a separate scale-net to predict the scale of each depth map. In order to obtain the pose and depth with absolute scale, PackNet-SfM\cite{packnet-sfm} introduces instantaneous velocity to calculate the adjacent frames displacement as weak velocity supervision. PackNet-SfM combines weak velocity supervision with the relative displacement estimate from PoseNet to construct velocity supervision loss. In order to introduce IMU data into unsupervised work, VIOlearner\cite{VIOLearner} used CNN to process IMU data and designed online error correction module to calculate multi-scale photometric errors. They interpolate the sparse point clouds to ground truth depth labels, and then they integrate the depth labels in photometric error with adjacent images. The performance of odometry is better than that of end-to-end VINet, but this work does not include depth estimation, so it can not reconstruct the scene in whole trajectory. Self-VIO\cite{almalioglu2022selfvio} transforms IMU data into inertial features through CNN and combines them with visual features to obtain fusion features. A fusion selection module is built to filter fusion features, and a GAN\cite{creswell2018generative} discriminator is designed to optimize the depth prediction to obtain high-frequency structure. At the same time, LSTM network is also used to comprehensively optimize the multi frame pose and depth predictions, Experimental results show that the performance of the proposed VIO is comparable to that of the mainstream model-based VIO. UnVIO\cite{UnVIO} introduces two optimization modules: intra-window optimization and inter-window optimization. It constructs the photometric error between two adjacent frames and smooth loss in intra window optimization. In inter window optimization, integrated information and pose prediction of odometry are used to construct trajectory consistency loss and 3D geometric consistency loss. Two kinds of consistency loss can constrain the global pose scale, and make pose prediction more consistent. However, the network cannot obtain pose results with absolute scale.
\section{Self-supervised Visual Egomotion and Depth Learning}
\subsection{Framework}
The overview framework of our proposed self-supervised ego-motion and depth learning is shown in Figure 1. Inspired by \cite{SfMLearner}, our framework also adopt double network design: PoseNet and DepthNet. Our PoseNet includes visual encoder, inertial encoder, attention based feature fusion module and pose estimation module. The framework inputs a sequence of images, and outputs the corresponding poses and depth maps. A pair of adjacent images are feed into ResNet18 based visual feature encoder to extract multi-state visual features, after which a LSTM network is used to model their temporal dependency. The visual features after LSTM are input into fully connected layers (FC layers) to produce the relative pose between the target frame and the source frame. DepthNet estimates the depth of each image in the image sequence. In addition, inertial data can be introduced into the framework to formulate visual-inertial PoseNet. Inertial feature extractor encodes inertial features from a sequence of IMUs. Then, an attention module is used to combine visual features and inertial features, which are further feed into LSTM. Our framework is trained in two steps: 1.Coarse Scale Recovery, it makes the PoseNet output pose with coarse absolute scale. 2. Coarse To Fine Scale Recovery, it refines the PoseNet pose prediction and updates the DepthNet parameters.
\subsubsection{Visual Encoder}
We use ResNet18 backbone as the visual feature encoder $f_\text{visual}$. Two adjacent images $\left \{\mathbf{I}_{i}, \mathbf{I}_{i+1}\right \} $ are combined on the RGB channel to obtain 6-dimensional tensors, which are inputted into the visual feature encoder to obtain visual features in 1/4 input image size. In order to simplify the network structure and improve the forward efficiency, we use global average pooling to transform ith frame visual features into a 128 dimensional feature vector $\mathbf{v}_i$.
\begin{equation}
\label{eq: visual features}
\mathbf{v}_i= f_\text{visual}\left \{ \mathbf{I}_{i}, \mathbf{I}_{i+1} \right \}
\end{equation}
\subsubsection{Inertial Encoder}
The data acquisition frequency of IMU is higher than that of camera, and it has high measurement accuracy in a short time. Between two adjacent images $\left \{\mathbf{I}_{i}, \mathbf{I}_{i+1}\right \} $, there exist j imu raw data ${M}^{0\cdots{j-1}}_{i\to{i+1}}$. The jth imu frame includes linear acceleration $\alpha_{j}$ and angular velocity $\omega_j$ in body coordinate, is one-dimensional time series data, so we introduce the Bi-directional LSTM network as inertial feature encoder $f_\text{inertial}$, which excels at processing time series data. The input dimension of two-layer Bi-directional LSTM network is 6, the number of hidden state $\left \{\mathcal{H}_{j-1},\mathcal{H}_{j} \right \} $is 128, and the dimension of jth frame output inertial feature vector $\mathbf{a}_j$ is 256.
\begin{equation}
\label{eq: IMU Raw data}
\mathbf{M}_{i\to{i+1} }=\begin{bmatrix}
\alpha^{0}_{i} & \omega^{0}_{i} \\
\cdots &\cdots \\
\alpha^{j-1}_{i+1}& \omega^{j-1}_{i+1}
\end{bmatrix}\in \mathbb{R}^{j\times 6}
\end{equation}
\begin{equation}
\label{eq: inertial features}
\mathbf{a}_i, \mathcal{H}_i =f_\text{inertial}\left \{({\alpha_i},{\omega_i} );\mathcal{H}_{i-1} \right \}
\end{equation}
\subsubsection{Attention based Feature Fusion Module}
There may be a few misalignment in the raw data, bias in the IMU data, and errors in the calibration of intrinsics and extrinsics parameters. If the features of the two kind of data are directly combined, the odometry performance is likely to be suboptimal. Even in the pure visual feature, image features with missing texture will weaken the performance of pose estimation. Inspired by \cite{chen2019selective}, we introduce the attention module in PoseNet. Therefore, poseNet can focus more on features that are conducive to improving the performance of pose estimation. We concatenate two kind of features $\left \{\mathbf{a_i}; \mathbf{v_i} \right \}$ to form the concatenated features. And then we use the attention mask \cite{chen2019selective} to filter the concatenated features. In the end we obtain the filtered fusion features $\mathbf{z_i}$.
\begin{equation}
\label{eq: filtered fusion features}
\mathbf{z_i} =f_\text{attention}\left \{\mathbf{a_i}; \mathbf{v_i} \right \}
\end{equation}
\subsubsection{Pose Estimation Module}
The filtered fusion features are brought into pose estimation module. In this module, there exist a LSTM network and a FC layer. The LSTM network integrates the filtered multi fusion features $\mathbf{z_{i,i\in{0\cdots n-1} }}$ in one sequence window, sequence length is $n$. We set the hidden layer nodes in LSTM network to 1024 and the high-level output features $\mathbf{ \bar{z}_{i}}$ dimension to 2048. After the LSTM network, we use a dropout layer with a coefficient of 0.2 to further prevent the possibility of over-fitting. We subsequently send the high-level output features to the FC layer to obtain 6-DOF relative poses $\mathbf{T^{i+1}_{i}}$.
\begin{equation}
\label{eq: Pose Estimation Module_lstm}
\mathbf{ \bar{z}_{i}} = \text{LSTM} \left \{\mathbf{{z}_{i}}\right \} _{i\in 0\cdots {n-1} }
\end{equation}
\begin{equation}
\label{eq: Pose Estimation Module_FC}
\mathbf{T^{i+1}_{i}} = \text{FC} \left \{\mathbf{ \bar{z}_{i}}\right \} _{i\in 0\cdots {n-1} }
\end{equation}
\subsubsection{DepthNet}
We use U-Net like DepthNet, which is divided into encoder and decoder. The encoder of the DepthNet adopts the ResNet18 backbone, which is pretrained on ImageNet data set. DepthNet encoder can extract the visual feature in input image. The decoder of the DepthNet combines the feature maps from multi-scale encoder layers. There exist multiple convolution layers and nearest neighbor interpolations in DepthNet decoder. Therefore, DepthNet decoder can upsample the visual feature to original image resolution. We take the DepthNet decoder output as the disparity map. In the end we use the reciprocal of the disparity map as the depth prediction.
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{depth_coarse.pdf}
\caption{coarse depth scale recovery}
\label{fig2:Generation of depth labels}
\end{figure}
\subsubsection{Photometric loss}
The pixel points of the same 3D projection in two adjacent frames have brightness constancy and spatial smoothness.
Given the estimated relative pose between the target frame $\mathbf{I}_t$ and the source frame $\mathbf{I}_s$, the depth of target frame $\mathbf{D}_{t}$ and camera intrinsic $\mathbf{K}$, we can generate the synthesized image $\mathbf{\hat{I}}_s$ by warping $\mathbf{I}_s$ into target-view:
\begin{equation}
\label{eq: photometric loss}
\mathbf{p}_s\sim \mathbf{K} \mathbf{T}_{t}^{s} \mathbf{D}_{t}(\mathbf{p}_t) \mathbf{K}^{-1} \mathbf{p}_{t},
\end{equation}
where $\mathbf{p}_s$ and $\mathbf{p}_t$ indicate pixel point in source frame and target frame.
This work aims to recover the scale of ego-motion and depth learning. If the odometry pose scale is not recovered, it is difficult for odometry to be applied in realistic scenes. The reasons why the monocular unsupervised odometry cannot recover the pose scale: 1. The monocular image cannot determine the absolute distance of the 3D point in the single image. 2. It is impossible to determine the scale of $\mathbf{T}_{t}^{s}$'s translation vector and $\mathbf{D}_{t}$ only with photometric loss as a constraint.
\subsection{Coarse Depth Scale Recovery via Photometric-loss-consistent Pretrained Model}
We firstly introduce a depth scale recovery method via photometric-loss-consistent pretrained model.
Unlike previous works\cite{VIOLearner}\cite{VIOcompletion}, we do not use dense depth completion results as supervisions. This is due to the fact that the performance of depth completion is closely related to the point clouds density. If point clouds are too sparse, its performance will be degraded. The dense depth completion methods can be divided into interpolation-based and learning-based methods. \cite{VIOLearner} completes the sparse point clouds by interpolation. However, the interpolated areas do not necessarily conform to the multi-view geometry, and thus the generated depths are not suitable to being applied in learning depth recovery.
The learning-based methods \cite{VIOcompletion,selfcompletion} learn to generate dense depths from sparse depths, which can mitigate such problems to some extent, but there still exist completion areas that do not conform to multi-view geometry, such as out of the LIDAR scanning range. And it is too complicated to remove outliers before introducing these completion-based depths as supervisions into photometric loss for scale recovery.
To handle the problem mentioned above, in our framework,
we propose to use a pre-trained DepthNet to produce the depth predictions, and the scale of depth prediction is recovered by comparing them with the corresponding point clouds.
The depth predictions from pre-trained model naturally conform to the multi-view geometry. The depth predictions from DepthNet are more smooth and complete, and there is no need to remove abnormal values in advance.
We use the scale-recovered depth prediction as the supervisions for the recovery of pose and depth estimates.
We firstly transform the point cloud vector $\mathbf{v}_{4\times1}$ from LIDAR to the pixel coordinate as the depth projection point $\mathbf{d}_{v}$.
\begin{equation}
\label{deqn_D_vel}
\mathbf{d}_{v} = \mathbf{P}_{c2} \mathbf{T}^\text{rect}_{c0} \mathbf{T}^{c0}_{v} \mathbf{v}_{4\times1}
\end{equation}
where $\mathbf{P}_{c2}$ is projection matrix, $\mathbf{T}^\text{rect}_{c0}$ is transformation matrix from camera $c0$ coordination to rectified camera $c0$ coordination, $\mathbf{T}^{c0}_{v}$ is transformation matrix from lidar $\mathbf{v}$ coordination to camera $c0$ coordination.
We form the depth projection image $\mathbf{D}_{v}$ from depth projection points$\mathbf{d}_{v}$.
\begin{figure*}[htp]
\centering
\includegraphics[width=16cm]{bidirection.pdf}
\caption{The calculation of forward loss and backward loss}
\label{fig3:The calculation of bi-directional loss}
\end{figure*}
Then we calculate the ratio between the median/mean value of depth projection images $\text{med}(\mathbf{D}_{v})$ and the median/mean value of depth predictions from pre-trained DepthNet $\text{med}(\mathbf{D}_{t})$.
The scale factor $\varepsilon$ is calculated as below,
\begin{equation}
\label{eq: scale factor}
\varepsilon = \frac{\text{med}(\mathbf{D}_{v})}{\text{med}(\mathbf{D}_{t})}
\end{equation}
Finally, the scale factor $\varepsilon$ is multiplied with the depth predictions to obtain the depths with absolute scale $\bar{\mathbf{D}}$.
\begin{equation}
\label{eq: coarse depth}
\bar{\mathbf{D}} = \varepsilon \cdot \mathbf{D}
\end{equation}
\subsection{Single-directional Coarse Pose Scale Recovery}
The depths with absolute scale are further leveraged to recover the pose scale of our self-supervised learning framework. We propose a two-stages process to achieve this.
The first stage aims to estimate coarse poses by exploiting the calculated coarse depths, and then the second stage refines the network estimates by jointly optimizing poses and depths.
Firstly, coarse poses are obtained by integrating the calculated coarse depths from Equation \ref{eq: coarse depth} into the Equation \ref{eq: photometric loss} to produce the synthesized images $\hat{\mathbf{I}}_s$ in target frame.
We calculate the difference between the wrapped images $\hat{\mathbf{I}}_s$ and the real image in target frame $\mathbf{I}_t$ to construct a photometric loss. Besides, SSIM loss is added to mitigate complex illumination changes \cite{SSIM}. We formulate our photometric loss $L_\text{photo}$ as below,
\begin{equation}
\label{deqn_pho}
\mathcal{L}_\text{pho}=\lambda_1 \cdot |\mathbf{I}_t-\mathbf{\hat{I}}_s| + \lambda_2 \cdot \frac{1-\text{SSIM}(\mathbf{I}_t, \mathbf{\hat{I}}_s)}{2},
\end{equation}
where $\lambda_1$ and $\lambda_2$ are set as 0.15 and 0.85.
The specific function of SSIM can refer to \cite{SSIM}. (please add SSIM function)
To encourage pose and depth predictions to be scale consistent, a 3D geometric consistency loss $L_\text{GC}$\cite{bian2019unsupervised} is introduced:
\begin{equation}
\label{deqn_GC}
\mathcal{L}_\text{GC} =\frac{|\mathbf{\hat{D}}_s-\mathbf{T}_{t}^{s} \mathbf{D}_t|}{\mathbf{\hat{D}}_s+\mathbf{T}_{t}^{s} \mathbf{D}_t},
\end{equation}
where $\mathbf{T}_{t}^{s} \mathbf{D}_t$ indicates the process of wrapping the target depth $\mathbf{D}_t$ into the source frame view using the relative pose $\mathbf{T}_{t}^{s}$. And $\mathbf{\hat{D}}_s$ is the interpolated source depth map aligning with $\mathbf{T}_{t}^{s} \mathbf{D}_t$.
Most existing unsupervised ego-motion networks optimize the learned pose and depth between adjacent frames.
To consider the temporal dependency of poses and depths in consecutive frames, our framework optimizes them in a sequence window, which further encourages the pose scale-recovery to be more smooth in a larger window size.
We set $i$ as the time step of the target frame, $i+1$ as the time step of source frame, and $n$ as the sequence window length. In each time step, we combine photometric loss $\mathcal{L}^{i}_\text{pho}$ and 3D geometric consistency loss $\mathcal{L}^{i}_\text{GC}$ to form an optimization loss.
Combining the total losses of $n$ frames, we form the single direction scale-recovery loss $\mathcal{L}^{i\to i+1,i\in n}$ as below,
\begin{equation}
\label{deqn_positive}
\mathcal{L}^{i\to i+1}_\text{forward}
=\lambda_3\cdot \sum_{i=1}^{n}{\mathcal{L}^{i}_\text{pho}} + \lambda_4\cdot \sum_{i=1}^{n}{\mathcal{L}^{i}_\text{GC}}
\end{equation}
where $\lambda_3$ and $\lambda_4$ are set as 1 and 0.5.
We use the calculated coarse depths $(\bar{\mathbf{D}}_t,\bar{\mathbf{D}}_s)$ as the target and source depth in this stage to construct a coarse scale recovery loss $\mathcal{L}_\text{coarse}$,
\begin{equation}
\label{deqn_1st.stage}
\mathcal{L}_\text{coarse}=\mathcal{L}^{i\to i+1}_\text{forward}|_{(\bar{\mathbf{D}}_t,\bar{\mathbf{D}}_s)}
\end{equation}
The model is trained for several epochs to obtain a coarse scale recovered PoseNet, and then the entire framework is further refined in the next stage for jointly optimizing PoseNet and DepthNet together.
\subsection{Bi-directional Coarse-to-Fine Pose Scale Recovery}
In the coarse scale recovery stage, the depths are calculated from pre-trained model and estimated scale factor. This stage considers to enable pose and depth learning jointly, which further refines pose and depth estimates.
Instead of using calculated depths, we use the depth predictions from DepthNet as the target and source depths to construct the photometric loss and 3D geometry consistency loss of coarse-to-fine scale recovery.
If the optimization is only on the forward-direction, the model encourages depth predictions to be overfit on one side, leading to unstable training. Therefore, we introduce a backward loss,
\begin{equation}
\label{deqn_negative}
\mathcal{L}^{i+1\to i}_\text{back}
=\lambda_3\cdot \sum_{i=1}^{n}{\mathcal{L}^{i}_\text{pho}} + \lambda_4\cdot \sum_{i=1}^{n}{\mathcal{L}^{i}_\text{GC}}
\end{equation}
Here, as shown in Figure 3, the relative pose and the order of target and source frame are inverse.
Furthermore, we introduce an edge-aware smoothness loss $\mathcal{L}_\text{smooth}$ to overcome the shortage of photometric loss in the low-texture regions \cite{Smooth}.
\begin{equation}
\label{deqn_ex2a}
\mathcal{L}_\text{smooth} = \sum_{p\in(u,v)}^{} (e^{-\bigtriangledown \mathbf{I}_t(p)}\cdot \mathbf{D}_t(p))^2
\end{equation}
where $p$ indicates pixel point in coordination $(u,v)$, and $\mathbf{I}_t(p)$ indicates the pixel value in target frame $\mathbf{I}_t$, $\bigtriangledown$ means the first derivative along two spatial directions on pixel coordination $(u,v)$.
Combining the forward $\mathcal{L}^{i\to i+1}$ loss, backward $\mathcal{L}^{i+1 \to i}$ loss and smoothness loss $\mathcal{L}_\text{smooth}$, the total loss for bi-directional scale recovery is formulated as:
\begin{equation}
\label{deqn_ex2a}
\mathcal{L}_\text{refine}=\lambda_5(\mathcal{L}^{i\to i+1}_\text{forward} +\mathcal{L}^{i+1\to i}_\text{back})|_{({\mathbf{D}_t},{\mathbf{D}_s})} +\lambda_6\cdot\mathcal{L}_\text{smooth}
\end{equation}
where we set $\lambda_5=1$ and $\lambda_6=0.1$. ${({\mathbf{D}_t},{\mathbf{D}_s})}$ are the learned depth from DepthNet.
\subsection{Discussion}
We have also tried to use coarse scaled depth predictions to directly train the overall framework, instead of dividing training process into two stages: coarse scale recovery and coarse to fine scale recovery. However, the experimental results show that the pose and depth scale cannot be recovered directly by introducing depth labels. We analyze that since the absolute-scale depth supervisions are non-learnable, at the beginning of training, the photometric loss formed by the depth supervision is large, and the loss reduction potential is small. The photometric loss formed by the learnable depth prediction will decrease with the training iteration, and the loss reduction potential is large. Besides, the direction of network training is consistent with the direction of loss reduction. Therefore, the scaled depth supervision can not help framework to recover absolute scale in direct training.
\section{Experiment}
This section introduces implementation details, and discusses the evaluation of pose and depth estimation of our proposed model above two datasets. In addition, ablation study is conducted to verify the effectiveness of the important modules in our model.
\begin{table*}
\centering
\caption{The pose performance on the KITTI dataset, "Scale" indicates whether the pose estimates are scaled or not.}
\renewcommand\arraystretch{1.3}
\begin{tabular}{ccccccccc}
\hline
\multicolumn{1}{c}{\multirow{2}{*}{Method}} & \multirow{2}{*}{Sensors} & \multirow{2}{*}{Scale} & \multicolumn{2}{c}{Seq. 09} & \multicolumn{2}{c}{Seq. 10} & \multicolumn{2}{c}{Avg} \\ \cline{4-9} & & & $t_{rel}$ & $r_{rel}$ & $t_{rel}$ & $r_{rel}$ & $t_{rel}$ & $r_{rel}$ \\ \hline
ORB-SLAM & Mono & x & 15.3 & \textbf{0.26} & 3.68 & \textbf{0.48} & 9.49 & \textbf{0.37} \\
Depth-VO-Feat & Stereo & x & 11.92& 3.60 &12.62 &3.43 &12.27 &3.515 \\
GeoNet & Mono & x &23.94 &9.81 &20.73 &9.10 &22.34 &9.46 \\
Monodepth2 &Mono & x & 18.12 & 3.86 & 12 & 5.34 & 15.06 & 4.6 \\
SfMLearner &Mono & x &17.84 &6.78 &37.91 &17.78 &27.875 &12.28 \\
SC &Mono & x & 8.62 & 3.05 & 7.81 & 4.9 & 8.22 & 3.98 \\
Ours(VO) &Mono & \checkmark & \textbf{7.90} & 1.23 & \textbf{6.55} & 2.05 & \textbf{7.23} & 1.64 \\
\hline
VINS &Mono+IMU & \checkmark & 41.47 & 2.41 & 20.35 & 2.73 & 30.91 & 2.57 \\
VINet &Mono+IMU & \checkmark & 11.83 & 3.00 & 8.60 & 4.39 & 10.22 & 3.70 \\
UnVIO &Mono+IMU & x & 4.41 & 0.92 & 6.42 & 0.63 & 5.41 & 1.55 \\
Ours(VIO) &Mono+IMU & \checkmark & 5.48 & 0.19 & 5.37 & 0.43 & 5.43 & 0.31 \\
Ours(VIO) (Scaled) &Mono+IMU & \checkmark & \textbf{3.65} & \textbf{0.19} & \textbf{4.42} & \textbf{0.43} & \textbf{4.04} & \textbf{0.31} \\\hline
\end{tabular}
\label{tb: pose kitti}
\end{table*}
\subsection{Training details}
We use PyTorch to implement our proposed network. Our model is trained and tested via a NVIDIA RTX3090 GPU. Adam is chosen as optimizer to recover the optimal parameters, whose attenuation coefficient is $\beta _1 = 0.9$, $\beta _2 = 0.999$. The model is trained for 200 epochs, and in each epoch there are 1000 random data sequences.
The model training follows a two-stages process: first, we use the pseudo depth labels to train the network for around 30-100 epochs, until a coarse pose estimation with absolute scale can be recovered; then, depth and pose networks are jointly trained to enable coarse-to-fine pose and depth estimation. The learning rate is chose as 1e-4 in the first stage and reduced to 1e-5 in the second stage.
\subsection{Datasets}
\subsubsection{KITTI Odometry Dataset}
The KITTI Raw dataset is a typical car-driving dataset. It provides RGB images, IMU, LIDAR point cloud data and ground-truth pose data. We use this dataset to generate pseudo depth labels and odometry network training data. The KITTI Raw dataset for odometry includes 10 sequences. We use Sequence 00-08 excluding Sequence 03 for training and evaluation, as the IMU data of Sequence 03 are missing, while Sequence 09 and 10 are used for testing. Since the IMU and LIDAR data in the KITTI Raw dataset are unsynchronized, we manually synchronize these data according to their timestamps. The frequency of IMU is 100Hz, while the frequency of images and the synchronized LIDAR point cloud data is 10Hz. The ground-truth pose is not involved in training process, but used in the testing process to evaluate the pose accuracy.
\subsubsection{MVSEC Dataset}
In order to evaluate our model performance at night, we select the Multi Vehicle Stereo Event Camera (MVSEC) dataset, which collected gray images and IMU data. The images were sampled at a frequency of 20Hz and IMU data were recorded at a frequency of 200Hz.
In our experiments, we use the data from several car-driving scenarios in daytime and at night, including two day-time sequences (Day1, Day2) and three night-time sequences (Evening1, Evening2, Evening3).
We used the Day1, Day2, Evening1, and Evening2 as training data, and the Evening3 as testing data.
From the figure \ref{MVSEC}, it can be seen that the lighting conditions at night impose challenges to the visual pose and depth estimation. The dataset also provides ground-truth pose, which is only used in the testing process for evaluation.
\begin{figure}[htb]
\centering
\subfloat[Day-time image]{\includegraphics[width=0.48\linewidth]{0000002195.jpeg}}
\hfill
\subfloat[Night-time image]{\includegraphics[width=0.48\linewidth]{0000000193.jpeg}}
\caption{The sample images of the MVSEC dataset: a) driving in daytime b) driving at night}
\label{MVSEC}
\end{figure}
\begin{figure}[!t]
\centering
\subfloat[KITTI Sequence 09]{\includegraphics[width=9.2cm]{092.pdf}%
\label{fig_first_case}}
\hfil
\subfloat[KITTI Sequence 10]{\includegraphics[width=9.2cm]{102.pdf}%
\label{fig_second_case}}
\caption{The trajectories of our proposed self-learning based VO and VIO model on Sequence 09 and 10 of the KITTI dataset, comparing with baselines. }
\label{fig: trajectory kitti}
\end{figure}
\subsection{Pose evaluation on the KITTI dataset}
We first come to evaluate our proposed VO/VIO models on the public-available KITTI dataset. Following the previous research, we use Sequence 00-08 for training and evaluation, and Sequence 09 and 10 for testing. The KITTI's official evaluation is adopted here. It is to calculate the RMSE of translation vector and rotation of the sequences with a length from 100m to 800m, and average them as the criterion of pose accuracy.
\begin{table*}
\centering
\caption{The depth evaluation on the KITTI dataset. These models are trained with Sequence 00-08, and tested with Sequence 09 and 10.}
\renewcommand\arraystretch{1.5}
\begin{tabular}{cccccccccc}
\hline
\multicolumn{1}{c}{\multirow{2}{*}{Method}} &\multirow{2}{*}{Sensors} & \multirow{2}{*}{Resolution}&\multirow{2}{*}{Scale} &\multicolumn{3}{c}{Error metric} & \multicolumn{3}{c}{Accuracy metric($\delta$ )} \\ \cline{5-10}
& & & & Abs Rel $\downarrow$ & Sq Rel $\downarrow$ & RMSE $\downarrow$ & (1.25) $\uparrow$ & (1.25\textasciicircum{}2) $\uparrow$ & (1.25\textasciicircum{}3) $\uparrow$\\ \hline
SfMLearner &Mono &$832\times256$ & x & 0.3272 & 3.1131 & 9.5216 & 0.4232 & 0.7010 & 0.8476 \\
SC &Mono &$832\times256$ & x & 0.1629 & 0.9644 & 4.9129 & 0.7760 & 0.9315 & 0.9773 \\
UnVIO &Mono+IMU &$832\times256$ & x & 0.1322 & 0.73005 & \textbf{4.2443} & 0.8324 & 0.9509 & 0.9821 \\
Ours1 &Mono &$832\times256$ & x & 0.1322 & 0.7244 & 4.2715 & 0.8313 & 0.9498 & 0.9816 \\
Ours2 &Mono &$832\times256$ & \checkmark & \textbf{0.1308} & 0.6980 & 4.2930 & 0.8318 & 0.9520 & 0.9836 \\
Ours3 &Mono+IMU &$832\times256$ & \checkmark & 0.1301 & \textbf{0.6942} & 4.2963 & \textbf{0.8341} & \textbf{0.9526} & \textbf{0.9837} \\
\hline
\end{tabular}
\label{tb: depth kitti}
\end{table*}
\begin{figure*}
\centering
\begin{minipage}[b]{0.08\linewidth}
\centering
\caption*{Input}\vspace{9pt}
\caption*{SfM}\vspace{9pt}
\caption*{SC}\vspace{9pt}
\caption*{Ours1}\vspace{9pt}
\caption*{Ours2}
\end{minipage}
\begin{minipage}[b]{0.15\linewidth}
\includegraphics[width=1\linewidth]{0000000289.jpg}\vspace{4pt}
\includegraphics[width=1\linewidth]{SFM-0289.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{SC-0289.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{OursVO-0289.png}
\includegraphics[width=1\linewidth]{Ours-0289.png}
\end{minipage}
\begin{minipage}[b]{0.15\linewidth}
\includegraphics[width=1\linewidth]{0000000341.jpg}\vspace{4pt}
\includegraphics[width=1\linewidth]{SFM-0341.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{SC-0341.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{OursVO-0341.png}
\includegraphics[width=1\linewidth]{Ours-0341.png}
\end{minipage}
\begin{minipage}[b]{0.15\linewidth}
\includegraphics[width=1\linewidth]{0000000513.jpg}\vspace{4pt}
\includegraphics[width=1\linewidth]{SFM-0513.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{SC-0513.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{OursVO-0513.png}
\includegraphics[width=1\linewidth]{Ours-0513.png}
\end{minipage}
\begin{minipage}[b]{0.15\linewidth}
\includegraphics[width=1\linewidth]{0000000528.jpg}\vspace{4pt}
\includegraphics[width=1\linewidth]{SFM-0528.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{SC-0528.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{OursVO-0528.png}
\includegraphics[width=1\linewidth]{Ours-0528.png}
\end{minipage}
\begin{minipage}[b]{0.15\linewidth}
\includegraphics[width=1\linewidth]{0000000588.jpg}\vspace{4pt}
\includegraphics[width=1\linewidth]{SFM-0588.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{SC-0588.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{OursVO-0588.png}
\includegraphics[width=1\linewidth]{Ours-0588.png}
\end{minipage}
\caption{The samples of depth estimation from SfM-Learner, SC and our proposed models}
\label{fig: depth kitti}
\end{figure*}
For vision only pose estimation, we compare with a mainstream traditional SLAM (i.e. ORB-SLAM) and five learning based VOs (i.e. Depth-VO-Feat, GeoNet, MonoDepth2, SfMLearner, SC).
Among the learning based baselines, Depth-VO, GeoNet, MonoDepth2, SfMLearner are scale-ambiguity, while SC is scale-consistent, but still loses its absolute scale. However, our proposed self-supervised VO can produce pose with global scale. From Table \ref{tb: pose kitti}, it is clear to see that our model outperforms all both traditional and learning based baselines in translation, and outperforms all the learning based VOs in rotation. Our VO model further improves SC 12.04\% in translation and 58.79\% in rotation averagely. Though the rotation estimation of ORB-SLAM
is still better than our VO model slightly, we show that by introducing inertial data into our model as learning based VIO, our VIO model shows superior performance in rotation than all the VO/VIO baselines.
Our VIO model is compared with a representative traditional VIO, i.e. VINS, and two learning based VIO, i.e. VINet and UnVIO. VINet is a supervised learning based VIO model using a combination of ConvNet and LSTM. UnVIO is an unsupervised learning based VIO model via novel view synthesis similar to our model, but it has no absolute scale metric. Clearly, our model significantly outperforms these baselines in both translation and rotation. Our VIO model improves the translation and rotation estimation of UnVIO around 25.32\% and 80\% respectively.
As illustrated in Figure \ref{fig: trajectory kitti}, the generated trajectories from our VO and VIO models are closer to the ground-truth trajectories, compared with other baselines, i.e. VINS-mono, MonoDepth2, SC, ORB-SLAM and VINet.
\subsection{Depth evaluation on the KITTI Dataset}
In addition to pose estimation, we also evaluate the depth evaluation on the KITTI dataset.
Different from the SC-SfMLearner that use KITTI raw dataset for training, we use Sequence 00-08 of KITTI as these sequences are with IMU data.
Table \ref{tb: depth kitti} shows the quantitative results of the depth estimation from our proposed models on the Sequence 09 and 10, comparing with three mainstream learning based depth estimation methods, i.e. SfMLearner, SC and UnVIO.
In Table \ref{tb: depth kitti}, we use following evaluation metrics for depth estimation, where $p$ indicates pixel point in coordination $(u, v)$, $\mathbf{D}$ and $\mathbf{D}_v$ indicate learned depth prediction from DepthNet and corresponding depth projection image. $a$ is the threshold factor. $N$ is the sum of pixel points in the depth image coordinate.
\begin{equation}
\label{deqn_depth_metrics}
\begin{aligned}
\text{Abs}\ \text{Rel} &=\frac{1}{N}\sum _{p}\frac{\left | \mathbf{D} (p)-\mathbf{D}_v (p) \right | }{\mathbf{D}_v (p)} \\
\text{Sq}\ \text{Rel} &=\frac{1}{N}\sum _{p}\frac{ (\mathbf{D} (p)-\mathbf{D}_v (p))^2 }{\mathbf{D}_v (p)} \\
\text{RMSE} &=\sqrt{\frac{1}{N}\sum_p(\mathbf{D} (p)-\mathbf{D}_v (p)) } \\
\delta (a) &= \frac{1}{N}\sum_p{\text{max}(\frac{\mathbf{D} _v(p)}{\mathbf{D} (p)},\frac{\mathbf{D} _(p)}{\mathbf{D}_v (p)} )<a}
\end{aligned}
\end{equation}
It indicates that our depth estimation models (Ours2 and Ours3) can produce depth in absolute scale and outperform other three baselines.
The introduction of IMU into depth estimation throws little influence upon depth estimation.
Figure \ref{fig: depth kitti} shows several depth images from monocular camera, generated by our proposed models and baselines. We can see that compared with other learning based depth estimation models, our models can produce depth maps with more delicate details, e.g., the poles can be clearly shown in our depth maps.
\begin{table}[h!]
\centering
\caption{A comparison of the scale factor estimation with or without our introduce scale recovery method.}
\renewcommand\arraystretch{1.5}
\begin{tabular}{cccc}
\hline
Method & Scale & $\mu$ & $\delta$ \\ \hline
Ours & x & 33.109 & 4.384 \\
Ours & \checkmark & 1.128 & 0.118 \\ \hline
\end{tabular}
\label{tb: scale factor}
\end{table}
Moreover, we continue to study the effectiveness of our introduced scale recovery method. Table \ref{tb: scale factor} shows the mean value and standard deviation value by comparing the scale factor of depth estimation with or without scale recovery. It can be seen that the averaged scale coefficient of depth estimation values is closer to 1 when adopted our proposed coarse-to-fine scale recovery method. If scale recovery is not performed, there is a large distance between the estimated scale-ambiguous depths and real depths.
This indicates that the introduction of scale recovery can not only enable PoseNet to output pose predictions with global absolute scale, but also help DepthNet output depth estimates closer to the real absolute scale metric.
\subsection{Pose and Depth Evaluation on the Night Scenes of MVSEC Dataset}
To evaluate our proposed model on very challenging
night scenes, which are in low-light conditions and relatively featureless, we selected the MVSEC dataset to conduct experiments.
A recent state-of-the-art monocular SLAM, (i.e. ORB-SLAM3), a representative unsupervised-learning based VO (i.e. SC-SfMLearner) and a supervised learning based (i.e. DeepVO) are chosen for a comparison.
The trajectories of each model on Evening3 are displayed in Figure \ref{fig: trajectory night}. It can be seen seen that the learning based VOs can robustly estimate vehicle ego-motion and generate trajectories in this challenging scene. However, the traditional model, ORB-SLAM3-Mono fails. This might be due to fact that ORB-SLAM3 Mono can extract and match enough feature points under low-light environment, and thus are not capable of performing effective feature tracking for pose estimation, leading to positioning failure. In contrast, deep learning networks excel at learning and extracting features
and thus enable learning-based methods to extract sufficient and reliable visual feature for positioning.
We use RMSE as the evaluation metric via Equation \ref{deqn_RMSE}, and further compare these quantitative results in Table \ref{tb: pose night},
\begin{equation}
\label{deqn_RMSE}
\text{Error} = \sqrt{\frac{1}{n}(\mathbf{x}_i-\mathbf{x}^\text{gt}_{i}) } \end{equation}
Where $\mathbf{x}_i$ represents the estimated three-dimensional location in $i th$ frame, and $\mathbf{x}^\text{gt}_{i}$ is the corresponding ground-truth location. The quantitative results show that the performance of our proposed VO model exceeds both unsupervised learning based VO, i.e. SC, and supervised learning based VO, i.e. DeepVO. It demonstrates that our framework can robustly and accurately estimate pose in low light environment.
\begin{table}[h!]
\centering
\caption{The pose estimation results in the night car-driving scenes.}
\setlength{\tabcolsep}{0.8mm}{
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{cccc}
\hline
\multicolumn{1}{c}{Method} &\multicolumn{1}{c}{Sensors}&\multicolumn{1}{c}{Scale} & \multicolumn{1}{c}{Error} \\ \hline
ORB-SLAM3 Mono & Mono & \checkmark & failed \\
SC-SfMLearner & Mono & x & 72.05 \\
DeepVO & Mono & \checkmark & 39.37 \\
Ours(VO) & Mono & \checkmark & 35.12 \\
Ours(VIO) & Mono+IMU & \checkmark & \textbf{19.62} \\
\hline
\end{tabular}}
\label{tb: pose night}
\end{table}
\begin{figure}[htp]
\centering
\includegraphics[width=9.2cm]{evening3plusvio.pdf}
\caption{The generated trajectories in the night car-driving scenes}
\label{fig: trajectory night}
\end{figure}
In addition, we qualitatively evaluate the depth estimation performance of our framework in the challenging night scene, comparing with SC.
Figure \ref{fig: depth night} shows input RGB image, depth maps from LIDAR data, depth estimation from SC, and depth estimation from our our model.
Clearly, SC does not perform well when it is in the area of strong light. SC will misleadingly estimates depth in a shorter distance in strong-light condition.
It might be because our model introduces a coarse-to-fine stage into depth estimation that enables DepthNet to overcome the influence of light condition, e.g. too strong or too dark.
\begin{figure*}
\centering
\begin{minipage}[b]{0.05\linewidth}
\centering
\caption*{Input}\vspace{58pt}
\caption*{GT}\vspace{44pt}
\caption*{SC}\vspace{34pt}
\caption*{Ours}
\end{minipage}
\begin{minipage}[b]{0.18\linewidth}
\includegraphics[width=1\linewidth]{RGB353.jpeg}\vspace{4pt}
\includegraphics[width=1\linewidth]{353.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{SC353b.jpg}\vspace{4pt}
\includegraphics[width=1\linewidth]{OursVO353.jpg}
\end{minipage}
\begin{minipage}[b]{0.18\linewidth}
\includegraphics[width=1\linewidth]{RGB2765.jpeg}\vspace{4pt}
\includegraphics[width=1\linewidth]{2765.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{SC2765b.jpg}\vspace{4pt}
\includegraphics[width=1\linewidth]{OursVO2765.jpg}
\end{minipage}
\begin{minipage}[b]{0.18\linewidth}
\includegraphics[width=1\linewidth]{RGB3112.jpeg}\vspace{4pt}
\includegraphics[width=1\linewidth]{3112.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{SC3112.jpg}\vspace{4pt}
\includegraphics[width=1\linewidth]{OursVO3112b.jpg}
\end{minipage}
\begin{minipage}[b]{0.18\linewidth}
\includegraphics[width=1\linewidth]{RGB3211.jpeg}\vspace{4pt}
\includegraphics[width=1\linewidth]{3211.png}\vspace{4pt}
\includegraphics[width=1\linewidth]{SC3211b.jpg}\vspace{4pt}
\includegraphics[width=1\linewidth]{OursVO3211.jpg}
\end{minipage}
\caption{The depth predictions from our model and baselines in low-light condition of driving at night.}
\label{fig: depth night}
\end{figure*}
\subsection{Ablation Study}
Finally, we conduct ablation study to verify the effectiveness of different modules in our framework. We use the KITTI dataset and learning based VIO model. The results of ablation study into framework modules are shown in Table \ref{tb: ablation}.
In order to test the impact of the attention module on pose estimation, we remove attention mask and directly input the fusion features of vision and inertial into the subsequent LSTM and PoseNet. We can see that the attention module can further improve the performance of pose network, especially rotation estimation. Also, the addition of IMU data into framework can enhance the network performance. This indicates the effectiveness of multimodal fusion, as inertial data contributes to rotation estimation more, and the fusion of features can compensate the shortcomings of different modalities.
In order to evaluate the effectiveness of LSTM network, we remove LSTM module, and input the fused features directly into PoseNet, showing that adding LSTM module enhances model performance.
\begin{table}[h!]
\centering
\caption{The ablation study into framework modules}
\setlength{\tabcolsep}{1mm}{
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{cccccccccc}
\hline
\multicolumn{1}{c}{\multirow{2}{*}{Method}} & \multirow{2}{*}{IMU} & \multirow{2}{*}{LSTM} & \multirow{2}{*}{Attention} & \multicolumn{2}{c}{Seq.09} & \multicolumn{2}{c}{Seq.10} & \multicolumn{2}{c}{Avg} \\ \cline{5-10}
\multicolumn{1}{c}{} & & & & $t_{rel}$ & $r_{rel}$ & $t_{rel}$ & $r_{rel}$ & \multicolumn{1}{c}{$t_{rel}$} & \multicolumn{1}{c}{$r_{rel}$} \\ \hline
Ours & & $\checkmark$ & $\checkmark$ & 8.84 & 1.31 & 7.34 & 1.98 & 8.09 & 1.64 \\
Ours & $\checkmark$ & & $\checkmark$ & 14.57 & 0.56 & 10.92 & 1.73 & 12.74 & 1.15 \\
Ours & $\checkmark$ & $\checkmark$ & & \textbf{5.06} & 0.29 & 6.33 & 0.57 & 5.70 & 0.43 \\
Ours & $\checkmark$ & $\checkmark$ & $\checkmark$ & 5.48 & \textbf{0.19} & \textbf{5.37} & \textbf{0.43} & \textbf{5.43} & \textbf{0.31} \\ \hline
\end{tabular}}
\label{tb: ablation}
\end{table}
Then, we study the impact of different sequence lengths on pose estimation, and the comparison results are shown in Table \ref{tb: sequence ablation}.
Compared with a short sequence length (3 frames),
we can see that a longer window length (a sequence of 5 frames) can improve both translation and rotation estimation accuracy, especially the rotation improve is more significant.
\begin{table}[h!]
\renewcommand\arraystretch{1.3}
\centering
\caption{The ablation study into sequence length}
\setlength{\tabcolsep}{2mm}{
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{cccccccc}
\hline
\multicolumn{1}{c}{\multirow{2}{*}{Method}} & \multirow{2}{*}{Window length} & \multicolumn{2}{c}{Seq.09} & \multicolumn{2}{c}{Seq.10} & \multicolumn{2}{c}{Avg} \\ \cline{3-8} & & $t_\text{rel}$ & $r_\text{rel}$ & $t_\text{rel}$ & $r_\text{rel}$ & $t_\text{rel}$ & $r_\text{rel}$ \\ \hline
Ours1 &3 & 6.42 & 0.45 & \textbf{4.90} & 0.55 & 5.66 & 0.5 \\
Ours2 &5 & \textbf{5.48} & \textbf{0.19} & 5.37 & 0.43 & \textbf{5.43} & \textbf{0.31} \\ \hline
\end{tabular}}
\label{tb: sequence ablation}
\end{table}
Finally, the influence of adding scaled depth predictions and learned depth on pose estimation is studied. The comparison results are shown in Table \ref{tb: supervision ablation}.
In order to verify the effectiveness of the learnable DepthNet on pose estimation, Ours1 follows coarse-to-fine scale recovery training, but it uses the scaled depth predictions instead of the learned depths from DepthNet, to form the total loss for bi-directional scale recovery, and the learning rate is consistent with Ours3. Although the scaled depth predictions must be higher in accuracy than the learned depths from DepthNet, the scaled depth predictions are unlearnable and cannot be optimized with the iteration of training. The potential for scaled pseudo depth labels to improve the pose estimation accuracy is thus limited.
In order to verify the effectiveness of the learned depth from DepthNet on pose estimation, Ours2 does not introduce any supervision in framework, but directly uses learned depth to form the total loss for bi-directional scale recovery, and the learning rate is 1e-4.
It can be seen that the performance Ours3 with our proposed training strategy is higher than Ours1 and Ours2.
\begin{table}[h!]
\centering
\caption{The ablation into supervision signal and learning rate, "C" indicates Coarse Scale Recovery, "F" indicates Coarse-to-fine Scale Recovery, "D" indicates scaled depth predictions, "-" means this stage does not exist, "LR" indicates learning rate.}
\setlength{\tabcolsep}{1.5mm}{
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{ccccccccccc}
\hline
\multicolumn{1}{c}{\multirow{2}{*}{Method}}&\multicolumn{2}{c}{Supervision} &\multicolumn{2}{c}{LR} & \multicolumn{2}{c}{Seq.09} & \multicolumn{2}{c}{Seq.10} & \multicolumn{2}{c}{Avg} \\ \cline{2-3} \cline{4-5} \cline{6-11}
&C &F &C &F & $t_\text{rel}$ & $r_\text{rel}$ & $t_\text{rel}$ & $r_\text{rel}$ & $t_\text{rel}$ & $r_\text{rel}$ \\ \hline
Ours1 &D &D &1e-4 &1e-5 & 8.01 & 1.07 & 9.06 & 1.45 & 8.54 & 1.26 \\
Ours2 & - & x &$\times$ &1e-4 & 6.06 & 0.66 & 7.65 & 2.25 & 6.86 & 1.45 \\
Ours3 &D & x &1e-4 & 1e-5 & \textbf{5.48} & \textbf{0.19} & \textbf{5.37} & \textbf{0.43} & \textbf{5.43} & \textbf{0.31} \\
\hline
\end{tabular}}
\label{tb: supervision ablation}
\end{table}
\section{Conclusion}
In this work, we propose a self-supervised learning based depth and egomotion estimation framework with a novel coarse-to-fine scale recovery strategy.
Our model can accurately and robustly output pose and depth with global scale metric, even in low-light conditions at night. We conducted extensive experiments to evaluate our model on two public datasets, i.e. KITTI and MVSEC. On both datasets, our model outperforms both traditional and learning based VOs/VIOs. Ablation study is also performed to verify the effectiveness of each module in our framework.
In the future, we will attempt to exploit IMU data for scale recovery and pose estimation that further contributes to the performance improvement.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,204 |
Removing Barriers to Online Gaming
Video games are more than a mere leisure activity. According to The AbleGamers Foundation, "Video games allow individuals with disabilities to experience situations that may be difficult or limited in the real world, provide social networking opportunities to maintain mental and emotional health, and participate in one of the world's largest pastimes."
Mark Barlet, founder of The AbleGamers Foundation definitely connects something here: video gaming, while a fun pastime and social equalizer also has benefits in in many fields – robotics, business, sports,
science, and medicine. They are such a popular pastime because they allow players to leave the mundane world behind and assume super powers and identities that they don't possess.
Imagine, however, that upon buying a new game – a substantial investment! – you start it up only to find that the sensitivity is too great or not enough, or that it won't run with any other input device than a standard keyboard and mouse, or that there are no 'save' points throughout the game. These and many more barriers rear their heads for a segment of the population. Luckily, AbleGamers exists to help gamers with a wide range of abilities.
Their website outlines their mission statement in three points.
AbleGamers Foundation engages in outreach to citizens who require barrier-free gaming, consults with with programmers, companies, and organizations in order to help them meet the criteria of accessibility from the 'ground up', and offer grants to individuals and companies that manufacture or modify devices and controllers that make gaming truly accessible for disabled gamers.
As an article in Tom's Guide points out, when buying an 'accessible' controller, there is no guarantee that it will be a good match for the gamer – one's disability is as individual as the person that it affects. Therefore, the modification and customization of controllers has been terribly expensive, until recently.
Ben Heckendorn, an ordinary citizen with an inquiring mind, has managed to modify an existing PlayStation 4 DualShock controller by using a 3D printer and rewiring the controller. The product gives excellent maneuverability to players that can only use one hand. Ben Heck, as he is known on his YouTube channel, produces the controllers for about $350 USD per piece due to how labor intensive they are to create. Connecting tinkerers like Heckendorn with charities like AbleGaming is a solid way to create and distribute accessible controllers that match the physical criteria of the individual gamer.
As far as AbleGaming is concerned, there exists one reason for their relentless pursuit of accessibility in the gaming world. As founder Mark Barlet says, "I believe that there is nothing more powerful for people with disabilities than the freedom that only videogames can provide. It is an art form that allows us to all run, jump, and be whatever we want to be." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 0 |
\section{Introduction}
An immersion $f:M\to N$ of a compact Riemannian submanifold $M$ into
a K\" ahler\ manifold $(N,\omega)$ is said to be Hamiltonian stationary Lagrangian\footnote{These
are also known as H-minimal or Hamiltonian minimal.} (which we will abbreviate to HSL) if it is
Lagrangian and its volume is
critical for Hamiltonian variations.
In \cite{Oh2} Oh derived the Euler-Lagrange equations in the following form.
Let $H$ be the mean curvature vector for $f$ and $\sigma_H$ the mean curvature 1-form
$f^*(H\rfloor\omega)$ on $M$. Oh showed that $M$ is Hamiltonian stationary if and only if
$\sigma_H$ is co-closed. When $N$ is K\" ahler-Einstein
$\sigma_H$ is necessarily closed for any Lagrangian submanifold. Therefore a compact
Lagrangian immersion $f:M\to N$ is Hamiltonian stationary if and only if $\sigma_H$ is a harmonic
1-form.
As well as being interesting in its own right, the HSL condition also arises in the study of
minimal Lagrangian submanifolds of KE
manifolds (and not just because this is the case $H=0$). In their study of Lagrangian
surfaces in K\" ahler-Einstein 4-manifolds which minimise volume in their homotopy class, Schoen \&
Wolfson \cite{SchW1} discovered that the minimiser may have isolated singularities whose
tangent cone is only HSL. These are cones over HSL curves in $\mathbb{CP}^1$ and classification of the
latter allowed Schoen \& Wolfson to compute exactly which of these could give tangent cones for the
singular minimisers.
This provides a compelling reason for studying HSL submanifolds in $\mathbb{CP}^n$, since cones over these in
$\mathbb{C}^{n+1}$ provide the models for isolated conical singularities of Lagrangian volume minimisers in
higher dimensional KE manifolds. There are examples of HSL submanifolds of $\mathbb{CP}^n$ which are not
minimal, mostly for the case $n=2$ (see, for example, \cite{CasHU,MaS,Mir1}) and all these examples are
highly symmetric. Because the study of minimal examples is much better developed throughout
the rest of this article we will use ``HSL'' exclusively to mean non-minimal Hamiltonian stationary Lagrangian\ submanifolds.
The majority examples of HSL surfaces in $\mathbb{CP}^2$ are immersed tori (for an example of a HSL Klein
bottle, see \cite{MirN}),
and it is for this case that H\' elein \& Romon \cite{HelR3} (see also Ma \cite{Ma})
showed that the equations governing HSL tori lie in the realm of integrable systems.
It was already known that for $H=0$ this approach gives a classification of minimal Lagrangian
tori \cite{McI03}
in terms of the associated spectral data (basically, an algebraic curve and a holomorphic
line bundle over it) and this led to an understanding of how
complicated the moduli space of minimal Lagrangian tori really is \cite{CarM,Has}.
Our aim here is to provide the analogous classification by spectral data for HSL tori. To be
precise, let us define what we mean by HSL spectral data.
\begin{defn}\label{defn:spectral_data}
A triple $(X,\lambda,\mathcal{L})$ will be called spectral data for a (non-minimal) HSL torus in $\mathbb{CP}^2$
when the following conditions hold.
\begin{enumerate}
\item $X$ must be a complete connected algebraic curve possessing a holomorphic involution $\tau$
with only two fixed points and a real involution $\rho$, such that $\tau$ and $\rho$ commute.
In particular, the genus $g$ of $X$ is even.
\item $\lambda$ must be a rational function on $X$ of degree $3$, non-constant on every irreducible
component of $X$, satisfying
\begin{equation}
\overline{\rho^*\lambda} = \bar\lambda^{-1},\quad \tau^*\lambda = -\lambda,
\end{equation}
and whose zeroes $P_1,P_2,P_3$ and poles $Q_1,Q_2,Q_3$ are distinct smooth points. $\tau$ must
fix one zero $P_3$ (and hence one pole $Q_3$) of $\lambda$. $\rho$ must fix every point over the
circle $|\lambda|=1$ and $\lambda$ must have no branch points on this circle.
\item A double periodicity condition, which only depends upon $(X,\lambda)$,
must be satisfied. The details of this are given in \S \ref{sec:Rec} below.
\item $\mathcal{L}\simeq\mathcal{O}_X(D)$ must be a line bundle of degree $g+2$ over $X$ and satisfy the linear
equivalences
\begin{equation}\label{eq:lequiv}
\rho_*D+D\sim R,\quad \tau_*D+D\sim R+P_3-Q_3,
\end{equation}
where $R$ is the ramification divisor of $\lambda$. Moreover, $D$ must contain no points on
$|\lambda|=1$.
\end{enumerate}
\end{defn}
Now our principal result can be stated as follows.
\begin{thm}\label{thm:main}
There is an essentially bijective correspondence between:
\begin{enumerate}
\item weakly conformal, immersed, (non-minimal) HSL tori in $\mathbb{CP}^2$, and;
\item triples of HSL spectral data $(X,\lambda,\mathcal{L})$.
\end{enumerate}
\end{thm}
The word ``essentially'' means we work with the natural equivalence classes for such data.
Equations \eqref{eq:lequiv} can also be written in terms of the canonical class of $X$ (see
\eqref{eq:leqcan} below).
The real value of this theorem lies in the way the spectral data parameterises
the set of all HSL tori, and what this can tell us about families of HSL tori. The construction
of explicit examples
is briefly discussed in \S\ref{sec:explicit}, where we show what must be done in the case where
$X$ is smooth, using the Riemann $\theta$-function. But such approaches are not very practical
for curves of genus $4$ or more (and for genera $0$ and $2$, where only rational functions or
elliptic functions are required, the examples can be obtained faster
using the direct methods found in \cite{CasHU,HelR3,MaS,Mir1}).
The spectral data itself is best parameterised by the branch
divisor $B=\lambda_*R$ of $\lambda$, which determines $(X,\lambda)$ more or less uniquely as a
$3$-fold branched cover of the Riemann sphere $\mathbb{C}_\infty$. The symmetries $\rho,\tau$ will be consequences of
symmetries of $B$. In particular, the \emph{existence} of a HSL torus with spectral curve $X$
depends only on $(X,\lambda)$ and is independent of condition (d). In fact
(d) can be satisfied whenever $(X,\lambda)$ satisfy (a) and (b) (see \S\ref{sec:Rec} below). When
equations \eqref{eq:lequiv}
can be satisfied they can be satisfied by a real family of divisors of dimension $g/2$.
However, it is non-trivial matter to demonstrate
that spectral data for non-minimal HSL tori exists for a given (even) spectral genus $g\geq 4$,
although this will presumably follow from a perturbation of the argument used for minimal
Lagrangian tori in \cite{CarM}. Here we discuss our expectations for the dimension of the moduli
space of HSL tori in \S\ref{sec:moduli}.
The form of Theorem \ref{thm:main} looks very similar to the corresponding
theorem for minimal Lagrangian tori \cite{McI03}, to which it reduces when the mean curvature is
zero (provided one interprets this limit carefully with respect to the double periodicity
condition: see \S\ref{sec:moduli}). However, to achieve this required careful
adaptation of the tools used for that case, because the non-minimal HSL equations are encoded as
the Maurer-Cartan
equations for a loop algebra valued $1$-form $\alpha_\zeta$ which has \emph{quadratic} dependence
on the spectral parameter $\zeta$ (see equation \eqref{eq:MCform}).
The success of the adaptation rested on two insights: (i) the natural vacuum solution for the
relevant dressing action does \emph{not} correspond to a HSL immersion, but only gives a
constant map into $\mathbb{CP}^2$, (ii) the loop algebra can be ``partially untwisted'' (see
\S\ref{sec:untwist}) to enable us to view $\alpha_\zeta$ as linear in $\zeta^2$.
To avoid assumptions about the
smoothness or irreducibility of the spectral curve we construct the spectral data from the
algebra of polynomial Killing\ fields (i.e, the spectral curve $X$ is a
ringed space and it carries a ``line bundle'' $\mathcal{L}$ as an $\mathcal{O}_X$-module), following
\cite{McI95,McI03}. In particular, our method of proof does not rule out the existence of
singular spectral curves. Indeed, this is a subtle point which is not well addressed in the
literature, since many spectral curve constructions force smoothness (by taking the normalisation)
or assume it without comment. We prefer to allow for the possibility of singular spectral curves
since this simplifies, amongst other things, the periodicity conditions.
We expect the general method of our approach will apply to other examples where the Maurer-Cartan
form is not linear in the spectral parameter (such as those in \cite{BurK,Khe,Ter}). In particular,
the appendix \ref{app:dressing} adapts the dressing construction of \cite{BurP2} to deal with the
general situation of a quadratic dependence on the spectral parameter.
By contrast, Mironov \cite{Mir2} gives an approach, specific to HSL, which uses a
different form for the equations.
\smallskip\noindent
\textit{Acknowledgments.} This article includes and extends the work of the first author
which appeared in his dissertation \cite{Hun}. These results were first announced
at the Symposium of the differential geometry of submanifolds at Valenciennes, July 2007
\cite{McIval}. The approach taken here in defining the Lagrangian angle was suggested to us
by Fran Burstall. We would also like to thank Fran Burstall and
David Calderbank for pointing out the inadequacies of the original attempt to
describe the dressing arguments contained in appendix \ref{app:dressing}.
\smallskip\noindent
\textit{Notational conventions.}
Throughout this article we denote $\mathbb{C}\setminus\{0\}$ by $\mathbb{C}^\times$. We will use $e_1,\ldots,e_n$ to
denote the standard (oriented orthonormal) basis
of $\mathbb{R}^n$ and $\mathbb{C}^n$. For any complex matrix $A$ its Hermitian transpose will be denoted by
$A^\dagger$. When $A$ is invertible we use $A^*$ to denote $(A^t)^{-1}$.
For any non-zero
vector $v\in\mathbb{C}^{n+1}$ we denote by $[v]\in\mathbb{CP}^n$ the line generated by $v$. For a real Lie algebra
$\mathfrak{g}$ with complexification $\mathfrak{g}^\mathbb{C}$ conjugation with respect to that real form will be denoted by
$\bar\xi$ for $\xi\in\mathfrak{g}$. For any $1$-form $\beta\in\Omega^1_M$ its type
decomposition will be denoted by $\beta'+\beta''$, where $\beta'\in\Omega_M^{(1,0)}$ and
$\beta''\in\Omega_M^{(0,1)}$. For the most part the surfaces we deal with are tori, which we will
write as $\mathbb{C}/\Gamma$ (for a lattice $\Gamma$) when we want to emphasize the complex structure,
or as $\mathbb{R}^2/\Gamma$ when we want to emphasize the real structure.
\section{The HSL condition via a loop of flat connexions.}
We begin by clarifying the notion of the \emph{Lagrangian angle} for Lagrangian submanifolds
of a K\" ahler-Einstein manifold. Let $(N,\omega)$ be a KE $2n$-manifold with K\" ahler form $\omega$
and volume form $\mathrm{vol}_N$. Let $\operatorname{Lag}(N)\to N$ be its bundle of oriented Lagrangian $n$-planes in $TN$.
To any oriented Lagrangian immersion $f:M\to N$ we can assign the Gauss map
\[
\gamma:M\to f^{-1}\operatorname{Lag}(N);\quad \gamma(p) = df(T_pM),
\]
and there is a unique smooth section $\Omega:M\to f^{-1}K_N$ for which at each point
$\Omega|\gamma(p)=\mathrm{vol}_N$: we will say $\Omega$ calibrates $\gamma(p)$. In that case
$\Omega$ must be a section of the unit circle subbundle $\mathcal{S}$ of $f^{-1}K_N$.
\begin{defn}
The unique section $\Omega:M\to \mathcal{S}\subset f^{-1}K_N$ of the unit circle
bundle $\mathcal{S}$ which calibrates the Gauss map $\gamma$ of $f$ at each point will be called
the \emph{Lagrangian angle} of $f$.
\end{defn}
When $N$ is K\" ahler-Einstein $\mathcal{S}$ is flat, since $K_N$
has curvature given, up to scale, by $\omega$. We define the Maslov form by
\begin{equation}\label{eq:Maslov}
\mu = -\frac{1}{i\pi}\Omega^*\alpha
\end{equation}
where $\alpha$ is the connexion 1-form for the induced connexion on
the principal circle bundle $S$. Thus $\mu$ is closed, since $\mathcal{S}$ is flat, and provides
the Maslov class $[\mu]\in H^1(M,\mathbb{R})$. A calculation originally due to Dazord \cite{Daz}
asserts that the mean curvature 1-form $\sigma_H = f^*(H\rfloor\omega)$ satisfies
\[
\sigma_H = \frac{\pi}{2}\mu.
\]
Thus $f$ is HSL if and only if it has co-closed, and therefore harmonic, Maslov form.
We focus now on the case where $N=\mathbb{CP}^2$. We consider $\mathbb{CP}^2$ as the homogeneous space $G/K$
for the projective unitary group $G=PU(3)$ (i.e., $SU(3)$ modulo its centre), where $K$ is the
stabiliser of the line $[e_3]$. To make definitions easier, elements of $PU(3)$ will
tend to be thought of as represented by elements of
$SU(3)$. Thus $K$ is the fixed point subgroup for the involution
$\sigma\in\operatorname{Aut}(G)$ defined by
\begin{equation}\label{eq:sigma}
\sigma(g) = \begin{pmatrix} -I_2 & 0 \\ 0 & 1\end{pmatrix} g \begin{pmatrix} -I_2 & 0 \\ 0 &
1\end{pmatrix},
\end{equation}
where $I_2$ is the $2\times 2 $ identity matrix. The corresponding reductive decomposition of
$\mathfrak{g}\simeq\mathfrak{su}(3)$ will be written $\mathfrak{g}=\mathfrak{k} +\mathfrak{m}$. Recall that the tangent space
$T^\mathbb{C}(G/K)$ can be identified with the quotient space $G\times_K\mathfrak{m}^\mathbb{C}$, where $K$
acts adjointly on $\mathfrak{m}^\mathbb{C}$.
Now define an order 4 outer automorphism $\tau\in\operatorname{Aut}(G)$ by
\begin{equation}\label{eq:tau}
\tau(g) = Sg^*S^{-1},\quad S = \begin{pmatrix} J & 0 \\ 0 & 1\end{pmatrix},\quad
J = \begin{pmatrix} 0 & 1\\ -1 & 0\end{pmatrix}.
\end{equation}
Since $\tau$ preserves the centre of $SU(3)$ it is well-defined on $PU(3)$. We will denote the fixed
point subgroup of $\tau$ by $G_0$: it is easy to check that $G_0\simeq SU(2)$. Of course, $G/G_0$ is a
4-symmetric space.
It can be shown (see \cite{BurK,McIval}) that it is isomorphic to the unit subbundle
of the canonical bundle $K_{\mathbb{CP}^2}$.
We are now in a position to introduce a preferred family of frames for Lagrangian immersions $f:M\to
G/K$. As we have explained above, any such immersion possesses a Lagrangian angle $\Omega:M\to G/G_0$,
thought of as a section of $f^{-1}K_{\mathbb{CP}^2}$.
\begin{defn}
Let $f:M\to G/K$ be a Lagrangian immersion and $\Omega:M\to G/G_0$ its Lagrangian angle. A frame
$F:\tilde M\to G$ will be called a \emph{Lagrangian frame} for $f$ if it also frames $\Omega$.
\end{defn}
Of course, Lagrangian frames are not unique: they may be right multiplied by any element of the gauge
group $\mathcal{G}_0=C^\infty(\tilde M,G_0)$.
Given a Lagrangian frame $F$ its left Maurer-Cartan form $\alpha=F^{-1}dF$ admits a decomposition
$\alpha_\mathfrak{k}+\alpha_\mathfrak{m}$ into summands taking values in the eigenspaces of $\sigma$.
Recall that
$\operatorname{Ad} F\cdot\alpha_\mathfrak{m}$ represents the tangent map $df$. Now let $\mathfrak{g}_j\subset \mathfrak{g}^\mathbb{C}$ denote the
$i^j$-eigenspace of $\tau$ (as the induced automorphism on $\mathfrak{g}^\mathbb{C}$) and notice that since
$\sigma=\tau^2$ we must have $\mathfrak{k}^\mathbb{C} = \mathfrak{g}_0+\mathfrak{g}_2$, and
$\mathfrak{m}^\mathbb{C} = \mathfrak{g}_{-1}+\mathfrak{g}_1$. To be explicit, these subspaces $\mathfrak{g}_j\subset\mathfrak{sl}_3(\mathbb{C})$ are
\begin{eqnarray}\label{eq:g_j}
& \mathfrak{g}_0 = \{\begin{pmatrix} Q & 0\\ 0&0\end{pmatrix}: Q\in\mathfrak{sl}_2(\mathbb{C})\},
&\mathfrak{g}_1 = \{\begin{pmatrix} O_2& u\\-i(Ju)^t & 0 \end{pmatrix}: u\in\mathbb{C}^2\},\notag \\
&\mathfrak{g}_{-1} = \bar\mathfrak{g}_1,& \mathfrak{g}_2 = \{\begin{pmatrix} aI_2 & 0 \\ 0 & -2a\end{pmatrix}:a\in\mathbb{C}\}.
\end{eqnarray}
Here we are using $O_2$ to denote the $2\times 2$ zero matrix.
Note in particular that $[\mathfrak{g}_0,\mathfrak{g}_2]$=0, hence the adjoint action of $G_0$ on $\mathfrak{g}_2$ is trivial.
Now let $\alpha_j$ denote the projection of $\alpha$ onto $\mathfrak{g}_j$ for $j=-1,0,1$, then
$\alpha_\mathfrak{m} = \alpha_{-1}+\alpha_1$ and the condition that $f$ is conformal Lagrangian is that
$\alpha^\prime_\mathfrak{m} = \alpha_{-1}$. The projection of $\alpha$ onto $\mathfrak{g}_2$
is, effectively, the Maslov form. We will write this projection as $\alpha_{-2}+\alpha_2$ where
$\alpha_{-2}''=0$ and $\alpha_2=\bar\alpha_{-2}$, so that their sum gives the type decomposition of
that projection. We will also fix the metric on $G/G_0$ so that by equation
\eqref{eq:Maslov}
\begin{equation}\label{eq:alpha2}
\alpha_{-2}+\alpha_2 = -i\pi\mu\begin{pmatrix} I_2 & 0 \\ 0 & -2\end{pmatrix}.
\end{equation}
From $\alpha$ we can form the \emph{extended Maurer-Cartan form}
\begin{equation}\label{eq:MCform}
\alpha_\zeta= \zeta^{-2}\alpha_{-2}+\zeta^{-1}\alpha_{-1}+\alpha_0+\zeta\alpha_1+\zeta^2\alpha_2.
\end{equation}
Here $\zeta$ is a rational parameter on the Riemann sphere $\mathbb{C}_\infty$, so that $\alpha_\zeta$ takes
values in $\mathfrak{g}^\mathbb{C}$ for $\zeta\in\mathbb{C}^\times$.
It possesses two symmetries, namely, a real symmetry and $\tau$-equivariance:
\begin{equation}\label{eq:real}
\overline{\alpha_{\zeta}} = \alpha_{\bar\zeta^{-1}},\quad
\tau(\alpha_\zeta) = \alpha_{i\zeta}.
\end{equation}
Consequently we may think of $\alpha_\zeta$ as taking values in
the twisted loop algebra $\Lambda^\tau\fg$ of $\tau$-equivariant real analytic maps $\xi_\zeta:S^1\to\mathfrak{g}^\mathbb{C}$
possessing the real symmetry. This is a real subalgebra of the complex algebra $\Lambda^\tau\fg^\C$ of
$\tau$-equivariant real analytic maps $\xi_\zeta:S^1\to\mathfrak{g}^\mathbb{C}$.
\begin{thm}[\cite{HelR2}]\label{thm:HelR}
Let $\alpha_\zeta$ be the extended Maurer-Cartan form, obtained from any Lagrangian frame,
for a weakly conformal Lagrangian immersion of a surface $f:M\to\mathbb{CP}^2$. Then $f$ is HSL if and only if
\begin{equation}\label{eq:MC}
d\alpha_\zeta+\frac{1}{2}[\alpha_\zeta\wedge\alpha_\zeta]=0.
\end{equation}
Conversely, suppose $\alpha_\zeta\in\Omega^1_{\tilde M}\otimes \Lambda^\tau\fg$ has the form \eqref{eq:MCform}
with $\alpha_{-2}''=0$, $\alpha_{-1}\neq 0$ and $\alpha_{-1}''=0$. If $\alpha_\zeta$ satisfies
\eqref{eq:MC} then there exists a \emph{based extended Lagrangian frame}
\begin{equation}\label{eq:based}
F_\zeta:\tilde M\to C^\omega(S^1,G),\qquad F_\zeta^{-1}dF_\zeta = \alpha_\zeta,\
F_\zeta(0)\in G_0.
\end{equation}
In that case $F_1$ is the Lagrangian frame for a weakly conformal HSL immersion
$f:\tilde M\to G/K$.
\end{thm}
The phrase ``weakly conformal'' acknowledges the possibility that $\alpha_{-1}$ (equally $df$,
and hence the induced metric) may
vanish at certain points, which we must allow for in the dressing construction: this is explained
in remark \ref{rem:branches} below. Such points will be branch points of the immersion
$f$.
Notice that since $\alpha_\zeta$ is holomorphic in $\zeta$ throughout $\mathbb{C}^\times$, every extended
Lagrangian frame $F_\zeta$ can be extended uniquely holomorphically in $\zeta$ to $\mathbb{C}^\times$, in
which case as a function of $z$ it takes values in $C^\omega(\mathbb{C}^\times,G^\mathbb{C})$. Since $\alpha_\zeta$ is
$\tau$-equivariant and $F_\zeta(0)=I$ we deduce that $F_\zeta$ is also $\tau$-equivariant.
\section{The spectral data for HSL tori.}
\subsection{Polynomial Killing fields.}
From now on we restrict our attention to the case where $M$ is a 2-torus with conformal class fixed by
$M=\mathbb{C}/\Gamma$, where $\Gamma$ is the period lattice, and fix a conformal coordinate $z$ on $\mathbb{C}$.
In this case HSL immersions of $\mathbb{C}/\Gamma$ into
$\mathbb{CP}^2$ are of finite type in the sense of integrable systems geometry \cite{HelR3}.
Let us recall what this means. First we need the notion of an adapted polynomial Killing field.
\begin{defn}
An \emph{adapted polynomial Killing\ field} for an extended Maurer-Cartan form
$\alpha_\zeta\in\Omega^1_\mathbb{C}\otimes\Lambda^\tau\fg$ is a smooth map $\xi_\zeta:\mathbb{C}/\Gamma\to\Lambda^\tau\fg$ satisfying
\begin{eqnarray}
d\xi_\zeta & = & [\xi_\zeta,\alpha_\zeta],\label{eq:Lax}\\
\xi_\zeta & = & \zeta^{-4d-2}\alpha_{-2}(\frac{\partial}{\partial z}) +
\zeta^{-4d-1}\alpha_{-1}(\frac{\partial}{\partial z}) + \ldots,\label{eq:adapted}
\end{eqnarray}
for some non-negative integer $d$. We say $\xi_\zeta$ has \emph{degree} $4d+2$.
\end{defn}
The second condition ensures that the first
condition is a pair of commuting Lax equations on a finite dimensional subspace of the
loop algebra, and the arguments of \cite{BurP1} can be adapted to show that this implies
Hamiltonian integrability (see the appendix \ref{app:dressing} below).
Now we recall that a HSL surface is of \emph{finite type} when it has an extended
Maurer-Cartan form
which admits an adapted polynomial Killing field. Since this property is gauge invariant for the
gauge group $\mathcal{G}_0$ it is a property of the Lagrangian angle and not the particular choice of
extended frame.
Adapted polynomial Killing\ fields are not unique: there are infinitely many linearly independent adapted
polynomial Killing\ fields.
More generally, if we drop condition \eqref{eq:adapted} we get a large collection
of polynomial Killing\ fields.
Polynomial Killing fields are the key to encoding the immersion as spectral data.
The approach we take constructs the spectral data arises from a commutative algebra $\mathcal{A}$
of polynomial Killing\ fields. To understand this algebra we adapt the idea of ``dressing the vacuum
solution'', as presented in \cite {BurP2}, to take account of quadratic dependence
of \eqref{eq:MCform} on $\zeta$.
\subsection{Dressing action.}
For any subset $\Delta\subset\mathbb{C}_\infty$ which is invariant under the real involution
$\zeta\mapsto\bar\zeta^{-1}$ we define
\[
\Lambda^\tau(\Delta,G) = \{g:\Delta\to G^\mathbb{C}:\text{$g$ is analytic},\
\bar{g_\zeta}=g_{\bar\zeta^{-1}},\ \tau(g_\zeta) = g_{i\zeta}\}.
\]
Because the extended Maurer-Cartan form $\alpha_\zeta$ is analytic in $\zeta$ on
$\mathbb{C}^\times$ we can think of every
extended Lagrangian frame $F_\zeta$ as a map from the universal cover $\mathbb{C}$ of our torus into
$\Lambda^\tau(\C^\times,G)$. Our dressing action
will be an action of a loop group on the space of HSL immersions of $\mathbb{C}$ into $\mathbb{CP}^2$.
We take the domain to be $\mathbb{C}$ because typically the dressing action does not preserve
periodicity. But dressing does preserve the Maslov form, so that to study dressing orbits of
tori we need only consider the case where $\mu$ is a constant real $1$-form on $\mathbb{C}$
(where constant means $\mu'=a\, dz$ for some complex constant $a$).
To each real constant $1$-form $\mu\in\Omega^1_\mathbb{C}$ define $\mathcal{E}(\mu)$ to be the space of based
extended Lagrangian frames for HSL immersions of $\mathbb{C}$ into $\mathbb{CP}^2$ with Maslov
form $\mu$, which is equivalent to saying the Maurer-Cartan form of each extended frame satisfies
\eqref{eq:alpha2}. We then define $\mathcal{L}(\mu)$ to be the space of Lagrangian angles for these
immersions. One
consequence of theorem \ref{thm:HelR} is that $\mathcal{L}(\mu)\simeq \mathcal{E}(\mu)/\mathcal{G}_0$. The dressing theory
exploits a loop group action on, in our case, a slight enlargement of $\mathcal{E}(\mu)$ to show that all
HSL tori of finite type and Maslov form $\mu$ lie in the same dressing orbit. The dressing action
is described as follows.
Fix a real number $0<\epsilon<1$ and define
\[
C_\epsilon = \{\zeta\in\mathbb{C}^\times: |\zeta|=\epsilon\ \textit{or}\ \epsilon^{-1}\}.
\]
This is a union of two circles. The closed annulus in $\mathbb{C}^\times$ bounded by $C_\epsilon$ will be called $E_\epsilon$ and
$I_\epsilon$ will denote the union of closed discs in $\mathbb{C}_\infty$ with boundary $C_e$. Notice that restriction to
the boundary $C_\epsilon$ allows us to consider both $\Lambda^\tau(E_\epsilon,G)$ and $\Lambda^\tau(I_\epsilon,G)$ as subgroups of $\Lambda^\tau(C_\epsilon,G)$: we shall do
this without further comment. Let $B\subset
G_0^\mathbb{C}$ denote the subgroup of all those matrices in $G_0^\mathbb{C}$ which are upper triangular with positive
real entries on the diagonal, and define
\[
\Lambda^\tau_B(I_\epsilon,G) = \{g_\zeta\in\Lambda^\tau(I_\epsilon,G):g_0\in B\}.
\]
We recall a fundamental factorisation result concerning the loop
group $\Lambda^\tau(C_\epsilon,G)$.
\begin{thm}[\cite{McIFW,DorPW}]\label{thm:factor}
The map
\[
\Lambda^\tau(E_\epsilon,G)\times\Lambda^\tau_B(I_\epsilon,G)\to\Lambda^\tau(C_\epsilon,G);\quad (g,h)\mapsto gh,
\]
is a diffeomorphism, hence every smooth
$\Lambda^\tau(C_\epsilon,G)$-valued function $g(z)$ admits a unique factorisation into the product $g_E(z)g_I(z)$ of smooth
functions with values in $\Lambda^\tau(E_\epsilon,G)$ and $\Lambda^\tau_B(I_\epsilon,G)$ respectively.
\end{thm}
As a consequence (see \cite[\S 2]{BurP1})
one obtains the dressing action of $\Lambda^\tau(I_\epsilon,G)$, for any $0<\epsilon<1$, on $\Lambda^\tau(\C^\times,G)$, defined by
\begin{equation}\label{eq:dressing}
\Lambda^\tau(I_\epsilon,G)\times\Lambda^\tau(\C^\times,G)\to \Lambda^\tau(\C^\times,G);\quad (g,h)\mapsto g\sharp h = (gh)_E.
\end{equation}
Burstall \& Pedit \cite{BurP1} also show that this action descends to the space $\Lambda^\tau(\C^\times,G)/G_0$ of cosets
of constant loops.
To apply this to extended Lagrangian frames we first note that, by a standard calculation which
exploits the factorisation theorem (cf.\ equation \eqref{eq:dressalpha} below),
whenever $F_\zeta\in\mathcal{E}(\mu)$, with Maurer-Cartan form $\alpha_\zeta$ \eqref{eq:MCform},
and $g\in\Lambda^\tau(I_\epsilon,G)$ the map $g\sharp F_\zeta:\mathbb{C}\to\Lambda^\tau(\C^\times,G)$ defined by
\begin{equation}\label{eq:Fdressing}
(g\sharp F_\zeta)(z) = g\sharp (F_\zeta(z))
\end{equation}
has the property that its Maurer-Cartan form has the shape
\begin{equation}\label{eq:degree2}
\zeta^{-2}\alpha_{-2}+\zeta^{-1}\beta_{-1}+\beta_0 +\zeta \beta_1 + \zeta^2 \alpha_2
\end{equation}
for some $1$-forms $\beta_j\in\Omega^1_\mathbb{C}\otimes \mathfrak{g}^\mathbb{C}$ satisfying $\beta_{-1}''=0$ and
$\bar\beta_j = \beta_{-j}$. In particular, this has the same coefficients of $\zeta^{\pm 2}$
as $\alpha_\zeta$ since the adjoint action of $G_0$ on $\mathfrak{g}_2$ is trivial. Thus $g\sharp F_\zeta$
will also be an extended Lagrangian frame for a weakly conformal immersion with the same Maslov form
provided $\beta_{-1}'$ is not identically zero. This
can fail and in fact $\beta_{-1}= 0$ precisely for what we will call the ``vacuum
solution''. The vacuum solution is the simplest non-trivial
solution of the equations \eqref{eq:MC} for which $\alpha_\zeta$ has the shape \eqref{eq:degree2},
but it does not produce a surface in $G/K$, so there is no corresponding HSL surface.
For this reason, we must introduce the strict enlargement
$\hat\mathcal{E}(\mu)$ of $\mathcal{E}(\mu)$ consisting of those $\Lambda^\tau(\C^\times,G)$-valued functions $H_\zeta(z)$ on $\mathbb{C}$ which
satisfy \eqref{eq:degree2}, with $\beta_{-1}''=0$ and \eqref{eq:Maslov}. Then \eqref{eq:Fdressing}
does describe an action of $\Lambda^\tau(I_\epsilon,G)$ on
$\hat\mathcal{E}(\mu)$, and by the remarks above this action descends to an action on the enlarged space
$\hat\mathcal{L}(\mu) = \hat\mathcal{E}(\mu)/\mathcal{G}_0$
\begin{defn}\label{defn:vacuum}
The \emph{vacuum solution} in $\hat\mathcal{E}(\mu)$ is the based extended frame
\begin{equation}\label{eq:Fvacuum}
F^\mu_\zeta = \exp(z\zeta^{-2}A +\bar z\zeta^2 \bar A),\quad
A\, dz = -i\pi\mu'\begin{pmatrix}I_2 & 0 \\ 0 & -2\end{pmatrix}.
\end{equation}
It has Maurer-Cartan form $\alpha^\mu_\zeta = \zeta^{-2} A\, dz +\zeta^2 \bar A\, d\bar z$.
\end{defn}
\begin{rem}
It is convenient, and makes comparison with \cite{HelR3} easier, for us to write
$-\mu=\bar\mu_0dz+\mu_0d\bar z$, in which case $e^{i(z\bar\mu_0+\bar z\mu_0)}$
is the Lagrangian angle function in the terminology of \cite{HelR3}.
\end{rem}
\begin{lem}\label{lem:vacuum}
Let $F_\zeta\in\hat\mathcal{E}(\mu)$ be such that its Maurer-Cartan form $\alpha_\zeta$ has $\alpha_{-1}=0$,
then $F_\zeta$ is gauge equivalent to $F^\mu_\zeta$.
\end{lem}
\begin{proof}
By definition of $\hat\mathcal{E}(\mu)$, $\alpha_{\pm 2}$ satisfy \eqref{eq:Maslov} and therefore
$[\alpha_{-2}\wedge\alpha_2]=0$. The Maurer-Cartan equations \eqref{eq:MC} then imply
\[
d\alpha_0 + \frac{1}{2}[\alpha_0\wedge\alpha_0]=0,
\]
Since we are working over $\mathbb{C}$ this means the term $\alpha_0$ is a pure gauge term, hence
$\alpha_\zeta$ is gauge equivalent to $\alpha^\mu_\zeta$.
\end{proof}
Notice that $F^\mu_1$ takes values in
$\exp(\mathfrak{g}_2)$, which is an $S^1$ subgroup of $K$, hence the vacuum solution corresponds to a constant
map of $\mathbb{C}$ into $\mathbb{CP}^2$.
We are now in a position to state the dressing theorem for HSL tori in $\mathbb{CP}^2$.
\begin{thm}\label{thm:dressing}
Let $f:\mathbb{C}/\Gamma\to\mathbb{CP}^2$ be a HSL torus with Maslov form $\mu$. Then it admits a
based extended frame $F_\zeta$ for which, for some $\epsilon >0$ and $g\in\Lambda^\tau(I_\epsilon,G)$,
$F_\zeta = g\sharp F^\mu_\zeta$.
Consequently, there exists $\chi:\mathbb{C}\to\Lambda^\tau_B(I_\epsilon,G)$ for which
\begin{equation}\label{eq:chi}
gF^\mu_\zeta = F_\zeta\chi.
\end{equation}
\end{thm}
The proof is in appendix \ref{app:dressing}.
\begin{rem}\label{rem:branches}
The HSL maps $f:\mathbb{R}^2\to\mathbb{CP}^2$ which are constructed by dressing might have branch points. To see
why, recall that branch points occur precisely at the points where $\alpha_{-1}$ vanishes. We can
express this in terms of the coefficients of the
factor $\chi=(I+\chi_1\zeta+\ldots)\chi_0$ in \eqref{eq:chi}. We have
\begin{equation}\label{eq:dressalpha}
\alpha_\zeta = F_\zeta^{-1}dF_\zeta = \operatorname{Ad}\chi_\zeta\cdot\alpha^\mu_\zeta -d\chi.\chi^{-1},
\end{equation}
and so
\[
\alpha_{-1} = \operatorname{Ad}\chi_0\cdot[\chi_1,A].
\]
Now $\ker(\operatorname{ad} A)\cap\mathfrak{g}_{-1}=\{0\}$, so that $\alpha_{-1}$ vanishes precisely where $\chi_1$
vanishes. This is at worst on a proper analytic subset of $\mathbb{R}^2$, but we know of no proof that for
tori it must be non-zero everywhere, so we cannot rule out the possibility of branch points. In the
case of HSL tori in $\mathbb{R}^4$ branch points can also occur \cite{McIR}. However, for minimal
Lagrangian tori in $\mathbb{CP}^2$ branch points cannot occur, since in that case the formula for
$\alpha_{-1}$ has
the form $\operatorname{Ad}\chi_0\cdot A$ (with the appropriate interpretation of $\chi_0$ and $A$ for that case),
and $\chi_0$ is invertible, hence nowhere zero, on $\mathbb{R}^2$.
\end{rem}
The previous theorem provides the following extremely useful characterisation of polynomial Killing\ fields.
\begin{lem}\label{lem:pkf}
A map $\xi:\mathbb{C}\to\Lambda^\tau\fg^\C$ is a polynomial Killing\ field for
$F_\zeta\in\hat\mathcal{E}(\mu)$ if and only if $\xi = \operatorname{Ad}\chi\cdot\eta$ for
some $\eta\in\Lambda^\tau\fg^\C$ for which $[A,\eta]=0$ and $\operatorname{Ad}\chi\cdot\eta$ is a Laurent polynomial in
$\zeta$ (of bounded degree as $z$ varies).
\end{lem}
The proof is a direct adaptation of lemma A.1 in \cite{McI98}, but the origin of the idea is the
Zakharov-Shabat dressing argument for zero curvature equations. The idea is that
$\eta=\operatorname{Ad}\chi^{-1}\cdot\xi$ satisfies
\[
d\eta + [\eta,\alpha_\zeta^\mu]=0,
\]
and from this it follows that $d\eta=0$ and $[A,\eta]=0$.
\subsection{The algebra generated by polynomial Killing\ fields.}
Our aim is to produce a unital commutative complex matrix algebra $\mathcal{A}$ which contains all the
$\tau$-equivariant polynomial Killing\ fields, and ideally $\mathcal{A}$ should be no larger than necessary. Since the
Lax equation \eqref{eq:Lax} above is preserved under
matrix multiplication it is possible to construct $\mathcal{A}$ so that its elements still satisfy
\eqref{eq:Lax}, but at the expense of losing $\tau$-equivariance. Equation \eqref{eq:Lax} is also
satisfied by any $z$-independent multiple of the identity $I\in\mathfrak{gl}_3(\mathbb{C})$, therefore it is natural
in the first instance to consider polynomial Killing\ fields taking values in the loop algebra
\[
\mathcal{R}_\epsilon = \{\eta:C_\epsilon\to \mathfrak{gl}_3(\mathbb{C})|\ \eta\ \text{is algebraic}\},
\]
where the terminology ``$\eta$ is algebraic'' means $\eta$ is analytic on $C_\epsilon$ and
extends meromorphically into $I_\epsilon$ where it may only have poles, if any, at $0$ and $\infty$.
All the $\tau$-equivariant polynomial Killing\ fields take values in $\mathcal{R}$. Moreover, a simple calculation
shows that if $\xi_\zeta,\eta_\zeta$ are $\tau$-equivariant polynomial Killing\ fields then
\[
\xi_{-\zeta}\eta_{-\zeta} = \tau^2(\xi_\zeta\eta_\zeta).
\]
Now we recall that $\tau^2 =\sigma$ \eqref{eq:sigma}, so we may
restrict our attention to polynomial Killing\ fields with values in the subalgebra
\[
\mathcal{R}_\epsilon^\sigma = \{\eta_\zeta\in\mathcal{R}: \sigma(\eta_\zeta) = \eta_{-\zeta}\}.
\]
We will use $\mathcal{K}^\sigma$ to denote the set of polynomial Killing fields
with values in $\mathcal{R}_\epsilon^\sigma$. This is a unital algebra which is independent of $\epsilon$: it contains
the identity matrix as well as
all our original polynomial Killing\ fields and their products. The Lax equation \eqref{eq:Lax}, together with the
algebraic properties of $\chi_\zeta$, ensures that $\operatorname{Ad}\chi_\zeta$ maps $\mathcal{K}^\sigma$ into
$\mathcal{R}_\epsilon^\sigma$. As a
consequence of the argument used to prove lemma \ref{lem:pkf}, a map $\xi:\mathbb{C}\to\mathcal{R}_\epsilon^\sigma$ which
is algebraic in $\zeta$ satisfies
the Lax equation \eqref{eq:Lax} if and only if $\eta=\operatorname{Ad}\chi^{-1}\cdot\xi$ satisfies $d\eta=0$ and
$[\eta,A]=0$. It follows that $\mathcal{K}^\sigma$ is non-abelian,
because the centraliser $\mathfrak{z}(A)$ of $A$ in $\mathfrak{gl}_3(\mathbb{C})$
is not abelian. To find a maximal abelian subalgebra
$\mathcal{A}^\sigma\subset\mathcal{K}^\sigma$ we fix maximal torus $\mathfrak{h}\subset\mathfrak{z}(A)$. It does not matter which
one, but for convenience we choose the maximal torus of diagonal matrices in $\mathfrak{gl}_3(\mathbb{C})$. We let
$\mathcal{H}_\epsilon^\sigma\subset\mathcal{R}_\epsilon^\sigma$ denote the elements of $\mathcal{R}_\epsilon^\sigma$ with values in $\mathfrak{h}$.
Now we can define
\[
\mathcal{A}^\sigma = \{\xi=\operatorname{Ad}\chi\cdot\eta:\eta\in\mathcal{H}_\epsilon^\sigma, \xi\ \text{Laurent poly.\ in}\ \zeta\}.
\]
This is a commutative unital $\mathbb{C}$-algebra of polynomial Killing\ fields.
In fact, $\mathcal{A}^\sigma$ is best thought of as a complex $1$-parameter family of subalgebras of
$\mathcal{R}_\epsilon^\sigma$:
\[
\mathcal{A}^\sigma(z) = \{\xi(z):\xi\in\mathcal{A}^\sigma\}\subset\mathcal{R}_\epsilon^\sigma.
\]
These are all isomorphic since the Lax equation \eqref{eq:Lax} is equivalent to the condition
\begin{equation}\label{eq:evolution}
\xi_\zeta(z) = \operatorname{Ad} F_\zeta(z)^{-1}\cdot\xi_\zeta(0).
\end{equation}
Moreover, the algebra homomorphism $\mathcal{A}^\sigma\to\mathcal{A}^\sigma(0)$ which maps $\xi(z)$ to
$\xi(0)$ is injective for the same reason, so we may identify each $\mathcal{A}^\sigma(z)$ with
$\mathcal{A}$ itself.
\subsection{Untwisting.}\label{sec:untwist}
The geometric interpretation of $\mathcal{A}^\sigma$ simplifies by eliminating
the $\sigma$-equivariance of these polynomial Killing\ fields using ``untwisting'', which is an algebra
isomorphism $\mathcal{R}_\epsilon^\sigma\simeq\mathcal{R}_{\epsilon^2}$. This ``partially untwists'' the
$\tau$-equivariant loops, i.e., they retain a symmetry which distinguishes them. Because we need
to work with the extended frame as well
as the polynomial Killing\ fields we must introduce this untwisting at the loop group level as well as the algebra
level.
First we define
\[
\kappa_\zeta = \begin{pmatrix} \zeta I_2 & 0 \\0 & 1\end{pmatrix},
\]
the significance being that $\kappa_\zeta$ is the $1$-parameter subgroup of $SU(3)$ for which
conjugation by $\kappa_{-1}$ yields the involution $\sigma$. By this property whenever $g_\zeta \in
\Lambda^\sigma(C_\epsilon,G)$ the quantity $\kappa_\zeta^{-1}g_\zeta\kappa_\zeta$ is a function of $\zeta^2$: we will
write this, for the moment, as
\begin{equation}\label{eq:untwist}
\hat g_\lambda = \kappa_\zeta^{-1}g_\zeta\kappa_\zeta,\quad \lambda = \zeta^2.
\end{equation}
At the loop group level, untwisting is the Lie group isomorphism
\[
\Lambda^\sigma(C_\epsilon,G)\to\Lambda(C_{\epsilon^2},G);\quad g_\zeta\mapsto \hat g_\lambda.
\]
We are mainly interested in the corresponding Lie algebra isomorphism, which can be extended
to $\mathcal{R}_\epsilon^\sigma$ and provides the algebra isomorphism
\begin{equation}\label{eq:untwistalg}
\mathcal{R}_\epsilon^\sigma\to\mathcal{R}_{\epsilon^2}:\quad \eta_\zeta\mapsto \hat\eta_\lambda =
\kappa_\zeta^{-1}\eta_\zeta\kappa_\zeta.
\end{equation}
The $\tau$-equivariant loops are characterised after untwisting by the symmetry
\begin{eqnarray}
\hat g_{-\lambda}& = & S_\lambda^{-1}\hat g_\lambda^*S_\lambda,\quad g_\zeta\in
\Lambda^\sigma(C_\epsilon,G),\label{eq:tau_untwisted}\\
\hat\eta_{-\lambda} & = & -S_\lambda^{-1}\hat\eta_\lambda^t S_\lambda,\quad
\eta_\zeta\in\mathcal{R}^\sigma,
\end{eqnarray}
where
\[
S_\lambda = \begin{pmatrix} 0 & i\lambda & 0 \\ -i\lambda & 0 & 0 \\ 0 &0&1\end{pmatrix}.
\]
For each $z\in\mathbb{C}$ we will define $\mathcal{A}(z)\subset\mathcal{R}_{\epsilon^2}$ to be the image of $\mathcal{A}(z)^\sigma$ under
untwisting: this is independent of $\epsilon$ because its elements are Laurent polynomials. Because of
equation \eqref{eq:evolution} we also need to consider the untwisted version
of the dressing factorisation \eqref{eq:chi}, which we write as
\begin{equation}\label{eq:dress_fact}
\hat g_\lambda\hat F^\mu_\lambda = \hat F_\lambda\hat \chi_\lambda.
\end{equation}
It is easy to check that
\[
\hat F^\mu_\lambda = \exp(z\lambda^{-1}A +\bar z\lambda\bar A),
\]
and that $g_\zeta$ belongs to $\Lambda^\tau(I_\epsilon,G)$ if and only if $\hat g_\lambda$ extends holomorphically into
the interior of $C_{\epsilon^2}$ and $\hat g_0$ has the shape
\begin{equation}\label{eq:shape}
\begin{pmatrix} a & * &*\\0&a^{-1}&*\\0&0&1\end{pmatrix},\quad a\in\mathbb{C}.
\end{equation}
From now on we will work exclusively in the untwisted setting. For this reason, the ``hat''
notation used above will be dropped.
\subsection{The spectral curve.}
The spectral data is the geometric realisation of the algebra $\mathcal{A}$ which encodes
the isomorphism class of $\mathcal{A}$, its symmetries and
the natural gradings $\mathcal{A}$ carries as an algebra of Laurent polynomials.
For ease of notation we introduce the algebra homomorphism
\begin{equation}\label{eq:h}
h:\mathcal{A}\to\mathcal{H}_{\epsilon^2};\quad \xi\mapsto \operatorname{Ad}\chi^{-1}\cdot\xi.
\end{equation}
We denote the $j$-th diagonal entry of the diagonal matrix $h(\xi)$ by $h_j(\xi)$: this is an
analytic function on $C_{\epsilon^2}$ which extends meromorphically into $I_{\epsilon^2}$. To aid our
discussion we will set $\mathcal{C} = C^\omega(C_{\epsilon^2},\mathbb{C})$.
To begin, notice that $\mathcal{A}$ contains a copy of the algebra $\mathcal{B} =
\mathbb{C}[\lambda^{-1},\lambda]$, as scalar multiples of the identity, and the
three prime ideals
\[
\mathcal{I}_j = \{\xi\in\mathcal{A}:h_j(\xi)=0\},\quad j=1,2,3,
\]
which may be trivial. It also possess two involutions $\hat\rho$ and $\hat\tau$,
the first real linear and the second complex linear, defined by
\begin{equation}\label{eq:rho_tau}
\hat\rho(\xi_\lambda) = (\xi_{\bar\lambda^{-1}})^\dagger,\quad
\hat\tau(\xi_\lambda) = \operatorname{Ad} S_\lambda^*\cdot(\xi_{-\lambda})^t.
\end{equation}
\begin{lem}\label{lem:rho_tau}
$\hat\rho$ and $\hat\tau$ are algebra automorphisms of $\mathcal{A}$ (over $\mathbb{R}$ and $\mathbb{C}$
respectively) which commute, and they each preserve the subalgebra $\mathcal{B}$. While $\hat\rho$
preserves $\mathcal{I}_1,\mathcal{I}_2,\mathcal{I}_3$, $\hat\tau$ preserves $\mathcal{I}_3$ and swaps $\mathcal{I}_1$ with $\mathcal{I}_2$.
\end{lem}
\begin{proof}
Both $-\hat\rho$ and $-\hat\tau$ are Lie algebra automorphisms which fix
the untwisted Maurer-Cartan form $\alpha_\lambda$. Hence if $\xi$ is any (untwisted) polynomial Killing\ field
then so is $-\hat\rho(\xi)$ and $-\hat\tau(\xi)$. Since the Lax equations \eqref{eq:Lax} are
linear in $\xi$ it follows that $\hat\rho(\xi)$ and $\hat\tau(\xi)$ are also polynomial Killing\ fields.
Further, it is easy to show that $S_\lambda^\dagger = S_{\bar\lambda^{-1}}$ and therefore
\[
\hat\rho\hat\tau(\xi_\lambda) =
\operatorname{Ad} (\bar S_{\bar\lambda^{-1}})^{-1}\cdot \bar\xi_{-\bar\lambda^{-1}}=
\operatorname{Ad} S_\lambda^*\cdot\bar\xi_{-\bar\lambda^{-1}} = \hat\tau\hat\rho(\xi_\lambda).
\]
Next, $\chi_\lambda$ possesses the symmetries
\begin{equation}\label{eq:group_symmetries}
\chi_{\bar\lambda^{-1}} = (\chi_{\lambda}^{-1})^\dagger,\quad
\chi_{-\lambda} = S_\lambda^{-1}\chi_\lambda^* S_\lambda.
\end{equation}
From these it follows that for any $\xi\in\mathcal{A}$
\[
\hat\rho(\xi) = \operatorname{Ad}\chi\cdot \hat\rho(h(\xi)),\quad
\hat\tau(\xi) = \operatorname{Ad}\chi\cdot \hat\tau(h(\xi)),
\]
where we have used the fact the the definitions of $\hat\rho$ and $\hat\tau$ apply equally well
to $\mathcal{H}_{\epsilon^2}$. It is easy to see from this that $\hat\rho$ and $\hat\tau$ preserve $\mathcal{B}$,
since $h(\mathcal{B}) = \mathcal{B}$. Clearly $\hat\rho(h(\xi))$ has a zero on its $j$-th diagonal entry if
and only if $h(\xi)$ does (i.e., $h_j(\xi)=0$), therefore $\hat\rho$ preserves $\mathcal{I}_j$.
Finally, on $\mathcal{H}_{\epsilon^2}$,
\[
\hat\tau(\begin{pmatrix} a(\lambda) &0&0\\ 0& b(\lambda) &0 \\
0&0&c(\lambda)\end{pmatrix}) =
\begin{pmatrix} b(-\lambda) &0&0\\0 & a(-\lambda) &0 \\
0&0&c(-\lambda)\end{pmatrix}
\]
and therefore $\hat\tau$ swaps $\mathcal{I}_1$ with $\mathcal{I}_2$ but fixes $\mathcal{I}_3$.
\end{proof}
The geometric realisation is that we have an affine curve $X_A=\operatorname{Spec}(\mathcal{A})$ with a natural
rational function $\lambda:X_A\to\mathbb{C}^\times$ (dual to $\mathcal{B}\subset\mathcal{A}$) and $X_A$ comes
equipped with a holomorphic involution $\tau$ and a real involution $\rho$, which commute.
This curve is reducible if any of the ideals $\mathcal{I}_j$ is non-trivial: the irreducible
components are isomorphic to $\operatorname{Spec}(\mathcal{A}_j)$. The involution $\tau$ maps $\operatorname{Spec}(\mathcal{A}_1)$ to
$\operatorname{Spec}(\mathcal{A}_2)$, which can be either distinct or identical (i.e., either $\mathcal{I}_1$ and $\mathcal{I}_2$
are distinct or identical) and $\lambda$ is non-constant on every irreducible
component of $X$.
We will define the spectral curve of $\mathcal{A}$ to be the completion $X$ of $X_A$ by smooth points.
Its irreducible component corresponding to the prime ideal $\mathcal{I}_j$ will be denoted by $X_j$.
Clearly
the points of completion must lie over $\lambda=0,\infty$. The nature
of this completion is determined by the valuations on its field of rational functions $\mathbb{C}(X)$
(equally, the field of fractions of $\mathcal{A}$) which correspond to measuring the degree of each
rational function at the points of completion. Because $\mathcal{A}$ need not be an integral domain we
must take a little care when defining these.
For any algebraic element $a\in \mathcal{C}$ define its degree about
zero $\deg_0(a)$ to be the degree at $\lambda=0$ of its meromorphic extension into $I_{\epsilon^2}$,
and similarly define its degree about infinity $\deg_\infty(a)$. Using these notions we define
\begin{eqnarray}\label{eq:nu}
\nu^0_j:\mathcal{A}-\mathcal{I}_j\to \mathbb{Z},&\quad& \nu^0_j(\xi) = \deg_0(h_j(\xi)), \\
\nu^\infty_j:\mathcal{A}-\mathcal{I}_j\to \mathbb{Z},&\quad& \nu^\infty_j(\xi) = \deg_\infty(h_j(\xi)).
\end{eqnarray}
It is easy to check that these six functions have the valuation properties:
\begin{enumerate}
\item $\nu(\xi_1\xi_2) = \nu(\xi_1)+\nu(\xi_2)$,
\item $\nu(\xi_1+\xi_2) \geq \min\{\nu(\xi_1),\nu(\xi_2)\}$.
\end{enumerate}
\begin{lem}\label{lem:fractions}
Each of $\nu_j^0$ and $\nu_j^\infty$ induces a valuation on the field of fractions $\mathcal{F}_j$
of the integral domain $\mathcal{A}_j$. Further,
\begin{equation}\label{eq:nu_rho_tau}
\nu_1^0\circ\hat\tau = \nu_2^0,\quad \nu_3^0\circ\hat\tau=\nu_3^0,\quad
\nu_j^0\circ\hat\rho = \nu_j^\infty.
\end{equation}
These valuations correspond to $6$ smooth points on $X$:
$P_j$ corresponding to $\nu_j^0$ and $Q_j$ corresponding to $\nu_j^\infty$. These points have
the properties
\begin{equation}\label{eq:points}
\lambda(P_j)=0,\ \lambda(Q_j)=\infty,\ Q_j=\rho P_j,\ P_2=\tau P_1,\ \tau P_3 = P_3
\end{equation}
Consequently the rational function $\lambda:X\to\mathbb{C}_\infty$ has degree $3$.
\end{lem}
\begin{proof}
An element of the field of fractions of $\mathcal{A}_j$ can be written $[\xi_1]/[\xi_2]$ where
$[\xi_1],[\xi_2]\in\mathcal{A}_j$ and $[\xi]$ denotes $\xi+\mathcal{I}_j$. We define
\[
\nu_j:\mathcal{F}_j-\{0\}\to\mathbb{Z};\quad \nu_j([\xi_1]/[\xi_2]) = \nu_j(\xi_1)-\nu_j(\xi_2),
\]
for $\nu_j$ either of $\nu_j^0$ or $\nu_j^\infty$.
It is easy to check that this is well-defined and retains the valuation properties. The
symmetries \eqref{eq:nu_rho_tau} follow from lemma \ref{lem:rho_tau} and the fact that
$\deg_0(a(\lambda)) = \deg_\infty(\bar a(\bar\lambda^{-1}))$ for any algebraic $a\in
C^\omega(C_{\epsilon^2},\mathbb{C}))$. The valuations $\nu_j^0,\nu_j^\infty$ correspond to smooth points
$P_j,Q_j\in X_j$. On $\mathbb{C}_\infty$ the points $0,\infty$ correspond to the
valuations $\deg_0,\deg_\infty$ on $\mathbb{C}(\lambda)$, which we identify with the field of fractions
of $\mathcal{B}$. Now on $\mathbb{C}(\lambda)\subset\mathcal{F}_j$ we have $\deg_0=\nu_0^j$ and
$\deg_\infty = \nu_\infty^j$,
so $\lambda(P_j)=0$ while $\lambda(Q_j)=\infty$. The symmetries in \eqref{eq:points} follow from
\eqref{eq:nu_rho_tau}. Finally, matrix multiplication of vectors makes
$\mathcal{B}\otimes\mathbb{C}^3$ a rank three $\mathcal{B}$-module. It is also clearly a faithful
$\mathcal{A}(0)$-module, and therefore $\mathcal{A}$ is at most rank three as a $\mathcal{B}$ module. It follows
that $\lambda$ is at most degree three. Since it possesses at least three points over $0$ it must
be exactly degree three. Finally, the symmetries \eqref{eq:points} follow easily from the proof
of lemma \ref{lem:rho_tau}.
\end{proof}
We can also import the argument from \cite[Prop.6]{McI95} to deduce that $X$ can only be
disconnected if $f$ is not linearly full. Since immersed Lagrangian surfaces in $\mathbb{CP}^2$ must be
linearly full, $X$ is connected. In particular, this disposes of the spectral curve for the vacuum
solution, since it is a disjoint union of three copies of the Riemann sphere.
We will use the term ``spectral curve'' to mean the totality of information
$X,\lambda,\tau,\rho$, which includes the points over $\lambda=0$ and $\lambda=\infty$.
Up to this point we have proved the following.
\begin{thm}
The spectral curve $(X,\lambda,\rho,\tau)$ for a HSL torus in $\mathbb{CP}^2$ is a complete
connected algebraic curve $X$ equipped with a degree $3$ rational function $\lambda$,
a real involution $\rho$
and a holomorphic involution $\tau$. The function $\lambda$ has symmetries
\begin{equation}\label{eq:lambda_symmetries}
\overline{\rho^*\lambda} = \bar\lambda^{-1},\quad \tau^*\lambda = -\lambda.
\end{equation}
Further, $\lambda$ has distinct zeroes (and therefore distinct poles) each of which is a
smooth point and only one of which is fixed by $\tau$.
\end{thm}
\subsection{The linear family of line bundles.}
While the spectral curve determines $\mathcal{A}$ it does not carry the information which
distinguishes $\mathcal{A}(0)$ from $\mathcal{A}(z)$: this is encoded in the representation of each
$\mathcal{A}(z)$ on its module $\mathcal{M} = \mathcal{B}\otimes\mathbb{C}^3$. The geometric realisation
of this is a rank one torsion free\footnote{In the case that $X$ is reducible ``torsion free''
means the restriction to each irreducible component is torsion-free.} coherent sheaf
$\mathcal{L}^z$ over $X$. From now on whenever we use the phrase ``line bundle'' it should be
interpreted to include maximal rank 1 torsion free coherent sheaves whenever $X$ is not smooth.
The term ``maximal'' means the sheaf is not the direct image of any sheaf on a less singular
curve than $X$, equally, $\operatorname{Hom}(\mathcal{L},\mathcal{L})$ is trivial.
To begin, for each $z$ the $\mathcal{A}(z)$-module $\mathcal{M}$ determines a line bundle $\mathcal{L}^z$ over the
affine curve $X_A$ in the usual way, i.e., the stalk $\mathcal{L}^z_P$ at $P\in X_A$ is the
localisation $\mathcal{M}(z)_P$ of $\mathcal{M}$ at the ideal $\mathfrak{m}(z)_P\in\operatorname{Spec}(\mathcal{A})$ corresponding to the
point $P$. Our task is to extend this to the completion $X$ in a natural way. The key is to use
filtrations of $\mathcal{M}$ which are compatible with the valuations determining the points of
completion. We proceed as follows.
First we introduce the linear map
\begin{equation}\label{eq:s}
s:\mathcal{M}\to\mathcal{C}\otimes\mathbb{C}^3;\quad v\mapsto \chi(z)^{-1}v.
\end{equation}
This depends upon $z$, but we will supress this fact in the notation since we will not have a
cause to explicitly refer to this dependence. Now
define $s_j(v)\in\mathcal{C}$ to mean the $j$-th entry in the vector $s(v)$. This is compatible
with the homomorphism $h$ in \eqref{eq:h}, since $s(\xi v) = h(\xi)s(v)$. Consequently the
subspace
\[
\mathcal{S}_j(z) = \{v\in\mathcal{M}:s_j(v)=0\}
\]
is an $\mathcal{A}$-submodule of $\mathcal{M}$ with the property that $\mathcal{I}_j\mathcal{M}\subset\mathcal{S}_j(z)$. Therefore
the quotient space $\mathcal{M}_j(z)=\mathcal{M}/\mathcal{S}_j(z)$ is an $\mathcal{A}_j(z)$-module.
On each of these we define two functions
\begin{eqnarray}\label{eq:mu}
\mu_j^0:\mathcal{M}_j(z)-\{0\}\to\mathbb{Z};&\quad &\mu_j^0([v]) = \deg_0(s_j(v)),\\
\mu_j^\infty:\mathcal{M}_j(z)-\{0\}\to\mathbb{Z};&\quad &\mu_j^\infty([v]) = \deg_\infty(s_j(v)),
\end{eqnarray}
where $[v]$ stands for $v+\mathcal{S}_j$. These have the properties
\[
\mu([v] + [w]) \geq \min\{\mu([v]),\mu([w])\},\quad \mu([\xi v]) = \nu([\xi]) + \mu([v]),
\]
whenever $[v],[w]\in\mathcal{M}_j(z)$ and $[\xi]\in\mathcal{A}_j(z)$. Because of these properties each
function extends naturally the module of fractions $\tilde\mathcal{M}_j(z) =
\mathcal{F}_j(z)\otimes_{\mathcal{A}_j(z)}\mathcal{M}_j(z)$. We define
\begin{eqnarray}\label{eq:LPQ}
\mathcal{L}^z_{P_j}& =& \{ v\in\tilde\mathcal{M}_j(z)-\{0\}:\mu_j^0(v)\geq 0\}\cup\{0\},\\
\mathcal{L}^z_{Q_j}& =& \{ v\in\tilde\mathcal{M}_j(z)-\{0\}:\mu_j^\infty(v)\geq 0\}\cup\{0\}.
\end{eqnarray}
It follows that by attaching these stalks over the appropriate points of $X$ we obtain a
sheaf of $\mathcal{O}_X$-modules which we define to be $\mathcal{L}^z$.
\begin{lem}\label{lem:caL}
For each $z_0\in\mathbb{C}$ the sheaf $\mathcal{L}^{z_0}$ is a maximal, rank one, torsion free coherent sheaf with
$h^0(\mathcal{L}^{z_0})=3$ and $h^1(\mathcal{L}^{z_0})=0$. Thus, by the Riemann-Roch theorem, it has degree
$\deg(\mathcal{L}^{z_0})=g+2$, where $g$ is the arithmetic genus of $X$.
\end{lem}
\begin{proof}
From the construction it is clear that $\mathcal{L}^{z_0}$ is a torsion free coherent sheaf. That it has
rank one follows from the fact that $\mathcal{M}$ is a rank three $\mathcal{B}$-module and $\lambda$ has
degree three. To see that it is maximal it suffices to show that the sheaf
$\operatorname{Hom}(\mathcal{L}^{z_0},\mathcal{L}^{z_0})$ is trivial. For any affine open $U\subset X$
consisting only of smooth points one knows that $\operatorname{Hom}(\mathcal{L}^{z_0}_U,\mathcal{L}^{z_0}_U)\simeq \mathcal{O}_U$,
(since $\mathcal{L}^{z_0}_U$ is locally free and rank one) so
since any singularities lie in $X_A$ it suffices to check that this for $U=X_A$, and this
amounts to checking that if $\eta\in\operatorname{Hom}_\mathcal{A}(z_0)(\mathcal{M},\mathcal{M})$ then $\eta$ represents
multiplication by an element of $\mathcal{A}(z_0)$. We use the natural embedding
\[
\operatorname{Hom}_\mathcal{A}(z_0)(\mathcal{M},\mathcal{M})\subset \operatorname{Hom}_\mathcal{B}(\mathcal{M},\mathcal{M})\simeq \mathfrak{gl}_3\otimes\mathcal{B}
\]
to view $\eta\in\mathfrak{gl}_3\otimes\mathcal{B}$. This must commute with every element of $\mathcal{A}(z_0)$, but
$\mathcal{A}(z_0)$ is a maximal abelian subalgebra of $\mathfrak{gl}_3\otimes\mathcal{B}$, hence $\eta\in\mathcal{A}(z_0)$.
The remainder of the lemma is equivalent to the assertion that the direct image
$\mathcal{E}^{z_0}=\lambda_*\mathcal{L}^{z_0}$ is a trivial rank three vector bundle over $\mathbb{C}_\infty$, since
$h^i(\mathcal{E}^{z_0})=h^i(\mathcal{L}^{z_0})$. We will prove this by
showing that if $\sigma_1^{z_0},\sigma_2^{z_0},\sigma_3^{z_0}$ are the three sections of
$\lambda_*\mathcal{L}^{z_0}$ over $\mathbb{C}^\times$ corresponding to the generators $e_1,e_2,e_3$ for
$\mathcal{M}$ over $\mathcal{B}$ (the standard basis vectors for $\mathbb{C}^3$) then they extend holomorphically to
$\mathbb{C}_\infty$ and span $H^0(\mathcal{L}^{z_0})$. By our construction of $\mathcal{L}^{z_0}$ over $\lambda=0,\infty$
this means we must consider each $\chi_\lambda^{-1}e_j$ about $\lambda=0,\infty$. But these are
the columns of matrix $\chi_\lambda^{-1}$, so each is holomorphic and
non-vanishing about $\lambda=0,\infty$. Therefore the only linear combinations over $\mathcal{B}$ of these
which remain holomorphic at both ends are those with constant coefficients. Hence
\begin{equation}\label{eq:Eframe}
H^0(\mathcal{L}^{z_0})=H^0(\mathcal{E}^{z_0}) = \mathrm{Span}_\mathbb{C}\{\sigma_1^{z_0},\sigma_2^{z_0},\sigma_3^{z_0}\}.
\end{equation}
\end{proof}
It will be useful for us to know later how to characterise the basis of sections $\sigma_j^z$.
\begin{lem}\label{lem:basis}
Up to scaling, the sections $\sigma_1^z$, $\sigma_2^z$ and $\sigma_3^z$ are the
unique global sections of, respectively, $\mathcal{L}^z(-P_2-P_3)$, $\mathcal{L}^z(-P_3-Q_1)$ and
$\mathcal{L}^z(-Q_1-Q_2)$.
\end{lem}
\begin{proof}
That these sections vanish at the points indicated follows directly from an inspection of the
columns of $\chi^{-1} e_j$ evaluated at $\lambda=0$ and $\infty$, which can be read off
\eqref{eq:shape} since this is the shape of $\chi_0$ and $\chi_\infty = \bar\chi_0^*$.
For the same reason, a constant vector $v\in\mathbb{C}^3$ corresponds to a global section vanishing at
$P_2+P_3$ only if $v$ is a scalar multiple of $e_1$. Similarly, the other two vanishing conditions
can only occur when $v=e_2$ and $v=e_3$ respectively.
\end{proof}
As a consequence we identify the direct image $\mathcal{E}(z)$ with the trivial bundle $\mathcal{E}$ over $\mathbb{C}_\infty$
using the frame $\sigma^z$ obtained from \eqref{eq:Eframe}.
Although we constructed $X$ and $\mathcal{L}^z$ purely algebraically, we can now invoke Serre's GAGA
principle and treat them as complex analytic objects too. This is essential if we wish to
understand the family $\mathcal{L}^z$, since it is not in general algebraic. This perspective allows
us to interpret the map $s$ as a trivialisation $\varphi^z$ for $\mathcal{L}^z$ over
$U=\lambda^{-1}(I_{\epsilon^2})$,
which we may assume without loss of generality contains no branch points of $\lambda$ and is a
disjoint union of open discs, one for each point $P_j$ or $Q_j$. For then,
by definition, $v\in\tilde\mathcal{M}_j$ represents a holomorphic section over $U$ if and only if
the components $s_j(v)$ of $\chi_\lambda^{-1}v$ are holomorphic
functions on $I_{\epsilon^2}$. Since $s$ is linear over $\mathcal{A}(z)$ this defines a trivialisation of
$\mathcal{L}^z$ over $U$. Because $U$ contains no branch points this pushes down to a trivialisation
$\Psi_U^z$ for the direct image $\mathcal{E}(z)$.
The transition relation between this and the frame $\sigma^z$ is clearly
\begin{equation}\label{eq:transition1}
\chi(z)\Psi_U^z = \sigma^z,\quad \text{over}\ I_{\epsilon^2}.
\end{equation}
This provides us with a direct link to the dressing factorisation \eqref{eq:dress_fact}, for in the
analytic category we can replace, away from $0,\infty$, the frame $\sigma^z$
by the frame $\Psi_A^z$ corresponding to the columns of $F(z)$,
i.e., $\Psi_A^z(\sigma_j) = F(z)e_j$, and therefore
\begin{equation}\label{eq:transition2}
g F^\mu(z)\Psi_U^z = \Psi_A^z,\quad\text{over}\ I_{\epsilon^2}-\{0,\infty\}.
\end{equation}
These transition relations are encoded in the spectral data in the following way. For the
statement of this proposition we assume $\mathcal{L}$ is a line bundle, for ease of exposition.
\begin{prop}\label{prop:L}
Let $R$ be the ramification divisor of $\lambda$ and write $\mathcal{L}^z=\mathcal{O}_X(D)$ for some positive
divisor $D$. Let $J_\mathbb{R}(X)$ denote the real subgroup $\{L\in\operatorname{Jac}(X):\overline{\rho^*L}\simeq L^{-1}\}$
of $\operatorname{Jac}(X)$. Finally, let $X^s$ be the variety of smooth points on $X$.
\begin{enumerate}
\item We have the linear equivalences
\begin{equation}\label{eq:L_symmetries}
\rho_*D+D\sim R,\quad \tau_*D+D\sim R+P_3-Q_3.
\end{equation}
Further, $\rho$ fixes every point $P\in X$ for which $|\lambda(P)|=1$ and $D$ has no support on
this unit circle.
\item
The map $\ell:\mathbb{C}\to J_\mathbb{R}(X)$ be given by $\ell(z)= \mathcal{L}^z\otimes(\mathcal{L}^0)^{-1}$ is a homomorphism of real
groups. It is completely characterised by
\begin{equation}\label{eq:ltangent}
\frac{\partial \ell}{\partial z}|_{z=0} =
\frac{3i\pi\bar\mu_0}{2}
\left(
\frac{\partial\mathfrak{A}_{P_1}}{\partial\lambda}(0)+
\frac{\partial\mathfrak{A}_{P_2}}{\partial\lambda}(0)
\right)
\end{equation}
Here $\mathfrak{A}_P:X^s\to\operatorname{Jac}(X)$ denotes the Abel map with base point $P$, i.e.,
$\mathfrak{A}_P(Q) = \mathcal{O}_X(Q-P)$.
\end{enumerate}
\end{prop}
\begin{rem}
(i) To interpret the right hand side of equation \eqref{eq:ltangent} we identify $T_eJ_\mathbb{R}(X)^\mathbb{C}$ with
$T_e\operatorname{Jac}(X)\simeq T_e^{1,0}\operatorname{Jac}(X)$.
Notice that $\ell_z$ actually takes values in a real subgroup of the Prym variety
$\operatorname{Prym}(X,\tau)\subset\operatorname{Jac}(X)$: since $\tau$ has precisely two fixed points ($P_3$ and $Q_3$) this
Prymian has dimension $g/2$ when $g$ is the arithmetic genus of $X$.\\
(ii) The role of the ramification divisor is to provide a coherent sheaf $\mathcal{D}=\mathcal{O}_X(R)$
over $X$ for which
\[
\lambda_*(\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{L},\mathcal{D})) = (\lambda_*\mathcal{L})^*.
\]
When $X$ is smooth we can take $\mathcal{D}=\mathcal{O}_X(R)$ and use the trace map $\mathbb{C}(X)\to\mathbb{C}(\lambda)$ as
part of the dual pairing (see \cite{McI95}). This trace map is essentially the same as the trace in
Serre duality (cf. \cite[II\S 12]{Ser}) so
when $X$ is singular it may be necessary to interpret $\mathcal{O}_X(R)$ as $K_X\otimes\lambda^*K_{\mathbb{C}_\infty}^{-1}$
(cf.\ Hartshorne \cite[Ch.IV,\S 2]{Har}), where $K_X$ is the dualising sheaf for $X$.
\end{rem}
\begin{proof}
The equivalence $D+\rho_*D\sim R$ is a direct consequence (see \cite{McI96}) of the real symmetry
of $\chi$ in \eqref{eq:group_symmetries}, from which we also obtain the conditions on $\rho$.
In particular, there is a rational function with divisor $D+\rho_*D-R$ which is real
and positive over the unit circle, which is equivalent to the
requirement that $D$ (or a representative in its linear equivalence class) have no points on the
unit circle.
The symmetry $\chi_{-\lambda} = S_\lambda^{-1}\chi_\lambda^*S_\lambda$
gives the second linear equivalence as follows. First, $\chi_\lambda^*$ is the transition function
for the dual bundle $(\lambda_*\mathcal{L})^*$, which is the direct image of $\mathcal{O}_X(R-D)$ (again, from
\cite{McI96}). Therefore that symmetry means
\[
S_\lambda\tau^*\Psi_U = \Psi_U^*
\]
where $\Psi_U^*$ is the trivialisation of $\mathcal{O}_X(R-D)$ over $U$ dual to $\Psi_U$. Hence any local
trivialisation of $\mathcal{O}_X(\tau_*D)$ over $U$ corresponds to a section of $\mathcal{O}_X(R-D)$ with divisor
$P_1+P_2-Q_1-Q_2$.
For part (ii), we first note that equation \eqref{eq:transition2} tells us that both $\Psi_U^0$ and
$F^\mu(z)\Psi_U^z$ are local trivialisations for $\lambda_*\mathcal{L}^0$, and since $F^\mu$ is diagonal
both come from local trivialisations of $\mathcal{L}^0$. Therefore $\mathcal{L}^z$ is obtained from $\mathcal{L}^0$ by
twisting it by a line bundle obtained by glueing the trivial bundles over $U$ and $X_A$ together
using the diagonal elements of $F^\mu(z)$ as the transition functions, with the $j$-th diagonal
providing the transition function over the punctured discs about $P_j,Q_j$. This gives a
$1$-cocycle for $\ell(z)$. Differentiating this cocycle gives us
\[
\frac{\partial \ell}{\partial z}|_{z=0} =
\frac{i\pi\bar\mu_0}{2}
\left(
\frac{\partial\mathfrak{A}_{P_1}}{\partial\lambda}(0)
+\frac{\partial\mathfrak{A}_{P_2}}{\partial\lambda}(0)
-2\frac{\partial\mathfrak{A}_{P_3}}{\partial\lambda}(0)
\right).
\]
But $(\partial\mathfrak{A}_{P_j}/\partial\lambda)(0)$ can be identified with the linear map
\[
H^0(\Omega_X)\to\mathbb{C};\quad \omega\mapsto\operatorname{Res}_{P_j}(\lambda^{-1})
\]
so from the residue theorem, since $P_1+P_2+P_3$ is the divisor of zeroes of $\lambda$, we deduce that
\[
\frac{\partial\mathfrak{A}_{P_1}}{\partial\lambda}(0)
+\frac{\partial\mathfrak{A}_{P_2}}{\partial\lambda}(0)
+\frac{\partial\mathfrak{A}_{P_3}}{\partial\lambda}(0)=0.
\]
Thus we obtain \eqref{eq:ltangent}.
\end{proof}
\subsection{Double periodicity.}\label{sec:double}
As well as the properties of the spectral curve $(X,\lambda)$ stated
above, the double periodicity of $f$ can be encoded into the spectral curve in exactly the same way
as was achieved for minimal Lagrangian tori \cite{McI03}. This involves introducing a particular
singularisation of $X$. Let $O_1,O_2,O_3$ be the three distinct smooth points lying over
$\lambda=1$. Define $X_\mathfrak{o}$ to be the singularisation of $X_\mathfrak{o}$ obtained by identifying together
these points (the singular curve for the modulus $\mathfrak{o}=O_1+O_2+O_3$ \cite{Ser}). For any smooth
point $P\in X_\mathfrak{o}$ let $\mathcal{A}^\mathfrak{o}_P:X_\mathfrak{o}^s\to\operatorname{Jac}(X_\mathfrak{o})$ denote the Abel map based at $P$. We can
define a homomorphism of real groups $\ell^\mathfrak{o}:\mathbb{R}^2\to J_\mathbb{R}(X_\mathfrak{o})$ by insisting that
\begin{equation}\label{eq:lotangent}
\frac{\partial \ell^\mathfrak{o}}{\partial z}|_{z=0} =
\frac{3i\pi\bar\mu_0}{2}
\left(
\frac{\partial\mathfrak{A}^\mathfrak{o}_{P_1}}{\partial\lambda}(0)+
\frac{\partial\mathfrak{A}^\mathfrak{o}_{P_2}}{\partial\lambda}(0)
\right)
\end{equation}
\begin{prop}
Let $(X,\lambda)$ be the spectral curve for a HSL torus $f:\mathbb{C}/\Gamma\to\mathbb{CP}^2$, then $\ell^\mathfrak{o}$ is
$\Gamma$-periodic.
\end{prop}
To prove this we begin by observing that the map $f$ is reconstructed from its spectral data in
exactly the same way as the minimal Lagrangian case (and, more generally, harmonic non-isotropic
tori in $\mathbb{CP}^n$), viz, the line $f(z)\in\mathbb{CP}^2$ is given by $[F_1(z)e_3]$ (modulo an isometry of
$\mathbb{CP}^2$) and therefore, by lemma \ref{lem:basis} corresponds to the line $H^0(\mathcal{L}_z(-Q_1-Q_2))$ in
$\P H^0(\mathcal{L}_z)$ and these projective spaces are identified together by the frame $F_1$.
This leaves only the final identification of $\P H^0(\mathcal{L}_0)$ with
$\mathbb{CP}^2$ unfixed, so $f$ is only recovered up to an isometry of $\mathbb{CP}^2$.
What we have here is an adaptation of the construction detailed in \cite{McI99} for harmonic
maps (see also \S\ref{sec:explicit} below). It follows that we can apply the tools described
there to describe our reconstruction of $f$.
In other words, the HSL torus $f:\mathbb{C}/\Gamma\to\mathbb{CP}^2$ factors through $\ell^\mathfrak{o}$:
\begin{equation}\label{eq:theta}
f:\mathbb{C}/\Gamma\stackrel{\ell^\mathfrak{o}}{\to}\operatorname{Jac}(X_\mathfrak{o})\stackrel{\theta}{\to}\mathbb{CP}^2,
\end{equation}
where $\theta$ is a certain rational map determined by choosing the divisor $Q_1+Q_2$
to single out the point in $\P H^0(\mathcal{L}_z)$ determined by the line $H^0(\mathcal{L}_z(-Q_1-Q_2))$.
More detail is given in \S\ref{sec:explicit} below.
In truth, to establish \eqref{eq:theta} we have to prove \emph{first} that $\ell^\mathfrak{o}$ is
$\Gamma$-periodic, since \emph{ a priori} it could be that the periodicity is due to $\theta$
rather than $\ell^\mathfrak{o}$. For this we argue as follows. The translated map $f^{w}(z) = f(z+w)$
determines the spectral data $(X,\lambda,\mathcal{L}^w)$ and therefore $\ell(z)$ is $\Gamma$-periodic. But
$\ell^\mathfrak{o}$ lifts $\ell$ to $\operatorname{Jac}(X_\mathfrak{o})$ (which is a $(\mathbb{C}^\times)^2$-bundle over $\operatorname{Jac}(X)$)
and from \cite[Lemma 1]{McI99} the holonomy of this lift gives a homorphism $h:\Gamma\to PU(3)$ for
which $f(z+\gamma) = h(\gamma)f(z)$ for all $z\in\mathbb{C}$ and each $\gamma\in\Gamma$. Since
$f(z+\gamma)=f(z)$ and $f$ is linearly full, $h(\gamma)$ is trivial for each $\gamma$,
whence $\ell^\mathfrak{o}$ is $\Gamma$-periodic.
\section{The reconstruction from spectral data.}
In this section we will complete the proof of theorem \ref{thm:main} by showing that we can start
with spectral data and construct a HSL torus $f$ in a manner
which reverses the procedure given above for extracting the spectral data from
$f$. We finish with some brief comments about the consequences of this classification for the
study of the moduli space of HSL tori.
\subsection{Reconstruction.}\label{sec:Rec}
Suppose we are given a triple of HSL spectral data $(X,\lambda,\mathcal{L})$ as per definition
\ref{defn:spectral_data}. Let us first show that provided $(X,\lambda)$ satisfy conditions
(a) and (b)
of that definition we can always find a line bundle $\mathcal{L}=\mathcal{O}_X(D)$ satisfying condition (d).
First let us rewrite equations \eqref{eq:lequiv} in terms of the canonical class $K_X$ of $X$.
Since $R\sim K_X+\sum_{j=1}^3(P_j+Q_j)$ a degree $g+2$ divisor $D>0$ satisfies \eqref{eq:lequiv} if
and only if $D'=D-Q_1-Q_2$ satisfies
\begin{equation}\label{eq:leqcan}
D'+\rho_*D'\sim K_X+P_3+Q_3,\qquad (\rho\tau)_*D'\sim D'.
\end{equation}
Further, since \eqref{eq:lequiv} implies $\dim H^0(\mathcal{L}(-Q_1-Q_2)=1$ we may assume $D'>0$. Notice
that $\deg(D')=g$ so that these equations reduce to those of Sharipov \cite{Sha} for minimal
Lagrangian tori.
\begin{rem}
We take this opportunity to point out an error in \cite[Lemma 2]{McI03}. The expression
for equation (15) in that reference should be
\[
\mu_*\mathcal{L}_z\simeq\mathcal{L}_z^{-1}\otimes\mathcal{O}_X(R+P_\infty-P_0),
\]
which, given the conventions of that paper, agrees with \cite{Sha}.
\end{rem}
\begin{lem}
Whenever $(X,\lambda)$ satisfy conditions (a) and (b) of definition \ref{defn:spectral_data} there
exists a degree $g$ divisor $D'>0$ satisfying \eqref{eq:leqcan}.
\end{lem}
\begin{proof}
Set $\iota=\rho\tau$ and let $V = \{k\in\mathbb{C}(X):(k)>-K_X-P_3-Q_3\}$. Since the class $K_X+P_3+Q_3$
is both $\rho$ and $\tau$ invariant $V$ admits two real linear involutions,
$\bar\rho(k)=\overline{\rho^*k}$ and $\bar\iota(k)=\overline{\iota^*k}$. Since $g$ is even
there exists a
non-zero $k\in V$ with divisor of zeroes of the form $D'+\rho_*D'$ where $\iota_*D'=D'$ if and only
if $k$ has a divisor of zeroes which is both $\rho$ and $\iota$ invariant.
The fixed points of $\bar\rho$ form a real subspace of $V$
of dimension $g$ which is preserved by the involution $\bar\iota$, so there exists a non-zero
$k\in V$ for which $\overline{\rho^*k}=k$ and $\overline{\iota^*k}=\pm k$.
\end{proof}
In light of \S\ref{sec:double} above we can give the details
of the double periodicity condition (c) in definition \ref{defn:spectral_data}. With the
notation of \S\ref{sec:double} let
$\ell^\mathfrak{o}:\mathbb{R}^2\to\operatorname{Jac}_\mathbb{R}(X_\mathfrak{o})$ be the unique
homomorphism of real groups satisfying
\begin{equation}\label{eq:lotangent2}
\frac{\partial\ell^\mathfrak{o}}{\partial w}|_{w=0}=\frac{3\pi i}{2}\left(
\frac{\partial\mathfrak{A}^\mathfrak{o}_{P_1}}{\partial\lambda}(0)+
\frac{\partial\mathfrak{A}^\mathfrak{o}_{P_2}}{\partial\lambda}(0)
\right).
\end{equation}
This equation is to be read in the same sense as \eqref{eq:ltangent}.
The periodicity condition (c) in definition \ref{defn:spectral_data} is that
$\ell^\mathfrak{o}$ must be doubly periodic.
A comparison between the equations \eqref{eq:lotangent2} and \eqref{eq:lotangent} shows us how to
obtain the Maslov form (and hence the mean curvature).
If $\Gamma$ denotes the period lattice for $\ell^\mathfrak{o}$ then \eqref{eq:lotangent2} equips
$\mathbb{R}^2/\Gamma$ with
the harmonic $1$-form $dw+d\bar w$ (pushed down to $\mathbb{R}^2/\Gamma$, where it is no longer exact).
This will give the negative of the Maslov form $\mu$.
Now we may state and prove the main result of this section, and so complete the proof of theorem
\ref{thm:main}.
\begin{prop}\label{prop:spectral}
To each triple $(X,\lambda,\mathcal{L})$ of HSL spectral data we can canonically assign a HSL torus
$f:\mathbb{C}/\Gamma\to \mathbb{CP}^2$, uniquely up to base point preserving isometries of $\mathbb{CP}^2$,
in a manner which reverses the construction of the spectral data from $f$.
\end{prop}
Although \eqref{eq:theta} provides a direct construction of the HSL torus from its spectral
data, we could not see a direct way of verifying that this map is HSL. So instead
we will prove this proposition by constructing a coset $gZ^\epsilon_A$ in
$\Lambda^\tau_B(I_\epsilon,G)/Z^\epsilon_A$ which corresponds
to a point in the dressing orbit $\mathcal{O}^\epsilon(\mu)$ and yields the Lagrangian angle $\Omega$ of $f$.
Because of the way we have specified equation \eqref{eq:lotangent2} we work with a normalised
extended frame for the vacuum solution,
\[
F^w_\zeta = \exp[i\pi(w\zeta^{-2}+\bar w\bar\zeta^2)\begin{pmatrix} I_2 & 0 \\ 0 & -2\end{pmatrix}].
\]
Once we have constructed a coset $gZ^\epsilon_A$ we obtain an extended frame $F_\zeta(w) =
g\sharp F^w_\zeta$ for a Lagrangian angle $\Omega:\mathbb{C}/\Gamma\to G/G_0$. The Maslov form of the
corresponding HSL torus is then $\mu=-(dw+d\bar w)$ as a harmonic $1$-form on $\mathbb{C}/\Gamma$.
\begin{proof}
We begin by choosing $\epsilon>0$ so that the union of open discs
\[
I_{\epsilon^2} = \{\lambda\in\mathbb{C}_\infty:|\lambda|<\epsilon^2\text{ or } |\lambda|>\epsilon^{-2}\}
\]
contains no branch points of $\lambda$. Let $\mathcal{E}$ denote $\lambda_*\mathcal{L}$. The reality conditions
on $\mathcal{L}$ are such that $\mathcal{E}$ is trivial \cite{McI96}: our aim is to produce a global trivialisation
$\Psi$ and another trivialisation $\Psi_I$ over $I_{\epsilon^2}$ so that the transition relation
\begin{equation}\label{eq:transition}
g_\lambda\Psi_I=\Psi,\quad \lambda\in I_{\epsilon^2}-\{0,\infty\},
\end{equation}
furnishes the coset $gZ^\epsilon_A$ in untwisted form.
From \cite{McI96} we know that, as a consequence of the reality conditions, $H^0(\mathcal{L})=H^0(\mathcal{E})$
possesses a Hermitian inner product $h(\sigma_1,\sigma_2)$
derived from the trace map $\operatorname{Tr}:\mathcal{O}_X(R)\to\mathcal{O}_{\mathbb{C}_\infty}$ \cite{McI95,McI99}. This inner
product is determined only up to a positive real scaling factor, but this will prove to be
sufficient\footnote{This lack of uniqueness reflects the fact that throughout the whole discussion
of the HSL condition we have only used the fact that the metric on $\mathbb{CP}^2$ is $PU(3)$-invariant,
which only fixes the metric up to scale.}.
Further,
the subspaces $H^0(\mathcal{L}(-P_2-P_3))$ and $H^0(\mathcal{L}(-Q_1-Q_2))$ are one dimensional (since the
triviality of $\mathcal{E}$ means neither $\mathcal{L}(-\sum P_j)$ nor $\mathcal{L}(-\sum Q_j)$ have non-trivial global
sections). Moreover, one knows that for $\sigma_1,\sigma_2\in H^0(\mathcal{L})$ their inner product
$h(\sigma_1,\sigma_2)$ is zero if $\sigma_1\overline{\rho_*\sigma_2}$ vanishes on any divisor
given by a fibre of $\lambda:X\to\mathbb{C}_\infty$. Using this property it is straightforward to check that
$H^0(\mathcal{L}(-P_2-P_3))$, $H^0(\mathcal{L}(-P_3-Q_1))$ and $H^0(\mathcal{L}(-Q_1-Q_2))$ are mutually perpendicular
and we can choose an $h$-unitary frame $\sigma_1$, $\sigma_2$ and $\sigma_3$ from those lines in
$H^0(\mathcal{L})$. These are fixed only up to common $\mathbb{R}^+$ scaling and individual $S^1$ rotation.
Together they provide a global trivialisation
\[
\Psi:\mathcal{E}\to \mathcal{O}_{\mathbb{C}_\infty}^3;\quad \Psi(\sigma_j)=e_j.
\]
Moreover, because $h$ is non-degenerate we have a canonical isomorphism
$\overline{\rho^*\mathcal{E}}\simeq\mathcal{E}^*$, where we are using $\rho$ to represent $\lambda\mapsto
\bar\lambda^{-1}$ on $\mathbb{C}_\infty$. Under this isomorphism $\overline{\rho^*\Psi}$ is
identified with $\Psi^*$, the trivialisation dual to $\Psi$.
Now set $U=\lambda^{-1}(I_{\epsilon^2})$. This is a disjoint union of three copies of $I_{\epsilon^2}$, since
$\lambda$ is locally biholomorphic about each pair $P_j,Q_j$. Therefore
any trivialisation $\psi$ for $\mathcal{L}$ over $U$ induces a
trivialisation $\Psi_I$ for $\mathcal{E}$ using the identification
\[
\mathcal{E}_{I_{\epsilon^2}}=\mathcal{L}_U\simeq(\mathcal{O}_{I_{\epsilon^2}})^3.
\]
Further, since we can change $\psi$ independently over each of the connected components of $U$, it
can be chosen so that $\overline{\rho^*\Psi_I}=\Psi_I^*$. Now consider the transition matrix
$g_\lambda$ in \eqref{eq:transition}.
It satisfies $\overline{g_{\bar\lambda^{-1}}}=g_\lambda^*$ because of the
reality conditions on $\Psi$ and $\Psi_I$. Its value $g_0$ at $\lambda=0$ is upper triangular
because of the zeroes of $\sigma_1$ and $\sigma_2$. Moreover we can adjust $\psi$ over $\lambda=0$
so that $g_0$ has the shape \eqref{eq:shape} with positive diagonal entries.
Our final task is to show that, by possibly placing further conditions on $\Psi$ and $\Psi_I$, we
also ensure the appropriate $\tau$-symmetry \eqref{eq:tau_untwisted} of $g_\lambda$.
To this end set $E=P_1+P_2-Q_1-Q_2$ and define
\[
\theta_1 = i\lambda^{-1}\sigma_2,\ \theta_2=-i\lambda^{-1}\sigma_1,\ \theta_3=\sigma_3,
\]
which we may think of as globally holomorphic sections of $\mathcal{F}=\lambda_*\mathcal{L}(E)$. Indeed, these
globally frame $\mathcal{F}$. Clearly
\[
\Psi(\theta_1) = i\lambda^{-1}e_2,\ \Psi(\theta_2) = -i\lambda^{-1}e_1,\ \Psi(\theta_3)=e_3,
\]
and therefore $(\tau^*S_\lambda)\Psi(\theta_j)=e_j$, i.e., $(\tau^*S_\lambda)\Psi$ globally
trivialises $\mathcal{F}$. By assumption
\[
\tau_*\mathcal{L}(E)\simeq \mathcal{O}_X(R)\otimes\mathcal{L}^{-1}
\]
and therefore $\tau^*\mathcal{F}\simeq\mathcal{E}^*$. We fix this isomorphism as
\begin{equation}\label{eq:iso}
\tau^*\mathcal{F}\to\mathcal{E}^*;\quad S_\lambda\tau^*\Psi\mapsto \Psi^*.
\end{equation}
Clearly $S_\lambda\tau^*\Psi_I$ is a trivialisation for $\tau^*\mathcal{F}$ over $I$. After our choices
above, we still have freedom to scale $\psi$ independently over $P_1$ and $P_2$ in such a way that
$S_\lambda\tau^*\Psi_I$ maps to $\Psi^*$ under \eqref{eq:iso}. Therefore from $g_\lambda\Psi_I=\Psi$ we
deduce
\begin{eqnarray*}
\tau^*g_\lambda\tau^*\Psi_I&=&\tau^*\Psi\\
\implies (S_\lambda\tau^*g_\lambda S_\lambda^{-1})S_\lambda\tau^*\Psi_I& = &S_\lambda\tau^*\Psi,\\
\implies (S_\lambda\tau^*g_\lambda S_\lambda^{-1})\Psi_I^* & = & \Psi^*.
\end{eqnarray*}
Hence $S_\lambda\tau^*g_\lambda S_\lambda^{-1}=g_\lambda^*$.
\end{proof}
\subsection{Explicit reconstruction.}\label{sec:explicit}
The method by which we have proved proposition \ref{prop:spectral} amounts to an algorithm for
performing the dressing construction of theorem \ref{thm:dressing} but, as mentioned above, to
construct the HSL torus this can be circumvented using results from \cite{McI99}, which tell us
that $f:\mathbb{C}/\Gamma\to\mathbb{CP}^2$ must factor through the generalised Jacobian $\operatorname{Jac}(X_\mathfrak{o})$ in the form
\eqref{eq:theta}. Recall that $\mathfrak{o}=O_1+O_2+O_3$ is the divisor of points lying over $\lambda=1$ and
$X_\mathfrak{o}$ is the singular obtained by identifying these three points together.
We will summarise here what this means in the case where $X$ is smooth: the
description for the case where $X$ may be singular simply requires more technical details.
First we describe the rational map $\theta:\operatorname{Jac}(X_\mathfrak{o})\to\mathbb{CP}^2$.
The general principle, regardless of whether or not $X$ is smooth, is that $\theta$ assigns to each
$\tilde L$ in an open subvariety $\mathcal{U}\subset \operatorname{Jac}(X_\mathfrak{o})$ the point in $\mathbb{CP}^2$ corresponding to
the line
\[
H^0(\mathcal{L}(-Q_1-Q_2)\otimes L)\subset H^0(\mathcal{L}\otimes L),
\]
where $L=\pi(\tilde L)\in\operatorname{Jac}(X)$ for
the fibration $\pi:\operatorname{Jac}(X_\mathfrak{o})\to\operatorname{Jac}(X)$. The open set $\mathcal{U}$ is the set on
which $H^0(\mathcal{L}(-Q_1-Q_2)\otimes L)$ has dimension $1$ and hence $\P H^0(\mathcal{L}\otimes L)\simeq \mathbb{CP}^2$:
it precisely the extra data carried by $\tilde L$ which determines this isomorphism once it has
been fixed for $\P H^0(\mathcal{L})$. When $X$ is smooth $\mathcal{U}$ is the preimage
of the complement of a translate of the $\Theta$-divisor on $\operatorname{Jac}(X)$. It was shown in
\cite{McI99} that this map is
algebraic, hence rational, and that the real subgroup
\[
J_\mathbb{R}(X_\mathfrak{o})=\{\tilde L\in \operatorname{Jac}(X_\mathfrak{o}):\overline{\rho^*\tilde L}\simeq \tilde L^{-1}\}
\]
lies inside $\mathcal{U}$. Thus $\theta$ is real analytic on $J_\mathbb{R}(X_\mathfrak{o})$, so that
$f=\theta\circ\ell^\mathfrak{o}$ is also real analytic. The reality conditions on $\mathcal{L}$ and $\tilde L$
are necessary to identify each $\P H^0(\mathcal{L}\otimes L)$ with $\mathbb{CP}^2$ as a Hermitian symmetric space.
Given a smooth HSL spectral curve $(X,\lambda)$ we can write $\theta$ in terms of the Riemann
$\Theta$-function for $X$. We first identify
\begin{equation}
\label{eq:Jac}
\operatorname{Jac}(X_\mathfrak{o})\cong H^0(\Omega_X(\mathfrak{o}))^*/H_1(X-\mathfrak{o},\mathbb{Z})\simeq
\mathbb{C}^{g+n}/\Lambda^\prime,
\end{equation}
where $\Omega_X(\mathfrak{o})$ is the sheaf of mermorphic differentials on $X$ with divisor of
poles no worse than $\mathfrak{o}$ and $\Lambda^\prime$ is a lattice on
$2g+2$ generators. We choose coordinates so that
$\pi:\operatorname{Jac}(X_\mathfrak{o})\to\operatorname{Jac}(X)$ is covered by the map
\[
\pi:\mathbb{C}^{g+n}\to\mathbb{C}^g;\quad \tilde W=(w_1,\ldots,w_{g+2})\mapsto W=(w_1,\ldots,w_g).
\]
These coordinates amount to making a choice of basis for $H^0(\Omega_X(\mathfrak{o}))$ consisting of a
basis $\omega_1,\ldots,\omega_g$ for $H^0(\Omega_X)$ and two meromorphic differentials
of the third kind $\omega_{g+j}\in H^0(\Omega_X(O_1+O_{j+1}))$. These can be normalised in the
standard manner with respect to a choice of ``a-cycles'' for the punctured curve
$X-\mathfrak{o}$, i.e., so that
\[
\int_{a_i}\omega_j=\delta_{ij},\ i,j=1,\ldots,g+2.
\]
The real symmetry of $X$ is such that the cycles and differentials can be chosen to allow
$\overline{\rho^*\omega_j} =-\omega_j$ and
$J_R(X_\mathfrak{o})\simeq \mathbb{R}^{g+2}/(\Lambda'\cap\mathbb{R}^{g+2})$. It is shown in \cite[p432]{McI99} that this
is a real torus.
Let $\Theta$ be the classical Riemann $\theta$-function on $\mathbb{C}^g$ corresponding
to the induced isomorphism $\operatorname{Jac}(X)\simeq \mathbb{C}^g/\pi(\Lambda^\prime)$.
Now let us define $\phi_0(\tilde W) = \Theta(W)$ and
for $j=1,2$ define
\[
\phi_j(\tilde W) = \exp(2\pi iw_{g+j})\Theta(W+\mathfrak{A}_{O_1}(O_{j+1})).
\]
Each $\phi_j$ represents a global holomorphic section of a line bundle over $\operatorname{Jac}(X_\mathfrak{o})$, namely,
the pullback by $\pi$ of (an appropriate translate of) the $\Theta$-line bundle over $J(X)$
\cite{McI99}.
Now let $D'$ be the unique positive divisor in the class $\mathcal{L}(-Q_1-Q_2)$. Let
$\kappa\in\mathbb{C}^g$ be the appropriate translation for which $\Theta(\mathfrak{A}_{O_1}(P)+\kappa)$ has divisor of
zeroes $D'$. We set $\tilde\kappa=(\kappa_1,\ldots,\kappa_g,0,0)\in\mathbb{C}^{g+2}$.
As a consequence of the reality condition on $\mathcal{L}$ in \eqref{eq:lequiv}, the set
\[
\mathcal{D}(\mathcal{L})=\{\text{divisors $D$ on}\ X-\mathfrak{o}:\mathcal{O}_X(D)\simeq\mathcal{L},\ D>0,\ D+\rho_*D\sim_\mathfrak{o} R\}
\]
is non-empty, where ``$\sim_\mathfrak{o}$'' means linear equivalence on $X_\mathfrak{o}$, i.e.,
$D+\rho_*D-R$ is the divisor of a rational function taking the value $1$ at each
point $O_1,O_2,O_3$. In fact $\mathcal{D}(\mathcal{L})$ is identifiable with
\[
\{\tilde\mathcal{L}\in\operatorname{Pic}_{g+2}(X_\mathfrak{o}):\pi(\tilde\mathcal{L}) = \mathcal{L},\
\tilde\mathcal{L}\otimes\overline{\rho^*\tilde\mathcal{L}}\simeq\mathcal{O}_{X_\mathfrak{o}}(R)\},
\]
which is a real slice of the fibre of $\pi:\operatorname{Pic}(X_\mathfrak{o})\to\operatorname{Pic}(X)$ over $\mathcal{L}$.
The kernel of the group homomorphism $\pi:J_\mathbb{R}(X_\mathfrak{o})\to J_\mathbb{R}(X)$ acts freely and
transitively on this set, hence $\mathcal{D}(\mathcal{L})\simeq S^1\times S^1$,
Choose some $D\in\mathcal{D}(\mathcal{L})$. Since $D\sim D'+Q_1+Q_2$
there is a rational function $k$ on $X$ with divisor $D-(D'+Q_1+Q_2)$
and $k(P)\Theta(\mathfrak{A}_{O_1}(P)+\kappa)$ has divisor
$D-Q_1-Q_2$. Notice it does not vanish at any $O_j$. Now we define
\begin{eqnarray*}
\theta:\operatorname{Jac}(X_\mathfrak{o})&\to&\mathbb{CP}^2\\
\theta(\tilde W\bmod\Lambda')& = &
[c_0\phi_0(\tilde W+\tilde\kappa),c_1\phi_1(\tilde W+\tilde\kappa),c_2\phi_2(\tilde W+\tilde\kappa)],
\end{eqnarray*}
where the constants $c_j$ are given by
\begin{equation}\label{eq:c_j}
c_j = \frac{1}{k(O_{j+1})\Theta(\mathfrak{A}_{O_1}(O_{j+1}) +\kappa)},\qquad j=0,1,2.
\end{equation}
Finally, the HSL torus $f:\mathbb{C}/\Gamma\to\mathbb{CP}^2$ corresponding to $(X,\lambda,\mathcal{L})$ is obtained
as the composition $f = \theta\circ\ell^\mathfrak{o}$. Different choices of $D\in\mathcal{D}(\mathcal{L})$ alter the
constants $c_j$ by unimodular multipliers, so that $f$ is determined up to an
isometry of $\mathbb{CP}^2$ by the data $(X,\lambda,\mathcal{L})$.
The explicit formula for the real homomorphism
$\ell^\mathfrak{o}:\mathbb{C}/\Gamma\to\operatorname{Jac}(X_\mathfrak{o})$ can be easily calculated from \eqref{eq:lotangent2} using the
standard observation that
\[
\frac{\partial\mathcal{A}_{P_i}}{\partial\lambda}(0) =
(\operatorname{Res}_{P_i}\lambda^{-1}\omega_1,\ldots,\operatorname{Res}_{P_i}\lambda^{-1}\omega_{g+2})\in\mathbb{C}^{g+2}.
\]
\begin{exam}\label{exam:g=0}
To illustrate this we will compute the HSL tori in $\mathbb{CP}^2$ which arise from the choice
$X=\mathbb{C}_\infty$. It is a consequence of \cite[\S 4.2]{McI99} that these will be $S^1\times S^1$-equivariant
(called ``homogeneous'' in \cite{HelR3}). Recall that this means $f:\mathbb{C}/\Gamma\to\mathbb{CP}^2$
possesses a real homomorphism $h:\mathbb{C}/\Gamma\to SU(3)$ for which $f(p)=h(p)f(0)$ for all $p\in\mathbb{C}/\Gamma$.
All such HSL tori were constructed explicitly (in non-conformal coordinates) in \cite[\S 5]{HelR3}.
Let $\zeta$ denote the natural rational parameter on $\mathbb{C}_\infty$ and, for a fixed
$a\in\mathbb{C}$ with $0<|a|<1$, set
\[
\lambda = \zeta\frac{(\zeta^2-a^2)}{(\bar a^2\zeta^2-1)}\frac{(\bar a^2-1)}{(1-a^2)}.
\]
This choice ensures that the involutions $\rho^*\zeta = \bar\zeta^{-1}$ and $\tau^*\zeta = -\zeta$
have the correct effects on $\lambda$ and that we can take $O_1$ to be $\zeta=1$. The other two
points $O_2,O_3$ over $\lambda=1$ are the two roots of the quadratic
\[
\zeta^2+\frac{1-|a|^4}{1-\bar a^2}\zeta + \frac{1-a^2}{1-\bar a^2}.
\]
We also take $\zeta(P_1)=a$, $\zeta(P_2)=-a$ and $\zeta(P_3)=0$ and so forth for the points $Q_j$.
The first homology of $\mathbb{C}_\infty-\mathfrak{o}$ is generated by positively oriented cycles $a_1,a_2$ encircling
$O_2$ and $O_3$ respectively and the dual basis of differentials is given by
\[
\omega_j = \frac{1}{2\pi i}(\frac{1}{\zeta-O_{j+1}}-\frac{1}{\zeta-O_1})d\zeta,\quad
j=1,2.
\]
Thus $\operatorname{Jac}(X_\mathfrak{o})\simeq\mathbb{C}^2/\mathbb{Z}^2$ and $J_\mathbb{R}(X_\mathfrak{o})\simeq \mathbb{R}^2/\mathbb{Z}^2$. We can take $\Theta=1$ and
thus
\[
\phi_0(W_1,W_2) = 1,\quad \phi_1(W_1,W_2)=\exp(2\pi iW_1),\quad \phi_2(W_1,W_2)=\exp(2\pi iW_2).
\]
The ramification divisor $R$ is the four point divisor of zeroes of
$d\lambda/d\zeta$, and is given by the roots of
\[
(\zeta^2-C_+)(\zeta^2-C_-),\qquad C_{\pm} = \frac{1}{2\bar a^2}(3-|a|^4\pm\sqrt{(|a|^4-1)(|a|^4-9)}.
\]
Let $R_1+R_2$ be the roots of $\zeta^2-C_+$, then $R=R_1+R_2+\rho_*R_1+\rho_*R_2$.
Since the genus is zero we can choose $\mathcal{L}$ to be any degree $2$ line bundle and choose
$D=R_1+R_2$. The rational function $k$ with divisor $R_1+R_2-Q_1-Q_2$ and normalised by $k(O_1)=1$
is given by
\[
k(\zeta) = \frac{(\zeta^2-C_+)}{(\bar a^2\zeta^2-1)}\frac{(\bar a^2 -1)}{(1-C_+)}
\]
Finally, define
\begin{eqnarray*}
U_1=\frac{\partial\mathcal{A}_{P_1}}{\partial\lambda}(0)& =
&\frac{1}{2\pi i}\frac{(1+a)(1-|a|^4)}{2a^2(1-\bar a^2)}
\left(\frac{O_2-1}{a-O_2},\frac{O_3-1}{a-O_3}\right),\\
U_2=\frac{\partial\mathcal{A}_{P_2}}{\partial\lambda}(0)& =
&\frac{1}{2\pi i}\frac{(1-a)(1-|a|^4)}{2a^2(1-\bar a^2)}
\left(\frac{O_2-1}{a+O_2},\frac{O_3-1}{a+O_3}\right),
\end{eqnarray*}
and set $U = (3\pi i/2)(U_1+U_2)\in\mathbb{C}^2$.
Then there is a maximal lattice $\Gamma\subset\mathbb{C}$ for which
\[
\ell^\mathfrak{o}:\mathbb{C}/\Gamma\to \mathbb{R}^2/\mathbb{Z}^2\simeq J_\mathbb{R}(X_\mathfrak{o});\quad w\bmod\Gamma\mapsto wU+\bar w\bar U\bmod\mathbb{Z}^2,
\]
and the HSL torus is given by
\begin{eqnarray}
f:\mathbb{C}/\Gamma&\to&\mathbb{CP}^2;\label{eq:homog}\\
f(w)& = &[1,c_1\exp(wA_1-\bar w \bar A_1),c_2\exp(wA_2- \bar w\bar A_2)] \notag
\end{eqnarray}
where $c_j=1/k(O_{j+1})$ and
\[
A_j = \frac{3\pi i(O_{j+1}^2-1)(1-|a|^4)}{4a(a^2-O_{j+1}^2)(1-\bar a^2)}.
\]
This gives a two real parameter family of conformally embedded homogeneous HSL tori, with
parameter $a$. To verify that $f$ is indeed HSL, let us point out that \eqref{eq:homog} is certainly a
homogeneous immersion of a torus, and the condition that a map of this form is both conformal and
Lagrangian can be shown to be
\begin{equation}
A_1^2|c_1|^2 + A_2^2|c_2|^2 + |c_1c_2|^2(A_2-A_1)^2=0.
\end{equation}
Numerical calculations verify that this holds for the quantities above. The HSL condition
follows since every homogeneous conformal Lagrangian torus has harmonic Maslov form.
Notice that the computation above must exclude $a=0$, the minimal
Lagrangian limit, because of the expression for $C_{\pm}$, $U_1$ and $U_2$.
However, when $a=0$ we can set $R_1+R_2= 2.\infty$ (since $Q_1=Q_2=Q_3=\infty$
is the ramification point over $\lambda=\infty$ in this case) and take
$U=(\partial\mathfrak{A}_{P_1}/\partial\zeta)(0)$. In this case the calculation simplifies greatly,
and does indeed produce the unique (up to isometries) homogeneous minimal Lagrangian torus in $\mathbb{CP}^2$.
\end{exam}
\subsection{Brief comments on moduli.}\label{sec:moduli}
The HSL spectral data given in definition \ref{defn:spectral_data} looks very similar
to that for a minimal Lagrangian torus. Indeed, the conditions (a), (b) and (d) reproduce the
conditions for a minimal Lagrangian torus \cite{McI03,Sha} when we force $P_1=P_2=P_3$ (and hence
$Q_1=Q_2=Q_3$). In that limit $\lambda$ is branched over $0$ and $\infty$, so it is no longer a
local parameter at those points. Therefore in this limit \eqref{eq:lotangent} must be interpreted
as a statement about tangent planes in the generalised Jacobian.
Nevertheless, one can think of the passage from minimal
Lagrangian to HSL as a trifurcation of the zeroes of $\lambda$. This
allow us to count the expected dimension of the moduli space of spectral curve pairs
$(X,\lambda)$ which admit a HSL torus. For minimal Lagrangian tori the expected dimension is zero
(so that generically no continuous deformations exist, see
\cite{McI03,CarM}). For HSL tori there are two free real parameters in the count, so we can expect
each HSL spectral curve pair $(X,\lambda)$ to be able to be deformed in two parameter families.
This is certainly consistent with what we know for the homogeneous tori in example \ref{exam:g=0}
above.
The spectral genus $g$ also gives us a measure of the deformation space for each HSL torus, for we
are able to move $\mathcal{L}$ continuously without breaking the double periodicity (indeed, this even
fixes the period lattice $\Gamma\subset\mathbb{C}$). Just as with the minimal Lagrangian case,
the symmetry restrictions on $\mathcal{L}$ oblige it to lie on a real slice of a translate of the Prym
variety $\operatorname{Prym}(X,\tau)$. Since $\tau$ has exactly two fixed points $g$ is even and this Prymian has
complex dimension $g/2$. This contributes $g/2-2$ real dimensions to the deformation space of a HSL
torus of spectral genus $g$, since we must remove the $2$-parameters corresponding to
translations of the base point. Altogether this predicts smooth $g/2$-dimensional families of
HSL tori, fibered by $g/2-2$-dimensional leaves (each of which will be a torus) consisting of
HSL tori of the same conformal type.
| {
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} | 5,206 |
A prova dos 3000 metros com obstáculos masculino do Campeonato da Europa de Atletismo de 2018 foi disputada entre os dias 7 e 9 de agosto de 2018 no Estádio Olímpico de Berlim em Berlim, na Alemanha.
Recordes
Antes desta competição, os recordes mundiais e do campeonato nesta prova eram os seguintes:
Medalhistas
Cronograma
Todos os horários são locais (UTC+2).
Resultados
Bateria
Qualificação: 5 atletas de cada bateria (Q) mais os 5 melhores qualificados (q).
Final
Ligações externas
Site da Associação Europeia de Atletismo
Campeonato da Europa de Atletismo de 2018 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,372 |
Q: What expression are these titles alluding to? There are two episode titles from two separate shows that are written similarly:
*
*From Castle: The mistress always spanks twice
*From Doc Martin: The GP always rings twice
Episode titles are usually clever plays on words, so I'm assuming the two titles are alluding to a common expression. What expression (or other cultural reference) are the titles making a reference to?
A: Probably from the novel 'The Postman Always Rings Twice' by James M Cain, published in 1934: http://en.wikipedia.org/wiki/The_Postman_Always_Rings_Twice_%28novel%29
| {
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Do Engineers Owe a Duty to Third Parties?
A Texas Court of Appeals, in USA Walnut Creek, DST v. Terracon Consultants, Inc., recently ruled that an engineer owed a duty to the buyer of an apartment complex, even though the engineer had no contractual relationship with the buyer. This is an expansion of the duty professionals owe on construction projects and could signal a change in the law.
In the case, Walnut Creek purchased a three year old apartment complex. A few years after taking possession, Walnut Creek noticed problems with the apartments, including cracking foundations, walls, breaking windows, and out of square door frames. Walnut Creek sued the developer and general contractor, alleging construction defects. The developer claimed that the engineer, Terracon, was at fault and Walnut Creek added Terracon to the lawsuit, asserting that Terracon was negligent in performing engineering services during construction. Terracon asked the court to dismiss the claim, arguing that it did not owe a duty to Walnut Creek. Walnut Creek in turn argued that engineers do owe a duty to subsequent owners. The trial court dismissed the case against the engineer and Walnut Creek appealed.
The appellate court reversed the trial court, finding that the engineer did owe a duty to subsequent purchasers. The court seemed persuaded by the allegations that the engineer actually created the construction defects which were the basis for the litigation.
Ultimately, the court ruled that the engineer did have a duty to the subsequent buyer and the lawsuit against the engineer could proceed.
Why You Should Care: Courts are split on whether an engineer owes a duty to anyone other than the entity or person with whom it was contracted. Nebraska and Iowa courts have generally held that engineers may not be sued by subsequent purchasers because the engineer does not owe a duty to the subsequent purchasers. But, the Walnut Creek case may signal a change in court opinions.
This is an issue that we will monitor, much like the definition of an occurrence in a CGL policy.
Craig Martin2018-04-23T23:55:11-05:00May 25th, 2015|Construction Contractor Advisor, Engineers and Architects Regulation Act| | {
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} | 6,730 |
// Package record has all client logic for recording and reporting events.
package record
| {
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} | 5,144 |
{"url":"http:\/\/mathhelpforum.com\/number-theory\/223712-number-series-possible-print.html","text":"# number of series possible\n\n\u2022 October 31st 2013, 12:40 AM\nnikhil\nnumber of series possible\nGreetings,\nstruck with the question that how many sequences of consecutive numbers exist such that on adding them we get 500.\n\nthanks\n\u2022 October 31st 2013, 04:03 AM\nPlato\nRe: number of series possible\nQuote:\n\nOriginally Posted by nikhil\nThe question that how many sequences of consecutive numbers exist such that on adding them we get 500.\n\nYou need to find how many integer pairs $(n,x)$ satisfy $\\sum\\limits_{k = 0}^n {(x + k)}=500~.$\n\nThat can be written as $\\left( {n + 1} \\right)x + \\frac{{n(n + 1)}}{2} = 500$.\n\u2022 October 31st 2013, 06:04 AM\nnikhil\nRe: number of series possible\nThanks Plato but am not able to find the integral solutions to this equation as x is itself a variable(1st term of sequence)\n\u2022 October 31st 2013, 06:54 AM\nSlipEternal\nRe: number of series possible\nAssume $x$ is a constant and solve for $n$:\n\n$n = \\dfrac{1}{2}\\left(\\sqrt{4x^2-4x+4001}-2x-1\\right)$\n\nSince both $x$ and $n$ must be integers, figure out for which values of $x$ $4x^2-4x+4001$ is an odd perfect square (it needs to be odd since otherwise, $n$ will not be an integer).\n\nEdit: Also $n$ must be nonnegative and $x\\le 500$... That leaves only 8 solutions.\n\nHere are the first two:\n$(n,x) = (0,500)$ or $(n,x) = (999,-499)$.\n\nEvery solution $(n,x)$ with $x>0$ will have another solution $(n+2x-1,1-x)$ since adding all of the numbers from $1-x$ to $x-1$ will give zero, then you just have the same $n$ integers from $x$ to $x+n-1$ from the initial solution. Hence, if you find three more solutions with $x>0$, then you have found all of the solutions.\n\u2022 October 31st 2013, 08:39 AM\nebaines\nRe: number of series possible\nYes, 8 solutions. To be honest I would have said only 4 solutions, not realizing until reading SlipEternal's post that each solution a, a+1, a+2,...a+n has a counterpart of -(a-1), -(a-2), -(a-3) ... a-3, a-2, a-1, a, a+1, a+2,...a+n. My approach is to check the value of the median number in the series of n integers to see if it \"makes sense\" - for n odd the median number must be an integer, and for n even the mediian must be an integer + 1\/2. This is because the starting number of the sequence of n terms is (500\/n)-(n-1)\/2. If this is an integer you have a sequence of n terms that adds to 500. I found four values for n that work, so using SlipEternal's idea that means there are a total of 8.\n\u2022 October 31st 2013, 11:26 AM\nnikhil\nRe: number of series possible\nthanks slipEternal and ebaines. SlipEternal could you plz explain how you got the equation will be odd perfect square 8 times or just give any online reference ( lyk Diophantine equation or smthin)\n\u2022 October 31st 2013, 11:43 AM\nSlipEternal\nRe: number of series possible\nebaines idea is easier to use. Solve for $x$ instead of solving for $n$. Then you have $x = \\dfrac{500}{n+1} - \\dfrac{n}{2}$. Now, let's consider cases where $x$ is an integer.\n\nCase 1: 2 divides n\nThen (n+1) must divide 500. Since (n+1) must be odd, we know $n+1 \\in \\{1,5,25,125\\}$ as those are the only odd factors of 500.\n\nCase 2: 2 does not divide n.\nThen n must be odd and n+1 must be even. But, n+1 cannot divide 500. It must divide 1,000, though. Moreover, $\\dfrac{1000}{n+1}$ must be odd. Since the only odd factors of 1,000 are the odd factors of 500, we have $\\dfrac{1000}{n+1} \\in \\{1,5,25,125\\}$.\n\nThose are all eight solutions.","date":"2014-07-26 01:54:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 31, \"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\": 0.856505811214447, \"perplexity\": 584.0338586768294}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-23\/segments\/1405997894931.59\/warc\/CC-MAIN-20140722025814-00021-ip-10-33-131-23.ec2.internal.warc.gz\"}"} | null | null |
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OTA Advisory Committee Members
Administrative Law Judge, Los Angeles
Andrew Wong received a B.A. from Stanford University, a J.D. from the University of Southern California, and an LL.M. (Taxation) from New York University. He served as a law clerk to the Honorable Nancy A. Becker during her term as Chief Justice of the Supreme Court of Nevada, and worked as an associate attorney in the Las Vegas office of Jones Vargas, where his practice focused on civil litigation, public finance, and state and local tax controversy. Prior to joining the Office of Tax Appeals, he was a tax counsel in the California Department of Tax and Fee Administration's Appeals Bureau.
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\section{Introduction}
The low-lying eigenspace of operators has
many important applications, including those in quantum chemistry,
numerical PDEs, and statistics. Given a $n \times n$ symmetric matrix
$H$, and denote its eigenvectors as $\{\Phi_i\}, i = 1, \ldots,
n$. The low-lying eigenspace is given by the span of the first $N$
(usually $N \ll n$) eigenvectors.
In many scenario, the real interest is the subspace itself, but not a
particular set of basis functions. In particular, we are interested in a
sparse representation of the eigenspace. The eigenvectors form a
natural basis set, but for oftentimes they are not sparse or localized
(consider for example the eigenfunctions of the free Laplacian
operator $-\Delta$ on a periodic box). This suggests asking for an alternative
sparse representation of the eigenspace.
In quantum chemistry, the low-lying eigenspace for a Hamiltonian
operator corresponds to the physically occupied space of electrons. In
this context, a localized class of basis functions of the low-lying
eigenspaces is called Wannier functions \cites{Wannier:1937,
Kohn:1959Wannier}. These functions provide transparent
interpretation and understandings of covalent bonds, polarizations,
\textit{etc.} of the electronic structure. These localized
representations are also the starting point and the essence for many
efficient algorithms for electronic structure calculations (see
e.g.~the review article~\cite{Goedecker:99}).
\subsection{Our contribution}
In this work, we propose a convex minimization principle for finding a
sparse representation of the low-lying eigenspace.
\begin{equation}\label{eqn:l1DensityMatrix}
\begin{aligned}
& \min_{P \in \mathbb{R}^{n\times n}} \tr(H P) + \frac{1}{\mu} \norm{P}_1\\
& \text{s.t.} \ P = P^{\mathrm{T}},\, \tr P = N,\, 0 \preceq P \preceq I,
\end{aligned}
\end{equation}
where $\norm{\cdot}_1$ is the entrywise $\ell_1$ matrix norm, $A
\preceq B$ denotes that $B - A$ is a positive semi-definite matrix,
and $\mu$ is a penalty parameter for entrywise sparsity. Here $H$ is an
$n\times n$ symmetric matrix, which is the (discrete) Hamiltonian in
the electronic structure context. The variational principle gives $P$
as a sparse representation of the projection operator onto the
low-lying eigenspace.
The key observation here is to use the matrix $P$ instead of the wave
functions $\Psi$. This leads to a convex variational
principle. Physically, this corresponds to looking for a sparse
representation of the density matrix. We also noted that in cases
where we expect degeneracy or near-degeneracy of eigenvalues of the
matrix $H$, the formulation in terms of the density matrix $P$ is more
natural, as it allows fractional occupation of states. This is a
further advantage besides the convexity.
Moreover, we design an efficient minimization algorithm based on split
Bregman iteration to solve the above variational problem. Starting
from any initial condition, the algorithm always converges to a
minimizer.
\subsection{Previous works}
There is an enormous literature on numerical algorithms for
Wannier functions and more generally sparse representation of
low-lying eigenspace. The influential work \cite{Marzari:1997}
proposed a minimization strategy within the occupied space to find
spatially localized Wannier functions (coined as ``maximally localized
Wannier functions'').
In \cite{E:2010PNAS}, the second author with his collaborators
developed a localized subspace iteration (LSI) algorithm to find
Wannier functions. The idea behind the LSI algorithm is to combine the
localization step with the subspace iteration method as an iterative
algorithm to find Wannier functions of an operator. The method has
been applied to electronic structure calculation in
\cite{Garcia-CerveraLuXuanE:09}. As \cite{Garcia-CerveraLuXuanE:09}
shows, due to the truncation step involved, the LSI algorithm does not
in general guarantee convergence.
As a more recent work in \cites{OzolinsLaiCaflischOsher:13}, $L_1$ regularization is proposed to
be used in the variational formulation of the Schr\"odinger equation
of quantum mechanics for creating compressed modes, a set of spatially
localized functions $\{\psi_i\}_{i=1}^N$ in $\mathbb{R}^d$ with
compact support.
\begin{equation}
\label{model:CMs}
E = \min_{\Psi_N} \sum_{j=1}^N \left( \frac{1}{\mu}\left| \psi_j \right|_1 + \langle \psi_j , \hat{H} \psi_j \rangle \right) \quad \mbox{\text{s.t.}} \quad \langle \psi_j, \psi_k \rangle = \delta_{jk},
\end{equation}
where $\hat{H} = -\frac{1}{2}\Delta + V(\mathrm{x})$ is the Hamilton operator
corresponding to potential $V(\mathrm{x})$, and the $L_1$ norm is defined
as $\left| \psi_j \right|_1 = \int | \psi_j | d\mathrm{x}$. This $L_1$
regularized variational approach describes a general formalism for
obtaining localized (in fact, compactly supported) solutions to a
class of mathematical physics PDEs, which can be recast as variational
optimization problems. Although an efficient algorithm based on a method of splitting
orthogonality constraints (SOC)~\cite{Lai:2014splitting} is designed to solve the above non-convex problem, it is still
a big challenge to theoretically analyze the convergence of the proposed the algorithm.
The key idea in the proposed convex formulation~\eqref{eqn:l1DensityMatrix} of the variational principle is
the use of the density matrix $P$. The density matrix is widely used
in electronic structure calculations, for example the density matrix
minimization algorithm \cite{LiNunesVanderbilt:93}. In this type of
algorithm, sparsity of density matrix is specified explicitly by
restricting the matrix to be a banded matrix. The resulting
minimization problem is then non-convex and found to suffer from many local
minimizers. Other electronic structure algorithms that use density
matrix include density matrix purification \cite{McWeeny:60}, Fermi
operator expansion algorithm \cite{BaroniGiannozzi:92}, just to name a
few.
From a mathematical point of view, the use of density matrix can be
viewed as similar to the idea of lifting, which has been recently used
in recovery problems \cite{CandesStrohmerVoroninski:13}. While a
nuclear norm is used in PhaseLift method
\cite{CandesStrohmerVoroninski:13} to enhance sparsity in terms of
matrix rank; we will use an entrywise $\ell_1$ norm to favor sparsity
in matrix entries.
\bigskip
The rest of the paper is organized as follows. We formulate and
explain the convex variational principle for finding localized
representations of the low-lying eigenspace in
Section~\ref{sec:formulation}. An efficient algorithm is proposed in
Section~\ref{sec:Algorithm} to solve the variational
principle, with numerical examples presented in Section~\ref{sec:numerics}. The convergence proof of the algorithm is given in Section~\ref{sec:proof}.
\section{Formulation}\label{sec:formulation}
Let us denote by $H$ a symmetric matrix \footnote{With obvious changes, our results generalize to the Hermitian case} coming from, for example, the
discretization of an effective Hamiltonian operator in electronic
structure theory. We are interested in a sparse representation of the
eigenspace corresponding to its low-lying eigenvalues. In physical
applications, this corresponds to the occupied space of a Hamiltonian;
in data analysis, this corresponds to the principal components (for
which we take the negative of the matrix so that the largest
eigenvalue becomes the smallest). We are mainly interested in physics
application here, and henceforth, we will mainly interpret the formulation and algorithms from a physical view point.
The Wannier functions, originally defined for periodic Schr\"odinger
operators, are spatially localized basis functions of the occupied
space. In \cite{OzolinsLaiCaflischOsher:13}, it was proposed to find the
spatially localized functions by minimizing the variational problem
\begin{equation}\label{eq:Psi}
\min_{\Psi \in \mathbb{R}^{n \times N},\, \Psi^{\mathrm{T}} \Psi = I} \tr (\Psi^{\mathrm{T}} H \Psi) + \frac{1}{\mu} \norm{\Psi}_{1}
\end{equation}
where $\norm{\Psi}_{1}$ denotes the entrywise $\ell_1$ norm of $\Psi$. Here $N$ is the number of Wannier functions and $n$ is the number of spatial degree of freedom (e.g. number of spatial grid points or basis functions).
The idea of the above minimization can be easily understood by looking at each term in the energy functional. The $\tr(\Psi^{\mathrm{T}} H \Psi)$ is the sum of the Ritz value in the space spanned by the columns of $\Psi$. Hence, without the $\ell_1$ penalty term, the minimization
\begin{equation}
\min_{\Psi \in \mathbb{R}^{n \times N},\, \Psi^{\mathrm{T}} \Psi = I} \tr (\Psi^{\mathrm{T}} H \Psi)
\end{equation}
gives the eigenspace corresponds to the first $N$ eigenvalues (here
and below, we assume the non-degeneracy that the $N$-th and $(N+1)$-th
eigenvalues of $H$ are different). While the $\ell_1$ penalty prefers
$\Psi$ to be a set of sparse vectors. The competition of the two terms gives a sparse representation of a subspace that is close to the eigenspace.
Due to the orthonormality constraint $\Psi^{\mathrm{T}} \Psi = I$, the minimization problem \eqref{eq:Psi} is not convex, which may result in troubles in finding the minimizer of the above minimization problem and also makes the proof of convergence difficult.
Here we take an alternative viewpoint, which gives a convex
optimization problem. The key idea is instead of $\Psi$, we consider $P
= \Psi \Psi^{\mathrm{T}} \in \mathbb{R}^{n\times n}$. Since the columns of $\Psi$
form an orthonormal set of vectors, $P$ is the projection operator
onto the space spanned by $\Psi$. In physical terms, if $\Psi$ are
the eigenfunctions of $H$, $P$ is then the density matrix which corresponds
to the Hamiltonian operator. For insulating systems, it is known that
the off-diagonal terms in the density matrix decay exponentially fast
\cites{Kohn:59, Panati:07, Cloizeaux:64a, Cloizeaux:64b, Nenciu:83,
Kivelson:82, NenciuNenciu:98, ELu:CPAM, ELu:13}.
We propose to look for a sparse approximation of the exact density matrix by solving the minimization
problem proposed in \eqref{eqn:l1DensityMatrix}.
The variational problem \eqref{eqn:l1DensityMatrix} is a convex relaxation of the non-convex variational problem
\begin{equation}\label{eq:Pnonconvex}
\begin{aligned}
& \min_{P \in \mathbb{R}^{n\times n}} \tr(H P) + \frac{1}{\mu} \norm{P}_1 \\
& \text{s.t.} \ P = P^{\mathrm{T}},\, \tr P = N,\, P = P^2,
\end{aligned}
\end{equation}
where the constraint $0 \preceq P \preceq I$ is replaced by the
idempotency constraint of $P$: $P = P^2$. The variational principle
\eqref{eq:Pnonconvex} can be understood as a reformulation of
\eqref{eq:Psi} using the density matrix as variable. The idempotency
condition $P = P^2$ is indeed the analog of the orthogonality
constraint $\Psi^{\mathrm{T}} \Psi = I$.
Note that $0 \preceq P \preceq I$ requires that the eigenvalues of $P$
(the occupation number in physical terms) are between $0$ and $1$,
while $P = P^2$ requires the eigenvalues are either $0$ or $1$. Hence,
the set
\begin{equation}
\mathcal{C} = \{ P: P = P^{\mathrm{T}},\, \tr P = N,\, 0 \preceq P \preceq I \}
\end{equation}
is the convex hull of the set
\begin{equation}
\mathcal{D} = \{ P: P = P^{\mathrm{T}},\, \tr P = N,\, P = P^2\}.
\end{equation}
Therefore \eqref{eqn:l1DensityMatrix} is indeed a convex relaxation of \eqref{eq:Pnonconvex}.
Without the $\ell_1$ regularization, the variational problems \eqref{eqn:l1DensityMatrix} and \eqref{eq:Pnonconvex} become
\begin{equation}\label{eq:P'}
\begin{aligned}
& \min_{P \in \mathbb{R}^{n\times n}} \tr(H P) \\
& \text{s.t.} \ P = P^{\mathrm{T}},\, \tr P = N,\, 0 \preceq P \preceq I,
\end{aligned}
\end{equation}
and
\begin{equation}\label{eq:Pnonconvex'}
\begin{aligned}
& \min_{P \in \mathbb{R}^{n\times n}} \tr(H P) \\
& \text{s.t.} \ P = P^{\mathrm{T}},\, \tr P = N,\, P = P^2.
\end{aligned}
\end{equation}
These two minimizations actually lead to the same result in the
non-degenerate case.
\begin{prop}\label{prop:equiv}
Let $H$ be a symmetric $n \times n$ matrix. Assume that the $N$-th
and $(N+1)$-th eigenvalues of $H$ are distinct, the minimizers of
\eqref{eq:P'} and \eqref{eq:Pnonconvex'} are the same.
\end{prop}
This is perhaps a folklore result in linear algebra, nevertheless we include
the short proof here for completeness.
\begin{proof}
It is clear that the unique minimizer of \eqref{eq:Pnonconvex'} is
given by the projection matrix on the first $N$ eigenvectors of $H$,
given by
\begin{equation*}
P_N = \sum_{i=1}^N v_i v_i^{\mathrm{T}}
\end{equation*}
where $\{v_i\}, i = 1, \ldots, n$ are the eigenvectors of $H$,
ordered according to their associated eigenvalues. Let us prove
that \eqref{eq:P'} is minimized by the same solution.
Assume $P$ is a minimizer of \eqref{eq:P'}, we calculate
\begin{equation}\label{eq:trhp}
\tr(HP) = \sum_{i=1}^n v_i^{\mathrm{T}} H P v_i
= \sum_{i=1}^n \lambda_i v_i^{\mathrm{T}} P v_i
= \sum_{i=1}^n \lambda_i \theta_i(P),
\end{equation}
where $\theta_i(P) = v_i^{\mathrm{T}} P v_i$. On the other hand, we have
\begin{equation*}
\tr(P) = \sum_{i=1}^n v_i^{\mathrm{T}} P v_i = \sum_{i=1}^n \theta_i(P) = N,
\end{equation*}
and $0 \leq \theta_i(P) \leq 1$ since $0 \preceq P \preceq
I$. Therefore, if we view \eqref{eq:trhp} as a variational problem
with respect to $\{\theta_i\}$, it is clear that the unique minimum
is achieved when
\begin{equation*}
\theta_i(P) =
\begin{cases}
1, & i \leq N; \\
0, & \text{otherwise}.
\end{cases}
\end{equation*}
We conclude the proof by noticing that the above holds if and only
if $P = P_N$.
\end{proof}
This result states that we can convexify the set of admissible matrices.
We remark that, somewhat surprisingly, this result also holds for the
Hartree-Fock theory \cite{Lieb:77} which can be vaguely understood as
a nonlinear eigenvalue problem. However the resulting variational problem is still non-convex for the Hartree-Fock theory.
Proposition~\ref{prop:equiv} implies that the variational principle \eqref{eqn:l1DensityMatrix} can be understood as an $\ell_1$ regularized version of the variational problem \eqref{eq:Pnonconvex'}. The equivalence no longer holds for \eqref{eqn:l1DensityMatrix} and \eqref{eq:Pnonconvex} with the $\ell_1$ regularization. The advantage of \eqref{eqn:l1DensityMatrix} over \eqref{eq:Pnonconvex} is that the former is a convex problem while the latter is not.
Coming back to the properties of the variational problem \eqref{eqn:l1DensityMatrix}. We note that while the objective function of \eqref{eqn:l1DensityMatrix} is convex, it is not strictly convex as the $\ell_1$-norm is not strictly convex and the trace term is linear. Therefore, in general, the minimizer of \eqref{eqn:l1DensityMatrix} is not unique.
\begin{example} Let $\mu \in \mathbb{R}_+$, $N = 1$ and
\begin{equation}
H =
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix},
\end{equation}
The non-uniqueness comes from the degeneracy of the Hamiltonian
eigenvalues. Any diagonal matrix $P$ with trace $1$ and non-negative
diagonal entries is a minimizer.
\end{example}
\begin{example} Let $\mu = 1$, $N = 1$ and
\begin{equation}
H =
\begin{pmatrix}
1 & 0 & 0 \\
0 & 2 & 2 \\
0 & 2 & 2
\end{pmatrix}
\end{equation}
The non-uniqueness comes from the competition between the trace term
and the $\ell_1$ regularization. The eigenvalues of $H$ are $0, 1$ and
$4$. Straightforward calculation shows that
\begin{equation}
P_0 =
\begin{pmatrix}
0 & 0 & 0 \\
0 & 1/2 & -1/2 \\
0 & -1/2 & 1/2
\end{pmatrix}
\end{equation}
which corresponds to the eigenvector $(0, \sqrt{2}/2,
-\sqrt{2}/2)^{\mathrm{T}}$ associated with eigenvalue $0$ and
\begin{equation}
P_1 =
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\end{equation}
which corresponds to the eigenvector $(1, 0, 0)^{\mathrm{T}}$ associated with
eigenvalue $1$ are both minimizers of the objective function $\tr(HP)
+ \norm{P}_{1}$. Actually, due to convexity, any convex combination of
$P_0$ and $P_1$ is a minimizer too.
\end{example}
It is an open problem under what assumptions that the uniqueness is guaranteed.
\section{Algorithm}\label{sec:Algorithm}
To solve the proposed minimization problem \eqref{eqn:l1DensityMatrix}, we design a fast algorithm based on split Bregman iteration~\cite{Goldstein:2009split}, which comes from the ideas of variables splitting and Bregman iteration~\cite{Osher:2005}. Bregman iteration has attained intensive attention due to its efficiency in many $\ell_1$ related constrained optimization problems~\cite{Yin:2008bregman,yin2013error}. With the help of auxiliary variables, split Bregman iteration iteratively approaches the original optimization problem by computation of several easy-to-solve subproblems. This algorithm popularizes the idea of using operator/variable splitting to solve optimization problems arising from information science. The equivalence of the split Bregman iteration to the alternating direction method of multipliers (ADMM), Douglas-Rachford splitting and augmented Lagrangian method can be found in \cite{Esser:2009CAM,Setzer:2009SSVMCV,Wu:2010SIAM}.
By introducing auxiliary variables $Q = P$ and $R=P$, the optimization problem \eqref{eqn:l1DensityMatrix} is equivalent to
\begin{equation}\label{eq:P_split}
\begin{aligned}
& \min_{P, Q, R \in \mathbb{R}^{n\times n}} \frac{1}{\mu} \norm{Q}_1 + \tr(H P) \\
& \text{s.t.} \ Q = P,\, R = P,\, \tr P = N,\,R = R^{\mathrm{T}},\, 0 \preceq R \preceq I,
\end{aligned}
\end{equation}
which can be iteratively solved by:
\begin{align}
(P^k,Q^k,R^k) &= \arg\min_{P, Q, R \in \mathbb{R}^{n\times n}} \frac{1}{\mu} \norm{Q}_1 + \tr(H P) + \frac{\lambda}{2} \|P - Q + b\|_F^2 + \frac{r}{2}\|P - R + d\|^2_F \\
& \qquad \text{s.t.} \qquad \tr P = N,\,R = R^{\mathrm{T}},\, 0 \preceq R \preceq I, \nonumber \label{eqn:PQR}\\
b^k &= b^{k-1} + P^k - Q^k \\
d^k &= d^{k-1} + P^k - R^k
\end{align}
where variables $b, d$ are essentially Lagrangian multipliers and parameters $r, \lambda$ control the penalty terms.
Solving $P^k, Q^k, R^k$ in \eqref{eqn:PQR} alternatively, we have the following algorithm.
\begin{algorithm}
\label{alg:CM_P}
Initialize $Q^0 = R^0 = P^0 \in\mathcal{C} , b^0 = d^0 = 0$
\While{``not converge"}{
\begin{enumerate}
\item $\displaystyle P^k = \arg\min_{P \in \mathbb{R}^{n\times n}}\tr(H P) + \frac{\lambda}{2} \|P - Q^{k-1} + b^{k-1}\|_F^2 + \frac{r}{2}\|P - R^{k-1} + d^{k-1}\|^2_F, ~ \text{s.t.} ~ \tr P = N $.
\item $\displaystyle Q^k = \arg\min_{Q \in \mathbb{R}^{n\times n}} \frac{1}{\mu}\norm{Q}_1+ \frac{\lambda}{2} \|P^k - Q + b^{k-1}\|_F^2 $.
\item $R^k = \displaystyle\arg\min_{R \in \mathbb{R}^{n\times n}} \frac{r}{2}\|P^{k} - R + d^{k-1}\|^2_F, \quad \text{s.t.} \quad R = R^{\mathrm{T}},\, 0 \preceq R \preceq I$.
\item $b^{k} = b^{k-1} + P^k - Q^k$.
\item $d^{k} = d^{k-1} + P^k - R^k$.
\end{enumerate}
}
\end{algorithm}
Note that the minimization problem in the steps of Algorithm~\ref{alg:CM_P} can be solved explicitly, as follows:
\begin{align}
P^k &= B^k - \frac{\tr (B^k) - N}{n}, \\
& \quad \text{where } \quad B^k = \frac{\lambda}{\lambda+r}(Q^{k-1} - b^{k-1}) + \frac{r}{\lambda+r}(R^{k-1} - d^{k-1}) - \frac{1}{\lambda +r} H \notag \\
Q^k &= \operatorname{Shrink}\left(P^k +b^{k-1},\frac{1}{\lambda\mu}
\right)
= \operatorname{sign}(P^k + b^{k-1})\max\left\{|P^{k} +b^{k-1}| - \frac{1}{\lambda\mu},0\right\} \\
R^k &= V\min\{\max\{D,0\},1\}V^T , \text{ where } [V,~ D] =
\operatorname{eig}(P^k + d^{k-1}).
\end{align}
Starting form any initial guess, the following theorem guarantees that the algorithm converges to one of the minimizers of the variational problem \eqref{eq:P_split}.
\begin{theorem}\label{thm:conv}
The sequence $\big\{(P^k, Q^k, R^k)\big\}_k$ generated by
Algorithm~\ref{alg:CM_P} from any starting point converges to a
minimum of the variational problem \eqref{eq:P_split}.
\end{theorem}
We will prove a slightly more general version of the above
(Theorem~\ref{thm:three}). The idea of the proof follows from the
general framework of analyzing split Bregman iteration,
\textit{i.e.}~alternating direction method of multipliers (ADDM), see
for example \cite{GlowinskiLeTallec:89}. The standard proof needs to
be generalized to cover the current case of ``two level splitting''
and the non-strictly convexity of the functionals. We defer the
detailed proof to Section~\ref{sec:proof}.
\section{Numerical results}\label{sec:numerics}
In this section, numerical experiments are presented to demonstrate
the proposed model \eqref{eqn:l1DensityMatrix} for density matrix
computation using algorithm \ref{alg:CM_P}. We illustrate our
numerical results in three representative cases, free electron model,
Hamiltonian with energy band gap and a non-uniqueness example of the
proposed optimization problem. All numerical experiments are
implemented by \textsf{MATLAB} in a PC with a 16G RAM and a 2.7 GHz CPU.
\subsection{1D Laplacian}
In the first example, we consider the proposed model for the free
electron case, in other words, we consider the potential free
Schr\"{o}dinger operator $-1/2\Delta$ defined on 1D domain $\Omega =
[0,~ 100]$ with periodic boundary condition. This model approximates
the behavior of valence electrons in a metallic solid with weak atomic
pseudopotentials.
In this case, the matrix $H$ is a central difference discretization of
$-1/2\Delta$ on $[0, ~100]$ with equally spaced $256$ points, and we
take $N = 10$.
Figure~\ref{fig:DensityMatrix_Lap}(a) illustrates the true density
matrix $\sum_{i=1}^{10} |\phi_i\rangle \langle \phi_i |$ obtained by
the first $10$ eigenfunctions of $H$. As the free Laplacian does not
have a spectral gap, the density matrix decays slowly in the
off-diagonal direction. Figure~\ref{fig:DensityMatrix_Lap}(b) and
(c) plot the density matrices obtained from the proposed model with
parameter $\mu = 10$ and $100$. Note that they are much localized than
the original density matrix. As $\mu$ gets larger, the variational
problem imposes a smaller penalty on the sparsity, and hence the
solution for $\mu = 100$ has a wider spread than that for $\mu = 10$.
\begin{figure}[ht]
\centering
\includegraphics[width=.5\linewidth]{Lap_DensityM_true.eps}\\
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{Lap_DensityM_10.eps}\\
\end{minipage}\hfill
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{Lap_DensityM_100.eps}
\end{minipage}\hfill
\caption{(a): The true density matrix obtained by the first 10
eigenfunctions of $H$. (b), (c): solutions of the density matrices
with $\mu = 10, 100$ respectively. \label{fig:DensityMatrix_Lap}}
\end{figure}
After we obtain the sparse representation of the density
matrix $P$, we can find localized Wannier functions as its action on
the delta functions, as plotted in Figure~\ref{fig:Projection_Lap} upper and lower pictures for $\mu = 10$ and $100$ respectively.
\begin{figure}[htbp]
\includegraphics[width=.8\linewidth]{Lap_Projection_10.eps}\\
\vspace{0.2cm}
\includegraphics[width=.8\linewidth]{Lap_Projection_100.eps}
\caption{Projection of Delta function $\delta(x - x_i)$ using density
matrices with $\mu = 10$ (upper) and $\mu = 100$ (lower)
respectively.}
\label{fig:Projection_Lap}
\end{figure}
To indicate the approximation behavior of the proposed model, we
consider the energy function approximation of $ \frac{1}{\mu}
\norm{P}_1 + \tr(H P) $ to $\sum_{i=1}^{10} \langle \phi_i | H |
\phi_i \rangle$ with different values of $\mu$. In addition, we define
$\sum_{i=1}^{10} \| \phi_i - P \phi_i\|^2$ as a measurement for the
space approximation of the density matrix $P$ to the lower eigen-space
$Span\{\phi_i\}_{i=1}^{10}$. Figure~\ref{fig:DensityFunApprox_Lap}
reports the energy approximation and the space approximation with
different values of $\mu$. Both numerical results suggest that the
proposed model will converge to the energy states of the
Schr\"{o}dinger operator. We also remark that even though the exact
density matrix is not sparse, a sparse approximation gives fairly good
results in terms of energy and space approximations.
\begin{figure}[h]
\centering
\includegraphics[width=.8\linewidth]{Lap_EnergyFunApprox.eps}\\
\includegraphics[width=.8\linewidth]{Lap_SpaceApprox.eps}\\
\caption{Upper: Energy approximation as a function of $\mu$. Lower: Space
approximation as a function of $\mu$.}
\label{fig:DensityFunApprox_Lap}
\end{figure}
\subsection{1D Hamiltonian operator with a band gap}
We then consider a modified Kronig--Penney (KP)
model~\cite{Kronig:1931quantum} for a one-dimensional insulator. The
original KP model describes the states of independent electrons in a
one-dimensional crystal, where the potential function $V(x)$ consists
of a periodic array of rectangular potential wells. We replace the
rectangular wells with inverted Gaussians so that the potential is
given by
\begin{equation*}
V(x) = -V_0\sum_{j=1}^{N_{\text{at}}} \exp\left[ -\frac{(x -
x_j)^2}{\delta^2} \right],
\end{equation*}
where $N_{\text{at}}$ gives the number of potential wells. In our
numerical experiments, we choose $N_{\text{at}} = 10$ and $x_j = 100 j
/ 11$ for $j = 1, \ldots, N_{\text{at}}$, and the domain is $[0, 100]$
with periodic boundary condition. The potential is plotted in
Figure~\ref{fig:V_KP}(a). For this given potential, the Hamiltonian
operator $H = - \tfrac{1}{2} \Delta + V(x)$ exhibits two low-energy
bands separated by finite gaps from the rest of the eigenvalue
spectrum (See Figure~\ref{fig:V_KP}(b)). Here a centered difference is
used to discretize the Hamiltonian operator.
\begin{figure}[ht]
\centering
\begin{minipage}{0.6\linewidth}
\includegraphics[width=1\linewidth]{KP_PotentialFun.eps}\\
\end{minipage}\hfill
\begin{minipage}{0.39\linewidth}
\includegraphics[width=.9\linewidth]{KP_LapEigs.eps}\\
\end{minipage}\hfill\\
\begin{minipage}{0.6\linewidth}
\centering (a)
\end{minipage}\hfill
\begin{minipage}{0.39\linewidth}
\centering (b)
\end{minipage}
\caption{(a): The potential function in the modified Kronig-Penney
model. (b): The spectrum of the (discretized) Hamiltonian operator.}
\label{fig:V_KP}
\end{figure}
We consider three choices of $N$ for this model: $N = 10$, $N = 15$
and $N = 20$. They correspond to three interesting physical situations
of the model, as explained below.
For $N = 10$, the first band of the Hamiltonian is occupied, and hence
the system has a spectral gap between the occupied and unoccupied
states. As a result, the associated density matrix is exponentially
localized, as shown in Figure~\ref{fig:DensityFunApprox_KP}(a). The
resulting sparse representation from the convex optimization is shown
in Figure~\ref{fig:DensityFunApprox_KP}(b) and (c) for $\mu = 10$ and
$100$ respectively. We see that the sparse representation agrees well
with the exact density matrix, as the latter is very localized. The
Wannier functions obtained by projection of delta functions are shown
in Figure~\ref{fig:Projection_KP}. As the system is an
insulator, we see that the localized representation converges quickly
to the exact answer when $\mu$ increases. This is further confirmed in
Figure~\ref{fig:DensityFunApprox_KP_energy} where the energy
corresponding to the approximated density matrix and space
approximation measurement $\sum_{i=1}^{10} \| \phi_i - P \phi_i\|^2$
are plotted as functions of $\mu$.
\begin{figure}[ht]
\centering
\includegraphics[width=.5\linewidth]{KP_DensityM_true.eps}\\
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{KP_DensityM_10.eps}\\
\end{minipage}\hfill
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{KP_DensityM_100.eps}\\
\end{minipage}\hfill
\caption{(a): The true density matrix obtained by the first 10 eigenfunctions of $H$. (b), (c): solutions of the density matrices with $\mu = 10, 100$ respectively.}
\label{fig:DensityFunApprox_KP}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=.7\linewidth]{KP_Projection_10.eps}\\
\centering
\includegraphics[width=.7\linewidth]{KP_Projection_100.eps}\\
\caption{Projection of Delta function $\delta(x - x_i)$ using density
matrices with $\mu = 10$ (upper) and $\mu = 100$ (lower)
respectively.}
\label{fig:Projection_KP}
\end{figure}
\begin{figure}[ht]
\includegraphics[width=.8\linewidth]{KP_EnergyFunApprox.eps}\\
\centering (a)\\
\includegraphics[width=.8\linewidth]{KP_SpaceApprox.eps}\\
\centering (b)
\caption{(a): Energy approximation as a function of $\mu$. (b): Space
approximation as a function of $\mu$.}
\label{fig:DensityFunApprox_KP_energy}
\end{figure}
Next we consider the case $N = 15$. The first band of $10$ eigenstates
of $H$ is occupied and the second band of $H$ is ``half-filled''. That
is we have only $5$ electrons occupying the $10$ eigenstates of
comparable eigenvalue of $H$. Hence, the system does not have a gap,
which is indicated by the slow decay of the density matrix shown in
Figure~\ref{fig:KP_N15}(a). Nevertheless, the algorithm with $\mu =
100$ gives a sparse representation of the density matrix, which
captures the feature of the density matrix near the diagonal, as shown in Figure~\ref{fig:KP_N15}(b). To understand better the resulting sparse representation, we diagonal the matrix $P$:
\begin{equation*}
P = \sum_{i} f_i \varphi_i \varphi_i^{\mathrm{T}}.
\end{equation*}
The eigenvalues $f_i$, known as the occupation number in the physics
literature, are sorted in the decreasing order. The first $40$
occupation numbers are shown in Figure~\ref{fig:KP_N15}(c). We have
$\sum_i f_i = \tr P = 15$, and we see that $\{f_i\}$ exhibits two
groups. The first $10$ occupation numbers are equal to $1$,
corresponding to the fact that the lowest $10$ eigenstates of the
Hamiltonian operator is occupied. Indeed, if we compare the
eigenvalues of the operator $PH$ with the eigenvalues of $H$, as in
Figure~\ref{fig:KP_N15}(d), we see that the first $10$ low-lying
states are well represented in $P$. This is further confirmed by the filtered density matrix $M_1$ given by the first $10$ eigenstates of $P$ as
\begin{equation*}
M_1 = \sum_{i=1}^{10} f_i \varphi_i \varphi_i^{\mathrm{T}},
\end{equation*}
plotted in Figure~\ref{fig:KP_N15}(e). It is clear that it is very
close to the exact density matrix corresponding to the first $10$
eigenfunctions of $H$, as plotted in
Figure~\ref{fig:DensityFunApprox_KP}(a). The next group of occupation
numbers in Figure~\ref{fig:KP_N15}(c) gets value close to $0.5$. This
indicates that those states are ``half-occupied'', matches very well
with the physical intuition. This is also confirmed by the state
energy shown in Figure~\ref{fig:KP_N15}(d). Note that due to the fact
these states are half filled, the perturbation in the eigenvalue by
the localization is much stronger. The corresponding filtered density
matrix
\begin{equation*}
M_2 = \sum_{i=11}^{20} f_i \varphi_i \varphi_i^{\mathrm{T}},
\end{equation*}
is shown in Figure~\ref{fig:KP_N15}(f).
For this example, we compare with the results obtained using the
variational principle \eqref{eq:Psi} as in
\cite{OzolinsLaiCaflischOsher:13} shown in
Figure~\ref{fig:CMs_KP_N15}. As the variational principle
\eqref{eq:Psi} is formulated with orbital functions $\Psi$, it does
not allow fractional occupations, in contrast with the one in terms of
the density matrix. Hence, the occupation number is either $1$ or $0$,
which is equivalent to the idempotency condition, as shown in
Figure~\ref{fig:CMs_KP_N15}(b). As a result, even though the states in
the second band have very similar energy, the resulting $\Psi$ are
forced to choose five states over the ten, as can be seen from the
Ritz value plotted in Figure~\ref{fig:CMs_KP_N15}(c). The solution is
quite degenerate in this case. Physically, what happens is that the
five electrons choose $5$ wells out of the ten to sit in (on top of
the state corresponding to the first band already in the well), as
shown from the corresponding density matrix in
Figure~\ref{fig:CMs_KP_N15}(a), or more clearly by the filtered
density matrix in Figure~\ref{fig:CMs_KP_N15}(d) for the five higher
energy states.
\begin{figure}[ht]
\centering
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{KP_ExactP_N15.eps}\\
\end{minipage}\hfill
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{KP_PMatrix_N15.eps}\\
\end{minipage}\hfill\\
\begin{minipage}{0.49\linewidth}
\includegraphics[width=.9\linewidth]{KP_P_EigenValue_N15.eps}\\
\end{minipage}\hfill
\begin{minipage}{0.49\linewidth}
\includegraphics[width=.9\linewidth]{KP_StateEnergy_N15.eps}\\
\end{minipage}\hfill\\
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{KP_DensityEigM1_N15.eps}\\
\end{minipage}\hfill
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{KP_DensityEigM2_N15.eps}\\
\end{minipage}
\caption{(a): The true density matrix corresponds to the first $15$
eigenfunctions of $H$. (b): The sparse representation $P$ of the
density matrix for $\mu = 100$. (c): The occupation number
(eigenvalues) of $P$. (d) The first $15$ eigenvalues of $PH$
compared with the eigenvalues of $H$. (e): The filtered density
matrix $M_1$ corresponds to the first $10$ eigenstates of $P$. (f)
The filtered density matrix $M_2$ corresponds to the next $10$
eigenstates of $P$.\label{fig:KP_N15}}
\end{figure}
\begin{figure}[ht]
\centering
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{CMs_KP_PMatrix_N15.eps}\\
\end{minipage}\hfill
\begin{minipage}{0.49\linewidth}
\includegraphics[width=.9\linewidth]{CMs_KP_P_EigenValue_N15.eps}\\
\end{minipage}\hfill\\
\begin{minipage}{0.49\linewidth}
\includegraphics[width=.9\linewidth]{CMs_KP_StateEnergy_N15.eps}\\
\end{minipage}\hfill
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{CMs_KP_DensityEigM2_N15.eps}\\
\end{minipage}
\caption{Results obtained by the first 15 Compressed modes $\Psi =
\{\psi_i\}_{i=1}^{15}$ for $\mu = 100$. (a): The density
representation $P$ given by $P = \Psi^T\Psi $. (b): The occupation
number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of
$\Psi^T H \Psi$ compared with the eigenvalues of $H$. (d) The
filtered density matrix $M_2$ corresponds to the $5$ states in the
second band.
\label{fig:CMs_KP_N15}}
\end{figure}
Finally, the $N = 20$ case corresponds to the physical situation that
the first two bands are all occupied. Note that as the band gap
between the second band from the rest of the spectrum is smaller than
the gap between the first two bands, the density matrix, while still
exponentially localized, has a slower off diagonal decay rate. The
exact density matrix corresponds to the first $20$ eigenfunctions of
$H$ is shown in Figure~\ref{fig:KP_N20}(a), and the localized
representation with $\mu = 100$ is given in
Figure~\ref{fig:KP_N20}(b). The occupation number is plotted in
Figure~\ref{fig:KP_N20}(c), indicates that the first $20$ states are
fully occupied, while the rest of the states are empty. This is
further confirmed by comparison of the eigenvalues given by $HP$ and
$H$, shown in Figure~\ref{fig:KP_N20}(d). In this case, we see that
physically, each well contains two states. Hence, if we look at the
electron density, which is diagonal of the density matrix, we see a
double peak in each well. Using the projection of delta functions, we
see that the sparse representation of the density matrix $P$
automatically locate the two localized orbitals centered at the two
peaks, as shown in Figure~\ref{fig:KP_N20}(e).
\begin{figure}[ht]
\centering
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{KP_ExactP_N20.eps}\\
\end{minipage}\hfill
\begin{minipage}{0.49\linewidth}
\includegraphics[width=1\linewidth]{KP_PMatrix_N20.eps}\\
\end{minipage}\hfill\\
\begin{minipage}{0.49\linewidth}
\includegraphics[width=.9\linewidth]{KP_P_EigenValue_N20.eps}\\
\end{minipage}\hfill
\begin{minipage}{0.49\linewidth}
\includegraphics[width=.9\linewidth]{KP_StateEnergy_N20.eps}\\
\end{minipage}\hfill\\
\centering
\includegraphics[width=1\linewidth]{KP_Projection_N20.eps}\\
\caption{(a): The true density matrix
corresponds to the first $20$ eigenfunctions of $H$. (b): The sparse
representation $P$ of the density matrix for $\mu = 100$. (c): The
occupation number (eigenvalues) of $P$. (d) The first $20$
eigenvalues of $PH$ compared with the eigenvalues of $H$. (e)
Projection of Delta function $\delta(x - x_i)$.}
\label{fig:KP_N20}
\end{figure}
\subsection{An example of non-unique minimizers}
Let us revisit the Example~$2$ in Section~\ref{sec:formulation} for
which the minimizers to the variational problem is
non-unique. Theorem~\ref{thm:conv} guarantees that the algorithm will
converge to some minimizer starting from any initial condition.
It is easy to check that in this case
\begin{equation}
P^{\ast} = Q^{\ast} = R^{\ast} =
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix},
\quad
b^{\ast} = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & -1 \\
0 & -1 & 1
\end{pmatrix}, \quad d^{\ast} = \begin{pmatrix}
0 & 0 & 0 \\
0 & -1 & -1 \\
0 & -1 & -1
\end{pmatrix}
\end{equation}
is a fixed point of the algorithm. In Figure~\ref{fig:DecayDist}, we
plot the sequence $\Big\{ \lambda \|b^{k} - b^{\ast}\|^2 + r \|d^{k} -
d^{\ast}\|^2 + \lambda \| Q^{k} - Q^{\ast} \|^2 + r \| R^{k} -
R^{\ast} \|^2 \Big\}_k$ for a randomly chosen initial data. We see
that the distance does not converge to $0$ as the algorithm converges
to another minimizer of the variational problem. Nonetheless, as will
be shown in the proof of Theorem~\ref{thm:conv} in
Section~\ref{sec:proof}, the sequence is monotonically non-increasing.
\begin{figure}[ht]
\centering
\includegraphics[width=.8\linewidth]{DecayDist.eps}
\caption{$\lambda \|b^{k} - b^{\ast}\|^2 + r \|d^{k} -
d^{\ast}\|^2 + \lambda \| Q^{k} - Q^{\ast} \|^2 + r \| R^{k} -
R^{\ast} \|^2$ as a function of $k$ for Algorithm~\ref{alg:CM_P}.}\label{fig:DecayDist}
\end{figure}
\section{Convergence of Algorithm~\ref{alg:CM_P}}
\label{sec:proof}
For ease of notation, we will prove the convergence of the algorithm for the following slightly generalized variational problem.
\begin{equation}\label{eq:three}
\begin{aligned}
& \min_{P,Q,R} f(P) + g(Q) + h(R) \\
& \text{s.t.} \; P = Q, P = R
\end{aligned}
\end{equation}
where $f$, $g$, and $h$ are proper convex functionals, but not
necessarily strictly convex. In particular, we will get \eqref{eq:P_split} if we set
\begin{align*}
& f(P) =
\begin{cases}
\tr(H P), & \text{if } \tr P = N; \\
+ \infty, & \text{otherwise},
\end{cases} \\
& g(Q) = \norm{Q}_1 / \mu \\
& h(R) =
\begin{cases}
0, & \text{if } R = R^{\mathrm{T}}, \text{and } 0 \preceq R \preceq I; \\
+ \infty, & \text{otherwise}.
\end{cases}
\end{align*}
The corresponding algorithm for \eqref{eq:three} is given by
\begin{algorithm}\label{alg:split2}
Initialize $P^0 = Q^0 = R^0, b^0 = d^0 = 0$
\While{``not converge"}{
\begin{enumerate}
\item $\displaystyle P^k = \arg\min_{P} f(P) + \frac{\lambda}{2} \norm{P - Q^{k-1} + b^{k-1}}^2 + \frac{r}{2} \norm{P - R^{k-1} + d^{k-1}}^2$,
\item $\displaystyle Q^k = \arg\min_{Q} g(Q) + \frac{\lambda}{2} \norm{P^k - Q + b^{k-1}}^2 $.
\item $R^k = \displaystyle\arg\min_{R} h(R) + \frac{r}{2}\norm{P^{k} - R + d^{k-1}}^2$.
\item $b^{k} = b^{k-1} + P^k - Q^k$.
\item $d^{k} = d^{k-1} + P^k - R^k$.
\end{enumerate}
}
\end{algorithm}
We define an augmented Lagrangian
\begin{multline}\label{eq:Lag}
\mc{L}(P,Q,R;b,d) = f(P) + g(Q) + h(R) + \frac{\lambda}{2}\|P - Q\|^2 + \lambda\langle P - Q, b\rangle \\
+ \frac{r}{2}\|P - R\|^2 + r\langle P - R, d\rangle
\end{multline}
\begin{defn}
We call $(P^{\ast}, Q^{\ast}, R^{\ast}; b^{\ast}, d^{\ast})$ a
\emph{saddle point} of the Lagrangian \eqref{eq:Lag}, if
\begin{equation} \label{eq:saddle}
\mc{L}(P^{\ast}, Q^{\ast}, R^{\ast}; b, d) \leq \mc{L}(P^{\ast}, Q^{\ast}, R^{\ast}; b^{\ast}, d^{\ast}) \leq \mc{L}(P, Q, R; b^{\ast}, d^{\ast})
\end{equation}
for any $(P, Q, R; b, d) \in \mathbb{R}^{n\times n} \times \mathbb{R}^{n\times n} \times \mathbb{R}^{n\times n}\times \mathbb{R}^{n\times n}\times \mathbb{R}^{n\times n}$.
\end{defn}
\begin{lemma} $(P^{*}, Q^{*}, R^{*})$ is a solution of the optimization problem \eqref{eq:three} if any only if there exist $b^{*}, d^{*} \in \mathbb{R}^{n\times n}$ such that $(P^{*}, Q^{*}, R^{*};b^{*}, d^{*})$ is a saddle point satisfying \eqref{eq:saddle}
\end{lemma}
\begin{proof} Given a saddle point $(P^{*}, Q^{*}, R^{*};b^{*}, d^{*})$ satisfying \eqref{eq:saddle}, it is clear that the first inequality in \eqref{eq:saddle} implies $P^{*} = Q^{*} = R^{*}$. Substitute $P = Q = R$ in the second inequality of \eqref{eq:saddle}, we can immediately have $(P^{*}, Q^{*}, R^{*})$ is a minimizer \eqref{eq:three}.
On the other hand, suppose $(P^{*}, Q^{*}, R^{*})$ is a solution of \eqref{eq:three}. The first inequality in \eqref{eq:saddle} holds since $P^{*} = Q^{*} = R^{*}$. Moreover, there exist $b^{*}, d^{*}$ such that
\begin{equation*}
-\lambda b^{*} - r d^{*} \in \partial f(P^{*}), \qquad \lambda b^{*} \in \partial g(Q^{*}), \qquad r d^{*} \in \partial h(R^{*})
\end{equation*}
which suggests, for any $P, Q, R \in \mathbb{R}^{n\times n}$
\begin{align*}
f(P^{*}) &\leq f(P) + \lambda \langle b^{*}, P - P^{*} \rangle + r \langle d^{*}, P - P^{*} \rangle \nonumber \\
g(Q^{*}) &\leq g(Q) - \lambda \langle b^{*}, Q - Q^{*} \rangle \nonumber \\
h(R^{*}) &\leq h(R) - r \langle d^{*}, R - R^{*} \rangle \nonumber
\end{align*}
The summation of the above three inequalities yield the second inequality in \eqref{eq:saddle}.
\end{proof}
\begin{theorem}\label{thm:three}
The sequence $\Big\{(P^k, Q^k, R^k)\Big\}_k$ generated by
Algorithm~\ref{alg:split2} from any starting point converges to a
minimum of the variational problem \eqref{eq:three}.
\end{theorem}
\begin{remark}
We remind the readers that the minimizers of the variational
principle \eqref{eq:three} might not be unique. In the non-unique
case, the above theorem states that any initial condition will
converge to some minimizer, while different initial condition might
give different minimizers.
\end{remark}
\begin{proof}
Let $(P^{\ast}, Q^{\ast}, R^{\ast})$ be an optimal solution of \eqref{eq:three}.
We introduce the short hand notations
\begin{equation}
\begin{split}
& \wb{P}^k = P^k - P^{*}, \quad \wb{Q}^k = Q^k - Q^{*}, \quad
\text{and} \quad \wb{R}^k = R^k - R^{*}. \\
& \wb{b}^k = b^k - b^{*}, \quad \wb{d}^k = d^k - d^{*}.
\end{split}
\end{equation}
From Step $4$ and $5$ in the algorithm, we get
\begin{equation}
\wb{b}^k = \wb{b}^{k-1} + \wb{P}^{k} - \wb{Q}^{k}, \quad \text{and} \quad
\wb{d}^k = \wb{d}^{k-1} + \wb{P}^{k} - \wb{R}^{k},
\end{equation}
and hence
\begin{equation} \label{eqn:bderror}
\begin{split}
& \|\wb{b}^{k-1}\|^2 - \|\wb{b}^{k}\|^2 = -2\langle \wb{b}^{k-1}, \wb{P}^{k} - \wb{Q}^{k} \rangle - \|\wb{P}^{k} - \wb{Q}^{k}\|^2 \\
& \norm{\wb{d}^{k-1}}^2 - \|\wb{d}^{k}\|^2 = - 2\langle
\wb{d}^{k-1}, \wb{P}^{k} - \wb{R}^{k} \rangle - \|\wb{P}^{k}
- \wb{R}^{k}\|^2
\end{split}
\end{equation}
Note that by optimality
\begin{align}
P^{*} = \arg\min_{P\in \mathcal{C}_P} \mathcal{L}(P,Q^{*},R^{*};b^{*},d^{*}) \\
Q^{*} = \arg\min_{Q\in \mathcal{C}_Q} \mathcal{L}(P^{*},Q,R^{*};b^{*},d^{*}) \\
R^{*} = \arg\min_{R\in \mathcal{C}_R} \mathcal{L}(P^{*},Q^{*},R;b^{*},d^{*})
\end{align}
Hence, for any $P, Q, R\in \mathbb{R}^{n\times n}$, we have
\begin{align}
& f(P) - f(P^{*}) + \lambda\langle P^{*} - Q^{*} + b^{*}, P - P^{*} \rangle + r \langle P^{*} - R^{*} + d^{*}, P - P^{*} \rangle \geq 0 \label{eqn:fPs}\\
& g(Q) - g(Q^{*}) + \lambda\langle Q^{*} - P^{*} - b^{*}, Q - Q^{*} \rangle \geq 0 \label{eqn:gQs} \\
& h(R) - h(R^{*}) + r \langle R^{*} - P^{*} - d^{*}, R - R^{*} \rangle \geq 0 \label{eqn:hRs}
\end{align}
According to the construction of $\{P^k, Q^k, R^k\}$, for any $P, Q, R \in \mathbb{R}^{n\times n}$, we have
\begin{align}
&
\begin{aligned}
f(P) - f(P^{k}) & + \lambda\langle P^{k} - Q^{k-1} + b^{k-1}, P - P^{k} \rangle \\
& + r \langle P^{k} - R^{k} + d^{k-1}, P - P^{k} \rangle \geq 0
\end{aligned} \label{eqn:fPk}\\
& g(Q) - g(Q^{k}) + \lambda\langle Q^{k}- P^{k} - b^{k-1}, Q - Q^{k} \rangle \geq 0 \label{eqn:gQk} \\
& h(R) - h(R^{k}) + r \langle R^{k} - P^{k} - d^{k-1}, R - R^{k}
\rangle \geq 0 \label{eqn:hRk}
\end{align}
Let $P = P^{k}$ in \eqref{eqn:fPs} and $P = P^{*}$ in \eqref{eqn:fPk}, their summation yields
\begin{equation}\label{eqn:fPsk}
\lambda\langle -\wb{P}^{k} + \wb{Q}^{k-1} - \wb{b}^{k-1}, \wb{P}^{k} \rangle + r \langle -\wb{P}^{k} + \wb{R}^{k-1} - \wb{d}^{k-1}, \wb{P}^{k} \rangle \geq 0
\end{equation}
Similarly, let $Q = Q^{k}$ in \eqref{eqn:gQs} and $Q = Q^{*}$ in \eqref{eqn:gQk}, and let $R = R^{k}$ in \eqref{eqn:hRs} and $R = R^{*}$ in \eqref{eqn:hRk}, we obtain
\begin{align}
& \lambda\langle -\wb{Q}^{k} + \wb{P}^{k} + \wb{b}^{k-1},
\wb{Q}^{k} \rangle \geq 0 \label{eqn:gQsk} \\
& r \langle -\wb{R}^{k} + \wb{P}^{k} + \wb{d}^{k-1}, \wb{R}^{k}
\rangle \geq 0 \label{eqn:hRsk}
\end{align}
The summation of \eqref{eqn:fPsk}, \eqref{eqn:gQsk}, and
\eqref{eqn:hRsk} yields
\begin{multline}
\lambda\langle - \wb{b}^{k-1}, \wb{P}^{k} \rangle + \lambda\langle
\wb{Q}^{k-1} -\wb{P}^{k} , \wb{P}^{k} \rangle + r \langle -
\wb{d}^{k-1}, \wb{P}^{k} \rangle + r \langle \wb{R}^{k-1} -\wb{P}^{k},
\wb{P}^{k} \rangle \\
+ \lambda \langle \wb{b}^{k-1}, \wb{Q}^{k} \rangle + \lambda\langle
\wb{P}^{k} - \wb{Q}^{k}, \wb{Q}^{k} \rangle + r \langle \wb{d}^{k-1},
\wb{R}^{k} \rangle + r \langle \wb{P}^{k} - \wb{R}^{k}, \wb{R}^{k}
\rangle \geq 0.
\end{multline}
This gives us, after organizing terms
\begin{multline}
-\lambda\langle \wb{b}^{k-1}, \wb{P}^{k} -\wb{Q}^{k} \rangle - \lambda \|\wb{Q}^{k} - \wb{P}^{k}\|^2 - \lambda\langle \wb{P}^{k} , \wb{Q}^{k} -\wb{Q}^{k-1} \rangle \\
-r\langle \wb{d}^{k-1}, \wb{P}^{k} -\wb{R}^{k} \rangle - r
\|\wb{R}^{k} - \wb{P}^{k}\|^2 - r \langle \wb{P}^{k},
\wb{R}^{k} -\wb{R}^{k-1} \rangle \geq 0
\end{multline}
Combining the above inequality with \eqref{eqn:bderror}, we have
\begin{equation}\label{eqn:Monotone1}
\begin{aligned}
\bigl(\lambda \|\wb{b}^{k-1}\|^2 + r \|\wb{d}^{k-1}\|^2 \bigr) & - \bigl(\lambda \|\wb{b}^{k}\|^2 + r \|\wb{d}^{k}\|^2 \bigr) \\
& = \lambda \bigl(-2\langle \wb{b}^{k-1}, \wb{P}^{k} - \wb{Q}^{k} \rangle - \|\wb{P}^{k} - \wb{Q}^{k}\|^2\bigr) \\
& \qquad + r \bigl(-2\langle \wb{d}^{k-1}, \wb{P}^{k} - \wb{R}^{k} \rangle - \|\wb{P}^{k} - \wb{R}^{k}\|^2 \bigr) \\
& \geq \lambda \|\wb{Q}^{k} - \wb{P}^{k}\|^2 + 2 \lambda\langle \wb{P}^{k}, \wb{Q}^{k} -\wb{Q}^{k-1}\rangle \\
& \qquad + r \|\wb{R}^{k} - \wb{P}^{k}\|^2 + 2 r \langle
\wb{P}^{k}, \wb{R}^{k} -\wb{R}^{k-1} \rangle
\end{aligned}
\end{equation}
Now, we calculate $ \langle \wb{P}^{k} , \wb{Q}^{k} -\wb{Q}^{k-1}\rangle$. It is clear that
\begin{multline}\label{eqn:ErrorPQ1}
\langle \wb{P}^{k} , \wb{Q}^{k} -\wb{Q}^{k-1}\rangle = \langle \wb{P}^{k} - \wb{P}^{k-1} , \wb{Q}^{k} -\wb{Q}^{k-1}\rangle + \langle \wb{P}^{k-1} - \wb{Q}^{k-1} , \wb{Q}^{k} -\wb{Q}^{k-1}\rangle \\
+ \langle \wb{Q}^{k-1} , \wb{Q}^{k} -\wb{Q}^{k-1}\rangle
\end{multline}
Note that $\displaystyle Q^{k-1} = \arg\min_{Q} g(Q) + \frac{\lambda}{2} \|Q - P^{k-1} - b^{k-2} \|^2 $. Thus, for any $Q\in \mathbb{R}^{n\times n}$, we have
\begin{eqnarray}
g(Q) - g(Q^{k-1}) + \lambda\langle Q^{k-1} - P^{k-1} - b^{k-2}, Q - Q^{k-1} \rangle \geq 0
\end{eqnarray}
In particular, let $Q = Q^{k}$, we have
\begin{eqnarray} \label{eqn:gQkk1}
g(Q^{k}) - g(Q^{k-1}) + \lambda\langle Q^{k-1} - P^{k-1} - b^{k-2}, Q^{k} - Q^{k-1} \rangle \geq 0
\end{eqnarray}
On the other hand, set $Q = Q^{k-1}$ in \eqref{eqn:gQk}, we get
\begin{eqnarray} \label{eqn:gQkk2}
g(Q^{k-1}) - g(Q^{k}) + \lambda\langle Q^{k}- P^{k} - b^{k-1}, Q^{k-1} - Q^{k} \rangle \geq 0
\end{eqnarray}
The summation of \eqref{eqn:gQkk1} and \eqref{eqn:gQkk2} yields
\begin{equation}
\langle b^{k-1} - b^{k-2}, Q^{k} - Q^{k-1} \rangle + \langle Q^{k-1} - Q^{k} + P^{k} - P^{k-1} , Q^{k} - Q^{k-1} \rangle \geq 0
\end{equation}
Note that $P^{k} - P^{k-1} = \wb{P}^{k} - \wb{P}^{k-1}, Q^{k} - Q^{k-1} = \wb{Q}^{k} - \wb{Q}^{k-1}, b^{k-1} - b^{k-2} = \wb{P}^{k-1} - \wb{Q}^{k-1}$,
thus we have
\begin{equation}
\langle \wb{P}^{k-1} - \wb{Q}^{k-1}, \wb{Q}^{k} - \wb{Q}^{k-1} \rangle + \langle \wb{P}^{k} - \wb{P}^{k-1} , \wb{Q}^{k} - \wb{Q}^{k-1} \rangle \geq \|\wb{Q}^{k} - \wb{Q}^{k-1}\|^2
\label{eqn:ErrorPQ2}
\end{equation}
Combine \eqref{eqn:ErrorPQ2} with \eqref{eqn:ErrorPQ1}, we have
\begin{equation}
\langle \wb{P}^{k}, \wb{Q}^{k} - \wb{Q}^{k-1} \rangle \geq \| \wb{Q}^{k} - \wb{Q}^{k-1} \|^2 + \langle \wb{Q}^{k-1}, \wb{Q}^{k} - \wb{Q}^{k-1} \rangle
\label{eqn:ErrorPQ3}
\end{equation}
Similarly, we have
\begin{equation}
\langle \wb{P}^{k}, \wb{R}^{k} - \wb{R}^{k-1} \rangle \geq \| \wb{R}^{k} - \wb{R}^{k-1} \|^2 + \langle \wb{R}^{k-1}, \wb{R}^{k} - \wb{R}^{k-1} \rangle
\label{eqn:ErrorPR3}
\end{equation}
Substitute \eqref{eqn:ErrorPQ3} and \eqref{eqn:ErrorPR3} into
\eqref{eqn:Monotone1}, we have
\begin{equation} \label{eqn:Monotone2}
\begin{aligned}
\bigl(\lambda \|\wb{b}^{k-1}\|^2 + r \|\wb{d}^{k-1}\|^2 ) & - (\lambda \|\wb{b}^{k}\|^2 + r \|\wb{d}^{k}\|^2 ) \\
& \geq \lambda \|\wb{Q}^{k} - \wb{P}^{k}\|^2 + 2 \lambda\langle \wb{P}^{k}, \wb{Q}^{k} - \wb{Q}^{k-1}\rangle \\
& \qquad + r \|\wb{R}^{k} - \wb{P}^{k}\|^2 + 2 r \langle \wb{P}^{k} , \wb{R}^{k} - \wb{R}^{k-1} \rangle \\
& \geq \lambda \|\wb{Q}^{k} - \wb{P}^{k}\|^2 + 2 \lambda
\bigl(\| \wb{Q}^{k} - \wb{Q}^{k-1} \|^2 + \langle \wb{Q}^{k-1}, \wb{Q}^{k} - \wb{Q}^{k-1} \rangle \bigr) \\
& \qquad + r \|\wb{R}^{k} - \wb{P}^{k}\|^2 + 2 r \bigl(\| \wb{R}^{k} - \wb{R}^{k-1} \|^2 + \langle \wb{R}^{k-1}, \wb{R}^{k} - \wb{R}^{k-1} \rangle\bigr) \\
&= \lambda \|\wb{Q}^{k} - \wb{P}^{k}\|^2 + \lambda \bigl(\| \wb{Q}^{k} \|^2 - \| \wb{Q}^{k-1} \|^2 + \| \wb{Q}^{k} - \wb{Q}^{k-1} \|^2 \bigr) \\
& \qquad + r \|\wb{R}^{k} - \wb{P}^{k}\|^2 + r \bigl(\|
\wb{R}^{k} \|^2 - \| \wb{R}^{k-1} \|^2 + \| \wb{R}^{k} -
\wb{R}^{k-1} \|^2 \bigr)
\end{aligned}
\end{equation}
which yields
\begin{multline} \label{eqn:Monotone3}
\bigl(\lambda \|\wb{b}^{k-1}\|^2 + r \|\wb{d}^{k-1}\|^2 + \lambda \| \wb{Q}^{k-1} \|^2 + r \| \wb{R}^{k-1} \|^2 \bigr) \\
- \bigl(\lambda \|\wb{b}^{k}\|^2 + r \|\wb{d}^{k}\|^2 + \lambda \| \wb{Q}^{k} \|^2 + r \| \wb{R}^{k} \|^2 \bigr) \\
\geq \lambda \|\wb{Q}^{k} - \wb{P}^{k}\|^2 + \lambda \| \wb{Q}^{k} -
\wb{Q}^{k-1} \|^2 + r \|\wb{R}^{k} - \wb{P}^{k}\|^2 + r \|
\wb{R}^{k} - \wb{R}^{k-1} \|^2
\end{multline}
This concludes that the sequence $\Big\{ \lambda \|\wb{b}^{k}\|^2 + r
\|\wb{d}^{k}\|^2 + \lambda \| \wb{Q}^{k} \|^2 + r \| \wb{R}^{k} \|^2
\Big\}_k$ is non-increasing and hence convergent.
This further implies,
\renewcommand{\labelenumi}{(\alph{enumi})}
\begin{enumerate}
\item $\{P^k\}_k, \{Q^k\}_k, \{R^k\}_k, \{b^k\}_k, \{d^k\}_k$ are all bounded sequences, and hence the sequences has limit points.
\item $\lim_{k\rightarrow\infty} \| Q^k - P^k\| = 0$ and $\lim_{k\rightarrow\infty} \| R^k - P^k\| = 0$.
\end{enumerate}
Therefore, the sequences have limit points. Let us denote $(\wt{P},
\wt{Q}, \wt{R}; \wt{b}, \wt{d})$ as a limit point, that is, a
subsequence converges
\begin{equation}
\lim_{j\to \infty} (P^{k_j}, Q^{k_j}, R^{k_j}; b^{k_j}, d^{k_j})
= (\wt{P}, \wt{Q}, \wt{R}; \wt{b}, \wt{d}).
\end{equation}
We now prove that $(\wt{P}, \wt{Q}, \wt{R})$ is a minimum of the variational problem \eqref{eq:three}, \textit{i.e.}
\begin{equation}\label{eq:limitPQR}
f(\wt{P}) + g(\wt{Q}) + h(\wt{R}) = \lim_{j\rightarrow \infty} f(P^{k_j}) + g(Q^{k_j}) + h(R^{k_j}) = f(P^{*}) + g(Q^{*}) + h(R^{*})
\end{equation}
First note that since $(P^{*}, Q^{*}, R^{*}; b^{*}, d^{*})$ is a saddle point, we have
\begin{multline}
f(P^{*}) + g(Q^{*}) + h(R^{*}) \leq f(P^{k_j}) + g(Q^{k_j}) + h(R^{k_j}) + \frac{\lambda}{2}\|P^{k_j} - Q^{k_j}\|^2 \\
+ \lambda\langle P^{k_j} - Q^{k_j}, b^{*} \rangle + \frac{r}{2}\|P^{k_j} - R^{k_j}\|^2 + r\langle P^{k_j} - R^{k_j}, d^{*} \rangle
\end{multline}
Taking the limit $j \to \infty$, we get
\begin{equation}
f(P^{*}) + g(Q^{*}) + h(R^{*}) \leq f(\wt{P}) + g(\wt{Q}) + h(\wt{R}).
\end{equation}
On the other hand, taking $P = P^{*}$, $Q = Q^{*}$, and $R = R^{*}$ in \eqref{eqn:fPk}--\eqref{eqn:hRk}, we get
\begin{align*}
f(P^{*}) & + g(Q^{*}) + h(R^{*}) \\
& \geq f(P^{k_j}) + g(Q^{k_j}) + h(R^{k_j}) - \lambda\langle P^{k_j} - Q^{k_j-1} + b^{k_j-1}, P^{*} - P^{k_j} \rangle \\
& \qquad - r \langle P^{k_j} - R^{k_j} + d^{k_j-1}, P^{*} - P^{k_j} \rangle - \lambda\langle Q^{k_j}- P^{k_j} - b^{k_j-1}, Q^{*} - Q^{k_j} \rangle \\
& \qquad - r \langle R^{k_j} - P^{k_j} - d^{k_j-1}, R^{*} - R^{k_j}
\rangle \\
&= f(P^{k_j}) + g(Q^{k_j}) + h(R^{k_j}) \\
& \qquad - \lambda\langle b^{k_j-1}, Q^{k_j} - P^{k_j} \rangle - \lambda\langle P^{k_j} - Q^{k_j-1} , P^{*} - P^{k_j} \rangle - \lambda\langle Q^{k_j}- P^{k_j} , Q^{*} - Q^{k_j} \rangle \\
& \qquad - r \langle d^{k_j-1}, R^{k_j} - P^{k_j} \rangle - r \langle
P^{k_j} - R^{k_j} , P^{*} - P^{k_j} \rangle - r \langle R^{k_j} -
P^{k_j} , R^{*} - R^{k_j} \rangle
\end{align*}
From \eqref{eqn:Monotone3}, we have $\{P^{k_j}\}, \{Q^{k_j}\}, \{R^{k_j}\}, \{b^{k_j}\}, \{d^{k_j}\}$ are all bounded sequences, and furthermore,
\begin{eqnarray}
\lim_{j\rightarrow\infty} \| Q^{k_j }- P^{k_j}\| = 0, \qquad \lim_{j\rightarrow\infty} \| Q^{k_j }- Q^{k_j - 1}\| = 0.\nonumber \\
\lim_{j\rightarrow\infty} \| R^{k_j} - P^{k_j}\| = 0, \qquad \lim_{j\rightarrow\infty} \| R^{k_j }- R^{k_j - 1}\| = 0. \nonumber
\end{eqnarray}
Taking the limit $j \to \infty$, we then get
\begin{equation}
f(P^{*}) + g(Q^{*}) + h(R^{*}) \geq f(\wt{P}) + g(\wt{Q}) + h(\wt{R}).
\end{equation}
Hence, the limit point is a minimizer of the variational principle.
Finally, repeating the derivation of \eqref{eqn:Monotone3} by replacing $(P^{*}, Q^{*}, R^{*})$ by $(\wt{P}, \wt{Q}, \wt{R})$, we get convergence of the whole sequence due to the monotonicity.
\end{proof}
\begin{bibdiv}
\begin{biblist}
\bib{BaroniGiannozzi:92}{article}{
author={Baroni, S.},
author={Giannozzi, P.},
title={Towards very large-scale electronic-structure calculations},
date={1992},
journal={Europhys. Lett.},
volume={17},
pages={547\ndash 552},
}
\bib{CandesStrohmerVoroninski:13}{article}{
author={Candes, E.~J.},
author={Strohmer, T.},
author={Voroninski, V.},
title={Phase{L}ift: exact and stable recovery from magnitude
measurements via convex programming},
date={2013},
journal={Comm. Pure Appl. Math.},
volume={66},
pages={1241\ndash 1274},
}
\bib{Cloizeaux:64b}{article}{
author={des Cloizeaux, J.},
title={Analytical properties of $n$-dimensional energy bands and
{W}annier functions},
date={1964},
journal={Phys. Rev.},
volume={135},
pages={A698\ndash A707},
}
\bib{Cloizeaux:64a}{article}{
author={des Cloizeaux, J.},
title={Energy bands and projection operators in a crystal: analytic and
asymptotic properties},
date={1964},
journal={Phys. Rev.},
volume={135},
pages={A685\ndash A697},
}
\bib{E:2010PNAS}{article}{
author={E, W.},
author={Li, T.},
author={Lu, J.},
title={Localized bases of eigensubspaces and operator compression},
date={2010},
journal={Proc Natl Acad Sci U S A},
volume={107},
number={1273--1278},
}
\bib{ELu:CPAM}{article}{
author={E, W.},
author={Lu, J.},
title={The electronic structure of smoothly deformed crystals:
{C}auchy-{B}orn rule for nonlinear tight-binding model},
date={2010},
journal={Comm. Pure Appl. Math.},
volume={63},
pages={1432\ndash 1468},
}
\bib{ELu:13}{article}{
author={E, W.},
author={Lu, J.},
title={The {K}ohn-{S}ham equation for deformed cyrstals},
date={2013},
journal={Mem. Amer. Math. Soc.},
volume={221},
number={1040},
}
\bib{Esser:2009CAM}{article}{
author={Esser, E.},
title={Applications of lagrangian-based alternating direction methods
and connections to split bregman},
date={2009},
journal={UCLA CAM Report (09-31)},
}
\bib{Garcia-CerveraLuXuanE:09}{article}{
author={Garcia-Cervera, C.~J.},
author={Lu, J.},
author={Xuan, Y.},
author={E, W.},
title={A liear scaling subspace iteration algorithm with optimally
localized non-orthogonal wave functions for {K}ohn-{S}ham density functional
theory},
date={2009},
journal={Phys. Rev. B},
volume={79},
pages={115110},
}
\bib{Goedecker:99}{article}{
author={Geodecker, S.},
title={Linear scaling electronic structure methods},
date={1999},
journal={Rev. Mod. Phys.},
volume={71},
pages={1085\ndash 1123},
}
\bib{GlowinskiLeTallec:89}{book}{
author={Glowinski, R.},
author={Le~Tallec, P.},
title={Augmented {L}agrangian and operator-splitting methods in
nonlinear mechanics},
publisher={SIAM},
address={Philadelphia},
date={1989},
}
\bib{Goldstein:2009split}{article}{
author={Goldstein, T.},
author={Osher, S.},
title={The split {B}regman method for {L}1-regularized problems},
date={2009},
journal={SIAM Journal on Imaging Sciences},
volume={2},
number={2},
pages={323\ndash 343},
}
\bib{Kivelson:82}{article}{
author={Kivelson, S.},
title={Wannier functions in one-dimensional disordered systems:
{A}pplication to fractionally charged solitons},
date={1982},
journal={Phys. Rev. B},
volume={26},
pages={4269\ndash 4277},
}
\bib{Kohn:1959Wannier}{article}{
author={Kohn, W},
title={{Analytic Properties of Bloch Waves and Wannier Functions}},
date={1959},
journal={Physical Review},
volume={115},
number={4},
pages={809\ndash 821},
}
\bib{Kohn:59}{article}{
author={Kohn, W.},
title={Analytic properties of {B}loch waves and {W}annier functions},
date={1959},
journal={Phys. Rev.},
volume={115},
pages={809\ndash 821},
}
\bib{Kronig:1931quantum}{article}{
author={Kronig, R de~L},
author={Penney, WG},
title={Quantum mechanics of electrons in crystal lattices},
date={1931},
journal={Proceedings of the Royal Society of London. Series A},
volume={130},
number={814},
pages={499\ndash 513},
}
\bib{Lai:2014splitting}{article}{
author={Lai, R.},
author={Osher, S.},
title={A splitting method for orthogonality constrained problems},
date={2014},
journal={Journal of Scientific Computing},
volume={58},
number={2},
pages={431\ndash 449},
}
\bib{LiNunesVanderbilt:93}{article}{
author={Li, X.-P.},
author={Nunes, R.~W.},
author={Vanderbilt, D.},
title={Density-matrix electronic-structure method with linear
system-size scaling},
date={1993},
journal={Phys. Rev. B},
volume={47},
pages={10891\ndash 10894},
}
\bib{Lieb:77}{article}{
author={Lieb, E.~H.},
author={Simon, B.},
title={The {H}artree-{F}ock theory for {C}oulomb systems},
date={1977},
journal={Commun. Math. Phys.},
volume={53},
pages={185\ndash 194},
}
\bib{Marzari:1997}{article}{
author={Marzari, Nicola},
author={Vanderbilt, David},
title={{Maximally localized generalized Wannier functions for composite
energy bands}},
date={1997},
journal={Physical Review B},
volume={56},
number={20},
pages={12847\ndash 12865},
}
\bib{McWeeny:60}{article}{
author={McWeeny, R.},
title={Some recent advances in density matrix theory},
date={1960},
journal={Rev. Mod. Phys.},
volume={32},
pages={335\ndash 369},
}
\bib{NenciuNenciu:98}{article}{
author={Nenciu, A.},
author={Nenciu, G.},
title={The existence of generalised {W}annier functions for
one-dimensional systems},
date={1998},
journal={Commun. Math. Phys.},
volume={190},
pages={541\ndash 548},
}
\bib{Nenciu:83}{article}{
author={Nenciu, G.},
title={Existence of the exponentially localised {W}annier functions},
date={1983},
journal={Commun. Math. Phys.},
volume={91},
pages={81\ndash 85},
}
\bib{Osher:2005}{article}{
author={Osher, S.},
author={Burger, M.},
author={Goldfarb, D.},
author={Xu, J.},
author={Yin, W.},
title={An iterative regularizatoin method for total variation-based
image restoration},
date={2005},
journal={Multiscale Model. Simul.},
volume={4},
pages={460\ndash 489},
}
\bib{OzolinsLaiCaflischOsher:13}{article}{
author={Ozolins, V.},
author={Lai, R.},
author={Caflisch, R.},
author={Osher, S.},
title={Compressed modes for variational problems in mathematics and
physics},
date={2013},
journal={Prol. Natl. Acad. Sci. USA},
volume={110},
pages={18368\ndash 18373},
}
\bib{Panati:07}{article}{
author={Panati, G.},
title={Trviality of {B}loch and {B}loch-{D}irac bundles},
date={2007},
journal={Ann. Henri Poincar\'e},
volume={8},
pages={995\ndash 1011},
}
\bib{Setzer:2009SSVMCV}{article}{
author={Setzer, S.},
title={Split bregman algorithm, douglas-rachford splitting and frame
shrinkage},
date={2009},
journal={Proceedings of the 2nd International Conference on Scale Space
and Variational Methods in Computer Vision,},
volume={LNCS, 5567},
}
\bib{Wannier:1937}{article}{
author={Wannier, G~H},
title={{The structure of electronic excitation levels in insulating
crystals}},
date={1937-08},
journal={Physical Review},
volume={52},
number={3},
pages={0191\ndash 0197},
}
\bib{Wu:2010SIAM}{article}{
author={Wu, C.},
author={Tai, X.},
title={Augmented lagrangian method, dual methods and split-bregman
iterations for {ROF}, vectorial {TV} and higher order models},
date={2010},
journal={SIAM J. Imaging Science},
volume={3},
number={3},
pages={300\ndash 339},
}
\bib{yin2013error}{article}{
author={Yin, W.},
author={Osher, S.},
title={Error forgetting of bregman iteration},
date={2013},
journal={Journal of Scientific Computing},
volume={54},
number={2-3},
pages={684\ndash 695},
}
\bib{Yin:2008bregman}{article}{
author={Yin, W.},
author={Osher, S.},
author={Goldfarb, D.},
author={Darbon, J.},
title={Bregman iterative algorithms for l1-minimization with
applications to compressed sensing},
date={2008},
journal={SIAM Journal on Imaging Sciences},
volume={1},
number={1},
pages={143\ndash 168},
}
\end{biblist}
\end{bibdiv}
\end{document}
In the above algorithm, the shrinkage of eigenvalues of a matrix is used. Let $M$ be a Hermitian matrix, we would like to compute
\begin{equation}
R = h(M)
\end{equation}
where $h(x) = (x \vee 0) \wedge 1$. The direct evaluation of this involves diagonalization of the matrix $M$. To get a more efficient algorithm, we use a polynomial expansion to approximate $h(M)$.
Let $[\lambda_{\min}, \lambda_{\max}]$ be an estimate of the spectrum width of $M$:
\begin{equation}
\lambda_{\min} \leq \min \spec(M) \quad \text{and} \quad \lambda_{\min} \geq \max \spec(M)
\end{equation}
It suffices to consider $\wt{h}(\wt{M})$ where
\begin{equation}
\wt{M} = \frac{M - (\lambda_{\min}+\lambda_{\max})/2}{\lambda_{\max} - \lambda_{\min}}
\end{equation}
and $\wt{h}$ is the corresponding rescaled version of $h$:
\begin{equation}
\wt{h}(x) = h\bigl[ (\lambda_{\max} - \lambda_{\min}) x + (\lambda_{\max} + \lambda_{\min})/2 \bigr].
\end{equation}
Due to the scaling, it suffices to approximate $\wt{h}$ on $[-1, 1]$. We propose to use a Chebyshev approximation of the function:
\begin{equation}
\wt{h}(x) = \sum_{k=0}^{N-1} c_k T_k(x).
\end{equation}
Hence, we just need to calculate $p(M)$ for some polynomial $p$. This can be done efficiently by using the fact that $M$ is (near) sparse.
Another possibility is to find a pole approximation of the function $\wt{h}$ as
\begin{equation}
\wt{h}(x) = \sum_{j=0}^{N-1} \frac{w_j}{x - x_j}
\end{equation}
and then use Newton-Schulz iteration to compute $(M - x_j)^{-1}$. It is not clear whether this would give a good approximation in the current case, as $\wt{h}$ is not analytic on $[-1, 1]$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,147 |
Discussion in 'Tutorials' started by admin, Apr 28, 2015.
Runar Finanger, Novelda Senior Marcom Manager, demonstrates the XeThru respiration module by monitoring his two year old son's breathing rate throughout the night.
I like the product and technology involved. I would like to know, how it shall behave in a noisy environment, e.g. Inside a moving vehicle.
The breathing person must be static relative to the sensor and the person must be sitting still, my guess is that steering and gearing will prevent the sensor to track the respiration most of the time.
See https://community.xethru.com/threads/respiraton-module-field-test.29/ for a similar discussion.
Here is our recent test result on breathing monitoring. Please take a look. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,618 |
\section{The bulk geometry for non-relativistic charged fluid} \label{sec:bulk}
In this section, we construct the bulk geometry that is dual to a non-relativistic conformal charged fluid theory living on its boundary. We begin with a locally boosted charged black brane solution in $4 + 1$ dimensional AdS space \cite{Banerjee:2008th,Erdmenger:2008rm}. We uplift the solution to 10 dimensions and then perform TsT transformations to obtain a new solution with a Schr\"odinger symmetry at the boundary. The effective five dimensional theory contains a metric, a mass-less gauge field, a massive gauge field and a dilaton. The mass-less gauge field corresponds to a global $U(1)$ current on the boundary.
\subsection{Locally boosted charged black brane geometry and
hydrodynamics}
The Einstein-Maxwell action with a gauge Chern-Simons term in $AdS_5$ space is given by,
\begin{eqnarray}\displaystyle S= {1\over 16 \pi G_5} \int d^5x \sqrt{-g} \left ( R + 12
-F_{ab}F^{ab} -\frac{4\k} 3 \epsilon^{abcde}\mathcal{C}_a\mathcal{G}_{bc}\mathcal{G}_{de}\right ) .
\end{eqnarray}
The $AdS$ radius has been set to 1. Here, $\mathcal{C}$ is an abelian gauge field
and
\begin{equation}
\label{eq:Gdef}
\mathcal{G}_{ab} = \partial_a\mathcal{C}_b-\partial_b\mathcal{C}_a
\end{equation}
is the corresponding field strength. The value of the parameter $\k$ is fixed : $\k=1/2\sqrt{3}$. However, we shall keep this parameter as $\k$ to trace the effect of this term in final results. Equations of motion obtained from this action are given by,
\begin{eqnarray}\displaystyle
\begin{split}
R_{ab} -\frac12 R g_{ab} - 6 g_{ab} -2 \left ( \mathcal{G}_{ac} \mathcal{G}_b^{c}
-\frac14 \mathcal{G}^2 g_{ab}\right ) &=0\\
\nabla_b\mathcal{G}^{ab}+ \k \epsilon^{abcde}\mathcal{G}_{bc}\mathcal{G}_{de} &= 0.
\end{split}
\end{eqnarray}
These equations of motion admit a solution for metric and gauge field, which in the boosted frame is given by (AdS-Reissner-Nordstrom black-brane solution)
\begin{eqnarray}
\label{eq:d3metricboost}
\begin{split}
ds^2 &=-2 u_\mu dx^\mu dr-r^2 f(r, q, m)
u_\mu u_\nu dx^\mu dx^\nu +r^2 (u_\mu u_\nu +\eta _{\mu\nu})dx^\mu
dx^\nu, \\
\mathcal{C} &=-\frac{\sqrt{3} q}{2 r^2} u_\m dx^\m
\end{split}
\end{eqnarray}
with,
\begin{eqnarray}
f(r,m,q)=1-\frac{m}{r^4}+\frac{q^2}{r^6}.
\end{eqnarray}
The Hawking temperature, chemical potential and physical charge
density of this black brane are given by
\begin{eqnarray}
T=\frac{1}{2 \pi}(2-q^2), \quad \phi= \sqrt{3}q/2 \quad
n=\frac{\sqrt{3}q}{4 \pi G_5}.
\end{eqnarray}
We choose the black hole horizon to be at 1. This implies a relation between the charge parameter $q$ and the mass parameter $m$ : $q^2=m-1$. This system is dual to a field theory with $U(1)$ gauge invariance living on the boundary of $AdS$ spacetime. Thermodynamic properties of this field theory can be computed from thermodynamic properties of charged black hole in the bulk.
Low energy fluctuations of a quantum field theory in local thermal equilibrium is best understood in terms of hydrodynamics. Different transport coefficients capture the response of the system against external probes. Such hydrodynamic properties of a boundary fluid system can be consistently studied holographically. Low energy fluctuations in the boundary correspond to low energy fluctuations of bulk parameters (velocity, temperature etc.). One can consistently solve the bulk geometry in presence of these perturbations order by order in derivative expansion \cite{Bhattacharyya:2008jc,Banerjee:2008th} by solving bulk equations of motion and construct stress tensor and other currents of the boundary system in presence of these perturbations using holographic dictionary \cite{Balasubramanian:1999re}. The stress tensor and currents are also written in terms of derivative expansion of velocities, temperature etc. First and higher derivative terms capture hydrodynamic properties of the system.
We consider velocities and temperature to be local function of boundary coordinates and expand them up to first order in derivative expansion \cite{Banerjee:2008th}
\begin{eqnarray}
\begin{split}
m&=m_0+x^{\mu} \partial_{\mu} m^{(0)},\quad
q=q_0+x^{\mu} \partial_{\mu} q^{(0)},\quad
u_i=x^{\mu} \partial_{\mu} u_i^{(0)}.
\end{split}
\end{eqnarray}
Here $u_i$ is $i^{th}$ component of velocity. Once we consider fluid velocities and temperature are local functions the metric and gauge field do not solve equations of motion any more. We need to add more terms to metric and gauge field to satisfy Einstein equation in the bulk up to first order in derivative expansion. This whole procedure was explicitly carried out in the \cite{Banerjee:2008th}. Here we write the final result in a covariant form.
\begin{eqnarray}\label{eq:5dchargedmetricglob}
\begin{split}
ds^2& =-2u_{\mu}dx^{\mu} dr-r^2 f(r,m,q)
u_{\mu}u_{\nu}dx^{\mu}dx^{\nu}+r^2 P_{\mu\nu}dx^{\mu}dx^{\nu}\\
& \qquad +2u_{\mu}dx^{\mu}r(u^{\lambda}\partial_{\lambda}u_{\nu}-
\frac{\partial_{\lambda}u^{\lambda}}{3
}u_{\nu}) dx^{\nu}+2 r^2 F_2(r,m)\sigma_{\mu\nu}dx^{\mu}dx^{\nu}\\
& \qquad-2 u_{\mu}dx^{\mu}(\frac{\sqrt{3}\kappa q^3}{ m r^4}l_{\nu}+6 q r^2
P_{\nu}^{\lambda} D_{\lambda}q F_1(r,m)) dx^{\nu}\\
\mathcal{C}&=\left (\frac{\sqrt{3} q}{2 r^2} u_{\mu}+\frac{\sqrt{3}\kappa q^2}{2 m
r^2}l_{\mu}-\frac{\sqrt{3}r^5}{2} P_{\nu}^{\lambda} D_{\lambda}q
F_1^{1,0}(r,m)\right ) dx^{\mu}
\end{split}
\end{eqnarray}
where,
\begin{eqnarray}
\begin{split}
P^{\nu\mu} &=\eta^{\nu\nu} +u^{\nu}u^{\mu}, \quad \sigma^{\mu\nu} = P^{\mu\a}P^{\nu\b}\partial_{(\a}u_{\b)} -\frac13P^{\mu\nu}\partial\cdot u\\
P^\l_\m D_{\lambda}q &= P^\l_\m \partial_\l q + 3 q u^\l \partial_l u_\m
\quad \text{is weyl covariant derivative and} \quad l^{\mu}
=\epsilon^{\nu\lambda\sigma\mu} u_{\nu}\partial_{\lambda}
u_{\sigma}\\
F_1(r,m)
&=\frac{1}{3}(1-\frac{m}{r^4}+\frac{q^2}{r^6})\int_{r}^{\infty} dp
\frac{1}{(1-\frac{m}{p^4}+\frac{q^2}{p^6})^2}(\frac{1}{p^8}-\frac{3}{4
p^7}(1+\frac{1}{m}))\\
F_2(r,m)&=\int_{r}^{\infty} dp
\frac{p(p^2+p+1)}{(p+1)(p^4+p^2-m+1)}.
\end{split}
\end{eqnarray}
The above solution has Poincar\'e isometry at the boundary. Calculating stress energy tensor and $U(1)$ charge current holographically \cite{Balasubramanian:1999re} one finds that these conserved currents describing a relativistic fluid up to first order in derivative expansion. The stress-energy tensor and charge current are given by,
\begin{eqnarray}\displaystyle
\begin{split}
T_{\mu\nu} & =(E+P)u_{\mu}u_{\nu}+P \eta_{\mu\nu} - \eta_{rel} \sigma_{\mu\nu}\\
J_{\mu} &= n u^{\mu} -\mathcal{D} P^{\nu}_{\mu} {D}_{\nu} n +\mho l_{\mu}
\end{split}
\end{eqnarray}
where, transport coefficients are given by
\begin{equation}\label{eq:holographictransportrel}
E=3P = {3m\over 16\pi G_5}, \quad \eta={1\over 16\pi G_5},\quad n={\sqrt3 q\over 4\pi G_5},\quad \mathcal{D} = {1+m \over 4m}, \quad \mho = {3\kappa q^2\over 16 \pi G_5 m}.
\end{equation}
The relation $E=3P$, which is the equation of state for relativistic fluid, implies that the relativistic fluid is conformal and hence, relativistic fluid has zero bulk viscosity.
In the next subsection we start with locally boosted five dimensional geometry (\ref{eq:5dchargedmetricglob}) and construct a new solution with asymptotic Schr\"odinger isometry by using a solution generating technique called $TsT$ transformation.
\subsection{TsT transformation of locally boosted charged black brane solution}
\label{sec:tst}
The TsT transformation is a solution generating technique in string theory \cite{Son:2008ye,Herzog:2008wg,Balasubramanian:2008dm,Adams:2008wt,Goldberger:2008vg,Maldacena:2008wh,Bobev:2009mw,Bobev:2009zf,Banerjee:2011jb,Imeroni:2009cs,Yamada:2008if,Kim:2010tf,Singh:2010rt}. Supergravity is a low energy solution of full string theory. A solution to supergravity equations of motion can be shown to be a solution to string theory. A symmetry transformation of this solution with respect to a supergravity symmetry does not give rise to a new solution in supergravity. However, if the transformation employed is a symmetry of full string theory then the transformed solution can be interpreted as a new solution of string theory. In case of the TsT transformations, T-dualities are performed along the isometry direction, whereas s-transformation mixes the dualised coordinate with another isometry direction. This transformation generically changes asymptotic of the resulting metric but is guaranteed to be a solution to the supergravity equations of motion.
Since TsT transformation is a symmetry of full string theory we first need to uplift the five dimensional metric (\ref{eq:5dchargedmetricglob}, \ref{eq:d3metricboost} ) to ten dimensions and embed in string theory solution \cite{Mazzucato:2008tr,Imeroni:2009cs,Banerjee:2011jb}. The full type IIB 10 dimensional solution is a obtained by a direct sum of five dimensional metric and a Sasaki-Einstein space and a five form field strength.
Denoting this five dimensional metric
\begin{equation}
ds^2_5 = g_{ab} dx^a dx^b, \quad \text{where $a,b$ are five
dimensional indices}
\end{equation}
the ten dimensional metric is given by
\begin{equation}
ds^2_{10} = g_{AB}dx^A dx^B = g_{ab} dx^a dx^b+ ds^2_{SE}
\end{equation}
where the Sasaki-Einstein manifold here is a five sphere, which can be
written as fibration over $\rm{CP^2}$
\begin{equation}
ds^2_{SE} = \left ( d\psi + \mathcal{P} -\frac2{\sqrt3} \mathcal{C}\right )^2 + ds_{\rm{CP^2}}^2\, .
\end{equation}
The one form $\mathcal{P}$ is given by
\begin{equation} \mathcal{P} = \frac{1}{3}
\left(\text{d$\chi $}_1+\text{d$\chi $}_2\right)-\sin ^2\alpha
\left(\text{d$\chi $}_2 \sin ^2\beta+\text{d$\chi $}_1 \cos
^2\beta\right),
\end{equation}
and
\begin{eqnarray}\displaystyle
ds_{\rm{CP^2}}^2 = d\alpha^2 &+& \sin^2\alpha d\beta^2
+ \sin^2 \alpha\cos^2\alpha(\cos^2\beta d\chi_1 + \sin^2\beta
d\chi_2)^2 \nonumber \\
&+& \sin^2\alpha \sin^2 \beta \cos^2 \beta (d\chi_1-d\chi_2)^2\, .
\end{eqnarray}
The ten dimensional geometry is supported by a five form field strength is given by
\begin{equation}
\mathcal{F}_5 = 2(1+*_{10})\left [ \left ( d \psi +P-\frac{2}{\sqrt{3}} \mathcal{C} \right ) \wedge J_2-\frac{1}{\sqrt{3}}*_5 \mathcal{G}\right ] \wedge J_2, \quad \text{where} \quad J_2 = \frac12 d\mathcal{P},
\end{equation}
a two form field strength $\mathcal{G}$ given by equation (\ref{eq:Gdef}) and a dilaton
\begin{equation}
\Phi=0.
\end{equation}
To obtain TsT transformed solution we first define two light-cone coordinates $x^{+}$ and $x^-$
\begin{equation}\label{eq:lccoord}
x^{+} = \gamma(v+z) \quad \text{and} \quad x^- = \frac1{2\gamma} (v-z)
\end{equation}
where $\gamma$ is twist parameter defined below. Thus we have two isometry direction $x^-$ and $\psi$. The TsT transformations, then, correspond to performing T-duality along $\psi$
direction, followed by a shift along $x^-$, i.e., $x^-\rightarrow x^- - \gamma\psi$. $\gamma$ is shift or twist parameter, we often set that to $1$. Finally we perform T-duality back along the $\psi$ direction.
To perform a TsT transformation we write down the five dimensional solution (\ref{eq:5dchargedmetricglob}) in the following form
\begin{equation}
ds^2 = -2 u_{\mu}dx^\m dr + \mathcal{S}_{\m\nu} dx^\m dx^\n, \quad \mathcal{C} = \mathcal{C}_\m dx^\m
\end{equation}
where,
\begin{eqnarray}
\begin{split}
\mathcal{S}_{\m\nu}&=-r^2 f u_{\mu}u_{\nu}dx^{\mu}dx^{\nu}+r^2 P_{\mu\nu}dx^{\mu}dx^{\nu}
+2u_{\mu}dx^{\mu}r(u^{\lambda}\partial_{\lambda}u_{\nu}-\frac{\partial_{\lambda}u^{\lambda}}{3 }u_{\nu})dx^{\nu}\\
&\quad +2 r^2 F_2(r,m)\sigma_{\mu\nu}dx^{\mu}dx^{\nu}
-2 u_{\mu}dx^{\mu}(\frac{\sqrt{3}\kappa q^3}{ m r^4}l_{\nu}+6 q r^2 P_{\nu}^{\lambda} D_{\lambda}q F_1(r,m)) dx^{\nu},\qquad\\
\mathcal{C}_\m &= \frac{\sqrt{3} q}{2 r^2} u_{\mu}+\frac{\sqrt{3}\kappa q^2}{2 m
r^2}l_{\mu}-\frac{\sqrt{3}r^5}{2} P_{\nu}^{\lambda} D_{\lambda}q
F_1^{1,0}(r,m).
\end{split}
\end{eqnarray}
Replacing $v$ and $z$ in terms of light cone coordinates $x^\pm$ this metric can be written as,
\begin{equation}
ds^2_5 = A_1(dx^-+K_1)^2 + ds_4^2
\end{equation}
where
\begin{eqnarray}\displaystyle
\begin{split}
A_1 & = \mathcal{S}_{--}, \quad K_1 = \frac1{\mathcal{S}_{--}} \left ( \mathcal{S}_{+-} dx^+ -u_- dr
+\mathcal{S}_{- x}dx +\mathcal{S}_{-y}dy \right ) \\
ds_4^2 &= -2u_{\bar a} dx^{\bar a} dr +\mathcal{S}_{\bar a\bar b} dx^{\bar a}dx^{\bar b} +\frac1{\mathcal{S}_{--}} [ (2u^-
\mathcal{S}_{--} -2 u^+ \mathcal{S}_{+-}) dx^+ dr \\
& \quad - (u_- dr - \mathcal{S}_{-i} dx^i)^2 -
(\mathcal{S}_{+-}dx^+ +2 \mathcal{S}_{+i} dx^i) \mathcal{S}_{+-} dx^+ ].
\end{split}
\end{eqnarray}
Here $x^{\bar a} = \{x^+, x, y\}$ and $x^i = \{x,y\}$. Therefore, the full
10-dimensional metric is given by, Then 10 dimensional metric, therefore, is given by,
\begin{eqnarray}\displaystyle
ds_{10}^2=A_1(dx^-+K_1)^2 + ds_4^2 + \left ( d\psi + \mathcal{P} -\frac2{\sqrt3} \mathcal{C} \right )^2 +
ds_{\rm{CP^2}}^2.
\end{eqnarray}
There is no NS-NS two form and dilaton to start with. Apart from that we have a five form field strength and a two form field strength in 10 dimensions given by equation (\ref{eq:tsttranchar}).
A TsT transformation of the above 10 dimensional solution will give rise to the following 10 dimensional solution\footnote{We denote all the $TsT$ transformed fields with ``hat''. The above solution is written string frame.}.
\begin{eqnarray}\displaystyle\label{eq:tsttranchar}
\begin{split}
d\hat s^2 &= \mathcal{M} A_1 (dx^-+K_1)^2 +\mathcal{M} \left ( d\psi + \mathcal{P}-\frac{2}{\sqrt{3}} \mathcal{C} \right )^2 + ds_8^2, \quad e^{2\hat \Phi} = \mathcal{M}, \\
\hat B_2 & = \mathcal{A} \wedge \left ( d \psi + \mathcal{P}-\frac{2}{\sqrt{3}} \mathcal{C}\right ),
\quad \hat \mathcal{F}_3 = g\wedge d\mathcal{P}, \quad g = \frac12 d\mathcal{C}_-,\\
\hat \mathcal{F}_5 &= \mathcal{F} + \hat B_2 \wedge \hat \mathcal{F}_3.
\end{split}
\end{eqnarray}
where,
\begin{eqnarray}\displaystyle
\mathcal{M} = (1+\gamma^2 A_1)^{-1} \quad \text{and} \quad \mathcal{A} = -\gamma \mathcal{M} A_1 (dx^-+K_1)
\end{eqnarray}
In addition to this there is a two form field strength $\mathcal{G}$ which remains unaltered under TsT transformation.
\subsection{Effective five dimensional action and boundary terms}
\label{sec:boundaryterm}
The $TsT$ transformed ten dimensional fields (\ref{eq:tsttranchar}) can be consistently truncated over $S^5$ \cite{Maldacena:2008wh,Adams:2009dm}. Upon KK reduction on $S^5$, the five-dimensional effective solution includes a metric, a massive vector $\mathcal{A}_M$ coming from reduction of $\hat B_2$ over $S^5$, a scalar and a massless vector $\mathcal{C}$, which was present in the 10D metric before the TsT. One can also obtain an one form from $\hat \mathcal{F}_3$. However, this one form is not an independent mode of excitation. It is completely fixed by the massive gauge field and the massless gauge field. Massless gauge field doesn't change with $TsT$ transformation and remains same as before. The reduced five dimensional fields in
Einstein's frame are given by \cite{Brattan:2010bw},
\begin{eqnarray}\displaystyle\label{eq:tstmetein}
\begin{split}
d\hat s_E^2 &= e^{4\Phi\over 3} A_1 (dx^-+K_1)^2 + e^{-{2\Phi\over 3}}
ds_4^2\\
\hat \mathcal{A} &= -\gamma \mathcal{M} A_1 (dx^-+K_1),\quad \hat \mathcal{C} = \mathcal{C}_\m dx^\m, \quad g = \frac12 d\mathcal{C}_-\\
e^{2\hat \Phi}& = \mathcal{M} e^{2\Phi}, \quad \mathcal{M} = (1+\gamma^2 A_1)^{-1}.
\end{split}
\end{eqnarray}
These five dimensional fields are solutions of equations of motions
(up to first order in derivative expansion) obtained from an effective
five dimensional action \cite{Brattan:2010bw}
\begin{eqnarray}
\label{eq:5deffactioncharged}
S_5=\frac{1}{16\pi G_5}\int d^5x \sqrt{-g}&&\Big[{R}-\frac{4}{3}(\partial_a \Phi)(\partial^a \Phi)-\frac{1}{4}e^{-8\Phi/3}\mathcal{W}_{ab}\mathcal{W}^{ab}-4\mathcal{A}_a \mathcal{A}^a+16 e^{2 \Phi/3}-4 e^{8 \Phi/3}\nonumber\\
&&-\frac{1}{3}e^{4\Phi/3}\mathcal{G}_{ab} \mathcal{G}^{ab}-4 e^{2\Phi}g_{a}g^{a}-e^{-4 \Phi/3}\mathcal{A}_{a}\mathcal{G}_{bc}\mathcal{A}^{a}\mathcal{G}^{bc}\nonumber\\
&&-\frac{1}{2}e^{-2\Phi/3}\left(-\frac{2}{\sqrt{3}}\mathcal{G}_{ab}-4 g_a \mathcal{A}_b\right)\left(-\frac{2}{\sqrt{3}}\mathcal{G}^{ab}-4 g^a \mathcal{A}^b\right)\nonumber\\
&&+ \frac{4\k}{3} \mathcal{G}_{ab}\mathcal{G}_{cd}\mathcal{C}_{e}\epsilon^{abcde}
\Big].
\end{eqnarray}
Unlike asymptotically $AdS$ spacetime the metric (\ref{eq:tstmetein}) has inhomogeneous asymptotic fall-off. Therefore, it is not very straight forward to apply Brown-York type analysis to obtain boundary stress tensor for this geometry. First of all, to obtain a well defined boundary stress tensor one needs to add a suitable boundary term $S_{b}$ (which includes counterterms also) to action (\ref{eq:5deffactioncharged}) such that on-shell variation of the total action is zero and boundary stress tensor is finite. The boundary action for uncharged case is given in \cite{Herzog:2008wg,Adams:2008wt}. In presence of $U(1)$ charge, one can add extra counterterms for finite answer \cite{Liu:2004it}. However, since the leading massless $U(1)$ gauge field $\mathcal{C}$ and its fluctuations fall sufficiently faster as we approach asymptotic boundary at $r\rightarrow \infty$, we do not need any extra counterterms for renormalization. One can also have a $\ln r$ divergence at boundary \cite{DHoker:2009ixq} but in our case (the fluctuations we are considering) such divergences do not appear. Therefore, the boundary counterterms terms are given by
\begin{eqnarray}
\begin{split}
\label{eq:actionb}
S_{ct} &=\frac{1}{16 \pi G_5}\int d^4\xi \sqrt{-h} \left [-6 +3 \Phi^2 +\mathcal{A}_{\mu}\mathcal{A}^{\mu} \right ] .
\end{split}
\end{eqnarray}
There are other terms, may be added to this boundary action \cite{Herzog:2008wg}, however those terms will not play any role in calculating boundary stress tensor complex and current. Hence we do not write them here.
\section{Holographic non-relativistic charged fluid}
\label{sec:stresstensor}
Following the holographic renormalization programme \cite{Balasubramanian:1999re,deHaro:2000vlm} in the gauge-gravity duality one can construct stress tensor for relativistic field theories from a well-defined action principle by varying the action with respect to the boundary data. In case of TsT transformed bulk geometries the asymptotic metric is not conformal to a flat spacetime. Non-relativistic time coordinate and space coordinate have different $r$ dependences, leads to degenerate boundary metric. It has been discussed in \cite{Ross:2009ar} how to obtain a consistent stress tensor complex for a non-relativistic system in presence of non-uniform $r$ dependence in boundary metric.
TsT transformed five dimensional bulk action (\ref{eq:5deffactioncharged}) includes a vector field a scalar field and massless gauge field. These fields will modify the definition of holographic stress tensor and current. We follow the prescription of \cite{Hollands:2005ya} and \cite{Ross:2009ar} (who extended the work of \cite{Hollands:2005ya} to non-relativistic spacetime) to determine boundary stress energy complex for charged Schr\"odinger field theory. On-shell variation of bulk action and boundary counterterm action with respect to boundary fields is given by
\begin{equation}
\delta S_5 +\delta S_{ct} = \frac1{16\pi G_5} \int d^4x\sqrt{-h} \left ( s_{\a\b}\delta h^{\a\b}
+ s_{\a} \d \mathcal{A}^\a +\tilde{s}_{\beta} \delta \mathcal{C}^{\beta}+ s_{\Phi}\d \Phi\right ),
\end{equation}
where,
\begin{eqnarray}
\begin{split}
s_{\alpha \beta}&=\pi_{\alpha\beta}+3
h_{\alpha\beta}+\mathcal{A}_\alpha
\mathcal{A}_\beta-\frac{1}{2} \mathcal{A}_\gamma \mathcal{A}^\gamma
h_{\alpha\beta}-\frac{3}{2}\Phi ^2
h_{\alpha\beta}\\
s_\alpha&=-n^\mu \mathcal{W}_{\mu\alpha} e ^{-8 \Phi /3}+2\mathcal{A}_\alpha, \quad
s_\Phi =-\frac{8}{3} n^\mu \partial_\mu \Phi + 3\Phi.
\end{split}
\end{eqnarray}
Coefficient $\tilde s_\a$ is given by,
\begin{eqnarray}
\begin{split}
\tilde{s}_{\alpha} &= -\left ( \frac4{3}e^{\frac{4\Phi}{3}} + \frac{8}{3} e^{\frac{-2\Phi}{3}} + 4e^{-\frac{4\Phi}{3}} \mathcal{A}_{b} \mathcal{A}^{b} \right ) n^{a}\mathcal{G}_{a \alpha}
-\frac{8}{\sqrt{3}} e^{-\frac{2\Phi}{3}} n^{a} \left ( g_{a} \mathcal{A}_{\alpha} - g_{\alpha} \mathcal{A}_a \right )\\
&\hspace{7.75cm} + \frac{8 \kappa}{3\sqrt{-h} }n^{a} \epsilon_{a\alpha b c d} \mathcal{C}^{b} \mathcal{G}^{cd}.
\end{split}
\end{eqnarray}
The main idea of \cite{Ross:2009ar} is that in presence of tensor fields, we have to consider the variation of boundary frame field $\hat{e}_\alpha^{(A)}$ (instead of boundary metric), holding the tangent space components $\phi^{[\mathcal{I}]}_{AB....}$ of the matter fields fixed where $A,B,..$ denote the tangent space indices and $\mathcal{I}$ are for matter fields species. This treatment will provide a stress tensor which will give us conserved charges using the definition of charges from stress tensor, which will generate the symmetry of
asymptotic spacetime. This charges are conserved up to terms derivatives of other fields.
Writing metric in terms of frame fields and vector fields in terms of tangent space indices we have
\begin{equation}
h^{\a\b} = e^\a_A e^\b_B \eta^{AB}, \quad \mathcal{A}^\a = e^\a_A \mathcal{A}^A, \quad \mathcal{C}^\a = e^\a_A \mathcal{C}^A.
\end{equation}
Following the analysis similar to \cite{Ross:2009ar} we define components of boundary charged non-relativistic stress energy complex and current as,
\begin{eqnarray}\displaystyle
\begin{split}
\delta S_b = \int d^4x\sqrt{-h} \left(T^{\alpha}_\b e^\b_A \delta
{e}_{\alpha}^{A} + s_\a e^\a_A \delta A^A + s_{\phi} \delta \phi + J_\a e^\a_A \delta \mathcal{C}^A\right).
\end{split}
\end{eqnarray}
where,
\begin{eqnarray}
\label{eq:nonstress1}
\begin{split}
\varepsilon &= T^+_{\ +}=2 s^+_{\ +} - s^+ \mathcal{A}_+-\tilde{s}^+ \mathcal{C}_+,\hspace{1.1cm}
\varepsilon ^i = T^i_{\ +}= 2 s^i_{\ +}-s^i \mathcal{A}_+-\tilde{s}^i \mathcal{C}_+,\\
\rho^i &= - T^i_{\ -}= - 2 s^i_{\ -}+s^i \mathcal{A}_- + \tilde{s}^i \mathcal{C}_-,\hspace{.7cm}
\pi^i_j = - T^i_{\ j}=-2 s^i_{\ j}+s^i \mathcal{A}_j+\tilde{s}^i \mathcal{C}_j,\\
\rho &= T^+_{\ -}=2 s^+_{\ -}-s^+ \mathcal{A}_- - \tilde{s}^+ \mathcal{C}_-,\quad
\hspace{.75cm}Q = J^+ = \tilde{s}^+ ,\quad j_i = J^i = \tilde{s}^i.
\end{split}
\end{eqnarray}
Thus addition of massless $U(1)$ field simply adds up a contribution to energy momentum complex like a massive gauge field. At the same time sources a global $U(1)$ charge current at the boundary : $J^0 $ and $J^i$ are charge density and charge current respectively.
Now we have all the terms required to do the computation of various components of stress energy tensor and current.
\subsection{Calculation of constitutive relations}
Writing mass and charge parameters ($m$ and $q$ respectively) of black hole in terms of temperature and chemical potential \cite{Son:2009tf}
\begin{eqnarray}\label{eq:mqrel}
\begin{split}
m&=\frac{\pi ^4T^4}{16} (\gamma +1)^3 (3 \gamma -1), \quad
q=\frac{\phi }{\sqrt{3}}\frac{\pi^2 T^2}{2} (\gamma +1)^2\\
\text{where} \quad \gamma &= \sqrt{1+{8\phi^2\over 3\pi^2 T^2}}.
\end{split}
\end{eqnarray}
Here $\phi$ is chemical potential and horizon radius $r_+$ is given by
\begin{equation}
r_+ = \frac{\pi}{2} T (\gamma+1).
\end{equation}
Since, in our holographic calculations, unperturbed horizon radius is set to $r_+=1$, the unperturbed values of the parameters satisfy
\begin{eqnarray}\displaystyle
m_0=1+q_0^2, \quad \phi_0 = \frac{\sqrt{3} q_0}{2}, \quad T_0 = \frac{2-q_0^2}{2\pi}, \quad \gamma_0 = \frac{2+q_0^2}{2-q_0^2}.
\end{eqnarray}
$m_0,\ q_0, T_0, \phi_0$ are unperturbed values of $m,\ q, T, \phi$ respectively.
Now we compute non-relativistic constitutive relations order by order in derivative expansion. We start with ideal ($zero^{th}$) order.
\subsubsection{Ideal order}
We use equation (\ref{eq:nonstress1}) to compute mass density, energy density, pressure and charge at ideal order. They are given by
\begin{eqnarray}\displaystyle\label{eq:idealorder}
\r &= 4m_0 (u^+)^2, \quad
p =m_0, \quad \varepsilon = m_0, \quad Q =4\sqrt{3} \ q_0u^+.
\end{eqnarray}
Following \cite{Banerjee:2014mka}, we define non-relativistic temperature and $U(1)$ chemical potential as
\begin{equation}
\tau = {T\over u^+}, \qquad \text{and} \qquad \m = {\phi \over u^+}.
\end{equation}
Entropy density of non-relativistic system at ideal order is given by the entropy density of black hole
\begin{equation}
s= \frac{r_+^2 u^+}{4G_5} = 4\pi u^+.
\end{equation}
We find that these ideal order variables satisfy non-relativistic Euler's relation
\begin{equation}
\tau s = \varepsilon+p - \mu Q - \varrho_m \rho
\end{equation}
where $\varrho_m$, mass chemical potential associated with mass density $\r$, is given by
\begin{eqnarray}
\label{eq:masschem}
\varrho_m= -\frac{1}{2 (u^+)^2}.
\end{eqnarray}
Also note that non-relativistic system has equation of state $\varepsilon-p=0$ at ideal order. This is because the parent relativistic theory was conformal and had equation of state $E=3P$. Using holographic results for relativistic fluid (equation \ref{eq:holographictransportrel}) we see our holographic computation matches with LCR computations (\ref{eq:LCRdictionary}) at ideal order : \(\varepsilon=\displaystyle\frac12(E-P)\) and \(p=P\). One can slao check that non-relativistic thermodynamic variables satisfy a constraint equation \(\varepsilon+p+\varrho_m\rho=0\) at ideal order.
\subsubsection{First order hydrodynamics}
We start with first order relativistic solution (\ref{eq:5dchargedmetricglob}) and write down the metric and gauge field in light-cone coordinates : $\{r, x^+, x^-, x, y\}$. We expand the velocity vector in derivative expansion in local rest frame and keep terms up to first order in derivative \cite{Banerjee:2008th}. We consider boundary indices ${\m,\n}$ running over
$\{+,-,x,y\}$. The velocity components are given by,
%
\begin{eqnarray}\displaystyle
\begin{split}
u^+ & = 1 + \epsilon \left ( x^+ \partial_+ u^+ + x^j \partial_j u^+\right ), \quad
u^i = \epsilon \left ( x^+ \partial_+ u^i + x^j \partial_j u^i \right ).
\end{split}
\end{eqnarray}
%
$u^-$ component of velocity is fixed by normalization condition
$u^2=-1$
%
\begin{equation}
u^- =\frac12 -\frac\e2 \left ( x^+ \partial_+ u^+ +x^i \partial_i u^i\right )
+\mathcal{O}(\epsilon^2).
\end{equation}
%
$\epsilon$ is a derivative counting parameter. Mass and charge parameters are given as
%
\begin{eqnarray}\displaystyle
\begin{split}
m &= m_0+ \epsilon(x^+\partial_+ m + x^i \partial_i m)\\
q &=q_0+ \epsilon( x^+\partial_+ q + x^i \partial_i q).
\end{split}
\end{eqnarray}
Writing the relativistic metric and gauge field in a local frame and in light-cone coordinates we perform TsT transformation as described in section \ref{sec:tst} to obtain five dimensional TsT transformed geometry (solution) as given by equation (\ref{eq:tstmetein}). The expressions are very lengthy, hence we refrain ourselves to produce those expression in the paper. As a consistency, one can check if these solutions satisfy effective five dimensional equations of motion obtained from the action (\ref{eq:5deffactioncharged}). We find that the TsT tranformed geoemtry satisfy 5D equations of motion provided
\begin{eqnarray}\label{eq:holoconstrain}
\begin{split}
\partial_+u^+ & =\frac{1}{2} \partial_iu^i, \quad \partial_+m=- 4 m_0 \partial_+u^+, \quad \partial_i m=4 m_0 \partial _+ u^i ,\quad
\partial_+q =-3 q_0 \partial_+u^+.
\end{split}
\end{eqnarray}
These relation can also be obtained by doing the light cone reduction of relativistic constraint relations obtained in \cite{Banerjee:2008th}.
Using the relations (\ref{eq:mqrel}), we express derivatives of mass and charge in terms of derivatives of temperature and chemical potential in the following way,
\begin{eqnarray}
\begin{split}
\partial_i m &= 4\pi \partial_i T + 4\sqrt3 q_0 \partial_i\phi \\
\text{and} \quad \partial_i q & = \frac{2 q_0 }{\gamma _0 T_0}\partial_iT + \frac{2(2+5q_0^2)}{\sqrt3(2+q_0^2)}\partial_i\phi.
\end{split}
\end{eqnarray}
Using the holographic constraints (\ref{eq:holoconstrain}) and equations in (\ref{eq:mqrel}) we find that
\begin{equation}
\partial_+\left ( \frac{\mu}{\tau}\right ) =0
\end{equation}
and
\begin{equation}
\partial_+ u^+ = -{1\over 2\tau} u^+\partial_+\tau.
\end{equation}
In non-relativistic constitutive relations we have terms proportional to $\partial_iu^+$. We identify this term with derivative of mass chemical potential. From equation (\ref{eq:masschem}) we have,
\begin{eqnarray}
{\partial_iu^+ \over u^+} = -{\partial_i \varrho_{m} \over 2 \varrho_m}.
\end{eqnarray}
Thus, the constitutive relations for non-relativistic fluid can be written as derivative expansion of temperature, mass-chemical potential and $U(1)$ charge-chemical potential. However, we define a new basis to write the constitutive relations in a more compact form. We use a reduced chemical potential
\begin{equation}
\partial_i\n \equiv \partial_i\left ( \frac{\mu}{\tau}\right )
\end{equation}
such that $\partial_+\nu =0$ and a redefined mass chemical potential
\begin{equation}
\partial_i \mu_m \equiv \partial_i u^+ = u^{+3} \partial_i\varrho_m.
\end{equation}
Hence, we display our constitutive relations as derivative expansion of $(\tau, \nu, \mu_m)$. From equation (\ref{eq:nonstress1}) we find\footnote{At first order, there will be some terms proportional to fluid velocity $v^i$. We have dropped those terms since we are working in a local frame.} non-relativistic stress tensor quantities energy density $\varepsilon$, energy current $\varepsilon^i$ , spatial stress tensor $\pi_{ij}$, mass density $\rho$ and mass current $\rho^i$. Here we have suppressed a factor of $16 \pi G_5$.
\begin{eqnarray}
\begin{split}
\r^i &= \r v^i, \quad
\pi^i_j = p \delta^i_j-\eta \sigma^i_j,\quad
\epsilon_i = \left ( \epsilon+p+\frac12 \r v^2\right ) v^i +\k_T \partial_i \t +\t \sigma \partial_i {\n} ,\\
j_i &= Q v_i +\k_q \partial_i \t +\t \sigma_q \partial_i {\n} +\l_q \partial_i \mu_m + \epsilon^{ij} (\tilde \k_q \partial_i \t +\t \tilde\sigma_q \partial_i {\n} +\tilde\l_q \partial_i \m_m)
\end{split}
\end{eqnarray}
with,
\begin{eqnarray}\displaystyle
\begin{split}
\r &= 4m_0 (u^+)^2,\quad
p =m_0,\quad
\varepsilon = m_0,\quad
v_i= {u_i\over u^+} + \frac{1}{4 m_0 \tau _0} \partial_i \t +\frac{\nu_0 \tau _0^2 }{2m_0^2} \partial_i\n, \\
Q & =J^0= 4 \sqrt{3} {q_0 u^+}+\Delta Q, \quad \Delta Q = - \frac{\l_q}{2\tau_0}\partial_+\tau+\frac{12 \kappa q_0^2}{m_0}\epsilon ^{{ij}} \partial _i u_j .
\end{split}
\end{eqnarray}
The transport coefficients in energy current are given by
\begin{eqnarray}
\label{eq:unchargedtransport}
\begin{split}
\eta &= u^+, \quad \kappa_T = -{1\over \tau_0}, \quad \sigma = -\frac{\sqrt3 q_0}{m_0} = -{Q u^+\over \rho}.
\end{split}
\end{eqnarray}
Transports appearing in charge current are given by,
\begin{eqnarray}
\begin{split}
\k_q &= \frac{\sqrt{3} q_0 \left(3 m_0+2\right)}{m_0 \tau _0},\qquad \sigma_q= \frac{\sqrt3 q_0 \tau_0 }{m_0}\kappa_q,\qquad \l_q = \frac{3\sqrt3 q_0(1+m_0)}{m_0}\\
\tilde{\kappa}_q &=- \frac{12 q_0^2 \kappa}{m_0 \ \tau_0}, \qquad \tilde{\sigma}_q =- \frac{12\sqrt3 \kappa q_0^3 }{m_0^2}, \qquad \tilde{\lambda}_q = -\frac{24 \kappa q_0^2}{m_0}.
\end{split}
\end{eqnarray}
Entropy density of the system receives a correction at first order. We denote that correction \cite{Banerjee:2014mka} by $\chi$
\begin{equation}
s = 4\pi u^+(1+\chi).
\end{equation}
Demanding that first law of thermodynamics will be satisfied at first order (in local frame), we have
\begin{eqnarray}\displaystyle
\tau s u^+\left [ \partial_+\chi + {\mu u^+\over \tau s} \partial_+ \Delta Q\right ] =0.
\end{eqnarray}
From this equation we find first order correction to entropy density
\begin{eqnarray}\displaystyle
s =4\pi u^+ \left ( 1- {\nu\over 4\pi} \Delta Q \right ).
\end{eqnarray}
One can check that with this correction to entropy, Euler relation is satisfied for
\begin{equation}
\label{eq:masschem1st}
\varrho_m= -\frac{1}{2 (u^+)^2},
\end{equation}
which means mass chemical potential does not receive any correction at first order and hence \(\varepsilon+p+\varrho_m\rho=0\) is also satisfied at first order in derivative expansion. This observation is in agreement with \cite{Banerjee:2014mka}.
\section{Discussion}
\label{sec:discussion}
In this paper we derive the constitutive relations of a charged Galilean\ fluid in holographic set up. We start with a five dimensional geometry whose low energy fluctuations correspond to a relativistic charged conformal fluid with a parity-odd term in global $U(1)$ current \cite{Banerjee:2008th,Erdmenger:2008rm}. Then we uplift the five dimensional solution to ten dimensions which fits with a particular configuration in type IIB string theory. Using TsT transformation we generate a new solution of type IIB theory with has Galilean\ isometry at the boundary. Upon reduction over $S^5$, the lower dimensional geometry serves as holographic dual of a Galilean\ fluid at the boundary. Following the work of \cite{Ross:2009ar}, with a little modification in presence of $U(1)$ charge, we compute boundary stress tensor complex which provides constitutive relations of a Galilean\ fluid.
One can also obtain a Galilean\ fluid by reducing the constitutive relations of a relativistic theory over light-cone. This direct approach produces set of constitutive relations for a lower dimensional Galilean\ fluid in terms of all the data of its mother theory \cite{Rangamani:2008gi,Brattan:2010bw,Banerjee:2014mka}. Similarly reducing the first law of thermodynamics and Euler relation for relativistic fluid one can recover first law and Euler relation for non-relativistic fluid but at the cost of an extra relation between the thermodynamic variables : $\varepsilon+p+\varrho_m\rho =0$. This extra constraint implies that the fluid we obtain under LCR is not a generic Galilean\ fluid rather a restricted class. See section \ref{sec:review}. A generic Galilean\ fluid can have more terms in constitutive relations \cite{Banerjee:2015uta,Banerjee:2015hra}. Therefore, we find it interesting to obtain the Galilean\ fluid holographically and check if holographic results produce anything different than LCR. However, it turns out that both the approaches result the same fluid. Thermodynamic variables, computed holographically, also satisfy an extra condition $\varepsilon+p+\varrho_m \rho =0$ up to first order in derivative expansion. Probably there is no surprise in our results, as TsT transformation also involves a reduction of the bulk geometry along a light-cone direction to another geometry with Galilean\ asymptotic. An explicit check of results enriches our understanding.
We start with a locally boosted baulk geometry which is dual to a relativistic charged fluid with scale invariance. Scale invariance implies that the relativistic fluid has equation of state $E-3P=0$ and zero bulk viscosity. LCR of this fluid produces a non-relativistic fluid with equation of state $\varepsilon-p=0$ and with zero bulk viscosity (incompressible flow). We see that holographic computations also produces the same result. For uncharged fluid, transport coefficients obtained holographically (equation \ref{eq:unchargedtransport}) exactly matches with transports obtained by LCR (\ref{eq:LCRdictionarytransport}).
In case of charged fluid there are some apparent differences between holographic results and LCR. This is because \cite{Banerjee:2014mka} used a different basis ($\tau, \nu$ and $p$) than what we considered in this paper. However, if we look at first order correction to physical charge density we see that $Q$ has a parity-odd correction \(\displaystyle Q = 4\sqrt3 {q_0 u^+}- \frac{\l_q}{2\tau_0}\partial_+\tau + \frac{12\kappa q_0^2}{m_0}\epsilon^{ij}\partial_jv_j\). This correction term is same as we get after LCR (equation \ref{eq:diffdeffornrflu}). One striking difference is that holographic construction restricts the fluid to have time independent reduced chemical potential $\nu$.
\paragraph{The extra constraint for non-relativistic system :} The extra constraint relation $\varepsilon+p+\varrho_m\rho =0$ for Galilean\ fluid when reduced holographically or by LCR is natural\footnote{We are thankful to Nabamita Banerjee and Akash Jain for a discussion on this issue.}. In $(d+1)$ dimensions, a conformal relativistic fluid has $d+1$ constitutive relations. The number of variables which appear in constitutive relations and thermodynamics is $d+4$ ($d$ velocities, temperature $T$, energy $E$, pressure $P$ and entropy $S$). Among these variables one can take fluid velocity $u^i$ and temperature T as fluid variables and write energy momentum tensor as derivative expansion of $u^i$ and $T$ by solving constitutive equation. To solve the system exactly we need to know other thermodynamic variables (energy, pressure and entropy) in terms of temperature. Therefore we need three more equations. For relativistic fluid these equations are the first law of thermodynamics, Euler relation and equation of state :
\begin{eqnarray}\displaystyle
\begin{split}
dP -SdT &= 0 \ \text{(1st law)},\\
E+P-ST &=0 \ \text{(Euler relation)}, \\
E-3P &=0 \ \text{(for conformal fluid)} .
\end{split}
\end{eqnarray}
Solving these equations for relativistic fluid we have $S\sim T^3$, $E\sim 3 T^4$ and $P\sim T^4$ (up to an overall constant factor). Now consider a non-relativistic fluid in $d-1$ space dimensions (since LCR of a $(d+1)$ dimensional relativistic theory gives a non-relativistic system in $d-1$ space dimensions). This system has $d+1$ constitute equations : one continuity equation, $d-1$ momentum conservation equations and one energy conservation equations and ariables are : $d-1$ velocities, temperature $\tau$, mass chemical potential $\varrho_m$, energy density $\varepsilon$, pressure $p$, entropy $s$ and mass density $\rho$. Using constitutive equations one can write mass current, energy current and stress tensor as derivative expansion of velocities, temperature and mass chemical potential. Therefore we are left with four thermodynamic variables $\rho, \varepsilon, s$ and $p$. To solve the system we have three equations as before :
\begin{eqnarray}\displaystyle
\begin{split}
dp -s d\tau - \rho d\varrho_m &= 0 \ \text{(1st law)},\\
\varepsilon+p-s\tau -\varrho_m \rho &=0 \ \text{(Euler relation)}
\end{split}
\end{eqnarray}
and equation of state. If we consider the non-relativistic fluid obtained from a relativistic system, then equation of state is given by $\varepsilon-p=0$. With this equation one can solve the above two equations (first law and Euler equation) and find : \(\displaystyle \varepsilon = p \sim \tau^{c_1}\varrho_m^{2-c_1} \) with $c_1+c_2 = 2$. To make it consistent with LCR (or holographic) results : \(\varepsilon=P, \ \tau = {T\over u^+} \ \text{and}\ \varrho_m=-{1\over 2(u^+)^2}\), we have $c_1=4$ and $c_2 =-2$. These values of $c_1$ and $c_2$ can also be fixed, in stead, imposing and extra condition $\varepsilon+p+\varrho_m\rho=0$. Therefore, when we synthesis a non-relativistic fluid from an exactly solvable parent relativistic fluid we get that extra equation.
As mentioned in introduction, a more general class of Galilean\ fluid can be obtained by light cone reduction of a modified relativistic system called "null fluid", a relativistic fluid with an extra null isometry direction \cite{Banerjee:2015uta,Banerjee:2015hra,Jain:2015jla,Jain:2016rlz,Banerjee:2016qxf}. A reduction of this system in its symmetry broken phase is equivalent to the non-relativistic fluid without any extra constraint relation. Thus we have freedom to choose this extra condition. LCR provides that relation, null fluid construction leaves that to us. In that sense, LCR deals with a particular type of fluid, where as null fluid is more general. It would, therefore, be interesting to find a bulk dual for null fluid.
Thus we find that holographic computation of non-relativistic constitutive relations are in one-to-one correspondence with those obtained by LCR up to first order in derivative expansion.
\section{Introduction and summary}\label{sec:intro}
Gauge/gravity duality plays an important role to study different properties of fluid dynamics. The duality has been used extensively to obtain holographic stress-tensor and other conserved quantities up to second order in derivative expansion for a fairly general class of relativistic fluids. Holographic study opens up a new paradigm in fluid dynamics where different parity-odd transports are present. Study of parity-odd hydrodynamics has become a very fascinating topics in recent years.
Most of the recent developments in fluid dynamics are focused to relativistic systems. Much
attention has not been paid to non-relativistic systems\footnote{Non-relativistic conformal field theories with Galilean\ group symmetry consists of the usual Galilean invariance, the scaling symmetry as well as the particle number symmetry \cite{Duval:1984cj,Duval:2008jg,Duval:2009vt}.}. Non-relativistic fluids, in particular, are interesting as they are expected to be realised in low energy experiments. A non-relativistic systems can be thought of an effective low energy (velocity) description of an underlying relativistic systems. Hence, it is natural to expect that the constitutive relations of a non-relativistic fluid, obtained as an effective description of a relativistic theory, also contain parity-odd terms. There are different ways to take non-relativistic limit of a relativistic system. It is well known that under discrete light-cone reduction (LCR), $(d+1,1)$ dimensional Poincar\'e algebra boils down to $d$ dimensional Galilean algebra. Since hydrodynamics is a low energy fluctuations of equilibrium quantum field theory, LCR of relativistic constitutive relations produces constitutive relations of a Galilean fluid in one lower dimension \cite{Rangamani:2008gi,Brattan:2010bw,Banerjee:2014mka}. All the thermodynamic and hydrodynamic data of non-relativistic fluid are completely fixed in terms of the corresponding relativistic fluid data. However \cite{Banerjee:2014mka} observed that the reduced non-relativistic fluid obtained by LCR follows some restricted class of thermodynamics i.e. thermodynamic variables satisfy an extra equation. This extra condition follows from the demand that reduced non-relativistic fluid satisfies first law of thermodynamics and Euler relation locally. This extra relation implies that LCR provides a particular kind of fluid which satisfies that equation locally. A more general class of non-relativistic fluid\footnote{See also \cite{Geracie:2015xfa}.} can be obtained by light cone reduction if we start with a modified relativistic system called "null fluid" \cite{Banerjee:2015uta,Banerjee:2015hra}.
Gauge/gravity duality also offers a platform to explore properties of fluids order by order in derivative expansion \cite{Bhattacharyya:2008jc,Banerjee:2008th,Erdmenger:2008rm}. It would be interesting to understand whether holographic technique will result a generic class of Galilean/ Galilean\ fluid in comparison to LCR. In this paper we {\it holographically} construct the constitutive relations for a non-relativistic charged fluid up to first order in derivative expansion using the fluid/gravity correspondence. A similar study was done by \cite{Rangamani:2008gi,Brattan:2010bw}, where the authors computed non-relativistic currents and transport coefficients using the {\it LCR dictionary} between relativistic fluid and non-relativistic fluid\footnote{LCR provides a dictionary between non-relativistic fluid variables, transports and those of non-relativistic fluid. See section \ref{sec:review} for details.}. They used holographic values of relativistic transports \cite{Bhattacharyya:2008jc,Banerjee:2008th,Erdmenger:2008rm} to find transports of non-relativistic fluid using the LCR dictionary. A direct computation of fluid constitutive relations from bulk was missing in their work. In this paper, we follow a completely different route. We start with dual holographic geometry of a relativistic charged conformal fluid \cite{Banerjee:2008th,Erdmenger:2008rm}. We uplift this five dimensional solution to ten dimensions to fit as a solution of type IIB string theory. Performing TsT transformations followed by a dimensional reduction over five sphere we obtain an effective five dimensional geometry whose asymptotic symmetry group is not Poincar\'e but Schr\"odinger. This effective five dimensional geometry can be written as a solution of equations of motion obtained from an effective five dimensional action \cite{Herzog:2008wg,Adams:2009dm,Brattan:2010bw}. We find the boundary action for this effective theory for well defined variation. Following \cite{Ross:2009ar}, we compute the constitutive relations for Schr\"odinger fluid from this boundary action .
In summary, there are two different paths to find constitutive relations for non-relativistic fluid dynamics holographically.
\begin{itemize}
\item Path I : Construct dual holographic geometry of a relativistic fluid and apply AdS/CFT dictionary to obtain conserved currents and transports of boundary system \cite{Banerjee:2008th,Erdmenger:2008rm}. Obtain non-relativistic currents and transports by using the LCR dictionary between relativistic fluid and Galilean\ fluid \cite{Rangamani:2008gi,Brattan:2010bw}.
\item Path II : Consider holographic spacetime for relativistic fluid \cite{Kovtun:2003wp,Son:2006em}. Apply $TsT$ transformation on this geometry to obtain bulk
dual of an asymptotically Galilean\ fluid. Construct conserved current of boundary theory using AdS/CFT prescription \cite{Ross:2009ar} and find out the transports from those currents.
\end{itemize}
\begin{figure}[h]
\centering
\includegraphics[width=12cm,height=6cm]{dig}
\caption{Light-Cone Reduction vs TsT Transformations.}
\label{fig:dig}
\end{figure}
It would be interesting to understand if these two paths are compatible to all orders in derivative expansion. In this paper, we follow the second path to obtain non-relativistic constitutive relations for a $U(1)$ charged fluid. We observe that path I and path II are same up to first order in derivative expansion. The non-relativistic fluid obtained by LCR has restricted thermodynamics : LCR of relativistic first law and Euler relation provides correct first law and Euler relation for non-relativistic system at the cost of a constraint relation between non-relativistic energy density, pressure and mass density. In this scenes, LCR provides a restricted thermodynamic system. While obtaining the constitutive relations using holography it turns out that the $'+'$ component of relativistic velocity $u^+(x)$ plays the role of mass chemical potential and terms proportional to derivative of mass chemical potential ($\partial_i u^+$) appear in constitutive relations. We explicitly check that the non-relativistic fluid, obtained holographically is same as that obtained by LCR. We compare our results with \cite{Banerjee:2014mka} and show that the holographic non-relativistic fluid also belong to the same restricted class up to first order in derivative expansion. We summarise our observations in a commutative diagram given in figure \ref{fig:dig}.
Organization of this paper is following. In section \ref{sec:review} we briefly review the main results of LCR of relativistic charged fluid. Construction of bulk geometry for non-relativistic charged fluid has been discussed in section \ref{sec:bulk}. The main results (constitutive relations and transport coefficients for non-relativistic charged fluid) of this paper are given in section \ref{sec:stresstensor}. Finally, we provide a vivid discussion on our results in section \ref{sec:discussion}.
\section{Light cone reduction of relativistic fluids}\label{sec:review}
In this section we discuss how non-relativistic constitutive relations and transports can be obtained from relativistic data \cite{Rangamani:2008gi,Brattan:2010bw,Banerjee:2014mka}. We start with relativistic constitutive equations and reduce these equations along light-cone direction. Non-relativistic quantities like energy density, pressure etc. are given in terms of relativistic stress tensor.
A conformal relativistic fluid in flat $(d+1,1)$ dimensional spacetime with $U(1)$ isometry is given by the following two conservation equations
\begin{eqnarray}
\begin{split}
\partial_{\mu}T^{\mu\nu}&=0, \\
\partial_{\mu} J^{\mu}&=0
\end{split}
\end{eqnarray}
where $T^{\mu\nu}$ is energy momentum tensor, given by
\begin{eqnarray}
\begin{split}
T^{\mu\nu} &= (E+P)u^{\mu}u^{\nu}+P \eta_{\mu\nu}+\Pi^{\mu\nu}
\end{split}
\end{eqnarray}
and $J^{\mu}$ is conserved $U(1)$ current,
\begin{eqnarray}
\begin{split}
J^{\mu} &= n u^{\mu}+ \Upsilon^{\mu}.
\end{split}
\end{eqnarray}
$E, P$ and $n$ are energy density, pressure and charge density respectively. $\Pi^{\mu\nu}$ and $J^{\mu}$ are dissipative part of energy-momentum tensor and charge current. Up to first order in derivative expansion they are given by,
\begin{eqnarray}
\begin{split}
\Pi^{\mu\nu} &= - \eta_{rel} P^{\mu\alpha}P^{\nu\beta}\left[
\partial_\alpha u_\beta + \partial_\beta u_\alpha -
\frac{2}{d+1} \eta_{\alpha\beta}\partial\cdot u \right], \quad P^{\mu\nu}=\eta^{\mu\nu}+u^\mu u^\nu \\
\Upsilon^\mu & =
-\varrho P^{\mu\nu}\partial_\nu \left(\frac{M}{T}\right)
- \gamma P^{\mu\nu}\partial_\nu T
+ \mho l^\mu , \quad l^\mu = \epsilon^{\mu \alpha \beta
\gamma}u_\alpha\partial\beta u_\gamma
\end{split}
\end{eqnarray}
where $T$ and $M$ are temperature and $U(1)$ chemical potential. $\eta_{rel}, \varrho, \gamma$ are shear viscosity, charge conductivity, thermal conductivity respectively. $\mho$ is a parity-odd transport. Since we are dealing with relativistic conformal fluid (i.e. $T^\mu_{\ \mu}=0$) its bulk viscosity coefficient is zero and the fluid satisfies an equation of state $E=3P$.
We are considering the fluid in a flat background (in absence of any external fields). The metric is given by
\begin{equation}
ds^2 = -(dx^0)^2 + (dx^{d+1})^2+\sum_{i=1}^{d}(dx^i)^2.
\end{equation}
We introduce the light-cone coordinates,
\begin{equation}
x^{\pm} = \frac{1}{\sqrt2}\left ( x^0 \pm x^{d+1} \right ).
\end{equation}
In the light-cone frame the metric can be written as,
\begin{equation}
ds^2 = - 2 dx^+dx^- + \sum_{i=1}^{d} (dx^i)^2 .
\end{equation}
Since, the symmetry algebra of the relativistic theory reduces to corresponding non-relativistic symmetry algebra upon light-cone reduction, the QFT in this light-cone frame evolves in light-cone time $x^+$ for fixed light-cone momentum $P_-$ and thus we obtain a system in $d + 1$ dimensions with non-relativistic invariance. This is known as discrete light-cone quantization of quantum field theories.
Since hydrodynamics is low energy fluctuation of equilibrium quantum field theory, light-cone reduction of relativistic constitutive equations boil down to the non-relativistic constitutive equations for a fluid in one lower dimension. We consider a relativistic anomalous fluid system in $(3+1)$ dimensions and reduce the constitutive equation over light-cone coordinates and obtain the corresponding non-relativistic equations for a fluid in one lower dimensions. We also find a mapping between the degrees of freedom of the $(d+2)$-dimensional fluid to the degrees of freedom of the $(d +1)$-dimensional fluid.
We denote the $d$ spatial coordinates with $x^i$. Metric components in light-cone coordinates are given by
\begin{equation} \label{E:LCRgij}
g^{ij}=\delta^{ij} \quad\text{and}\quad g^{+-}=-1,
\end{equation}
rest are zero. Gradient operator is given by,
\begin{equation} \label{E:partialsDef}
\partial_{\mu}=\{ \partial_+,\partial_-,\partial_{i} \}
\quad\text{and}\quad \partial^{\mu}=\{
-\partial_-,-\partial_+,\partial_{i} \}.
\end{equation}
We shall reduce the theory along the $x^-$ direction, and consider $x^+$ to be the non-relativistic time. We consider only solutions to the relativistic equations that do not depend on $x^-$; that is, all derivatives $\partial_-$ vanish.
The reduction of relativistic conservation equations for energy-momentum and charge current are given by:
\begin{eqnarray}
\begin{split}
\partial_{+}T^{++}+\partial_{i}T^{+i}&=0, \quad
\partial_{+}T^{+-}+\partial_{i}T^{i-}=0\\
\partial_{+}T^{+j}+\partial_{i}T^{ij}&=0, \quad
\partial_+J^+ + \partial_i J^i=0.
\end{split}
\end{eqnarray}
These equations are similar to non-relativistic equations
\begin{eqnarray}
\begin{split}
\partial_t \rho+ \partial_i(\rho v^i)& =0,\quad
\partial_t (\rho v^i)+ \partial_i(t^{ij})=0,\\
\partial_t (\epsilon+ \frac{1}{2} \rho v^2)+ \partial_i(j^i) &=0,\quad
\partial_t Q+ \partial_i(j^i_q)=0
\end{split}
\end{eqnarray}
under following identifications
\begin{eqnarray} \label{Eq:identification}
\begin{split}
T^{++} & = \rho, \quad T^{i+}=\rho v^i, \quad
T^{+-} = \varepsilon + \frac{1}{2}\rho \mathbf{v}^2, \quad
T^{i-} =\varepsilon^i, \quad T^{ij} = t^{ij}, \\
& \hspace{3cm}J^{+} = Q, \qquad J^{i}=j^i_q.
\end{split}
\end{eqnarray}
We use these mappings to find the following relations between relativistic and non-relativistic parameters.
\begin{eqnarray}\label{eq:LCRdictionary}
\begin{split}
\rho &= (E+P)(u^+)^2,\quad
v^i = \frac{u^i}{u^+} -\frac{\eta}{\rho}\mathbf{Y}^i, \quad p = P ,
\quad \varepsilon =\frac{1}{2}(E-P)\\
\varepsilon^i &= \left(\epsilon + p +\frac{1}{2}\rho \mathbf{v}^2\right)v^i
-\eta \sigma^{ik} v_k -\kappa_T \partial^i \t - \t \sigma \partial^i\left(\frac{\mu}{\t}\right)
\end{split}
\end{eqnarray}
and
\begin{eqnarray}
\begin{split}
Q& =u^+n + u^+ \vp, \\
j^i &= Q v^i - \kappa_q \partial^i \t
- \t \sigma_q \partial^i\left (\frac{\mu}{\t}\right )
-\tilde m\partial^i p
+ \tilde \k_q \epsilon^{ij}\partial_j \t
+ \t \tilde{\sigma}_q\epsilon^{ij} \partial_j\left (\frac{\mu}{\t}\right )
-\bar m\epsilon^{ij} \partial_j p
\end{split}
\end{eqnarray}
where,
\begin{eqnarray}\label{eq:diffdeffornrflu}
\begin{split}
\mathbf{Y}^\mu &=\left( \partial^\mu u^+ - \frac{u^+\partial^\mu P}{E+P} \right),\quad
\vp =-\varrho u^\nu\partial_\nu \left(\frac{M}{T}\right) - \gamma u^\nu\partial_\nu T
- \mho \epsilon^{ij}u^+\nabla_i v_j.
\end{split}
\end{eqnarray}
Non-relativistic temperature, $U(1)$ chemical potential and charge density are given by
\begin{equation} \label{E:tIden}
\t =\frac{T}{u^+}, \qquad \mu = \frac{M}{u^+}, \qquad Q =
n u^+ .
\end{equation}
Non-relativistic transport coefficients are given by
\begin{eqnarray}\displaystyle\label{eq:LCRdictionarytransport}
\begin{split}
\eta &= \eta_{rel} u^+, \quad \kappa_T = \frac{\eta}{\t u^+}, \quad \sigma = \eta_{rel} \frac{Q}{\rho}, \quad \k_q = \frac{\k Q}{2(\varepsilon + p)}, \quad \t \sigma_q = \left[\varrho + \frac{\t \sigma Q}{2(\epsilon + p)} - \tilde m Q \t\right],\\
\tilde m &= \frac{\gamma \t u^+}{2(\varepsilon + p)}, \quad \tilde\k_q = \k_T\frac{\mho (u^+)^2}{\eta}, \quad \tilde\sigma_q = \sigma \frac{\mho (u^+)^2}{\eta}, \quad \bar m = \frac{2 \mho (u^+)^2 }{\rho}.
\end{split}
\end{eqnarray}
The above relations are the dictionary between relativistic and non-relativistic fluid data. Using fluid-gravity correspondence one can compute constitutive relations and transports for relativistic fluid \cite{Banerjee:2008th,Erdmenger:2008rm} and then using the above dictionary find transports for non-relativistic fluid \cite{Rangamani:2008gi,Brattan:2010bw}.
One can also check that starting from relativistic thermodynamic relations : $dE= TdS + M dn$ (first law) and $E+P = S T + n M$ (Euler relation) one obtains usual non-relativistic thermodynamics : $d\varepsilon=\tau ds+\mu dQ+\varrho_m d\rho$ (first law) and $\varepsilon+p = s\tau + \varrho_m \rho + \mu Q$ (Euler equation) if $\varepsilon+p+\varrho_m \rho =0$. The last equation puts a further restriction on non-relativistic thermodynamics.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,273 |
Gilnockie Tower is a 16th-century tower house, located at the hamlet of Hollows, 2.3 km north of Canonbie, in Dumfriesshire, south-west Scotland. The tower is situated on the west bank of the River Esk. It was originally known as Hollows Tower.
Gilnockie Castle is a separate, but nearby site.
History
The name Gilnockie is from the Scottish Gaelic Geal Cnocan meaning 'Little White Hill'. Hollows was built around 1520 by Johnnie Armstrong, famous Border outlaw and younger brother of Thomas Armstrong of Mangerton. In 1528, the tower was burned by Sir Christopher Dacre, English Warden of the Western Marches, and in 1530 Johnnie and 50 followers were hanged by James V, after being tricked into joining a hunting party, an event recorded in the ballad "Johnnie Armstrong".
The tower was rebuilt, but was damaged again by English raids in the 1540s, only to be rebuilt again with a new parapet walk, and a beacon stance on the gable.
Restoration
In 1978, the tower was a roofless ruin, when it was bought by Major T.C.R. Armstrong-Wilson, who undertook a full restoration. It was re-roofed, and floors were reconstructed at four levels. Authentic oak doors were fitted to all rooms. The interior was plastered out, and electricity and water taken into the building. The tower is a Category A listed building, and all work was carried out in consultation with the Scottish Development Department (Ancient Monuments). In January 2015, the tower was closed to the public and major repairs were commenced. The principal behind these repairs was to take the building back to as near possible of how it would have looked in the 16th century. It was also necessary to ensure that the building met the safety standards required of a 21st century visitor centre. The repairs were completed in 2018 and on 1 April 2018, Gilnockie Tower and the Clan Armstrong Centre were opened to the public. The building now consists of an authentic clan leaders house complete with furnished Grand Hall and Master Bedroom. The tower also houses the Clan Armstrong Museum, previously located in the Episcopal Church in Langholm. Many artefacts relating to the Clan Armstrong are on display and there is a special section devoted to Neil Alden Armstrong, the First Man on the Moon.
In 2019, Gilnockie Tower was awarded a 4 star rating from Visit Scotland as a visitor centre. It is internationally recognised as the ancestral home of the Armstrong Clan.
The tower
Gilnockie Tower is a simple rubble-built tower house of four storeys plus an attic, measuring around 10 by 7.6 metres at the base. The basement comprises a vaulted cellar, with gun loops to south, west and north. A spiral stair in the south-west corner leads up to the first floor, devoted to a hall. Above this are two further rooms, with the attic space above between the crow-stepped gables. At the top of the wall, corbels show the presence of a parapet walk. During the repair programme carried out between 2015 and 2018, the walkway was sympathetically brought up to date to meet safety standards. This included the installation of a safety fence. A notable feature is the beacon stance, corbelled out from the south gable at the highest point of the building.
The oldest part of the tower is thought to be the large stone by the doorway into the basement. Carvings of spirals and a key-like symbol are believed to date from the 2nd millennium BC, with the slab having been reused in the building.
Gilnockie Castle
The site now known as Gilnockie Castle lies near Canonbie at the east end of Gilnockie Bridge, which crosses the Esk in Hollows, just 500 m to the south east (). Today, only an earthwork remains, and there is some doubt as to whether a tower stood there, although it is possible that the earlier tower destroyed in 1528 was located there. It is associated with Johnnie Armstrong, Laird of Gilnockie.
See also
William Armstrong, one-time owner of the castle
References
Coventry, Martin The Castles of Scotland (3rd Edition), Goblinshead, 2001
Gifford, John The Buildings of Scotland: Dumfries and Galloway, Penguin, 1996
Maxwell-Irving, A. M. T. (2000) The Border Towers of Scotland, Creedon Publications
Salter, Mike The Castles of South West Scotland, Folly Publications, 1993
National Monuments Record of Scotland Site Reference NY37NE 3.0 (Gilnockie Tower)
NMRS Site Reference NY37NE 4.0 (Gilnockie Castle)
External links
Prehistoric Rock Art in Gilnockie Tower
Spiral Stairs
https://www.gilnockietower.co.uk Gilnockie Tower Reiver Centre, Canonbie
Castles in Dumfries and Galloway
Category A listed buildings in Dumfries and Galloway
Listed castles in Scotland
Tower houses in Scotland | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,847 |
Rabbi Ponders Security for Her Congregation at Har Shalom
Peter Christian Published: May 1, 2019
With anti Semitic violence outbreaks in this country and around the world, Missoula's Jewish Community, Har Shalom sits on a busy Russell Street, headed by Rabbi Laurie Franklin who is pondering the idea of security for her small but close-knit congregation.
"There have been ongoing discussions throughout the entire Jewish community about what we should be doing at this time and there's no one view," said Rabbi Franklin. "We're all united in that we all want to be safe and we want our places of worship to be inviting and open, and so we have to balance those qualities and then do what we have to do. It's a crazy world where we have to think about the possibility of locking our doors and the possibility of using specialized equipment to protect ourselves."
Franklin said a campaign is underway to replace their aging doors with sturdy doors with heavier glass and what are called 'panic bars' for security.
"We now also have a code-activated keypad on one of our doors so those who want to have access to the building," she said. "It's our responsibility to keep everyone in the building as safe as possible with regard for their humanity and with love. There is a watchfulness that we're all engaged in as part of the community. If we don't recognize someone, literally encountering them and saying hello so that we can know who's in the place and common sense things like that."
Franklin said there is a school that operates in the Har Shalom building during the week.
"We do have a Montessori School under our roof which we are blessed to have and so the keypad and code system is largely for their protection," she said. "What's interesting is that a couple of years ago, we were all walking around saying 'we don't want to do this', and yet in a very short time it has become very sadly our 'new normal'".
Franklin said that Har Shalom is a member of the Interfaith Collaborative with many of the Christian Churches but was not aware of the church security seminar this weekend at Christian Life Center. She said that Har Shalom has a close relationship with the First Christian Church just a few blocks away on Russell Street.
Filed Under: Anti-Semitism, Har Shalom Jewish Congregation, Security
Man Threatens St. Patrick's Hospital Staff With a Knife
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Butte Courthouse Increases Security For Murder Trial
Sen. Steve Daines Pushes to Protect American Energy Security
Montana State to Increase Security at Brawl of the Wild on Saturday
Daines Calls on Obama to Not Accept Syrian Refugees, Fix the "Root Cause" of the Crisis in Syria
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Belgium Steps Up Rail Security – U.S. Airman Being Treated at French Hospital
Senator Steve Daines Encourages President Barack Obama to Ensure Agreement Ends Nuke Threat with Iran | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,049 |
#pragma once
#include <aws/macie2/Macie2_EXPORTS.h>
#include <aws/core/utils/memory/stl/AWSString.h>
#include <aws/core/utils/memory/stl/AWSVector.h>
namespace Aws
{
namespace Macie2
{
class Macie2EndpointRules
{
public:
static Aws::String GetRulesAsString();
static const Aws::Vector<char> Rules;
};
} // namespace Macie2
} // namespace Aws
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,880 |
Some would say that we should never, ever tinker with the trinity of lettuce, tomato and onion. And if we had only one burger per year, maybe that would work. But since we've got a long summer ahead, we have more than enough time to expand our burger repertoire. Here are some of favorite burger combinations and updates.
Cedar planking isn't just for salmon. Bobby grills Cedar-Planked Burgers (pictured above) for a complex smokiness. If you ask him, it'll be the first thing you taste.
Before you squeeze it out of a bottle, take a step back. With a little simmer action, your very own homemade ketchup is just a stove pot away. For a blend perfectly tailored to the burger, Guy's Killer Inside-Out Burger with Worcestershire-Tomato Ketchup is spiked with fresh garlic, dill and even a little mustard seed.
Next time you're grilling up a few burgers, reach for Chinese barbecue in a bottle, hoisin sauce. During the burgers last few moments on the grill, simply brush it on for Hoisin Burgers. Even if ketchup and mustard are part of your typical regime, this cucumber-topped twist is sweet, salty and just begging for a swipe of Sriracha mayo.
We're accustomed to a good chili dog, but how about loading up a cheeseburger? Use leftover chili for a double dose of beef in the form of Loaded Chili Cheeseburgers.
You may be used to topping your burger with cheese, but topping it with cured meat like Genoa salami makes for a nice salty kick. Food Network Magazine's Italian Burgers are themed in the most refined sense, coming with balsamic-tossed arugula, mushrooms and fresh ricotta cheese. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,357 |
Kærlighedens melodi er en dansk komediefilm fra 1959 instrueret af Bent Christensen efter manuskript af Arvid Müller. Filmen bærende roller spilles af datidens populære sangduo Nina & Frederik. I filmen medvirker Louis Armstrong og hans orkester.
Filmen handler om det unge musikalske par, Susy og Peter (spillet af Nina & Frederik), der må gå meget igennem for kærligheden og musikken.
Medvirkende (udvalg)
Nina van Pallandt
Frederik van Pallandt
Preben Mahrt
Clara Østø
Gunnar Lauring
Else Marie Hansen
Holger Juul Hansen
Chr. Arhoff
Eksterne henvisninger
Komediefilm fra Danmark
Danske film fra 1959
Film fra Danmark (administrationskategori) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,259 |
Q: ios - Swift 2.2 - SpriteKit - sublassing SKSpriteNode class For a puzzle game project on iOS, I try to subclass the SKSpriteNode class from SpriteKit with the following code:
class SKPuzzle: SKSpriteNode {
var name2:String = "";
}
I need to add other variables in SKSpriteNode like another name (name2) in this case. Here is the use I made of the class in a class type SKScene:
class GameScene: SKScene {
let background = SKSpriteNode(imageNamed: "BW")
var selectedNode = SKPuzzle()
override init(size: CGSize) {
super.init(size: size)
let imageNames = [sheet.Puzzle13() , sheet.Puzzle19(),sheet.Puzzle30(),
sheet.Puzzle11(), sheet.Puzzle29(), sheet.Puzzle35() ]
for i in 0..<imageNames.count {
let imageName = imageNames[i]
let sprite = SKPuzzle(texture: imageName)
sprite.name = kAnimalNodeName
sprite.name2 = "\(i)"
let offsetFraction = (CGFloat(i) + 1.0)/(CGFloat(imageNames.count) + 1.0)
sprite.position = CGPoint(x: size.width * offsetFraction, y: size.height / 2)
sprite.zPosition = 1
background.addChild(sprite)
}
}
I have the sprite object from the subclass SKPuzzle wich contains the new variable name2.
sprite.name2 = "\(i)"
The problem I have is the variable selectedNode (created with)
var selectedNode = SKPuzzle()
used later in the program contain always a nil value for the data name and name2. When I click on the jigsaw parts of the game, I get the following error:
fatal error: unexpectly found nil while unwrapping an optional value in the following function:
func panForTranslation(translation : CGPoint) {
let position = selectedNode.position
if selectedNode.name! == kAnimalNodeName {
selectedNode.position = CGPoint(x: position.x + translation.x * 2, y: position.y + translation.y * 2)
}
}
selectedNode seems containing only nil values. The code is working fine when I just use the SKSpriteNode but failed with my SKPuzzle class.
Here is the whole code of the program:
import SpriteKit
import UIKit
private let kAnimalNodeName = "puzzle"
private let kdancing = "dancing"
class SKPuzzle: SKSpriteNode {
var name2:String = "";
}
class GameScene: SKScene {
let background = SKSpriteNode(imageNamed: "BW")
var selectedNode = SKPuzzle()
var selectedVideo = SKVideoNode()
override init(size: CGSize) {
super.init(size: size)
// 1
self.background.name = kdancing
self.background.anchorPoint = CGPointZero
background.zPosition = 0
self.addChild(background)
//background.play()
// 2
let sheet = Statiques()
let sprite_dancing1 = SKSpriteNode(texture: sheet.Dancing1())
let sprite_dancing2 = SKSpriteNode(texture: sheet.Dancing2())
sprite_dancing1.name = kdancing
sprite_dancing2.name = kdancing
sprite_dancing1.position = CGPoint(x: 837, y: 752)
sprite_dancing1.zPosition = 1
sprite_dancing2.position = CGPoint(x: 1241, y: 752)
sprite_dancing2.zPosition = 1
background.addChild(sprite_dancing1)
background.addChild(sprite_dancing2)
let imageNames = [sheet.Puzzle13() , sheet.Puzzle19(), sheet.Puzzle30(), sheet.Puzzle11(), sheet.Puzzle29(), sheet.Puzzle35() ]
for i in 0..<imageNames.count {
let imageName = imageNames[i]
let sprite = SKPuzzle(texture: imageName)
sprite.name = kAnimalNodeName
sprite.name2 = "\(i)"
let offsetFraction = (CGFloat(i) + 1.0)/(CGFloat(imageNames.count) + 1.0)
sprite.position = CGPoint(x: size.width * offsetFraction, y: size.height / 2)
sprite.zPosition = 1
background.addChild(sprite)
}
}
required init?(coder aDecoder: NSCoder) {
fatalError("init(coder:) has not been implemented")
}
override func touchesBegan(touches: Set<UITouch>, withEvent event: UIEvent?) {
for touch: AnyObject in touches {
let positionInScene = touch.locationInNode(self)
selectNodeForTouch(positionInScene)
}
}
override func didMoveToView(view: SKView) {
let gestureRecognizer = UIPanGestureRecognizer(target: self, action: Selector("handlePanFrom:"))
self.view!.addGestureRecognizer(gestureRecognizer)
}
func handlePanFrom(recognizer : UIPanGestureRecognizer) {
if recognizer.state == .Began {
var touchLocation = recognizer.locationInView(recognizer.view)
touchLocation = self.convertPointFromView(touchLocation)
self.selectNodeForTouch(touchLocation)
} else if recognizer.state == .Changed {
var translation = recognizer.translationInView(recognizer.view!)
translation = CGPoint(x: translation.x, y: -translation.y)
self.panForTranslation(translation)
recognizer.setTranslation(CGPointZero, inView: recognizer.view)
} else if recognizer.state == .Ended {
}
}
func degToRad(degree: Double) -> CGFloat {
return CGFloat(degree / 180.0 * M_PI)
}
func selectNodeForTouch(touchLocation : CGPoint) {
// 1
let touchedNode = self.nodeAtPoint(touchLocation)
if touchedNode is SKPuzzle {
// 2
if !selectedNode.isEqual(touchedNode) {
selectedNode.removeAllActions()
selectedNode.runAction(SKAction.rotateToAngle(0.0, duration: 0.1))
//selectedNode = touchedNode as! SKSpriteNode
// 3
if touchedNode.name! == kAnimalNodeName {
let sequence = SKAction.sequence([SKAction.rotateByAngle(degToRad(-4.0), duration: 0.1),
SKAction.rotateByAngle(0.0, duration: 0.1),
SKAction.rotateByAngle(degToRad(4.0), duration: 0.1)])
selectedNode.runAction(SKAction.repeatActionForever(sequence))
}
}
}
}
func panForTranslation(translation : CGPoint) {
let position = selectedNode.position
if selectedNode.name! == kAnimalNodeName {
selectedNode.position = CGPoint(x: position.x + translation.x * 2, y: position.y + translation.y * 2)
}
}
}
Thanks by advance for your help, I'm beginning to code with Swift on iOS.
A: When selectedNode is created by SKPuzzle(), name is nil.
You have to set selectedNode.name to some value in init method.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,638 |
Rockwell Automation announces acquisition of Odos Imaging
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About Rockwell Automation
Rockwell Automation is a company dedicated to industrial automation and information, making its customers more productive and the world more sustainable. Headquartered in Milwaukee, Wis., Rockwell Automation employs approximately 22,000 people serving customers in more than 80 countries.
Vision, Factory Automation, Advancing Automation using IIoT and Industry 4.0 Concepts, Systems Integration, Sensors & Industrial I/O
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,574 |
2017-2018 Arizona Exemplary Title I Program Awards!
The school vision included providing an exceptional education for all students through nurturing individual greatness via purposeful, rigorous instruction and a focus on student leadership development. Our mission is to provide a collaborative environment focused on the development of innovative leaders with strong character while ensuring learning for all.
• Focus on using data to ensure students understand the why, what and how of learning through universal design for learning practices.
Inclusive schools are conducive to student learning, fulfillment, and well-being, as well as professional satisfaction, morale, and effectiveness. The school met the challenge of ensuring students physical safety needs were met by examining procedures and revamping their Comprehensive Emergency Plan.
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As a result, they implemented Kids at Hope. The Kids at Hope's vision is that every child is afforded the belief, guidance and encouragement that creates a sense of hope and optimism, supported by a course of action needed to experience success.
(4) be engaged in the community".
• Structures to support families and the community: Parent Resource Center, WIC, SW Behavioral Health on site, Literacy and STEAM nights.
• Every student receives a backpack at the beginning of the year with supplies and four books at their Literacy night. Not some students, ALL 710 students.
The Arizona Department of Education, the Educator and School Excellence Unit and the office of Arts Education would like to thank and acknowledge the Desert Edge High School students for their artful creations and the support of our 2017-2018 Title I "Exemplary School" Awards! | {
"redpajama_set_name": "RedPajamaC4"
} | 3,128 |
{"url":"http:\/\/mathhelpforum.com\/calculus\/282394-c4-further-differentiation.html","text":"1. ## C4 Further Differentiation\n\nQ: if the velocity v is given by the formula$\\displaystyle v = \\frac{u}{(1+ks)}$ where u is the initial velocity, s is the distance and k is a constant prove that the accleration varies as $\\displaystyle v^{3}$\n\nmy attempt:\n$\\displaystyle v \\frac{u}{(1+ks)}=\\frac{a}{b}$\n$\\displaystyle \\frac{\\mathrm{d}v}{\\mathrm{d}s}=\\frac{b\\frac{\\math rm{d}a}{\\mathrm{d} s}-a\\frac{\\mathrm{d} b}{\\mathrm{d} s}}{b^2}$\n$\\displaystyle a=u; \\frac{\\mathrm{d} a}{\\mathrm{d} s} =0$\n$\\displaystyle b = 1+ks; \\frac{\\mathrm{d} b}{\\mathrm{d} s}= k$\n\n$\\displaystyle \\frac{\\mathrm{d} v}{\\mathrm{d} s}= \\frac{((1+ks)(0)-u(k))}{(1+ks)^2}b$\n\n$\\displaystyle \\frac{\\mathrm{d} v}{\\mathrm{d} s}= \\frac{-uk}{(1+ks)^2}$\n\nwe know that a = v . dv\/ds\nthus; $\\displaystyle a = \\frac{u}{(1+ks)} \\times \\frac{-uk}{(1+ks)^2}$\n\n$\\displaystyle a = \\frac{-u^{2}k}{(1+ks)^2}$\n\nnote $\\displaystyle s = \\frac{u-v}{kv}$ sub s into a therefore;\n$\\displaystyle a = -\\frac{kv^3}{u}$\n\nthus as v tend to infinity, $\\displaystyle v^{3}$ tends to infinity thus acceleration varies as $\\displaystyle v^{3}$\n\nI want to know if I am right?\n\nAlso the bold part is the correct reasoning?\n\n2. ## Re: C4 Further Differentiation\n\nOriginally Posted by bigmansouf\nQ: if the velocity v is given by the formula$\\displaystyle v = \\frac{u}{(1+ks)}$ where u is the initial velocity, s is the distance and k is a constant prove that the accleration varies as $\\displaystyle v^{3}$\n\nmy attempt:\n$\\displaystyle v \\frac{u}{(1+ks)}=\\frac{a}{b}$\n$\\displaystyle \\frac{\\mathrm{d}v}{\\mathrm{d}s}=\\frac{b\\frac{\\math rm{d}a}{\\mathrm{d} s}-a\\frac{\\mathrm{d} b}{\\mathrm{d} s}}{b^2}$\n$\\displaystyle a=u; \\frac{\\mathrm{d} a}{\\mathrm{d} s} =0$\n$\\displaystyle b = 1+ks; \\frac{\\mathrm{d} b}{\\mathrm{d} s}= k$\n\n$\\displaystyle \\frac{\\mathrm{d} v}{\\mathrm{d} s}= \\frac{((1+ks)(0)-u(k))}{(1+ks)^2}b$\n\n$\\displaystyle \\frac{\\mathrm{d} v}{\\mathrm{d} s}= \\frac{-uk}{(1+ks)^2}$\n\nwe know that a = v . dv\/ds\nthus; $\\displaystyle a = \\frac{u}{(1+ks)} \\times \\frac{-uk}{(1+ks)^2}$\n\n$\\displaystyle a = \\frac{-u^{2}k}{(1+ks)^2}$\n\nnote $\\displaystyle s = \\frac{u-v}{kv}$ sub s into a therefore;\n$\\displaystyle a = -\\frac{kv^3}{u}$\n\nthus as v tend to infinity, $\\displaystyle v^{3}$ tends to infinity thus acceleration varies as $\\displaystyle v^{3}$\n\nI want to know if I am right?\n\nAlso the bold part is the correct reasoning?\nYou've got your dv\/ds correct but I'd advise using \"a\" in any problem that uses acceleration. It could be very confusing.\n\nYou derived the (correct) relationship between a and v. But you don't need to let anything go to infinity, a is already proportional to v^3 without that. All we need to do to finish the problem is to derive a = (constant)v^3 and we already have that.\n\n-Dan\n\n3. ## Re: C4 Further Differentiation\n\n$\\dfrac{d}{dt} \\bigg[v = v_0 (1+ks)^{-1} \\bigg]$\n\n$\\dfrac{dv}{dt} = -v_0(1+ks)^{-2} \\cdot k \\cdot \\dfrac{ds}{dt}$\n\n$a = -\\dfrac{v_0}{1+ks} \\cdot \\dfrac{k}{1+ks} \\cdot v$\n\n$a = -\\dfrac{v_0}{1+ks} \\cdot \\dfrac{k}{v_0} \\cdot \\dfrac{v_0}{1+ks} \\cdot v$\n\n$a = -\\dfrac{k}{v_0} \\cdot v^3$\n\nletting the constant $b = -\\dfrac{k}{v_0} \\implies a = bv^3$\n\nacceleration varies directly with velocity cubed.\n\n4. ## Re: C4 Further Differentiation\n\nwell you got the right answer in that\n\n$a(t) = -\\dfrac{k(v(t))^3}{u}$\n\nthus\n\n$a(t) \\sim (v(t))^3$\n\nYou don't need to bring limits into this\n\n5. ## Re: C4 Further Differentiation\n\nOriginally Posted by topsquark\nYou've got your dv\/ds correct but I'd advise using \"a\" in any problem that uses acceleration. It could be very confusing.\n\nYou derived the (correct) relationship between a and v. But you don't need to let anything go to infinity, a is already proportional to v^3 without that. All we need to do to finish the problem is to derive a = (constant)v^3 and we already have that.\n\n-Dan\nthank you\n\n6. ## Re: C4 Further Differentiation\n\nOriginally Posted by romsek\nwell you got the right answer in that\n\n$a(t) = -\\dfrac{k(v(t))^3}{u}$\n\nthus\n\n$a(t) \\sim (v(t))^3$\n\nYou don't need to bring limits into this\nthank you\n\n7. ## Re: C4 Further Differentiation\n\nOriginally Posted by Cervesa\n$\\dfrac{d}{dt} \\bigg[v = v_0 (1+ks)^{-1} \\bigg]$\n\n$\\dfrac{dv}{dt} = -v_0(1+ks)^{-2} \\cdot k \\cdot \\dfrac{ds}{dt}$\n\n$a = -\\dfrac{v_0}{1+ks} \\cdot \\dfrac{k}{1+ks} \\cdot v$\n\n$a = -\\dfrac{v_0}{1+ks} \\cdot \\dfrac{k}{v_0} \\cdot \\dfrac{v_0}{1+ks} \\cdot v$\n\n$a = -\\dfrac{k}{v_0} \\cdot v^3$\n\nletting the constant $b = -\\dfrac{k}{v_0} \\implies a = bv^3$\n\nacceleration varies directly with velocity cubed.","date":"2019-03-24 10:46:47","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\": 0.9241060018539429, \"perplexity\": 1290.3114095602862}, \"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-2019-13\/segments\/1552912203438.69\/warc\/CC-MAIN-20190324103739-20190324125739-00062.warc.gz\"}"} | null | null |
\section{Introduction}
\vspace{-0.5em}
A multivariate Gaussian distribution $q(x)$ can be expressed in the information form as \cite{WalkSum1}
$$
q(x)\propto\exp\Big\{
-\frac{1}{2}x^TJx + h^Tx
\Big\},
$$
where $J$, termed as the information matrix, is a symmetric, positive definite matrix ($J\succ 0$) and $h$ is the potential vector.
Conditional independence relationship among variables in $x$ can be viewed via an undirected graph also known as the
Gaussian Markov random field (GMRF).
To construct the GMRF, the underlying Gaussian distribution is first factorized as
\begin{equation}\label{MRF-eqn}
q\left( x\right)
\propto
\prod_{i\in \mathcal{V}} \psi_{i} \left( x_i \right)
\prod_{J_{i,j}\neq 0; i\leq j } \psi_{i,j} \left( x_i, {x}_j\right),
\end{equation}
where
\vspace{-0.5em}
\begin{equation}\label{pairwise}
\psi_{i} \left(x_i\right) = \exp\{-\frac{1}{2}J_{i,i}x^2_i + h_ix_i\}
\quad
\textrm{and} \quad
\psi_{i,j} \left(x_i, x_j\right) = \exp\{-x_iJ_{i,j}x_j\}.
\end{equation}
The associated GMRF contains a set of vertices corresponding to each random variable $x_i$, and each vertex for $x_i$ is associated with the \textit{node potential function} $\psi_{i} \left(x_i\right)$. If $J_{i,j}\neq 0$, there is an edge between $x_i$ and $x_j$, which is associated with the \textit{edge potential function} $\psi_{i} \left(x_i, x_j\right)$.
Fig. 1(a) is an example of the GMRF corresponding to an instance of $J$ in (\ref{J}).
For inference problems, given a joint Gaussian distribution $q(x)$, one is often interested in computing the marginal distribution for each variable $x_i$ in $x$, which is equivalent to determining the mean vector
$\mu := \mathbb E\{x\} $ and diagonal elements of the covariance matrix $P := \mathbb E\{(x-\mu)(x-\mu)^T\} $ with
\begin{equation}\label{1}
\mu = J^{-1}h \quad \textrm{and} \quad P = J^{-1}.
\end{equation}
However, directly inverting $J$ has computational complexity order
$\mathcal{O}\left(M^3\right)$ with $M$ being the dimension of $x$.
Moreover, in large-scale networks with random variables $x_i$ associated with different agents in the network, computing $J^{-1}$ at a fusion center may also suffers from large communication overhead, heavy computation burden, and be susceptible to central agent failure.
Dealing with highly distributed data has been recognized by the U.S. National Research Council as one of the big challenges for processing big data \cite{MassiveData}.
Therefore, distributed processing to infer $\mu$ and diagonal elements of $P$ that
only requires local communication and local computation is important for problems arising in distributed networks \cite{du2014TSP, du2016convergence}.
Gaussian belief propagation (GBP)
\cite{DiagnalDominant} provides an efficient way for computing the marginal mean in (\ref{1}).
Although with great empirical success \cite{Murphy}, and as recognized in different research areas, it is known that a major challenge that hinders GBP is the lack of convergence guarantees in loopy networks.
Convergence of other forms of loopy BP are analyzed in \cite{chertkov2006loop, NIPS2016_6318, gomez2007truncating,NIPS2012_4649,Ihler05, Martin13}, but these analyses are not directly
applicable to GBP.
All previous convergence analyses of GBP have focused on GMRFs \cite{DiagnalDominant,WalkSum1,minsum09}, where the the joint distribution is factorized according to (\ref{MRF-eqn}).
Recently, a distributed convergence condition for both GMRF and Gaussian linear model is proposed in \cite{du2017verification}.
In \cite{WalkSum1},
based on the fact that $(J^{-1})_{i,j}$ can be interpreted as the sum of the weights of all the walks from $j$ to $i$ on the corresponding GMRF, a sufficient convergence condition, given by the spectrum radius
$\rho(|I - J|)<1$, is obtained, commonly known as walk-summable property.
We emphasize two important points here. First, restricting to GMRFs, the recursive update structure of the GBP obtained on the basis of the usual GMRF factorization (\ref{MRF-eqn}) could be different from the recursive updates obtained using other types of factorizations such as those based on the distributed linear Gaussian model representation of the GMRF studied in this paper. Secondly, and importantly, there exist GMRF scenarios (see example below) in which the information matrix $J$ fails to satisfy the walk-summable property, although, a GBP update based on a different factorization of the GMRF (specifically, based on the distributed linear Gaussian model representation of the GMRF as studied in this paper) may be
obtained that is shown to yield convergence. Before proceeding further, consider the following GMRF in which the information matrix $J$ is given by
\begin{equation}\label{J}
J=\left[
\begin{matrix}
\begin{smallmatrix}
1&\frac{1}{3\sqrt{2}}& \frac{1}{\sqrt{3}} & \frac{\sqrt{2}}{3} \\
\frac{1}{3\sqrt{2}}&1&0&\frac{1}{3} \\
\frac{1}{\sqrt{3}}&0&1&\frac{1}{\sqrt{6}}\\
\frac{\sqrt{2}}{3}&\frac{1}{3}&\frac{1}{\sqrt{6}}&1
\end{smallmatrix}
\end{matrix}
\right],
\end{equation}
which satisfies $J\succ 0$ and $J = J^T$.
Since $\rho(|I-J|)=1.0754$, which is non walk-summable \cite{WalkSum1}, the convergence condition in \cite{WalkSum1} is inconclusive as to whether GBP converges for this GMRF.
Rather than the GMRF factorization,
we study the GBP convergence of this example by employing a different factorization method based on the linear Gaussian model representation.
To this end, we rewrite
$J$ as
$J = A^TR^{-1}A + W^{-1}$, where
$
A=\left[
\begin{matrix}
\begin{smallmatrix}
\frac{2}{\sqrt{6}}& 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\
\frac{{1}}{\sqrt{6}}&\frac{{1}}{\sqrt{3}}&0&0 \\
0&\frac{{1}}{\sqrt{3}}&0&\frac{{1}}{\sqrt{3}}
\end{smallmatrix}
\end{matrix}
\right]$,
$
W=\left[
\begin{matrix}
\begin{smallmatrix}
6& 0 & 0 & 0 \\
0& 3&0&0 \\
0&0&2&0\\
0&0&0&3
\end{smallmatrix}
\end{matrix}
\right],
$
and $R = I$.
Note that $J^{-1}=(A^TR^{-1}A + W^{-1})^{-1}$ is the covariance matrix for the joint posterior distribution of the linear Gaussian model
\begin{equation}\label{linearC}
y = A {x} + {z}
\end{equation}
with ${z}\sim \mathcal{N}\left( {z}| {0}, {R}\right)$ and $x\sim \mathcal{N}\left( {x}| {0}, W\right)$.
Consequently,
to perform the marginalization equivalently, GBP is employed on an alternative
factorization of $q(x)$, given by
\vspace{-0.5em}
\begin{equation}\label{jointpost1}
q\left( x\right)
\propto
p\left( {y}_1| {x}_1,{x}_2,{x}_3 \right)
p\left( {y}_2| {x}_1,{x}_2 \right)
p\left( {y}_3| {x}_2,{x}_3 \right)
\prod_{n=1}^3
p\left(x_n\right).
\end{equation}
This factorization stems from the products of the local likelihood functions and local prior distributions associated with (\ref{linearC}).
This can be
expressed by a
factor graph,
where every variable $ {x}_i$ is represented by a circle (called a variable node)
and the probability distribution of a variable or a group of variables is represented by a square (called a factor node).
A variable node is connected to a factor node if the variable is an argument of that particular factor.
For example, Fig.~1(b) shows the factor graph representation for the GMRF in Fig.~1(a)
\footnote{Factor graphs represent the same Markov relationship among variables as in the GMRFs with the same $q(x)$ \cite[Chapter~9]{mezard2009information}.}.
In Theorem \ref{ref}, we successfully show that, for a factor graph
that is the union of a forest and a single loop, as in Fig.~1(b), GBP always converges to the exact $\mu_i$.
This is in sharp contrast to the fact that for the same joint distribution and with factorization based on the classical GMRF representation in (\ref{MRF-eqn}), existing conditions and analyses are inconclusive as to whether GBP will converge or not.
We note that the factor graph based representation of GMRFs and distributed linear Gaussian models as in (\ref{linearC})-(\ref{jointpost1})
arise
in a variety of areas including image interpolation \cite{image-interp}, cooperative localization \cite{Win}, distributed beamforming \cite{A2}, distributed synchronization \cite{JianClock, JianCFO, du2017proactive, du2013fully}, fast solver for system of linear equations \cite{A4}, factor analysis learning \cite{A6}, sparse Bayesian learning \cite{A7}, and peer-to-peer rating in social networks \cite{A9}, in which it is of interest to compute the $\mu_{i}$ in a distributed fashion.
Recently, \cite{giscard2016exact} proposes an exact inference method to compute $\mu$ and $J^{-1}$ based on path sum on the GMRF with arbitrary topology. GBP, by contrast, only targets to compute $\mu$ and the diagonal elements of $J^{-1}$, the parameters of the associated marginal distributions.
Different from GBP that only requires local computation and communication, \cite{giscard2016exact} requires summing over all the simple paths and simple cycles on the graph, which requires centralized processing for computing all the simple paths and simple cycles as well as centralized scheduling.
\begin{figure}\label{SystemModel}
\centering
\mbox{\subfigure[]{\epsfig{figure=./mrf.PNG,width=1.2in}}\label{Network} }
\quad
\mbox{ \subfigure[]{\epsfig{figure=./fg.PNG,width=1.8in}}\label{MRF} }
\caption{(a) The GMRF corresponding to $J$ in (\ref{J}) with the factorization following (\ref{MRF-eqn});
(b) The factor graph corresponding to $J$ in (\ref{J}) with the factorization following (\ref{jointpost}).
}
\end{figure}
In this paper, we first study analytically the convergence of
GBP for the linear Gaussian models.
The distributed algorithm based on GBP involves only local computation and local communication, and thus, scales with the network size.
We prove
convergence of the
resulting GBP distributed algorithm.
Specifically, by establishing certain contractive properties of the dynamical system modeling the distributed inverse covariance updates under the Birkhoff metric, we show that,
with arbitrary nonnegative initial message inverse variance,
the belief variance for each variable converges at a geometric rate to a unique positive limit.
We
also demonstrate that
there is a better choice
for initializing the message inverse variance
than the commonly used zero initial condition
to achieve faster convergence.
Further, we prove, under the proposed necessary and sufficient convergence condition, the belief mean
converges to the optimal value.
In particular, we show that for a graph that is given by the union of a single loop and a forest, GBP always converges.
\section{Distributed Inference with GBP on Linear Gaussian Model}
\label{headings}
Consider a general connected network
of $M$ agents, with $\mathcal{V}=\left\{1,\ldots, M\right\}$ denoting the set of agents.
Agent $n$ has an unknown random variable $x_n$ and makes a local linear observation
${y}_n = \sum_{i\in n\cup\mathcal{I}\left(n\right)}
{A}_{n,i}{x}_i + {z}_n$,
where
$\mathcal{I}\left(n\right)$ denotes the set of neighbors of agent $n$,
and
${A}_{n,i}$ is a known coefficient\footnote{This model also allows two neighboring agents to share a common observation \cite{du2017pairwise}, e.g., $y_i = y_j = G(x_i-x_j)+z_i$. Our analysis applies to this case.}.
The prior distribution for $x_i$ is Gaussian, i.e., ${x}_i\sim \mathcal{N}\left({x}_i|{0},{W}_{i}\right)$,
and ${z}_n$ is the additive noise with distribution ${z}_n\sim \mathcal{N}\left({z}_n|{0},{R}_n\right)$
\footnote{If the $y_n$ is noiseless, it would represent linear equality constraints among the variables, i.e., ${y}_n = \sum_{i\in n\cup\mathcal{I}\left(n\right)}
{A}_{n,i}{x}_i$, which conflicts with $J>0$ that we assume. Thus, we assume $R_n>0$ in this paper.}.
It is assumed that
$p\left({x}_i, {x}_j\right)=p\left({x}_i\right)p\left({x}_j\right)$
and
$p\left({z}_i,{z}_j\right)
=p\left({z}_i\right)p\left({z}_j\right)$ for $i\neq j$.
The goal is to learn ${x}_i$, based on ${y}_n$, $p\left({x}_i\right)$, and $p\left({z}_n\right)$.
The joint distribution $p\left( x\right)p\left( {y}| x\right)$ can be written as the product of the prior distribution and the local likelihood function as
\begin{equation}\label{jointpost}
p\left( x\right)p\left( {y}| x\right) =
\prod_{n\in \mathcal{V}}
p\left(x_n\right)
\prod_{n\in \mathcal{V}}
\underbrace{p\left( {y}_n| \left\{ {x}_i\right\}_{i\in n\cup\mathcal{I}\left(n\right)} \right)}_{:= f_n }.
\end{equation}
We derive the Gaussian
BP algorithm over the corresponding factor graph to learn $ x_n$ for all $n\in \mathcal V$. It
involves two types of messages:
one is the message
from a variable node $ x_j$ to its neighboring factor node $f_n$, defined as
\begin{equation} \label{BPv2f1}
m^{\left(\ell\right)}_{j \to f_n}\left( x_j\right)
= p\left( x_j\right)
\prod_{f_k\in \mathcal B(j)\setminus f_n}m^{\left(\ell-1\right)}_{f_k\to j}\left( x_j\right),
\end{equation}
where $\mathcal B\left(j\right)$ denotes the set of neighbouring factor nodes of $ x_j$,
and $m^{\left(\ell-1\right)}_{f_k\to j}\left( x_j\right)$ is the message from $f_k$ to $ x_j$ at $(\ell-1)$-iteration.
The second type of
message is from a factor node $f_n$ to a neighboring variable node $ {x}_i$, defined as
\begin{equation}\label{BPf2v1}
m^{\left(\ell\right)}_{f_n \to i}\left( {x}_i\right)
= \int \cdots \int
f_n \times \!
\prod_{j\in\mathcal B\left(f_n\right)\setminus i} m^{ \left(\ell\right)}_{j \to f_n}\left( x_j\right)
\,\mathrm{d}\left\{ x_j\right\}_{j\in\B\left(f_n\right)\setminus i},
\end{equation}
where $\mathcal B\left(f_n\right)$ denotes the set of neighboring variable nodes of $f_n$.
The process iterates between equations (\ref{BPv2f1}) and (\ref{BPf2v1}).
At each iteration $\ell$, the approximate marginal distribution, also referred to as belief, on $ {x}_i$ is computed locally at $ {x}_i$ as
\begin{equation} \label{BPbelief}
b_{\textrm{BP}}^{\left(\ell\right)}\left( {x}_i\right)
= p\left( {x}_i\right) \prod_{ f_n\in \mathcal B\left(i\right)} m^{\left(\ell\right)}_{ f_n \to i}\left( {x}_i\right).
\end{equation}
Let
the initial messages at each variable node and factor node be
in Gaussian function forms as
$m^{\left(0\right)}_{f_n \to i}\left( {x}_i\right)\propto
\exp
\big\{-\frac{1}{2}
|| {x}_i- {v}^{\left(0\right)}_{f_n\to i}||^2
_{{J}^{\left(0\right)}_{f_n\to i}}
\big\}$
and $
m^{\left(0\right)}_{j \to f_n}\left( x_j\right)\propto
\exp\big\{-\frac{1}{2}|| x_j- {v}^{\left(0\right)}_{j\to f_n}||^2_{ {J}^{\left(0\right)}_{j \to f_n}}\big\}.$
It is evident that the general expression for the message from variable node $j$ to factor node $f_n$ is
$ m^{\left(\ell\right)}_{j \to f_n}\left( x_j\right) \propto
\exp
\big\{-\frac{1}{2}
|| x_j- {v}^{\left(\ell\right)}_{j\to f_n}||^2
_{ {J}^{\left(\ell\right)}_{j\to f_n}}
\big\}$,
with
\begin{equation} \label{v2fV}
{J}^{\left(\ell\right)}_{j \to f_n}
= {W}_j^{-1} +
\sum_{f_k\in\B\left(j\right)\setminus f_n}
{J}_{f_k\to j}^{\left(\ell-1\right)},
\quad
{v}^{\left(\ell\right)}_{j\to f_n}=
\left[{J}^{\left(\ell\right)}_{j\to f_n}\right]^{-1}
\sum_{f_k\in\B\left(j\right)\setminus f_n}
{J}_{f_k\to j}^{\left(\ell-1\right)}
{v}^{\left(\ell-1\right)}_{f_k\to j},
\end{equation}
where $ {J}_{f_k\to j}^{\left(\ell-1\right)}$ and $ {v}_{f_k\to j}^{\left(\ell-1\right)}$ are the message inverse variance and mean received at variable node $j$ at the $\left(\ell-1\right)$-$\textrm{th}$ iteration, respectively.
Furthermore,
the message from factor node $f_n$ to variable node $i$ is given by
$ m^{\left(\ell\right)}_{f_n \to i}\left( {x}_i\right)\propto
\exp
\big\{-\frac{1}{2}
|| {x}_i- {v}^{\left(\ell\right)}_{f_n\to i}||^2
_{ {J}^{\left(\ell\right)}_{f_n\to i}}
\big\}$,
with
\begin{equation}\label{Cov}
{J}^{\left(\ell\right)}_{f_n\to i}
=
{A}_{n,i}^2
\Big[ {R}_n
+\!\!\!\!\!\!\!\!
\sum_{j\in\B\left(f_n\right)\setminus i}\!\!\!\! {A}_{n,j}^2 \left[{J}^{\left(\ell\right)}_{j\to f_n}\right]^{-1} \Big]^{-1}\!\!\!,
\quad
{v}^{\left(\ell\right)}_{f_n\to i}
=
{A}_{n,i}^{-1}
\Big( {y}_n-\!\!\!\!\!\!\!\!\sum_{j\in\B\left(f_n\right)\setminus i}
\!\!\!\! {A}_{n,j}
{v}^{\left(\ell\right)}_{j\to f_n}\Big).
\end{equation}
For this factor graph based approach, according to the message updating procedure (\ref{v2fV}) and (\ref{Cov}), message exchange is only needed between neighboring nodes.
For example, the messages transmitted from node $n$ to its neighboring node $i$ are $m_{f_n\to i}^{\left(\ell\right)}\left( {x}_i\right)$ and $m_{n\to f_i}^{\left(\ell\right)}\left( x_n\right)$.
Thus, the message passing scheme given in (\ref{BPv2f1}) and (\ref{BPf2v1}) conforms with the network topology.
Furthermore, if the messages $m_{j\to f_n}^{\left(\ell\right)}\left( x_j\right)$
and $m_{f_n\to i}^{\left(\ell\right)}\left( {x}_i\right)$ exist for all $\ell$,
the messages are Gaussian, therefore only the corresponding mean and inverse of variance are needed to be exchanged.
Finally, according to the definition of belief in (\ref{BPbelief}),
$b_{\textrm{BP}}^{\left(\ell\right)}\left( {x}_i\right)$ at iteration $\ell$ is computed as
$ {b}_{\textrm{BP}}^{\left(\ell\right)}\left( {x}_i\right)
=p\left( {x}_i\right)\prod_{f_n\in\mathcal B\left(i\right)} m_{f_n\to i}^{\left(\ell\right)}\left( {x}_i\right)
\propto \mathcal{N}\big( {x}_i|
{\mu}_i^{\left(\ell\right)}, {P}_i^{\left(\ell\right)}\big)$,
where
\begin{equation} \label{beliefcov}
{P}_i^{\left(\ell\right)} =
\Big[ {W}_i^{-1}
+\sum_{f_n\in\B\left(i\right)}
{J}_{f_n\to i}^{\left(\ell\right)}\Big]^{-1}
\quad
\textrm{and}\quad
{\mu}_i^{\left(\ell\right)}= {P}_i^{\left(\ell\right)}\Big[
\sum_{f_n\in\B\left(i\right)}
{J}_{f_n\to i}^{\left(\ell\right)} {v}^{\left(\ell\right)}_{f_n\to i}\Big].
\end{equation}
The iterative computation
starts by initializing
$ {J}_{f_k\to j}^{(0)}$ and
$ {v}_{f_k\to j}^{\left(0\right)}
$ (\ref{v2fV}) for all $k\in \mathcal V$ and $j\in \mathcal B(f_k)$;
it terminates when message (\ref{v2fV}) and (\ref{Cov}) converges to a fixed value or the maximum number of iterations is reached.
\section{GBP Convergence Analysis}
\label{others}
\vspace{-0.5em}
A challenge with GBP for large-scale networks is determining whether it converges or not.
In particular, it
is generally known that, if the factor graph contains cycles, the GBP algorithm may
diverge.
Thus, determining convergence conditions for the GBP algorithm is very important.
Sufficient conditions for the convergence of GBP in loopy graphs are available in \cite{DiagnalDominant, WalkSum1}.
However, these conditions are derived based on the classical GMRF based factorization of the joint distribution in the form of (\ref{pairwise}).
This differs from the model considered in this paper, where the factor
$f_n$ follows (\ref{jointpost}), which leads to intrinsically different recursive equations.
More specifically, the recursive equations
(\ref{v2fV}) and (\ref{Cov}) have different properties from recursive equations (7) and (8) in \cite{WalkSum1}.
Thus, the convergence results in \cite{DiagnalDominant, WalkSum1} cannot be applied to the GBP for the linear Gaussian model.
Due to the recursive updating of $m_{j\to f_n}^{\left(\ell\right)}\left( x_j\right)$ and $m_{f_n\to i}^{\left(\ell\right)}\left( {x}_i\right)$ in (\ref{v2fV}) and (\ref{Cov}), the message evolution can be simplified by combining these two messages into one.
By substituting
$ {J}^{\left(\ell\right)}_{j \to f_n}$ in (\ref{v2fV}) into (\ref{Cov}), the updating of the message variance inverse can be written as
\vspace{-0.5em}
\begin{equation}\label{CovFunc}
{J}_{f_n\to i}^{\left(\ell\right)}
=\!
{A}_{n,i}^2 \bigg[ {R}_n\!
+\! \!\!\!\!\!\!\sum_{j\in\B\left(f_n\right)\setminus i}
\!\!\!\!\!\!\!\!
{A}^2_{n,j}
(
{W}_{j}^{-1} \!\!+\!\!\!\!\!\!\!
\sum_{f_k\in\B\left(j\right)\setminus f_n}
\!\!\!\!\!\!\!\!
{J}_{f_k\to j}^{\left(\ell-1\right)}
)^{-1}
\bigg]^{-1}\!\!\!\!
:=\!
\mathcal{F}_{n\to i}
\left(\left\{
{J}_{f_k\to j}^{\left(\ell-1\right)}\right\}_{\left(f_k, j\right)\in \mathcal{\widetilde{B}}\left(f_n, i\right)}
\right),
\end{equation}
where $\mathcal{\widetilde{B}}\left(f_n, i\right)=\left\{\left(f_k, j\right) | j \in \B\left(f_n\right)\setminus i, f_k\in \B\left(j\right)\setminus f_n\right\}$.
Observing that ${J}_{f_n\to i}^{\left(\ell\right)}$ in (\ref{CovFunc}) is independent of $ {v}^{\left(\ell\right)}_{j\to f_n} $ and $ {v}^{\left(\ell\right)}_{f_n\to i}$ in (\ref{v2fV}) and (\ref{Cov}),
we can first focus on the convergence property of $ {J}_{f_n\to i}^{\left(\ell\right)}$ alone and then later on the convergence of $ {v}^{\left(\ell\right)}_{f_n\to i}$.
Once we have the convergence characterization of
${J}_{f_n\to i}^{\left(\ell\right)}$ and
$ {v}^{\left(\ell\right)}_{f_n\to i}$, we will go back and investigate the convergence of belief variances and means in (\ref{beliefcov}).
\vspace{-0.5em}
\subsection{Convergence Analysis of Message Inverse Variance }\label{Convariance}
To efficiently represent the updates of all message inverse variances, we {{introduce} the following definitions.
Let
${ {J}}^{\left(\ell-1\right)}
\triangleq
\texttt{Bdiag}
\Big(\left\{ {J}_{f_n\to i}^{\left(\ell-1\right)}\right\}_{n\in \mathcal{V},i\in \B\left(f_n\right)}\Big)$ be
a diagonal matrix with diagonal elements
being the message inverse variances in the network at iteration $\ell-1$
with index arranged in ascending order first on $n$ and then on $i$.
Using the definition of $ {J}^{\left(\ell-1\right)}$, the term $\sum_{f_k \in \mathcal B\left(j\right) \backslash f_n} {J}_{f_k\rightarrow j}^{\left(\ell-1\right)} $ in (\ref{CovFunc}) can be written as ${\Xi}_{n,j} J^{\left(\ell-1\right)} {\Xi}_{n,j}^T$, where ${\Xi}_{n,j}$ selects appropriate components from $ J^{\left(\ell-1\right)}$ to form the summation.
Further, define $ {H}_{n,i}=\left[\left\{ {A}_{n,j} \right\}_{j\in B\left(f_n\right) \backslash i}\right]$,
${\Psi}_{n,i} = \texttt{Bdiag} \Big( \left\{ {W}_j ^{-1} \right\}_{j\in B\left(f_n\right) \backslash i} \Big)$ and
$ {K}_{n,i}=\texttt{Bdiag} \left(\left\{ {\Xi}_{n,j} \right\}_{j\in B\left(f_n\right) \backslash i}\right) $, each with components arranged in an ascending order on $j$. Then (\ref{CovFunc}) can be written as
\vspace{-0.5em}
\begin{equation}\label{CovFunc3}
{J}^{\left(\ell\right)}_{f_n\rightarrow i}= {A}_{n,i}^2\left\{ {R}_n+ {H}_{n,i}^2\left[ \Psi_{n,i} + {K}_{n,i}^2 \left( {I}_{|\mathcal{B}\left(f_n\right)|-1} \otimes {J}^{\left(\ell-1\right)}\right) \right]^{-1} \right\} ^{-1}.
\end{equation}
Now, define the function $\mathcal{F}\triangleq\left\{\mathcal{F}_{1\to k}, \ldots, \mathcal{F}_{n\to i}, \ldots, \mathcal{F}_{n \to M}\right\}$ by
${ {J}}^{\left(\ell\right)} = \mathcal{F}\left({ {J}}^{\left(\ell-1\right)}\right) $.
By stacking $ {J}_{f_n\to i}^{\left(\ell\right)}$ on the left side of
(\ref{CovFunc3}) for all $n$ and $i$ as the diagonal matrix $ {J}^{\left(\ell\right)}$, we obtain
\begin{equation}\label{CovFunc5}
{J}^{\left(\ell\right)}
= {A}^T \Big \{ {\Omega}+ {H}\left[{\Psi} + {K} \left({I}_\varphi \otimes {J}^{\left(\ell-1\right)}\right) {K}^T \right]^{-1} {H}^T \Big\} ^{-1} {A}, := \mathcal F\left( {J}^{\left(\ell-1\right)}\right),
\end{equation}
where $ {A}$, $ {H}$,
${\Psi}$, and $ {K}$ are diagonal matrices with elements $ {A}_{n,i}$, $ {H}_{n,i}$, ${\Psi}_{n,i} $, and $ {K}_{n,i}$, respectively, arranged in ascending order, first on $n$ and then on $i$ (i.e., the same order as $ {J}^{\left(\ell\right)}_{f_n \rightarrow i}$ in $ {J}^{\left(\ell\right)}$).
Furthermore,
$\varphi={\sum _{n=1} ^M |\mathcal B\left(f_n\right)|\left(|\mathcal B\left(f_n\right)|-1\right)}$
and
${\Omega}$ is a block diagonal matrix with diagonal blocks $ {I} _{|B\left(f_n\right)|} \otimes {R}_n$ with ascending order on $n$, where $\otimes$ denotes the matrix Kronecker product.
We first present some properties of the updating operator $\mathcal{F}\left(\cdot\right)$,
which may be readily verified.
\begin{proposition} \label{P_FUN}
The updating operator $\mathcal{F}\left(\cdot\right)$ satisfies the following properties with
respect to the partial order induced by the cone of positive semidefinite matrices.
\end{proposition}
\noindent P \ref{P_FUN}.1:
$\mathcal{F}\left( {J}^{\left(\ell\right)}\right) \succeq \mathcal{F}\left( {J}^{\left(\ell-1\right)}\right)$, if $ {J}^{\left(\ell\right)} \succeq {J}^{\left(\ell-1\right)}\succeq {0}$.
\noindent P \ref{P_FUN}.2: $\alpha\mathcal{F}\left( {J}^{\left(\ell\right)}\right) \succ \mathcal{F}\left(\alpha {J}^{\left(\ell\right)}\right)$
and
$\mathcal{F}\left(\alpha^{-1} {J}^{\left(\ell\right)}\right) \succ \alpha^{-1}\mathcal{F}\left( {J}^{\left(\ell\right)}\right)$, if $ {J}^{\left(\ell\right)} \succ {0}$ and $\alpha>1$.
\noindent P \ref{P_FUN}.3:
Define
$ {U}\triangleq {A}^T {\Omega}^{-1} {A}$
and $ {L}\triangleq
{A}^T \left[ {\Omega}+ {H}{\Psi}^{-1} {H}^T \right] ^{-1} {A}$.
With arbitrary $ {J}^{\left(0\right)}\succeq {0}$,
$\mathcal{F}\left( {J}^{\left(\ell\right)}\right)$ is bounded as
$ {U} \succeq \mathcal{F}\left( {J}^{\left(\ell\right)}\right)\succeq {L}\succ {0}$ for $\ell\geq 1$.
Note that we use $X \succeq 0$ in the paper to denote $X$ is a positive semi-definite matrix.
Based on the above properties of $\mathcal{F}\left(\cdot\right)$, we establish the convergence of $J$ \cite{du2017var}.
\begin{theorem} \label{guarantee}
The matrix sequence
$\left\{ {J}^{\left(\ell\right)}\right\}_{l=0,1,\ldots}$ defined by (\ref{CovFunc5}) converges to a unique positive definite matrix
for any initial $ {J}^{\left(0\right)}\succeq 0$.
\end{theorem}
Proof Outline.
The set $\left[ {L}, {U}\right] $ is a compact set.
Further, according to P \ref{P_FUN}.3, for arbitrary
$ {J}^{\left(0\right)}\succeq {0}$, $\mathcal{F}$ maps $\left[ {L}, {U}\right] $ into itself starting from
$\ell\geq 1$.
Since $\left[ {L}, {U}\right]$ is also a convex set,
the continuous function $\mathcal F$ maps a compact convex subset of the Banach space of positive definite matrices into itself.
Therefore, the mapping $\mathcal F$ has a fixed point in $\left[ {L}, {U}\right]$ according to Brouwer's Fixed-Point Theorem.
The uniqueness of the fixed point can be proved by contradiction assuming there are more than one fixed point. Leveraging the properties of $\mathcal F(\cdot)$ in Proposition 1, we can show that
${J}^{\left(\ell\right)}_{l=0,1,\ldots}$ defined by (\ref{CovFunc5}) converges to a unique positive definite matrix
for any initial covariance matrix.
According to Theorem \ref{guarantee}, $ J^{\left(\ell\right)}_{f_n\to i}$ converges if all initial message inverse variances are nonnegative, i.e., ${J}^{\left(0\right)}_{f_n\to i}\geq {0}$
for all $i \in \mathcal V$ and $f_n \in \mathcal B\left(i\right)$.
{Notice that, for the GMRF model, the message inverse variance does not necessarily converge for all initial non-negative values.}
Moreover, due to the computation of ${J}^{\left(\ell\right)}_{f_n\to i}$ being independent of the local observations $ {y}_n$,
as long as the network topology does not change, the converged value $ {J}^{\ast}_{f_n\to i}$ can be precomputed offline and stored at each node, and there is no need to re-compute ${J}^{\ast}_{f_n\to i}$ even if $ {y}_n$ varies.
Another fundamental question is how fast the convergence is, and this is the focus of the discussion below.
Since the convergence of a dynamical system is often studied with respect to the part metric \cite{PartBook},
in the following, we
start by introducing the part metric.
\begin{definition}\label{mydef}
Part (Birkhoff) Metric \cite{PartBook}:
For arbitrary square matrices $ {X}$ and $ {Y}$ with the same dimension,
if there exists
$\alpha\geq 1$ such that $\alpha {X} \succeq {Y} \succeq \alpha^{-1} {X} $,
$ {X} $ and $ {Y}$
are called the parts,
and
$ \mathrm{d} \left( {X}, {Y}\right)\triangleq
\inf \left\{\log \alpha: \alpha {X} \succeq {Y}\succeq \alpha^{-1} {X}, \alpha \geq 1\right\}$
defines a metric called the part metric.
\end{definition}
Next, we will show that
$\left\{ {J}^{\left(\ell\right)}\right\}_{l=1,..}$
converges at a geometric rate with respect to the part metric in $\mathcal J$ that is constructed as
\begin{equation}\label{partset}
\mathcal J =\left\{ {J}^{\left(\ell\right)}| {U} \succeq {J}^{\left(\ell\right)} \succeq {J}^{\ast}+ \epsilon {I}\right\}
\cup
\left\{ {J}^{\left(\ell\right)}|
{J}^{\ast}- \epsilon {I} \succeq {J}^{\left(\ell\right)} \succeq {L}\right\},
\end{equation}
where $\epsilon>0 $ is a scalar and can be arbitrarily small.
\begin{theorem}\label{RateCov}
Let the initial message inverse variance nonnegative, i.e., ${J}^{\left(0\right)}_{f_n\to i}\geq {0}$. Then
the sequence $\left\{ {J}^{\left(\ell\right)}\right\}_{\ell=0,1,\ldots}$ converges at a geometric rate with respect to the part metric in $\mathcal J$.
\end{theorem}
\noindent {Proof}:
{Consider two matrices $ {J}^{\left(\ell\right)} \in \mathcal C$, and $ {J}^{\ast} \not\in \mathcal C$.
According to Definition \ref{mydef},
we have
$ \mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\triangleq
\inf \left\{\log\alpha: \alpha {J}^{\left(\ell\right)} \succeq {J}^{\ast}\succeq \alpha^{-1} {J}^{\left(\ell\right)}\right\}$.
Since $\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)$ is the smallest number satisfying $\alpha {J}^{\left(\ell\right)} \succeq {J}^{\ast} \succeq \alpha^{-1} {J}^{\left(\ell\right)}$, this is equivalent to
$ \exp\left\{\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\right\} {J}^{\left(\ell\right)}
\succeq
{J}^{\ast}
\succeq
\exp\left\{-\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\right\}
{J}^{\left(\ell\right)}.$
Applying P \ref{P_FUN}.1 to this equation, we have
$\mathcal F\left(\exp\left\{\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\right\}
{J}^{\left(\ell\right)}\right)
\succeq
\mathcal F\left( {J}^{\ast}\right)
\succeq
\mathcal F\left(\exp\left\{-\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\right\}
{J}^{\left(\ell\right)}\right).
$
Then applying P~\ref{P_FUN}.2 and
considering that $\exp\left\{\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\right\}>1$ and
$\exp\left\{-\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\right\}<1$, we obtain
$ \exp\left\{\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\right\}
\mathcal F\left( {J}^{\left(\ell\right)}\right)
\succ
\mathcal F\left( {J}^{\ast}\right)
\succ
\exp\left\{-\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\right\}
\mathcal F\left( {J}^{\left(\ell\right)}\right)$.
Notice that, for arbitrary positive definite matrices $ {X}$ and $ {Y}$, if $ {X}-k {Y}\succ {0}$, then, by definition, we have $ {x}^T {X} {x}
-k {x}^T {Y} {x}
> {0}$ with $x\neq 0$.
Then, there must exist $o>0$ that is small enough such that
$ {x}^T {X} {x}
-\left(k+o\right)
{x}^T {Y} {x}
> {0}$
or equivalently
$ {X}
\succ \left(k+o\right)
{Y}$.
Thus, as $\exp{\left(\cdot\right)}$ is a continuous function, there must exist some $\triangle\mathrm{d}>0$ such that
$
\exp\left\{-\triangle\mathrm{d}+\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\right\}
\mathcal F\left( {J}^{\left(\ell\right)}\right)
\succ
\mathcal F\left( {J}^{\ast}\right)
\succ
\exp\left\{\triangle\mathrm{d}-
\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)\right\}
\mathcal F\left( {J}^{\left(\ell\right)}\right).\nonumber
$
Now, using the definition of the part metric, the above equation is equivalent to
$ -\triangle\mathrm{d}+\mathrm{d}
\left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)
\geq
\mathrm{d} \left(\mathcal F\left( {J}^{\left(\ell\right)}\right), \mathcal F\left( {J}^{\ast}\right)\right)$.
Hence, we obtain
$\mathrm{d} \left(\mathcal F\left( {J}^{\left(\ell\right)}\right), \mathcal F\left( {C}^{\ast}\right)\right)
<
\mathrm{d}
\left( {J}^{\left(\ell\right)}, {J}^{\ast}\right)$.
}
This result holds for any $ {J}^{\left(\ell\right)} \in \mathcal C$,
$\mathrm{d} \left(\mathcal{F}\left( {J}^{\left(\ell\right)}\right), \mathcal{F}\left( {J}^{\ast}\right)\right) <
c
\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right) $,
where $c=\sup_{ {J}^{(\ell)}\in \mathcal{J}} \frac{\mathrm{d}\left(\mathcal{F}\left( {J}^{(\ell)}\right) , \mathcal{F}\left( {J}^{\ast}\right)\right) }{\mathrm{d} \left( {J}^{(\ell)}, {J}^{\ast}\right)}<1$.
Consequently,
we have
\begin{equation}\label{rate}
\mathrm{d} \left( {J}^{\left(\ell\right)}, {J}^{\ast}\right) <
c^{\ell}
\mathrm{d} \left( {J}^{\left(0\right)}, {J}^{\ast}\right).
\end{equation}
Thus the sequence $\left\{ {J}^{\left(\ell\right)}\right\}_{\ell=1,\ldots}$ converges
at a geometric rate with respect to the part metric. \QEDA
The physical meaning of Theorem \ref{RateCov} is that the distance between $ {J}^{\left(\ell\right)}$ and $ {J}^{\ast}$ decreases exponentially before $ {J}^{\left(\ell\right)}$ enters the neighborhood of $ {J}^{\ast}$, which can be chosen to be arbitrarily small.
Note that, for GBP based on the usual GMRF based representation and factorization, the message inverse variance convergence rate is still unknown .
Moreover, we can further study how to choose the initial value
$ {J}^{\left(0\right)}$ so that $ {J}^{\left(\ell\right)}$ converges faster.
Since $\mathcal F$ is a monotonic function,
with
$ {0} \preceq {J}^{\left(0\right)}\preceq {L}$,
$ {J}^{\left(\ell\right)}$ is a monotonically increasing sequence, and
$ {J}^{\left(\ell\right)}$ converges most rapidly with
$ {J}^{\left(0\right)}= {L}$.
Likewise, with
$ {J}^{\left(0\right)}\succeq {U}$,
$ {J}^{\left(\ell\right)}$ is a monotonically decreasing sequence, and
$ {J}^{\left(\ell\right)}$ converges most rapidly
with
$ {J}^{\left(0\right)}=
{U}$.
It is common practice in the GBP literature to set the initial message inverse variance to $ {0}$, i.e.,
${J}^{\left(0\right)}_{f_n\to i}= {0}$
\cite{DiagnalDominant, WalkSum1}.
The above analysis, by contrast, reveals that there is a better
choice to guarantee faster convergence.
\vspace{-0.5em}
\subsection{Convergence Analysis of Message Mean}
According to Theorems \ref{guarantee} and \ref{RateCov},
as long as we choose ${J}_{f_k\to j}^{\left(0\right)}\geq {0}$
for all $j\in \mathcal V$ and $f_k\in \mathcal B\left(j\right)$,
${J}_{f_k\to j}^{\left(\ell\right)}$
converges at a geometric rate to a unique positive value
${J}_{f_k\to j}^{\ast}$.
Furthermore, according to (\ref{v2fV}),
$
{J}^{\left(\ell\right)}_{j \to f_n}$ also converges to a positive value once
${J}_{f_k\to j}^{\left(\ell\right)}$ converges, and the converged value is denoted by
$
{J}^{\ast}_{j \to f_n}$.
Then for arbitrary initial value
${v}^{\left(0\right)}_{f_k\to j}$, the evolution of
${v}^{\left(\ell\right)}_{j\to f_n}$ in
(\ref{v2fV}) can be written in terms of the converged message inverse variances, which is
\begin{equation}\label{v2fm36}
{v}^{\left(\ell\right)}_{j\to f_n}=
\left[{J}^{\ast}_{j\to f_n}\right]^{-1}
\sum_{f_k\in\B\left(j\right)\setminus f_n}
{J}_{f_k\to j}^{\ast}
{v}^{\left(\ell-1\right)}_{f_k\to j}.
\end{equation}
\vspace{-0.5em}
Using (\ref{Cov}), and
replacing indices $j$, $i$, $n$ with $z$, $j$, $k$ respectively,
${v}^{\left(\ell-1\right)}_{f_k\to j}$ is given by
\begin{equation}\label{f2vmm37}
\begin{split}
{v}^{\left(\ell-1\right)}_{f_k\to j}
=
\left[{J}_{f_k\to j}^{\ast}\right]^{-1}
{A}_{k,j}^T
\Big[{{R}_k
+
\sum_{z\in\B\left(f_k\right)\setminus j} {A}_{k,z}^2
[{J}^{\ast}_{z\to f_k}]^{-1}
} \Big]^{-1}
\Big({y}_k-\sum_{z\in\B\left(f_k\right)\setminus j} {A}_{k,z}
{v}^{\left(\ell-1\right)}_{z\to f_k}\Big).
\end{split}
\end{equation}
Let ${M}_{k,j} = {R}_k
+
\sum_{z\in\B\left(f_k\right)\setminus j} {A}_{k,z}^2
[{J}^{\ast}_{z\to f_k}]^{-1}$.
Substituting (\ref{f2vmm37}) into
(\ref{v2fm36}), we have
$ {v}^{\left(\ell\right)}_{j\to f_n}=
{b}_{j \to f_n}-
\sum_{f_k\in\B\left(j\right)\setminus f_n}
\sum_{z\in\B\left(f_k\right)\setminus j}
[{J}^{\ast}_{j\to f_n}]^{-1}
{A}_{k,j}^T {M}_{k,j}^{-1}
{A}_{k,z}
{v}^{\left(\ell-1\right)}_{z\to f_k}$,
where $ {b}_{j\to f_n}=
[{J}^{\ast}_{j\to f_n}]^{-1}
\sum_{f_k\in\B\left(j\right)\setminus f_n}
{A}_{k,j} {M}_{k,j}^{-1}
{y}_k $.
The above equation can be further written in compact form as
\begin{equation}\label{v2fm4}
{v}^{\left(\ell\right)}_{j\to f_n}=
{b}_{j \to f_n} -
{Q}_{j \to f_n}
{v}^{\left(\ell-1\right)},
\end{equation}
with the column vector $ {v}^{\left(\ell-1\right)}$
containing $ {v}^{\left(\ell-1\right)}_{z\to f_k}$ for all $z\in \mathcal V$ and $f_k\in \mathcal B\left(z\right)$ as subvector
with ascending index first on $z$ and then on $k$.
The matrix $ {Q}_{j \to f_n}$
is a row vector with components
$ {C}^{\ast}_{j\to f_n}
{A}_{k,j}^T {M}_{k,j}^{-1}
{A}_{k,z}$
if $f_k\in\B\left(j\right)\setminus f_n$ and ${z\in\B\left(f_k\right)\setminus j}$, and $ {0}$ otherwise.
Let $ {Q}$ be the matrix that stacks $ {Q}_{j \to f_n}$ first on $j$ and then on $ n$,
and $ {b}$ be the vector containing
$ {b}_{j \to f_n}$ with the same stacking order as $ {Q}_{j \to f_n}$. We have
\begin{equation}\label{meanvectorupdate}
{v}^{\left(\ell\right)} =
- {Q} {v}^{\left(\ell-1\right)} + {b}.
\end{equation}
It is well known that, for arbitrary initial value $ {v}^{\left(0\right)}$, $ {v}^{\left(\ell\right)}$ converges
if and only if the spectral radius $\rho\left( {Q}\right)<1$.
As $ {v}^{\left(\ell\right)}$ depends on the convergence of $ {J}^{\left(\ell\right)}$, we have the following result.
\begin{theorem} \label{meanvector}
The vector sequence
$\left\{ {v}^{\left(\ell\right)}\right\}_{l=0,1,\ldots}$ defined by (\ref{meanvectorupdate}) converges to a unique value
for any initial value $\left\{ {v}^{\left(0\right)}\right\}$ and initial $ {J}^{\left(0\right)}\succeq 0$ if and only if $\rho\left( {Q}\right)<1$.
\end{theorem}
Though the above analysis provides a necessary and sufficient condition for the convergence of $ {v}^{\left(\ell\right)}_{j\to f_n}$, the condition $\rho\left( {Q}\right)<1$ needs to be checked before using the distributed algorithm with equations (\ref{v2fV}) and (\ref{Cov}).
In the sequel, we will show that
$\rho \left( {Q}\right) <1$ for a
union of a forest and a single loop
factor graph; thus GBP converges in such a topology.
Although \cite{Weiss2000}
shows the convergence of BP on the GMRF with a single loop, the analysis method cannot be applied here since the factorization of the joint distribution in (\ref{jointpost}) is different from the pairwise model in \cite{Weiss2000}.
\begin{theorem}\label{ref}
For a connected factor graph that is the union of a forest and a single loop,
with arbitrary $ {J}^{\left(0\right)}_{f_n\rightarrow i}\geq {0}$ for all $i\in \mathcal V$ and $f_n\in \mathcal B\left(i\right)$, the message inverse variance ${J}^{\left(\ell\right)}_{f_n\rightarrow i}$ and mean
$ {v}^{\left(\ell\right)}_{f_n\rightarrow i}$ is guaranteed to
converge to their corresponding unique points.
\end{theorem}
Proof Outline.
Note that, for a single loop factor graph that is the union of a forest and a single loop, there are two kinds of nodes.
One is the factors/variables in the loop; the other is the factors/variables on the chains/trees but outside the loop.
Then message from a variable node to a neighboring factor node on the graph can be categorized into three groups:
1) messages on a tree/chain passing towards the loop;
2) messages on a tree/chain passing away from the loop; and
3) messages in the loop.
According to (\ref{BPv2f1}), computation of the messages in the first group does not depend on messages in the loop and is thus convergence guaranteed.
Also, from the definition of message computation in
(\ref{BPv2f1}),
if messages in the third group converge, the second group messages should also converge.
Therefore, we focus on showing the convergence of messages in the third group.
By observing that in the single loop case $QQ^T$ is a diagonal matrix and showing that the diagonal elements are all smaller than $1$, we can show that $\rho(Q)<1$. Therefore, for a factor graph that is the union of a forest and a single loop, GBP algorithm converges.
\vspace{-0.5em}
\subsection{Convergence Analysis of Belief Variance and Mean }\label{C}
According to the first equation in (\ref{beliefcov}), as the computation of the belief variance $ {P}_i^{\left(\ell\right)}$ depends on $
{J}_{f_n\to i}^{\left(\ell\right)}$,
using
Theorems \ref{guarantee}
and \ref{RateCov}, we have the following important
corollary that reveals the convergence and uniqueness property of $ {P}_i^{\left(\ell\right)}$.
\vspace{-0.5em}
\begin{corollary} \label{C-converge-iff2}
With arbitrary initial message inverse variance ${J}^{\left(0\right)}_{f_n\to i} \geq {0}$ for all $i\in \mathcal V$ and $f_n \in \mathcal B\left(i\right)$,
the belief variance $ {P}_i^{\left(\ell\right)}$ converges to a unique positive value
at a geometric rate with respect to the part metric in $\mathcal J$, where $\mathcal J$ is defined in (\ref{partset}).
\end{corollary}
\vspace{-0.5em}
Proof outline. Since $P_i^{(\ell)}$ is a nonlinear function of ${J}^{\left(\ell\right)}_{f_n\to i}$ according to (\ref{beliefcov}), convergence analysis of $P_i^{(\ell)}$ is not trivial.
In \cite{du2016convergence}, we provide
some additional properties of the part metric to facilitate the proof of the convergence rate of $P_i^{(\ell)}$.
On the other hand, the computation of the belief mean ${\mu}_i^{\left(\ell\right)}$ depends on the belief variance $ {P}_i^{\left(\ell\right)}$ and on the message mean $ {v}^{\left(\ell\right)}_{f_n\to i}$ as shown in (\ref{beliefcov}).
Thus, under the same condition as in Theorem \ref{meanvector}, ${\mu}^{\left(\ell\right)}_i$ is convergence guaranteed.
Moreover, it is shown in \cite[Appendix]{MRFtoFG} that, for GBP over a factor graph, the converged value of the belief mean equals its optimal value.
Together with the convergence guaranteed topology
shown in Theorem \ref{ref}, we have the following theorem.
\vspace{-0.5em}
\begin{theorem} \label{mean-if2}
With arbitrary $ {J}_{f_n\to i}^{\left(0\right)}\geq 0$
and $ v^{\left(0\right)}_{f_n\to i}$ for all $i\in \mathcal V$ and $f_n \in \mathcal B\left(i\right)$,
the mean ${\mu}_i^{\left(\ell\right)}$ in (\ref{beliefcov}) converges
to the optimal value ${ \mu}_i$ in (\ref{1})
if and only if $\rho\left( {Q}\right)<1$,
where $ {Q}$ is defined in (\ref{meanvectorupdate}). Furthermore, the condition $\rho\left( {Q}\right)<1$ is guaranteed to hold when the
factor graph is the union of a forest and a single loop.
\end{theorem}
\vspace{-1em}
\section{Conclusions}
\vspace{-1em}
In this paper, we have studied the convergence properties of Gaussian belief propagation (GBP) that applies to joint distributions
factorized on the basis of linear Gaussian
models.
This kind of factorization leads to GBP recursive computation structure that is different from existing work, in which GBP is directly applied to Gaussian Markov random fields (GMRFs) and factorizations based on the GMRFs.
As demonstrated with an example, there are
GMRF scenarios in which existing GBP convergence conditions based on the classical GMRF based factorization fail to apply.
But when the underlying GMRF is represented as a linear Gaussian model and factorized according to the latter,
the resulting GBP converges as shown in this paper.
For GBP applied to joint distributions
factorized based on linear Gaussian
models,
we have shown analytically that,
with arbitrary nonnegative initial message inverse variance,
the belief variance for each variable converges at a geometric rate to a unique positive value.
We have
demonstrated that to guarantee faster convergence
there is a better choice
for the initial value
than the commonly used all-zero initial condition.
Moreover, we have presented a necessary and sufficient condition for convergence under which the belief mean
converges to its optimal value.
We have also shown that GBP always converges
when the underlying graph is the union of a forest and a single loop.
\small
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,598 |
Pont Cesti (en italià: Ponte Cestio) és un antic pont de Roma que es conserva i que connecta una illa del Tíber amb el Janícul. El va erigir el cònsol Gai Cesti Gal en temps de l'emperador Tiberi. Però com que un pont d'aquesta envergadura era difícil que el construïs un personatge privat en temps de l'Imperi, es creu que en realitat va ser iniciat al en època republicana.
Al segle IV el Pont Cesti va ser reconstruït pels emperadors Valentinià I, Valent i Flavi Gracià i se li va canviar el nom l'any 370 anomenant-lo Pons Gratiani . El pont es va reconstruir utilitzant peperino, amb un revestiment de travertí. Una part del material de reconstrucció procedia del pòrtic que s'havia enderrocat del proper Teatre de Marcel.
Referències
Edificis de Roma
Cesti | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,889 |
{"url":"https:\/\/www.physicsforums.com\/threads\/help-calc-maximization.11028\/","text":"# Help Calc Maximization\n\n1. Dec 15, 2003\n\n### Timc150\n\nHelp!!! Calc Maximization\n\nThe question goes like this!! Take a sphere of radius R and insert a cone w\/ max volume. ANd i have no idea why i am doin!!!! HELP!!!!!\n\n2. Dec 15, 2003\n\n### himanshu121\n\nRelate the base radius with the height of cone and then to radius of sphere\nu will get\n\n$$r_{base radius}=h\\tan\\theta$$\n$$h=2R\\cos^2\\theta$$\n\n$$V=\\frac{\\pi r^2h}{3}$$\n\nConvert h,r in terms of theta and\nMax. V w.r.t $$\\theta$$","date":"2018-02-24 18:58:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8709877729415894, \"perplexity\": 4324.500263261645}, \"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-2018-09\/segments\/1518891815918.89\/warc\/CC-MAIN-20180224172043-20180224192043-00495.warc.gz\"}"} | null | null |
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Copyright © 2019 mapmyhotel.com. Proudly powered by WordPress. | {
"redpajama_set_name": "RedPajamaC4"
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La Supercoppa di Germania 2020 (ufficialmente DFL-Supercup 2020) è stata la ventunesima edizione della Supercoppa di Germania.
Si è svolta il 30 settembre 2020 all'Allianz Arena di Monaco di Baviera tra il , vincitore della Bundesliga 2019-2020 e della Coppa di Germania 2019-2020 e il , secondo classificato in campionato.
Il trofeo è stato vinto dal Bayern Monaco, che ha battuto il Borussia Dortmund per 3-2. I bavaresi hanno conseguito l'ottavo successo nella competizione, migliorando il record di vittorie che già detenevano.
Per la prima volta nella storia della competizione la partita è stata diretta da un arbitro donna, la tedesca Bibiana Steinhaus, alla sua ultima direzione in carriera.
Partecipanti
Tabellino
Note
2020
Competizioni sportive a Monaco di Baviera | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,877 |
\section{Introduction}
Combinatorial optimization problems (COPs) arise in many research areas and applications. They appear in many forms and variants like vehicle routing\cite{toth2002vehicle}, scheduling\cite{taillard1993benchmarks} and constraint satisfaction\cite{tsang2014foundations} problems but share some general properties. One of these properties is that most COPs are proven to be NP-hard which makes their solution very complex and time consuming.
Over the years many different solution approaches were proposed.
\textit{Exact methods} like \textit{branch-and-bound}\cite{lawler1966branch} attempt to find the global optimum of a COP based on smart and efficient ways of searching the solution space. While they are often able to find the optimal solution to small scale COPs in reasonable time, they require a significant amount of time to tackle larger instances of sizes relevant for practical applications.
For that reason, \textit{heuristic methods} were proposed which usually cannot guarantee to find the global optimum but sometimes can define a lower bound on the performance, which can be achieved at a minimum, and have shown good empirical performance.
One common and well-known heuristic method is \textit{local search} (LS)\cite{aarts2003local}. The main concept of LS is to iteratively explore the search space of candidate solutions in the close neighborhood of the current solution by applying small (local) changes.
The simplest local search procedure is employing a hill climbing routine which greedily applies local changes that improve the current solution. However, this approach can easily get stuck in bad local optima from which it cannot escape anymore.
Therefore, LS is commonly used in combination with \textit{meta-heuristics} which enable the procedure to escape from local optima and achieve better final performance.
The meta-heuristics introduced in section \ref{ss:rw_metaheuristics} are well established in the optimization community and have demonstrated very good performance on a plenitude of different COPs.
Since a few years however, there is an increasing interest in leveraging methods from the growing field of machine learning (ML) to solve COPs, as recent developments in neural network architectures, involving the Transformer\cite{vaswani2017attention} and Graph Neural Networks (GNNs)\cite{wu2020comprehensive}, have led to significant progress in this domain.
Most work based on ML is concerned with auto-regressive approaches to construct feasible solutions\cite{vinyals2015pointer,kool2018attention,hu2017solving,zhang2020learning}, but there is also some work which focuses on iterative improvement\cite{chen2019learning,lu2019learning,wu2021learning} or exact solutions\cite{gasse2019exact,nair2020solving} for COPs.
While the existing iterative improvement approaches share some similarities with meta-heuristics and local search, the exact formulation of the methods is often problem specific and misses the generality of the LS framework as well as a clear definition of the
intervention points which can be used to control the meta-heuristic procedure for LS.
In this work we present a consistent formulation of learned meta-heuristics in local search procedures and show on two representative problems how it can be successfully applied. In our experiments we compare our method to well-known meta-heuristics commonly used with LS at the example of capacitated vehicle routing (CVRP) and job shop scheduling (JSSP).
The results show that our GNN-based learned meta-heuristic consistently outperforms these methods in terms of speed and final performance. In further experiments we also establish our method in the context of existing ML solution approaches.
\ \\
\noindent
\textbf{Contributions}
\begin{enumerate}
\item We identify and describe three independent algorithmic aspects
of local search for combinatorial optimization, each with
several alternatives, and formalize their sequential selection
during an iterative search run as Markov Decision Process.
\item We design a deep graph neural network as policy model for this MDP,
yielding a learned controller for local search called NeuroLS.
\item We provide ample experimental evidence that NeuroLS outperforms
both, well-known general purpose local search controllers from
the Operations Research literature (so called meta-heuristics)
as well as earlier machine learning-based approaches.
\end{enumerate}
\section*{Appendices}
\subsection{Reinforcement Learning Algorithm}\label{appx:rl}
Let $\pi$ be a policy, then the discounted sum of future rewards is given by the random variable
\begin{equation}
Z^{\pi}(s, a) = \sum_{t=0}^{\infty} \gamma^t R(s_t, a_t)
\end{equation}
with $s_t \sim P(s_{t-1}, a_{t+1})$. Then the action-value function $Q(s, a) = \mathbb{E}[Z^{\pi}(s, a)]$ can be written in terms of the Bellman equation as
\begin{equation}
Q^{\pi}(s_t, a_t) = \mathbb{E}[R(s_t, a_t)] + \gamma \mathbb{E}[Q^{\pi}(s_{t+1}, a_{t+1})].
\end{equation}
Furthermore, the real action-value function can be approximated by a parameterized value estimator $Q_{\theta}(s, a)$ which is trained via minimization of the squared error of the temporal difference defined as
\begin{equation}
\text{TDE} = \left[ r_t + \gamma \max_{a' \in \mathcal{A}} Q_{\theta}(s_{t+1}, a') - Q_{\theta}(s_t, a_t) \right]^2
\end{equation}
based on an $\epsilon$-greedy policy.
Distributional RL\cite{bellemare} aims to replace the scalar value function $Q^{\pi}$ with a distribution over the returns $Z^{\pi}$.
For the distributional case there exist analogous formulations for the Bellman equation and the corresponding distributional Bellman optimality operator.
Implicit Quantile Networks\cite{dabney} are used to transform random samples from a base distribution to the respective quantile values of a target distribution, in this case the distribution over the returns. It is formulated as a risk-sensitive greedy policy (w.r.t.\ a distortion measure $\beta$) represented by a weighted sum over the quantiles:
\begin{equation}
\pi_{\beta}(s) = \argmax_{a\in \mathcal{A}} Q_{\beta}(s, a).
\end{equation}
Then, assuming two samples $x, x' \sim \mathcal{U}([0, 1])$, the respective TD error is defined as
\begin{equation}
\text{TDE}_{x, x'} = r_t + \gamma Z_{x'}(s_{t+1}, \pi_{\beta}(s_{t+1})) - Z_{x}(s_t, a_t)
\end{equation}
and the corresponding loss is estimated over multiple i.i.d. samples $x_i, x'_j \sim \mathcal{U}([0, 1])$.
\subsection{Training and Inference Setup}\label{appx:training}
For training we use the Adam optimizer\cite{kingma} with a learning rate of 0.0005.
We use $\epsilon=0.95$ for our $\epsilon$-greedy policy which we discount over the course of training to a final value of $\epsilon=0.05$. During inference we set $\epsilon=0.0$. Moreover, we use $n$-step returns with $n=3$ and a discount factor $\gamma=0.99$. The target network of our DQN is updated ever 500 steps. We use a replay buffer of size 32000 and prioritized experience replay\cite{schaul} with $\alpha=0.6$ and $\beta=0.4$.
All MLPs in our model have a hidden dimension of 128 and we set the embedding dimension of all components to $d_{\text{emb}}=128$. The IQN has a hidden dimension of 256. In the GNN encoder we use $L^{\text{stat}}=3$ and $L^{\text{dyna}}=2$ layers.
We train all our models for 80 epochs with 19200 transitions each. All experiments are run on an i7-7700K CPU (4.20GHz) and a NVIDIA GeForce 1080TI GPU.
Moreover, all our networks as well as the ML-based baselines are implemented in PyTorch\cite{paszke} and we use the tianshou library\cite{weng} for an efficient RL implementation.
The meta-heuristics use different hyper-parameter configurations which where tuned on the validation set consisting of 512 generated instances.
The exact configurations for all methods can be found in the configuration files in our github repository.
\subsection{JSSP Solver}\label{appx:jssp_solver}
To solve the JSSP we implement a local search solver in python. We implement several different swap and insertion heuristics to provide a suitable number of LS operators and corresponding neighborhoods. The first one is the CT operator proposed by \cite{laarhoven} which tries to swap adjacent nodes in critical blocks. Next, we have the CET\cite{nowicki} and ECET\cite{kuhpfahl} moves which respectively either swap only nodes at the start or the end of a critical block or both simultaneously. The final implemented operator is the CEI move introduced by \cite{balas}, which shifts a node in a critical block to a new position within the same block. \cite{kuhpfahl} provide a nice visualization and explanation of how all these operators work.
In order to quickly evaluate how good some moves are, without applying them and completely recalculating the resulting makespan, we use the potential estimation strategies by \cite{taillard} and \cite{murovec} which are computed in $O(m)$ where $m$ is the number of nodes between the initial and the final position of the moved nodes. Finally, our perturbation operator is a sequence of random CT moves without any evaluation of potentials, somehow similar to the shaking operator in \cite{sevkli}.
The initial solution for our iterative solver is created by the FDD/MWKR priority dispatching rule introduced in \cite{sels}, which we implement as a stochastic variant for restarting as proposed in \cite{lourenco}.
We will provide the implementation of our solver together with our model code.
\subsection{Additional Results}\label{appx:results}
We provide more detailed results for different methods and several different numbers of iterations in table \ref{appx:tab:results_jssp} for the JSSP and table \ref{appx:tab:results_cvrp} for the CVRP.
\begin{table*}[th]
\caption{
Results of PDRs and local search methods running for 50/100/200 iterations on the Taillard benchmark. For instance sizes 50x15, 50x20 and 100x20 we use the NeuroLS model trained on instances of size 30x15 and 30x20 respectively. Percentages are the gap to the best known upper bound. Best gap is marked in \textbf{bold}.
}
\label{appx:tab:results_jssp}
\centering
\begin{tabular}{l|ccccc|ccc|r}
\toprule
\textbf{Model} & \multicolumn{8}{c|}{\textbf{Instance size}} & \textbf{Avg} \\
\hline
& 15x15 & 20x15 & 20x20 & 30x15 & 30x20 & 50x15 & 50x20 & 100x20 & \\
\midrule
\multicolumn{9}{c}{\textbf{PDRs}} \\
RND & 28.90\% & 36.81\% & 31.71\% & 34.97\% & 36.30\% & 22.86\% & 28.05\% & 16.47\% & 29.51\% \\
FIFO & 23.92\% & 31.50\% & 27.42\% & 31.20\% & 32.14\% & 21.15\% & 23.62\% & 13.06\% & 25.50\% \\
SPT & 27.14\% & 32.92\% & 25.30\% & 34.54\% & 34.59\% & 23.81\% & 24.96\% & 15.39\% & 27.33\% \\
MWKR & 18.72\% & 23.94\% & 21.52\% & 23.91\% & 24.97\% & 16.76\% & 17.54\% & 8.33\% & 19.46\% \\
MOPNR & 20.54\% & 23.99\% & 21.60\% & 22.44\% & 25.10\% & 17.32\% & 17.96\% & 9.24\% & 19.77\% \\
FDD & 22.56\% & 24.79\% & 22.10\% & 23.40\% & 27.37\% & 17.48\% & 18.11\% & 8.14\% & 20.49\% \\
$\frac{\text{FDD}}{\text{MWKR}}$ & 17.50\% & 21.30\% & 20.11\% & 21.39\% & 24.01\% & 15.55\% & 15.95\% & 7.75\% & 17.95\% \\
\midrule
\multicolumn{9}{c}{\textbf{Local Search}} \\
\multicolumn{9}{l}{\textit{50 iterations}} \\
SA & 14.61\% & 18.54\% & 18.87\% & 18.65\% & 22.80\% & 13.63\% & 14.24\% & 7.13\% & 16.06\% \\
SA$_{\text{restart}}$ & 14.61\% & 18.54\% & 18.87\% & 18.65\% & 22.90\% & 13.63\% & 14.24\% & 7.15\% & 16.08\% \\
ILS & 13.03\% & 15.52\% & 15.14\% & 16.76\% & 19.67\% & 13.01\% & 13.18\% & 7.00\% & 14.16\% \\
ILS+SA & 13.59\% & 16.77\% & 16.48\% & 17.35\% & 20.03\% & 13.36\% & 13.66\% & 7.12\% & 14.79\% \\
VNS & 11.78\% & 14.78\% & 15.67\% & 16.91\% & 19.69\% & 12.64\% & 12.94\% & 6.71\% & 13.89\% \\
NLS$_{\text{A}}$ & \textbf{11.03}\% & 14.98\% & 14.78\% & \textbf{16.04}\% & \textbf{18.96}\% & 12.84\% & 12.89\% & 6.66\% & \textbf{13.52}\% \\
NLS$_{\text{AN}}$ & 11.27\% & \textbf{14.66}\% & \textbf{14.46}\% & 16.27\% & 19.27\% & 12.76\% & 12.79\% & 6.82\% & 13.54\% \\
NLS$_{\text{ANP}}$ & 11.67\% & 16.80\% & 18.60\% & 16.05\% & 19.39\% & \textbf{12.11}\% & \textbf{12.47}\% & \textbf{6.61}\% & 14.21\% \\
\midrule
\multicolumn{9}{l}{\textit{100 iterations}} \\
SA & 13.92\% & 17.01\% & 17.16\% & 17.53\% & 21.59\% & 12.50\% & 13.11\% & 6.61\% & 14.93\% \\
SA$_{\text{restart}}$ & 13.77\% & 17.01\% & 17.57\% & 17.62\% & 21.78\% & 12.54\% & 13.22\% & 6.75\% & 15.03\% \\
ILS & 11.57\% & 13.57\% & 13.85\% & 16.07\% & 18.72\% & 12.65\% & 12.15\% & 6.72\% & 13.16\% \\
ILS+SA & 13.32\% & 16.05\% & 15.38\% & 16.93\% & 19.74\% & 13.07\% & 13.43\% & 7.08\% & 14.37\% \\
VNS & 9.96\% & 13.71\% & 14.51\% & 15.77\% & 18.69\% & 11.64\% & 11.92\% & 6.26\% & 12.81\% \\
NLS$_{\text{A}}$ & \textbf{9.76}\% & 13.33\% & 13.02\% & 15.29\% & 17.94\% & 11.81\% & 11.96\% & 6.33\% & 12.43\% \\
NLS$_{\text{AN}}$ & 10.32\% & \textbf{13.18}\% & \textbf{12.95}\% & \textbf{14.91}\% & 17.78\% & 11.87\% & 12.02\% & 6.22\% & \textbf{12.41}\% \\
NLS$_{\text{ANP}}$ & 10.49\% & 16.32\% & 15.24\% & 15.35\% & \textbf{17.64}\% & \textbf{11.62}\% & \textbf{11.76}\% & \textbf{6.09}\% & 13.06\% \\
\midrule
\multicolumn{9}{l}{\textit{200 iterations}} \\
SA & 13.12\% & 15.92\% & 15.58\% & 15.73\% & 20.02\% & 11.35\% & 12.22\% & 6.01\% & 13.74\% \\
SA$_{\text{restart}}$ & 13.56\% & 16.25\% & 16.80\% & 16.83\% & 21.25\% & 11.62\% & 12.60\% & 6.12\% & 14.38\% \\
ILS & 9.55\% & 12.49\% & 12.39\% & 15.28\% & 17.51\% & 11.82\% & 11.51\% & 6.42\% & 12.12\% \\
ILS+SA & 12.50\% & 14.83\% & 14.70\% & 16.41\% & 18.75\% & 12.81\% & 12.60\% & 6.88\% & 13.69\% \\
VNS & 9.04\% & 13.07\% & 12.69\% & 14.65\% & 17.81\% & 10.88\% & 11.48\% & \textbf{5.65}\% & 11.91\% \\
NLS$_{\text{A}}$ & \textbf{7.74}\% & 12.16\% & \textbf{11.54}\% & 14.13\% & 16.35\% & 11.01\% & 11.25\% & 5.90\% & \textbf{11.26}\% \\
NLS$_{\text{AN}}$ & 8.74\% & \textbf{11.39}\% & 11.64\% & 14.10\% & 16.53\% & 10.95\% & 11.18\% & 5.84\% & 11.29\% \\
NLS$_{\text{ANP}}$ & 10.44\% & 16.29\% & 13.83\% & \textbf{13.61}\% & \textbf{16.06}\% & \textbf{10.62}\% & \textbf{10.88}\% & 5.73\% & 12.18\% \\
\midrule
\multicolumn{9}{c}{\textbf{ML-based}} \\
L2D\cite{zhang} & 25.92\% & 30.03\% & 31.58\% & 32.88\% & 33.64\% & 22.35\% & 26.37\% & 13.64\% & 27.05\% \\
L2S\cite{park_a} & 20.12\% & 24.83\% & 29.25\% & 24.59\% & 31.91\% & 15.89\% & 21.39\% & 9.26\% & 22.16\% \\
SN\cite{park_b} & 15.32\% & 19.43\% & 17.23\% & 18.95\% & 23.75\% & 13.83\% & 13.56\% & 6.67\% & 16.09\% \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[th]
\caption{
Results of state-of-the-art machine learning based methods and local search approaches (200/500/1000 iterations) on the Uchoa benchmark\cite{uchoa_bench}. For all instances sizes we use the NeuroLS model trained on instances of size 100. Percentages are the average gap to the best known upper bound. Best gap is marked in \textbf{bold}.
}
\label{appx:tab:results_cvrp}
\centering
\begin{tabular}{l|rrrrrrrr|r}
\toprule
\textbf{Model} & \multicolumn{8}{c|}{\textbf{Instance Group}} & \textbf{Avg} \\
\hline
& \multicolumn{2}{c}{\textit{n100}} & \multicolumn{2}{c}{\textit{n150}} & \multicolumn{2}{c}{\textit{n200}} & \multicolumn{2}{c|}{\textit{n250}} & \\
& cost & time & cost & time & cost & time & cost & time & \\
\midrule
\multicolumn{10}{c}{\textbf{constructive}} \\
POMO\cite{kwon} (single) & 23.49\% & 0.3 & 15.65\% & 0.4 & 25.86\% & 0.4 & 22.93\% & 0.4 & 21.98\% \\
POMO\cite{kwon} & 17.24\% & 0.3 & 12.81\% & 0.4 & 20.52\% & 0.4 & 15.14\% & 0.4 & 16.43\% \\
POMO\cite{kwon} (aug) & 6.32\% & \textbf{0.1} & 9.41\% & \textbf{0.1} & 13.47\% & \textbf{0.1} & 9.21\% & \textbf{0.1} & 9.60\% \\
\midrule
\multicolumn{10}{c}{\textbf{iterative (200)}} \\
ORT\cite{perron} GLS & 6.91\% & 9.4 & 10.46\% & 10.9 & \textbf{7.80}\% & 13.4 & 12.12\% & 5.0 & 9.32\% \\
ORT\cite{perron} TS & 6.78\% & 39.0 & 10.55\% & 109.5 & 7.85\% & 126.6 & 12.13\% & 7.0 & 9.33\% \\
SA & 6.92\% & 0.7 & 5.79\% & 1.3 & 16.63\% & 2.1 & 5.92\% & 3.3 & 8.81\% \\
SA$_{\text{restart}}$ & 6.96\% & 0.7 & 5.79\% & 1.3 & 16.67\% & 2.0 & 5.99\% & 3.0 & 8.85\% \\
ILS & 6.56\% & 0.8 & 5.96\% & 1.5 & 16.73\% & 2.4 & 6.08\% & 3.7 & 8.83\% \\
ILS+SA & 6.94\% & 0.6 & 6.01\% & 1.2 & 16.72\% & 1.9 & 6.04\% & 2.8 & 8.93\% \\
VNS & 7.99\% & 0.4 & 6.55\% & 0.7 & 17.27\% & 0.9 & 6.36\% & 1.4 & 9.54\% \\
\hline
DACT\cite{ma} & 13.19\% & 15.1 & 20.26\% & 21.2 & 16.91\% & 27.2 & 24.93\% & 33 & 18.82\% \\
DACT\cite{ma} (aug) & 11.52\% & 16.6 & 17.75\% & 23.1 & 15.34\% & 29.8 & 21.86\% & 37.8 & 16.62\% \\
NLS$_{\text{A}}$ & 5.43\% & 1.5 & 5.23\% & 2.4 & 15.97\% & 3.6 & 5.22\% & 5.3 & 7.96\% \\
NLS$_{\text{AN}}$ & \textbf{5.42}\% & 1.7 & \textbf{4.90}\% & 2.7 & 15.85\% & 4.0 & \textbf{5.08}\% & 5.9 & \textbf{7.81}\% \\
NLS$_{\text{ANP}}$ & \textbf{5.42}\% & 1.7 & \textbf{4.90}\% & 2.7 & 15.85\% & 4.0 & \textbf{5.08}\% & 6.1 & \textbf{7.81}\% \\
\midrule
\multicolumn{10}{c}{\textbf{iterative (500)}} \\
ORT\cite{perron} GLS & 5.69\% & 39.6 & 7.91\% & 64.9 & \textbf{5.06}\% & 83 & 8.66\% & 88.7 & 6.83\% \\
ORT\cite{perron} TS & 5.33\% & 525.8 & 7.44\% & 1267.1 & 5.16\% & 1021.3 & 8.54\% & 633.4 & \textbf{6.62}\% \\
SA & 6.92\% & 1.7 & 5.79\% & 3.1 & 16.63\% & 4.7 & 5.92\% & 7.4 & 8.81\% \\
SA$_{\text{restart}}$ & 6.91\% & 1.7 & 5.79\% & 3.1 & 16.67\% & 4.7 & 5.99\% & 6.9 & 8.84\% \\
ILS & 6.22\% & 2.0 & 5.70\% & 3.8 & 16.66\% & 6.2 & 6.08\% & 9.2 & 8.67\% \\
ILS+SA & 6.78\% & 1.6 & 5.66\% & 2.8 & 16.72\% & 4.4 & 6.04\% & 6.5 & 8.80\% \\
VNS & 7.99\% & 1.1 & 6.08\% & 1.8 & 17.27\% & 2.4 & 6.36\% & 3.5 & 9.42\% \\
\hline
DACT\cite{ma} & 11.09\% & 37.5 & 16.95\% & 53.3 & 16.70\% & 67.5 & 23.02\% & 84.8 & 16.94\% \\
DACT\cite{ma} (aug) & 10.55\% & 42.2 & 15.02\% & 59.1 & 14.76\% & 76.0 & 20.85\% & 94.0 & 15.29\% \\
NLS$_{\text{A}}$ & 5.04\% & 3.3 & 4.56\% & 5.6 & 15.61\% & 8.5 & 4.87\% & 13.0 & 7.52\% \\
NLS$_{\text{AN}}$ & \textbf{4.80}\% & 3.9 & \textbf{4.30}\% & 6.4 & 15.70\% & 9.6 & \textbf{4.78}\% & 14.4 & \underline{7.40}\% \\
NLS$_{\text{ANP}}$ & \textbf{4.80}\% & 3.8 & \textbf{4.30}\% & 6.5 & 15.70\% & 9.6 & \textbf{4.78}\% & 14.4 & \underline{7.40}\% \\
\midrule
\multicolumn{10}{c}{\textbf{iterative (1000)}} \\
ORT\cite{perron} GLS & 4.84\% & 96.3 & 6.89\% & 160.4 & \textbf{4.16}\% & 235.7 & 6.91\% & 290.4 & \textbf{5.70}\% \\
ORT\cite{perron} TS & \textbf{4.17}\% & 899.9 & 5.70\% & 2945.4 & 4.30\% & 2752.1 & 9.76\% & 2752.1 & 5.98\% \\
SA & 6.92\% & 3.2 & 5.79\% & 5.9 & 16.63\% & 9.0 & 5.92\% & 13.6 & 8.81\% \\
SA$_{\text{restart}}$ & 6.91\% & 3.2 & 5.79\% & 5.8 & 16.67\% & 8.9 & 5.99\% & 13.3 & 8.84\% \\
ILS & 5.73\% & 4.0 & 5.51\% & 7.6 & 16.66\% & 12.0 & 6.08\% & 18.4 & 8.50\% \\
ILS+SA & 6.78\% & 3.2 & 5.66\% & 5.7 & 16.72\% & 8.8 & 6.04\% & 13.0 & 8.80\% \\
VNS & 7.71\% & 2.1 & 6.08\% & 3.7 & 17.27\% & 4.8 & 6.36\% & 7.1 & 9.35\% \\
\hline
DACT\cite{ma} & 10.24\% & 79.6 & 15.32\% & 110.7 & 15.34\% & 138.1 & 21.59\% & 171.1 & 15.62\% \\
DACT\cite{ma} (aug) & 10.06\% & 82.9 & 14.19\% & 114.5 & 14.07\% & 148.2 & 20.21\% & 188.0 & 14.63\% \\
NLS$_{\text{A}}$ & \underline{4.33}\% & 6.5 & 4.12\% & 11.0 & 15.37\% & 16.7 & 4.60\% & 25.2 & 7.10\% \\
NLS$_{\text{AN}}$ & 4.40\% & 8.6 & \textbf{3.89}\% & 14.3 & 15.42\% & 21.6 & \textbf{4.48}\% & 32.4 & \underline{7.05}\% \\
NLS$_{\text{ANP}}$ & 4.40\% & 7.5 & \textbf{3.89}\% & 12.5 & 15.42\% & 18.9 & \textbf{4.48}\% & 28.5 & \underline{7.05}\% \\
\bottomrule
\end{tabular}
\end{table*}
\clearpage
\subsection{Detailed Benchmark Results}\label{appx:benchmark}
In this last section we give the per instance results on the Taillard and Uchoa benchmark datasets for different methods and different number of iterations.
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_11}
\centering
\begin{tabular}{ll|c|rrrrrrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{7}{c}{\textbf{PDRs}} \\
& & & RND & FIFO & SPT & MWKR & MOPNR & FDD & $\frac{\text{FDD}}{\text{MWKR}}$\\
15x15 & \multicolumn{9}{c}{} \\
\midrule
ta01 & cost & 1231 & 1655 & 1486 & 1462 & 1491 & 1438 & 1439 & 1417 \\
& time & & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\
ta02 & cost & 1244 & 1682 & 1486 & 1450 & 1449 & 1452 & 1447 & 1413 \\
& time & & 0.04 & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\
ta03 & cost & 1218 & 1596 & 1461 & 1526 & 1426 & 1418 & 1618 & 1423 \\
& time & & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\
ta04 & cost & 1175 & 1478 & 1575 & 1730 & 1382 & 1457 & 1453 & 1442 \\
& time & & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\
ta05 & cost & 1224 & 1672 & 1457 & 1618 & 1494 & 1448 & 1532 & 1431 \\
& time & & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\
ta06 & cost & 1238 & 1463 & 1528 & 1522 & 1369 & 1486 & 1453 & 1398 \\
& time & & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\
ta07 & cost & 1227 & 1650 & 1497 & 1434 & 1470 & 1456 & 1543 & 1368 \\
& time & & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\
ta08 & cost & 1217 & 1467 & 1496 & 1563 & 1491 & 1482 & 1425 & 1429 \\
& time & & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\
ta09 & cost & 1274 & 1605 & 1642 & 1622 & 1541 & 1594 & 1627 & 1603 \\
& time & & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\
ta10 & cost & 1241 & 1572 & 1600 & 1697 & 1476 & 1582 & 1524 & 1516 \\
& time & & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\
20x15 & \multicolumn{9}{c}{} \\
\midrule
ta11 & cost & 1357 & 1748 & 1701 & 1865 & 1685 & 1665 & 1766 & 1663 \\
& time & & 0.05 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\
ta12 & cost & 1367 & 1957 & 1810 & 1667 & 1692 & 1739 & 1631 & 1712 \\
& time & & 0.05 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\
ta13 & cost & 1342 & 1715 & 1882 & 1843 & 1690 & 1642 & 1619 & 1644 \\
& time & & 0.05 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\
ta14 & cost & 1345 & 1746 & 1795 & 1635 & 1563 & 1662 & 1687 & 1553 \\
& time & & 0.05 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\
ta15 & cost & 1339 & 1927 & 1788 & 1835 & 1700 & 1682 & 1659 & 1674 \\
& time & & 0.05 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\
ta16 & cost & 1360 & 1911 & 1696 & 1965 & 1584 & 1638 & 1681 & 1608 \\
& time & & 0.05 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\
ta17 & cost & 1462 & 2141 & 1899 & 2059 & 1851 & 1856 & 1950 & 1816 \\
& time & & 0.05 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\
ta18 & cost & 1396 & 1872 & 1833 & 1770 & 1751 & 1710 & 1721 & 1684 \\
& time & & 0.05 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\
ta19 & cost & 1332 & 1777 & 1716 & 1789 & 1696 & 1651 & 1659 & 1590 \\
& time & & 0.05 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\
ta20 & cost & 1348 & 1878 & 1827 & 1713 & 1703 & 1677 & 1658 & 1611 \\
& time & & 0.05 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_12}
\centering
\begin{tabular}{ll|c|rrrrrrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{7}{c}{\textbf{PDRs}} \\
& & & RND & FIFO & SPT & MWKR & MOPNR & FDD & $\frac{\text{FDD}}{\text{MWKR}}$\\
20x20 & \multicolumn{9}{c}{} \\
\midrule
ta21 & cost & 1642 & 2075 & 2089 & 2175 & 2044 & 1964 & 1962 & 1930 \\
& time & & 0.07 & 0.06 & 0.06 & 0.06 & 0.06 & 0.06 & 0.05 \\
ta22 & cost & 1600 & 1985 & 2146 & 1893 & 1914 & 1905 & 1921 & 1928 \\
& time & & 0.07 & 0.06 & 0.06 & 0.05 & 0.05 & 0.06 & 0.06 \\
ta23 & cost & 1557 & 2194 & 2010 & 1990 & 1954 & 1922 & 1945 & 1833 \\
& time & & 0.08 & 0.06 & 0.06 & 0.06 & 0.05 & 0.06 & 0.06 \\
ta24 & cost & 1644 & 2092 & 1989 & 2073 & 1982 & 1918 & 1903 & 1946 \\
& time & & 0.07 & 0.06 & 0.06 & 0.06 & 0.06 & 0.05 & 0.06 \\
ta25 & cost & 1595 & 2172 & 2160 & 1950 & 1905 & 1912 & 1959 & 1901 \\
& time & & 0.08 & 0.06 & 0.06 & 0.06 & 0.05 & 0.06 & 0.06 \\
ta26 & cost & 1643 & 2231 & 2182 & 1955 & 1967 & 2018 & 1958 & 1997 \\
& time & & 0.07 & 0.06 & 0.06 & 0.06 & 0.08 & 0.06 & 0.05 \\
ta27 & cost & 1680 & 2094 & 2112 & 2096 & 2091 & 2160 & 2080 & 2140 \\
& time & & 0.07 & 0.06 & 0.06 & 0.06 & 0.07 & 0.06 & 0.06 \\
ta28 & cost & 1603 & 2420 & 1980 & 1968 & 1997 & 1952 & 2032 & 1875 \\
& time & & 0.07 & 0.06 & 0.06 & 0.06 & 0.06 & 0.06 & 0.06 \\
ta29 & cost & 1625 & 2112 & 1999 & 2166 & 1860 & 1899 & 2081 & 1948 \\
& time & & 0.08 & 0.06 & 0.06 & 0.06 & 0.06 & 0.06 & 0.06 \\
ta30 & cost & 1584 & 1926 & 1941 & 1999 & 1939 & 2017 & 1906 & 1927 \\
& time & & 0.08 & 0.05 & 0.07 & 0.06 & 0.06 & 0.05 & 0.06 \\
30x15 & \multicolumn{9}{c}{} \\
\midrule
ta31 & cost & 1764 & 2207 & 2277 & 2335 & 2134 & 2143 & 2240 & 2116 \\
& time & & 0.08 & 0.06 & 0.06 & 0.07 & 0.06 & 0.06 & 0.06 \\
ta32 & cost & 1784 & 2411 & 2322 & 2432 & 2211 & 2188 & 2308 & 2273 \\
& time & & 0.08 & 0.06 & 0.06 & 0.06 & 0.06 & 0.08 & 0.06 \\
ta33 & cost & 1791 & 2527 & 2481 & 2428 & 2349 & 2308 & 2340 & 2278 \\
& time & & 0.08 & 0.06 & 0.06 & 0.06 & 0.07 & 0.06 & 0.07 \\
ta34 & cost & 1829 & 2635 & 2507 & 2534 & 2245 & 2187 & 2158 & 2168 \\
& time & & 0.08 & 0.06 & 0.06 & 0.06 & 0.07 & 0.07 & 0.06 \\
ta35 & cost & 2007 & 2612 & 2478 & 2497 & 2226 & 2255 & 2195 & 2225 \\
& time & & 0.08 & 0.06 & 0.07 & 0.06 & 0.07 & 0.06 & 0.08 \\
ta36 & cost & 1819 & 2379 & 2393 & 2364 & 2365 & 2307 & 2236 & 2186 \\
& time & & 0.1 & 0.06 & 0.07 & 0.06 & 0.07 & 0.07 & 0.06 \\
ta37 & cost & 1771 & 2393 & 2254 & 2664 & 2151 & 2190 & 2233 & 2232 \\
& time & & 0.08 & 0.06 & 0.06 & 0.06 & 0.06 & 0.06 & 0.06 \\
ta38 & cost & 1673 & 2344 & 2355 & 2201 & 2076 & 2179 & 2163 & 1991 \\
& time & & 0.08 & 0.06 & 0.07 & 0.06 & 0.07 & 0.06 & 0.06 \\
ta39 & cost & 1795 & 2494 & 2352 & 2465 & 2220 & 2135 & 2189 & 2154 \\
& time & & 0.08 & 0.06 & 0.06 & 0.06 & 0.07 & 0.07 & 0.06 \\
ta40 & cost & 1669 & 2160 & 2069 & 2166 & 2205 & 2028 & 2029 & 2109 \\
& time & & 0.08 & 0.07 & 0.06 & 0.06 & 0.06 & 0.06 & 0.06 \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_13}
\centering
\begin{tabular}{ll|c|rrrrrrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{7}{c}{\textbf{PDRs}} \\
& & & RND & FIFO & SPT & MWKR & MOPNR & FDD & $\frac{\text{FDD}}{\text{MWKR}}$\\
30x20 & \multicolumn{9}{c}{} \\
\midrule
ta41 & cost & 2005 & 2863 & 2556 & 2463 & 2620 & 2538 & 2445 & 2537 \\
& time & & 0.11 & 0.08 & 0.09 & 0.09 & 0.09 & 0.09 & 0.09 \\
ta42 & cost & 1937 & 2664 & 2648 & 2773 & 2424 & 2448 & 2567 & 2388 \\
& time & & 0.11 & 0.08 & 0.09 & 0.08 & 0.08 & 0.09 & 0.09 \\
ta43 & cost & 1846 & 2468 & 2442 & 2412 & 2348 & 2425 & 2269 & 2354 \\
& time & & 0.11 & 0.09 & 0.09 & 0.08 & 0.09 & 0.09 & 0.09 \\
ta44 & cost & 1979 & 2622 & 2687 & 2906 & 2493 & 2419 & 2402 & 2423 \\
& time & & 0.11 & 0.08 & 0.09 & 0.09 & 0.09 & 0.09 & 0.09 \\
ta45 & cost & 2000 & 2577 & 2607 & 2734 & 2524 & 2522 & 2511 & 2450 \\
& time & & 0.11 & 0.09 & 0.08 & 0.09 & 0.09 & 0.09 & 0.09 \\
ta46 & cost & 2004 & 2778 & 2606 & 2650 & 2447 & 2498 & 2646 & 2525 \\
& time & & 0.11 & 0.09 & 0.08 & 0.08 & 0.08 & 0.09 & 0.09 \\
ta47 & cost & 1889 & 2682 & 2471 & 2531 & 2282 & 2286 & 2660 & 2400 \\
& time & & 0.11 & 0.09 & 0.09 & 0.09 & 0.09 & 0.09 & 0.09 \\
ta48 & cost & 1941 & 2630 & 2606 & 2620 & 2356 & 2371 & 2420 & 2280 \\
& time & & 0.11 & 0.09 & 0.09 & 0.09 & 0.09 & 0.09 & 0.09 \\
ta49 & cost & 1961 & 2602 & 2604 & 2652 & 2382 & 2399 & 2492 & 2431 \\
& time & & 0.12 & 0.08 & 0.09 & 0.09 & 0.09 & 0.09 & 0.09 \\
ta50 & cost & 1923 & 2673 & 2520 & 2483 & 2474 & 2469 & 2406 & 2376 \\
& time & & 0.11 & 0.08 & 0.09 & 0.09 & 0.09 & 0.09 & 0.09 \\
50x15 & \multicolumn{9}{c}{} \\
\midrule
ta51 & cost & 2760 & 3518 & 3705 & 3856 & 3435 & 3567 & 3263 & 3486 \\
& time & & 0.14 & 0.11 & 0.11 & 0.11 & 0.11 & 0.11 & 0.11 \\
ta52 & cost & 2756 & 3429 & 3368 & 3266 & 3394 & 3278 & 3332 & 3276 \\
& time & & 0.14 & 0.11 & 0.11 & 0.11 & 0.1 & 0.1 & 0.11 \\
ta53 & cost & 2717 & 3296 & 3199 & 3273 & 3090 & 3115 & 3188 & 3059 \\
& time & & 0.15 & 0.1 & 0.1 & 0.1 & 0.1 & 0.11 & 0.11 \\
ta54 & cost & 2839 & 3498 & 3221 & 3167 & 3294 & 3265 & 3223 & 3053 \\
& time & & 0.17 & 0.1 & 0.1 & 0.1 & 0.11 & 0.11 & 0.11 \\
ta55 & cost & 2679 & 3367 & 3267 & 3289 & 3198 & 3279 & 3174 & 3103 \\
& time & & 0.16 & 0.1 & 0.11 & 0.11 & 0.11 & 0.11 & 0.11 \\
ta56 & cost & 2781 & 3385 & 3326 & 3554 & 3134 & 3100 & 3293 & 3302 \\
& time & & 0.14 & 0.1 & 0.11 & 0.1 & 0.11 & 0.11 & 0.11 \\
ta57 & cost & 2943 & 3444 & 3654 & 3745 & 3261 & 3335 & 3420 & 3298 \\
& time & & 0.14 & 0.11 & 0.11 & 0.11 & 0.11 & 0.11 & 0.12 \\
ta58 & cost & 2885 & 3481 & 3370 & 3459 & 3380 & 3420 & 3382 & 3311 \\
& time & & 0.14 & 0.11 & 0.11 & 0.11 & 0.1 & 0.11 & 0.12 \\
ta59 & cost & 2655 & 3352 & 3357 & 3341 & 3080 & 3139 & 3121 & 3156 \\
& time & & 0.14 & 0.1 & 0.1 & 0.1 & 0.1 & 0.11 & 0.11 \\
ta60 & cost & 2723 & 3309 & 3137 & 3393 & 3122 & 3044 & 3191 & 3006 \\
& time & & 0.15 & 0.1 & 0.11 & 0.11 & 0.1 & 0.11 & 0.11 \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_14}
\centering
\begin{tabular}{ll|c|rrrrrrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{7}{c}{\textbf{PDRs}} \\
& & & RND & FIFO & SPT & MWKR & MOPNR & FDD & $\frac{\text{FDD}}{\text{MWKR}}$\\
50x20 & \multicolumn{9}{c}{} \\
\midrule
ta61 & cost & 2868 & 3523 & 3690 & 3587 & 3343 & 3329 & 3303 & 3366 \\
& time & & 0.22 & 0.15 & 0.15 & 0.15 & 0.16 & 0.15 & 0.15 \\
ta62 & cost & 2869 & 3810 & 3672 & 3487 & 3423 & 3417 & 3415 & 3437 \\
& time & & 0.2 & 0.15 & 0.17 & 0.15 & 0.16 & 0.15 & 0.16 \\
ta63 & cost & 2755 & 3408 & 3320 & 3277 & 3233 & 3226 & 3247 & 3139 \\
& time & & 0.19 & 0.16 & 0.15 & 0.15 & 0.16 & 0.15 & 0.15 \\
ta64 & cost & 2702 & 3547 & 3161 & 3627 & 3173 & 3118 & 3138 & 3122 \\
& time & & 0.19 & 0.16 & 0.16 & 0.16 & 0.16 & 0.15 & 0.15 \\
ta65 & cost & 2725 & 3559 & 3263 & 3313 & 3429 & 3287 & 3267 & 3298 \\
& time & & 0.2 & 0.15 & 0.15 & 0.15 & 0.15 & 0.16 & 0.15 \\
ta66 & cost & 2845 & 3589 & 3594 & 3595 & 3287 & 3366 & 3309 & 3324 \\
& time & & 0.2 & 0.15 & 0.16 & 0.15 & 0.15 & 0.22 & 0.16 \\
ta67 & cost & 2825 & 3779 & 3612 & 3401 & 3305 & 3411 & 3374 & 3251 \\
& time & & 0.22 & 0.17 & 0.15 & 0.15 & 0.15 & 0.16 & 0.16 \\
ta68 & cost & 2784 & 3626 & 3380 & 3440 & 3203 & 3276 & 3380 & 3157 \\
& time & & 0.2 & 0.15 & 0.16 & 0.16 & 0.15 & 0.15 & 0.16 \\
ta69 & cost & 3071 & 3653 & 3675 & 3811 & 3550 & 3526 & 3655 & 3530 \\
& time & & 0.21 & 0.16 & 0.16 & 0.15 & 0.15 & 0.16 & 0.15 \\
ta70 & cost & 2995 & 3922 & 3790 & 3998 & 3482 & 3590 & 3500 & 3352 \\
& time & & 0.2 & 0.17 & 0.15 & 0.16 & 0.15 & 0.15 & 0.15 \\
100x20 & \multicolumn{9}{c}{} \\
\midrule
ta71 & cost & 5464 & 6436 & 6203 & 6377 & 5979 & 5938 & 6102 & 5863 \\
& time & & 0.42 & 0.33 & 0.33 & 0.32 & 0.31 & 0.32 & 0.32 \\
ta72 & cost & 5181 & 6002 & 5672 & 6080 & 5583 & 5639 & 5543 & 5539 \\
& time & & 0.43 & 0.36 & 0.35 & 0.32 & 0.32 & 0.31 & 0.32 \\
ta73 & cost & 5568 & 6605 & 6302 & 6371 & 6047 & 6161 & 6098 & 6233 \\
& time & & 0.41 & 0.36 & 0.33 & 0.33 & 0.31 & 0.34 & 0.32 \\
ta74 & cost & 5339 & 6026 & 5903 & 6283 & 5669 & 5634 & 5727 & 5628 \\
& time & & 0.43 & 0.33 & 0.32 & 0.34 & 0.32 & 0.32 & 0.32 \\
ta75 & cost & 5392 & 6801 & 6361 & 6208 & 5997 & 6212 & 6122 & 6062 \\
& time & & 0.42 & 0.33 & 0.36 & 0.33 & 0.32 & 0.33 & 0.32 \\
ta76 & cost & 5342 & 6096 & 6213 & 6583 & 5902 & 5936 & 5799 & 5851 \\
& time & & 0.44 & 0.34 & 0.33 & 0.33 & 0.32 & 0.33 & 0.32 \\
ta77 & cost & 5436 & 5972 & 5836 & 5915 & 5905 & 5829 & 5706 & 5736 \\
& time & & 0.42 & 0.32 & 0.31 & 0.32 & 0.31 & 0.31 & 0.32 \\
ta78 & cost & 5394 & 6280 & 6331 & 6156 & 5738 & 5895 & 5793 & 5762 \\
& time & & 0.44 & 0.32 & 0.33 & 0.32 & 0.32 & 0.34 & 0.32 \\
ta79 & cost & 5358 & 6186 & 6006 & 6029 & 5743 & 5652 & 5598 & 5667 \\
& time & & 0.44 & 0.35 & 0.33 & 0.32 & 0.32 & 0.33 & 0.32 \\
ta80 & cost & 5183 & 6088 & 5839 & 5912 & 5565 & 5720 & 5538 & 5476 \\
& time & & 0.44 & 0.33 & 0.32 & 0.31 & 0.31 & 0.31 & 0.33 \\
\bottomrule
\end{tabular}
\end{table*}
\clearpage
\newgeometry{top=30mm, bottom=25mm, left=45mm, right=45mm}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_21}
\centering
\begin{tabular}{ll|rrr|rrr|rrr|rrr|rrr|rrr|rrr|rrr}
\textbf{Inst.} & \multicolumn{24}{c}{\textbf{LS}} \\
& & \multicolumn{3}{l}{SA} & \multicolumn{3}{l}{SA$_{\text{re}}$} & \multicolumn{3}{l}{ILS} & \multicolumn{3}{l}{ILS+SA} & \multicolumn{3}{l}{VNS} & \multicolumn{3}{l}{NLS$_{\text{A}}$} & \multicolumn{3}{l}{NLS$_{\text{AN}}$} & \multicolumn{3}{l}{NLS$_{\text{ANP}}$} \\
15x15 & & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 \\
\midrule
ta01 & cost & 1378 & 1378 & 1378 & 1378 & 1378 & 1378 & 1365 & 1352 & 1352 & 1365 & 1365 & 1365 & 1361 & 1361 & 1361 & 1346 & 1346 & 1320 & 1359 & 1339 & 1331 & 1365 & 1365 & 1350 \\
& time & 0.39 & 0.78 & 1.57 & 0.39 & 0.81 & 1.66 & 0.43 & 0.87 & 1.88 & 0.42 & 0.78 & 1.5 & 1 & 2.18 & 4.74 & 0.88 & 1.62 & 2.96 & 1.03 & 1.89 & 4.01 & 1.07 & 1.94 & 3.75 \\
ta02 & cost & 1387 & 1387 & 1386 & 1387 & 1387 & 1387 & 1376 & 1351 & 1332 & 1376 & 1376 & 1376 & 1361 & 1326 & 1326 & 1333 & 1311 & 1311 & 1369 & 1369 & 1326 & 1369 & 1369 & 1369 \\
& time & 0.43 & 0.79 & 1.58 & 0.4 & 0.82 & 1.64 & 0.47 & 0.97 & 1.93 & 0.43 & 0.9 & 1.77 & 0.96 & 2.02 & 4.51 & 0.87 & 1.56 & 3.07 & 1.02 & 1.92 & 3.67 & 1.04 & 1.94 & 3.85 \\
ta03 & cost & 1391 & 1391 & 1383 & 1391 & 1391 & 1391 & 1333 & 1332 & 1321 & 1353 & 1333 & 1333 & 1373 & 1330 & 1330 & 1334 & 1333 & 1318 & 1333 & 1333 & 1333 & 1353 & 1339 & 1339 \\
& time & 0.46 & 0.78 & 1.59 & 0.4 & 0.82 & 1.88 & 0.43 & 0.95 & 1.96 & 0.35 & 0.82 & 1.72 & 1.01 & 2.11 & 4.63 & 0.84 & 1.57 & 3.09 & 0.96 & 1.86 & 3.94 & 1.01 & 1.87 & 3.69 \\
ta04 & cost & 1426 & 1426 & 1393 & 1426 & 1412 & 1405 & 1396 & 1396 & 1349 & 1416 & 1416 & 1396 & 1371 & 1350 & 1343 & 1377 & 1347 & 1287 & 1364 & 1354 & 1321 & 1364 & 1364 & 1358 \\
& time & 0.42 & 0.79 & 1.66 & 0.43 & 0.82 & 1.64 & 0.43 & 0.88 & 1.86 & 0.39 & 0.8 & 1.51 & 0.93 & 1.99 & 4.74 & 0.86 & 1.55 & 2.94 & 1.04 & 1.92 & 3.75 & 1.08 & 2.11 & 3.76 \\
ta05 & cost & 1364 & 1360 & 1345 & 1364 & 1360 & 1360 & 1364 & 1322 & 1292 & 1379 & 1370 & 1356 & 1333 & 1330 & 1329 & 1356 & 1334 & 1323 & 1356 & 1328 & 1292 & 1356 & 1328 & 1328 \\
& time & 0.4 & 0.79 & 1.6 & 0.4 & 0.8 & 1.65 & 0.42 & 0.9 & 1.88 & 0.35 & 0.71 & 1.55 & 0.93 & 2.09 & 4.9 & 0.94 & 1.57 & 3 & 1.04 & 1.89 & 3.62 & 1.07 & 1.94 & 3.74 \\
ta06 & cost & 1392 & 1368 & 1329 & 1392 & 1368 & 1365 & 1363 & 1363 & 1330 & 1370 & 1370 & 1363 & 1355 & 1355 & 1329 & 1345 & 1335 & 1311 & 1362 & 1349 & 1326 & 1355 & 1328 & 1361 \\
& time & 0.39 & 0.78 & 1.66 & 0.39 & 0.8 & 1.64 & 0.42 & 0.85 & 1.82 & 0.38 & 0.72 & 1.46 & 1.09 & 2.19 & 4.68 & 0.86 & 1.53 & 2.97 & 1.03 & 1.92 & 3.8 & 0.99 & 1.83 & 3.84 \\
ta07 & cost & 1315 & 1315 & 1315 & 1315 & 1315 & 1315 & 1309 & 1294 & 1284 & 1310 & 1309 & 1294 & 1315 & 1315 & 1304 & 1294 & 1294 & 1270 & 1309 & 1309 & 1309 & 1294 & 1294 & 1294 \\
& time & 0.38 & 0.78 & 1.61 & 0.38 & 0.82 & 1.63 & 0.4 & 0.83 & 1.75 & 0.36 & 0.73 & 1.47 & 1.16 & 2.19 & 4.6 & 0.84 & 1.56 & 2.95 & 1.02 & 1.86 & 3.77 & 1.01 & 1.92 & 3.7 \\
ta08 & cost & 1384 & 1381 & 1380 & 1384 & 1381 & 1379 & 1395 & 1387 & 1342 & 1395 & 1395 & 1395 & 1366 & 1350 & 1350 & 1337 & 1333 & 1305 & 1332 & 1310 & 1295 & 1332 & 1332 & 1324 \\
& time & 0.38 & 0.79 & 1.6 & 0.4 & 0.79 & 1.59 & 0.47 & 1.01 & 2.03 & 0.43 & 0.88 & 1.77 & 1.01 & 2.12 & 4.79 & 0.91 & 1.97 & 3.09 & 0.96 & 1.84 & 3.6 & 0.99 & 1.87 & 4.02 \\
ta09 & cost & 1587 & 1533 & 1533 & 1587 & 1533 & 1533 & 1531 & 1531 & 1487 & 1534 & 1531 & 1531 & 1478 & 1458 & 1390 & 1500 & 1492 & 1461 & 1492 & 1492 & 1461 & 1540 & 1493 & 1484 \\
& time & 0.39 & 0.79 & 1.58 & 0.4 & 0.84 & 1.61 & 0.46 & 0.96 & 2.03 & 0.43 & 0.86 & 1.79 & 0.93 & 2.08 & 4.69 & 0.86 & 1.52 & 2.9 & 0.97 & 2.02 & 3.64 & 1.02 & 1.95 & 4.22 \\
ta10 & cost & 1461 & 1461 & 1459 & 1461 & 1456 & 1442 & 1458 & 1383 & 1373 & 1461 & 1461 & 1416 & 1424 & 1338 & 1338 & 1422 & 1364 & 1334 & 1398 & 1374 & 1369 & 1395 & 1366 & 1365 \\
& time & 0.38 & 0.78 & 1.58 & 0.38 & 0.81 & 1.63 & 0.46 & 0.98 & 1.88 & 0.42 & 0.88 & 1.81 & 1.01 & 2.03 & 4.49 & 0.93 & 1.61 & 3.07 & 1.02 & 1.9 & 3.68 & 1.02 & 1.82 & 3.68 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_22}
\centering
\begin{tabular}{ll|rrr|rrr|rrr|rrr|rrr|rrr|rrr|rrr}
\textbf{Inst.} & \multicolumn{24}{c}{\textbf{LS}} \\
& & \multicolumn{3}{l}{SA} & \multicolumn{3}{l}{SA$_{\text{re}}$} & \multicolumn{3}{l}{ILS} & \multicolumn{3}{l}{ILS+SA} & \multicolumn{3}{l}{VNS} & \multicolumn{3}{l}{NLS$_{\text{A}}$} & \multicolumn{3}{l}{NLS$_{\text{AN}}$} & \multicolumn{3}{l}{NLS$_{\text{ANP}}$} \\
20x15 & & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 \\
\midrule
ta11 & cost & 1615 & 1615 & 1579 & 1615 & 1615 & 1615 & 1601 & 1579 & 1556 & 1611 & 1609 & 1579 & 1611 & 1579 & 1553 & 1575 & 1555 & 1555 & 1579 & 1568 & 1549 & 1608 & 1611 & 1611 \\
& time & 0.55 & 1 & 2.12 & 0.51 & 1.09 & 2.23 & 0.66 & 1.28 & 2.68 & 0.6 & 1.17 & 2.47 & 1.42 & 2.86 & 5.97 & 1.38 & 2.08 & 4.44 & 1.16 & 2.26 & 5.08 & 1.31 & 3.21 & 6.79 \\
ta12 & cost & 1642 & 1642 & 1611 & 1642 & 1642 & 1597 & 1571 & 1507 & 1485 & 1575 & 1571 & 1571 & 1594 & 1532 & 1509 & 1537 & 1512 & 1510 & 1542 & 1523 & 1472 & 1594 & 1594 & 1573 \\
& time & 0.49 & 1.01 & 2.12 & 0.51 & 1.07 & 2.24 & 0.52 & 1.07 & 2.29 & 0.47 & 0.95 & 2.05 & 1.3 & 2.66 & 5.72 & 0.97 & 2.11 & 4.35 & 1.12 & 2.29 & 4.5 & 1.31 & 3.08 & 7.32 \\
ta13 & cost & 1605 & 1556 & 1556 & 1605 & 1556 & 1556 & 1559 & 1548 & 1532 & 1564 & 1559 & 1558 & 1559 & 1559 & 1539 & 1541 & 1539 & 1496 & 1559 & 1541 & 1530 & 1556 & 1559 & 1559 \\
& time & 0.53 & 1.03 & 2.14 & 0.55 & 1.03 & 2.23 & 0.52 & 1.14 & 2.42 & 0.47 & 0.96 & 2.17 & 1.24 & 2.87 & 6.01 & 0.99 & 1.95 & 4.28 & 1.25 & 2.37 & 4.84 & 1.21 & 3.03 & 6.86 \\
ta14 & cost & 1497 & 1488 & 1484 & 1497 & 1488 & 1488 & 1489 & 1470 & 1470 & 1492 & 1492 & 1484 & 1492 & 1492 & 1487 & 1476 & 1474 & 1474 & 1473 & 1474 & 1448 & 1494 & 1480 & 1480 \\
& time & 0.5 & 1.02 & 2.13 & 0.5 & 1.02 & 2.22 & 0.6 & 1.17 & 2.41 & 0.56 & 1.13 & 2.56 & 1.26 & 2.87 & 6.13 & 1.11 & 2 & 4.41 & 1.16 & 2.53 & 4.82 & 1.27 & 2.94 & 6.67 \\
ta15 & cost & 1652 & 1553 & 1552 & 1652 & 1553 & 1553 & 1576 & 1547 & 1525 & 1598 & 1573 & 1554 & 1530 & 1530 & 1530 & 1556 & 1553 & 1526 & 1557 & 1551 & 1512 & 1624 & 1624 & 1624 \\
& time & 0.5 & 1 & 2.12 & 0.5 & 1.01 & 2.19 & 0.51 & 1.14 & 2.35 & 0.47 & 0.96 & 2.14 & 0.97 & 2.6 & 5.78 & 1 & 1.9 & 4.54 & 1.18 & 2.26 & 4.83 & 1.36 & 3.17 & 7.22 \\
ta16 & cost & 1605 & 1599 & 1599 & 1605 & 1599 & 1599 & 1565 & 1517 & 1508 & 1565 & 1565 & 1547 & 1550 & 1547 & 1543 & 1559 & 1559 & 1536 & 1539 & 1539 & 1539 & 1560 & 1560 & 1560 \\
& time & 0.49 & 1.01 & 2.14 & 0.5 & 1.07 & 2.23 & 0.62 & 1.21 & 2.57 & 0.55 & 1.16 & 2.44 & 1.26 & 2.74 & 5.96 & 1.07 & 2.25 & 4.5 & 1.23 & 2.44 & 4.87 & 1.19 & 3.26 & 7.19 \\
ta17 & cost & 1766 & 1760 & 1711 & 1766 & 1760 & 1711 & 1696 & 1663 & 1640 & 1751 & 1749 & 1696 & 1661 & 1661 & 1661 & 1742 & 1631 & 1628 & 1683 & 1606 & 1594 & 1745 & 1710 & 1738 \\
& time & 0.49 & 0.99 & 2.11 & 0.6 & 1.01 & 2.12 & 0.51 & 1.1 & 2.33 & 0.47 & 0.95 & 2.09 & 1.01 & 2.67 & 5.88 & 1.12 & 1.96 & 4.19 & 1.12 & 2.13 & 4.58 & 1.39 & 3.23 & 6.77 \\
ta18 & cost & 1644 & 1620 & 1611 & 1644 & 1620 & 1611 & 1663 & 1649 & 1630 & 1663 & 1663 & 1663 & 1620 & 1571 & 1571 & 1663 & 1606 & 1578 & 1663 & 1619 & 1560 & 1663 & 1641 & 1636 \\
& time & 0.5 & 1.01 & 2.09 & 0.57 & 1.02 & 2.2 & 0.64 & 1.23 & 2.42 & 0.57 & 1.16 & 2.34 & 1.17 & 2.53 & 5.79 & 1.15 & 1.99 & 4.27 & 1.33 & 2.39 & 4.71 & 1.4 & 3.24 & 6.83 \\
ta19 & cost & 1581 & 1581 & 1563 & 1581 & 1581 & 1581 & 1542 & 1516 & 1507 & 1586 & 1553 & 1516 & 1516 & 1516 & 1507 & 1535 & 1535 & 1516 & 1545 & 1526 & 1518 & 1566 & 1566 & 1566 \\
& time & 0.51 & 1.04 & 2.15 & 0.5 & 1.08 & 2.3 & 0.53 & 1.07 & 2.34 & 0.46 & 0.97 & 2.11 & 1.07 & 2.7 & 5.84 & 1.07 & 2.1 & 4.48 & 1.24 & 2.49 & 4.93 & 1.53 & 3.41 & 7.1 \\
ta20 & cost & 1572 & 1555 & 1555 & 1572 & 1555 & 1555 & 1504 & 1504 & 1499 & 1532 & 1504 & 1504 & 1532 & 1532 & 1532 & 1509 & 1503 & 1488 & 1509 & 1500 & 1480 & 1531 & 1531 & 1524 \\
& time & 0.5 & 1.03 & 2.15 & 0.57 & 1.04 & 2.2 & 0.5 & 1.12 & 2.37 & 0.46 & 0.98 & 2.24 & 1.13 & 2.75 & 5.98 & 0.98 & 1.88 & 4.34 & 1.16 & 2.45 & 4.66 & 1.23 & 2.91 & 6.57 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_23}
\centering
\begin{tabular}{ll|rrr|rrr|rrr|rrr|rrr|rrr|rrr|rrr}
\textbf{Inst.} & \multicolumn{24}{c}{\textbf{LS}} \\
& & \multicolumn{3}{l}{SA} & \multicolumn{3}{l}{SA$_{\text{re}}$} & \multicolumn{3}{l}{ILS} & \multicolumn{3}{l}{ILS+SA} & \multicolumn{3}{l}{VNS} & \multicolumn{3}{l}{NLS$_{\text{A}}$} & \multicolumn{3}{l}{NLS$_{\text{AN}}$} & \multicolumn{3}{l}{NLS$_{\text{ANP}}$} \\
20x20 & & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 \\
\midrule
ta21 & cost & 1930 & 1880 & 1865 & 1930 & 1930 & 1904 & 1835 & 1813 & 1795 & 1848 & 1835 & 1824 & 1871 & 1867 & 1858 & 1835 & 1819 & 1806 & 1835 & 1800 & 1803 & 1898 & 1863 & 1867 \\
& time & 0.65 & 1.34 & 2.83 & 0.7 & 1.44 & 2.95 & 0.67 & 1.54 & 3.09 & 0.66 & 1.33 & 2.66 & 1.59 & 3.66 & 7.8 & 1.27 & 2.41 & 4.57 & 1.41 & 2.8 & 5.93 & 2.13 & 3.74 & 7.01 \\
ta22 & cost & 1914 & 1871 & 1867 & 1914 & 1871 & 1871 & 1798 & 1796 & 1787 & 1812 & 1798 & 1796 & 1808 & 1808 & 1781 & 1789 & 1786 & 1772 & 1829 & 1774 & 1793 & 1911 & 1848 & 1821 \\
& time & 0.66 & 1.33 & 2.74 & 0.65 & 1.32 & 2.83 & 0.65 & 1.46 & 3.11 & 0.59 & 1.3 & 2.82 & 1.46 & 3.71 & 7.53 & 1.19 & 2.38 & 4.75 & 1.36 & 2.76 & 6.16 & 2.28 & 3.7 & 7.01 \\
ta23 & cost & 1811 & 1788 & 1786 & 1811 & 1791 & 1791 & 1745 & 1738 & 1722 & 1745 & 1745 & 1742 & 1796 & 1796 & 1757 & 1745 & 1745 & 1736 & 1765 & 1729 & 1736 & 1793 & 1804 & 1761 \\
& time & 0.66 & 1.33 & 2.81 & 0.66 & 1.39 & 2.89 & 0.74 & 1.72 & 3.18 & 0.69 & 1.52 & 3.03 & 1.97 & 3.89 & 7.71 & 1.24 & 2.43 & 4.77 & 1.52 & 2.86 & 6.36 & 2.15 & 4.01 & 6.86 \\
ta24 & cost & 1931 & 1897 & 1883 & 1931 & 1931 & 1931 & 1915 & 1838 & 1777 & 1937 & 1917 & 1882 & 1917 & 1880 & 1810 & 1882 & 1809 & 1809 & 1859 & 1820 & 1794 & 1946 & 1852 & 1825 \\
& time & 0.76 & 1.32 & 2.81 & 0.66 & 1.45 & 2.95 & 0.69 & 1.39 & 2.94 & 0.65 & 1.33 & 2.6 & 1.82 & 3.67 & 7.57 & 1.36 & 2.57 & 5 & 1.38 & 2.72 & 5.8 & 2.21 & 3.74 & 6.9 \\
ta25 & cost & 1852 & 1842 & 1793 & 1852 & 1842 & 1793 & 1796 & 1778 & 1778 & 1862 & 1796 & 1780 & 1796 & 1796 & 1796 & 1844 & 1790 & 1775 & 1787 & 1788 & 1772 & 1901 & 1812 & 1758 \\
& time & 0.68 & 1.35 & 2.76 & 0.67 & 1.35 & 2.81 & 0.69 & 1.48 & 3.33 & 0.61 & 1.35 & 2.77 & 1.54 & 3.57 & 7.81 & 1.26 & 2.51 & 5.1 & 1.46 & 2.68 & 5.87 & 2.36 & 3.86 & 6.66 \\
ta26 & cost & 1997 & 1977 & 1932 & 1997 & 1944 & 1942 & 1918 & 1906 & 1870 & 1935 & 1918 & 1918 & 1940 & 1879 & 1879 & 1918 & 1893 & 1855 & 1890 & 1894 & 1857 & 1925 & 1932 & 1874 \\
& time & 0.65 & 1.32 & 2.73 & 0.81 & 1.35 & 2.87 & 0.76 & 1.58 & 3.36 & 0.73 & 1.69 & 3.06 & 1.84 & 3.46 & 7.74 & 1.3 & 2.42 & 5 & 1.41 & 2.86 & 5.75 & 2.11 & 3.81 & 6.87 \\
ta27 & cost & 2116 & 2050 & 2004 & 2116 & 2050 & 2004 & 2050 & 2010 & 2008 & 2075 & 2049 & 2050 & 1990 & 1990 & 1969 & 2008 & 1990 & 1915 & 2022 & 1978 & 1928 & 2094 & 1993 & 1983 \\
& time & 0.7 & 1.33 & 2.72 & 0.66 & 1.47 & 2.76 & 0.77 & 1.45 & 3.1 & 0.62 & 1.23 & 2.53 & 1.23 & 3.33 & 7.48 & 1.28 & 2.45 & 4.96 & 1.47 & 2.75 & 5.82 & 2.33 & 3.84 & 6.7 \\
ta28 & cost & 1868 & 1866 & 1859 & 1868 & 1866 & 1866 & 1832 & 1816 & 1779 & 1855 & 1848 & 1827 & 1860 & 1860 & 1808 & 1823 & 1805 & 1783 & 1810 & 1814 & 1777 & 1871 & 1805 & 1828 \\
& time & 0.78 & 1.36 & 2.77 & 0.66 & 1.45 & 2.92 & 0.84 & 1.62 & 3.78 & 0.73 & 1.49 & 3.1 & 1.8 & 3.91 & 7.62 & 1.27 & 2.35 & 4.67 & 1.33 & 2.84 & 5.99 & 2.32 & 3.76 & 7.01 \\
ta29 & cost & 1911 & 1899 & 1845 & 1911 & 1911 & 1909 & 1856 & 1842 & 1787 & 1882 & 1876 & 1856 & 1882 & 1827 & 1780 & 1846 & 1813 & 1785 & 1844 & 1813 & 1801 & 1915 & 1840 & 1831 \\
& time & 0.69 & 1.34 & 2.75 & 0.76 & 1.51 & 2.9 & 0.74 & 1.48 & 3.24 & 0.72 & 1.49 & 2.8 & 1.76 & 3.69 & 7.67 & 1.26 & 2.33 & 4.65 & 1.34 & 2.82 & 5.84 & 2.2 & 3.71 & 6.83 \\
ta30 & cost & 1895 & 1879 & 1859 & 1895 & 1879 & 1879 & 1876 & 1876 & 1874 & 1887 & 1879 & 1876 & 1848 & 1816 & 1788 & 1874 & 1828 & 1803 & 1871 & 1858 & 1795 & 1927 & 1888 & 1862 \\
& time & 0.7 & 1.36 & 2.77 & 0.68 & 1.42 & 2.91 & 0.68 & 1.47 & 3.18 & 0.62 & 1.24 & 2.58 & 1.45 & 3.27 & 7.2 & 1.32 & 2.45 & 4.94 & 1.44 & 2.9 & 5.78 & 2.29 & 3.77 & 7.1 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_24}
\centering
\begin{tabular}{ll|rrr|rrr|rrr|rrr|rrr|rrr|rrr|rrr}
\textbf{Inst.} & \multicolumn{24}{c}{\textbf{LS}} \\
& & \multicolumn{3}{l}{SA} & \multicolumn{3}{l}{SA$_{\text{re}}$} & \multicolumn{3}{l}{ILS} & \multicolumn{3}{l}{ILS+SA} & \multicolumn{3}{l}{VNS} & \multicolumn{3}{l}{NLS$_{\text{A}}$} & \multicolumn{3}{l}{NLS$_{\text{AN}}$} & \multicolumn{3}{l}{NLS$_{\text{ANP}}$} \\
30x15 & & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 \\
\midrule
ta31 & cost & 2089 & 2081 & 2049 & 2089 & 2089 & 2089 & 2085 & 2077 & 2058 & 2089 & 2089 & 2077 & 2072 & 2072 & 2026 & 2072 & 2049 & 2001 & 2072 & 2067 & 2046 & 2081 & 2079 & 2048 \\
& time & 0.81 & 1.56 & 3.12 & 0.76 & 1.62 & 3.53 & 0.91 & 1.83 & 3.65 & 0.84 & 1.57 & 3.35 & 1.93 & 4.2 & 8.78 & 1.43 & 2.72 & 5.26 & 1.64 & 3.45 & 7 & 1.77 & 4.24 & 6.77 \\
ta32 & cost & 2148 & 2124 & 2065 & 2148 & 2124 & 2065 & 2125 & 2125 & 2125 & 2125 & 2125 & 2125 & 2125 & 2125 & 2125 & 2102 & 2065 & 2054 & 2125 & 2084 & 2058 & 2120 & 2087 & 2031 \\
& time & 0.77 & 1.55 & 3.16 & 0.77 & 1.71 & 3.5 & 0.96 & 2.17 & 4.39 & 0.78 & 1.73 & 3.73 & 1.64 & 4.06 & 8.94 & 1.32 & 2.57 & 5.76 & 1.53 & 2.91 & 6.7 & 1.53 & 3.39 & 6.94 \\
ta33 & cost & 2228 & 2200 & 2158 & 2228 & 2200 & 2172 & 2187 & 2154 & 2138 & 2255 & 2194 & 2154 & 2182 & 2134 & 2101 & 2169 & 2157 & 2133 & 2206 & 2130 & 2125 & 2169 & 2153 & 2099 \\
& time & 0.8 & 1.6 & 3.15 & 0.82 & 1.7 & 3.47 & 0.89 & 1.91 & 3.83 & 0.81 & 1.72 & 3.48 & 1.24 & 2.96 & 7.22 & 1.27 & 2.58 & 5.92 & 1.46 & 3.11 & 6.7 & 1.27 & 3.2 & 6.16 \\
ta34 & cost & 2082 & 2082 & 2067 & 2082 & 2082 & 2082 & 2076 & 2076 & 2070 & 2076 & 2076 & 2076 & 2076 & 2076 & 2066 & 2076 & 2076 & 2046 & 2067 & 2050 & 2055 & 2056 & 2056 & 2056 \\
& time & 0.72 & 1.71 & 3.09 & 0.76 & 1.69 & 3.48 & 0.97 & 2.16 & 3.89 & 0.82 & 1.77 & 3.51 & 2.06 & 4.26 & 9 & 1.46 & 2.95 & 5.66 & 1.62 & 3.33 & 7.12 & 1.66 & 3.86 & 6.71 \\
ta35 & cost & 2201 & 2173 & 2146 & 2201 & 2163 & 2163 & 2159 & 2138 & 2109 & 2178 & 2178 & 2178 & 2178 & 2159 & 2159 & 2159 & 2159 & 2130 & 2155 & 2122 & 2122 & 2149 & 2149 & 2122 \\
& time & 0.76 & 1.67 & 3.11 & 0.83 & 1.75 & 3.58 & 0.94 & 2.06 & 3.84 & 0.82 & 1.7 & 3.15 & 2.04 & 4.25 & 9.07 & 1.52 & 2.74 & 5.36 & 1.69 & 3.1 & 6.86 & 1.79 & 3.86 & 7.27 \\
ta36 & cost & 2154 & 2150 & 2140 & 2154 & 2154 & 2154 & 2150 & 2150 & 2150 & 2150 & 2150 & 2150 & 2150 & 2112 & 2112 & 2136 & 2136 & 2113 & 2135 & 2104 & 2118 & 2132 & 2105 & 2064 \\
& time & 0.78 & 1.6 & 3.15 & 0.79 & 1.87 & 3.65 & 0.83 & 1.88 & 3.63 & 0.75 & 1.49 & 2.88 & 2.11 & 3.98 & 8.77 & 1.48 & 2.89 & 5.59 & 1.6 & 3.17 & 6.93 & 1.51 & 3.5 & 6.32 \\
ta37 & cost & 2201 & 2149 & 2121 & 2201 & 2149 & 2121 & 2131 & 2108 & 2084 & 2131 & 2131 & 2108 & 2079 & 2079 & 2060 & 2084 & 2084 & 2075 & 2101 & 2060 & 2060 & 2118 & 2095 & 2075 \\
& time & 0.74 & 1.56 & 3.08 & 0.77 & 1.69 & 3.26 & 0.96 & 1.93 & 3.76 & 0.85 & 1.76 & 3.27 & 1.61 & 3.82 & 8.54 & 1.31 & 2.51 & 5.57 & 1.58 & 3.04 & 6.79 & 1.73 & 3.73 & 6.64 \\
ta38 & cost & 1991 & 1978 & 1961 & 1991 & 1991 & 1991 & 1961 & 1946 & 1911 & 1961 & 1961 & 1952 & 1974 & 1974 & 1919 & 1971 & 1946 & 1919 & 1938 & 1976 & 1889 & 1933 & 1933 & 1917 \\
& time & 0.71 & 1.55 & 3.12 & 0.83 & 1.75 & 3.58 & 0.85 & 1.71 & 3.66 & 0.69 & 1.43 & 2.85 & 2.22 & 4.54 & 8.97 & 1.58 & 2.88 & 5.59 & 1.6 & 3.58 & 6.59 & 1.53 & 3.79 & 7.25 \\
ta39 & cost & 2123 & 2100 & 2074 & 2123 & 2100 & 2074 & 2051 & 2043 & 2043 & 2066 & 2051 & 2043 & 2056 & 2041 & 2041 & 2043 & 2043 & 2036 & 2056 & 2043 & 2043 & 2056 & 2046 & 1991 \\
& time & 0.78 & 1.57 & 3.15 & 0.77 & 1.67 & 3.5 & 0.83 & 1.85 & 4.15 & 0.83 & 1.73 & 3.51 & 1.6 & 3.69 & 8.53 & 1.42 & 2.72 & 5.68 & 1.5 & 3.19 & 7.29 & 1.56 & 3.57 & 7.26 \\
ta40 & cost & 2023 & 2004 & 1937 & 2023 & 2004 & 2004 & 1977 & 1961 & 1949 & 1977 & 1977 & 1977 & 2037 & 1954 & 1915 & 1961 & 1925 & 1924 & 1959 & 1935 & 1911 & 1962 & 1947 & 1936 \\
& time & 0.74 & 1.57 & 3.16 & 0.82 & 1.74 & 3.64 & 0.92 & 1.9 & 3.94 & 0.78 & 1.69 & 3.62 & 1.99 & 3.83 & 8.26 & 1.46 & 2.69 & 5.56 & 1.48 & 3.02 & 7.02 & 1.47 & 3.48 & 7.26 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_25}
\centering
\begin{tabular}{ll|rrr|rrr|rrr|rrr|rrr|rrr|rrr|rrr}
\textbf{Inst.} & \multicolumn{24}{c}{\textbf{LS}} \\
& & \multicolumn{3}{l}{SA} & \multicolumn{3}{l}{SA$_{\text{re}}$} & \multicolumn{3}{l}{ILS} & \multicolumn{3}{l}{ILS+SA} & \multicolumn{3}{l}{VNS} & \multicolumn{3}{l}{NLS$_{\text{A}}$} & \multicolumn{3}{l}{NLS$_{\text{AN}}$} & \multicolumn{3}{l}{NLS$_{\text{ANP}}$} \\
30x20 & & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 \\
\midrule
ta41 & cost & 2490 & 2465 & 2429 & 2490 & 2465 & 2465 & 2427 & 2405 & 2404 & 2465 & 2427 & 2405 & 2465 & 2442 & 2442 & 2423 & 2414 & 2384 & 2434 & 2413 & 2372 & 2430 & 2408 & 2351 \\
& time & 0.95 & 2.02 & 4.12 & 1.08 & 2.12 & 4.37 & 1.03 & 2.28 & 4.92 & 1.07 & 2.16 & 4.52 & 2.42 & 5.13 & 11.14 & 1.68 & 3.39 & 7.39 & 1.88 & 4.1 & 8.26 & 1.88 & 3.89 & 7.94 \\
ta42 & cost & 2355 & 2331 & 2309 & 2374 & 2354 & 2307 & 2341 & 2330 & 2302 & 2341 & 2341 & 2333 & 2315 & 2277 & 2277 & 2291 & 2291 & 2254 & 2295 & 2295 & 2285 & 2291 & 2241 & 2233 \\
& time & 0.98 & 2.03 & 4.16 & 1.29 & 2.15 & 4.3 & 1.25 & 2.61 & 5.16 & 1.08 & 2.26 & 4.7 & 2.38 & 5.13 & 11.29 & 1.63 & 3.39 & 7.29 & 1.93 & 4.14 & 8.44 & 1.96 & 3.95 & 8.61 \\
ta43 & cost & 2354 & 2342 & 2308 & 2354 & 2296 & 2282 & 2297 & 2248 & 2247 & 2297 & 2297 & 2248 & 2268 & 2251 & 2216 & 2259 & 2228 & 2208 & 2272 & 2253 & 2201 & 2265 & 2232 & 2182 \\
& time & 1.01 & 2.08 & 4.15 & 1.23 & 2.1 & 4.38 & 1.11 & 2.33 & 5.37 & 0.99 & 2.21 & 4.52 & 2.1 & 4.91 & 10.7 & 1.67 & 3.26 & 7.21 & 1.83 & 3.88 & 7.97 & 1.98 & 3.92 & 8.57 \\
ta44 & cost & 2399 & 2388 & 2346 & 2399 & 2399 & 2399 & 2317 & 2317 & 2304 & 2318 & 2317 & 2317 & 2318 & 2309 & 2309 & 2317 & 2305 & 2260 & 2308 & 2308 & 2291 & 2303 & 2301 & 2280 \\
& time & 0.98 & 2.02 & 4.15 & 1.09 & 2.14 & 4.7 & 1.08 & 2.41 & 5.5 & 1.02 & 2.2 & 4.53 & 2.05 & 4.97 & 11.03 & 1.62 & 3.45 & 7.16 & 1.81 & 4.1 & 8.42 & 1.89 & 3.9 & 8.55 \\
ta45 & cost & 2421 & 2362 & 2344 & 2421 & 2362 & 2362 & 2278 & 2251 & 2241 & 2278 & 2274 & 2241 & 2278 & 2278 & 2243 & 2278 & 2262 & 2241 & 2278 & 2248 & 2228 & 2278 & 2244 & 2241 \\
& time & 0.98 & 2.04 & 4.16 & 1.09 & 2.09 & 4.31 & 0.87 & 2.07 & 4.6 & 0.91 & 2.04 & 4.28 & 1.2 & 4.21 & 9.81 & 1.41 & 3.14 & 6.65 & 1.42 & 3.46 & 8.23 & 1.53 & 3.41 & 8.04 \\
ta46 & cost & 2525 & 2489 & 2432 & 2525 & 2525 & 2525 & 2424 & 2395 & 2320 & 2437 & 2424 & 2395 & 2433 & 2366 & 2366 & 2424 & 2358 & 2324 & 2437 & 2371 & 2322 & 2437 & 2366 & 2324 \\
& time & 0.97 & 2.03 & 4.36 & 1.25 & 2.2 & 4.63 & 0.94 & 2.04 & 4.31 & 0.88 & 2.01 & 3.82 & 2.08 & 4.55 & 10.59 & 1.57 & 3.22 & 6.46 & 1.9 & 3.75 & 7.93 & 2 & 3.58 & 7.76 \\
ta47 & cost & 2350 & 2308 & 2259 & 2350 & 2308 & 2274 & 2281 & 2269 & 2252 & 2295 & 2295 & 2269 & 2269 & 2269 & 2237 & 2252 & 2246 & 2192 & 2247 & 2209 & 2209 & 2281 & 2235 & 2213 \\
& time & 0.98 & 2.21 & 4.17 & 1.21 & 2.05 & 4.9 & 1.14 & 2.42 & 5.16 & 1.01 & 2.21 & 4.69 & 1.92 & 4.96 & 10.59 & 1.63 & 3.37 & 7.16 & 1.87 & 3.51 & 7.99 & 2.16 & 3.71 & 8.22 \\
ta48 & cost & 2279 & 2264 & 2242 & 2279 & 2264 & 2257 & 2253 & 2243 & 2227 & 2256 & 2256 & 2243 & 2264 & 2227 & 2227 & 2256 & 2233 & 2217 & 2268 & 2230 & 2230 & 2268 & 2253 & 2225 \\
& time & 0.96 & 2.16 & 4.15 & 1.06 & 2.01 & 4.69 & 1.1 & 2.5 & 5.71 & 1.1 & 2.22 & 4.59 & 2.73 & 5.38 & 11.69 & 1.73 & 3.51 & 6.86 & 2.3 & 4.14 & 8.7 & 2.43 & 4.3 & 8.92 \\
ta49 & cost & 2398 & 2395 & 2395 & 2398 & 2398 & 2398 & 2368 & 2351 & 2335 & 2368 & 2368 & 2356 & 2352 & 2352 & 2352 & 2356 & 2356 & 2333 & 2359 & 2356 & 2314 & 2378 & 2353 & 2324 \\
& time & 0.99 & 2.07 & 4.18 & 1.25 & 2.11 & 4.44 & 1.12 & 2.39 & 5.15 & 1 & 2.2 & 4.59 & 1.71 & 4.69 & 10.86 & 1.6 & 3.4 & 6.8 & 1.85 & 4.1 & 8.14 & 2.13 & 3.97 & 8.73 \\
ta50 & cost & 2357 & 2348 & 2321 & 2357 & 2357 & 2357 & 2332 & 2324 & 2264 & 2332 & 2332 & 2331 & 2359 & 2356 & 2286 & 2323 & 2288 & 2257 & 2342 & 2266 & 2253 & 2332 & 2289 & 2242 \\
& time & 0.99 & 2.14 & 4.1 & 1.24 & 2.18 & 5.1 & 1.01 & 2.3 & 4.76 & 0.96 & 1.97 & 4.05 & 2.83 & 5.8 & 10.91 & 1.72 & 3.19 & 6.62 & 2.14 & 3.68 & 7.95 & 2.18 & 3.86 & 9.01 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_26}
\centering
\begin{tabular}{ll|rrr|rrr|rrr|rrr|rrr|rrr|rrr|rrr}
\textbf{Inst.} & \multicolumn{24}{c}{\textbf{LS}} \\
& & \multicolumn{3}{l}{SA} & \multicolumn{3}{l}{SA$_{\text{re}}$} & \multicolumn{3}{l}{ILS} & \multicolumn{3}{l}{ILS+SA} & \multicolumn{3}{l}{VNS} & \multicolumn{3}{l}{NLS$_{\text{A}}$} & \multicolumn{3}{l}{NLS$_{\text{AN}}$} & \multicolumn{3}{l}{NLS$_{\text{ANP}}$} \\
50x15 & & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 \\
\midrule
ta51 & cost & 3444 & 3381 & 3320 & 3444 & 3381 & 3320 & 3391 & 3355 & 3333 & 3391 & 3391 & 3355 & 3326 & 3287 & 3283 & 3391 & 3314 & 3273 & 3387 & 3300 & 3281 & 3373 & 3361 & 3281 \\
& time & 1.34 & 2.65 & 5.51 & 1.37 & 2.71 & 5.37 & 1.46 & 2.9 & 6.31 & 1.33 & 2.81 & 5.66 & 1.71 & 4.38 & 12.36 & 2.25 & 4.22 & 8.48 & 2.23 & 4.31 & 9.56 & 2.29 & 4.83 & 9.99 \\
ta52 & cost & 3254 & 3225 & 3210 & 3254 & 3225 & 3210 & 3228 & 3224 & 3198 & 3276 & 3240 & 3228 & 3276 & 3193 & 3160 & 3267 & 3231 & 3188 & 3240 & 3231 & 3161 & 3231 & 3231 & 3205 \\
& time & 1.33 & 2.62 & 5.25 & 1.31 & 2.64 & 5.21 & 1.48 & 3.16 & 6.63 & 1.48 & 2.88 & 5.98 & 3.72 & 5.69 & 13.07 & 2.55 & 4.95 & 9.53 & 2.91 & 5.23 & 10.11 & 2.52 & 5.6 & 10.37 \\
ta53 & cost & 2998 & 2987 & 2933 & 2998 & 2998 & 2998 & 3000 & 2994 & 2994 & 3014 & 3005 & 2994 & 2994 & 2968 & 2967 & 3010 & 2980 & 2974 & 2994 & 2990 & 2974 & 2983 & 2983 & 2974 \\
& time & 1.27 & 2.58 & 5.19 & 1.33 & 2.7 & 5.55 & 1.51 & 3.31 & 6.81 & 1.41 & 2.89 & 5.85 & 2.63 & 5.84 & 13.65 & 2.39 & 4.54 & 9.45 & 2.43 & 5.06 & 10.75 & 2.43 & 5.3 & 10.43 \\
ta54 & cost & 2989 & 2944 & 2935 & 2989 & 2944 & 2943 & 3011 & 2997 & 2956 & 3011 & 3011 & 3011 & 2981 & 2961 & 2941 & 2953 & 2923 & 2914 & 2960 & 2930 & 2910 & 2961 & 2924 & 2924 \\
& time & 1.26 & 2.81 & 5.25 & 1.24 & 2.68 & 5.2 & 1.66 & 3.36 & 6.82 & 1.39 & 2.9 & 5.84 & 3.09 & 6.54 & 13.93 & 1.95 & 3.94 & 8.34 & 2.26 & 4.39 & 9.73 & 2.56 & 4.89 & 9.94 \\
ta55 & cost & 3071 & 3057 & 3037 & 3071 & 3057 & 3037 & 3063 & 3059 & 3059 & 3065 & 3063 & 3059 & 3063 & 3052 & 3026 & 3059 & 3027 & 3027 & 3063 & 3044 & 3026 & 3059 & 3027 & 3026 \\
& time & 1.32 & 2.75 & 5.41 & 1.3 & 2.7 & 5.49 & 1.59 & 3.42 & 6.97 & 1.4 & 3.16 & 5.94 & 2.87 & 6.42 & 13.61 & 2.24 & 4.57 & 8.9 & 2.45 & 4.96 & 10.46 & 2.47 & 4.99 & 10.28 \\
ta56 & cost & 3177 & 3102 & 3068 & 3177 & 3102 & 3068 & 3117 & 3117 & 3062 & 3150 & 3117 & 3117 & 3117 & 3117 & 3087 & 3116 & 3082 & 3082 & 3140 & 3096 & 3062 & 3078 & 3078 & 3015 \\
& time & 1.31 & 2.68 & 5.38 & 1.3 & 2.69 & 5.34 & 1.51 & 3.31 & 6.36 & 1.38 & 3.07 & 5.73 & 2.43 & 6.21 & 13.08 & 2.24 & 4.48 & 9.21 & 2.37 & 4.95 & 10.48 & 2.11 & 5 & 9.71 \\
ta57 & cost & 3248 & 3228 & 3213 & 3248 & 3228 & 3213 & 3266 & 3266 & 3239 & 3266 & 3266 & 3266 & 3266 & 3243 & 3213 & 3266 & 3266 & 3219 & 3239 & 3225 & 3207 & 3221 & 3213 & 3204 \\
& time & 1.32 & 2.62 & 5.4 & 1.29 & 2.65 & 5.36 & 1.66 & 3.62 & 7.27 & 1.41 & 2.94 & 5.88 & 3.27 & 6.43 & 13.91 & 2.49 & 4.86 & 9.54 & 2.4 & 5.02 & 10.7 & 2.21 & 4.91 & 9.99 \\
ta58 & cost & 3243 & 3223 & 3217 & 3243 & 3223 & 3201 & 3249 & 3223 & 3217 & 3249 & 3249 & 3249 & 3237 & 3217 & 3176 & 3261 & 3229 & 3206 & 3249 & 3219 & 3223 & 3208 & 3204 & 3185 \\
& time & 1.27 & 2.59 & 5.18 & 1.28 & 2.54 & 5.29 & 1.71 & 3.38 & 6.95 & 1.43 & 2.91 & 5.96 & 3.17 & 6.47 & 13.38 & 2.48 & 4.71 & 9.23 & 2.71 & 5.31 & 11.32 & 2.39 & 5.25 & 10.29 \\
ta59 & cost & 3119 & 3083 & 3007 & 3119 & 3083 & 3007 & 3015 & 3006 & 2982 & 3015 & 3015 & 3006 & 3015 & 2988 & 2974 & 3015 & 3009 & 2961 & 3015 & 3004 & 2953 & 3015 & 2973 & 2945 \\
& time & 1.32 & 2.68 & 5.37 & 1.34 & 2.61 & 5.32 & 1.4 & 3.21 & 6.43 & 1.28 & 2.82 & 5.73 & 2.2 & 5.66 & 12.98 & 1.98 & 4.39 & 8.41 & 2.13 & 4.63 & 10.53 & 2.23 & 4.51 & 9.12 \\
ta60 & cost & 2977 & 2974 & 2945 & 2977 & 2974 & 2964 & 3006 & 3006 & 2977 & 3006 & 3006 & 3006 & 2968 & 2940 & 2929 & 2961 & 2953 & 2948 & 2991 & 2991 & 2977 & 2968 & 2968 & 2926 \\
& time & 1.31 & 2.64 & 5.4 & 1.31 & 2.61 & 5.37 & 1.77 & 3.62 & 7.16 & 1.5 & 3.05 & 6.12 & 2.71 & 5.68 & 13.28 & 2.27 & 4.42 & 9.24 & 2.67 & 5.43 & 11.57 & 2.33 & 5.5 & 9.58 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_27}
\centering
\begin{tabular}{ll|rrr|rrr|rrr|rrr|rrr|rrr|rrr|rrr}
\textbf{Inst.} & \multicolumn{24}{c}{\textbf{LS}} \\
& & \multicolumn{3}{l}{SA} & \multicolumn{3}{l}{SA$_{\text{re}}$} & \multicolumn{3}{l}{ILS} & \multicolumn{3}{l}{ILS+SA} & \multicolumn{3}{l}{VNS} & \multicolumn{3}{l}{NLS$_{\text{A}}$} & \multicolumn{3}{l}{NLS$_{\text{AN}}$} & \multicolumn{3}{l}{NLS$_{\text{ANP}}$} \\
50x20 & & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 \\
\midrule
ta61 & cost & 3252 & 3236 & 3205 & 3252 & 3236 & 3217 & 3215 & 3196 & 3175 & 3215 & 3196 & 3196 & 3204 & 3192 & 3192 & 3208 & 3167 & 3143 & 3208 & 3196 & 3181 & 3208 & 3188 & 3173 \\
& time & 1.71 & 3.5 & 7 & 1.71 & 3.47 & 7.24 & 1.55 & 3.38 & 7.09 & 1.55 & 3.31 & 6.47 & 2.25 & 6.33 & 16.52 & 2.26 & 4.47 & 9.85 & 2.2 & 5.39 & 12.45 & 2.21 & 5.56 & 13.09 \\
ta62 & cost & 3368 & 3337 & 3299 & 3368 & 3337 & 3299 & 3321 & 3321 & 3317 & 3321 & 3321 & 3321 & 3329 & 3329 & 3308 & 3329 & 3321 & 3296 & 3329 & 3295 & 3290 & 3322 & 3290 & 3269 \\
& time & 1.66 & 3.44 & 7.02 & 1.67 & 3.42 & 7.19 & 1.72 & 3.59 & 7.99 & 1.7 & 3.28 & 6.5 & 3.31 & 8.29 & 18.25 & 2.79 & 5.74 & 12.01 & 2.96 & 5.69 & 12.79 & 3.09 & 5.89 & 13.02 \\
ta63 & cost & 3095 & 3090 & 3065 & 3095 & 3090 & 3090 & 3076 & 3046 & 3030 & 3076 & 3076 & 3054 & 3076 & 3033 & 3025 & 3058 & 3049 & 3019 & 3058 & 3058 & 3011 & 3053 & 3028 & 3009 \\
& time & 1.73 & 3.45 & 6.76 & 1.74 & 3.52 & 7.35 & 1.85 & 3.89 & 8.4 & 1.68 & 3.77 & 7.63 & 3.08 & 6.68 & 16.19 & 2.65 & 5.65 & 10.94 & 2.54 & 6.16 & 11.96 & 2.7 & 5.77 & 12.45 \\
ta64 & cost & 3084 & 3066 & 3057 & 3084 & 3066 & 3056 & 3080 & 3058 & 2996 & 3080 & 3080 & 3080 & 3080 & 3069 & 3009 & 3074 & 3028 & 2998 & 3071 & 3047 & 3001 & 3055 & 3045 & 3003 \\
& time & 1.7 & 3.5 & 7.02 & 1.71 & 3.49 & 7.19 & 1.72 & 3.77 & 7.68 & 1.54 & 3.16 & 6.71 & 3.9 & 7.86 & 16.36 & 2.74 & 5.02 & 10.43 & 2.97 & 5.8 & 13.44 & 2.59 & 6.48 & 12.73 \\
ta65 & cost & 3257 & 3188 & 3150 & 3257 & 3188 & 3150 & 3224 & 3182 & 3150 & 3270 & 3252 & 3188 & 3190 & 3148 & 3119 & 3213 & 3166 & 3140 & 3219 & 3169 & 3111 & 3209 & 3195 & 3103 \\
& time & 1.65 & 3.35 & 6.95 & 1.65 & 3.35 & 6.92 & 1.91 & 3.93 & 8.12 & 1.87 & 3.79 & 7.79 & 3.72 & 8.05 & 17.31 & 2.6 & 5.12 & 10.67 & 2.92 & 5.88 & 12 & 2.64 & 5.9 & 11.58 \\
ta66 & cost & 3264 & 3204 & 3171 & 3264 & 3204 & 3171 & 3227 & 3166 & 3163 & 3273 & 3273 & 3163 & 3179 & 3166 & 3166 & 3202 & 3170 & 3166 & 3194 & 3157 & 3157 & 3166 & 3166 & 3150 \\
& time & 1.7 & 3.52 & 7.15 & 1.7 & 3.52 & 7.33 & 1.89 & 3.61 & 8.25 & 1.85 & 3.71 & 7.82 & 2.73 & 7.1 & 17.28 & 2.48 & 6.32 & 11.58 & 2.5 & 5.25 & 12.52 & 2.38 & 6.07 & 13.64 \\
ta67 & cost & 3242 & 3203 & 3190 & 3242 & 3203 & 3190 & 3195 & 3181 & 3168 & 3211 & 3194 & 3184 & 3224 & 3155 & 3155 & 3211 & 3152 & 3137 & 3180 & 3169 & 3158 & 3185 & 3142 & 3122 \\
& time & 1.7 & 3.46 & 6.76 & 1.7 & 3.46 & 6.99 & 1.84 & 4.06 & 8.43 & 1.74 & 3.68 & 8.01 & 4.51 & 7.11 & 17.44 & 2.83 & 5.45 & 11.31 & 2.78 & 6.12 & 13.15 & 2.89 & 5.43 & 12.47 \\
ta68 & cost & 3115 & 3097 & 3071 & 3115 & 3127 & 3127 & 3114 & 3042 & 3029 & 3115 & 3114 & 3114 & 3114 & 3035 & 3031 & 3091 & 3083 & 3039 & 3101 & 3062 & 3021 & 3068 & 3029 & 3015 \\
& time & 1.64 & 3.28 & 6.78 & 1.68 & 3.47 & 7.2 & 1.72 & 3.4 & 7.27 & 1.57 & 3.14 & 6.3 & 4.39 & 5.95 & 15.78 & 2.6 & 5.6 & 10.97 & 2.91 & 5.8 & 11.68 & 2.63 & 5.47 & 12.5 \\
ta69 & cost & 3487 & 3434 & 3411 & 3487 & 3434 & 3429 & 3443 & 3427 & 3408 & 3472 & 3459 & 3431 & 3431 & 3410 & 3408 & 3427 & 3413 & 3413 & 3427 & 3416 & 3407 & 3427 & 3409 & 3407 \\
& time & 1.73 & 3.47 & 6.97 & 1.71 & 3.51 & 7.33 & 1.98 & 4.19 & 8.68 & 1.82 & 3.82 & 7.7 & 3.1 & 7 & 16.89 & 2.56 & 5.4 & 12.13 & 2.75 & 6.05 & 13.65 & 2.92 & 6.1 & 13.69 \\
ta70 & cost & 3325 & 3313 & 3294 & 3325 & 3313 & 3294 & 3292 & 3276 & 3275 & 3292 & 3292 & 3292 & 3292 & 3292 & 3292 & 3292 & 3292 & 3288 & 3289 & 3289 & 3281 & 3292 & 3292 & 3283 \\
& time & 1.69 & 3.45 & 6.85 & 1.67 & 3.43 & 6.88 & 1.7 & 3.55 & 7.48 & 1.52 & 3.12 & 6.4 & 3.9 & 8.62 & 18.89 & 2.7 & 5.86 & 11.83 & 3.14 & 6.58 & 13.89 & 3.11 & 6.63 & 14.51 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Taillard benchmark\cite{taillard_bench}.}
\label{appx:tab:results_ta_28}
\centering
\begin{tabular}{ll|rrr|rrr|rrr|rrr|rrr|rrr|rrr|rrr}
\textbf{Inst.} & \multicolumn{24}{c}{\textbf{LS}} \\
& & \multicolumn{3}{l}{SA} & \multicolumn{3}{l}{SA$_{\text{re}}$} & \multicolumn{3}{l}{ILS} & \multicolumn{3}{l}{ILS+SA} & \multicolumn{3}{l}{VNS} & \multicolumn{3}{l}{NLS$_{\text{A}}$} & \multicolumn{3}{l}{NLS$_{\text{AN}}$} & \multicolumn{3}{l}{NLS$_{\text{ANP}}$} \\
& & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 & 50 & 100 & 200 \\
\multicolumn{2}{l}{100x20} & \multicolumn{23}{c}{}\\
\midrule
ta71 & cost & 5855 & 5822 & 5813 & 5855 & 5855 & 5807 & 5855 & 5855 & 5855 & 5855 & 5855 & 5855 & 5848 & 5820 & 5810 & 5832 & 5817 & 5811 & 5855 & 5821 & 5811 & 5842 & 5810 & 5799 \\
& time & 3.79 & 7.22 & 14.21 & 4.11 & 8.33 & 15.63 & 4.85 & 9.83 & 19.8 & 3.94 & 8.1 & 16.26 & 9.97 & 18.47 & 38.01 & 6.06 & 11.69 & 24.48 & 7.48 & 13.23 & 26.77 & 7.17 & 12.63 & 25.61 \\
ta72 & cost & 5531 & 5528 & 5490 & 5531 & 5531 & 5531 & 5499 & 5474 & 5460 & 5499 & 5499 & 5480 & 5496 & 5496 & 5458 & 5480 & 5471 & 5446 & 5480 & 5465 & 5448 & 5480 & 5445 & 5431 \\
& time & 3.61 & 7.19 & 14.75 & 3.58 & 7.99 & 16.08 & 3.8 & 8.16 & 17.58 & 3.66 & 7.74 & 15.61 & 5.56 & 15.53 & 31.26 & 4.27 & 10.56 & 22.76 & 4.5 & 11.51 & 26.33 & 4.69 & 10.54 & 24.12 \\
ta73 & cost & 6178 & 6142 & 6091 & 6178 & 6142 & 6091 & 6173 & 6173 & 6173 & 6173 & 6173 & 6173 & 6148 & 6102 & 6032 & 6118 & 6079 & 6079 & 6173 & 6090 & 6042 & 6149 & 6149 & 6123 \\
& time & 3.56 & 7.04 & 14.28 & 3.56 & 7.1 & 14.44 & 4.2 & 9.11 & 19.5 & 3.65 & 7.85 & 16.33 & 6.3 & 14.01 & 28.39 & 4.69 & 10.25 & 24.22 & 6.26 & 12.2 & 25.01 & 5.91 & 12.78 & 27.29 \\
ta74 & cost & 5590 & 5561 & 5537 & 5590 & 5561 & 5537 & 5556 & 5534 & 5534 & 5556 & 5556 & 5536 & 5556 & 5523 & 5502 & 5556 & 5531 & 5495 & 5556 & 5534 & 5504 & 5556 & 5534 & 5524 \\
& time & 3.53 & 7.1 & 14.54 & 3.61 & 7.14 & 14.5 & 3.86 & 7.8 & 17.69 & 3.5 & 7.67 & 15.61 & 6.04 & 11.97 & 29.92 & 4.9 & 10.18 & 20.99 & 5.39 & 11.32 & 24.27 & 5.44 & 11.94 & 25.85 \\
ta75 & cost & 6000 & 5965 & 5914 & 6000 & 5965 & 5914 & 5966 & 5928 & 5909 & 5966 & 5966 & 5937 & 5957 & 5911 & 5911 & 5966 & 5940 & 5900 & 5966 & 5966 & 5966 & 5966 & 5937 & 5903 \\
& time & 3.57 & 7.16 & 14.67 & 3.59 & 7.2 & 14.93 & 3.74 & 7.84 & 17.02 & 3.35 & 7.47 & 15.51 & 5.1 & 9.92 & 30.23 & 4.8 & 10.97 & 21.83 & 5.05 & 12.37 & 27.38 & 5.06 & 11.31 & 25.26 \\
ta76 & cost & 5812 & 5782 & 5755 & 5812 & 5782 & 5755 & 5836 & 5836 & 5775 & 5836 & 5836 & 5836 & 5789 & 5758 & 5694 & 5795 & 5785 & 5736 & 5798 & 5770 & 5715 & 5795 & 5728 & 5701 \\
& time & 3.83 & 7.16 & 15.01 & 3.66 & 7.4 & 14.89 & 4.86 & 10.01 & 19.1 & 3.91 & 7.99 & 16.19 & 7.16 & 14.52 & 29.82 & 5.64 & 11.39 & 22 & 6.48 & 12.65 & 23.72 & 5.67 & 10.47 & 22.87 \\
ta77 & cost & 5682 & 5629 & 5585 & 5682 & 5629 & 5585 & 5718 & 5690 & 5644 & 5736 & 5718 & 5718 & 5677 & 5647 & 5593 & 5644 & 5624 & 5593 & 5662 & 5598 & 5552 & 5636 & 5596 & 5593 \\
& time & 3.55 & 7.15 & 15.47 & 3.64 & 7.26 & 14.6 & 4.66 & 9.18 & 17.58 & 3.97 & 8.23 & 16.1 & 7.83 & 15.29 & 31.8 & 5.5 & 10.43 & 22.2 & 6.08 & 10.8 & 22.99 & 5.56 & 11.25 & 25.24 \\
ta78 & cost & 5732 & 5719 & 5688 & 5732 & 5719 & 5688 & 5726 & 5726 & 5724 & 5756 & 5756 & 5726 & 5726 & 5724 & 5723 & 5749 & 5724 & 5695 & 5756 & 5726 & 5724 & 5756 & 5726 & 5724 \\
& time & 3.63 & 7.27 & 15.32 & 3.62 & 7.25 & 15.19 & 4.37 & 9.33 & 19.33 & 3.92 & 8.36 & 16.4 & 9.17 & 19.11 & 39.43 & 6.62 & 12.62 & 24.92 & 7.36 & 13.07 & 27.1 & 7.78 & 13.6 & 27.5 \\
ta79 & cost & 5640 & 5622 & 5586 & 5640 & 5622 & 5586 & 5641 & 5600 & 5584 & 5641 & 5641 & 5641 & 5616 & 5590 & 5547 & 5649 & 5649 & 5649 & 5649 & 5606 & 5606 & 5606 & 5586 & 5531 \\
& time & 3.72 & 7.44 & 14.32 & 3.57 & 7.25 & 14.52 & 4.55 & 9.26 & 18.53 & 3.89 & 8.07 & 16.64 & 7.73 & 16.25 & 33.4 & 6.18 & 12.55 & 24.73 & 7.1 & 13.58 & 27.64 & 6.34 & 12.9 & 25.11 \\
ta80 & cost & 5464 & 5433 & 5421 & 5474 & 5474 & 5445 & 5445 & 5445 & 5445 & 5457 & 5457 & 5445 & 5445 & 5445 & 5421 & 5440 & 5431 & 5419 & 5421 & 5421 & 5421 & 5420 & 5413 & 5405 \\
& time & 3.62 & 7.05 & 14.61 & 3.99 & 7.81 & 15.33 & 4.21 & 9.42 & 19.11 & 3.92 & 8.15 & 16.19 & 8.47 & 18.83 & 36.55 & 5.83 & 11.28 & 24.77 & 5.32 & 12.84 & 27.61 & 5.28 & 11.65 & 25.99 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\restoregeometry
\clearpage
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench}.}
\label{appx:tab:results_uchoa_11}
\centering
\begin{tabular}{ll|c|rrrrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{3}{l}{\textbf{POMO}\cite{kwon}} & \textbf{VRPH} & \\
n100 & & & single & multi & aug=8 & Sweep & Savings \\
\midrule
X-n101-k25 & cost & 27591 & 29627 & 29025 & 28596 & 35807 & 29674 \\
& k & & 28 & 27 & 27 & 30 & 30 \\
& time & & 0.29 & 0.3 & 0.04 & 0 & 0 \\
X-n106-k14 & cost & 26362 & 27521 & 27140 & 26849 & 34811 & 27442 \\
& k & & 14 & 14 & 14 & 14 & 14 \\
& time & & 0.28 & 0.29 & 0.04 & 0 & 0 \\
X-n110-k13 & cost & 14971 & 15850 & 15316 & 15278 & 20435 & 15883 \\
& k & & 14 & 13 & 13 & 14 & 13 \\
& time & & 0.28 & 0.29 & 0.04 & 0 & 0 \\
X-n115-k10 & cost & 12747 & 13523 & 13192 & 13192 & 18727 & 13585 \\
& k & & 10 & 10 & 10 & 11 & 11 \\
& time & & 0.28 & 0.29 & 0.04 & 0 & 0 \\
X-n120-k6 & cost & 13332 & 14077 & 13846 & 13614 & 33166 & 14745 \\
& k & & 6 & 6 & 6 & 6 & 6 \\
& time & & 0.28 & 0.29 & 0.04 & 0 & 0 \\
X-n125-k30 & cost & 55539 & 79994 & 74465 & 59504 & 71695 & 60561 \\
& k & & 41 & 38 & 32 & 36 & 34 \\
& time & & 0.31 & 0.32 & 0.04 & 0 & 0 \\
X-n129-k18 & cost & 28940 & 29576 & 29220 & 29220 & 48717 & 30407 \\
& k & & 18 & 18 & 18 & 19 & 18 \\
& time & & 0.3 & 0.3 & 0.04 & 0 & 0 \\
X-n134-k13 & cost & 10916 & 11803 & 11601 & 11377 & 19640 & 11415 \\
& k & & 14 & 14 & 14 & 14 & 14 \\
& time & & 0.3 & 0.3 & 0.04 & 0 & 0 \\
X-n139-k10 & cost & 13590 & 14714 & 14147 & 13900 & 22174 & 14726 \\
& k & & 11 & 11 & 10 & 11 & 11 \\
& time & & 0.3 & 0.3 & 0.04 & 0 & 0 \\
X-n143-k7 & cost & 15700 & 16195 & 16138 & 16138 & 41910 & 17664 \\
& k & & 7 & 7 & 7 & 7 & 7 \\
& time & & 0.3 & 0.31 & 0.04 & 0 & 0 \\
X-n148-k46 & cost & 43448 & 72057 & 64407 & 52085 & 61758 & 48048 \\
& k & & 77 & 66 & 52 & 57 & 53 \\
& time & & 0.34 & 0.35 & 0.05 & 0 & 0 \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench}.}
\label{appx:tab:results_uchoa_12}
\centering
\begin{tabular}{ll|c|rrrrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{3}{l}{\textbf{POMO}\cite{kwon}} & \textbf{VRPH} & \\
n150 & & & single & multi & aug=8 & Sweep & Savings \\
\midrule
X-n153-k22 & cost & 21220 & 25709 & 24591 & 23833 & 31639 & 22649 \\
& k & & 27 & 26 & 26 & 29 & 26 \\
& time & & 0.67 & 0.39 & 0.05 & 0 & 0 \\
X-n157-k13 & cost & 16876 & 17544 & 17331 & 17164 & 25353 & 19171 \\
& k & & 13 & 13 & 13 & 15 & 15 \\
& time & & 0.33 & 0.43 & 0.04 & 0 & 0 \\
X-n162-k11 & cost & 14138 & 15142 & 14844 & 14813 & 24262 & 15708 \\
& k & & 11 & 11 & 11 & 11 & 11 \\
& time & & 0.32 & 0.37 & 0.04 & 0 & 0 \\
X-n167-k10 & cost & 20557 & 22039 & 21564 & 21390 & 47099 & 22004 \\
& k & & 10 & 10 & 10 & 10 & 10 \\
& time & & 0.33 & 0.35 & 0.04 & 0 & 0 \\
X-n172-k51 & cost & 45607 & 61768 & 60867 & 55635 & 61324 & 48634 \\
& k & & 69 & 66 & 56 & 63 & 57 \\
& time & & 0.37 & 0.39 & 0.05 & 0 & 0 \\
X-n176-k26 & cost & 47812 & 55315 & 53233 & 52723 & 79892 & 52866 \\
& k & & 28 & 28 & 28 & 33 & 29 \\
& time & & 0.35 & 0.36 & 0.05 & 0 & 0 \\
X-n181-k23 & cost & 25569 & 26255 & 26101 & 26101 & 33162 & 26561 \\
& k & & 23 & 23 & 23 & 23 & 23 \\
& time & & 0.35 & 0.38 & 0.05 & 0 & 0 \\
X-n186-k15 & cost & 24145 & 25673 & 24683 & 24683 & 48964 & 25690 \\
& k & & 15 & 15 & 15 & 15 & 15 \\
& time & & 0.35 & 0.39 & 0.05 & 0 & 0 \\
X-n190-k8 & cost & 16980 & 20053 & 19642 & 18550 & 40624 & 17816 \\
& k & & 8 & 8 & 8 & 8 & 8 \\
& time & & 0.55 & 0.38 & 0.05 & 0 & 0 \\
X-n195-k51 & cost & 44225 & 51016 & 49771 & 48308 & 60623 & 45976 \\
& k & & 56 & 55 & 54 & 62 & 55 \\
& time & & 0.41 & 0.46 & 0.06 & 0 & 0 \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench}.}
\label{appx:tab:results_uchoa_13}
\centering
\begin{tabular}{ll|c|rrrrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{3}{l}{\textbf{POMO}\cite{kwon}} & \textbf{VRPH} & \\
n200 & & & single & multi & aug=8 & Sweep & Savings \\
\midrule
X-n200-k36 & cost & 58578 & 63316 & 61811 & 61512 & 72574 & 61911 \\
& k & & 37 & 37 & 37 & 39 & 38 \\
& time & & 0.39 & 0.38 & 0.06 & 0 & 0 \\
X-n204-k19 & cost & 19565 & 21581 & 20853 & 20657 & 33405 & 21004 \\
& k & & 20 & 19 & 19 & 19 & 19 \\
& time & & 0.37 & 0.37 & 0.06 & 0 & 0 \\
X-n209-k16 & cost & 30656 & 33346 & 32316 & 32145 & 66836 & 32793 \\
& k & & 16 & 16 & 16 & 17 & 16 \\
& time & & 0.36 & 0.38 & 0.05 & 0 & 0 \\
X-n214-k11 & cost & 10856 & 17074 & 14923 & 14129 & 25760 & 11949 \\
& k & & 20 & 15 & 14 & 12 & 12 \\
& time & & 0.38 & 0.38 & 0.06 & 0 & 0 \\
X-n219-k73 & cost & 117595 & 164917 & 157869 & 148715 & 199383 & 172876 \\
& k & & 104 & 101 & 87 & 109 & 109 \\
& time & & 0.44 & 0.44 & 0.07 & 0 & 0 \\
X-n223-k34 & cost & 40437 & 42884 & 42162 & 42161 & 56590 & 42639 \\
& k & & 34 & 35 & 35 & 37 & 36 \\
& time & & 0.4 & 0.4 & 0.06 & 0 & 0 \\
X-n228-k23 & cost & 25742 & 30048 & 28727 & 27596 & 44883 & 27296 \\
& k & & 24 & 24 & 23 & 29 & 25 \\
& time & & 0.38 & 0.4 & 0.06 & 0 & 0 \\
X-n233-k16 & cost & 19230 & 27044 & 25061 & 20983 & 33749 & 20585 \\
& k & & 17 & 17 & 17 & 17 & 17 \\
& time & & 0.38 & 0.4 & 0.06 & 0 & 0 \\
X-n237-k14 & cost & 27042 & 29262 & 29041 & 28835 & 71726 & 30571 \\
& k & & 14 & 14 & 14 & 14 & 14 \\
& time & & 0.37 & 0.4 & 0.07 & 0 & 0 \\
X-n242-k48 & cost & 82751 & 85689 & 85463 & 85325 & 123830 & 89918 \\
& k & & 48 & 48 & 48 & 55 & 52 \\
& time & & 0.4 & 0.41 & 0.07 & 0 & 0.01 \\
X-n247-k50 & cost & 37274 & 76015 & 67888 & 50926 & 56494 & 41935 \\
& k & & 94 & 81 & 58 & 65 & 59 \\
& time & & 0.43 & 0.47 & 0.08 & 0 & 0.01 \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench}.}
\label{appx:tab:results_uchoa_14}
\centering
\begin{tabular}{ll|c|rrrrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{3}{l}{\textbf{POMO}\cite{kwon}} & \textbf{VRPH} & \\
n250 & & & single & multi & aug=8 & Sweep & Savings \\
\midrule
X-n251-k28 & cost & 38684 & 41465 & 40784 & 40466 & 60078 & 40285 \\
& k & & 28 & 28 & 28 & 29 & 28 \\
& time & & 0.43 & 0.43 & 0.07 & 0 & 0.01 \\
X-n256-k16 & cost & 18839 & 39858 & 29117 & 22825 & 28514 & 20518 \\
& k & & 35 & 23 & 19 & 17 & 17 \\
& time & & 0.42 & 0.41 & 0.08 & 0 & 0.01 \\
X-n261-k13 & cost & 26558 & 30107 & 29300 & 28842 & 70523 & 28638 \\
& k & & 13 & 13 & 13 & 13 & 13 \\
& time & & 0.39 & 0.4 & 0.07 & 0 & 0.01 \\
X-n266-k58 & cost & 75478 & 81464 & 80122 & 80122 & 115703 & 81046 \\
& k & & 60 & 60 & 60 & 67 & 64 \\
& time & & 0.41 & 0.44 & 0.08 & 0 & 0.01 \\
X-n270-k35 & cost & 35291 & 39142 & 38535 & 38367 & 48710 & 37181 \\
& k & & 36 & 37 & 37 & 38 & 37 \\
& time & & 0.41 & 0.45 & 0.08 & 0 & 0.01 \\
X-n275-k28 & cost & 21245 & 28811 & 25833 & 23360 & 35048 & 24177 \\
& k & & 32 & 29 & 28 & 31 & 31 \\
& time & & 0.41 & 0.42 & 0.08 & 0 & 0.01 \\
X-n280-k17 & cost & 33503 & 36961 & 36503 & 35835 & 88461 & 35951 \\
& k & & 17 & 17 & 17 & 21 & 18 \\
& time & & 0.41 & 0.41 & 0.08 & 0 & 0.01 \\
X-n284-k15 & cost & 20215 & 47812 & 38875 & 25404 & 44890 & 21924 \\
& k & & 35 & 24 & 16 & 15 & 15 \\
& time & & 0.44 & 0.45 & 0.09 & 0 & 0.01 \\
X-n289-k60 & cost & 95151 & 104199 & 101303 & 99850 & 146744 & 99808 \\
& k & & 66 & 64 & 63 & 70 & 65 \\
& time & & 0.44 & 0.54 & 0.09 & 0 & 0.01 \\
X-n294-k50 & cost & 47161 & 58954 & 55642 & 55011 & 67614 & 49307 \\
& k & & 54 & 53 & 53 & 56 & 53 \\
& time & & 0.43 & 0.47 & 0.1 & 0 & 0.01 \\
X-n298-k31 & cost & 34231 & 39956 & 37929 & 37363 & 57847 & 35996 \\
& k & & 32 & 32 & 32 & 33 & 32 \\
& time & & 0.43 & 0.44 & 0.09 & 0 & 0.01 \\
\bottomrule
\end{tabular}
\end{table*}
\newgeometry{top=30mm, bottom=25mm, left=40mm, right=40mm}
\clearpage
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (200 iterations).}
\label{appx:tab:results_uchoa_21}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n101-k25 & cost & 27591 & 28906 & 29804 & 29181 & 29243 & 28845 & 29243 & 29674 & 30927 & 29562 & 28506 & 28444 & 28444 \\
& k & & 26 & 26 & 30 & 30 & 27 & 30 & 30 & 27 & 28 & 27 & 27 & 27 \\
& time & & 2.09 & 5.73 & 0.59 & 0.47 & 0.5 & 0.43 & 0.27 & 12.37 & 13.36 & 1.07 & 1.23 & 1.25 \\
X-n106-k14 & cost & 26362 & 27182 & 27182 & 27099 & 27099 & 27442 & 27099 & 27442 & 28323 & 28085 & 26928 & 27057 & 27057 \\
& k & & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 \\
& time & & 2.53 & 5.52 & 0.45 & 0.48 & 0.54 & 0.44 & 0.27 & 12.89 & 14.01 & 1.17 & 1.31 & 1.36 \\
X-n110-k13 & cost & 14971 & 16266 & 15956 & 15883 & 15883 & 15883 & 15883 & 15883 & 17071 & 16376 & 15883 & 15738 & 15738 \\
& k & & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 \\
& time & & 11.93 & 86.13 & 0.55 & 0.5 & 0.58 & 0.46 & 0.3 & 13.7 & 14.48 & 1.15 & 1.36 & 1.48 \\
X-n115-k10 & cost & 12747 & 13169 & 13169 & 13567 & 13585 & 13553 & 13585 & 13585 & 15535 & 14454 & 13381 & 13382 & 13382 \\
& k & & 10 & 10 & 11 & 11 & 11 & 11 & 11 & 10 & 10 & 11 & 11 & 11 \\
& time & & 4.13 & 5.92 & 0.69 & 0.62 & 0.83 & 0.56 & 0.42 & 14.61 & 14.97 & 1.39 & 1.58 & 1.67 \\
X-n120-k6 & cost & 13332 & 13988 & 13981 & 14643 & 14655 & 14745 & 14655 & 14745 & 15713 & 16316 & 14745 & 14487 & 14487 \\
& k & & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\
& time & & 30.55 & 8.72 & 0.77 & 0.79 & 0.9 & 0.68 & 0.55 & 14.53 & 16.03 & 1.49 & 1.68 & 1.75 \\
X-n125-k30 & cost & 55539 & 58333 & 58352 & 59239 & 59239 & 58461 & 59239 & 60561 & 60585 & 59489 & 57805 & 57774 & 57774 \\
& k & & 31 & 31 & 34 & 34 & 32 & 34 & 34 & 31 & 31 & 32 & 32 & 32 \\
& time & & 1.88 & 1.46 & 0.65 & 0.66 & 0.74 & 0.65 & 0.36 & 14.99 & 17.04 & 1.4 & 1.63 & 1.75 \\
X-n129-k18 & cost & 28940 & 31063 & 30827 & 30407 & 30407 & 30407 & 30407 & 30407 & 33014 & 32825 & 30407 & 30407 & 30407 \\
& k & & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 \\
& time & & 11.15 & 30.97 & 0.73 & 0.67 & 0.77 & 0.6 & 0.36 & 15.68 & 17.41 & 1.46 & 1.67 & 1.74 \\
X-n134-k13 & cost & 10916 & 12514 & 12520 & 11415 & 11415 & 11415 & 11415 & 11415 & 13240 & 13182 & 11415 & 11415 & 11415 \\
& k & & 13 & 13 & 14 & 14 & 14 & 14 & 14 & 15 & 14 & 14 & 14 & 14 \\
& time & & 3.76 & 10.06 & 0.86 & 0.79 & 0.94 & 0.7 & 0.4 & 16 & 18.18 & 1.64 & 1.85 & 2.03 \\
X-n139-k10 & cost & 13590 & 14995 & 14560 & 14726 & 14726 & 14726 & 14726 & 14726 & 16436 & 15261 & 14635 & 14552 & 14552 \\
& k & & 10 & 10 & 11 & 11 & 11 & 11 & 11 & 13 & 11 & 10 & 10 & 10 \\
& time & & 13.36 & 261.77 & 0.93 & 0.82 & 0.98 & 0.76 & 0.43 & 16.6 & 18.05 & 1.65 & 1.93 & 1.9 \\
X-n143-k7 & cost & 15700 & 17544 & 17258 & 17598 & 17598 & 17126 & 17544 & 17664 & 20067 & 19960 & 16932 & 17243 & 17243 \\
& k & & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 \\
& time & & 10.22 & 7.38 & 1.12 & 1.09 & 1.29 & 1.03 & 0.82 & 17.08 & 18.57 & 2.01 & 2.24 & 2.17 \\
X-n148-k46 & cost & 43448 & 47362 & 47362 & 47593 & 47593 & 47789 & 47606 & 48048 & 46923 & 47938 & 46788 & 46891 & 46891 \\
& k & & 46 & 46 & 53 & 53 & 51 & 53 & 53 & 49 & 48 & 51 & 49 & 49 \\
& time & & 11.84 & 5.44 & 0.79 & 0.81 & 1.02 & 0.78 & 0.45 & 18.02 & 20.12 & 1.65 & 1.96 & 2.02 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (200 iterations).}
\label{appx:tab:results_uchoa_22}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n153-k22 & cost & 21220 & 24812 & 24812 & 22597 & 22597 & 22649 & 22597 & 22649 & 26294 & 25312 & 22587 & 22548 & 22548 \\
& k & & 25 & 25 & 26 & 26 & 26 & 26 & 26 & 26 & 25 & 26 & 26 & 26 \\
& time & & 0.74 & 0.76 & 1.12 & 1.01 & 1.27 & 0.93 & 0.49 & 19.78 & 20.39 & 2.09 & 2.24 & 2.21 \\
X-n157-k13 & cost & 16876 & 17257 & 17286 & 18963 & 18963 & 19050 & 19171 & 19171 & 18556 & 18352 & 18782 & 18810 & 18810 \\
& k & & 13 & 13 & 15 & 15 & 15 & 15 & 15 & 13 & 13 & 15 & 15 & 15 \\
& time & & 13.77 & 10.37 & 0.96 & 1.01 & 1.2 & 0.86 & 0.49 & 19.44 & 20.86 & 1.98 & 2.16 & 2.08 \\
X-n162-k11 & cost & 14138 & 14782 & 14920 & 14898 & 14898 & 15227 & 14898 & 15708 & 17539 & 16034 & 14958 & 14847 & 14847 \\
& k & & 11 & 11 & 11 & 11 & 11 & 11 & 11 & 12 & 12 & 11 & 11 & 11 \\
& time & & 11.89 & 629.62 & 1.26 & 1.22 & 1.36 & 1.21 & 0.6 & 19.65 & 21.05 & 2.1 & 2.41 & 2.71 \\
X-n167-k10 & cost & 20557 & 22629 & 22678 & 21656 & 21656 & 22004 & 22004 & 22004 & 25399 & 25062 & 21695 & 21836 & 21836 \\
& k & & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 \\
& time & & 17.69 & 7.74 & 1.4 & 1.44 & 1.64 & 1.16 & 0.72 & 19.95 & 22.11 & 2.37 & 2.64 & 2.67 \\
X-n172-k51 & cost & 45607 & 50686 & 50779 & 47924 & 47924 & 47790 & 47924 & 47921 & 55282 & 51166 & 47125 & 46846 & 46846 \\
& k & & 52 & 52 & 57 & 57 & 54 & 57 & 57 & 55 & 54 & 53 & 54 & 54 \\
& time & & 8.46 & 71.76 & 1.1 & 1.08 & 1.32 & 1.04 & 0.62 & 21 & 24.02 & 2.09 & 2.45 & 2.79 \\
X-n176-k26 & cost & 47812 & 55614 & 55614 & 51382 & 51382 & 50872 & 51433 & 51783 & 61411 & 60630 & 51241 & 50822 & 50822 \\
& k & & 27 & 27 & 29 & 29 & 28 & 29 & 29 & 29 & 29 & 28 & 28 & 28 \\
& time & & 0.66 & 0.63 & 1.43 & 1.39 & 1.54 & 1.25 & 0.64 & 21.13 & 23.29 & 2.46 & 2.68 & 2.78 \\
X-n181-k23 & cost & 25569 & 26346 & 26277 & 26355 & 26355 & 26561 & 26352 & 26561 & 27557 & 27939 & 26324 & 26301 & 26301 \\
& k & & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 \\
& time & & 33.52 & 274.17 & 1.17 & 1.19 & 1.4 & 1.17 & 0.54 & 21.94 & 23.81 & 2.38 & 2.56 & 2.66 \\
X-n186-k15 & cost & 24145 & 26063 & 26073 & 25690 & 25690 & 25690 & 25690 & 25690 & 28652 & 28979 & 25595 & 25493 & 25493 \\
& k & & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 16 & 15 & 15 & 15 \\
& time & & 12.37 & 91.63 & 1.5 & 1.35 & 1.71 & 1.27 & 0.66 & 22.14 & 24.14 & 2.59 & 2.89 & 2.99 \\
X-n190-k8 & cost & 16980 & 17949 & 17949 & 17742 & 17742 & 17816 & 17742 & 17816 & 19990 & 20064 & 17709 & 17689 & 17689 \\
& k & & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 \\
& time & & 7.86 & 6.1 & 2.16 & 2.1 & 2.37 & 1.87 & 1.67 & 23.28 & 24.99 & 3.4 & 3.59 & 3.58 \\
X-n195-k51 & cost & 44225 & 49975 & 49975 & 45976 & 45976 & 45976 & 45976 & 45976 & 52594 & 52788 & 45612 & 45520 & 45520 \\
& k & & 52 & 52 & 55 & 55 & 55 & 55 & 55 & 53 & 54 & 54 & 54 & 54 \\
& time & & 2.03 & 2.06 & 1.37 & 1.3 & 1.6 & 1.23 & 0.66 & 23.5 & 26.25 & 2.5 & 2.92 & 3.01 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (200 iterations).}
\label{appx:tab:results_uchoa_23}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n200-k36 & cost & 58578 & 61080 & 61080 & 60902 & 60913 & 61051 & 60920 & 61911 & 64205 & 64746 & 60373 & 60337 & 60337 \\
& k & & 37 & 37 & 38 & 38 & 38 & 38 & 38 & 37 & 38 & 37 & 37 & 37 \\
& time & & 4.99 & 9.32 & 1.57 & 1.49 & 1.65 & 1.35 & 0.66 & 24.31 & 26.05 & 2.76 & 3.03 & 3.12 \\
X-n204-k19 & cost & 19565 & 21828 & 21922 & 20906 & 20970 & 21004 & 21004 & 21004 & 24188 & 23259 & 20925 & 20587 & 20587 \\
& k & & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 20 & 19 & 19 & 19 & 19 \\
& time & & 25.3 & 12.72 & 1.71 & 1.58 & 2.02 & 1.55 & 0.7 & 25.05 & 28.41 & 2.88 & 3.26 & 3.41 \\
X-n209-k16 & cost & 30656 & 32553 & 32724 & 32793 & 32793 & 32793 & 32793 & 32793 & 35254 & 36933 & 32636 & 32602 & 32602 \\
& k & & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 \\
& time & & 10.64 & 59.37 & 1.77 & 1.71 & 2.18 & 1.6 & 0.81 & 25.14 & 27.97 & 3.25 & 3.52 & 3.63 \\
X-n214-k11 & cost & 10856 & 12498 & 12507 & 11910 & 11949 & 11949 & 11949 & 11949 & 14626 & 14231 & 11756 & 11915 & 11915 \\
& k & & 11 & 11 & 12 & 12 & 12 & 12 & 12 & 11 & 12 & 12 & 12 & 12 \\
& time & & 16.36 & 17.37 & 2.36 & 2.07 & 2.65 & 1.94 & 1.21 & 26.41 & 27.61 & 3.79 & 4.06 & 4.1 \\
X-n219-k73 & cost & 117595 & 118423 & 118373 & 172876 & 172876 & 172876 & 172876 & 172876 & 120142 & 120293 & 172876 & 172876 & 172876 \\
& k & & 73 & 73 & 109 & 109 & 109 & 109 & 109 & 73 & 73 & 109 & 109 & 109 \\
& time & & 77.09 & 1280.43 & 1.41 & 1.39 & 1.79 & 1.31 & 0.72 & 26.31 & 28.92 & 2.81 & 3.22 & 3.35 \\
X-n223-k34 & cost & 40437 & 44320 & 44320 & 42343 & 42343 & 42639 & 42299 & 42639 & 51570 & 47983 & 42190 & 42230 & 42230 \\
& k & & 34 & 34 & 36 & 36 & 36 & 36 & 36 & 35 & 34 & 35 & 35 & 35 \\
& time & & 1.57 & 1.61 & 1.82 & 1.81 & 2.17 & 1.74 & 0.8 & 27.24 & 28.34 & 3.28 & 3.68 & 3.55 \\
X-n228-k23 & cost & 25742 & 32143 & 32143 & 27076 & 27076 & 27296 & 27105 & 27296 & 35231 & 33820 & 27064 & 26996 & 26996 \\
& k & & 23 & 23 & 25 & 25 & 25 & 25 & 25 & 25 & 26 & 24 & 24 & 24 \\
& time & & 1.49 & 1.58 & 2.36 & 2.29 & 2.61 & 2.19 & 0.93 & 28.31 & 30.94 & 3.78 & 4.12 & 4.18 \\
X-n233-k16 & cost & 19230 & 23128 & 23128 & 20478 & 20585 & 20585 & 20585 & 20585 & 27821 & 25915 & 20432 & 20385 & 20385 \\
& k & & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 20 & 18 & 17 & 17 & 17 \\
& time & & 2.08 & 2.25 & 2.41 & 2.38 & 3.05 & 2.19 & 1.15 & 29.32 & 30.83 & 4.31 & 4.74 & 4.99 \\
X-n237-k14 & cost & 27042 & 29175 & 29175 & 30341 & 30341 & 30571 & 30433 & 30571 & 32205 & 32948 & 30003 & 29828 & 29828 \\
& k & & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14.00 & 14 & 14 & 14 \\
& time & & 4.81 & 4.86 & 2.46 & 2.45 & 2.95 & 2.32 & 1.19 & 29.15 & 31.69 & 4.09 & 4.75 & 4.59 \\
X-n242-k48 & cost & 82751 & 87069 & 87069 & 88179 & 88179 & 88320 & 88179 & 88696 & 95420 & 94,852.00 & 86973 & 87097 & 87097 \\
& k & & 48 & 48 & 52 & 52 & 52 & 52 & 52 & 48 & 48.00 & 50 & 50 & 50 \\
& time & & 1.64 & 1.66 & 2.08 & 2.07 & 2.48 & 2.13 & 0.95 & 29.05 & 32.75 & 3.75 & 4.19 & 4.06 \\
X-n247-k50 & cost & 37274 & 44134 & 44134 & 40022 & 40022 & 39246 & 40123 & 40508 & 48507 & 46782 & 39514 & 39349 & 39349 \\
& k & & 52 & 52 & 59 & 59 & 55 & 59 & 59 & 54 & 54 & 55 & 55 & 55 \\
& time & & 1.02 & 0.99 & 2.68 & 2.71 & 3.01 & 2.4 & 1.19 & 28.92 & 33.92 & 4.4 & 4.88 & 5.06 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (200 iterations).}
\label{appx:tab:results_uchoa_24}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n251-k28 & cost & 38684 & 40417 & 40424 & 40279 & 40285 & 40285 & 40285 & 40285 & 42297 & 42545 & 40285 & 40285 & 40285 \\
& k & & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 \\
& time & & 14.7 & 32.3 & 2.63 & 2.23 & 2.82 & 2.09 & 0.95 & 30.17 & 34.44 & 4.04 & 4.73 & 4.5 \\
X-n256-k16 & cost & 18839 & 21363 & 21361 & 20518 & 20518 & 20518 & 20518 & 20518 & 23949 & 22469 & 20390 & 20185 & 20185 \\
& k & & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 \\
& time & & 5.47 & 5.27 & 3.11 & 2.66 & 3.94 & 2.49 & 1.24 & 31.66 & 35.04 & 4.98 & 5.31 & 5.54 \\
X-n261-k13 & cost & 26558 & 30665 & 30665 & 28638 & 28638 & 28638 & 28638 & 28638 & 36604 & 34448 & 28638 & 28638 & 28638 \\
& k & & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 14 & 13 & 13 & 13 \\
& time & & 4.07 & 4.2 & 3.51 & 3.33 & 4.12 & 3 & 1.86 & 31.61 & 36.04 & 5.49 & 6.09 & 6.72 \\
X-n266-k58 & cost & 75478 & 80064 & 80064 & 80553 & 80553 & 80792 & 80657 & 80979 & 83468 & 83310 & 79913 & 79308 & 79308 \\
& k & & 59 & 59 & 64 & 64 & 64 & 64 & 64 & 61 & 61 & 62 & 61 & 61 \\
& time & & 6.33 & 5.7 & 2.34 & 2.36 & 2.85 & 2.22 & 1.04 & 32 & 36.59 & 4.54 & 4.94 & 5.28 \\
X-n270-k35 & cost & 35291 & 39152 & 39152 & 37181 & 37181 & 37181 & 37181 & 37181 & 40547 & 40415 & 36862 & 36529 & 36529 \\
& k & & 36 & 36 & 37 & 37 & 37 & 37 & 37 & 37 & 37 & 36 & 36 & 36 \\
& time & & 5.67 & 5.22 & 2.78 & 2.36 & 3.13 & 2.39 & 1.08 & 32.6 & 37.08 & 4.59 & 5.21 & 5.53 \\
X-n275-k28 & cost & 21245 & 22562 & 22591 & 23907 & 23907 & 24177 & 24177 & 24177 & 26052 & 26044 & 23961 & 23776 & 23776 \\
& k & & 28 & 28 & 31 & 31 & 31 & 31 & 31 & 31 & 32 & 31 & 31 & 31 \\
& time & & 9.36 & 15.92 & 2.87 & 2.82 & 3.33 & 2.56 & 1.08 & 32.71 & 37.01 & 4.86 & 5.49 & 5.66 \\
X-n280-k17 & cost & 33503 & 40878 & 40878 & 35951 & 35951 & 35951 & 35951 & 35951 & 46179 & 45719 & 35531 & 35760 & 35760 \\
& k & & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 17 & 18 & 18 \\
& time & & 1.7 & 1.69 & 4.5 & 3.64 & 4.95 & 3.52 & 2 & 33.58 & 38.56 & 6.47 & 7.23 & 7.27 \\
X-n284-k15 & cost & 20215 & 23695 & 23695 & 21924 & 21924 & 21924 & 21924 & 21924 & 26201 & 26436 & 21891 & 21757 & 21757 \\
& k & & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 16 & 15 & 15 & 15 \\
& time & & 1.98 & 1.87 & 4.01 & 3.91 & 4.78 & 3.48 & 2.12 & 33.8 & 39.2 & 6.55 & 7.24 & 7.84 \\
X-n289-k60 & cost & 95151 & 103936 & 103936 & 98648 & 98824 & 98743 & 98686 & 99808 & 117218 & 113473 & 97618 & 98070 & 98070 \\
& k & & 62 & 62 & 65 & 65 & 65 & 65 & 65 & 62 & 63 & 63 & 63 & 63 \\
& time & & 2.06 & 1.51 & 3.39 & 2.99 & 3.62 & 2.9 & 1.29 & 34.1 & 39.93 & 5.35 & 6.21 & 5.92 \\
X-n294-k50 & cost & 47161 & 54834 & 54834 & 49169 & 49307 & 49307 & 49307 & 49307 & 66687 & 62413 & 48934 & 48947 & 48947 \\
& k & & 51 & 51 & 53 & 53 & 53 & 53 & 53 & 59 & 54 & 52 & 52 & 52 \\
& time & & 1.57 & 1.48 & 3.51 & 2.99 & 3.61 & 2.77 & 1.31 & 35.26 & 39.9 & 5.39 & 6.22 & 6.08 \\
X-n298-k31 & cost & 34231 & 42908 & 42908 & 35996 & 35996 & 35996 & 35996 & 35996 & 48423 & 46636 & 35617 & 35767 & 35767 \\
& k & & 31 & 31 & 32 & 32 & 32 & 32 & 32 & 31 & 32 & 32 & 32 & 32 \\
& time & & 1.76 & 2.15 & 3.58 & 3.18 & 4.1 & 3.03 & 1.37 & 35.89 & 41.82 & 5.74 & 6.44 & 7.07 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (500 iterations).}
\label{appx:tab:results_uchoa_31}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n101-k25 & cost & 27591 & 28725 & 29430 & 29181 & 29211 & 28845 & 29243 & 29674 & 29561 & 29554 & 28458 & 28309 & 28309 \\
& k & & 26 & 26 & 30 & 30 & 27 & 30 & 30 & 27 & 27 & 27 & 27 & 27 \\
& time & & 18.79 & 134.68 & 1.08 & 1.06 & 1.31 & 1.01 & 0.71 & 31.32 & 34.52 & 2.32 & 2.75 & 2.74 \\
X-n106-k14 & cost & 26362 & 27182 & 27182 & 27099 & 27099 & 27442 & 27099 & 27442 & 28052 & 28085 & 26875 & 26747 & 26747 \\
& k & & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 \\
& time & & 20.28 & 38.39 & 1.15 & 1.19 & 1.34 & 1.14 & 0.77 & 32.58 & 36.42 & 2.62 & 2.96 & 2.79 \\
X-n110-k13 & cost & 14971 & 15698 & 15568 & 15883 & 15883 & 15883 & 15883 & 15883 & 16389 & 16170 & 15612 & 15738 & 15738 \\
& k & & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 \\
& time & & 36.63 & 199.29 & 1.23 & 1.21 & 1.4 & 1.11 & 0.73 & 32.8 & 37.45 & 2.57 & 3.04 & 3.06 \\
X-n115-k10 & cost & 12747 & 13169 & 13138 & 13567 & 13585 & 13553 & 13585 & 13585 & 15033 & 13868 & 13362 & 13382 & 13382 \\
& k & & 10 & 10 & 11 & 11 & 11 & 11 & 11 & 11 & 10 & 11 & 11 & 11 \\
& time & & 31.74 & 917.03 & 1.56 & 1.58 & 1.84 & 1.4 & 0.98 & 33.91 & 38.87 & 3.07 & 3.58 & 3.54 \\
X-n120-k6 & cost & 13332 & 13899 & 13730 & 14643 & 14655 & 14745 & 14655 & 14745 & 15713 & 16047 & 14554 & 14456 & 14456 \\
& k & & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\
& time & & 85.32 & 740.25 & 1.76 & 1.89 & 2.11 & 1.61 & 1.43 & 35.76 & 40.68 & 3.27 & 4 & 3.85 \\
X-n125-k30 & cost & 55539 & 57951 & 57916 & 59239 & 59239 & 57845 & 59239 & 60561 & 60115 & 59489 & 57597 & 57549 & 57549 \\
& k & & 31 & 31 & 34 & 34 & 31 & 34 & 34 & 32 & 31 & 32 & 32 & 32 \\
& time & & 25.07 & 145 & 1.57 & 1.6 & 1.91 & 1.53 & 0.95 & 38 & 43.3 & 3.11 & 3.71 & 3.69 \\
X-n129-k18 & cost & 28940 & 31061 & 30448 & 30407 & 30407 & 30407 & 30407 & 30407 & 32572 & 32832 & 30302 & 30407 & 30407 \\
& k & & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 \\
& time & & 37.55 & 171.38 & 1.71 & 1.58 & 1.99 & 1.54 & 0.95 & 38.36 & 43.12 & 3.2 & 3.82 & 3.79 \\
X-n134-k13 & cost & 10916 & 11964 & 12152 & 11415 & 11415 & 11415 & 11415 & 11415 & 13140 & 13182 & 11374 & 11320 & 11320 \\
& k & & 13 & 13 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 13 & 13 & 13 \\
& time & & 28.02 & 183.34 & 1.9 & 1.91 & 2.23 & 1.78 & 1.06 & 40.06 & 45.47 & 3.75 & 4.3 & 4.22 \\
X-n139-k10 & cost & 13590 & 14489 & 14457 & 14726 & 14726 & 14619 & 14726 & 14726 & 15616 & 15261 & 14635 & 14514 & 14514 \\
& k & & 10 & 10 & 11 & 11 & 10 & 11 & 11 & 13 & 11 & 10 & 10 & 10 \\
& time & & 47.74 & 1779.76 & 2.06 & 2.11 & 2.49 & 1.84 & 1.15 & 41.82 & 46.71 & 3.78 & 4.48 & 4.43 \\
X-n143-k7 & cost & 15700 & 16975 & 16675 & 17598 & 17598 & 17126 & 17115 & 17664 & 20067 & 19499 & 16847 & 16696 & 16696 \\
& k & & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 \\
& time & & 40.74 & 1105.7 & 2.51 & 2.53 & 3.16 & 2.54 & 2.18 & 41.76 & 47.04 & 4.63 & 5.25 & 5.27 \\
X-n148-k46 & cost & 43448 & 47003 & 46473 & 47593 & 47494 & 47621 & 47606 & 48048 & 46059 & 46905 & 46788 & 46653 & 46653 \\
& k & & 46 & 46 & 53 & 53 & 50 & 53 & 53 & 48 & 48 & 51 & 49 & 49 \\
& time & & 63.94 & 369.25 & 1.88 & 1.97 & 2.33 & 1.84 & 1.24 & 46.1 & 50.66 & 3.71 & 4.55 & 4.6 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (500 iterations).}
\label{appx:tab:results_uchoa_32}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n153-k22 & cost & 21220 & 22507 & 22670 & 22597 & 22597 & 22649 & 22597 & 22649 & 25709 & 24538 & 22587 & 22495 & 22495 \\
& k & & 25 & 25 & 26 & 26 & 26 & 26 & 26 & 25 & 24 & 26 & 25 & 25 \\
& time & & 19.98 & 327.65 & 2.55 & 2.52 & 3.16 & 2.3 & 1.25 & 46.37 & 51.29 & 4.78 & 5.23 & 5.5 \\
X-n157-k13 & cost & 16876 & 17238 & 17286 & 18963 & 18963 & 19050 & 19171 & 19171 & 18556 & 18352 & 18782 & 18696 & 18696 \\
& k & & 13 & 13 & 15 & 15 & 15 & 15 & 15 & 13 & 13 & 15 & 15 & 15 \\
& time & & 47.88 & 564.82 & 2.31 & 2.32 & 2.93 & 2.11 & 1.19 & 48.52 & 52.49 & 4.51 & 5.04 & 5.55 \\
X-n162-k11 & cost & 14138 & 14782 & 14746 & 14898 & 14898 & 15215 & 14898 & 15708 & 17524 & 16034 & 14958 & 14847 & 14847 \\
& k & & 11 & 11 & 11 & 11 & 11 & 11 & 11 & 12 & 12 & 11 & 11 & 11 \\
& time & & 79.83 & 3074.87 & 2.97 & 2.81 & 3.45 & 2.66 & 1.53 & 49.46 & 54.97 & 5.01 & 5.74 & 5.8 \\
X-n167-k10 & cost & 20557 & 22561 & 22394 & 21656 & 21656 & 22004 & 22004 & 22004 & 24880 & 25062 & 21695 & 21836 & 21836 \\
& k & & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 \\
& time & & 101 & 2112.88 & 3.25 & 3.24 & 3.96 & 2.82 & 1.93 & 50.64 & 55.89 & 5.53 & 6.4 & 6.45 \\
X-n172-k51 & cost & 45607 & 49782 & 48519 & 47924 & 47924 & 47790 & 47924 & 47921 & 50753 & 49283 & 47114 & 46846 & 46846 \\
& k & & 52 & 52 & 57 & 57 & 54 & 57 & 57 & 54 & 53 & 53 & 54 & 54 \\
& time & & 48.91 & 393.94 & 2.57 & 2.54 & 3.38 & 2.44 & 1.45 & 53.19 & 58.27 & 4.83 & 5.75 & 5.83 \\
X-n176-k26 & cost & 47812 & 53100 & 52794 & 51382 & 51382 & 50185 & 50454 & 50472 & 59133 & 57878 & 49978 & 49851 & 49851 \\
& k & & 26 & 26 & 29 & 29 & 28 & 29 & 29 & 29 & 28 & 28 & 28 & 28 \\
& time & & 13.46 & 8.37 & 3.09 & 3.04 & 3.79 & 3.01 & 1.6 & 55.44 & 59.67 & 5.76 & 6.44 & 6.51 \\
X-n181-k23 & cost & 25569 & 26346 & 26243 & 26355 & 26355 & 26561 & 26352 & 26561 & 27479 & 27924 & 26265 & 26201 & 26201 \\
& k & & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 \\
& time & & 97.27 & 1256.51 & 2.74 & 2.89 & 3.41 & 2.63 & 1.39 & 55.36 & 62 & 5.44 & 6.08 & 6.34 \\
X-n186-k15 & cost & 24145 & 25692 & 25910 & 25690 & 25690 & 25690 & 25690 & 25690 & 28652 & 28709 & 25215 & 25278 & 25278 \\
& k & & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 16 & 15 & 15 & 15 \\
& time & & 74.41 & 1137.01 & 3.38 & 3.31 & 4.24 & 3.03 & 1.62 & 56.36 & 62.7 & 6.16 & 7.19 & 7.12 \\
X-n190-k8 & cost & 16980 & 17864 & 17925 & 17742 & 17742 & 17816 & 17742 & 17816 & 19552 & 20064 & 17709 & 17689 & 17689 \\
& k & & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 \\
& time & & 94.32 & 3646.07 & 4.82 & 4.89 & 6.08 & 4.49 & 4.49 & 58.07 & 66.91 & 8.16 & 9.19 & 9.14 \\
X-n195-k51 & cost & 44225 & 49181 & 49248 & 45976 & 45976 & 45976 & 45976 & 45976 & 51860 & 50913 & 45460 & 45302 & 45302 \\
& k & & 52 & 52 & 55 & 55 & 55 & 55 & 55 & 54 & 55 & 54 & 54 & 54 \\
& time & & 71.58 & 148.58 & 3.22 & 3.14 & 3.97 & 2.94 & 1.66 & 59.75 & 66.44 & 5.87 & 7.23 & 6.75 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (500 iterations).}
\label{appx:tab:results_uchoa_33}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n200-k36 & cost & 58578 & 60475 & 60865 & 60902 & 60913 & 61051 & 60920 & 61911 & 64112 & 64640 & 60316 & 60315 & 60315 \\
& k & & 37 & 37 & 38 & 38 & 38 & 38 & 38 & 37 & 37 & 37 & 37 & 37 \\
& time & & 62.28 & 1629.3 & 3.45 & 3.38 & 4.07 & 3.23 & 1.65 & 60.76 & 67.26 & 6.5 & 7.43 & 7.26 \\
X-n204-k19 & cost & 19565 & 20897 & 21330 & 20906 & 20970 & 21004 & 21004 & 21004 & 24187 & 23045 & 20481 & 20587 & 20587 \\
& k & & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 \\
& time & & 129.9 & 1503.12 & 3.84 & 3.74 & 4.83 & 3.61 & 1.77 & 61.87 & 67.71 & 6.84 & 7.8 & 8.12 \\
X-n209-k16 & cost & 30656 & 32419 & 32324 & 32793 & 32793 & 32793 & 32793 & 32793 & 35254 & 36933 & 32636 & 32602 & 32602 \\
& k & & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 \\
& time & & 104.84 & 1009.65 & 4.18 & 4.12 & 5.86 & 3.84 & 2.03 & 62.16 & 70.68 & 7.42 & 8.47 & 8.5 \\
X-n214-k11 & cost & 10856 & 12498 & 12627 & 11910 & 11949 & 11949 & 11949 & 11949 & 14626 & 14186 & 11711 & 11739 & 11739 \\
& k & & 11 & 11 & 12 & 12 & 12 & 12 & 12 & 11 & 12 & 12 & 12 & 12 \\
& time & & 63.84 & 1975 & 5.29 & 5.12 & 6.83 & 4.74 & 3.12 & 63.67 & 72.79 & 9.12 & 10.12 & 9.78 \\
X-n219-k73 & cost & 117595 & 118420 & 118029 & 172876 & 172876 & 172876 & 172876 & 172876 & 120127 & 120293 & 172876 & 172876 & 172876 \\
& k & & 73 & 73 & 109 & 109 & 109 & 109 & 109 & 73 & 73 & 109 & 109 & 109 \\
& time & & 209.5 & 2412.98 & 3.34 & 3.3 & 4.43 & 3.21 & 1.72 & 66.32 & 73.6 & 6.61 & 7.71 & 7.74 \\
X-n223-k34 & cost & 40437 & 42810 & 42810 & 42343 & 42343 & 42639 & 42299 & 42639 & 51486 & 48485 & 42073 & 41981 & 41981 \\
& k & & 34 & 34 & 36 & 36 & 36 & 36 & 36 & 36 & 35 & 34 & 35 & 35 \\
& time & & 42.6 & 680.88 & 4.29 & 4.25 & 5.51 & 3.92 & 2.23 & 68 & 77.52 & 7.66 & 8.81 & 8.86 \\
X-n228-k23 & cost & 25742 & 29817 & 29648 & 27076 & 27076 & 27296 & 27105 & 27296 & 34924 & 32931 & 26995 & 26974 & 26974 \\
& k & & 23 & 23 & 25 & 25 & 25 & 25 & 25 & 27 & 25 & 24 & 24 & 24 \\
& time & & 12.83 & 530.02 & 5.21 & 5.3 & 7.67 & 4.71 & 2.37 & 68.55 & 77.19 & 9.06 & 10.18 & 10.35 \\
X-n233-k16 & cost & 19230 & 20530 & 20618 & 20478 & 20585 & 20585 & 20585 & 20585 & 27468 & 25457 & 20066 & 20385 & 20385 \\
& k & & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 22 & 18 & 17 & 17 & 17 \\
& time & & 72.48 & 286.46 & 5.66 & 5.58 & 7.63 & 5.45 & 2.9 & 72.53 & 79.31 & 10.52 & 11.45 & 11.52 \\
X-n237-k14 & cost & 27042 & 28836 & 28928 & 30341 & 30341 & 30571 & 30433 & 30571 & 32205 & 32948 & 30003 & 29828 & 29828 \\
& k & & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14.00 & 14 & 14 & 14 & 14 \\
& time & & 136.25 & 790.7 & 5.92 & 5.79 & 7.43 & 5.44 & 3 & 69.62 & 80.9 & 9.96 & 11.19 & 11.28 \\
X-n242-k48 & cost & 82751 & 86419 & 86408 & 88179 & 88179 & 88320 & 88179 & 88696 & 95,260.00 & 94031 & 86682 & 86843 & 86843 \\
& k & & 48 & 48 & 52 & 52 & 52 & 52 & 52 & 48.00 & 48 & 50 & 50 & 50 \\
& time & & 73.45 & 411.85 & 4.77 & 4.77 & 6.15 & 4.72 & 2.37 & 75.44 & 83.39 & 8.88 & 10.52 & 10.13 \\
X-n247-k50 & cost & 37274 & 40372 & 40372 & 40022 & 40022 & 38876 & 40123 & 40508 & 48507 & 46088 & 39207 & 39349 & 39349 \\
& k & & 52 & 52 & 59 & 59 & 53 & 59 & 59 & 54 & 54 & 54 & 55 & 55 \\
& time & & 4.72 & 4.87 & 5.87 & 6.18 & 7.42 & 5.76 & 2.7 & 74.11 & 85.21 & 10.54 & 11.76 & 11.85 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (500 iterations).}
\label{appx:tab:results_uchoa_34}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n251-k28 & cost & 38684 & 40322 & 40215 & 40279 & 40285 & 40285 & 40285 & 40285 & 42297 & 42545 & 40071 & 40233 & 40233 \\
& k & & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 \\
& time & & 107.2 & 1223.66 & 5.39 & 5.25 & 7.02 & 5.1 & 2.56 & 78.34 & 85.31 & 9.84 & 11.13 & 11.63 \\
X-n256-k16 & cost & 18839 & 19856 & 20274 & 20518 & 20518 & 20518 & 20518 & 20518 & 23910 & 22469 & 20084 & 20021 & 20021 \\
& k & & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 \\
& time & & 136.78 & 54.78 & 6.84 & 6.33 & 8.53 & 6.18 & 3.33 & 79.97 & 89.58 & 11.68 & 13.22 & 13.29 \\
X-n261-k13 & cost & 26558 & 29596 & 29476 & 28638 & 28638 & 28638 & 28638 & 28638 & 36604 & 34448 & 28601 & 28638 & 28638 \\
& k & & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 14 & 13 & 13 & 13 \\
& time & & 88.99 & 272.73 & 8.25 & 7.91 & 11 & 7.23 & 4.97 & 83.13 & 89.54 & 13.37 & 15.25 & 15.23 \\
X-n266-k58 & cost & 75478 & 78318 & 78291 & 80553 & 80553 & 80792 & 80657 & 80979 & 83468 & 83296 & 79722 & 79150 & 79150 \\
& k & & 59 & 59 & 64 & 64 & 64 & 64 & 64 & 61 & 61 & 61 & 61 & 61 \\
& time & & 81.38 & 1296.57 & 5.41 & 5.57 & 7.1 & 5.23 & 2.68 & 81.35 & 89.48 & 10.64 & 11.96 & 12.21 \\
X-n270-k35 & cost & 35291 & 37521 & 37615 & 37181 & 37181 & 37181 & 37181 & 37181 & 40547 & 40415 & 36445 & 36492 & 36492 \\
& k & & 36 & 36 & 37 & 37 & 37 & 37 & 37 & 37 & 37 & 36 & 36 & 36 \\
& time & & 94.71 & 109.27 & 5.91 & 5.81 & 7.54 & 5.39 & 2.53 & 80.68 & 90.59 & 11.27 & 12.53 & 12.89 \\
X-n275-k28 & cost & 21245 & 22434 & 22528 & 23907 & 23907 & 24177 & 24177 & 24177 & 26052 & 25589 & 23865 & 23776 & 23776 \\
& k & & 28 & 28 & 31 & 31 & 31 & 31 & 31 & 31 & 31 & 31 & 31 & 31 \\
& time & & 135.55 & 1880.31 & 6.61 & 6.34 & 8.08 & 5.84 & 2.78 & 87.4 & 93.01 & 12.02 & 13.32 & 13.5 \\
X-n280-k17 & cost & 33503 & 41455 & 41442 & 35951 & 35951 & 35951 & 35951 & 35951 & 46179 & 45719 & 35531 & 35760 & 35760 \\
& k & & 17 & 17 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 17 & 18 & 18 \\
& time & & 26.14 & 23.69 & 9.63 & 8.83 & 11.71 & 8.15 & 4.73 & 84.99 & 94.03 & 16.18 & 17.94 & 17.62 \\
X-n284-k15 & cost & 20215 & 22010 & 22049 & 21924 & 21924 & 21924 & 21924 & 21924 & 25621 & 26436 & 21891 & 21456 & 21456 \\
& k & & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 16 & 15 & 15 & 15 \\
& time & & 92.56 & 531.23 & 9.77 & 9.02 & 12.05 & 8.63 & 5.18 & 85.8 & 96.51 & 16.52 & 17.86 & 17.19 \\
X-n289-k60 & cost & 95151 & 102712 & 102712 & 98648 & 98824 & 98743 & 98686 & 99808 & 114545 & 112830 & 97618 & 97654 & 97654 \\
& k & & 61 & 61 & 65 & 65 & 65 & 65 & 65 & 62 & 63 & 63 & 62 & 62 \\
& time & & 10.86 & 9.97 & 7.29 & 6.83 & 9.19 & 6.64 & 3.31 & 89.92 & 100.13 & 13.59 & 15.21 & 14.49 \\
X-n294-k50 & cost & 47161 & 50946 & 50466 & 49169 & 49307 & 49307 & 49307 & 49307 & 61648 & 59086 & 48629 & 48764 & 48764 \\
& k & & 51 & 51 & 53 & 53 & 53 & 53 & 53 & 55 & 55 & 51 & 52 & 52 \\
& time & & 90.68 & 925.2 & 7.69 & 6.91 & 9.02 & 6.29 & 3.3 & 88.93 & 102.38 & 13.4 & 14.7 & 14.82 \\
X-n298-k31 & cost & 34231 & 39855 & 39395 & 35996 & 35996 & 35996 & 35996 & 35996 & 48235 & 46590 & 35617 & 35767 & 35767 \\
& k & & 31 & 31 & 32 & 32 & 32 & 32 & 32 & 38 & 37 & 32 & 32 & 32 \\
& time & & 111.07 & 640.51 & 8.43 & 7.62 & 10.08 & 7.15 & 3.35 & 92.33 & 103.76 & 14.34 & 15.48 & 16.04 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (1000 iterations).}
\label{appx:tab:results_uchoa_41}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n101-k25 & cost & 27591 & 28725 & 28614 & 29181 & 29211 & 28845 & 29243 & 29674 & 29322 & 29315 & 28414 & 28261 & 28261 \\
& k & & 26 & 26 & 30 & 30 & 27 & 30 & 30 & 27 & 27 & 27 & 26 & 26 \\
& time & & 52.8 & 231.94 & 2.03 & 2 & 2.39 & 2.04 & 1.35 & 67.91 & 67.28 & 4.75 & 5.88 & 5.31 \\
X-n106-k14 & cost & 26362 & 27182 & 26789 & 27099 & 27099 & 27190 & 27099 & 27442 & 27885 & 28069 & 26656 & 26747 & 26747 \\
& k & & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 \\
& time & & 45.25 & 165.72 & 2.21 & 2.37 & 2.67 & 2.31 & 1.44 & 67.19 & 70.63 & 5.37 & 6.68 & 5.7 \\
X-n110-k13 & cost & 14971 & 15545 & 15408 & 15883 & 15883 & 15883 & 15883 & 15883 & 16144 & 15954 & 15612 & 15568 & 15568 \\
& k & & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 \\
& time & & 71.76 & 267.08 & 2.33 & 2.3 & 2.77 & 2.31 & 1.42 & 71.38 & 72.5 & 5.16 & 7.43 & 5.92 \\
X-n115-k10 & cost & 12747 & 13151 & 13138 & 13567 & 13585 & 13551 & 13585 & 13585 & 14863 & 13766 & 13361 & 13321 & 13321 \\
& k & & 10 & 10 & 11 & 11 & 10 & 11 & 11 & 11 & 10 & 11 & 11 & 11 \\
& time & & 84.82 & 1538.97 & 2.93 & 2.85 & 3.89 & 2.91 & 1.97 & 74.94 & 75.52 & 6.35 & 7.85 & 6.94 \\
X-n120-k6 & cost & 13332 & 13738 & 13730 & 14643 & 14655 & 14745 & 14655 & 14745 & 15656 & 16047 & 14120 & 14386 & 14386 \\
& k & & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\
& time & & 196.8 & 1143.29 & 3.4 & 3.27 & 4.53 & 3.37 & 2.93 & 75.3 & 82 & 6.49 & 8.27 & 7.55 \\
X-n125-k30 & cost & 55539 & 57766 & 57567 & 59239 & 59239 & 57494 & 59239 & 59837 & 60100 & 59407 & 57023 & 57240 & 57240 \\
& k & & 30 & 31 & 34 & 34 & 31 & 34 & 34 & 32 & 31 & 31 & 31 & 31 \\
& time & & 57.32 & 369.91 & 2.92 & 3.01 & 3.62 & 3.08 & 1.83 & 79.55 & 84.21 & 5.92 & 8.32 & 7.14 \\
X-n129-k18 & cost & 28940 & 31061 & 30406 & 30407 & 30407 & 30407 & 30407 & 30407 & 32387 & 32832 & 30287 & 30276 & 30276 \\
& k & & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 \\
& time & & 89.46 & 388.64 & 3.18 & 3.06 & 3.92 & 3.22 & 1.88 & 82.23 & 85.99 & 6.3 & 8.63 & 7.78 \\
X-n134-k13 & cost & 10916 & 11615 & 12095 & 11415 & 11415 & 11415 & 11415 & 11415 & 13128 & 12696 & 11374 & 11320 & 11320 \\
& k & & 13 & 13 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 13 & 13 & 13 \\
& time & & 77.84 & 364.38 & 3.61 & 3.65 & 4.49 & 3.66 & 2.04 & 86.3 & 88.15 & 7.26 & 9.6 & 8.23 \\
X-n139-k10 & cost & 13590 & 14261 & 14385 & 14726 & 14726 & 14411 & 14726 & 14726 & 14990 & 15261 & 14308 & 14368 & 14368 \\
& k & & 10 & 10 & 11 & 11 & 10 & 11 & 11 & 11 & 11 & 10 & 10 & 10 \\
& time & & 118.8 & 3041.7 & 3.93 & 3.72 & 4.81 & 3.79 & 2.22 & 88.16 & 92.42 & 7.31 & 10.15 & 8.78 \\
X-n143-k7 & cost & 15700 & 16871 & 16676 & 17598 & 17598 & 17126 & 17115 & 17664 & 19614 & 19499 & 16686 & 16605 & 16605 \\
& k & & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 \\
& time & & 121.62 & 1827.76 & 4.85 & 4.77 & 6.48 & 5 & 4.16 & 89.59 & 93.87 & 9.31 & 11.86 & 10.39 \\
X-n148-k46 & cost & 43448 & 45953 & 45293 & 47593 & 47494 & 47157 & 47606 & 48048 & 46003 & 46760 & 46678 & 46612 & 46612 \\
& k & & 46 & 46 & 53 & 53 & 50 & 53 & 53 & 48 & 48 & 49 & 49 & 49 \\
& time & & 142.41 & 559.28 & 3.72 & 3.71 & 4.9 & 3.77 & 2.38 & 93.46 & 98.81 & 7.25 & 10.07 & 8.84 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (1000 iterations).}
\label{appx:tab:results_uchoa_42}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n153-k22 & cost & 21220 & 22441 & 22876 & 22597 & 22597 & 22328 & 22597 & 22649 & 24792 & 24260 & 22486 & 22495 & 22495 \\
& k & & 25 & 24 & 26 & 26 & 25 & 26 & 26 & 24 & 24 & 25 & 25 & 25 \\
& time & & 95.8 & 851.36 & 4.7 & 4.55 & 6.15 & 4.51 & 2.44 & 99.17 & 100.44 & 9.18 & 12.26 & 10.27 \\
X-n157-k13 & cost & 16876 & 17209 & 17298 & 18963 & 18963 & 19050 & 19171 & 19171 & 18556 & 18352 & 18782 & 18696 & 18696 \\
& k & & 13 & 13 & 15 & 15 & 15 & 15 & 15 & 13 & 13 & 15 & 15 & 15 \\
& time & & 119.01 & 957.66 & 4.56 & 4.73 & 5.69 & 4.31 & 2.37 & 99.23 & 103.26 & 8.86 & 11.51 & 9.95 \\
X-n162-k11 & cost & 14138 & 14782 & 14487 & 14898 & 14898 & 15029 & 14898 & 15708 & 17063 & 16012 & 14775 & 14677 & 14677 \\
& k & & 11 & 11 & 11 & 11 & 11 & 11 & 11 & 12 & 12 & 11 & 11 & 11 \\
& time & & 179.75 & 8348.19 & 5.3 & 5.3 & 6.84 & 5.25 & 3.09 & 102.87 & 103.87 & 9.69 & 12.98 & 11.18 \\
X-n167-k10 & cost & 20557 & 22365 & 22176 & 21656 & 21656 & 22004 & 22004 & 22004 & 24511 & 25062 & 21695 & 21836 & 21836 \\
& k & & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 \\
& time & & 215.65 & 6219.87 & 5.93 & 5.86 & 7.94 & 5.73 & 3.76 & 106.48 & 108.99 & 11 & 14.01 & 12.35 \\
X-n172-k51 & cost & 45607 & 49782 & 47940 & 47924 & 47924 & 47790 & 47924 & 47921 & 50319 & 49216 & 47052 & 46762 & 46762 \\
& k & & 52 & 52 & 57 & 57 & 54 & 57 & 57 & 54 & 53 & 53 & 53 & 53 \\
& time & & 179.88 & 927.46 & 5.01 & 4.84 & 6.94 & 5.01 & 3.1 & 109.6 & 115.82 & 9.34 & 13.02 & 11.41 \\
X-n176-k26 & cost & 47812 & 51466 & 51233 & 51382 & 51382 & 50150 & 50454 & 50472 & 57753 & 57029 & 49403 & 49416 & 49416 \\
& k & & 26 & 26 & 29 & 29 & 27 & 29 & 29 & 28 & 27 & 27 & 27 & 27 \\
& time & & 60.6 & 151.31 & 5.89 & 5.74 & 7.51 & 5.98 & 3.1 & 112 & 116.58 & 12.01 & 14.46 & 12.52 \\
X-n181-k23 & cost & 25569 & 26346 & 26243 & 26355 & 26355 & 26561 & 26352 & 26561 & 27408 & 27568 & 26045 & 25999 & 25999 \\
& k & & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 \\
& time & & 197.02 & 2715.74 & 5.23 & 5.29 & 7.02 & 5.19 & 2.9 & 113.25 & 119.85 & 10.61 & 13.73 & 12.02 \\
X-n186-k15 & cost & 24145 & 25667 & 25910 & 25690 & 25690 & 25690 & 25690 & 25690 & 28368 & 28709 & 25150 & 25179 & 25179 \\
& k & & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 16 & 16 & 15 & 15 & 15 \\
& time & & 184.65 & 2340.14 & 6.56 & 6.32 & 8.42 & 6.06 & 3.33 & 120.08 & 124.02 & 11.85 & 15.5 & 13.79 \\
X-n190-k8 & cost & 16980 & 17831 & 17758 & 17742 & 17742 & 17816 & 17742 & 17816 & 19552 & 20064 & 17709 & 17689 & 17689 \\
& k & & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 \\
& time & & 191.15 & 6243.92 & 9.2 & 9.38 & 11.82 & 8.72 & 9.29 & 121.17 & 123.99 & 15.85 & 19.78 & 17.63 \\
X-n195-k51 & cost & 44225 & 48330 & 46999 & 45976 & 45976 & 45976 & 45976 & 45976 & 51250 & 50167 & 45441 & 45167 & 45167 \\
& k & & 52 & 52 & 55 & 55 & 55 & 55 & 55 & 54 & 54 & 53 & 53 & 53 \\
& time & & 180.69 & 697.97 & 6.52 & 5.91 & 7.95 & 6.06 & 3.41 & 123.37 & 128.23 & 11.44 & 15.66 & 13.95 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (1000 iterations).}
\label{appx:tab:results_uchoa_43}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n200-k36 & cost & 58578 & 60473 & 60810 & 60902 & 60913 & 61051 & 60920 & 61911 & 64112 & 64677 & 60290 & 60315 & 60315 \\
& k & & 37 & 37 & 38 & 38 & 38 & 38 & 38 & 37 & 37 & 37 & 37 & 37 \\
& time & & 156.86 & 1853.73 & 6.43 & 6.18 & 8.23 & 6.2 & 3.39 & 125.59 & 130.62 & 12.69 & 16.04 & 13.9 \\
X-n204-k19 & cost & 19565 & 20840 & 21045 & 20906 & 20970 & 21004 & 21004 & 21004 & 23895 & 22868 & 20481 & 20587 & 20587 \\
& k & & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 \\
& time & & 297.17 & 2850.43 & 7.32 & 7.21 & 9.61 & 7.17 & 3.69 & 129.61 & 135.19 & 13.58 & 18.25 & 15.7 \\
X-n209-k16 & cost & 30656 & 32268 & 32324 & 32793 & 32793 & 32793 & 32793 & 32793 & 35254 & 36933 & 32592 & 32602 & 32602 \\
& k & & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 16 \\
& time & & 249.75 & 2389.62 & 8.28 & 7.84 & 10.83 & 7.67 & 4.32 & 130.94 & 137.71 & 14.93 & 19.17 & 16.7 \\
X-n214-k11 & cost & 10856 & 12499 & 12281 & 11910 & 11949 & 11949 & 11949 & 11949 & 14626 & 13879 & 11711 & 11654 & 11654 \\
& k & & 11 & 11 & 12 & 12 & 12 & 12 & 12 & 11 & 13 & 12 & 12 & 12 \\
& time & & 169.87 & 3546.43 & 10.34 & 10.27 & 13.23 & 9.56 & 6.6 & 132.5 & 145.47 & 18.29 & 22.44 & 19.64 \\
X-n219-k73 & cost & 117595 & 118420 & 118029 & 172876 & 172876 & 172876 & 172876 & 172876 & 120127 & 120053 & 172876 & 172876 & 172876 \\
& k & & 73 & 73 & 109 & 109 & 109 & 109 & 109 & 73 & 73 & 109 & 109 & 109 \\
& time & & 384.66 & 6480.44 & 6.62 & 6.26 & 8.72 & 6.42 & 3.62 & 135.41 & 145.65 & 13.35 & 18.17 & 15.29 \\
X-n223-k34 & cost & 40437 & 42810 & 42810 & 42343 & 42343 & 42639 & 42299 & 42639 & 51322 & 47618 & 41724 & 41981 & 41981 \\
& k & & 34 & 34 & 36 & 36 & 36 & 36 & 36 & 36 & 35 & 34 & 35 & 35 \\
& time & & 166.89 & 1242.28 & 8.22 & 8.04 & 10.63 & 7.8 & 4.07 & 137.28 & 149.75 & 15.37 & 19.77 & 17.54 \\
X-n228-k23 & cost & 25742 & 27759 & 28008 & 27076 & 27076 & 27296 & 27105 & 27296 & 33420 & 32341 & 26714 & 26974 & 26974 \\
& k & & 23 & 23 & 25 & 25 & 25 & 25 & 25 & 25 & 25 & 23 & 24 & 24 \\
& time & & 123.99 & 1675.92 & 9.97 & 9.87 & 13.32 & 9.81 & 4.86 & 141.83 & 153.01 & 17.35 & 21.95 & 20.49 \\
X-n233-k16 & cost & 19230 & 20483 & 20215 & 20478 & 20585 & 20585 & 20585 & 20585 & 25884 & 25427 & 20066 & 20212 & 20212 \\
& k & & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 18 & 18 & 17 & 17 & 17 \\
& time & & 296.93 & 2849.93 & 10.98 & 10.7 & 15.49 & 10.97 & 5.79 & 134.9 & 151.9 & 20.5 & 26.24 & 23.36 \\
X-n237-k14 & cost & 27042 & 28748 & 28850 & 30341 & 30341 & 30571 & 30433 & 30571 & 32205 & 32791 & 30003 & 29828 & 29828 \\
& k & & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 & 14 \\
& time & & 370.75 & 3986.04 & 11.01 & 10.93 & 14.81 & 10.62 & 6.09 & 147.2 & 156.38 & 19.54 & 25.89 & 22.12 \\
X-n242-k48 & cost & 82751 & 85879 & 86408 & 88179 & 88179 & 88320 & 88179 & 88696 & 95260 & 93756 & 86584 & 86612 & 86612 \\
& k & & 48 & 48 & 52 & 52 & 52 & 52 & 52 & 48 & 48 & 50 & 50 & 50 \\
& time & & 213 & 1207.66 & 9.26 & 9.15 & 12.39 & 9.25 & 4.7 & 154.18 & 160.67 & 17.66 & 22.76 & 20 \\
X-n247-k50 & cost & 37274 & 39090 & 39136 & 40022 & 40022 & 38876 & 40123 & 40508 & 45682 & 45488 & 38869 & 38530 & 38530 \\
& k & & 51 & 51 & 59 & 59 & 53 & 59 & 59 & 55 & 54 & 54 & 53 & 53 \\
& time & & 163.3 & 2191 & 11.08 & 11.18 & 14.8 & 11.02 & 5.58 & 149.93 & 163.7 & 20.67 & 26.65 & 23.43 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\begin{landscape}
\begin{table*}[th]
\caption{Detailed Results on the Uchoa benchmark\cite{uchoa_bench} (1000 iterations).}
\label{appx:tab:results_uchoa_44}
\centering
\begin{tabular}{ll|c|rr|rrrrr|rr|rrr}
\textbf{Instance} & & \textbf{BKS} & \multicolumn{2}{l}{\textbf{ORTools}\cite{perron}} & \multicolumn{5}{l}{\textbf{Meta-VRPH}} & \multicolumn{2}{l}{\textbf{DACT}\cite{ma}} & \multicolumn{3}{l}{\textbf{NeuroLS}} \\
n100 & & & GLS & TS & SA & SA$_{\text{re}}$ & ILS & ILS+SA & VNS & aug=1 & aug=8 & NLS$_{\text{A}}$ & NLS$_{\text{AN}}$ & NLS$_{\text{ANP}}$ \\
\midrule
X-n251-k28 & cost & 38684 & 40320 & 40215 & 40279 & 40285 & 40285 & 40285 & 40285 & 42297 & 42540 & 40015 & 40062 & 40062 \\
& k & & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 & 28 \\
& time & & 269.72 & 2767.31 & 10.44 & 10.37 & 13.78 & 9.92 & 5.15 & 155.26 & 166.65 & 19.54 & 25.48 & 22.31 \\
X-n256-k16 & cost & 18839 & 19464 & 20274 & 20518 & 20518 & 20518 & 20518 & 20518 & 23292 & 22172 & 19845 & 20021 & 20021 \\
& k & & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 \\
& time & & 422.55 & 1168.37 & 12.65 & 12.26 & 17.36 & 12.32 & 6.45 & 154.69 & 172.04 & 22.68 & 29.79 & 26.32 \\
X-n261-k13 & cost & 26558 & 29094 & 29476 & 28638 & 28638 & 28638 & 28638 & 28638 & 35139 & 34448 & 28587 & 28638 & 28638 \\
& k & & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 13 & 14 & 13 & 13 & 13 \\
& time & & 212.79 & 2006.18 & 15.18 & 14.79 & 21.12 & 14.42 & 10.11 & 162.02 & 176.41 & 26.86 & 34.45 & 29.91 \\
X-n266-k58 & cost & 75478 & 78063 & 78291 & 80553 & 80553 & 80792 & 80657 & 80979 & 83349 & 83296 & 79497 & 79132 & 79132 \\
& k & & 59 & 59 & 64 & 64 & 64 & 64 & 64 & 61 & 61 & 61 & 61 & 61 \\
& time & & 234.45 & 1746.06 & 10.62 & 10.44 & 13.96 & 10.26 & 5.38 & 159.12 & 182.19 & 20.78 & 26.8 & 24.02 \\
X-n270-k35 & cost & 35291 & 37516 & 36999 & 37181 & 37181 & 37181 & 37181 & 37181 & 40547 & 40165 & 36265 & 36288 & 36288 \\
& k & & 36 & 36 & 37 & 37 & 37 & 37 & 37 & 37 & 38 & 36 & 36 & 36 \\
& time & & 283.61 & 2120.87 & 11.57 & 11.08 & 15.03 & 10.83 & 5.3 & 171.85 & 185 & 21.98 & 28.38 & 24.37 \\
X-n275-k28 & cost & 21245 & 22434 & 22377 & 23907 & 23907 & 24177 & 24177 & 24177 & 25964 & 25355 & 23865 & 23776 & 23776 \\
& k & & 28 & 28 & 31 & 31 & 31 & 31 & 31 & 30 & 31 & 31 & 31 & 31 \\
& time & & 374.72 & 6795.72 & 12.19 & 11.95 & 15.98 & 11.47 & 5.62 & 165.78 & 185.29 & 24.17 & 29.22 & 26.23 \\
X-n280-k17 & cost & 33503 & 37558 & 38819 & 35951 & 35951 & 35951 & 35951 & 35951 & 44726 & 45269 & 35531 & 35760 & 35760 \\
& k & & 17 & 17 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 17 & 18 & 18 \\
& time & & 205.65 & 1134.45 & 17.03 & 16.58 & 23.44 & 16.28 & 9.76 & 177.71 & 191.53 & 30.76 & 39.78 & 34.03 \\
X-n284-k15 & cost & 20215 & 21719 & 21984 & 21924 & 21924 & 21924 & 21924 & 21924 & 25621 & 26089 & 21688 & 21448 & 21448 \\
& k & & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 16 & 15 & 15 & 15 \\
& time & & 328 & 2745.15 & 17.66 & 17.19 & 24.49 & 16.9 & 10.51 & 180.56 & 195.5 & 31.57 & 40.34 & 34.41 \\
X-n289-k60 & cost & 95151 & 100823 & 101783 & 98648 & 98824 & 98743 & 98686 & 99808 & 113817 & 112830 & 97481 & 97182 & 97182 \\
& k & & 61 & 61 & 65 & 65 & 65 & 65 & 65 & 62 & 63 & 62 & 62 & 62 \\
& time & & 216.52 & 621.48 & 13.77 & 13.17 & 18.27 & 13.13 & 6.31 & 185.42 & 200.77 & 25.49 & 32.6 & 30.98 \\
X-n294-k50 & cost & 47161 & 50946 & 50429 & 49169 & 49307 & 49307 & 49307 & 49307 & 60510 & 58229 & 48494 & 48350 & 48350 \\
& k & & 51 & 51 & 53 & 53 & 53 & 53 & 53 & 55 & 54 & 51 & 51 & 51 \\
& time & & 341.68 & 1550.54 & 13.73 & 13.47 & 18.64 & 12.86 & 6.48 & 183.91 & 206.46 & 25.57 & 33.7 & 29.83 \\
X-n298-k31 & cost & 34231 & 39244 & 38074 & 35996 & 35996 & 35996 & 35996 & 35996 & 47477 & 46191 & 35617 & 35683 & 35683 \\
& k & & 31 & 31 & 32 & 32 & 32 & 32 & 32 & 36 & 34 & 32 & 32 & 32 \\
& time & & 304.55 & 1806.41 & 15.27 & 14.85 & 20.63 & 14.3 & 6.75 & 185.8 & 205.82 & 27.66 & 35.56 & 31.47 \\
\bottomrule
\end{tabular}
\end{table*}
\end{landscape}
\restoregeometry
\clearpage
\section{Preliminaries}
\subsection{Problem Formulation}
A Combinatorial Optimization Problem $\Omega$ is defined on its domain $D_{\Omega}$ which is the set of its instances $x \in D_{\Omega}$. A COP instance $x$ is usually given by a pair ($S_{\Omega}$, $f_{\Omega}$) of the solution space $S$ consisting of all feasible solutions to $\Omega$ and a corresponding cost function $f: S \to \IR$.
Combinatorial optimization problems normally are either to be minimized or maximized. In this paper we consider all COPs to be problems for which the cost of a corresponding objective function has to be minimized. The main concern is to find a solution $s^* \in S$ representing a \textit{global optimum}, i.e.\ $f(s^*) \leq f(s) \ \forall s \in S$.
\subsection{Local Search}
LS is a heuristic search method which is based on the concept of neighborhoods. A neighborhood $\mathcal{N}(s) \subseteq S$ of solution $s \in S$ represents a set of solutions which are somehow close to $s$. This ``closeness'' is defined by the neighborhood function $\mathcal{N}^{\phi}$ w.r.t.\ some problem specific operator $\phi \in \Phi_{\Omega}$ (e.g.\ an exchange of nodes in a routing problem).
Moreover, we always consider $s$ to be part of its own neighborhood, i.e.\ $s \in \mathcal{N}(s)$. Then the \textit{local optimum} $\hat{s}$ in the neighborhood $\mathcal{N}$ satisfies $f(\hat{s}) \leq f(s) \ \forall s \in \mathcal{N}(\hat{s})$. A general LS procedure (see algorithm \ref{alg:ls}) iterates through the neighborhood $\mathcal{N}(s)$ of the current solution $s$ until it finds the local optimum $\hat{s}$.
\SetKwComment{Comment}{// }{ }
\SetKwFunction{acpt}{accept}
\SetKwFunction{stp}{stop}
\SetKwInOut{Input}{input}
\DontPrintSemicolon
\begin{algorithm}[ht]
\caption{Local Search}\label{alg:ls}
\Input{
cost function $f$, solution $s$, neighborhood function $\mathcal{N}$,\\
acceptance rule $\acpt$, stopping rule $\stp$,
}
\While{not $\stp(s)$
}{
find $s' \in \mathcal{N}(s)$ for which $\acpt(s, s')$\\
$s \gets s'$
}
\Return{s}
\end{algorithm}
\subsection{Meta-Heuristics}
Meta-heuristics wrap an LS procedure to enable it to escape from local optima and to explore the solution space more efficiently. Some of the most common meta-heuristic strategies were described in section \ref{ss:rw_metaheuristics}. Algorithm \ref{alg:mh} describes a general formulation of a single solution meta-heuristic procedure.
Each particular strategy takes different decisions about restarting or perturbing the current solution, configuring the local search and accepting intermediate candidate solutions $s'$ (see algorithm \ref{alg:ls}). Some decisions can be fixed and are treated as hyper-parameters for some methods. For example, an SA procedure normally does not select a specific neighborhood but just decides about acceptance during the LS. Other approaches like VNS greedily accept all improving moves but apply a perturbation and select a new operator neighborhood every time a local optimum has been reached.
\SetKwComment{Comment}{// }{ }
\SetKwFunction{constr}{construct}
\SetKwFunction{ls}{LocalSearch}
\SetKwFunction{nbh}{GetNeighborhood}
\SetKwFunction{ops}{GetOperator}
\SetKwFunction{acpt}{accept}
\SetKwFunction{stp}{stop}
\SetKwFunction{rest}{restart}
\SetKwFunction{pert}{perturb}
\SetKwInOut{Input}{input}
\DontPrintSemicolon
\begin{algorithm}[ht]
\caption{Meta-Heuristic (Single Solution)}\label{alg:mh}
\Input{Solution space $S$, cost function $f$, stopping criterion}
$s \gets \constr(S)$ \Comment*[r]{Construct initial solution}
\While{not stopping criterion}{
$s \gets \pert(S, s)$ \Comment*[r]{Decide if to perturb/restart}
$\mathcal{N} \gets \nbh(S, s)$ \Comment*[r]{Define search neighborhood}
$s \gets \ls(f, s, \mathcal{N}, \acpt, \stp)$ \Comment*[r]{Execute local search}
}
\Return{s}
\end{algorithm}
\section{Related Work}
There is a plenitude of approaches and algorithms to tackle combinatorial optimization problems.
One common heuristic method is \textit{Local Search} (LS)\cite{aarts2003local}.
In the classical discrete optimization literature it is often embedded into a meta-heuristic procedure to escape local optima.
\subsection{Construction Heuristics}
Construction algorithms are concerned with finding a first feasible solution for a given COP. They usually start with an empty solution and consecutively assign values to the respective decision variables to construct a full solution. Well-known methods are e.g.\ the \textit{Savings} heuristic\cite{clarke1964scheduling} for vehicle routing problems or \textit{priority dispatching rules} (PDR)\cite{blackstone1982state} for scheduling.
Most improvement and meta-heuristic methods require a feasible initial solution from which they can start to improve.
\subsection{Meta-Heuristics}\label{ss:rw_metaheuristics}
Meta-heuristics are the go-to method for complex discrete optimization problems.
They effectively balance the exploration of the solution space (escaping local optima) and the exploitation of promising solutions.
There are two major types of methods:\\
\noindent
\textbf{Single Solution Methods} Single solution methods include many well-known approaches
which are used in combination with LS.
During the search they only maintain a single solution at a time which is iteratively changed and adapted.
\textit{Simulated Annealing} (SA) is a probabilistic acceptance strategy based on the notion of controlled cooling of materials first proposed in \cite{kirkpatrick1983optimization}. The idea is to also accept solutions which are worse than the best solution found so far to enable exploration of the search space but with a decreasing probability to increasingly focus on exploitation the further the search advances.
\textit{Iterated Local Search} (ILS)\cite{lourencco2019iterated} alternates between a diversification and an intensification phase. In the diversification step the current solution is perturbed
while the alternating intensification step executes a greedy local search with a particular neighborhood.
\textit{Variable Neighborhood Search} (VNS)\cite{mladenovic1997variable} employs a similar diversification step but focuses on a systematic control of the LS neighborhood in the intensification phase, by changing the type of the applied LS move frequently after each perturbation in a predefined order.
\textit{Tabu Search} (TS) was proposed by Glover\cite{glover1986future} and is based on the possible acceptance of non-improving moves and a so called tabu list, a kind of filter preventing moves which would keep the search stuck in local optima. This list acts as a memory which in the simplest case stores the solutions of the last $k$ iterations and prevents changes which would move the current solution back to solutions encountered in recent steps.
Instead of using a tabu list, \textit{Guided Local Search} (GLS)\cite{voudouris2010guided} relies on penalties for different moves to guide the search. These penalties are often based on problem specific features in the solution, e.g.\ the edges between nodes in a routing problem, and are added to the original objective function of the problem when the LS gets stuck.
Methods include different evolutionary algorithms, particle swarm optimization and other bio-inspired algorithms such as ant colony optimization \cite{gendreau2010handbook}.
Their main idea is based on different adaption, selection and recombination schemes to refine the solution pool during search in order to find better solutions.
While a learned population based meta-heuristic is interesting and potentially promising, in this paper we focus on the impact and effectiveness of a learned single solution approach.
More information on advanced meta-heuristics and their extensions can be found in \cite{gendreau2010handbook}.
\subsection{Machine Learning based Methods}\label{ss:rw_ml}
In recent years also many approaches employing ML methods have been proposed. While some work \cite{vinyals2015pointer,joshi2019efficient,thyssens2022supervised} relies on supervised learning, most current state-of-the-art methods use reinforcement learning (RL).
A large fraction of the work focuses on auto-regressive models which learn to sequentially construct feasible solutions from scratch.
Such methods have been proposed for many common COPs including the TSP\cite{bello2016neural}, CVRP\cite{nazari2018reinforcement,kool2018attention,kwon2020pomo}, CVRP-TW\cite{falkner2020learning} and JSSP\cite{zhang2020learning,park2021learning,hottung2021efficient,park2021schedulenet}.
The second type of methods is concerned with improvement approaches.
Chen et al.\cite{chen2019learning} design a model to rewrite sub-sequences of the problem solution for the CVRP and JSSP based on a component which selects a specific element of the solution and a second component parameterizing a heuristic move to change that part of the solution. However, their model is limited to specific problem settings with a fixed number of jobs and machines or customers while our approach works seamlessly for different problem sizes, as we show in the experiments in section \ref{ss:results}.
In contrast, the authors in \cite{lu2019learning} learn a policy that selects a specific LS move at each iteration.
However, their method incurs prohibitively large computation times and for this reason is not competitive with any of the recent related work \cite{kwon2020pomo,kool2021deep,ma2021learning}.
The authors in \cite{hottung2019neural} learn a repair operator to re-construct heuristically destroyed solutions in a Large Neighborhood Search.
Finally, da Costa et al.\cite{d2020learning} learn a model to select node pairs for 2-opt moves in the TSP while
Wu et al.\cite{wu2021learning} learn a similar pair-wise selection scheme for 2-opt, node swap and relocation moves in TSP and CVRP.
The authors in \cite{ma2021learning} further improve on the method in \cite{wu2021learning} by introducing the Dual-aspect collaborative Transformer (DACT) model. Although there exist advanced inference approaches\cite{hottung2021efficient} to further improve the performance of auto-regressive ML methods on COPs, here we focus on the vanilla inference via greedy decoding or sampling.
While these methods share some similarities with our approach, they ignore the importance of being able to reject unpromising moves to escape local optima, whereas our approach specifically focuses on this important decision point to effectively control the search.
Moreover, our approach is also able to learn when to apply a particular perturbation if the rejection of a sequence of moves is not sufficient for exploration.
A detailed overview of the current machine learning approaches to COPs is given in \cite{bengio2021machine,mazyavkina2021reinforcement}.
\section{Proposed Method}
\subsection{Intervention Points of Meta-Heuristics for Local Search}
The application of meta-heuristic strategies to an underlying LS involves several points of intervention, at which decisions can be done to help the search escape local optima in order to guide it towards good solutions. In the following, we define three such intervention points that have a significant impact on the search:
\begin{enumerate}
\item \textit{Acceptance}:
The first intervention point is the acceptance of candidate solutions $s'$ in a local search step (algorithm \ref{alg:ls}, line 2). A simple hill climbing heuristic is completely greedy and only accepts improving moves, which often leaves the search stuck in local optima very quickly. In contrast, other approaches like SA will also accept non-improving moves with some probability.
\item \textit{Neighborhood}:
The second possible decision a meta-heuristic can make is the selection of a particular operator $\phi$ that defines the neighborhood function $\mathcal{N}^{\phi}$ for the LS (algorithm \ref{alg:mh}, line 4). Possible operators for scheduling or routing problems could for example be a node exchange. While many standard approaches like SA and ILS only use one particular neighborhood which they treat as a hyper-parameter, VNS is an example for a method that selects a different operator at each step.
\item \textit{Perturbation}:
Many meta-heuristics like ILS and VNS employ perturbations $\psi \in \Psi_{\Omega}$ to the current solution to move to different regions of the search space and escape particularly persistent local optima. Such a perturbation can simply be a restart from a new stochastically constructed initial solution, a random permutation of (a part of) the current solution or a random sequence of node exchanges.
The decision \textit{when} to employ such a perturbation (algorithm \ref{alg:mh}, line 5) is usually
done w.r.t.\ a specific pre-defined number of steps without improvement.
\end{enumerate}
\subsection{Meta-Heuristics as Markov Decision Process}\label{ss_mh_mdp}
In this section we formulate meta-heuristics in terms of an MDP\cite{sutton2018reinforcement} to enable the use of RL approaches to learn a parameterized policy to replace them. In general an MDP is given by a tuple
$(\mathcal{S}, \mathcal{A}, \mathcal{P}(s_t, a_t), \mathcal{R}(s_t, a_t))$ representing the set of states $\mathcal{S}$, the set of actions $\mathcal{A}$, a transition probability function $\mathcal{P}(s_t, a_t)$ and a reward function $\mathcal{R}(s_t, a_t)$. For our method we define these entities in terms of meta-heuristic decisions as follows:
\ \\
\noindent
\textbf{States} \
We define the state $s_t$ of the problem at time step $t$ with slight abuse of notation as the solution $s$ at time step $t$, combined with 1) its cost $f(s)$, 2) the cost $f(\hat{s}_t)$ of the best solution found so far, 3) the last acceptance decision, 4) the last operator used, 5) current time step $t$, 6) number of LS steps without improvement and 7) the number of perturbations or restarts.
\ \\
\noindent
\textbf{Actions} \
Depending on the policy we want to train, we define the action set as the combinatorial space of
\begin{enumerate}
\item \textit{Acceptance} decisions: a boolean decision variable of either accepting or rejecting the last LS step
\begin{equation}
\mathcal{A}_{\text{A}} \coloneqq \{0, 1\},
\end{equation}
\item \textit{Acceptance}-\textit{Neighborhood} decisions: the joint space of the acceptance of the last move and the set of possible operators $\phi \in \Phi$ to apply in the next step
\begin{equation}
\mathcal{A}_{\text{AN}} \coloneqq \{0, 1\} \times \Phi,
\end{equation}
\item \textit{Acceptance}-\textit{Neighborhood}-\textit{Perturbation} decisions: the joint space of acceptance and the combined sets of operators $\phi \in \Phi$ and perturbations $\psi \in \Psi$
\begin{equation}
\mathcal{A}_{\text{ANP}} \coloneqq \{0, 1\} \times \{\Phi \cup \Psi\}.
\end{equation}
\end{enumerate}
\ \\
\noindent
\textbf{Transitions} \
The transition probability function $\mathcal{P}(s_t, a_t)$ models the state transition from state $s_t$ to the next state $s_{t+1}$ depending on action $a_t$ representing the acceptance decision, the next operator $\phi$
and a possible perturbation or restart (part of the problem state or action in case of $\mathcal{A}_{\text{ANP}}$).
\ \\
\noindent
\textbf{Rewards} \
The reward function $\mathcal{R}(s_t, a_t)$ defines the reward for a transition from state $s_t$ to the next state $s_{t+1}$.
Here we define the reward $r_t$ of the transition as the relative improvement of the last LS step (defined by action $a_t$) w.r.t.\ the cost of the best solution found until $t$ and clamped at 0 to avoid negative rewards:
\begin{equation}
r_t \coloneqq \max(f(\hat{s}_t) - f(s_{t+1}), 0)
\end{equation}
\ \\
\noindent
\textbf{Policy} \
We employ Deep Q-Learning\cite{van2016deep} and parameterize the learned policy $\pi_{\theta}$ via a softmax over the corresponding Q-function $Q_{\theta}(s_t, a_t)$ which is represented in turn by our GNN-based encoder-decoder model with trainable parameters $\theta$:
\begin{equation}
\pi_{\theta}(a_t \mid s_t) = \frac{\exp(Q_{\theta}(s_t, a_t))}{\sum_{\mathcal{A}} \exp(Q_{\theta}(s_t, a_t))}.
\end{equation}
\subsection{Model Architecture}
In this section we describe our encoder and decoder models to parameterize $Q_{\theta}(s_t, a_t)$.
Many COPs like routing and scheduling problems have an underlying graph structure which can be used when encoding these problems and that provides an effective inductive bias for the corresponding learning methods.
In general we assume a graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ with the set of nodes $\mathcal{V}$, $N = |\mathcal{V}|$ and the set of directed edges $\mathcal{E} \subseteq \{(i, j) \subseteq \mathcal{V} \}$.
Moreover, we assume an original node feature matrix $X \in \IR^{N \times d_{\text{in}}}$ and edge features $e_{i, j} \in \IR$ for each edge $(i,j)$.
To leverage this structural information many authors have used Recurrent Neural Networks (RNN)\cite{vinyals2015pointer,bello2016neural,chen2019learning}, Transformers\cite{vaswani2017attention,kool2018attention,kwon2020pomo,ma2021learning} or Graph Neural Networks (GNN)\cite{joshi2019efficient,d2020learning,zhang2020learning}.
\ \\
\noindent
\textbf{Encoder} \\
For our model we employ the 1-GNN operator of \cite{morris2019weisfeiler} which is able to work with edge weights if they are present. One such layer
is defined as
\begin{align}
h^{(l)}_i &= \GNN^{(l)}(h^{(l-1)}_i) \nonumber \\
&= \sigma\big( \MLP^{(l)}_1(h^{(l-1)}_i) + \MLP^{(l)}_2(\sum_{j\in \mathcal{H}(i)} e_{j, i} \cdot h^{(l-1)}_j) \big),
\end{align}
where $h^{(l-1)}_i \in \IR^{1 \times d_{\text{emb}}}$ is the latent feature embedding of node $i$ at the previous layer $l-1$, $\mathcal{H}(i)$ is the 1-hop graph neighborhood of node $i$, $\MLP^{(l)}_1$ and $\MLP^{(l)}_2$ are Multi-Layer Perceptrons $\MLP: \IR^{d_{\text{emb}}} \to \IR^{d_{\text{emb}}}$ and $\sigma()$ is a suitable element-wise non-linearity, for which we use GeLU\cite{hendrycks2016gaussian}.
Furthermore, we add residual connections and layer normalization\cite{ba2016layer} to each layer.
In the first layer the latent feature vector $h^{(0)}_i$ is created by feeding the original node features $x_i$ into an $\MLP: \IR^{d_{\text{in}}} \to \IR^{d_{\text{emb}}}$:
\begin{equation}
h^{(0)}_i = \MLP^{(0)}(x_i).
\end{equation}
and another $\MLP^{(L)}: \IR^{d_{\text{emb}}} \to \IR^{d_{\text{emb}}}$ is placed at the end of the GNN stack.
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{fig/architecture.pdf}
\caption{
Visualization of the NeuroLS model architecture.
}
\label{fig:arch}
\end{figure}
In order to further leverage structural information, we introduce 3 stages to compute the latent embeddings. The first stage uses the edge set $\mathcal{E}^{\text{stat}}$ of the \textit{static} node neighborhood graph, e.g.\ the $K$ nearest neighbors graph for the CVRP or the Directed Acyclic Graph (DAG) representing the order of operations for each job in scheduling problems. This edge set is fixed and does not change throughout the search.
In contrast, the second stage utilizes the edge set $\mathcal{E}^{\text{dyna}}$ representing e.g.\ the tours in routing or the machine graph in scheduling, which is \textit{dynamic} and usually changes in every LS step.
Our proposed network architecture consists of $L^{\text{stat}}$ GNN layers for the static graph, followed by $L^{\text{dyna}}$ layers which propagate over the dynamic graph. Finally, we add another layer, again using the static edges, to consolidate the dynamic information over the static graph, leading to a total of $L = L^{\text{stat}} + L^{\text{dyna}} +1$ GNN layers.
The final stage serves to refine the embedding via aggregation based on the dynamic information of group membership which is present in the solution. Each node normally belongs to one of $K$ (not necessarily disjoint) groups $\mathcal{M}_k$ of the solution, e.g.\ to a particular tour or machine. Following this idea we pool the final embeddings $\omega^{\text{node}}_i$ from the GNN stack w.r.t.\ their membership and feed them through another MLP:
\begin{equation}
\omega^{\text{grp}}_k = \MLP^{\text{grp}}\big([\MAX(\omega^{\text{node}}_i \mid i \in \mathcal{M}_k); \MEAN(\omega^{\text{node}}_i \mid i \in \mathcal{M}_k)]\big),
\end{equation}
with max and mean pooling $\IR^{N \times d_{\text{emb}}} \to \IR^{K \times d_{\text{emb}}}, K << N$ over the node dimension $N$, $\mathcal{M}_k$ as membership to the $k$-th group ($k$-th tour, $k$-th machine, etc.) and $[\ ;]$ representing concatenation in the embedding dimension $d_{\text{emb}}$.
Finally, the additional features of the state representation (current cost, best cost, last acceptance, etc.) described in the last section, are concatenated and projected by a simple linear layer to create an additional latent feature vector $\omega^{\text{feat}} \in \IR^{d_{\text{emb}}}$.
\ \\
\noindent
\textbf{Decoder} \\
Our decoder takes the different embeddings created in the encoder, aggregates the node embeddings $\omega^{\text{node}} \in \IR^{N \times d_{\text{emb}}}$ and group embeddings $\omega^{\text{grp}} \in \IR^{K \times d_{\text{emb}}}$ via a simple mean over the node and group dimension and concatenates them with the feature embedding $\omega^{\text{feat}} \in \IR^{d_{\text{emb}}}$.
This representation is the input to a final 2-layer MLP regression head $\IR^{3*d_{\text{emb}}} \to \IR^{|\mathcal{A}|}$ which outputs the value predictions of the Q-function. The full architecture is shown in figure \ref{fig:arch}.
\subsection{Reinforcement Learning Algorithm}
To train our policy model we employ Double Deep Q-Learning\cite{van2016deep} with $n$-step returns\cite{sutton2018reinforcement}
and Implicit Quantile Networks (IQN)\cite{dabney2018implicit}. IQNs enable a distributional formulation of Q-Learning where the Deep Q-Network is trained w.r.t.\ an underlying value distribution represented by a learned quantile function instead of single point estimates. In order to represent this learned quantile function IQNs are introduced as a small additional neural network which is jointly trained to transform samples from a base distribution (e.g.\ uniform) to the respective quantile values of the target distribution, i.e.\ the distribution over the returns. Further details can be found in appendix \ref{appx:rl}.
\section{Experiments}
\subsection{Applications}
\noindent
\textbf{Job Shop Scheduling Problem (JSSP)}
The JSSP is concerned with scheduling a number of jobs $J$ on a set of $K$ machines denoted by $M$. Each job consists of a sequence of operations $O_{ij}$ with fixed processing times $p_{ij}$ which need to be processed in a predefined order. In the simplest problem variant every job has exactly one operation on each machine. A solution to the problem consists of the exact order of the operations on all machines.
In this paper we choose to minimize the \textit{makespan}, which is the longest time span from start of the first operation until the end of the last one to finish, corresponding to the longest path in the respective DAG representation of the problem. In the following we denote the size of a JSSP instances as $|J|\times |M|$. We follow \cite{zhang2020learning} in creating instances for training and validation by sampling processing times from a uniform distribution.
\ \\
\noindent
\textbf{Capacitated Vehicle Routing Problem (CVRP)}
The CVRP consists of a set of $N$ customer nodes and a depot node. It is concerned with serving the demands $q_i$ of the customer nodes in tours starting and ending at the depot node by employing $K$ homogeneous vehicles with a fixed capacity $Q > 0$. The tour of vehicle $k \in K$ is a sequence of indices w.r.t.\ a subset of all customers nodes representing the order in which vehicle $k$ visits the respective nodes. A set of feasible tours serving all customer nodes represents a solution to the problem, whereas the objective is to minimize the total length of all tours. We follow \cite{kool2018attention} in creating the training and validation sets by generating instances with coordinates uniformly sampled in the unit square.
\subsection{Setup}
For the JSSP we implement a custom LS solver in python. It implements four different node moves and a perturbation operator based on a sequence of such moves. As construction heuristic and for restarts we implement several stochastic variants of common PDRs (see appendix \ref{appx:jssp_solver} for more details).
The LS for the CVRP uses the C++ based open source solver VRPH\cite{groer2010library} for which we implement a custom wrapper and interface to expose and support the necessary intervention points.
VRPH includes several different LS moves and perturbation operators including 2-opt, 3-opt and different exchange and point moves.
We provide our model code and that of our JSSP solver in our github repository\footnote{\url{https://github.com/jokofa/NeuroLS}}.
Preliminary experiments showed that the CET and 2-opt moves performed best for the JSSP and CVRP respectively, when used for meta-heuristics which do not select the operator. Thus, we employ these operators in our main experiments.
We train all our models for 80 epochs with 19200 transitions each and pick the model checkpoint with the best validation performance.
Hyperparameters for NeuroLS and all meta-heuristics are tuned on a validation set consisting of 512 generated random instances. This is in contrast to most classical approaches which fine tune their hyper-parameters directly on the benchmark dataset. We argue that our approach facilitates a better and more objective comparison between these methods. Furthermore, we fix the random seed for all experiments.
We train 3 different types of policies for NeuroLS, one for each of the action spaces described in section \ref{ss_mh_mdp}. In the experiments we denote these policies as NLS$_{\text{A}}$, NLS$_{\text{AN}}$ and NLS$_{\text{ANP}}$.
As analyzed in \cite{wu2021learning}, random solutions do not provide a good starting point for improvement approaches. For that reason we initialize the solutions of NeuroLS and the meta-heuristics with the FDD/MWKR PDR\cite{sels2012comparison} for scheduling and the well known savings construction heuristic\cite{clarke1964scheduling} for the CVRP.
Further details on training and hyper-parameters can be found in Appendix \ref{appx:training}.
\subsection{Results}\label{ss:results}
\begin{table*}[th]
\caption{
Results of state-of-the-art machine learning based construction methods and local search approaches (100 iterations) on the Taillard benchmark\cite{taillard1993benchmarks}. For instances of size 50x15, 50x20 and 100x20 we use the NeuroLS model trained on instances of size 30x15 and 30x20 respectively. Percentages are the average gap to the best known upper bound. Best gap is marked in \textbf{bold}.
}
\label{tab:results_jssp}
\centering
\begin{tabular}{l|rrrrr|rrr|r}
\toprule
\textbf{Model} & \multicolumn{8}{c|}{\textbf{Instance size}} & \textbf{Avg} \\
\hline
& 15x15 & 20x15 & 20x20 & 30x15 & 30x20 & 50x15 & 50x20 & 100x20 & \\
\midrule
\multicolumn{10}{c}{\textbf{ML-based}} \\
L2D\cite{zhang2020learning} & 25.92\% & 30.03\% & 31.58\% & 32.88\% & 33.64\% & 22.35\% & 26.37\% & 13.64\% & 27.05\% \\
L2S\cite{park2021learning} & 20.12\% & 24.83\% & 29.25\% & 24.59\% & 31.91\% & 15.89\% & 21.39\% & 9.26\% & 22.16\% \\
SN\cite{park2021schedulenet} & 15.32\% & 19.43\% & 17.23\% & 18.95\% & 23.75\% & 13.83\% & 13.56\% & 6.67\% & 16.09\% \\
\midrule
\multicolumn{10}{c}{\textbf{Meta-heuristic + Local Search}} \\
SA & 13.92\% & 17.01\% & 17.16\% & 17.53\% & 21.59\% & 12.50\% & 13.11\% & 6.61\% & 14.93\% \\
SA$_{\text{restart}}$ & 13.77\% & 17.01\% & 17.57\% & 17.62\% & 21.78\% & 12.54\% & 13.22\% & 6.75\% & 15.03\% \\
ILS & 11.57\% & 13.57\% & 13.85\% & 16.07\% & 18.72\% & 12.65\% & 12.15\% & 6.72\% & 13.16\% \\
ILS+SA & 13.32\% & 16.05\% & 15.38\% & 16.93\% & 19.74\% & 13.07\% & 13.43\% & 7.08\% & 14.37\% \\
VNS & 9.96\% & 13.71\% & 14.51\% & 15.77\% & 18.69\% & 11.64\% & 11.92\% & 6.26\% & 12.81\% \\
\midrule
\multicolumn{10}{c}{\textbf{NeuroLS}} \\
NLS$_{\text{A}}$ & \textbf{9.76}\% & 13.33\% & 13.02\% & 15.29\% & 17.94\% & 11.81\% & 11.96\% & 6.33\% & 12.43\% \\
NLS$_{\text{AN}}$ & 10.32\% & \textbf{13.18}\% & \textbf{12.95}\% & \textbf{14.91}\% & 17.78\% & 11.87\% & 12.02\% & 6.22\% & \textbf{12.41}\% \\
NLS$_{\text{ANP}}$ & 10.49\% & 16.32\% & 15.24\% & 15.35\% & \textbf{17.64}\% & \textbf{11.62}\% & \textbf{11.76}\% & \textbf{6.09}\% & 13.06\% \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[th]
\caption{
Results of state-of-the-art machine learning based methods and local search approaches (200 iterations) on the Uchoa benchmark\cite{uchoa2017new}. For all instances sizes we use the NeuroLS model trained on instances of size 100. Percentages are the average gap to the best known solution. Best gap is marked in \textbf{bold}.
}
\label{tab:results_cvrp}
\centering
\begin{tabular}{l|rrrrrrrr|r}
\toprule
\textbf{Model} & \multicolumn{8}{c|}{\textbf{Instance Group}} & \textbf{Avg} \\
\hline
& \multicolumn{2}{c}{\textit{n100}} & \multicolumn{2}{c}{\textit{n150}} & \multicolumn{2}{c}{\textit{n200}} & \multicolumn{2}{c|}{\textit{n250}} & \\
& cost & time & cost & time & cost & time & cost & time & \\
\midrule
\multicolumn{10}{c}{\textbf{ML-based}} \\
POMO\cite{kwon2020pomo} & 17.24\% & 0.3 & 12.81\% & 0.4 & 20.52\% & 0.4 & 15.14\% & 0.4 & 16.43\% \\
POMO\cite{kwon2020pomo} (aug) & 6.32\% & \textbf{0.1} & 9.41\% & \textbf{0.1} & 13.47\% & \textbf{0.1} & 9.21\% & \textbf{0.1} & 9.60\% \\
DACT\cite{ma2021learning} & 13.19\% & 15.1 & 20.26\% & 21.2 & 16.91\% & 27.2 & 24.93\% & 33 & 18.82\% \\
DACT\cite{ma2021learning} (aug) & 11.52\% & 16.6 & 17.75\% & 23.1 & 15.34\% & 29.8 & 21.86\% & 37.8 & 16.62\% \\
\midrule
\multicolumn{10}{c}{\textbf{Meta-heuristic + Local Search}} \\
ORT\cite{ortools} GLS & 6.91\% & 9.4 & 10.46\% & 10.9 & \textbf{7.80}\% & 13.4 & 12.12\% & 5.0 & 9.32\% \\
ORT\cite{ortools} TS & 6.78\% & 39.0 & 10.55\% & 109.5 & 7.85\% & 126.6 & 12.13\% & 7.0 & 9.33\% \\
SA & 6.92\% & 0.7 & 5.79\% & 1.3 & 16.63\% & 2.1 & 5.92\% & 3.3 & 8.81\% \\
SA$_{\text{restart}}$ & 6.96\% & 0.7 & 5.79\% & 1.3 & 16.67\% & 2.0 & 5.99\% & 3.0 & 8.85\% \\
ILS & 6.56\% & 0.8 & 5.96\% & 1.5 & 16.73\% & 2.4 & 6.08\% & 3.7 & 8.83\% \\
ILS+SA & 6.94\% & 0.6 & 6.01\% & 1.2 & 16.72\% & 1.9 & 6.04\% & 2.8 & 8.93\% \\
VNS & 7.99\% & 0.4 & 6.55\% & 0.7 & 17.27\% & 0.9 & 6.36\% & 1.4 & 9.54\% \\
\midrule
\multicolumn{10}{c}{\textbf{NeuroLS}} \\
NLS$_{\text{A}}$ & 5.43\% & 1.5 & 5.23\% & 2.4 & 15.97\% & 3.6 & 5.22\% & 5.3 & 7.96\% \\
NLS$_{\text{AN}}$ & \textbf{5.42}\% & 1.7 & \textbf{4.90}\% & 2.7 & 15.85\% & 4.0 & \textbf{5.08}\% & 5.9 & \textbf{7.81}\% \\
NLS$_{\text{ANP}}$ & \textbf{5.42}\% & 1.7 & \textbf{4.90}\% & 2.7 & 15.85\% & 4.0 & \textbf{5.08}\% & 6.1 & \textbf{7.81}\% \\
\bottomrule
\end{tabular}
\end{table*}
\ \\
\noindent
\textbf{JSSP} We evaluate all methods on the well-known benchmark dataset of Taillard\cite{taillard1993benchmarks}. It consists of 80 instances of size 15x15 up to 100x20. We compare our model against common meta-heuristic baselines including SA, SA with restarts, ILS, ILS with SA acceptance and VNS. Moreover, we report the results of three recent state-of-the-art ML-based approaches: \textit{Learning to dispatch} (L2D)\cite{zhang2020learning}, \textit{Learning to schedule} (L2S)\cite{park2021learning} and \textit{ScheduleNet} (SN)\cite{park2021schedulenet}.
We follow \cite{zhang2020learning} in training different models for problem sizes 15x15 up to 30x20 and apply the 30x15 and 30x20 model to larger instances of size 50x15, 50x20 and 100x20 to evaluate its generalization capacity. All models are trained for 100 LS iterations but evaluated for 50-200. The aggregated results per group of same size are shown in table \ref{tab:results_jssp} (results per instance can be found in appendix \ref{appx:benchmark}). Results are reported as percentage gaps to the best known solution.
First of all, the results show that NeuroLS is able to outperform all other meta-heuristics on all sizes of instances. The different policies differ in how well they work for different problem sizes. While NLS$_{\text{A}}$ works best for the smallest 15x15 instances, it is outperformed by NLS$_{\text{AN}}$ on medium sized instances and NLS$_{\text{ANP}}$ achieves the best results for large instances. This is to some extent expected, since the effect of specific LS operators and more precise perturbations is greater for larger instances while this does not seem to be necessary for rather small instances.
VNS is the best of the meta-heuristic approaches which is able to beat NLS$_{\text{ANP}}$ on the smaller instances and NLS$_{\text{A}}$ and NLS$_{\text{AN}}$ at least on some of the larger ones.
In general, the iterative methods based on LS achieve much better results than the ML-based auto-regressive methods, which can be seen by the large improvements that can be achieved in just 100 iterations, reducing the gap by an average of 3.7\% compared to the best ML-method SN\cite{park2021schedulenet}.
\ \\
\noindent
\textbf{CVRP} For the CVRP we use the recent benchmark dataset of Uchoa et al.\cite{uchoa2017new} and select all instance up to 300 customer nodes. We define four groups of instances with 100-149 nodes as \textit{n100}, 150-199 nodes as \textit{n150}, 200-249 as \textit{n200} and 250-299 as \textit{n250}. We compare against the same meta-heuristics mentioned above and additionally to GLS and TS provided by the OR-Tools (ORT) library\cite{ortools}. Furthermore, we compare to the recent state-of-the-art ML approaches POMO\cite{kwon2020pomo} and DACT\cite{ma2021learning} which outperformed all other ML methods mentioned in section \ref{ss:rw_ml} in their experiments and provide open source code, which is why we consider them to be sufficient for a suitable comparison.
Since most ML-based methods (POMO, DACT, etc.) do not respect the maximum vehicle constraint for all instances of the Uchoa benchmark, we follow\cite{ma2021learning} in removing this constraint and treat the benchmark dataset as a highly diverse test set w.r.t.\ the distributions of customers and number of required vehicles.
This is also consistent with the general goal of the ML-based methods, which is not to achieve the best known results but to find sufficiently good results in reasonable time. In this case we control the computational resources spent on the search by specifying a particular number of iterations for the search. Furthermore, we evaluate all models with a batch size of 1.
The results presented in table \ref{tab:results_cvrp} show that NeuroLS is also able to outperform the meta-heuristic approaches mentioned above on all instance groups. Moreover, our approach also outperforms the best state-of-the-art ML-based methods on all groups but \textit{n200}, where \cite{kwon2020pomo} and \cite{ma2021learning} with additional instance augmentations (aug) outperform NeuroLS by a small margin. The OR-Tools implementation of GLS and TS outperforms our method only on the \textit{n200} instances, although they require prohibitively large runtimes (wall-clock time) several magnitudes larger than our approach. In terms of runtimes we also outperform DACT\cite{ma2021learning} by a magnitude, while the learned auto-regressive construction method POMO\cite{kwon2020pomo} is the fastest over all.
Finally, the experiment results also show that our method is well able to generalize to problem sizes as well as numbers of iterations unseen in training. We show this on the JSSP instances of size 50x15, 50x20 and 100x20 and basically on all Uchoa instances for the CVRP, which are larger than the 100 node instances used during training.
\section{Conclusion}
In this paper we identify three important intervention points in meta-heuristics for LS on COPs and incorporate them in a Markov Decision Process. We then design a GNN-based controller model which is trained with RL to parameterize three different types of learned meta-heuristics. The resulting methods learn to control the local search by deciding about acceptance, neighborhood selection and perturbations. In comprehensive experiments on two common COPs in scheduling and vehicle routing, NeuroLS outperforms several well-known meta-heuristics as well as state-of-the-art ML-based approaches, confirming the efficacy of our method.
For future work we consider to replace the problem graph representation with a graph representation of the corresponding local search graph, in which every node represents a feasible solution together with its respective cost. A Curriculum learning strategy (cp. \cite{ma2021learning}) could also yield further improvements regarding performance and generalization.
\subsubsection{Acknowledgements}
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 789 |
{"url":"https:\/\/www.experts-exchange.com\/questions\/28579549\/Importing-invoice-lines-in-an-array-in-powershell.html","text":"Go Premium for a chance to win a PS4. Enter to Win\n\nx\n\u2022 Status: Solved\n\u2022 Priority: Medium\n\u2022 Security: Public\n\u2022 Views: 124\n\n# Importing invoice lines in an array in powershell\n\nHello,\n\nI need to script the import of invoices into an array.\nThe problem is that the \"table\" is not always the same....\nI have been trying to do it manually with notepad and just with find and replace create char delimited file.\nThe only way I see it working is to start at the end:\n\nso first put the \"SR\" or \"OS\" in a column, then the total price if it exists, then the price, then the amount, then the description and then the partnumber.\n\nI have no idea if it is possible due to the bad source...\n\nso my output would be 6 columns with data\n\nTHanks\ninvoice-lines.txt\n0\nHans de Jongh\n\u2022 4\n\u2022 4\n\u2022 3\n\u2022 +1\n2 Solutions\n\nSystems EngineerCommented:\nCould adjust the data source? \u00a0What I mean is, could the rows with the 2 missing columns, have \"0\" entered if no entry exists. \u00a0If the data is coming out of a database, can the 2 fields in the table be defaulted to have an automatic entry of \"0\" when a record is created?\n\nThis would make every row have 6 columns. \u00a0If make it only a little easier to import.\n\nOtherwise... You could read in a line and split it on the separator 'space'. \u00a0Then trim the whitespace, so there are no extra blank characters on the field. \u00a0Check to see if the split destination array has either 4 or 6 elements, if there are 6 elements, just copy that data into a new array. \u00a0If the array has 4 elements, move the data into the new array but add the missing fields and define them as \"0\".\n\nNow you'll have the final array with 6 elements, and all values are populated.\n\nDan\n0\n\nAuthor Commented:\nhi,\n\nno these are converted pdf's. so no adjusting possible.\nNo I thought about that aswell but the problem is that sometimes in the text you find whitepaces of 6 elements so that doesn't work either...\n0\n\nCommented:\nRegex is very powerful. \u00a0Luckily there appears to be enough of a pattern to match here. \u00a0It was a bit of a pain that the description can include a block of spaces. \u00a0Anyway, here's what I have.\n$pattern = \"(?<part>^[0-9]{3}-[a-z0-9]{4,5}) +(?<desc>.+?) {7,}(?<qty>[0-9]{1,3}) +(?:(?<price>[0-9.,]+) +(?<total>[0-9.,]+) +)?(?<code>(SR|OS))$\"\nGet-Content .\\temp\\invoice-lines.txt | Where {$_ -match$pattern} | ForEach \n{\nNew-Object PsObject -Property ([ordered]@{\nPart = $Matches.part Description =$Matches.Desc\nQuantity = [int32]$Matches.qty Price =$Matches.price\nTotal = $Matches.total Code =$Matches.code\n})\n}\n\nI might even replace line 6 with the following to remove the blocks of spaces from the description.\n Description = $Matches.Desc -replace \" {2,}\",\" \" 0 Systems EngineerCommented: So here's a partial solution. The code below reads the file into a variable, then goes thru each line and replaces anything that match 2 or more space with a pipe. $SourceData = gc source.txt -totalcount 10\nforeach ($row in$SourceData)\n{\n$row = [regex]::replace($row,\"\\b[ ]{2,}\\b\",'|')\n$row } This leaves 6 lines with an issue because there is only 1 space between the first field and second. Dan 0 Systems EngineerCommented: The example above is for test and only looks at the first 10 lines. Limits the data for testing. Only thing left is to try to find lines with only 1 space between the first field and second. Knowing that the second field can contain a phrase that may contain single spaces between words. Dan 0 C++ DeveloperCommented: Similar solution as footech, who had to post while I was assembling something @*#. Doesn't rely on PowerShell 4: cls Get-Content c:\\temp\\ee\\invoice-lines.txt | ? {$_ -match '(?<PartNo>\\S*)\\s+(?<Description>.*)\\s\\s+(?<Amount>\\d+)\\s+(?<Prize>[0-9.]+,\\d\\d)?\\s+(?<Total>[0-9.]+,\\d\\d)?\\s+(?<SR>..)'} |\n% {\nNew-Object PsObject -Property @{\nPartNo = $matches.PartNo Description =$matches.Description\nAmount = $matches.Amount Prize =$matches.Prize\nTotal = $matches.Total SR =$matches.SR\n}\n} |\nselect PartNo, Description, Amount, Prize, Total, SR | ft -AutoSize\n`\n0\n\nAuthor Commented:\nthanks everybody, I guess I give the points to the first who solved this? (sorry)\n0\n\nAuthor Commented:\ncheck 10000 lines and not a single failure!\n0\n\nCommented:\n:)\nI find it a bit interesting to see the differing regex patterns that others use to parse the same input.\n\nI know Qlemo knows this, but for others who may not, you can adjust mine so that it doesn't rely on PowerShell 3+ (not just 4) just by removing [ordered] from line 4. \u00a0You would then have a use a Select statement like Qlemo did to order the columns if so desired.\n0\n\nCommented:\nIt's a judgement call. \u00a0When you have multiple working solutions posted at about the same time, I think it's appropriate to split points. \u00a0As a topic adviser, Qlemo could probably advise you better on this than I.\n0\n\nSystems EngineerCommented:\nInteresting parsing challenge! \u00a0Glad the issue was resolved and I get to take away a nice regex lesson.\n\nDan\n0\n\nAuthor Commented:\nso how can I get back the points and give them away (split them)\nin a 70\/30 split\n0\n\n## Featured Post\n\n\u2022 4\n\u2022 4\n\u2022 3\n\u2022 +1\nTackle projects and never again get stuck behind a technical roadblock.","date":"2017-11-24 06:21:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5156586766242981, \"perplexity\": 3383.7569931311023}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934807089.35\/warc\/CC-MAIN-20171124051000-20171124071000-00294.warc.gz\"}"} | null | null |
Radial & Flat ceiling reflectors are used to scatter and redirect noise energy to eliminate dead spots and provide a uniform listening area throughout the room. Reflectors can be positioned within the ceiling to provide multi-level ceilings. Constructed of rigid E-glass and molded into one piece shapes to combine high impact strength and hardness with light weight. Essi offers standard 9 Radial Shaped sizes and 2 Flat Shaped sizes. Standard White Gel Coat, Paintable Gel Coat or Fabric finishes provides design coordination with other interior products. Optional (IA) Internal Absorber increases absorption at all frequencies or (IL) Internal Liner increases reflection at all frequencies. The 3 point suspension system allows for easy installation and field adjustment. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,354 |
10379 Лейк Плесід (10379 Lake Placid) — астероїд головного поясу, відкритий 18 липня 1996 року.
Тіссеранів параметр щодо Юпітера — 3,115.
Примітки
Див. також
Список астероїдів (10301-10400)
Посилання
Інформація про малі планети на сайті minorplanetcenter.net
Астрономічні об'єкти, відкриті 1996
Головний пояс астероїдів | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,204 |
{"url":"https:\/\/riksun.riken.go.jp\/CTAN\/info\/latex2e-help-texinfo\/latex2e.html","text":"# LaTeX2e unofficial reference manual (July 2021)\n\nThis is an unofficial reference manual for LaTeX. See below for the Table of Contents. If you want a tutorial then please instead visit learnlatex.org or this list.\n\nThis manual has two versions. One has separate web pages for each section or subsection. It's also available as a single web page and as a pdf.\n\nThis document is not official. It has not been reviewed by the LaTeX maintainers. Our goal is to cover all (non-private) LaTeX commands. Your comments and contributions, including bug reports, are very welcome. See our project page for more, including license information and information on how you can contribute to this manual as well as mirror it.\n\nNext: , Up: (dir) \u00a0 [Contents][Index]\n\n# LaTeX2e: An unofficial reference manual\n\nThis document is an unofficial reference manual (version of July 2021) for LaTeX2e, a document preparation system.\n\nNext: , Previous: , Up: Top \u00a0 [Contents][Index]\n\nThis is an unofficial reference manual for the LaTeX2e document preparation system, which is a macro package for the TeX typesetting program (see Overview).\n\nThis document\u2019s home page is https:\/\/latexref.xyz; it has separate web pages for each topic. Alternatively. https:\/\/latexref.xyz\/dev\/latex2e.html has the entire document on a single page. For other output formats, the sources, and plenty more information, see https:\/\/latexref.xyz\/dev\/.\n\nIn this document, we will mostly just use \u2018LaTeX\u2019 rather than \u2018LaTeX2e\u2019, since the previous version of LaTeX\u00a0(2.09) was frozen decades ago.\n\nLaTeX is maintained by a group of volunteers (https:\/\/latex-project.org). The official documentation written by the LaTeX project is available from their web site. The present document is completely unofficial and has not been written or reviewed by the LaTeX maintainers. Do not send bug reports or anything else about this document to them. Instead, please send all comments to latexrefman@tug.org.\n\nThis document is a reference, not a tutorial. There is a vast array of other information available about LaTeX, at all levels. Here are a few introductions.\n\nhttps:\/\/ctan.org\/pkg\/latex-doc-ptr\n\nTwo pages of recommended references to LaTeX documentation.\n\nhttps:\/\/ctan.org\/pkg\/first-latex-doc\n\nWriting your first document, with a bit of both text and math.\n\nhttps:\/\/ctan.org\/pkg\/lshort\n\nA longer introduction to LaTeX, translated to many languages.\n\nhttps:\/\/tug.org\/begin.html\n\nIntroduction to the TeX system, including LaTeX, with further references.\n\nNext: , Previous: , Up: Top \u00a0 [Contents][Index]\n\n## 2 Overview of LaTeX\n\nLaTeX is a system for typesetting documents. It was originally created by Leslie Lamport in 1984, but has been maintained by a group of volunteers for many years now (https:\/\/latex-project.org). It is widely used, particularly but not exclusively for mathematical and technical documents.\n\nA LaTeX user writes an input file containing text to be typeset along with interspersed commands. The default encoding for the text is UTF-8 (as of 2018). The commands specify, for example, how the text should be formatted.\n\nLaTeX is implemented as a set of related so-called \u201cmacros\u201d which use Donald\u00a0E. Knuth\u2019s TeX typesetting program or one of its derivatives, collectively known as \u201cengines\u201d. Thus, the user produces output, typically PDF, by giving the input file to a TeX engine. (The following sections describe all this in more detail.)\n\nThe term LaTeX is also sometimes used to mean the language in which the input document is marked up, that is, to mean the set of commands available to a LaTeX user.\n\nThe name LaTeX is short for \u201cLamport TeX\u201d. It is pronounced LAH-teck or LAY-teck, or sometimes LAY-tecks. Inside a document, produce the logo with \\LaTeX. Where use of the logo is not sensible, such as in plain text, write it as \u2018LaTeX\u2019.\n\nNext: , Up: Overview \u00a0 [Contents][Index]\n\n### 2.1 Starting and ending\n\nLaTeX files have a simple global structure, with a standard beginning and ending. This is a small example.\n\n\\documentclass{article}\n\\begin{document}\nHello, \\LaTeX\\ world.\n\\end{document}\n\n\nEvery LaTeX document has a \\begin{document} line and an \\end{document} line.\n\nHere, the \u2018article\u2019 is the document class. It is implemented in a file article.cls. You can use any document class on your system. A few document classes are defined by LaTeX itself, and vast array of others are widely available. See Document classes.\n\nYou can include other LaTeX commands between the \\documentclass and the \\begin{document} commands. This area is called the preamble.\n\nThe \\begin{document}, \\end{document} pair defines an environment; the \u2018document\u2019 environment (and no others) is required in all LaTeX documents (see document). LaTeX make available to you many environments that are documented here (see Environments). Many more are available to you from external packages, most importantly those available at CTAN (see CTAN).\n\nThe following sections discuss how to produce PDF or other output from a LaTeX input file.\n\nNext: , Previous: , Up: Overview \u00a0 [Contents][Index]\n\n### 2.2 Output files\n\nLaTeX produces a main output file and at least two auxiliary files. The main output file\u2019s name ends in either .dvi or .pdf.\n\n.dvi\n\nIf LaTeX is invoked with the system command latex then it produces a DeVice Independent file, with extension .dvi. You can view this file with a command such as xdvi, or convert it to a PostScript .ps file with dvips or to a Portable Document Format .pdf file with dvipdfmx. The contents of the file can be dumped in human-readable form with dvitype. A vast array of other DVI utility programs are available (https:\/\/mirror.ctan.org\/dviware).\n\n.pdf\n\nIf LaTeX is invoked via the system command pdflatex, among other commands (see TeX engines), then the main output is a Portable Document Format (PDF) file. Typically this is a self-contained file, with all fonts and images included.\n\nLaTeX always produces at least two additional files.\n\n.log\n\nThis transcript file contains summary information such as a list of loaded packages. It also includes diagnostic messages and perhaps additional information for any errors.\n\n.aux\n\nAuxiliary information is used by LaTeX for things such as cross references. For example, the first time that LaTeX finds a forward reference\u2014a cross reference to something that has not yet appeared in the source\u2014it will appear in the output as a doubled question mark ??. When the referred-to spot does eventually appear in the source then LaTeX writes its location information to this .aux file. On the next invocation, LaTeX reads the location information from this file and uses it to resolve the reference, replacing the double question mark with the remembered location.\n\nLaTeX may produce yet more files, characterized by the filename ending. These include a .lof file that is used to make a list of figures, a .lot file used to make a list of tables, and a .toc file used to make a table of contents (see Table of contents etc.). A particular class may create others; the list is open-ended.\n\nNext: , Previous: , Up: Overview \u00a0 [Contents][Index]\n\n### 2.3 TeX engines\n\nLaTeX is a large set of commands that is executed by a TeX program (see Overview). Such a set of commands is called a format, and is embodied in a binary .fmt file, which can be read much more quickly than the corresponding TeX source.\n\nThis section gives a terse overview of the TeX programs that are commonly available (see also Command line interface).\n\nlatex\npdflatex\n\nIn TeX Live (https:\/\/tug.org\/texlive), if LaTeX is invoked via either the system command latex or pdflatex, then the pdfTeX engine is run (https:\/\/ctan.org\/pkg\/pdftex). When invoked as latex, the main output is a .dvi file; as pdflatex, the main output is a .pdf file.\n\npdfTeX incorporates the e-TeX extensions to Knuth\u2019s original program (https:\/\/ctan.org\/pkg\/etex), including additional programming features and bi-directional typesetting, and has plenty of extensions of its own. e-TeX is available on its own as the system command etex, but this is plain TeX (and produces .dvi).\n\nIn other TeX distributions, latex may invoke e-TeX rather than pdfTeX. In any case, the e-TeX extensions can be assumed to be available in LaTeX.\n\nlualatex\n\nIf LaTeX is invoked via the system command lualatex, the LuaTeX engine is run (https:\/\/ctan.org\/pkg\/luatex). This program allows code written in the scripting language Lua (http:\/\/luatex.org) to interact with TeX\u2019s typesetting. LuaTeX handles UTF-8 Unicode input natively, can handle OpenType and TrueType fonts, and produces a .pdf file by default. There is also dvilualatex to produce a .dvi file.\n\nxelatex\n\nIf LaTeX is invoked with the system command xelatex, the XeTeX engine is run (https:\/\/tug.org\/xetex). Like LuaTeX, XeTeX natively supports UTF-8 Unicode and TrueType and OpenType fonts, though the implementation is completely different, mainly using external libraries instead of internal code. XeTeX produces a .pdf file as output; it does not support DVI output.\n\nInternally, XeTeX creates an .xdv file, a variant of DVI, and translates that to PDF using the (x)dvipdfmx program, but this process is automatic. The .xdv file is only useful for debugging.\n\nplatex\nuplatex\n\nThese commands provide significant additional support for Japanese and other languages; the u variant supports Unicode. See https:\/\/ctan.org\/pkg\/ptex and https:\/\/ctan.org\/pkg\/uptex.\n\nAs of 2019, there is a companion -dev command and format for all of the above:\n\ndvilualatex-dev\nlatex-dev\nlualatex-dev\npdflatex-dev\nplatex-dev\nuplatex-dev\nxelatex-dev\n\nThese are candidates for an upcoming LaTeX release. The main purpose is to find and address compatibility problems before an official release.\n\nThese -dev formats make it easy for anyone to help test documents and code: you can run, say, pdflatex-dev instead of pdflatex, without changing anything else in your environment. Indeed, it is easiest and most helpful to always run the -dev versions instead of bothering to switch back and forth. During quiet times after a release, the commands will be equivalent.\n\nThese are not daily snapshots or untested development code. They undergo the same extensive regression testing by the LaTeX team before being released.\n\nFor more information, see \u201cThe LaTeX release workflow and the LaTeX dev formats\u201d by Frank Mittelbach, TUGboat 40:2, https:\/\/tug.org\/TUGboat\/tb40-2\/tb125mitt-dev.pdf.\n\nNext: , Previous: , Up: Overview \u00a0 [Contents][Index]\n\n### 2.4 LaTeX command syntax\n\nIn the LaTeX input file, a command name starts with a backslash character, \\. The name itself then consists of either (a)\u00a0a string of letters or (b)\u00a0a single non-letter.\n\nLaTeX commands names are case sensitive so that \\pagebreak differs from \\Pagebreak (the latter is not a standard command). Most command names are lowercase, but in any event you must enter all commands in the same case as they are defined.\n\nA command may be followed by zero, one, or more arguments. These arguments may be either required or optional. Required arguments are contained in curly braces, {...}. Optional arguments are contained in square brackets, [...]. Generally, but not universally, if the command accepts an optional argument, it comes first, before any required arguments.\n\nInside of an optional argument, to use the character close square bracket\u00a0(]) hide it inside curly braces, as in\u00a0\\item[closing bracket {]}]. Similarly, if an optional argument comes last, with no required argument after it, then to make the first character of the following text be an open square bracket, hide it inside curly braces.\n\nLaTeX has the convention that some commands have a * form that is related to the form without a *, such as \\chapter and \\chapter*. The exact difference in behavior varies from command to command.\n\nThis manual describes all accepted options and *-forms for the commands it covers (barring unintentional omissions, a.k.a. bugs).\n\nAs of the 2020-10-01 release of LaTeX, the expl3 and xparse packages are part of the LaTeX2e format. They provide a completely different underlying programming language syntax. We won\u2019t try to cover them in this document; see the related package documentation and other LaTeX manuals.\n\nNext: , Previous: , Up: Overview \u00a0 [Contents][Index]\n\n### 2.5 Environment\n\nSynopsis:\n\n\\begin{environment-name}\n...\n\\end{environment-name}\n\n\nAn environment is an area of LaTeX source, inside of which there is a distinct behavior. For instance, for poetry in LaTeX put the lines between \\begin{verse} and \\end{verse}.\n\n\\begin{verse}\nThere once was a man from Nantucket \\\\\n...\n\\end{verse}\n\n\nSee Environments, for a list of environments. Particularly notable is that every LaTeX document must have a document environment, a \\begin{document} ... \\end{document} pair.\n\nThe environment-name at the beginning must exactly match that at the end. This includes the case where environment-name ends in a star\u00a0(*); both the \\begin and \\end texts must include the star.\n\nEnvironments may have arguments, including optional arguments. This example produces a table. The first argument is optional (and causes the table to be aligned on its top row) while the second argument is required (it specifies the formatting of columns).\n\n\\begin{tabular}[t]{r|l}\n... rows of table ...\n\\end{tabular}\n\n\nPrevious: , Up: Overview \u00a0 [Contents][Index]\n\n### 2.6 CTAN: The Comprehensive TeX Archive Network\n\nThe Comprehensive TeX Archive Network, CTAN, is the TeX and LaTeX community\u2019s repository of free material. It is a set of Internet sites around the world that offer material related to LaTeX for download. Visit CTAN on the web at https:\/\/ctan.org.\n\nThis material is organized into packages, discrete bundles that typically offer some coherent functionality and are maintained by one person or a small number of people. For instance, many publishers have a package that allows authors to format papers to that publisher\u2019s specifications.\n\nIn addition to the massive holdings, the ctan.org web site offers features such as search by name or by functionality.\n\nCTAN is not a single host, but instead is a set of hosts, one of which is the so-called \u201cmaster\u201d. The master host actively manages the material, for instance, by accepting uploads of new or updated packages. For many years, it has been hosted by the German TeX group, DANTE e.V.\n\nOther sites around the world help out by mirroring, that is, automatically syncing their collections with the master site and then in turn making their copies publicly available. This gives users close to their location better access and relieves the load on the master site. The list of mirrors is at https:\/\/ctan.org\/mirrors.\n\nNext: , Previous: , Up: Top \u00a0 [Contents][Index]\n\n## 3 Document classes\n\nThe document\u2019s overall class is defined with this command, which is normally the first command in a LaTeX source file.\n\n\\documentclass[options]{class}\n\n\nThe following document class names are built into LaTeX. (Many other document classes are available as separate packages; see Overview.)\n\narticle\n\nFor a journal article, a presentation, and miscellaneous general use.\n\nbook\n\nFull-length books, including chapters and possibly including front matter, such as a preface, and back matter, such as an appendix (see Front\/back matter).\n\nletter\n\nMail, optionally including mailing labels (see Letters).\n\nreport\n\nFor documents of length between an article and a book, such as technical reports or theses, which may contain several chapters.\n\nslides\n\nFor slide presentations\u2014rarely used nowadays. The beamer package is perhaps the most prevalent (https:\/\/ctan.org\/pkg\/beamer). See beamer template, for a small template for a beamer document.\n\nStandard options are described in the next section.\n\nNext: , Up: Document classes \u00a0 [Contents][Index]\n\n### 3.1 Document class options\n\nYou can specify global options or class options to the \\documentclass command by enclosing them in square brackets. To specify more than one option, separate them with a comma.\n\n\\documentclass[option1,option2,...]{class}\n\n\nHere is the list of the standard class options.\n\nAll of the standard classes except slides accept the following options for selecting the typeface size (default is 10pt):\n\n10pt 11pt 12pt\n\n\nAll of the standard classes accept these options for selecting the paper size (these show height by width):\n\na4paper\n\n210 by 297mm (about 8.25 by 11.75\u00a0inches)\n\na5paper\n\n148 by 210mm (about 5.8 by 8.3\u00a0inches)\n\nb5paper\n\n176 by 250mm (about 6.9 by 9.8\u00a0inches)\n\nexecutivepaper\n\n7.25 by 10.5\u00a0inches\n\nlegalpaper\n\n8.5 by 14\u00a0inches\n\nletterpaper\n\n8.5 by 11\u00a0inches (the default)\n\nWhen using one of the engines pdfLaTeX, LuaLaTeX, or XeLaTeX (see TeX engines), options other than letterpaper set the print area but you must also set the physical paper size. One way to do that is to put \\pdfpagewidth=\\paperwidth and \\pdfpageheight=\\paperheight in your document\u2019s preamble.\n\nThe geometry package provides flexible ways of setting the print area and physical page size.\n\nMiscellaneous other options:\n\ndraft\nfinal\n\nMark (draft) or do not mark (final) overfull boxes with a black box in the margin; default is final.\n\nfleqn\n\nPut displayed formulas flush left; default is centered.\n\nlandscape\n\nSelects landscape format; default is portrait.\n\nleqno\n\nPut equation numbers on the left side of equations; default is the right side.\n\nopenbib\n\nUse \u201copen\u201d bibliography format.\n\ntitlepage\nnotitlepage\n\nSpecifies whether there is a separate page for the title information and for the abstract also, if there is one. The default for the report class is titlepage, for the other classes it is notitlepage.\n\nThe following options are not available with the slides class.\n\nonecolumn\ntwocolumn\n\nTypeset in one or two columns; default is onecolumn.\n\noneside\ntwoside\n\nSelects one- or two-sided layout; default is oneside, except that in the book class the default is twoside.\n\nFor one-sided printing, the text is centered on the page. For two-sided printing, the \\evensidemargin (\\oddsidemargin) parameter determines the distance on even (odd) numbered pages between the left side of the page and the text\u2019s left margin, with \\oddsidemargin being 40% of the difference between \\paperwidth and \\textwidth, and \\evensidemargin is the remainder.\n\nopenright\nopenany\n\nDetermines if a chapter should start on a right-hand page; default is openright for book, and openany for report.\n\nThe slides class offers the option clock for printing the time at the bottom of each note.\n\nNext: , Previous: , Up: Document classes \u00a0 [Contents][Index]\n\nLoad a package pkg, with the package options given in the comma-separated list options, as here.\n\n\\usepackage[options]{pkg}.\n\n\nTo specify more than one package you can separate them with a comma, as in \\usepackage{pkg1,pkg2,...}, or use multiple \\usepackage commands.\n\nAny options given in the \\documentclass command that are unknown to the selected document class are passed on to the packages loaded with \\usepackage.\n\nPrevious: , Up: Document classes \u00a0 [Contents][Index]\n\n### 3.3 Class and package construction\n\nYou can create new document classes and new packages. For instance, if your memos must satisfy some local requirements, such as a standard header for each page, then you could create a new class smcmemo.cls and begin your documents with \\documentclass{smcmemo}.\n\nWhat separates a package from a document class is that the commands in a package are useful across classes while those in a document class are specific to that class. Thus, a command to set page headers is for a package while a command to make the page headers say Memo from the SMC Math Department is for a class.\n\nInside of a class or package file you can use the at-sign @ as a character in command names without having to surround the code containing that command with \\makeatletter and \\makeatother. See \\makeatletter & \\makeatother. This allow you to create commands that users will not accidentally redefine. Another technique is to preface class- or package-specific commands with some string to prevent your class or package from interfering with others. For instance, the class smcmemo might have commands \\smc@tolist, \\smc@fromlist, etc.\n\n#### 3.3.1 Class and package structure\n\nA class file or package file typically has four parts.\n\n1. In the identification part, the file says that it is a LaTeX package or class and describes itself, using the \\NeedsTeXFormat and \\ProvidesClass or \\ProvidesPackage commands.\n2. The preliminary declarations part declares some commands and can also load other files. Usually these commands will be those needed for the code used in the next part. For example, an smcmemo class might be called with an option to read in a file with a list of people for the to-head, as \\documentclass[mathto]{smcmemo}, and therefore needs to define a command \\newcommand{\\setto}[1]{\\def\\@tolist{#1}} used in that file.\n3. In the handle options part the class or package declares and processes its options. Class options allow a user to start their document as \\documentclass[option list]{class name}, to modify the behavior of the class. An example is when you declare \\documentclass[11pt]{article} to set the default document font size.\n4. Finally, in the more declarations part the class or package usually does most of its work: declaring new variables, commands and fonts, and loading other files.\n\nHere is a starting class file, which should be saved as stub.cls where LaTeX can find it, for example in the same directory as the .tex file.\n\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesClass{stub}[2017\/07\/06 stub to start building classes from]\n\\DeclareOption*{\\PassOptionsToClass{\\CurrentOption}{article}}\n\\ProcessOptions\\relax\n\n\nIt identifies itself, handles the class options via the default of passing them all to the article class, and then loads the article class to provide the basis for this class\u2019s code.\n\nFor more, see the official guide for class and package writers, the Class Guide, at https:\/\/www.latex-project.org\/help\/documentation\/clsguide.pdf (much of the descriptions here derive from this document), or the tutorial https:\/\/www.tug.org\/TUGboat\/tb26-3\/tb84heff.pdf.\n\nPrevious: , Up: Class and package construction \u00a0 [Contents][Index]\n\n#### 3.3.2 Class and package commands\n\nThese are the commands designed to help writers of classes or packages.\n\n\\AtBeginDvi{specials}\n\nSave in a box register things that are written to the .dvi file at the beginning of the shipout of the first page of the document.\n\n\\AtEndOfClass{code}\n\\AtEndOfPackage{code}\n\nHook to insert code to be executed when LaTeX finishes processing the current class or package. You can use these hooks multiple times; the code will be executed in the order that you called it. See also \\AtBeginDocument.\n\n\\CheckCommand{cmd}[num][default]{definition}\n\\CheckCommand*{cmd}[num][default]{definition}\n\n\n\\ClassError{class name}{error text}{help text}\n\\PackageError{package name}{error text}{help text}\n\\ClassWarning{class name}{warning text}\n\\PackageWarning{package name}{warning text}\n\\ClassWarningNoLine{class name}{warning text}\n\\PackageWarningNoLine{package name}{warning text}\n\\ClassInfo{class name}{info text}\n\\PackageInfo{package name}{info text}\n\\ClassInfoNoLine{class name}{info text}\n\\PackageInfoNoLine{package name}{info text}\n\nProduce an error message, or warning or informational messages.\n\nFor \\ClassError and \\PackageError the message is error text, followed by TeX\u2019s ? error prompt. If the user then asks for help by typing h, they see the help text.\n\nThe four warning commands are similar except that they write warning text on the screen with no error prompt. The four info commands write info text only in the transcript file. The NoLine versions do not show the number of the line generating the message, while the other versions do show that number.\n\nTo format the messages, including the help text: use \\protect to stop a command from expanding, get a line break with \\MessageBreak, and get a space with \\space when a space character does not allow it, like after a command. Note that LaTeX appends a period to the messages.\n\n\\CurrentOption\n\nExpands to the name of the currently-being-processed option. Can only be used within the code argument of either \\DeclareOption or \\DeclareOption*.\n\n\\DeclareOption{option}{code}\n\\DeclareOption*{code}\n\nMake an option available to a user to invoke in their \\documentclass command. For example, the smcmemo class could have an option \\documentclass[logo]{smcmemo} allowing users to put the institutional logo on the first page. The class file must contain \\DeclareOption{logo}{code} (and later, \\ProcessOptions).\n\nIf you request an option that has not been declared, by default this will produce a warning like Unused global option(s): [badoption]. Change this behavior with the starred version \\DeclareOption*{code}. For example, many classes extend an existing class, using a command such as \\LoadClass{article}, and for passing extra options to the underlying class use code such as this.\n\n\\DeclareOption*{%\n\\PassOptionsToClass{\\CurrentOption}{article}%\n}\n\n\nAnother example is that the class smcmemo may allow users to keep lists of memo recipients in external files. Then the user could invoke \\documentclass[math]{smcmemo} and it will read the file math.memo. This code handles the file if it exists and otherwise passes the option to the article class.\n\n\\DeclareOption*{\\InputIfFileExists{\\CurrentOption.memo}{}{%\n\\PassOptionsToClass{\\CurrentOption}{article}}}\n\n\\DeclareRobustCommand{cmd}[num][default]{definition}\n\\DeclareRobustCommand*{cmd}[num][default]{definition}\n\n\n\n\n1. They use the low-level e-TeX protection mechanism rather than the higher level LaTeX \\protect mechanism, so they do not incur the slight loss of performance mentioned above, and\n2. They make the same distinction between \\new\u2026, \\renew\u2026, and \\provide\u2026, as the standard commands, so they do not just make a log message when you redefine cmd that already exists, in that case you need to use either \\renew\u2026 or \\provide\u2026 or you get an error.\n\\IfFileExists{filename}{true code}{false code}\n\\InputIfFileExists{filename}{true code}{false code}\n\nExecute true code if LaTeX finds the file file name or false code otherwise. In the first case it executing true code and then inputs the file. Thus the command\n\n\\IfFileExists{img.pdf}{%\n\n\nwill include the graphic img.pdf if it is found and otherwise give a warning.\n\nThis command looks for the file in all search paths that LaTeX uses, not only in the current directory. To look only in the current directory do something like \\IfFileExists{.\/filename}{true code}{false code}. If you ask for a filename without a .tex extension then LaTeX will first look for the file by appending the .tex; for more on how LaTeX handles file extensions see \\input.\n\n\\LoadClass[options list]{class name}[release date]\n\\LoadClassWithOptions{class name}[release date]\n\nLoad a class, as with \\documentclass[options list]{class name}[release info]. An example is \\LoadClass[twoside]{article}.\n\nThe options list, if present, is a comma-separated list. The release date is optional. If present it must have the form YYYY\/MM\/DD.\n\nIf you request a release date and the date of the package installed on your system is earlier, then you get a warning on the screen and in the log like this.\n\nYou have requested, on input line 4, version 2038\/01\/19' of\ndocument class article, but only version 2014\/09\/29 v1.4h\nStandard LaTeX document class' is available.\n\n\nThe command version \\LoadClassWithOptions uses the list of options for the current class. This means it ignores any options passed to it via \\PassOptionsToClass. This is a convenience command that lets you build classes on existing ones, such as the standard article class, without having to track which options were passed.\n\n\\ExecuteOptions{options-list}\n\nFor each option option in the options-list, in order, this command executes the command \\ds@option. If this command is not defined then that option is silently ignored.\n\nIt can be used to provide a default option list before \\ProcessOptions. For example, if in a class file you want the default to be 11pt fonts then you could specify \\ExecuteOptions{11pt}\\ProcessOptions\\relax.\n\n\\NeedsTeXFormat{format}[format date]\n\nSpecifies the format that this class must be run under. Often issued as the first line of a class file, and most often used as: \\NeedsTeXFormat{LaTeX2e}. When a document using that class is processed, the format name given here must match the format that is actually being run (including that the format string is case sensitive). If it does not match then execution stops with an error like \u2018This file needs format LaTeX2e' but this is xxx'.\n\nTo specify a version of the format that you know to have certain features, include the optional format date on which those features were implemented. If present it must be in the form YYYY\/MM\/DD. If the format version installed on your system is earlier than format date then you get a warning like this.\n\nYou have requested release 2038\/01\/20' of LaTeX, but only\nrelease 2016\/02\/01' is available.\n\n\\OptionNotUsed\n\nAdds the current option to the list of unused options. Can only be used within the code argument of either \\DeclareOption or \\DeclareOption*.\n\n\\PassOptionsToClass{option list}{class name}\n\\PassOptionsToPackage{option list}{package name}\n\nAdds the options in the comma-separated list option list to the options used by any future \\RequirePackage or \\usepackage command for package package name or the class class name.\n\nThe reason for these commands is: you may load a package any number of times with no options but if you want options then you may only supply them when you first load the package. Loading a package with options more than once will get you an error like Option clash for package foo. (LaTeX throws an error even if there is no conflict between the options.)\n\nIf your own code is bringing in a package twice then you can collapse that to once, for example replacing the two \\RequirePackage[landscape]{geometry} and \\RequirePackage[margins=1in]{geometry} with the single command \\RequirePackage[landscape,margins=1in]{geometry}.\n\nHowever, imagine that you are loading firstpkg and inside that package it loads secondpkg, and you need the second package to be loaded with option draft. Then before doing the first package you must queue up the options for the second package, like this.\n\n\\PassOptionsToPackage{draft}{secondpkg}\n\\RequirePackage{firstpkg}\n\n\n(If firstpkg.sty loads an option in conflict with what you want then you may have to alter its source.)\n\nThese commands are useful for general users as well as class and package writers. For instance, suppose a user wants to load the graphicx package with the option draft and also wants to use a class foo that loads the graphicx package, but without that option. The user could start their LaTeX file with \\PassOptionsToPackage{draft}{graphicx}\\documentclass{foo}.\n\n\\ProcessOptions\n\\ProcessOptions*\\@options\n\nExecute the code for each option that the user has invoked. Include it in the class file as \\ProcessOptions\\relax (because of the existence of the starred command).\n\nOptions come in two types. Local options have been specified for this particular package in the options argument of \\PassOptionsToPackage{options}, \\usepackage[options], or \\RequirePackage[options]. Global options are those given by the class user in \\documentclass[options] (If an option is specified both locally and globally then it is local.)\n\nWhen \\ProcessOptions is called for a package pkg.sty, the following happens:\n\n1. For each option option so far declared with \\DeclareOption, it looks to see if that option is either a global or a local option for pkg. If so then it executes the declared code. This is done in the order in which these options were given in pkg.sty.\n2. For each remaining local option, it executes the command \\ds@option if it has been defined somewhere (other than by a \\DeclareOption); otherwise, it executes the default option code given in \\DeclareOption*. If no default option code has been declared then it gives an error message. This is done in the order in which these options were specified.\n\nWhen \\ProcessOptions is called for a class it works in the same way except that all options are local, and the default code for \\DeclareOption* is \\OptionNotUsed rather than an error.\n\nThe starred version \\ProcessOptions* executes the options in the order specified in the calling commands, rather than in the order of declaration in the class or package. For a package this means that the global options are processed first.\n\n\\ProvidesClass{class name}[release date brief additional information]\n\\ProvidesClass{class name}[release date]\n\\ProvidesPackage{package name}[release date brief additional information]\n\\ProvidesPackage{package name}[release date]\n\nIdentifies the class or package, printing a message to the screen and the log file.\n\nWhen you load a class or package, for example with \\documentclass{smcmemo} or \\usepackage{test}, LaTeX inputs a file. If the name of the file does not match the class or package name declared in it then you get a warning. Thus, if you invoke \\documentclass{smcmemo}, and the file smcmemo.cls has the statement \\ProvidesClass{xxx} then you get a warning like You have requested document class smcmemo', but the document class provides 'xxx'. This warning does not prevent LaTeX from processing the rest of the class file normally.\n\nIf you include the optional argument then you must include a date, before any spaces, of the form YYYY\/MM\/DD. The rest of the optional argument is free-form, although it traditionally identifies the class, and is written to the screen during compilation and to the log file. Thus, if your file smcmemo.cls contains the line \\ProvidesClass{smcmemo}[2008\/06\/01 v1.0 SMC memo class] and your document\u2019s first line is \\documentclass{smcmemo} then you will see Document Class: smcmemo 2008\/06\/01 v1.0 SMC memo class.\n\nThe date in the optional argument allows class and package users to ask to be warned if the version of the class or package is earlier than release date. For instance, a user could enter \\documentclass{smcmemo}[2018\/10\/12] or \\usepackage{foo}[[2017\/07\/07]] to require a class or package with certain features by specifying that it must be released no earlier than the given date. (Although, in practice package users only rarely include a date, and class users almost never do.)\n\n\\ProvidesFile{filename}[additional information]\n\nDeclare a file other than the main class and package files, such as configuration files or font definition files. Put this command in that file and you get in the log a string like File: test.config 2017\/10\/12 config file for test.cls for filename equal to \u2018test.config\u2019 and additional information equal to \u20182017\/10\/12 config file for test.cls\u2019.\n\n\\RequirePackage[option list]{package name}[release date]\n\\RequirePackageWithOptions{package name}[release date]\n\nLoad a package, like the command \\usepackage (see Additional packages). The LaTeX development team strongly recommends use of these commands over Plain\u00a0TeX\u2019s \\input; see the Class Guide. An example is \\RequirePackage[landscape,margin=1in]{geometry}.\n\nThe option list, if present, is a comma-separated list. The release date, if present, must have the form YYYY\/MM\/DD. If the release date of the package as installed on your system is earlier than release date then you get a warning like You have requested, on input line 9, version 2017\/07\/03' of package jhtest, but only version 2000\/01\/01' is available.\n\nThe \\RequirePackageWithOptions version uses the list of options for the current class. This means it ignores any options passed to it via \\PassOptionsToClass. This is a convenience command to allow easily building classes on existing ones without having to track which options were passed.\n\nThe difference between \\usepackage and \\RequirePackage is small. The \\usepackage command is intended for the document file while \\RequirePackage is intended for package and class files. Thus, using \\usepackage before the \\documentclass command causes LaTeX to give error like \\usepackage before \\documentclass, but you can use \\RequirePackage there.\n\nNext: , Previous: , Up: Top \u00a0 [Contents][Index]\n\n## 4 Fonts\n\nLaTeX comes with powerful font capacities. For one thing, its New Font Selection Scheme allows you to work easily with the font families in your document (for instance, see\u00a0Font styles). And, LaTeX documents can use most fonts that are available today, including versions of Times Roman, Helvetica, Courier, etc. (Note, though, that many fonts do not have support for mathematics.)\n\nThe first typeface in the TeX world was the Computer Modern family, developed by Donald Knuth. It is the default for LaTeX documents and is still the most widely used. But changing to another font often only involves a few commands. For instance, putting the following in your preamble gives you a Palatino-like font, which is handsome and more readable online than many other fonts, while still allowing you to typeset mathematics. (This example is from Michael Sharpe, https:\/\/math.ucsd.edu\/~msharpe\/RcntFnts.pdf.)\n\n\\usepackage[osf]{newpxtext} % osf for text, not math\n\\usepackage{cabin} % sans serif\n\\usepackage[varqu,varl]{inconsolata} % sans serif typewriter\n\\usepackage[bigdelims,vvarbb]{newpxmath} % bb from STIX\n\\usepackage[cal=boondoxo]{mathalfa} % mathcal\n\n\nIn addition, the xelatex or lualatex engines allow you to use any fonts on your system that are in OpenType or TrueType format (see TeX engines).\n\nThe LaTeX Font Catalogue (https:\/\/tug.org\/FontCatalogue) shows font sample graphics and copy-and-pasteable source to use many fonts, including many with support for mathematics. It aims to cover all Latin alphabet free fonts available for easy use with LaTeX.\n\nMore information is also available from the TeX Users Group, at https:\/\/www.tug.org\/fonts\/.\n\nNext: , Up: Fonts \u00a0 [Contents][Index]\n\n### 4.1 fontenc package\n\nSynopsis:\n\n\\usepackage[font_encoding]{fontenc}\n\n\nor\n\n\\usepackage[font_encoding1, font_encoding2, ...]{fontenc}\n\n\nSpecify the font encodings. A font encoding is a mapping of the character codes to the font glyphs that are used to typeset your output.\n\nThis package only applies if you use the pdflatex engine (see TeX engines). If you use the xelatex or lualatex engine then instead use the fontspec package.\n\nTeX\u2019s original font family, Computer Modern, has a limited character set. For instance, to make common accented characters you must use \\accent (see \\accent) but this disables hyphenation. TeX users have agreed on a number of standards to access the larger sets of characters provided by modern fonts. If you are using pdflatex then this in the preamble\n\n\\usepackage[T1]{fontenc}\n\n\ngives you support for the most widespread European languages, including French, German, Italian, Polish, and others. In particular, if you have words with accented letters then LaTeX will hyphenate them and your output can be copied and pasted. (The optional second line allows you to directly enter accented characters into your source file.)\n\nIf you are using an encoding such as T1 and the characters appear blurry or do not magnify well then your fonts may be bitmapped, sometimes called raster or Type\u00a03. You want vector fonts. Use a package such as lmodern or cm-super to get a font that extends LaTeX\u2019s default using vector fonts.\n\nFor each font_encoding given as an option but not already declared, this package loads the encoding definition files, named font_encodingenc.def. It also sets \\encodingdefault to be the last encoding in the option list.\n\nThese are the common values for font_encoding.\n\nOT1\n\nThe original encoding for TeX. Limited to mostly English characters.\n\nOMS, OML\n\nMath symbols and math letters encoding.\n\nT1\n\nTeX text extended. Sometimes called the Cork encoding for the Users Group meeting where it was developed. Gives access to most European accented characters. The most common option for this package.\n\nTS1\n\nText Companion encoding.\n\nLaTeX\u2019s default is to load OML, T1, OT1, and then OMS, and set the default to OT1.\n\nEven if you do not use accented letters, you may need to specify a font encoding if your font requires it.\n\nIf you use T1\u00a0encoded fonts other than the default Computer Modern family then you may need to load the package that selects your fonts before loading fontenc, to prevent the system from loading any T1\u00a0encoded fonts from the default.\n\nThe LaTeX team reserve encoding names starting with: \u2018T\u2019 for the standard text encodings with 256 characters, \u2018TS\u2019 for symbols that extend the corresponding T encodings, \u2018X\u2019 for test encodings, \u2018M\u2019 for standard math encodings with 256 characters, \u2018A\u2019 for special applications, \u2018OT\u2019 for standard text encodings with 128 characters, and \u2018OM\u2019 for standard math encodings with 128 characters (\u2018O\u2019 stands for \u2018obsolete\u2019).\n\nThis package provides a number of commands, detailed below. Many of them are encoding-specific, so if you have defined a command that works for one encoding but the current encoding is different then the command is not in effect.\n\nNext: , Up: fontenc package \u00a0 [Contents][Index]\n\n#### 4.1.1 \\DeclareFontEncoding\n\nSynopsis:\n\n\\DeclareFontEncoding{encoding}{text-settings}{math-settings}\n\n\nDeclare the font encoding encoding. It also saves the value of encoding in \\LastDeclaredEncoding (see \\LastDeclaredEncoding).\n\nThe file t1enc.def contains this line (followed by many others).\n\n\\DeclareFontEncoding{T1}{}{}\n\n\nThe text-settings are the commands that LaTeX will run every time it switches from one encoding to another with the \\selectfont or \\fontencoding command. The math-settings are the commands that LaTeX will use whenever the font is accessed as a math alphabet.\n\nLaTeX ignores any space characters inside text-settings and math-settings, to prevent unintended spaces in the output.\n\nIf you invent an encoding you should pick a two or three letter name starting with \u2018L\u2019 for \u2018local\u2019, or \u2018E\u2019 for \u2018experimental\u2019.\n\nNote that output encoding files may be read several times by LaTeX so using, e.g., \\newcommand may cause an error. In addition, such files should contain \\ProvidesFile line (see Class and package commands).\n\nNote also that you should use the \\...Default commands only in a package, not in the encoding definition files, since those files should only contain declarations specific to that encoding.\n\nNext: , Previous: , Up: fontenc package \u00a0 [Contents][Index]\n\n#### 4.1.2 \\DeclareTextAccent\n\nSynopsis:\n\n\\DeclareTextAccent{cmd}{encoding}{slot}\n\n\nDefine an accent, to be put on top of other glyphs, in the encoding encoding at the location slot.\n\nThis line from t1enc.def declares that to make a circumflex accent as in \\^A, the system will put the accent in slot\u00a02 over the \u2018A\u2019 character, which is represented in ASCII as\u00a065. (This holds unless there is a relevant DeclareTextComposite or \\DeclareTextCompositeCommand declaration; see \\DeclareTextComposite.)\n\n\\DeclareTextAccent{\\^}{T1}{2}\n\n\nIf cmd has already been defined then \\DeclareTextAccent does not give an error but it does log the redefinition in the transcript file.\n\nNext: , Previous: , Up: fontenc package \u00a0 [Contents][Index]\n\n#### 4.1.3 \\DeclareTextAccentDefault\n\nSynopsis:\n\n\\DeclareTextAccentDefault{cmd}{encoding}\n\n\nIf there is an encoding-specific accent command cmd but there is no associated \\DeclareTextAccent for that encoding then this command will pick up the slack, by saying to use it as described for encoding.\n\nFor example, to make the encoding OT1 be the default encoding for the accent \\\", declare this.\n\n\\DeclareTextAccentDefault{\\\"}{OT1}\n\n\nIf you issue a \\\" when the current encoding does not have a definition for that accent then LaTeX will use the definition from OT1\n\nThat is, this command is equivalent to this call (see \\UseTextSymbol & \\UseTextAccent).\n\n\\DeclareTextCommandDefault[1]{cmd}\n{\\UseTextAccent{encoding}{cmd}{#1}}\n\n\nNote that \\DeclareTextAccentDefault works for any one-argument fontenc command, not just the accent command.\n\n#### 4.1.4 \\DeclareTextCommand & \\ProvideTextCommand\n\nSynopsis, one of:\n\n\\DeclareTextCommand{cmd}{encoding}{defn}\n\\DeclareTextCommand{cmd}{encoding}[nargs]{defn}\n\\DeclareTextCommand{cmd}{encoding}[nargs][optargdefault]{defn}\n\n\nor one of:\n\n\\ProvideTextCommand{cmd}{encoding}{defn}\n\\ProvideTextCommand{cmd}{encoding}[nargs]{defn}\n\\ProvideTextCommand{cmd}{encoding}[nargs][optargdefault]{defn}\n\n\nDefine the command cmd, which will be specific to one encoding. The command name cmd must begin with a backslash, \\. These commands can only appear in the preamble. Redefining cmd does not cause an error. The defined command will be robust even if the code in defn is fragile (see \\protect).\n\nFor example, the file t1enc.def contains this line.\n\n\\DeclareTextCommand{\\textperthousand}{T1}{\\%\\char 24 }\n\n\nWith that, you can express parts per thousand.\n\n\\usepackage[T1]{fontenc} % in preamble\n...\nLegal limit is $$0.8$$\\textperthousand.\n\n\nIf you change the font encoding to OT1 then you get an error like \u2018LaTeX Error: Command \\textperthousand unavailable in encoding OT1\u2019.\n\nThe \\ProvideTextCommand variant does the same, except that it does nothing if cmd is already defined. The \\DeclareTextSymbol command is faster than this one for simple slot-to-glyph association (see \\DeclareTextSymbol)\n\n\nNext: , Previous: , Up: fontenc package \u00a0 [Contents][Index]\n\n#### 4.1.5 \\DeclareTextCommandDefault & \\ProvideTextCommandDefault \n\nSynopsis:\n\n\\DeclareTextCommandDefault{cmd}{defn}\n\n\nor:\n\n\\ProvideTextCommandDefault{cmd}{defn}\n\n\nGive a default definition for cmd, for when that command is not defined in the encoding currently in force. This default should only use encodings known to be available.\n\nThis makes \\copyright available.\n\n\\DeclareTextCommandDefault{\\copyright}{\\textcircled{c}}\n\n\nIt uses only an encoding (OMS) that is always available.\n\nThe \\DeclareTextCommandDefault should not occur in the encoding definition files since those files should declare only commands for use when you select that encoding. It should instead be in a package.\n\nAs with the related non-default commands, the \\ProvideTextCommandDefault has exactly the same behavior as \\DeclareTextCommandDefault except that it does nothing if cmd is already defined (see \\DeclareTextCommand & \\ProvideTextCommand). So, packages can use it to provide fallbacks that other packages can improve upon.\n\n#### 4.1.6 \\DeclareTextComposite\n\nSynopsis:\n\n\\DeclareTextComposite{cmd}{encoding}{simple_object}{slot}\n\n\nAccess an accented glyph directly, that is, without having to put an accent over a separate character.\n\nThis line from t1enc.def means that \\^o will cause LaTeX to typeset lowercase\u00a0\u2018o\u2019 by taking the character directly from location 224 in the font.\n\n\\DeclareTextComposite{\\^}{T1}{o}{244}\n\n\nSee fontenc package, for a list of common encodings. The simple_object should be a single character or a single command. The slot argument is usually a positive integer represented in decimal (although octal or hexadecimal are possible). Normally cmd has already been declared for this encoding, either with \\DeclareTextAccent or with a one-argument \\DeclareTextCommand. In t1enc.def, the above line follows the \\DeclareTextAccent{\\^}{T1}{2} command.\n\nNext: , Previous: , Up: fontenc package \u00a0 [Contents][Index]\n\n#### 4.1.7 \\DeclareTextCompositeCommand\n\nSynopsis:\n\n\\DeclareTextCompositeCommand{cmd}{encoding}{arg}{code}\n\n\nA more general version of \\DeclareTextComposite that runs arbitrary code with cmd.\n\nThis allows accents on \u2018i\u2019 to act like accents on dotless\u00a0i, \\i.\n\n\\DeclareTextCompositeCommand{\\'}{OT1}{i}{\\'\\i}\n\n\nSee fontenc package, for a list of common encodings. Normally cmd will have already been declared with \\DeclareTextAccent or as a one argument \\DeclareTextCommand.\n\nNext: , Previous: , Up: fontenc package \u00a0 [Contents][Index]\n\n#### 4.1.8 \\DeclareTextSymbol\n\nSynopsis:\n\n\\DeclareTextSymbol{cmd}{encoding}{slot}\n\n\nDefine a symbol in the encoding encoding at the location slot. Symbols defined in this way are for use in text, not mathematics.\n\nFor example, this line from t1enc.def declares the number of the glyph to use for \u00ab, the left guillemet.\n\n\\DeclareTextSymbol{\\guillemotleft}{T1}{19}\n\n\nThe command \\DeclareTextCommand{\\guillemotleft}{T1}{\\char 19} has the same effect but is slower (see \\DeclareTextCommand & \\ProvideTextCommand).\n\nSee fontenc package, for a list of common encodings. The slot can be specified in decimal, or octal (as in '023), or hexadecimal (as in \"13), although decimal has the advantage that single quote or double quote could be redefined by another package.\n\nIf cmd has already been defined then \\DeclareTextSymbol does not give an error but it does log the redefinition in the transcript file.\n\nNext: , Previous: , Up: fontenc package \u00a0 [Contents][Index]\n\n#### 4.1.9 \\DeclareTextSymbolDefault\n\nSynopsis:\n\n\\DeclareTextSymbolDefault{cmd}{encoding}\n\n\nIf there is an encoding-specific symbol command cmd but there is no associated \\DeclareTextSymbol for that encoding, then this command will pick up the slack, by saying to get the symbol as described for encoding.\n\nFor example, to declare that if the current encoding has no meaning for \\textdollar then use the one from OT1, declare this.\n\n\\DeclareTextSymbolDefault{\\textdollar}{OT1}\n\n\nThat is, this command is equivalent to this call (see \\UseTextSymbol & \\UseTextAccent).\n\n\\DeclareTextCommandDefault{cmd}\n{\\UseTextSymbol{encoding}{cmd}}\n\n\nNote that \\DeclareTextSymbolDefault can be used to define a default for any zero-argument fontenc command.\n\nNext: , Previous: , Up: fontenc package \u00a0 [Contents][Index]\n\n#### 4.1.10 \\LastDeclaredEncoding\n\nSynopsis:\n\n\\LastDeclaredEncoding\n\n\nGet the name of the most recently declared encoding. The \\DeclareFontEncoding command stores the name so that it can be retrieved with this command (see \\DeclareFontEncoding).\n\nThis relies on \\LastDeclaredEncoding rather than give the name of the encoding explicitly.\n\n\\DeclareFontEncoding{JH1}{}{}\n\\DeclareTextAccent{\\'}{\\LastDeclaredEncoding}{0}\n\n\nPrevious: , Up: fontenc package \u00a0 [Contents][Index]\n\n#### 4.1.11 \\UseTextSymbol & \\UseTextAccent\n\nSynopsis:\n\n\\UseTextSymbol{encoding}{cmd}\n\n\nor:\n\n\\UseTextAccent{encoding}{cmd}{text}\n\n\nUse a symbol or accent not from the current encoding.\n\nIn general, to use a fontenc command in an encoding where it is not defined, and if the command has no arguments, then you can use it like this:\n\n\\UseTextSymbol{OT1}{\\ss}\n\n\nwhich is equivalent to this (note the outer braces form a group, so LaTeX reverts back to the prior encoding after the \\ss):\n\n{\\fontencoding{OT1}\\selectfont\\ss}\n\n\nSimilarly, to use a fontenc command in an encoding where it is not defined, and if the command has one argument, you can use it like this:\n\n\\UseTextAccent{OT1}{\\'}{a}\n\n\nwhich is equivalent to this (again note the outer braces forming a group):\n\n{fontencoding{OT1}\\selectfont\\'{\\fontencoding{enc_in_use}\\selectfont a}}\n\n\nHere, enc_in_use is the encoding in force before this sequence of commands, so that \u2018a\u2019 is typeset using the current encoding and only the accent is taken from OT1.\n\nNext: , Previous: , Up: Fonts \u00a0 [Contents][Index]\n\n### 4.2 Font styles\n\nThe following type style commands are supported by LaTeX.\n\nIn the table below the listed commands, the \\text... commands, are used with an argument as in \\textit{text}. This is the preferred form. But shown after it in parenthesis is the corresponding declaration form, which is often useful. This form takes no arguments, as in {\\itshape text}. The scope of the declaration form lasts until the next type style command or the end of the current group. In addition, each has an environment form such as \\begin{itshape}...\\end{itshape}, which we\u2019ll describe further at the end of the section.\n\nThese commands, in any of the three forms, are cumulative; for instance you can get bold sans serif by saying either of \\sffamily\\bfseries or \\bfseries\\sffamily.\n\nOne advantage of these commands is that they automatically insert italic corrections if needed (see \\\/). Specifically, they insert the italic correction unless the following character is in the list \\nocorrlist, which by default consists of period and comma. To suppress the automatic insertion of italic correction, use \\nocorr at the start or end of the command argument, such as \\textit{\\nocorr text} or \\textsc{text \\nocorr}.\n\n\\textrm (\\rmfamily)\n\nRoman.\n\n\\textit (\\itshape)\n\nItalics.\n\n\\textmd (\\mdseries)\n\nMedium weight (default).\n\n\\textbf (\\bfseries)\n\nBoldface.\n\n\\textup (\\upshape)\n\nUpright (default).\n\n\\textsl (\\slshape)\n\nSlanted.\n\n\\textsf (\\sffamily)\n\nSans serif.\n\n\\textsc (\\scshape)\n\nSmall caps.\n\n\\texttt (\\ttfamily)\n\nTypewriter.\n\n\\textnormal (\\normalfont)\n\nMain document font.\n\nAlthough it also changes fonts, the \\emph{text} command is semantic, for text to be emphasized, and should not be used as a substitute for \\textit. For example, \\emph{start text \\emph{middle text} end text} will result in the start text and end text in italics, but middle text will be in roman.\n\nLaTeX also provides the following commands, which unconditionally switch to the given style, that is, are not cumulative. They are used as declarations: {\\cmd...} instead of \\cmd{...}.\n\n(The unconditional commands below are an older version of font switching. The earlier commands are an improvement in most circumstances. But sometimes an unconditional font switch is what is needed.)\n\n\\bf\n\nSwitch to bold face.\n\n\\cal\n\nSwitch to calligraphic letters for math.\n\n\\it\n\nItalics.\n\n\\rm\n\nRoman.\n\n\\sc\n\nSmall caps.\n\n\\sf\n\nSans serif.\n\n\\sl\n\nSlanted (oblique).\n\n\\tt\n\nTypewriter (monospace, fixed-width).\n\nThe \\em command is the unconditional version of \\emph.\n\nThe following commands are for use in math mode. They are not cumulative, so \\mathbf{\\mathit{symbol}} does not create a boldface and italic symbol; instead, it will just be in italics. This is because typically math symbols need consistent typographic treatment, regardless of the surrounding environment.\n\n\\mathrm\n\nRoman, for use in math mode.\n\n\\mathbf\n\nBoldface, for use in math mode.\n\n\\mathsf\n\nSans serif, for use in math mode.\n\n\\mathtt\n\nTypewriter, for use in math mode.\n\n\\mathit\n(\\mit)\n\nItalics, for use in math mode.\n\n\\mathnormal\n\nFor use in math mode, e.g., inside another type style declaration.\n\n\\mathcal\n\nCalligraphic letters, for use in math mode.\n\nIn addition, the command \\mathversion{bold} can be used for switching to bold letters and symbols in formulas. \\mathversion{normal} restores the default.\n\nFinally, the command \\oldstylenums{numerals} will typeset so-called \u201cold-style\u201d numerals, which have differing heights and depths (and sometimes widths) from the standard \u201clining\u201d numerals, which all have the same height as uppercase letters. LaTeX\u2019s default fonts support this, and will respect \\textbf (but not other styles; there are no italic old-style numerals in Computer Modern). Many other fonts have old-style numerals also; sometimes package options are provided to make them the default. FAQ entry: https:\/\/www.texfaq.org\/FAQ-osf.\n\nNext: , Previous: , Up: Fonts \u00a0 [Contents][Index]\n\n### 4.3 Font sizes\n\nThe following standard type size commands are supported by LaTeX. The table shows the command name and the corresponding actual font size used (in points) with the \u201810pt\u2019, \u201811pt\u2019, and \u201812pt\u2019 document size options, respectively (see Document class options).\n\nCommand10pt11pt12pt\n\\tiny566\n\\scriptsize788\n\\footnotesize8910\n\\small91010.95\n\\normalsize (default)1010.9512\n\\large121214.4\n\\Large14.414.417.28\n\\LARGE17.2817.2820.74\n\\huge20.7420.7424.88\n\\Huge24.8824.8824.88\n\nThe commands are listed here in declaration (not environment) form, since that is how they are typically used. For example.\n\n\\begin{quotation} \\small\nThe Tao that can be named is not the eternal Tao.\n\\end{quotation}\n\n\nHere, the scope of the \\small lasts until the end of the quotation environment. It would also end at the next type style command or the end of the current group, so you could enclose it in curly braces {\\small This text is typeset in the small font.}.\n\nAn environment form of each of these commands is also defined; for instance, \\begin{tiny}...\\end{tiny}. However, in practice this form can easily lead to unwanted spaces at the beginning and\/or end of the environment without careful consideration, so it\u2019s generally less error-prone to stick to the declaration form.\n\n(Aside: Technically, due to the way LaTeX defines \\begin and \\end, nearly every command that does not take an argument technically has an environment form. But in almost all cases, it would only cause confusion to use it. The reason for mentioning the environment form of the font size declarations specifically is that this particular use is not rare.)\n\nPrevious: , Up: Fonts \u00a0 [Contents][Index]\n\n### 4.4 Low-level font commands\n\nThese commands are primarily intended for writers of macros and packages. The commands listed here are only a subset of the available ones.\n\n\\fontencoding{encoding}\n\nSelect the font encoding, the encoding of the output font. There are a large number of valid encodings. The most common are OT1, Knuth\u2019s original encoding for Computer Modern (the default), and T1, also known as the Cork encoding, which has support for the accented characters used by the most widespread European languages (German, French, Italian, Polish and others), which allows TeX to hyphenate words containing accented letters. For more, see https:\/\/ctan.org\/pkg\/encguide.\n\n\\fontfamily{family}\n\nSelect the font family. The web page https:\/\/tug.org\/FontCatalogue\/ provides one way to browse through many of the fonts easily used with LaTeX. Here are examples of some common families.\n\n pag Avant Garde fvs Bitstream Vera Sans pbk Bookman bch Charter ccr Computer Concrete cmr Computer Modern cmss Computer Modern Sans Serif cmtt Computer Modern Typewriter pcr Courier phv Helvetica fi4 Inconsolata lmr Latin Modern lmss Latin Modern Sans lmtt Latin Modern Typewriter pnc New Century Schoolbook ppl Palatino ptm Times uncl Uncial put Utopia pzc Zapf Chancery\n\\fontseries{series}\n\nSelect the font series. A series combines a weight and a width. Typically, a font supports only a few of the possible combinations. Some common combined series values include:\n\n m Medium (normal) b Bold c Condensed bc Bold condensed bx Bold extended\n\nThe possible values for weight, individually, are:\n\n ul Ultra light el Extra light l Light sl Semi light m Medium (normal) sb Semi bold b Bold eb Extra bold ub Ultra bold\n\nThe possible values for width, individually, are (the meaning and relationship of these terms varies with individual typefaces):\n\n uc Ultra condensed ec Extra condensed c Condensed sc Semi condensed m Medium sx Semi expanded x Expanded ex Extra expanded ux Ultra expanded\n\nWhen forming the series string from the weight and width, drop the m that stands for medium weight or medium width, unless both weight and width are m, in which case use just one (\u2018m\u2019).\n\n\\fontshape{shape}\n\nSelect font shape. Valid shapes are:\n\n n Upright (normal) it Italic sl Slanted (oblique) sc Small caps ui Upright italics ol Outline\n\nThe two last shapes are not available for most font families, and small caps are often missing as well.\n\n\\fontsize{size}{skip}\n\nSet the font size and the line spacing. The unit of both parameters defaults to points (pt). The line spacing is the nominal vertical space between lines, baseline to baseline. It is stored in the parameter \\baselineskip. The default \\baselineskip for the Computer Modern typeface is 1.2 times the \\fontsize. Changing \\baselineskip directly is inadvisable since its value is reset every time a size change happens; instead use \\baselinestretch. (see \\baselineskip & \\baselinestretch).\n\n\\linespread{factor}\n\nEquivalent to \\renewcommand{\\baselinestretch}{factor}, and therefore must be followed by \\selectfont to have any effect. Best specified in the preamble, or use the setspace package, as just described.\n\n\\selectfont\n\nThe effects of the font commands described above do not happen until \\selectfont is called, as in \\fontfamily{familyname}\\selectfont. It is often useful to put this in a macro:\n\\newcommand*{\\myfont}{\\fontfamily{familyname}\\selectfont}\n\n\\usefont{enc}{family}{series}{shape}\n\nThe same as invoking \\fontencoding, \\fontfamily, \\fontseries and \\fontshape with the given parameters, followed by \\selectfont. For example:\n\n\\usefont{ot1}{cmr}{m}{n}\n\n\nNext: , Previous: , Up: Top \u00a0 [Contents][Index]\n\n## 5 Layout\n\nCommands for controlling the general page layout.\n\nNext: , Up: Layout \u00a0 [Contents][Index]\n\n### 5.1 \\onecolumn\n\nSynopsis:\n\n\\onecolumn\n\n\nStart a new page and produce single-column output. If the document is given the class option onecolumn then this is the default behavior (see Document class options). This command is fragile (see \\protect).\n\nNext: , Previous: , Up: Layout \u00a0 [Contents][Index]\n\n### 5.2 \\twocolumn\n\nSynopses:\n\n\\twocolumn\n\\twocolumn[prelim one column text]\n\n\nStart a new page and produce two-column output. If the document is given the class option twocolumn then this is the default (see Document class options). This command is fragile (see \\protect).\n\nIf the optional prelim one column text argument is present, it is typeset in one-column mode before the two-column typesetting starts.\n\nThese parameters control typesetting in two-column output:\n\n\\columnsep\n\nThe distance between columns. The default is 35pt. Change it with a command such as \\setlength{\\columnsep}{40pt}. You must change it before the two column mode starts; in the preamble is a good place.\n\n\\columnseprule\n\nThe width of the rule between columns. The default is 0pt, meaning that there is no rule. Otherwise, the rule appears halfway between the two columns. Change it with a command such as \\setlength{\\columnseprule}{0.4pt}, before the two-column mode starts.\n\n\\columnwidth\n\nThe width of a single column. In one-column mode this is equal to \\textwidth. In two-column mode by default LaTeX sets the width of each of the two columns, \\columnwidth, to be half of \\textwidth minus \\columnsep.\n\nIn a two-column document, the starred environments table* and figure* are two columns wide, whereas the unstarred environments table and figure take up only one column (see figure and see table). LaTeX places starred floats at the top of a page. The following parameters control float behavior of two-column output.\n\n\\dbltopfraction\n\nThe maximum fraction at the top of a two-column page that may be occupied by two-column wide floats. The default is 0.7, meaning that the height of a table* or figure* environment must not exceed 0.7\\textheight. If the height of your starred float environment exceeds this then you can take one of the following actions to prevent it from floating all the way to the back of the document:\n\n\u2022 Use the [tp] location specifier to tell LaTeX to try to put the bulky float on a page by itself, as well as at the top of a page.\n\u2022 Use the [t!] location specifier to override the effect of \\dbltopfraction for this particular float.\n\u2022 Increase the value of \\dbltopfraction to a suitably large number, to avoid going to float pages so soon.\n\nYou can redefine it, as with \\renewcommand{\\dbltopfraction}{0.9}.\n\n\\dblfloatpagefraction\n\nFor a float page of two-column wide floats, this is the minimum fraction that must be occupied by floats, limiting the amount of blank space. LaTeX\u2019s default is 0.5. Change it with \\renewcommand.\n\n\\dblfloatsep\n\nOn a float page of two-column wide floats, this length is the distance between floats, at both the top and bottom of the page. The default is 12pt plus2pt minus2pt for a document set at 10pt or 11pt, and 14pt plus2pt minus4pt for a document set at 12pt.\n\n\\dbltextfloatsep\n\nThis length is the distance between a multi-column float at the top or bottom of a page and the main text. The default is 20pt plus2pt minus4pt.\n\n\\dbltopnumber\n\nOn a float page of two-column wide floats, this counter gives the maximum number of floats allowed at the top of the page. The LaTeX default is 2.\n\nThis example uses \\twocolumn\u2019s optional argument of to create a title that spans the two-column article:\n\n\\documentclass[twocolumn]{article}\n\\newcommand{\\authormark}[1]{\\textsuperscript{#1}}\n\\begin{document}\n\\twocolumn[{% inside this optional argument goes one-column text\n\\centering\n\n(Aside: The construct $$math$$ from Plain\u00a0TeX is sometimes mistakenly used as a synonym for displaymath. It is not a synonym, and is not officially supported in LaTeX at all; doesn\u2019t support the fleqn option (see Document class options), has different vertical spacing, and doesn\u2019t perform consistency checks.)\n\nThe output from this example is centered and alone on its line.\n\n\\begin{displaymath}\n\\int_1^2 x^2\\,dx=7\/3\n\\end{displaymath}\n\n\nAlso, the integral sign is larger than the inline version $$\\int_1^2 x^2\\,dx=7\/3$$ produces.\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.6 document\n\nThe document environment encloses the entire body of a document. It is required in every LaTeX document. See Starting and ending.\n\nNext: , Up: document \u00a0 [Contents][Index]\n\n#### 8.6.1 \\AtBeginDocument\n\nSynopsis:\n\n\\AtBeginDocument{code}\n\n\nSave code and execute it when \\begin{document} is executed, at the very end of the preamble. The code is executed after the font selection tables have been set up, so the normal font for the document is the current font. However, the code is executed as part of the preamble so you cannot do any typesetting with it.\n\nYou can issue this command more than once; the successive code lines will be executed in the order that you gave them.\n\nPrevious: , Up: document \u00a0 [Contents][Index]\n\n#### 8.6.2 \\AtEndDocument\n\nSynopsis:\n\n\\AtEndDocument{code}\n\n\nSave code and execute it near the end of the document. Specifically, it is executed when \\end{document} is executed, before the final page is finished and before any leftover floating environments are processed. If you want some of the code to be executed after these two processes then include a \\clearpage at the appropriate point in code.\n\nYou can issue this command more than once; the successive code lines will be executed in the order that you gave them.\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.7 enumerate\n\nSynopsis:\n\n\\begin{enumerate}\n\\item[optional label of first item] text of first item\n\\item[optional label of second item] text of second item\n...\n\\end{enumerate}\n\n\nEnvironment to produce a numbered list of items. The format of the label numbering depends on the nesting level of this environment; see below. The default top-level numbering is \u20181.\u2019, \u20182.\u2019, etc. Each enumerate list environment must have at least one item; having none causes the LaTeX error \u2018Something's wrong--perhaps a missing \\item\u2019.\n\nThis example gives the first two finishers in the 1908 Olympic marathon. As a top-level list the labels would come out as \u20181.\u2019 and \u20182.\u2019.\n\n\\begin{enumerate}\n\\item Johnny Hayes (USA)\n\\item Charles Hefferon (RSA)\n\\end{enumerate}\n\n\nStart list items with the \\item command (see \\item). If you give \\item an optional argument by following it with square brackets, as in \\item[Interstitial label], then the next item will continue the interrupted sequence (see \\item). That is, you will get labels like \u20181.\u2019, then \u2018Interstitial label\u2019, then \u20182.\u2019. Following the \\item is optional text, which may contain multiple paragraphs.\n\nEnumerations may be nested within other enumerate environments, or within any paragraph-making environment such as itemize (see itemize), up to four levels deep. This gives LaTeX\u2019s default for the format at each nesting level, where 1 is the top level, the outermost level.\n\n1. arabic number followed by a period: \u20181.\u2019, \u20182.\u2019,\u00a0\u2026\n2. lowercase letter inside parentheses: \u2018(a)\u2019, \u2018(b)\u2019\u00a0\u2026\n3. lowercase roman numeral followed by a period: \u2018i.\u2019, \u2018ii.\u2019,\u00a0\u2026\n4. uppercase letter followed by a period: \u2018A.\u2019, \u2018B.\u2019,\u00a0\u2026\n\nThe enumerate environment uses the counters \\enumi through \\enumiv (see Counters).\n\nFor other major LaTeX labeled list environments, see description and itemize. For information about list layout parameters, including the default values, and for information about customizing list layout, see list. The package enumitem is useful for customizing lists.\n\n\n\\renewcommand{\\labelenumi}{\\textbf{\\Alph{enumi}}}\n\\begin{enumerate}\n\\item Shows as boldface A\n\\item Shows as boldface B\n\\end{enumerate}\n\n\nFor a list of counter-labeling commands see \\alph \\Alph \\arabic \\roman \\Roman \\fnsymbol.\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.8 eqnarray\n\nThe eqnarray environment is obsolete. It has infelicities, including spacing that is inconsistent with other mathematics elements. (See \u201cAvoid eqnarray!\u201d by Lars Madsen https:\/\/tug.org\/TUGboat\/tb33-1\/tb103madsen.pdf). New documents should include the amsmath package and use the displayed mathematics environments provided there, such as the align environment. We include a description only for completeness and for working with old documents.\n\nSynopsis:\n\n\\begin{eqnarray}\nfirst formula left &first formula middle &first formula right \\\\\n...\n\\end{eqnarray}\n\n\nor\n\n\\begin{eqnarray*}\nfirst formula left &first formula middle &first formula right \\\\\n...\n\\end{eqnarray*}\n\n\nDisplay a sequence of equations or inequalities. The left and right sides are typeset in display mode, while the middle is typeset in text mode.\n\nIt is similar to a three-column array environment, with items within a row separated by an ampersand\u00a0(&), and with rows separated by double backslash\u00a0 \\\\). The starred form of line break (\\\\*) can also be used to separate equations, and will disallow a page break there (see \\\\).\n\nThe unstarred form eqnarray places an equation number on every line (using the equation counter), unless that line contains a \\nonumber command. The starred form eqnarray* omits equation numbering, while otherwise being the same.\n\nThe command \\lefteqn is used for splitting long formulas across lines. It typesets its argument in display style flush left in a box of zero width.\n\nThis example shows three lines. The first two lines make an inequality, while the third line has not entry on the left side.\n\n\\begin{eqnarray*}\n\\lefteqn{x_1+x_2+\\cdots+x_n} \\\\\n&\\leq &y_1+y_2+\\cdots+y_n \\\\\n&= &z+y_3+\\cdots+y_n\n\\end{eqnarray*}\n\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.9 equation\n\nSynopsis:\n\n\nmathematical text\n\n\n\nThe same as a displaymath environment (see displaymath) except that LaTeX puts an equation number flush to the right margin. The equation number is generated using the equation counter.\n\nYou should have no blank lines between and , or LaTeX will tell you that there is a missing dollar sign.\n\nThe package amsmath package has extensive displayed equation facilities. New documents should include this package.\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.10 figure\n\nSynopsis:\n\n\\begin{figure}[placement]\nfigure body\n\\caption[loftitle]{title} % optional\n\\label{label} % optional\n\\end{figure}\n\n\nor:\n\n\\begin{figure*}[placement]\nfigure body\n\\caption[loftitle]{title} % optional\n\\label{label} % optional\n\\end{figure*}\n\n\nFigures are for material that is not part of the normal text. An example is material that you cannot have split between two pages, such as a graphic. Because of this, LaTeX does not typeset figures in sequence with normal text but instead \u201cfloats\u201d them to a convenient place, such as the top of a following page (see Floats).\n\nThe figure body can consist of imported graphics (see Graphics), or text, LaTeX commands, etc. It is typeset in a parbox of width \\textwidth.\n\nThe possible values of placement are h for \u2018here\u2019, t for \u2018top\u2019, b for \u2018bottom\u2019, and p for \u2018on a separate page of floats\u2019. For the effect of these options on the float placement algorithm, see Floats.\n\nThe starred form figure* is used when a document is in double-column mode (see \\twocolumn). It produces a figure that spans both columns, at the top of the page. To add the possibility of placing at a page bottom see the discussion of placement b in Floats.\n\nThe label is optional; it is used for cross references (see Cross references). The optional \\caption command specifies caption text for the figure. By default it is numbered. If loftitle is present, it is used in the list of figures instead of title (see Table of contents etc.).\n\nThis example makes a figure out of a graphic. LaTeX will place that graphic and its caption at the top of a page or, if it is pushed to the end of the document, on a page of floats.\n\n\\usepackage{graphicx} % in preamble\n...\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{CTANlion.png}\n\\caption{The CTAN lion, by Duane Bibby}\n\\end{figure}\n\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.11 filecontents\n\nSynopsis:\n\n\\begin{filecontents}[option]{filename}\ntext\n\\end{filecontents}\n\n\nor\n\n\\begin{filecontents*}[option]{filename}\ntext\n\\end{filecontents*}\n\n\nCreate a file named filename in the current directory (or the output directory, if specified; see output directory) and write text to it. By default, an existing file is not overwritten.\n\nThe unstarred version of the environment filecontents prefixes the content of the created file with a header of TeX comments; see the example below. The starred version filecontents* does not include the header.\n\nThe possible options are:\n\nforce\noverwrite\n\nOverwrite an existing file.\n\nnoheader\n\nOmit the header. Equivalent to using filecontents*.\n\nnosearch\n\nOnly check the current directory (and the output directory, if specified) for an existing file, not the entire search path.\n\nThese options were added in a 2019 release of LaTeX.\n\nThis environment can be used anywhere in the preamble, although it often appears before the \\documentclass command. It is commonly used to create a .bib or other such data file from the main document source, to make the source file self-contained. Similarly, it can be used to create a custom style or class file, again making the source self-contained.\n\nFor example, this document:\n\n\\documentclass{article}\n\\begin{filecontents}{JH.sty}\n\\newcommand{\\myname}{Jim Hef{}feron}\n\\end{filecontents}\n\\usepackage{JH}\n\\begin{document}\nArticle by \\myname.\n\\end{document}\n\n\nproduces this file JH.sty:\n\n%% LaTeX2e file JH.sty'\n%% generated by the filecontents' environment\n%% from source test' on 2015\/10\/12.\n%%\n\\newcommand{\\myname}{Jim Hef{}feron}\n\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.12 flushleft\n\nSynopsis:\n\n\\begin{flushleft}\nline1 \\\\\nline2 \\\\\n...\n\\end{flushleft}\n\n\nAn environment that creates a paragraph whose lines are flush to the left-hand margin, and ragged right. If you have lines that are too long then LaTeX will linebreak them in a way that avoids hyphenation and stretching or shrinking interword spaces. To force a new line use a double backslash, \\\\. For the declaration form see\u00a0\\raggedright.\n\nThis creates a box of text that is at most 3 inches wide, with the text flush left and ragged right.\n\n\\noindent\\begin{minipage}{3in}\n\\begin{flushleft}\nA long sentence that will be broken by \\LaTeX{}\nat a convenient spot. \\\\\nAnd, a fresh line forced by the double backslash.\n\\end{flushleft}\n\\end{minipage}\n\n\nUp: flushleft \u00a0 [Contents][Index]\n\n#### 8.12.1 \\raggedright\n\nSynopses:\n\n{\\raggedright ... }\n\n\nor\n\n\\begin{environment} \\raggedright\n...\n\\end{environment}\n\n\nA declaration which causes lines to be flush to the left margin and ragged right. It can be used inside an environment such as quote or in a parbox. For the environment form see\u00a0flushleft.\n\nUnlike the flushleft environment, the \\raggedright command does not start a new paragraph; it only changes how LaTeX formats paragraph units. To affect a paragraph unit\u2019s format, the scope of the declaration must contain the blank line or \\end command that ends the paragraph unit.\n\nHere \\raggedright in each second column keeps LaTeX from doing very awkward typesetting to fit the text into the narrow column. Note that \\raggedright is inside the curly braces {...} to delimit its effect.\n\n\\begin{tabular}{rp{2in}}\nTeam alpha &{\\raggedright This team does all the real work.} \\\\\nTeam beta &{\\raggedright This team ensures that the water\ncooler is never empty.} \\\\\n\\end{tabular}\n\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.13 flushright\n\n\\begin{flushright}\nline1 \\\\\nline2 \\\\\n...\n\\end{flushright}\n\n\nAn environment that creates a paragraph whose lines are flush to the right-hand margin and ragged left. If you have lines that are too long to fit the margins then LaTeX will linebreak them in a way that avoids hyphenation and stretching or shrinking inter-word spaces. To force a new line use a double backslash, \\\\. For the declaration form see\u00a0\\raggedleft.\n\nFor an example related to this environment, see\u00a0flushleft, where one just have mutatis mutandis to replace flushleft by flushright.\n\nUp: flushright \u00a0 [Contents][Index]\n\n#### 8.13.1 \\raggedleft\n\nSynopses:\n\n{\\raggedleft ... }\n\n\nor\n\n\\begin{environment} \\raggedleft\n...\n\\end{environment}\n\n\nA declaration which causes lines to be flush to the right margin and ragged left. It can be used inside an environment such as quote or in a parbox. For the environment form see\u00a0flushright.\n\nUnlike the flushright environment, the \\raggedleft command does not start a new paragraph; it only changes how LaTeX formats paragraph units. To affect a paragraph unit\u2019s format, the scope of the declaration must contain the blank line or \\end command that ends the paragraph unit.\n\nFor an example related to this environment, see\u00a0\\raggedright, where one just have mutatis mutandis to replace \\raggedright by \\raggedleft.\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.14 itemize\n\nSynopsis:\n\n\\begin{itemize}\n\\item[optional label of first item] text of first item\n\\item[optional label of second item] text of second item\n...\n\\end{itemize}\n\n\nProduce an unordered list, sometimes called a bullet list. There must be at least one \\item within the environment; having none causes the LaTeX error \u2018Something's wrong--perhaps a missing \\item\u2019.\n\nThis gives a two-item list.\n\n\\begin{itemize}\n\\item Pencil and watercolor sketch by Cassandra\n\\item Rice portrait\n\\end{itemize}\n\n\nWith the default locale\u2014without loading e.g. babel package with another language than USenglish\u2014as a top-level list each label would come out as a bullet, \u2022. The format of the labeling depends on the nesting level; see below.\n\nStart list items with the \\item command (see \\item). If you give \\item an optional argument by following it with square brackets, as in \\item[Optional label], then by default Optional label will appear in bold and be flush right, so it could extend into the left margin. For labels that are flush left see the description environment. Following the \\item is the text of the item, which may be empty or contain multiple paragraphs.\n\nUnordered lists can be nested within one another, up to four levels deep. They can also be nested within other paragraph-making environments, such as enumerate (see enumerate).\n\nThe itemize environment uses the commands \\labelitemi through \\labelitemiv to produce the default label (note the convention of lowercase roman numerals at the end of the command names that signify the nesting level). These are the default marks at each level.\n\n1. \u2022 (bullet, from \\textbullet)\n2. -- (bold en-dash, from \\normalfont\\bfseries\\textendash)\n3. * (asterisk, from \\textasteriskcentered)\n4. . (vertically centered dot, rendered here as a period, from \\textperiodcentered)\n\nChange the labels with \\renewcommand. For instance, this makes the first level use diamonds.\n\n\\renewcommand{\\labelitemi}{$\\diamond$}\n\n\nThe distance between the left margin of the enclosing environment and the left margin of the itemize list is determined by the parameters \\leftmargini through \\leftmarginvi. (This also uses the convention of using lowercase roman numerals a the end of the command name to denote the nesting level.) The defaults are: 2.5em in level 1 (2em in two-column mode), 2.2em in level 2, 1.87em in level 3, and 1.7em in level 4, with smaller values for more deeply nested levels.\n\nFor other major LaTeX labeled list environments, see description and enumerate. The itemize, enumerate and description environment use the same list layout parameters. For a description, including the default values, and for information about customizing list layout, see list. The package enumitem is useful for customizing lists.\n\nThis example greatly reduces the margin space for outermost itemized lists.\n\n\\setlength{\\leftmargini}{1.25em} % default 2.5em\n\n\nEspecially for lists with short items, it may be desirable to elide space between items. Here is an example defining an itemize* environment with no extra spacing between items, or between paragraphs within a single item (\\parskip is not list-specific, see \\parindent & \\parskip):\n\n\\newenvironment{itemize*}%\n{\\begin{itemize}%\n\\setlength{\\itemsep}{0pt}%\n\\setlength{\\parsep}{0pt}}%\n\\setlength{\\parskip}{0pt}}%\n{\\end{itemize}}\n\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.15 letter environment: writing letters\n\nThis environment is used for creating letters. See Letters.\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.16 list\n\nSynopsis:\n\n\\begin{list}{labeling}{spacing}\n\\item[optional label of first item] text of first item\n\\item[optional label of second item] text of second item\n...\n\\end{list}\n\n\nAn environment for constructing lists.\n\nNote that this environment does not typically appear in the document body. Most lists created by LaTeX authors are the ones that come standard: the description, enumerate, and itemize environments (see description, enumerate, and itemize).\n\nInstead, the list environment is most often used in macros. For example, many standard LaTeX environments that do not immediately appear to be lists are in fact constructed using list, including quotation, quote, and center (see quotation & quote, see center).\n\nThis uses the list environment to define a new custom environment.\n\n\\newcounter{namedlistcounter} % number the items\n\\newenvironment{named}\n{\\begin{list}\n{Item~\\Roman{namedlistcounter}.} % labeling\n{\\usecounter{namedlistcounter} % set counter\n\\setlength{\\leftmargin}{3.5em}} % set spacing\n}\n{\\end{list}}\n\n\\begin{named}\n\\item Shows as Item~I.''\n\\item[Special label.] Shows as Special label.''\n\\item Shows as Item~II.''\n\\end{named}\n\n\nThe mandatory first argument labeling specifies the default labeling of list items. It can contain text and LaTeX commands, as above where it contains both \u2018Item\u2019 and \u2018\\Roman{\u2026}\u2019. LaTeX forms the label by putting the labeling argument in a box of width \\labelwidth. If the label is wider than that, the additional material extends to the right. When making an instance of a list you can override the default labeling by giving \\item an optional argument by including square braces and the text, as in the above \\item[Special label.]; see \\item.\n\nThe mandatory second argument spacing has a list of commands. This list can be empty. A command that can go in here is \\usecounter{countername} (see \\usecounter). Use this to tell LaTeX to number the items using the given counter. The counter will be reset to zero each time LaTeX enters the environment, and the counter is incremented by one each time LaTeX encounters an \\item that does not have an optional argument.\n\nAnother command that can go in spacing is \\makelabel, which constructs the label box. By default it puts the contents flush right. Its only argument is the label, which it typesets in LR mode (see Modes). One example of changing its definition is that to the above named example, before the definition of the environment add \\newcommand{\\namedmakelabel}[1]{\\textsc{#1}}, and between the \\setlength command and the parenthesis that closes the spacing argument also add \\let\\makelabel\\namedmakelabel. Then the labels will be typeset in small caps. Similarly, changing the second code line to \\let\\makelabel\\fbox puts the labels inside a framed box. Another example of the \\makelabel command is below, in the definition of the redlabel environment.\n\nAlso often in spacing are commands to redefine the spacing for the list. Below are the spacing parameters with their default values. (Default values for derived environments such as itemize can be different than the values shown here.) See also the figure that follows the list. Each is a length (see Lengths). The vertical spaces are normally rubber lengths, with plus and minus components, to give TeX flexibility in setting the page. Change each with a command such as \\setlength{itemsep}{2pt plus1pt minus1pt}. For some effects these lengths should be zero or negative.\n\n\\itemindent\n\nExtra horizontal space indentation, beyond leftmargin, of the first line each item. Its default value is 0pt.\n\n\\itemsep\n\nVertical space between items, beyond the \\parsep. The defaults for the first three levels in LaTeX\u2019s \u2018article\u2019, \u2018book\u2019, and \u2018report\u2019 classes at 10 point size are: 4pt plus2pt minus1pt, \\parsep (that is, 2pt plus1pt minus1pt), and \\topsep (that is, 2pt plus1pt minus1pt). The defaults at 11 point are: 4.5pt plus2pt minus1pt, \\parsep (that is, 2pt plus1pt minus1pt), and \\topsep (that is, 2pt plus1pt minus1pt). The defaults at 12 point are: 5pt plus2.5pt minus1pt, \\parsep (that is, 2.5pt plus1pt minus1pt), and \\topsep (that is, 2.5pt plus1pt minus1pt).\n\n\\labelsep\n\nHorizontal space between the label and text of an item. The default for LaTeX\u2019s \u2018article\u2019, \u2018book\u2019, and \u2018report\u2019 classes is 0.5em.\n\n\\labelwidth\n\nHorizontal width. The box containing the label is nominally this wide. If \\makelabel returns text that is wider than this then the first line of the item will be indented to make room for this extra material. If \\makelabel returns text of width less than or equal to \\labelwidth then LaTeX\u2019s default is that the label is typeset flush right in a box of this width.\n\nThe left edge of the label box is \\leftmargin+\\itemindent-\\labelsep-\\labelwidth from the left margin of the enclosing environment.\n\nThe default for LaTeX\u2019s \u2018article\u2019, \u2018book\u2019, and \u2018report\u2019 classes at the top level is \\leftmargini-\\labelsep, (which is 2em in one column mode and 1.5em in two column mode). At the second level it is \\leftmarginii-\\labelsep, and at the third level it is \\leftmarginiii-\\labelsep. These definitions make the label\u2019s left edge coincide with the left margin of the enclosing environment.\n\n\\leftmargin\n\nHorizontal space between the left margin of the enclosing environment (or the left margin of the page if this is a top-level list), and the left margin of this list. It must be non-negative.\n\nIn the standard LaTeX document classes, a top-level list has this set to the value of \\leftmargini, while a list that is nested inside a top-level list has this margin set to \\leftmarginii. More deeply nested lists get the values of \\leftmarginiii through \\leftmarginvi. (Nesting greater than level five generates the error message \u2018Too deeply nested\u2019.)\n\nThe defaults for the first three levels in LaTeX\u2019s \u2018article\u2019, \u2018book\u2019, and \u2018report\u2019 classes are: \\leftmargini is 2.5em (in two column mode, 2em), \\leftmarginii is 2.2em, and \\leftmarginiii is 1.87em.\n\n\\listparindent\n\nHorizontal space of additional line indentation, beyond \\leftmargin, for second and subsequent paragraphs within a list item. A negative value makes this an \u201coutdent\u201d. Its default value is 0pt.\n\n\\parsep\n\nVertical space between paragraphs within an item. The defaults for the first three levels in LaTeX\u2019s \u2018article\u2019, \u2018book\u2019, and \u2018report\u2019 classes at 10 point size are: 4pt plus2pt minus1pt, 2pt plus1pt minus1pt, and 0pt. The defaults at 11 point size are: 4.5pt plus2pt minus1pt, 2pt plus1pt minus1pt, and 0pt. The defaults at 12 point size are: 5pt plus2.5pt minus1pt, 2.5pt plus1pt minus1pt, and 0pt.\n\n\\partopsep\n\nVertical space added, beyond \\topsep+\\parskip, to the top and bottom of the entire environment if the list instance is preceded by a blank line. (A blank line in the LaTeX source before the list changes spacing at both the top and bottom of the list; whether the line following the list is blank does not matter.)\n\nThe defaults for the first three levels in LaTeX\u2019s \u2018article\u2019, \u2018book\u2019, and \u2018report\u2019 classes at 10 point size are: 2pt plus1 minus1pt, 2pt plus1pt minus1pt, and 1pt plus0pt minus1pt. The defaults at 11 point are: 3pt plus1pt minus1pt, 3pt plus1pt minus1pt, and 1pt plus0pt minus1pt). The defaults at 12 point are: 3pt plus2pt minus3pt, 3pt plus2pt minus2pt, and 1pt plus0pt minus1pt.\n\n\\rightmargin\n\nHorizontal space between the right margin of the list and the right margin of the enclosing environment. Its default value is 0pt. It must be non-negative.\n\n\\topsep\n\nVertical space added to both the top and bottom of the list, in addition to \\parskip (see \\parindent & \\parskip). The defaults for the first three levels in LaTeX\u2019s \u2018article\u2019, \u2018book\u2019, and \u2018report\u2019 classes at 10 point size are: 8pt plus2pt minus4pt, 4pt plus2pt minus1pt, and 2pt plus1pt minus1pt. The defaults at 11 point are: 9pt plus3pt minus5pt, 4.5pt plus2pt minus1pt, and 2pt plus1pt minus1pt. The defaults at 12 point are: 10pt plus4pt minus6pt, 5pt plus2.5pt minus1pt, and 2.5pt plus1pt minus1pt.\n\nThis shows the horizontal and vertical distances.\n\nThe lengths shown are listed below. The key relationship is that the right edge of the bracket for h1 equals the right edge of the bracket for h4, so that the left edge of the label box is at h3+h4-(h0+h1).\n\nv0\n\n\\topsep + \\parskip if the list environment does not start a new paragraph, and \\topsep+\\parskip+\\partopsep if it does\n\nv1\n\n\\parsep\n\nv2\n\n\\itemsep+\\parsep\n\nv3\n\nSame as v0. (This space is affected by whether a blank line appears in the source above the environment; whether a blank line appears in the source below the environment does not matter.)\n\nh0\n\n\\labelwidth\n\nh1\n\n\\labelsep\n\nh2\n\n\\listparindent\n\nh3\n\n\\leftmargin\n\nh4\n\n\\itemindent\n\nh5\n\n\\rightmargin\n\nThe list\u2019s left and right margins, shown above as h3 and h5, are with respect to the ones provided by the surrounding environment, or with respect to the page margins for a top-level list. The line width used for typesetting the list items is \\linewidth (see Page layout parameters). For instance, set the list\u2019s left margin to be one quarter of the distance between the left and right margins of the enclosing environment with \\setlength{\\leftmargin}{0.25\\linewidth}.\n\nPage breaking in a list structure is controlled by the three parameters below. For each, the LaTeX default is -\\@lowpenalty, that is, -51. Because it is negative, it somewhat encourages a page break at each spot. Change it with, e.g., \\@beginparpenalty=9999; a value of 10000 prohibits a page break.\n\n\\@beginparpenalty\n\nThe page breaking penalty for breaking before the list (default -51).\n\n\\@itempenalty\n\nThe page breaking penalty for breaking before a list item (default -51).\n\n\\@endparpenalty\n\nThe page breaking penalty for breaking after a list (default -51).\n\nThe package enumitem is useful for customizing lists.\n\nThis example has the labels in red. They are numbered, and the left edge of the label lines up with the left edge of the item text. See \\usecounter.\n\n\\usepackage{color}\n\\newcounter{cnt}\n\\newcommand{\\makeredlabel}[1]{\\textcolor{red}{#1.}}\n\\newenvironment{redlabel}\n{\\begin{list}\n{\\arabic{cnt}}\n{\\usecounter{cnt}\n\\setlength{\\labelwidth}{0em}\n\\setlength{\\labelsep}{0.5em}\n\\setlength{\\leftmargin}{1.5em}\n\\setlength{\\itemindent}{0.5em} % equals \\labelwidth+\\labelsep\n\\let\\makelabel=\\makeredlabel\n}\n}\n{\\end{list}}\n\n\nNext: , Up: list \u00a0 [Contents][Index]\n\n#### 8.16.1 \\item: An entry in a list\n\nSynopsis:\n\n\\item text of item\n\n\nor\n\n\\item[optional-label] text of item\n\n\nAn entry in a list. The entries are prefixed by a label, whose default depends on the list type.\n\nBecause the optional label is surrounded by square brackets \u2018[...]\u2019, if you have an item whose text starts with [, you have to hide the bracket inside curly braces, as in: \\item {[} is an open square bracket; otherwise, LaTeX will think it marks the start of an optional label.\n\nSimilarly, if the item does have the optional label and you need a close square bracket inside that label, you must hide it in the same way: \\item[Close square bracket, {]}]. See LaTeX command syntax.\n\nIn this example the enumerate list has two items that use the default label and one that uses the optional label.\n\n\\begin{enumerate}\n\\item Moe\n\\item[sometimes] Shemp\n\\item Larry\n\\end{enumerate}\n\n\nThe first item is labelled \u20181.\u2019, the second item is labelled \u2018sometimes\u2019, and the third item is labelled \u20182.\u2019. Because of the optional label in the second item, the third item is not labelled\u00a0\u20183.\u2019.\n\nPrevious: , Up: list \u00a0 [Contents][Index]\n\n#### 8.16.2 trivlist: A restricted form of list\n\nSynopsis:\n\n\\begin{trivlist}\n...\n\\end{trivlist}\n\n\nA restricted version of the list environment, in which margins are not indented and an \\item without an optional argument produces no text. It is most often used in macros, to define an environment where the \\item command is part of the environment\u2019s definition. For instance, the center environment is defined essentially like this:\n\n\\newenvironment{center}\n{\\begin{trivlist}\\centering\\item\\relax}\n{\\end{trivlist}}\n\n\nUsing trivlist in this way allows the macro to inherit some common code: combining vertical space of two adjacent environments; detecting whether the text following the environment should be considered a new paragraph or a continuation of the previous one; adjusting the left and right margins for possible nested list environments.\n\nSpecifically, trivlist uses the current values of the list parameters (see list), except that \\parsep is set to the value of \\parskip, and \\leftmargin, \\labelwidth, and \\itemindent are set to zero.\n\nThis example outputs the items as two paragraphs, except that (by default) they have no paragraph indent and are vertically separated.\n\n\\begin{trivlist}\n\\item The \\textit{Surprise} is not old; no one would call her old.\n\\item She has a bluff bow, lovely lines.\n\\end{trivlist}\n\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.17 math\n\nSynopsis:\n\n\\begin{math}\nmath\n\\end{math}\n\n\nThe math environment inserts given math material within the running text. $$...$$ and $...$ are synonyms. See Math formulas.\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.18 minipage\n\nSynopses:\n\n\\begin{minipage}{width}\ncontents\n\\end{minipage}\n\n\nor\n\n\\begin{minipage}[position][height][inner-pos]{width}\ncontents\n\\end{minipage}\n\n\nPut contents into a box that is width wide. This is like a small version of a page; it can contain its own footnotes, itemized lists, etc. (There are some restrictions, including that it cannot have floats.) This box will not be broken across pages. So minipage is similar to \\parbox (see \\parbox) but here you can have paragraphs.\n\nThis example will be 3\u00a0inches wide, and has two paragraphs.\n\n\\begin{minipage}{3in}\nStephen Kleene was a founder of the Theory of Computation.\n\nHe was a student of Church, wrote three influential texts,\nwas President of the Association for Symbolic Logic,\nand won the National Medal of Science.\n\\end{minipage}\n\n\nSee below for a discussion of the paragraph indent inside a minipage.\n\nThe required argument width is a rigid length (see Lengths). It gives the width of the box into which contents are typeset.\n\nThere are three optional arguments, position, height, and inner-pos. You need not include all three. For example, get the default position and set the height with \\begin{minipage}[c][2.54cm]{\\columnwidth} contents \\end{minipage}. (Get the natural height with an empty argument, [].)\n\nThe optional argument position governs how the minipage vertically aligns with the surrounding material.\n\nc\n\n(synonym m) Default. Positions the minipage so its vertical center lines up with the center of the adjacent text line.\n\nt\n\nMatch the top line in the minipage with the baseline of the surrounding text (plain TeX\u2019s \\vtop).\n\nb\n\nMatch the bottom line in the minipage with the baseline of the surrounding text (plain TeX\u2019s \\vbox).\n\nTo see the effects of these, contrast running this\n\n---\\begin{minipage}[c]{0.25in}\nfirst\\\\ second\\\\ third\n\\end{minipage}\n\n\nwith the results of changing c to b or\u00a0t.\n\nThe optional argument height is a rigid length (see Lengths). It sets the height of the minipage. You can enter any value larger than, or equal to, or smaller than the minipage\u2019s natural height and LaTeX will not give an error or warning. You can also set it to a height of zero or a negative value.\n\nThe final optional argument inner-pos controls the placement of contents inside the box. These are the possible values are (the default is the value of position).\n\nt\n\nPlace contents at the top of the box.\n\nc\n\nPlace it in the vertical center.\n\nb\n\nPlace it at the box bottom.\n\ns\n\nStretch contents out vertically; it must contain vertically stretchable space.\n\nThe inner-pos argument makes sense when the height option is set to a value larger than the minipage\u2019s natural height. To see the effect of the options, run this example with the various choices in place of b.\n\nText before\n\\begin{center}\n---\\begin{minipage}[c][3in][b]{0.25\\textwidth}\nfirst\\\\ second\\\\ third\n\\end{minipage}\n\\end{center}\nText after\n\n\nBy default paragraphs are not indented in a minipage. Change that with a command such as \\setlength{\\parindent}{1pc} at the start of contents.\n\nFootnotes in a minipage environment are handled in a way that is particularly useful for putting footnotes in figures or tables. A \\footnote or \\footnotetext command puts the footnote at the bottom of the minipage instead of at the bottom of the page, and it uses the \\mpfootnote counter instead of the ordinary footnote counter (see Counters).\n\nThis puts the footnote at the bottom of the table, not the bottom of the page.\n\n\\begin{center} % center the minipage on the line\n\\begin{minipage}{2.5in}\n\\begin{center} % center the table inside the minipage\n\\begin{tabular}{ll}\n\\textsc{Monarch} &\\textsc{Reign} \\\\ \\hline\nElizabeth II &63 years\\footnote{to date} \\\\\nVictoria &63 years \\\\\nGeorge III &59 years\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\end{center}\n\n\nIf you nest minipages then there is an oddness when using footnotes. Footnotes appear at the bottom of the text ended by the next \\end{minipage} which may not be their logical place.\n\nThis puts a table containing data side by side with a map graphic. They are vertically centered.\n\n% siunitx to have the S column specifier,\n% which aligns numbers on their decimal point.\n\\usepackage{siunitx}\n\\newcommand*{\\vcenteredhbox}[1]{\\begin{tabular}{@{}c@{}}#1\\end{tabular}}\n...\n\\begin{center}\n\\vcenteredhbox{\\includegraphics[width=0.3\\textwidth]{nyc.png}}\n\\hspace{0.1\\textwidth}\n\\begin{minipage}{0.5\\textwidth}\n\\begin{tabular}{r|S}\n% \\multicolumn to remove vertical bar between column headers\n\\multicolumn{1}{r}{Borough} &\n% braces to prevent siunitx from misinterpreting the\n% period as a decimal separator\n{Pop. (million)} \\\\ \\hline\nThe Bronx &1.5 \\\\\nBrooklyn &2.6 \\\\\nManhattan &1.6 \\\\\nQueens &2.3 \\\\\nStaten Island &0.5\n\\end{tabular}\n\\end{minipage}\n\\end{center}\n\n\nNext: , Previous: , Up: Environments \u00a0 [Contents][Index]\n\n### 8.19 picture\n\nSynopses:\n\n\\begin{picture}(width,height)\npicture command\n\\end{picture}\n\n\nor\n\n\\begin{picture}(width,height)(xoffset,yoffset)\npicture command\n\\end{picture}\n\n\nWhere there may be any number of picture command\u2019s.\n\nAn environment to create simple pictures containing lines, arrows, boxes, circles, and text. This environment is not obsolete, but new documents typically use much more powerful graphics creation systems, such as TikZ, PSTricks, MetaPost, or Asymptote. None of these are covered in this document; see CTAN.\n\nTo start, here\u2019s an example showing the parallelogram law for adding vectors.\n\n\\setlength{\\unitlength}{1cm}\n\\begin{picture}(6,6) % picture box will be 6cm wide by 6cm tall\n\\put(0,0){\\vector(2,1){4}} % for every 2 over this vector goes 1 up\n\\put(2,1){\\makebox(0,0)[l]{\\ first leg}}\n\\put(4,2){\\vector(1,2){2}}\n\\put(5,4){\\makebox(0,0)[l]{\\ second leg}}\n\\put(0,0){\\vector(1,1){6}}\n\\put(3,3){\\makebox(0,0)[r]{sum\\ }}\n\\end{picture}\n\n\nThe picture environment has one required argument, a pair of positive real numbers (width,height). Multiply these by the value \\unitlength to get the nominal size of the output, i.e. the space that LaTeX reserves on the output page. This nominal size need not be how large the picture really is; LaTeX will draw things from the picture outside the picture\u2019s box.\n\nThis environment also has an optional argument (xoffset,yoffset). It is used to shift the origin. Unlike most optional arguments, this one is not contained in square brackets. As with the required argument, it consists of a pair of two real numbers, but these may also be negative or null. Multiply these by \\unitlength to get the coordinates of the point at the lower-left corner of the picture.\n\nFor example, if \\unitlength has been set to 1mm, the command\n\n\\begin{picture}(100,200)(10,20)\n\n\nproduces a box of width 100 millimeters and height 200 millimeters. The picture\u2019s origin is the point (10mm,20mm) and so the lower-left corner is there, and the upper-right corner is at (110mm,220mm). When you first draw a picture you typically omit the optional argument, leaving the origin at the lower-left corner. If you then want to modify your picture by shifting everything, you can just add the appropriate optional argument.\n\nEach picture command tells LaTeX where to put something by providing its position. A position is a pair such as (2.4,-5) giving the x- and y-coordinates. A coordinate is a not a length, it is a real number (it may have a decimal point or a minus sign). It specifies a length in multiples of the unit length \\unitlength, so if \\unitlength has been set to 1cm, then the coordinate 2.54 specifies a length of 2.54 centimeters.\n\nLaTeX\u2019s default for \\unitlength is 1pt. It is a rigid length (see Lengths). Change it with the \\setlength command (see \\setlength). Make this change only outside of a picture environment.\n\nThe picture environment supports using standard arithmetic expressions as well as numbers.\n\nCoordinates are given with respect to an origin, which is by default at the lower-left corner of the picture. Note that when a position appears as an argument, as with \\put(1,2){...}, it is not enclosed in braces since the parentheses serve to delimit the argument. Also, unlike in some computer graphics systems, larger y-coordinates are further up the page, for example, y = 1 is above y = 0.\n\nThere are four ways to put things in a picture: \\put, \\multiput, \\qbezier, and \\graphpaper. The most often used is \\put. This\n\n\\put(11.3,-0.3){...}\n\n\nplaces the object with its reference point at coordinates (11.3,-0.3). The reference points for various objects will be described below. The \\put command creates an LR box (see Modes). Anything that can go in an \\mbox (see \\mbox & \\makebox) can go in the text argument of the \\put command. The reference point will be the lower left corner of the box. In this picture\n\n\\setlength{\\unitlength}{1cm}\n...\\begin{picture}(1,1)\n\\put(0,0){\\line(1,0){1}}\n\\put(0,0){\\line(1,1){1}}\n\\end{picture}\n\n\nthe three dots are just slightly left of the point of the angle formed by the two lines. (Also, \\line(1,1){1} does not call for a line of length one; rather the line has a change in the x coordinate of 1.)\n\nThe \\multiput, qbezier, and graphpaper commands are described below.\n\nYou can also use this environment to place arbitrary material at an exact location. For example:\n\n\\usepackage{color,graphicx} % in preamble\n...\n\\begin{center}\n\\setlength{\\unitlength}{\\textwidth}\n\\begin{picture}(1,1) % leave space, \\textwidth wide and tall\n\\put(0,0){\\includegraphics[width=\\textwidth]{desertedisland.jpg}}\n\\put(0.25,0.35){\\textcolor{red}{X Treasure here}}\n\\end{picture}\n\\end{center}\n\n\nThe red\u00a0X will be precisely a quarter of the \\textwidth from the left margin, and 0.35\\textwidth up from the bottom of the picture. Another example of this usage is to put similar code in the page header to get repeat material on each of a document\u2019s pages.\n\nNext: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.1 \\put\n\nSynopsis:\n\n\\put(xcoord,ycoord){content}\n\n\nPlace content at the coordinate (xcoord,ycoord). See the discussion of coordinates and \\unitlength in picture. The content is processed in LR mode (see Modes) so it cannot contain line breaks.\n\nThis includes the text into the picture.\n\n\\put(4.5,2.5){Apply the \\textit{unpoke} move}\n\n\nThe reference point, the location (4.5,2.5), is the lower left of the text, at the bottom left of the \u2018A\u2019.\n\nNext: , Previous: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.2 \\multiput\n\nSynopsis:\n\n\\multiput(x,y)(delta_x,delta_y){num-copies}{obj}\n\n\nCopy obj a total of num-copies times, with an increment of delta_x,delta_y. The obj first appears at position (x,y), then at (x+\\delta_x,y+\\delta_y), and so on.\n\nThis draws a simple grid with every fifth line in bold (see also \\graphpaper).\n\n\\begin{picture}(10,10)\n\\linethickness{0.05mm}\n\\multiput(0,0)(1,0){10}{\\line(0,1){10}}\n\\multiput(0,0)(0,1){10}{\\line(1,0){10}}\n\\linethickness{0.5mm}\n\\multiput(0,0)(5,0){3}{\\line(0,1){10}}\n\\multiput(0,0)(0,5){3}{\\line(1,0){10}}\n\\end{picture}\n\n\nNext: , Previous: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.3 \\qbezier\n\nSynopsis:\n\n\\qbezier(x1,y1)(x2,y2)(x3,y3)\n\\qbezier[num](x1,y1)(x2,y2)(x3,y3)\n\n\nDraw a quadratic Bezier curve whose control points are given by the three required arguments (x1,y1), (x2,y2), and (x3,y3). That is, the curve runs from (x1,y1) to (x3,y3), is quadratic, and is such that the tangent line at (x1,y1) passes through (x2,y2), as does the tangent line at (x3,y3).\n\nThis draws a curve from the coordinate (1,1) to (1,0).\n\n\\qbezier(1,1)(1.25,0.75)(1,0)\n\n\nThe curve\u2019s tangent line at (1,1) contains (1.25,0.75), as does the curve\u2019s tangent line at (1,0).\n\nThe optional argument num gives the number of calculated intermediate points. The default is to draw a smooth curve whose maximum number of points is \\qbeziermax (change this value with \\renewcommand).\n\nThis draws a rectangle with a wavy top, using \\qbezier for that curve.\n\n\\begin{picture}(8,4)\n\\put(0,0){\\vector(1,0){8}} % x axis\n\\put(0,0){\\vector(0,1){4}} % y axis\n\\put(2,0){\\line(0,1){3}} % left side\n\\put(4,0){\\line(0,1){3.5}} % right side\n\\qbezier(2,3)(2.5,2.9)(3,3.25)\n\\qbezier(3,3.25)(3.5,3.6)(4,3.5)\n\\thicklines % below here, lines are twice as thick\n\\put(2,3){\\line(4,1){2}}\n\\put(4.5,2.5){\\framebox{Trapezoidal Rule}}\n\\end{picture}\n\n\nNext: , Previous: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.4 \\graphpaper\n\nSynopsis:\n\n\\graphpaper(x_init,y_init)(x_dimen,y_dimen)\n\\graphpaper[spacing](x_init,y_init)(x_dimen,y_dimen)\n\n\nDraw a coordinate grid. Requires the graphpap package. The grid\u2019s origin is (x_init,y_init). Grid lines come every spacing units (the default is 10). The grid extends x_dimen units to the right and y_dimen units up. All arguments must be positive integers.\n\nThis make a grid with seven vertical lines and eleven horizontal lines.\n\n\\usepackage{graphpap} % in preamble\n...\n\\begin{picture}(6,20) % in document body\n\\graphpaper[2](0,0)(12,20)\n\\end{picture}\n\n\nThe lines are numbered every ten units.\n\nNext: , Previous: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.5 \\line\n\nSynopsis:\n\n\\line(x_run,y_rise){travel}\n\n\nDraw a line. It slopes such that it vertically rises y_rise for every horizontal x_run. The travel is the total horizontal change\u2014it is not the length of the vector, it is the change in x. In the special case of vertical lines, where (x_run,y_rise)=(0,1), the travel gives the change in y.\n\nThis draws a line starting at coordinates (1,3).\n\n\\put(1,3){\\line(2,5){4}}\n\n\nFor every over 2, this line will go up 5. Because travel specifies that this goes over 4, it must go up\u00a010. Thus its endpoint is (1,3)+(4,10)=(5,13). In particular, note that travel=4 is not the length of the line, it is the change in x.\n\nThe arguments x_run and y_rise are integers that can be positive, negative, or zero. (If both are 0 then LaTeX treats the second as 1.) With \\put(x_init,y_init){\\line(x_run,y_rise){travel}}, if x_run is negative then the line\u2019s ending point has a first coordinate that is less than x_init. If y_rise is negative then the line\u2019s ending point has a second coordinate that is less than y_init.\n\nIf travel is negative then you get LaTeX Error: Bad \\line or \\vector argument.\n\nStandard LaTeX can only draw lines with a limited range of slopes because these lines are made by putting together line segments from pre-made fonts. The two numbers x_run and y_rise must have integer values from -6 through\u00a06. Also, they must be relatively prime, so that (x_run,y_rise) can be (2,1) but not (4,2) (if you choose the latter then instead of lines you get sequences of arrowheads; the solution is to switch to the former). To get lines of arbitrary slope and plenty of other shapes in a system like picture, see the package pict2e (https:\/\/ctan.org\/pkg\/pict2e). Another solution is to use a full-featured graphics system such as TikZ, PSTricks, MetaPost, or Asymptote.\n\nNext: , Previous: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.6 \\linethickness\n\nSynopsis:\n\n\\linethickness{dim}\n\n\nDeclares the thickness of subsequent horizontal and vertical lines in a picture to be dim, which must be a positive length (see Lengths). It differs from \\thinlines and \\thicklines in that it does not affect the thickness of slanted lines, circles, or ovals.\n\nNext: , Previous: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.7 \\thinlines\n\nDeclaration to set the thickness of subsequent lines, circles, and ovals in a picture environment to be 0.4pt. This is the default thickness, so this command is unnecessary unless the thickness has been changed with either \\linethickness or \\thicklines.\n\nNext: , Previous: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.8 \\thicklines\n\nDeclaration to set the thickness of subsequent lines, circles, and ovals in a picture environment to be 0.8pt. See also \\linethickness and \\thinlines. This command is illustrated in the Trapezoidal Rule example of picture.\n\nNext: , Previous: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.9 \\circle\n\nSynopsis:\n\n\\circle{diameter}\n\\circle*{diameter}\n\n\nProduces a circle with a diameter as close as possible to the specified one. The *\u00a0form produces a filled-in circle.\n\nThis draws a circle of radius 6, centered at (5,7).\n\n\\put(5,7){\\circle{6}}\n\n\nThe available radii for \\circle are, in points, the even numbers from 2 to 20, inclusive. For \\circle* they are all the integers from 1 to 15.\n\nNext: , Previous: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.10 \\oval\n\nSynopsis:\n\n\\oval(width,height)\n\\oval(width,height)[portion]\n\n\nProduce a rectangle with rounded corners. The optional argument portion allows you to produce only half or a quarter of the oval. For half an oval take portion to be one of these.\n\nt\n\ntop half\n\nb\n\nbottom half\n\nr\n\nright half\n\nl\n\nleft half\n\nProduce only one quarter of the oval by setting portion to tr, br, bl, or tl.\n\nThis draws the top half of an oval that is 3 wide and 7 tall.\n\n\\put(5,7){\\oval(3,7)[t]}\n\n\nThe (5,7) is the center of the entire oval, not just the center of the top half.\n\nThese shapes are not ellipses. They are rectangles whose corners are made with quarter circles. These circles have a maximum radius of 20pt (see \\circle for the sizes). Thus large ovals are just boxes with a small amount of corner rounding.\n\nNext: , Previous: , Up: picture \u00a0 [Contents][Index]\n\n#### 8.19.11 \\shortstack\n\nSynopsis:\n\n\\shortstack[position]{line 1 \\\\ ... }\n\n\nProduce a vertical stack of objects.\n\nThis labels the y axis.\n\n\\put(0,0){\\vector(1,0){4}} % x axis\n\\put(0,0){\\vector(0,1){2}} % y\n\\put(-0.25,2){\\makebox[0][r]{\\shortstack[r]{$y$\\\\ axis}}}\n\n\nFor a short stack, the reference point is the lower left of the stack. In the above example the \\mbox & \\makebox puts the stack flush right in a zero width box so in total the short stack sits slightly to the left of the y\u00a0axis.\n\nThe valid positions are:\n\nr\n\nMake objects flush right\n\nl\n\nMake objects flush left\n\nc\n\nCenter objects (default)\n\n\nThe two mathematics modes are similar, but there are some differences. One involves the placement of subscripts and superscripts; in display math mode they are further apart and in inline math mode they are closer together.\n\nSometimes you want the display math typographical treatment to happen in the inline math mode. For this, the \\displaystyle declaration forces the size and style of the formula to be that of displaymath. Thus $$\\displaystyle \\sum_{n=0}^\\infty x_n$$ will have the limits above and below the summation sign, not next to it. Another example is that\n\n\\begin{tabular}{r|cc}\n\\textsc{Name} &\\textsc{Series} &\\textsc{Sum} \\\\ \\hline\nArithmetic &$$a+(a+b)+(a+2b)+\\cdots+(a+(n-1)b)$$\n&$$na+(n-1)n\\cdot\\frac{b}{2}$$ \\\\\nGeometric &$$a+ab+ab^2+\\cdots+ab^{n-1}$$\n&$$\\displaystyle a\\cdot\\frac{1-b^n}{1-b}$$ \\\\\n\\end{tabular}\n\n\nbecause it has no \\displaystyle, the \u2018Arithmetic\u2019 line\u2019s fraction will be scrunched. But, because of its \\displaystyle, the \u2018Geometric\u2019 line\u2019s fraction will be easy to read, with characters the same size as in the rest of the line.\n\nThe American Mathematical Society has made freely available a set of packages that greatly expand your options for writing mathematics, amsmath and amssymb (also be aware of the mathtools package that is an extension to, and loads, amsmath). New documents that will have mathematical text should use these packages. Descriptions of these packages is outside the scope of this document; see their documentation on CTAN.\n\nNext: , Up: Math formulas \u00a0 [Contents][Index]\n\n### 16.1 Subscripts & superscripts\n\nSynopsis (in math mode or display math mode), one of:\n\nbase^exp\nbase^{exp}\n\n\nor, one of:\n\nbase_exp\nbase_{exp}\n\n\nMake exp appear as a superscript of base (with the caret character,\u00a0^) or a subscript (with underscore,\u00a0_).\n\nIn this example the 0\u2019s and 1\u2019s are subscripts while the 2\u2019s are superscripts.\n\n$$(x_0+x_1)^2 \\leq (x_0)^2+(x_1)^2$$\n\n\nTo have the subscript or superscript contain more than one character, surround the expression with curly braces, as in e^{-2x}. This example\u2019s fourth line shows curly braces used to group an expression for the exponent.\n\n\\begin{displaymath}\n(3^3)^3=27^3=19\\,683\n3^{(3^3)}=3^{27}=7\\,625\\,597\\,484\\,987\n\\end{displaymath}\n\n\nLaTeX knows how to handle a superscript on a superscript, or a subscript on a subscript, or supers on subs, or subs on supers. So, expressions such as e^{x^2} and x_{i_0} give correct output. Note the use in those expressions of curly braces to give the base a determined exp. If you enter $$3^3^3$$ then you get \u2018Double superscript\u2019.\n\nLaTeX does the right thing when something has both a subscript and a superscript. In this example the integral has both. They come out in the correct place without any author intervention.\n\n\\begin{displaymath}\n\\int_{x=a}^b f'(x)\\,dx = f(b)-f(a)\n\\end{displaymath}\n\n\nNote the parentheses around x=a to make the entire expression a subscript.\n\nTo put a superscript or subscript before a symbol, use a construct like {}_t K^2. The empty curly braces {} give the subscript something to attach to and keeps it from accidentally attaching to a prior symbols.\n\nUsing the subscript or superscript character outside of math mode or display math mode, as in the expression x^2, will get you the error \u2018Missing $inserted\u2019. A common reason to want subscripts outside of a mathematics mode is to typeset chemical formulas. There are packages for that, such as mhchem; see CTAN. Next: , Previous: , Up: Math formulas [Contents][Index] ### 16.2 Math symbols LaTeX provides almost any mathematical or technical symbol that anyone uses. For example, if you include $\\pi$ in your source, you will get the pi symbol \u03c0. See the \u201cComprehensive LaTeX Symbol List\u201d package at https:\/\/ctan.org\/pkg\/comprehensive. Here is a list of commonly-used symbols. It is by no means exhaustive. Each symbol is described with a short phrase, and its symbol class, which determines the spacing around it, is given in parenthesis. Unless said otherwise, the commands for these symbols can be used only in math mode. To redefine a command so that it can be used whatever the current mode, see \\ensuremath. \\| \u2225 Parallel (relation). Synonym: \\parallel. \\aleph \u2135 Aleph, transfinite cardinal (ordinary). \\alpha \u03b1 Lowercase Greek letter alpha (ordinary). \\amalg \u2a3f Disjoint union (binary) \\angle \u2220 Geometric angle (ordinary). Similar: less-than sign < and angle bracket \\langle. \\approx \u2248 Almost equal to (relation). \\ast \u2217 Asterisk operator, convolution, six-pointed (binary). Synonym: *, which is often a superscript or subscript, as in the Kleene star. Similar: \\star, which is five-pointed, and is sometimes used as a general binary operation, and sometimes reserved for cross-correlation. \\asymp \u224d Asymptotically equivalent (relation). \\backslash \\ Backslash (ordinary). Similar: set minus \\setminus, and \\textbackslash for backslash outside of math mode. \\beta \u03b2 Lowercase Greek letter beta (ordinary). \\bigcap \u22c2 Variable-sized, or n-ary, intersection (operator). Similar: binary intersection \\cap. \\bigcirc \u26aa Circle, larger (binary). Similar: function composition \\circ. \\bigcup \u22c3 Variable-sized, or n-ary, union (operator). Similar: binary union \\cup. \\bigodot \u2a00 Variable-sized, or n-ary, circled dot operator (operator). \\bigoplus \u2a01 Variable-sized, or n-ary, circled plus operator (operator). \\bigotimes \u2a02 Variable-sized, or n-ary, circled times operator (operator). \\bigtriangledown \u25bd Variable-sized, or n-ary, open triangle pointing down (operator). \\bigtriangleup \u25b3 Variable-sized, or n-ary, open triangle pointing up (operator). \\bigsqcup \u2a06 Variable-sized, or n-ary, square union (operator). \\biguplus \u2a04 Variable-sized, or n-ary, union operator with a plus (operator). (Note that the name has only one p.) \\bigvee \u22c1 Variable-sized, or n-ary, logical-or (operator). \\bigwedge \u22c0 Variable-sized, or n-ary, logical-and (operator). \\bot \u22a5, Up tack, bottom, least element of a partially ordered set, or a contradiction (ordinary). See also \\top. \\bowtie \u22c8 Natural join of two relations (relation). \\Box \u25a1 Modal operator for necessity; square open box (ordinary). Not available in plain TeX. In LaTeX you need to load the amssymb package. \\bullet \u2022 Bullet (binary). Similar: multiplication dot \\cdot. \\cap \u2229 Intersection of two sets (binary). Similar: variable-sized operator \\bigcap. \\cdot \u22c5 Multiplication (binary). Similar: Bullet dot \\bullet. \\chi \u03c7 Lowercase Greek chi (ordinary). \\circ \u2218 Function composition, ring operator (binary). Similar: variable-sized operator \\bigcirc. \\clubsuit \u2663 Club card suit (ordinary). \\complement \u2201, Set complement, used as a superscript as in $S^\\complement$ (ordinary). Not available in plain TeX. In LaTeX you need to load the amssymb package. Also used: $S^{\\mathsf{c}}$ or $\\bar{S}$. \\cong \u2245 Congruent (relation). \\coprod \u2210 Coproduct (operator). \\cup \u222a Union of two sets (binary). Similar: variable-sized operator \\bigcup. \\dagger \u2020 Dagger relation (binary). \\dashv \u22a3 Dash with vertical, reversed turnstile (relation). Similar: turnstile \\vdash. \\ddagger \u2021 Double dagger relation (binary). \\Delta \u0394 Greek uppercase delta, used for increment (ordinary). \\delta \u03b4 Greek lowercase delta (ordinary). \\Diamond \u25c7 Large diamond operator (ordinary). Not available in plain TeX. In LaTeX you need to load the amssymb package. \\diamond \u22c4 Diamond operator (binary). Similar: large diamond \\Diamond, circle bullet \\bullet. \\diamondsuit \u2662 Diamond card suit (ordinary). \\div \u00f7 Division sign (binary). \\doteq \u2250 Approaches the limit (relation). Similar: geometrically equal to \\Doteq. \\downarrow \u2193 Down arrow, converges (relation). Similar: \\Downarrow double line down arrow. \\Downarrow \u21d3 Double line down arrow (relation). Similar: \\downarrow single line down arrow. \\ell \u2113 Lowercase cursive letter l (ordinary). \\emptyset \u2205 Empty set symbol (ordinary). The variant form is \\varnothing. \\epsilon \u03f5 Lowercase lunate epsilon (ordinary). Similar to Greek text letter. More widely used in mathematics is the script small letter epsilon \\varepsilon \u03b5. Related: the set membership relation \\in \u2208. \\equiv \u2261 Equivalence (relation). \\eta \u03b7 Lowercase Greek letter (ordinary). \\exists \u2203 Existential quantifier (ordinary). \\flat \u266d Musical flat (ordinary). \\forall \u2200 Universal quantifier (ordinary). \\frown \u2322 Downward curving arc (ordinary). \\Gamma \u0393 uppercase Greek letter (ordinary). \\gamma \u03b3 Lowercase Greek letter (ordinary). \\ge \u2265 Greater than or equal to (relation). This is a synonym for \\geq. \\geq \u2265 Greater than or equal to (relation). This is a synonym for \\ge. \\gets \u2190 Is assigned the value (relation). Synonym: \\leftarrow. \\gg \u226b Much greater than (relation). Similar: much less than \\ll. \\hbar \u210f Planck constant over two pi (ordinary). \\heartsuit \u2661 Heart card suit (ordinary). \\hookleftarrow \u21a9 Hooked left arrow (relation). \\hookrightarrow \u21aa Hooked right arrow (relation). \\iff \u27f7 If and only if (relation). It is \\Longleftrightarrow with a \\thickmuskip on either side. \\Im \u2111 Imaginary part (ordinary). See: real part \\Re. \\imath Dotless i; used when you are putting an accent on an i (see Math accents). \\in \u2208 Set element (relation). See also: lowercase lunate epsilon \\epsilon\u03f5 and small letter script epsilon \\varepsilon. \\infty \u221e Infinity (ordinary). \\int \u222b Integral (operator). \\iota \u03b9 Lowercase Greek letter (ordinary). \\Join \u2a1d Condensed bowtie symbol (relation). Not available in Plain TeX. \\jmath Dotless j; used when you are putting an accent on a j (see Math accents). \\kappa \u03ba Lowercase Greek letter (ordinary). \\Lambda \u039b uppercase Greek letter (ordinary). \\lambda \u03bb Lowercase Greek letter (ordinary). \\land \u2227 Logical and (binary). Synonym: \\wedge. See also logical or \\lor. \\langle \u27e8 Left angle, or sequence, bracket (opening). Similar: less-than <. Matches \\rangle. \\lbrace { Left curly brace (opening). Synonym: \\{. Matches \\rbrace. \\lbrack [ Left square bracket (opening). Synonym: [. Matches \\rbrack. \\lceil \u2308 Left ceiling bracket, like a square bracket but with the bottom shaved off (opening). Matches \\rceil. \\le \u2264 Less than or equal to (relation). This is a synonym for \\leq. \\leadsto \u21dd Squiggly right arrow (relation). To get this symbol outside of math mode you can put \\newcommand*{\\Leadsto}{\\ensuremath{\\leadsto}} in the preamble and then use \\Leadsto instead. \\Leftarrow \u21d0 Is implied by, double-line left arrow (relation). Similar: single-line left arrow \\leftarrow. \\leftarrow \u2190 Single-line left arrow (relation). Synonym: \\gets. Similar: double-line left arrow \\Leftarrow. \\leftharpoondown \u21bd Single-line left harpoon, barb under bar (relation). \\leftharpoonup \u21bc Single-line left harpoon, barb over bar (relation). \\Leftrightarrow \u21d4 Bi-implication; double-line double-headed arrow (relation). Similar: single-line double headed arrow \\leftrightarrow. \\leftrightarrow \u2194 Single-line double-headed arrow (relation). Similar: double-line double headed arrow \\Leftrightarrow. \\leq \u2264 Less than or equal to (relation). This is a synonym for \\le. \\lfloor \u230a Left floor bracket (opening). Matches: \\floor. \\lhd \u25c1 Arrowhead, that is, triangle, pointing left (binary). For the normal subgroup symbol you should load amssymb and use \\vartriangleleft (which is a relation and so gives better spacing). \\ll \u226a Much less than (relation). Similar: much greater than \\gg. \\lnot \u00ac Logical negation (ordinary). Synonym: \\neg. \\longleftarrow \u27f5 Long single-line left arrow (relation). Similar: long double-line left arrow \\Longleftarrow. \\longleftrightarrow \u27f7 Long single-line double-headed arrow (relation). Similar: long double-line double-headed arrow \\Longleftrightarrow. \\longmapsto \u27fc Long single-line left arrow starting with vertical bar (relation). Similar: shorter version \\mapsto. \\longrightarrow \u27f6 Long single-line right arrow (relation). Similar: long double-line right arrow \\Longrightarrow. \\lor \u2228 Logical or (binary). Synonym: \\vee. See also logical and \\land. \\mapsto \u21a6 Single-line left arrow starting with vertical bar (relation). Similar: longer version \\longmapsto. \\mho \u2127 Conductance, half-circle rotated capital omega (ordinary). \\mid \u2223 Single-line vertical bar (relation). A typical use of \\mid is for a set \\{\\, x \\mid x\\geq 5 \\,\\}. Similar: \\vert and | produce the same single-line vertical bar symbol but without any spacing (they fall in class ordinary) and you should not use them as relations but instead only as ordinals, i.e., footnote symbols. For absolute value, see the entry for \\vert and for norm see the entry for \\Vert. \\models \u22a8 Entails, or satisfies; double turnstile, short double dash (relation). Similar: long double dash \\vDash. \\mp \u2213 Minus or plus (relation). \\mu \u03bc Lowercase Greek letter (ordinary). \\nabla \u2207 Hamilton\u2019s del, or differential, operator (ordinary). \\natural \u266e Musical natural notation (ordinary). \\ne \u2260 Not equal (relation). Synonym: \\neq. \\nearrow \u2197 North-east arrow (relation). \\neg \u00ac Logical negation (ordinary). Synonym: \\lnot. Sometimes instead used for negation: \\sim. \\neq \u2260 Not equal (relation). Synonym: \\ne. \\ni \u220b Reflected membership epsilon; has the member (relation). Synonym: \\owns. Similar: is a member of \\in. \\not Long solidus, or slash, used to overstrike a following operator (relation). Many negated operators are available that don\u2019t require \\not, particularly with the amssymb package. For example, \\notin is typographically preferable to \\not\\in. \\notin \u2209 Not an element of (relation). Similar: not subset of \\nsubseteq. \\nu \u03bd Lowercase Greek letter (ordinary). \\nwarrow \u2196 North-west arrow (relation). \\odot \u2299 Dot inside a circle (binary). Similar: variable-sized operator \\bigodot. \\oint \u222e Contour integral, integral with circle in the middle (operator). \\Omega \u03a9 uppercase Greek letter (ordinary). \\omega \u03c9 Lowercase Greek letter (ordinary). \\ominus \u2296 Minus sign, or dash, inside a circle (binary). \\oplus \u2295 Plus sign inside a circle (binary). Similar: variable-sized operator \\bigoplus. \\oslash \u2298 Solidus, or slash, inside a circle (binary). \\otimes \u2297 Times sign, or cross, inside a circle (binary). Similar: variable-sized operator \\bigotimes. \\owns \u220b Reflected membership epsilon; has the member (relation). Synonym: \\ni. Similar: is a member of \\in. \\parallel \u2225 Parallel (relation). Synonym: \\|. \\partial \u2202 Partial differential (ordinary). \\perp \u27c2 Perpendicular (relation). Similar: \\bot uses the same glyph but the spacing is different because it is in the class ordinary. \\Phi \u03a6 Uppercase Greek letter (ordinary). \\phi \u03d5 Lowercase Greek letter (ordinary). The variant form is \\varphi \u03c6. \\Pi \u03a0 uppercase Greek letter (ordinary). \\pi \u03c0 Lowercase Greek letter (ordinary). The variant form is \\varpi \u03d6. \\pm \u00b1 Plus or minus (binary). \\prec \u227a Precedes (relation). Similar: less than <. \\preceq \u2aaf Precedes or equals (relation). Similar: less than or equals \\leq. \\prime \u2032 Prime, or minute in a time expression (ordinary). Typically used as a superscript: $f^\\prime$; $f^\\prime$ and $f'$ produce the same result. An advantage of the second is that $f'''$ produces the desired symbol, that is, the same result as $f^{\\prime\\prime\\prime}$, but uses rather less typing. You can only use \\prime in math mode. Using the right single quote ' in text mode produces a different character (apostrophe). \\prod \u220f Product (operator). \\propto \u221d Is proportional to (relation) \\Psi \u03a8 uppercase Greek letter (ordinary). \\psi \u03c8 Lowercase Greek letter (ordinary). \\rangle \u27e9 Right angle, or sequence, bracket (closing). Similar: greater than >. Matches:\\langle. \\rbrace } Right curly brace (closing). Synonym: \\}. Matches \\lbrace. \\rbrack ] Right square bracket (closing). Synonym: ]. Matches \\lbrack. \\rceil \u2309 Right ceiling bracket (closing). Matches \\lceil. \\Re \u211c Real part, real numbers, cursive capital R (ordinary). Related: double-line, or blackboard bold, R \\mathbb{R}; to access this, load the amsfonts package. \\restriction \u21be, Restriction of a function (relation). Synonym: \\upharpoonright. Not available in plain TeX. In LaTeX you need to load the amssymb package. \\revemptyset \u29b0, Reversed empty set symbol (ordinary). Related: \\varnothing. Not available in plain TeX. In LaTeX you need to load the stix package. \\rfloor \u230b Right floor bracket, a right square bracket with the top cut off (closing). Matches \\lfloor. \\rhd \u25c1 Arrowhead, that is, triangle, pointing right (binary). For the normal subgroup symbol you should instead load amssymb and use \\vartriangleright (which is a relation and so gives better spacing). \\rho \u03c1 Lowercase Greek letter (ordinary). The variant form is \\varrho \u03f1. \\Rightarrow \u21d2 Implies, right-pointing double line arrow (relation). Similar: right single-line arrow \\rightarrow. \\rightarrow \u2192 Right-pointing single line arrow (relation). Synonym: \\to. Similar: right double line arrow \\Rightarrow. \\rightharpoondown \u21c1 Right-pointing harpoon with barb below the line (relation). \\rightharpoonup \u21c0 Right-pointing harpoon with barb above the line (relation). \\rightleftharpoons \u21cc Right harpoon up above left harpoon down (relation). \\searrow \u2198 Arrow pointing southeast (relation). \\setminus \u29f5 Set difference, reverse solidus or reverse slash, like \\ (binary). Similar: backslash \\backslash and also \\textbackslash outside of math mode. \\sharp \u266f Musical sharp (ordinary). \\Sigma \u03a3 uppercase Greek letter (ordinary). \\sigma \u03c3 Lowercase Greek letter (ordinary). The variant form is \\varsigma \u03c2. \\sim \u223c Similar, in a relation (relation). \\simeq \u2243 Similar or equal to, in a relation (relation). \\smallint \u222b Integral sign that does not change to a larger size in a display (operator). \\smile \u2323 Upward curving arc, smile (ordinary). \\spadesuit \u2660 Spade card suit (ordinary). \\sqcap \u2293 Square intersection symbol (binary). Similar: intersection cap. \\sqcup \u2294 Square union symbol (binary). Similar: union cup. Related: variable-sized operator \\bigsqcup. \\sqsubset \u228f, Square subset symbol (relation). Similar: subset \\subset. Not available in plain TeX. In LaTeX you need to load the amssymb package. \\sqsubseteq \u2291 Square subset or equal symbol (binary). Similar: subset or equal to \\subseteq. \\sqsupset \u2290, Square superset symbol (relation). Similar: superset \\supset. Not available in plain TeX. In LaTeX you need to load the amssymb package. \\sqsupseteq \u2292 Square superset or equal symbol (binary). Similar: superset or equal \\supseteq. \\star \u22c6 Five-pointed star, sometimes used as a general binary operation but sometimes reserved for cross-correlation (binary). Similar: the synonyms asterisk * and \\ast, which are six-pointed, and more often appear as a superscript or subscript, as with the Kleene star. \\subset \u2282 Subset (occasionally, is implied by) (relation). \\subseteq \u2286 Subset or equal to (relation). \\succ \u227b Comes after, succeeds (relation). Similar: is less than >. \\succeq \u2ab0 Succeeds or is equal to (relation). Similar: less than or equal to \\leq. \\sum \u2211 Summation (operator). Similar: Greek capital sigma \\Sigma. \\supset \u2283 Superset (relation). \\supseteq \u2287 Superset or equal to (relation). \\surd \u221a Radical symbol (ordinary). The LaTeX command \\sqrt{...} typesets the square root of the argument, with a bar that extends to cover the argument. \\swarrow \u2199 Southwest-pointing arrow (relation). \\tau \u03c4 Lowercase Greek letter (ordinary). \\theta \u03b8 Lowercase Greek letter (ordinary). The variant form is \\vartheta \u03d1. \\times \u00d7 Primary school multiplication sign (binary). See also \\cdot. \\to \u2192 Right-pointing single line arrow (relation). Synonym: \\rightarrow. \\top \u22a4 Top, greatest element of a partially ordered set (ordinary). See also \\bot. \\triangle \u25b3 Triangle (ordinary). \\triangleleft \u25c1 Not-filled triangle pointing left (binary). Similar: \\lhd. For the normal subgroup symbol you should load amssymb and use \\vartriangleleft (which is a relation and so gives better spacing). \\triangleright \u25b7 Not-filled triangle pointing right (binary). For the normal subgroup symbol you should instead load amssymb and use \\vartriangleright (which is a relation and so gives better spacing). \\unlhd \u22b4 Left-pointing not-filled underlined arrowhead, that is, triangle, with a line under (binary). For the normal subgroup symbol load amssymb and use \\vartrianglelefteq (which is a relation and so gives better spacing). \\unrhd \u22b5 Right-pointing not-filled underlined arrowhead, that is, triangle, with a line under (binary). For the normal subgroup symbol load amssymb and use \\vartrianglerighteq (which is a relation and so gives better spacing). \\Uparrow \u21d1 Double-line upward-pointing arrow (relation). Similar: single-line up-pointing arrow \\uparrow. \\uparrow \u2191 Single-line upward-pointing arrow, diverges (relation). Similar: double-line up-pointing arrow \\Uparrow. \\Updownarrow \u21d5 Double-line upward-and-downward-pointing arrow (relation). Similar: single-line upward-and-downward-pointing arrow \\updownarrow. \\updownarrow \u2195 Single-line upward-and-downward-pointing arrow (relation). Similar: double-line upward-and-downward-pointing arrow \\Updownarrow. \\upharpoonright \u21be, Up harpoon, with barb on right side (relation). Synonym: \\restriction. Not available in plain TeX. In LaTeX you need to load the amssymb package. \\uplus \u228e Multiset union, a union symbol with a plus symbol in the middle (binary). Similar: union \\cup. Related: variable-sized operator \\biguplus. \\Upsilon \u03a5 uppercase Greek letter (ordinary). \\upsilon \u03c5 Lowercase Greek letter (ordinary). \\varepsilon \u03b5 Small letter script epsilon (ordinary). This is more widely used in mathematics than the non-variant lunate epsilon form \\epsilon \u03f5. Related: set membership \\in. \\vanothing \u2205, Empty set symbol. Similar: \\emptyset. Related: \\revemptyset. Not available in plain TeX. In LaTeX you need to load the amssymb package. \\varphi \u03c6 Variant on the lowercase Greek letter (ordinary). The non-variant form is \\phi \u03d5. \\varpi \u03d6 Variant on the lowercase Greek letter (ordinary). The non-variant form is \\pi \u03c0. \\varrho \u03f1 Variant on the lowercase Greek letter (ordinary). The non-variant form is \\rho \u03c1. \\varsigma \u03c2 Variant on the lowercase Greek letter (ordinary). The non-variant form is \\sigma \u03c3. \\vartheta \u03d1 Variant on the lowercase Greek letter (ordinary). The non-variant form is \\theta \u03b8. \\vdash \u22a2 Provable; turnstile, vertical and a dash (relation). Similar: turnstile rotated a half-circle \\dashv. \\vee \u2228 Logical or; a downwards v shape (binary). Related: logical and \\wedge. Similar: variable-sized operator \\bigvee. \\Vert \u2016 Vertical double bar (ordinary). See Delimiters, for how to use the mathtools package to create flexibly-sized norm symbols. \\vert | Single line vertical bar (ordinary). For \u201csuch that\u201d, as in the definition of a set, use \\mid because it is a relation. See Delimiters, for how to use the mathtools package to create flexibly-sized absolute-value symbols. \\wedge \u2227 Logical and (binary). Synonym: \\land. See also logical or \\vee. Similar: variable-sized operator \\bigwedge. \\wp \u2118 Weierstrass p (ordinary). \\wr \u2240 Wreath product (binary). \\Xi \u039e uppercase Greek letter (ordinary). \\xi \u03be Lowercase Greek letter (ordinary). \\zeta \u03b6 Lowercase Greek letter (ordinary). The following symbols are most often used in plain text but LaTeX provides versions to use in mathematical text. \\mathdollar Dollar sign in math mode:$.\n\n\\mathparagraph\n\nParagraph sign (pilcrow) in math mode, \u00b6.\n\n\\mathsection\n\n\\mathsterling\n\n\\mathunderscore\n\nUnderscore in math mode: _.\n\nNext: , Up: Math symbols \u00a0 [Contents][Index]\n\n#### 16.2.1 Arrows\n\nThese are the arrows that come with standard LaTeX. The latexsym and amsfonts packages contain many more.\n\nSymbolCommand\n\\Downarrow\n\\downarrow\n\\hookleftarrow\n\\hookrightarrow\n\\leftarrow\n\\Leftarrow\n\\Leftrightarrow\n\\leftrightarrow\n\\longleftarrow\n\\Longleftarrow\n\\longleftrightarrow\n\\Longleftrightarrow\n\\longmapsto\n\\Longrightarrow\n\\longrightarrow\n\\mapsto\n\\nearrow\n\\nwarrow\n\\Rightarrow\n\\rightarrow, or \\to\n\\searrow\n\\swarrow\n\\uparrow\n\\Uparrow\n\\updownarrow\n\\Updownarrow\n\nAn example of the difference between \\to and \\mapsto is: $$f\\colon D\\to C$$ given by $$n\\mapsto n^2$$.\n\nFor commutative diagrams there are a number of packages, including tikz-cd and amscd.\n\nNext: , Previous: , Up: Math symbols \u00a0 [Contents][Index]\n\n#### 16.2.2 \\boldmath & \\unboldmath\n\nSynopsis (used in paragraph mode or LR mode):\n\n\\boldmath $$math$$\n\n\nor\n\n\\unboldmath $$math$$\n\n\nDeclarations to change the letters and symbols in math to be in a bold font, or to countermand that and bring back the regular (non-bold) default. They must be used when not in math mode or display math mode (see Modes). Both commands are fragile (see \\protect).\n\nIn this example each \\boldmath command takes place inside an \\mbox,\n\n\n\nIf you want a reserved character to be printed as itself, in the text body font, for all but the final three characters in that list simply put a backslash\u00a0\\ in front of the character. Thus, typing \\$1.23 will produce $1.23 in your output.\n\nAs to the last three characters, to get a tilde in the text body font use \\~{} (omitting the curly braces would result in the next character receiving a tilde accent). Similarly, to get a text body font circumflex use \\^{}. To get a backslash in the font of the text body, enter \\textbackslash{}.\n\nTo produce the reserved characters in a typewriter font use \\verb!! as below (the double backslash\u00a0\\\\ is only there to split the lines in the output).\n\n\\begin{center}\n\\# \\$\\% \\& \\{ \\} \\_ \\~{} \\^{} \\textbackslash \\\\ \\verb!#$ % & { } _ ~ ^ \\!\n\\end{center}\n\n\nNext: , Previous: , Up: Special insertions \u00a0 [Contents][Index]\n\n### 23.2 Upper and lower case\n\nSynopsis:\n\n\\uppercase{text}\n\\lowercase{text}\n\\MakeUppercase{text}\n\\MakeLowercase{text}\n\n\nChange the case of characters. The TeX primitive commands \\uppercase and \\lowercase are set up by default to work only with the 26 letters a\u2013z and A\u2013Z. The LaTeX commands \\MakeUppercase and \\MakeLowercase commands also change characters accessed by commands such as \\ae or \\aa. The commands \\MakeUppercase and \\MakeLowercase are robust but they have moving arguments (see \\protect).\n\nThese commands do not change the case of letters used in the name of a command within text. But they do change the case of every other Latin letter inside the argument text. Thus, \\MakeUppercase{Let $y=f(x)$} produces \u2018LET Y=F(X)\u2019. Another example is that the name of an environment will be changed, so that \\MakeUppercase{\\begin{tabular} ... \\end{tabular}} will produce an error because the first half is changed to \\begin{TABULAR}.\n\nLaTeX uses the same fixed table for changing case throughout a document, The table used is designed for the font encoding T1; this works well with the standard TeX fonts for all Latin alphabets but will cause problems when using other alphabets.\n\nTo change the case of text that results from a macro inside text you need to do expansion. Here the \\Schoolname produces \u2018COLLEGE OF MATHEMATICS\u2019.\n\n\\newcommand{\\schoolname}{College of Mathematics}\n\\newcommand{\\Schoolname}{\\expandafter\\MakeUppercase\n\\expandafter{\\schoolname}}\n\n\nThe textcase package brings some of the missing feature of the standard LaTeX commands \\MakeUppercase and \\MakeLowerCase.\n\nTo uppercase only the first letter of words, you can use the package mfirstuc.\n\nHandling all the casing rules specified by Unicode, e.g., for non-Latin scripts, is a much bigger job than anything envisioned in the original TeX and LaTeX. It has been implemented in the expl3 package as of 2020. The article \u201cCase changing: From TeX primitives to the Unicode algorithm\u201d, (Joseph Wright, TUGboat\u00a041:1, https:\/\/tug.org\/TUGboat\/tb41-1\/tb127wright-case.pdf), gives a good overview of the topic, past and present.\n\nNext: , Previous: , Up: Special insertions \u00a0 [Contents][Index]\n\n### 23.3 Symbols by font position\n\nYou can access any character of the current font using its number with the \\symbol command. For example, the visible space character used in the \\verb* command has the code decimal 32 in the standard Computer Modern typewriter font, so it can be typed as \\symbol{32}.\n\nYou can also specify numbers in octal (base 8) by using a ' prefix, or hexadecimal (base 16) with a \" prefix, so the visible space at 32 decimal could also be written as \\symbol{'40} or \\symbol{\"20}.\n\nNext: , Previous: , Up: Special insertions \u00a0 [Contents][Index]\n\n### 23.4 Text symbols\n\nLaTeX provides commands to generate a number of non-letter symbols in running text. Some of these, especially the more obscure ones, are not available in OT1. As of the LaTeX February 2020 release, all symbols are available by default; before that, it was necessary to use the textcomp package for some (technically, those in the TS1 font encoding).\n\n\\copyright\n\\textcopyright\n\n\\dag\n\n\u2020 The dagger symbol (in text).\n\n\\ddag\n\n\u2021 The double dagger symbol (in text).\n\n\\LaTeX\n\nThe LaTeX logo.\n\n\\LaTeXe\n\nThe LaTeX2e logo.\n\n\\guillemotleft (\u00ab)\n\\guillemotright (\u00bb)\n\\guilsinglleft (\u2039)\n\\guilsinglright (\u203a)\n\n\u00ab, \u00bb, \u2039, \u203a Double and single angle quotation marks, commonly used in French.\n\n\\ldots\n\\dots\n\\textellipsis\n\n\u2026 An ellipsis (three dots at the baseline): \\ldots and \\dots also work in math mode.\n\n\\lq\n\n\u2018 Left (opening) quote.\n\n\\P\n\\textparagraph\n\n\u00b6 Paragraph sign (pilcrow).\n\n\\pounds\n\\textsterling\n\n\u00a3 English pounds sterling.\n\n\\quotedblbase (\u201e)\n\\quotesinglbase (\u201a)\n\n\u201e and \u201a Double and single quotation marks on the baseline.\n\n\\rq\n\n\u2019 Right (closing) quote.\n\n\\S\n\\textsection\n\n\u00a7 Section sign.\n\n\\TeX\n\nThe TeX logo.\n\n\\textasciicircum\n\n^ ASCII circumflex.\n\n\\textasciitilde\n\n~ ASCII tilde.\n\n\\textasteriskcentered\n\n* Centered asterisk.\n\n\\textbackslash\n\n\\ Backslash.\n\n\\textbar\n\n| Vertical bar.\n\n\\textbardbl\n\n\u23f8 Double vertical bar.\n\n\\textbigcircle\n\n\u25ef, Big circle symbol.\n\n\\textbraceleft\n\n{ Left brace.\n\n\\textbraceright\n\n} Right brace.\n\n\\textbullet\n\n\u2022 Bullet.\n\n\\textcircled{letter}\n\n\u24b6, Circle around letter.\n\n\\textcompwordmark\n\\textcapitalcompwordmark\n\\textascendercompwordmark\n\nUsed to separate letters that would normally ligature. For example, f\\textcompwordmark i produces \u2018fi\u2019 without a ligature. This is most useful in non-English languages. The \\textcapitalcompwordmark form has the cap height of the font while the \\textascendercompwordmark form has the ascender height.\n\n\\textdagger\n\n\u2020 Dagger.\n\n\\textdaggerdbl\n\n\u2021 Double dagger.\n\n\\textdollar (or \\$)$ Dollar sign.\n\n\\textemdash (or ---)\n\n\u2014 Em-dash. Used for punctuation, usually similar to commas or parentheses, as in \u2018The playoffs---if you're lucky enough to make the playoffs---are more like a sprint.\u2019 Conventions for spacing around em-dashes vary widely.\n\n\\textendash (or --)\n\n\u2013 En-dash. Used for ranges, as in \u2018see pages 12--14\u2019.\n\n\\texteuro\n\nThe Euro currency symbol: \u20ac.\n\nFor an alternative glyph design, try the eurosym package; also, most fonts nowadays come with their own Euro symbol (Unicode U+20AC).\n\n\\textexclamdown (or !)\n\n\u00a1 Upside down exclamation point.\n\n\\textfiguredash\n\nDash used between numerals, Unicode U+2012. Defined in the June 2021 release of LaTeX. When used in pdfTeX, approximated by an en-dash; with a Unicode engine, either typesets the glyph if available in the current font, or writes the usual \u201cMissing character\u201d warning to the log file.\n\n\\textgreater\n\n> Greater than symbol.\n\n\\texthorizontalbar\n\nHorizontal bar character, Unicode U+2015. Defined in the June 2021 release of LaTeX. Behavior as with \\textfiguredash above; the pdfTeX approximation is an em-dash.\n\n\\textless\n\n< Less than symbol.\n\n\\textleftarrow\n\n\u2190, Left arrow.\n\n\\textnonbreakinghyphen\n\nNon-breaking hyphen character, Unicode U+2011. Defined in the June 2021 release of LaTeX. Behavior as with \\textfiguredash above; the pdfTeX approximation is a regular ASCII hyphen (with breaks disallowed after).\n\n\\textordfeminine\n\\textordmasculine\n\n\u00aa, \u00ba Feminine and masculine ordinal symbols.\n\n\\textperiodcentered\n\n\u00b7 Centered period.\n\n\\textquestiondown (or ?)\n\n\u00bf Upside down question mark.\n\n\\textquotedblleft (or )\n\n\u201c Double left quote.\n\n\\textquotedblright (or '')\n\n\u201d Double right quote.\n\n\\textquoteleft (or )\n\n\u2018 Single left quote.\n\n\\textquoteright (or ')\n\n\u2019 Single right quote.\n\n\\textquotesingle\n\n', Straight single quote. (From TS1 encoding.)\n\n\\textquotestraightbase\n\\textquotestraightdblbase\n\nSingle and double straight quotes on the baseline.\n\n\\textregistered\n\n\u00ae Registered symbol.\n\n\\textrightarrow\n\n\u2192, Right arrow.\n\n\\textthreequartersemdash\n\n\ufe58, \u201cThree-quarters\u201d em-dash, between en-dash and em-dash.\n\n\\texttrademark\n\n\\texttwelveudash\n\n\ufe58, \u201cTwo-thirds\u201d em-dash, between en-dash and em-dash.\n\n\\textunderscore\n\n_ Underscore.\n\n\\textvisiblespace\n\n\u2423, Visible space symbol.\n\nNext: , Previous: , Up: Special insertions \u00a0 [Contents][Index]\n\n### 23.5 Accents\n\nLaTeX has wide support for many of the world\u2019s scripts and languages, provided through the core babel package, which supports pdfLaTeX, XeLaTeX and LuaLaTeX. The polyglossia package provides similar support with the latter two engines.\n\nThis section does not cover that support. It only lists the core LaTeX commands for creating accented characters. The \\capital... commands shown here produce alternative forms for use with capital letters. These are not available with OT1.\n\nBelow, to make them easier to find, the accents are all illustrated with lowercase \u2018o\u2019.\n\nNote that \\i produces a dotless i, and \\j produces a dotless j. These are often used in place of their dotted counterparts when they are accented.\n\n\\\"\n\\capitaldieresis\n\n\u00f6 Umlaut (dieresis).\n\n\\'\n\\capitalacute\n\n\u00f3 Acute accent.\n\n\\.\n\n\u022f Dot accent.\n\n\\=\n\\capitalmacron\n\n\u014d Macron (overbar) accent.\n\n\\^\n\\capitalcircumflex\n\n\u00f4 Circumflex (hat) accent.\n\n\\\n\\capitalgrave\n\n\u00f2 Grave accent.\n\n\\~\n\\capitaltilde\n\n\u00f1 Tilde accent.\n\n\\b\n\no_ Bar accent underneath.\n\nRelated to this, \\underbar{text} produces a bar under text. The argument is always processed in LR mode (see Modes). The bar is always a fixed position under the baseline, thus crossing through descenders. See also \\underline in Math miscellany.\n\n\\c\n\\capitalcedilla\n\n\u00e7 Cedilla accent underneath.\n\n\\d\n\\capitaldotaccent\n\n\u1ecd Dot accent underneath.\n\n\\H\n\\capitalhungarumlaut\n\n\u0151 Long Hungarian umlaut accent.\n\n\\k\n\\capitalogonek\n\n\u01eb Ogonek. Not available in the OT1 encoding.\n\n\\r\n\\capitalring\n\no* Ring accent.\n\n\\t\n\\capitaltie\n\\newtie\n\\capitalnewtie\n\noo[ Tie-after accent. The \\newtie form is centered in its box.\n\n\\u\n\\capitalbreve\n\n\u014f Breve accent.\n\n\\v\n\\capitalcaron\n\n\u01d2 H\u00e1\u010dek (check, caron) accent.\n\nUp: Accents \u00a0 [Contents][Index]\n\n#### 23.5.1 \\accent\n\nSynopsis:\n\n\\accent number character\n\n\nA TeX primitive command used to generate accented characters from accent marks and letters. The accent mark is selected by number, a numeric argument, followed by a space and then a character argument constructs the accented character in the current font.\n\nThese are accented \u2018e\u2019 characters.\n\n\\accent18 e\n\\accent20 e\n\\accent21 e\n\\accent22 e\n\\accent23 e\n\n\nThe first is a grave, the second is breve, etc.\n\nThe position of the accent is determined by the font designer and so the outcome of \\accent use may differ between fonts. In LaTeX it is desirable to have glyphs for accented characters rather than building them using \\accent. Using glyphs that already contain the accented characters (as in T1 encoding) allows correct hyphenation whereas \\accent disables hyphenation (specifically with OT1 font encoding where accented glyphs are absent).\n\nThere can be an optional font change between number and character. Note also that this command sets the \\spacefactor to 1000 (see \\spacefactor).\n\nAn unavoidable characteristic of some Cyrillic letters and the majority of accented Cyrillic letters is that they must be assembled from multiple elements (accents, modifiers, etc.) while \\accent provides for a single accent mark and a single letter combination. There are also cases where accents must appear between letters that \\accent does not support. Still other cases exist where the letters I and J have dots above their lowercase counterparts that conflict with dotted accent marks. The use of \\accent in these cases will not work as it cannot analyze upper\/lower case.\n\nNext: , Previous: , Up: Special insertions \u00a0 [Contents][Index]\n\nHere are the basic LaTeX commands for inserting letters beyond A\u2013Z that extend the Latin alphabet, used primarily in languages other than English.\n\n\\aa\n\\AA\n\n\u00e5 and \u00c5.\n\n\\ae\n\\AE\n\n\u00e6 and \u00c6.\n\n\\dh\n\\DH\n\nIcelandic letter eth: \u00f0 and \u00d0. Not available with OT1 encoding, you need the fontenc package to select an alternate font encoding, such as T1.\n\n\\dj\n\\DJ\n\nCrossed d and D, a.k.a. capital and small letter d with stroke. Not available with OT1 encoding, you need the fontenc package to select an alternate font encoding, such as T1.\n\n\\ij\n\\IJ\n\nij and IJ (except somewhat closer together than appears here).\n\n\\l\n\\L\n\n\u0142 and \u0141.\n\n\\ng\n\\NG\n\nLappish letter eng, also used in phonetics.\n\n\\o\n\\O\n\n\u00f8 and \u00d8.\n\n\\oe\n\\OE\n\n\u0153 and \u0152.\n\n\\ss\n\\SS\n\n\u00df and SS.\n\n\\th\n\\TH\n\nIcelandic letter thorn: \u00fe and \u00de. Not available with OT1 encoding, you need the fontenc package to select an alternate font encoding, such as T1.\n\nNext: , Previous: , Up: Special insertions \u00a0 [Contents][Index]\n\n### 23.7 inputenc package\n\nSynopsis:\n\n\\usepackage[encoding-name]{inputenc}\n\n\nDeclare the input file\u2019s text encoding to be encoding-name. The default, if this package is not loaded, is UTF-8. Technically, specifying the encoding name is optional, but in practice it is not useful to omit it.\n\nIn a computer file, the characters are stored according to a scheme called the encoding. There are many different encodings. The simplest is ASCII, which supports 95 printable characters, not enough for most of the world\u2019s languages. For instance, to typeset the a-umlaut character \u00e4 in an ASCII-encoded LaTeX source file, the sequence \\\"a is used. This would make source files for anything but English hard to read; even for English, often a more extensive encoding is more convenient.\n\nThe modern encoding standard, in some ways a union of the others, is UTF-8, one of the representations of Unicode. This is the default for LaTeX since 2018.\n\nThe inputenc package is how LaTeX knows what encoding is used. For instance, the following command explicitly says that the input file is UTF-8 (note the lack of a dash).\n\n\\usepackage[utf8]{inputenc}\n\n\nCaution: use inputenc only with the pdfTeX engine (see TeX engines). (The XeTeX and LuaTeX engines assume that the input file is UTF-8 encoded.) If you invoke LaTeX with either the xelatex command or the lualatex command, and try to declare a non-UTF-8 encoding with inputenc, such as latin1, then you will get the error inputenc is not designed for xetex or luatex.\n\nAn inputenc package error such as Invalid UTF-8 byte \"96 means that some of the material in the input file does not follow the encoding scheme. Often these errors come from copying material from a document that uses a different encoding than the input file; this one is a left single quote from a web page using latin1 inside a LaTeX input file that uses UTF-8. The simplest solution is to replace the non-UTF-8 character with its UTF-8 equivalent, or use a LaTeX equivalent command or character.\n\nIn some documents, such as a collection of journal articles from a variety of authors, changing the encoding in mid-document may be necessary. Use the command \\inputencoding{encoding-name}. The most common values for encoding-name are: ascii, latin1, latin2, latin3, latin4, latin5, latin9, latin10, and\u00a0utf8.\n\nNext: , Previous: , Up: Special insertions \u00a0 [Contents][Index]\n\n### 23.8 \\rule\n\nSynopsis, one of:\n\n\\rule{width}{thickness}\n\\rule[raise]{width}{thickness}\n\n\nProduce a rule, a filled-in rectangle.\n\nThis example produces a rectangular blob, sometimes called a Halmos symbol, or just \u201cqed\u201d, often used to mark the end of a proof:\n\n\\newcommand{\\qedsymbol}{\\rule{0.4em}{2ex}}\n\n\nThe amsthm package includes this command, with a somewhat different-looking symbol.\n\nThe mandatory arguments give the horizontal width and vertical thickness of the rectangle. They are rigid lengths (see Lengths). The optional argument raise is also a rigid length, and tells LaTeX how much to raise the rule above the baseline, or lower it if the length is negative.\n\nThis produces a line, a rectangle that is wide but not tall.\n\n\\noindent\\rule{\\textwidth}{0.4pt}\n\n\nThe line is the width of the page and 0.4\u00a0points tall. This line thickness is common in LaTeX.\n\nA rule that has zero width, or zero thickness, will not show up in the output, but can cause LaTeX to change the output around it. See \\strut, for examples.\n\nPrevious: , Up: Special insertions \u00a0 [Contents][Index]\n\n### 23.9 \\today\n\nSynopsis:\n\n\\today\n\n\nProduce today\u2019s date in the format \u2018month dd, yyyy\u2019. An example of a date in that format is \u2018July 4, 1976\u2019.\n\nMultilingual packages such as babel or polyglossia, or classes such as lettre, will localize \\today. For example, the following will output \u20184 juillet 1976\u2019:\n\n\\year=1976 \\month=7 \\day=4\n\\documentclass{minimal}\n\\usepackage[french]{babel}\n\\begin{document}\n\\today\n\\end{document}\n\n\n\\today uses the counters \\day, \\month, and \\year (see \\day & \\month & \\year).\n\nA number of package on CTAN work with dates. One is datetime package which can produce a wide variety of date formats, including ISO standards.\n\nThe date is not updated as the LaTeX process runs, so in principle the date could be incorrect by the time the program finishes.\n\nNext: , Previous: , Up: Top \u00a0 [Contents][Index]\n\n## 24 Splitting the input\n\nLaTeX lets you split a large document into several smaller ones. This can simplify editing or allow multiple authors to work on the document. It can also speed processing.\n\nRegardless of how many separate files you use, there is always one root file, on which LaTeX compilation starts. This shows such a file with five included files.\n\n\\documentclass{book}\n\\includeonly{ % comment out lines below to omit compiling\npref,\nchap1,\nchap2,\nappend,\nbib\n}\n\\begin{document}\n\\frontmatter\n\\include{pref}\n\\mainmatter\n\\include{chap1}\n\\include{chap2}\n\\appendix\n\\include{append}\n\\backmatter\n\\include{bib}\n\\end{document}\n\n\nThis will bring in material from pref.tex, chap1.tex, chap2.tex, append.tex, and bib.tex. If you compile this file, and then comment out all of the lines inside \\includeonly{...} except for chap1, and compile again, then LaTeX will only process the material in the first chapter. Thus, your output will appear more quickly and be shorter to print. However, the advantage of the \\includeonly command is that LaTeX will retain the page numbers and all of the cross reference information from the other parts of the document so these will appear in your output correctly.\n\nSee Larger book template, for another example of \\includeonly.\n\nNext: , Up: Splitting the input \u00a0 [Contents][Index]\n\n### 24.1 \\endinput\n\nSynopsis:\n\n\\endinput\n\n\nWhen you \\include{filename}, inside filename.tex the material after \\endinput will not be included. This command is optional; if filename.tex has no \\endinput then LaTeX will read all of the file.\n\nFor example, suppose that a document\u2019s root file has \\input{chap1} and this is chap1.tex.\n\n\\chapter{One}\nThis material will appear in the document.\n\\endinput\nThis will not appear.\n\n\nThis can be useful for putting documentation or comments at the end of a file, or for avoiding junk characters that can be added if the file is transmitted in the body of an email. It is also useful for debugging: one strategy to localize errors is to put \\endinput halfway through the included file and see if the error disappears. Now, knowing which half contains the error, moving \\endinput to halfway through that area further narrows down the location. This process rapidly finds the offending line.\n\nAfter reading \\endinput, LaTeX continues to read to the end of the line, so something can follow this command and be read nonetheless. This allows you, for instance, to close an \\if... with a \\fi.\n\nNext: , Previous: , Up: Splitting the input \u00a0 [Contents][Index]\n\n### 24.2 \\include & \\includeonly\n\nSynopsis:\n\n\\includeonly{ % in document preamble\n...\nfilename,\n...\n}\n...\n\\include{filename} % in document body\n\n\nBring material from the external file filename.tex into a LaTeX document.\n\nThe \\include command does three things: it executes \\clearpage (see \\clearpage & \\cleardoublepage), then it inputs the material from filename.tex into the document, and then it does another \\clearpage. This command can only appear in the document body.\n\nThe \\includeonly command controls which files will be read by LaTeX under subsequent \\include commands. Its list of filenames is comma-separated. It must appear in the preamble or even earlier, e.g., the command line; it can\u2019t appear in the document body.\n\nThis example root document, constitution.tex, brings in three files, preamble.tex, articles.tex, and amendments.tex.\n\n\\documentclass{book}\n\\includeonly{\npreamble,\narticles,\namendments\n}\n\\begin{document}\n\\include{preamble}\n\\include{articles}\n\\include{amendments}\n\\end{document}\n\n\nThe file preamble.tex contains no special code; you have just excerpted the chapter from consitution.tex and put it in a separate file just for editing convenience.\n\n\\chapter{Preamble}\nWe the People of the United States,\nin Order to form a more perfect Union, ...\n\n\nRunning LaTeX on constitution.tex makes the material from the three files appear in the document but also generates the auxiliary files preamble.aux, articles.aux, and amendments.aux. These contain information such as page numbers and cross-references (see Cross references). If you now comment out \\includeonly\u2019s lines with preamble and amendments and run LaTeX again then the resulting document shows only the material from articles.tex, not the material from preamble.tex or amendments.tex. Nonetheless, all of the auxiliary information from the omitted files is still there, including the starting page number of the chapter.\n\nIf the document preamble does not have \\includeonly then LaTeX will include all the files you call for with \\include commands.\n\nThe \\include command makes a new page. To avoid that, see \\input (which, however, does not retain the auxiliary information).\n\nSee Larger book template, for another example using \\include and \\includeonly. That example also uses \\input for some material that will not necessarily start on a new page.\n\nFile names can involve paths.\n\n\\documentclass{book}\n\\includeonly{\nchapters\/chap1,\n}\n\\begin{document}\n\\include{chapters\/chap1}\n\\end{document}\n\n\nTo make your document portable across distributions and platforms you should avoid spaces in the file names. The tradition is to instead use dashes or underscores. Nevertheless, for the name \u2018amo amas amat\u2019, this works under TeX Live on GNU\/Linux:\n\n\\documentclass{book}\n\\includeonly{\n\"amo\\space amas\\space amat\"\n}\n\\begin{document}\n\\include{\"amo\\space amas\\space amat\"}\n\\end{document}\n\n\nand this works under MiKTeX on Windows:\n\n\\documentclass{book}\n\\includeonly{\n{\"amo amas amat\"}\n}\n\\begin{document}\n\\include{{\"amo amas amat\"}}\n\\end{document}\n\n\nYou cannot use \\include inside a file that is being included or you get \u2018LaTeX Error: \\include cannot be nested.\u2019 The \\include command cannot appear in the document preamble; you will get \u2018LaTeX Error: Missing \\begin{document}\u2019.\n\nIf a file that you \\include does not exist, for instance if you \\include{athiesm} but you meant \\include{atheism}, then LaTeX does not give you an error but will warn you \u2018No file athiesm.tex.\u2019 (It will also create athiesm.aux.)\n\nIf you \\include the root file in itself then you first get \u2018LaTeX Error: Can be used only in preamble.\u2019 Later runs get \u2018TeX capacity exceeded, sorry [text input levels=15]\u2019. To fix this, you must remove the inclusion \\include{root} but also delete the file root.aux and rerun LaTeX.\n\nPrevious: , Up: Splitting the input \u00a0 [Contents][Index]\n\n### 24.3 \\input\n\nSynopsis:\n\n\\input{filename}\n\n\nLaTeX processes the file as if its contents were inserted in the current file. For a more sophisticated inclusion mechanism see \\include & \\includeonly.\n\nIf filename does not end in \u2018.tex\u2019 then LaTeX first tries the filename with that extension; this is the usual case. If filename ends with \u2018.tex\u2019 then LaTeX looks for the filename as it is.\n\nFor example, this\n\n\\input{macros}\n\n\nwill cause LaTeX to first look for macros.tex. If it finds that file then it processes its contents as thought they had been copy-pasted in. If there is no file of the name macros.tex then LaTeX tries the name macros, without an extension. (This may vary by distribution.)\n\nTo make your document portable across distributions and platforms you should avoid spaces in the file names. The tradition is to instead use dashes or underscores. Nevertheless, for the name \u2018amo amas amat\u2019, this works under TeX Live on GNU\/Linux:\n\n\\input{\"amo\\space amas\\space amat\"}\n\n\nand this works under MiKTeX on Windows:\n\n\\input{{\"amo amas amat\"}}\n\n\nNext: , Previous: , Up: Top \u00a0 [Contents][Index]\n\n## 25 Front\/back matter\n\nNext: , Up: Front\/back matter \u00a0 [Contents][Index]\n\nSynopsis, one of:\n\n\\tableofcontents\n\\listoffigures\n\\listoftables\n\n\nProduce a table of contents, or list of figures, or list of tables. Put the command in the input file where you want the table or list to go. You do not type the entries; for example, typically the table of contents entries are automatically generated from the sectioning commands \\chapter, etc.\n\nThis example illustrates the first command, \\tableofcontents. LaTeX will produce a table of contents on the book\u2019s first page.\n\n\\documentclass{book}\n% \\setcounter{tocdepth}{1}\n\\begin{document}\n\\tableofcontents\\newpage\n...\n\\chapter{...}\n...\n\\section{...}\n...\n\\subsection{...}\n...\n\\end{document}\n\n\nUncommenting the second line would cause that table to contain chapter and section listings but not subsection listings, because the \\section command has level\u00a01. See Sectioning, for level numbers of the sectioning units. For more on the tocdepth see Sectioning\/tocdepth.\n\nAnother example of the use of \\tableofcontents is in Larger book template.\n\nIf you want a page break after the table of contents, write a \\newpage command after the \\tableofcontents command, as above.\n\nTo make the table of contents, LaTeX stores the information in an auxiliary file named root-file.toc (see Splitting the input). For example, this LaTeX file test.tex\n\n\\documentclass{article}\n\\begin{document}\n\\tableofcontents\\newpage\n\\section{First section}\n\\subsection{First subsection}\n...\n\n\nwrites these lines to test.toc.\n\n\\contentsline {section}{\\numberline {1}First section}{2}\n\\contentsline {subsection}{\\numberline {1.1}First subsection}{2}\n\n\nEach line contains a single command, \\contentsline (see \\contentsline). The first argument, the section or subsection, is the sectioning unit. The second argument has two components. The hook \\numberline determines how the sectioning number, 1 or 1.1, appears in the table of contents (see \\numberline). The remainder of the second argument of \\contentsline, \u2018First section\u2019 or \u2018First subsection\u2019, is the sectioning title text. Finally, the third argument, \u20182\u2019, is the page number on which this sectioning unit starts.\n\nTo typeset these lines, the document class provides \\l@section-unit commands such as \\l@section{text}{pagenumber} and \\l@subsection{text}{pagenumber}. These commands often use the \\@dottedtocline command (see \\@dottedtocline).\n\nA consequence of LaTeX\u2019s strategy of using auxiliary files is that to get the correct information in the document you must run LaTeX twice, once to store the information and the second time to retrieve it. In the ordinary course of writing a document authors run LaTeX a number of times, but you may notice that the first time that you compile a new document, the table of contents page will be empty except for its \u2018Contents\u2019 header. Just run LaTeX again.\n\nThe commands \\listoffigures and \\listoftables produce a list of figures and a list of tables. Their information is stored in files with extension .lof and .lot. They work the same way as \\tableofcontents but the latter is more common, so we use it for most examples.\n\nYou can manually add material to the table of contents, the list of figures, and the list of tables. For instance, add a line about a section to the table of contents with \\addcontentsline{toc}{section}{text}. (see \\addcontentsline). Add arbitrary material, that is, non-line material, with \\addtocontents, as with the command \\addtocontents{lof}{\\protect\\vspace{2ex}}, which adds vertical space to the list of figures (see \\addtocontents).\n\nLines in the table of contents, the list of figures, and the list of tables, have four parts. First is an indent. Next is a box into which sectioning numbers are placed, and then the third box holds the title text, such as \u2018First section\u2019. Finally there is a box up against the right margin, inside of which LaTeX puts the page number box. For the indent and the width of the number box, see \\@dottedtocline. The right margin box has width \\@tocrmarg and the page number is flush right in that space, inside a box of width \\@pnumwidth. By default \\@tocrmarg is 2.55em and \\@pnumwidth is 1.55em. Change these as with \\renewcommand{\\@tocrmarg}{3.5em}.\n\nCTAN has many packages for the table of contents and lists of figures and tables (see CTAN). The package tocloft is convenient for adjusting some aspects of the default such as spacing. And, tocbibbind will automatically add the bibliography, index, etc. to the table of contents.\n\nTo change the header for the table of contents page, do something like these commands before you call \\tableofcontents, etc.\n\n\\renewcommand{\\contentsname}{Table of Contents}\n\\renewcommand{\\listfigurename}{Plots}\n\\renewcommand{\\listtablename}{Specifications}\n\n\nInternationalization packages such as babel or polyglossia will change these headers depending on the chosen base language.\n\n#### 25.1.1 \\@dottedtocline\n\nSynopsis:\n\n\\@dottedtocline{section-level-num}{indent}{numwidth}{text}{pagenumber}\n\n\nUsed internally by LaTeX to format an entry line in the table of contents, list of figures, or list of tables. Authors do not directly enter \\@dottedtocline commands.\n\nThis command is typically used by \\l@section, \\l@subsection, etc., to format the content lines. For example, the article.cls file contains these definitions:\n\n\\newcommand*\\l@section{\\@dottedtocline{1}{1.5em}{2.3em}}\n\\newcommand*\\l@subsection{\\@dottedtocline{2}{3.8em}{3.2em}}\n\\newcommand*\\l@subsubsection{\\@dottedtocline{3}{7.0em}{4.1em}}\n\n\nIn this example, \\@dottedcline appears to have been given only three arguments. But tracing the internal code shows that it picks up the final text and pagenumber arguments in the synopsis from a call to \\contentsline.\n\n\nIn the default book class, LaTeX does not use dotted leaders for the Part and Chapter table entries, and in the default article class it does not use dotted leaders for Section entries.\n\n#### 25.1.2 \\addcontentsline\n\nSynopsis:\n\n\\addcontentsline{ext}{unit}{text}\n\n\nAdd an entry to the auxiliary file with extension ext.\n\n\\addcontentsline{toc}{section}{\\protect\\textbf{Appendices}}\n\n\nIt will appear at the same indentation level as the sections, will be in boldface, and will be assigned the page number associated with the point where it appears in the input file.\n\nThe \\addcontentsline command writes information to the file root-name.ext. It writes that information as the text of the command \\contentsline{unit}{text}{num}, where num is the current value of counter unit (see \\contentsline). The most common case is the table of contents and there num is the page number of the first page of unit.\n\nThis command is invoked by the sectioning commands \\chapter, etc., and also by \\caption inside a float environment. But it is also used by authors. For example, an author writing a book whose style is to have an unnumbered preface may use the starred \\chapter*. But that command leaves out table of contents information, which can be entered manually, as here.\n\n\\chapter*{Preface}\n\n\nIn the .toc file LaTeX will put the line \\contentsline {chapter}{\\numberline {}Preface}{3}; note that the page number \u20183\u2019 is automatically generated by the system, not entered manually.\n\nAll of the arguments for \\addcontentsline are required.\n\next\n\nTypically one of the strings toc for the table of contents, lof for the list of figures, or lot for the list of tables. The filename extension of the information file.\n\nunit\n\nA string that depends on the value of the ext argument:\n\ntoc\n\nFor the table of contents, this is the name of a sectional unit: part, chapter, section, subsection, etc.\n\nlof\n\nFor the list of figures: figure.\n\nlot\n\nFor the list of tables: table.\n\ntext\n\nThe text of the entry. You must \\protect any commands that are fragile (see \\protect).\n\nThe \\addcontentsline command has an interaction with \\include (see \\include & \\includeonly). If you use them at the same level, as with \\addcontentsline{...}{...}{...}\\include{...} then lines in the table of contents can come out in the wrong order. The solution is to move \\addcontentsline into the file being included.\n\nIf you use a unit that LaTeX does not recognize, as here\n\n\\addcontentsline{toc}{setcion}{\\protect\\textbf{Appendices}}\n\n\nthen you don\u2019t get an error but the formatting in the table of contents will not make sense.\n\n#### 25.1.3 \\addtocontents\n\nSynopsis:\n\n\\addtocontents{ext}{text}\n\n\nAdd text, which may be text or formatting commands, directly to the auxiliary file with extension ext. This is most commonly used for the table of contents so that is the discussion here, but it also applies to the list of figures and list of tables.\n\n\\tableofcontents\\newpage\n\n\nThis puts the word \u2018Page\u2019, in boldface, above the column of page numbers and after the header.\n\n\\tableofcontents\n\\chapter{...}\n\n\nThis adds a line announcing work by a new author.\n\n\\addtocontents{toc}{%\n\\protect\\vspace{2ex}\n\\textbf{Chapters by N. Other Author}\\par}\n\n\nThe difference between \\addtocontents and \\addcontentsline is that the latter is strictly for lines, such as with a line giving the page number for the start of a new subset of the chapters. As the above examples show, \\addtocontents is for material such as spacing.\n\nThe \\addtocontents command has two arguments. Both are required.\n\next\n\nTypically one of: toc for the table of contents, lof for the list of figures, or lot for the list of tables. The extension of the file holding the information.\n\ntext\n\nThe text, and possibly commands, to be written.\n\nThe sectioning commands such as \\chapter use the \\addcontentsline command to store information. This command creates lines in the .toc auxiliary file containing the \\contentsline command (see \\addcontentsline). In contrast, the command \\addtocontents puts material directly in that file.\n\nThe \\addtocontents command has an interaction with \\include (see \\include & \\includeonly). If you use them at the same level, as with \\addtocontents{...}{...}\\include{...} then lines in the table of contents can come out in the wrong order. The solution is to move \\addtocontents into the file being included.\n\n#### 25.1.4 \\contentsline\n\nSynopsis:\n\n\\contentsline{unit}{text}{pagenumber}\n\n\nUsed internally by LaTeX to typeset an entry of the table of contents, list of figures, or list of tables (see Table of contents etc.). Authors do not directly enter \\contentsline commands.\n\nUsually adding material to these lists is done automatically by the commands \\chapter, \\section, etc. for the table of contents, or by the \\caption command inside of a \\figure or \\table environment (see figure and see table). Thus, where the base file is thesis.tex, and contains the declaration \\tableofcontents, the command \\chapter{Chapter One} produces something like this in the file thesis.toc.\n\n\\contentsline {chapter}{\\numberline {1}Chapter One}{3}\n\n\nIf the file contains the declaration \\listoffigures then a figure environment involving \\caption{Test} will produce something like this in thesis.lof.\n\n\\contentsline {figure}{\\numberline {1.1}{\\ignorespaces Test}}{6}\n\n\nTo manually add material, use \\addcontentsline{filetype}{unit}{text}, where filetype is toc, lof, or lot (see \\addcontentsline).\n\nFor manipulating how the \\contentline material is typeset, see the tocloft package.\n\nNote that the hyperref package changes the definition of \\contentsline (and \\addcontentsline) to add more arguments, to make hyperlinks. This is the source of the error Argument of \\contentsline has an extra }. Fix this error by deleting the .toc or .lof or .lot file, and running LaTeX again.\n\n#### 25.1.5 \\nofiles\n\nSynopsis:\n\n\\nofiles\n\n\nPrevent LaTeX from writing any auxiliary files. The only output will be the .log and .pdf (or .dvi) files. This command must go in the preamble.\n\nBecause of the \\nofiles command this example will not produce a .toc file.\n\n\\documentclass{book}\n\\nofiles\n\\begin{document}\n\\tableofcontents\\newpage\n\\chapter{...}\n...\n\n\nLaTeX will not erase any existing auxiliary files, so if you insert the \\nofiles command after you have run the file and gotten a .toc then the table of contents page will continue to show the old information.\n\n#### 25.1.6 \\numberline\n\nSynopsis:\n\n\\numberline{number}\n\n\nTypeset its argument flush left in a box. This is used in a \\contentsline command to typeset the section number (see \\contentsline).\n\nFor example, this line in a .toc file causes the 1 to be typeset flush left.\n\n\\contentsline {subsection}{\\numberline {1.1}Motivation}{2}\n\n\nBy default, LaTeX typesets the section numbers in a box of length \\@tempdima. That length is set by the commands \\l@section, \\l@subsection, etc. Put section numbers inside a natural-width box with \\renewcommand{\\numberline}[1]{#1~}.\n\nThis command is fragile, so you may need to precede it with \\protect (see \\protect). An example is the use of \\protect in the command \\addcontentsline{toc}{section}{\\protect\\numberline{}Summary} to get the \\numberline into this command in the .toc file: \\contentsline {section}{\\numberline {}Summary}{6} (the page number \u20186\u2019 is automatically added by LaTeX; see \\addcontentsline).\n\nNext: , Previous: , Up: Front\/back matter \u00a0 [Contents][Index]\n\n### 25.2 Indexes\n\nThis document has an index.\n\n\\documentclass{article}\n\\usepackage{makeidx} \\makeindex\n...\n\\begin{document}\n...\nRecall Wilson's Theorem: \\index{Wilson's Theorem}\na number $$n>1$$ is prime if and only if the factorial of $$n-1$$\nis congruent to $$-1$$ modulo~$$n$$.\n...\n\\printindex\n...\n\n\nThe \\usepackage{makeidx} and \\makeindex in the preamble bring in the relevant commands.\n\nProducing an index is a three stage process. First, in the document body you declare index entries with the \\index command (see \\index). When you run LaTeX, the \\index writes its information to an auxiliary file root-name.idx. Next, to alphabetize and to do other manipulations you run an external command, typically makeindex or xindy (see makeindex). These output a file root-name.ind. Finally, you bring the information back into your document and typeset it with the \\printindex command (see \\printindex).\n\nThere are many packages in the area of indexing. The showidx package causes each index entries to be shown in the margin on the page where the entry appears. This can help in preparing the index. The multind package, among others, supports multiple indexes. See also the TeX FAQ entry on this topic, https:\/\/www.texfaq.org\/FAQ-multind, and the CTAN topic, https:\/\/ctan.org\/topic\/index-multi.\n\nNext: , Up: Indexes \u00a0 [Contents][Index]\n\n#### 25.2.1 \\index\n\nSynopsis:\n\n\\index{index-entry-string}\n\n\nDeclare an entry in the index. This command is fragile (see \\protect).\n\nFor example, as described in Indexes, one way to get an index from what\u2019s below is to compile the document with pdflatex test, then process the index entries with makeindex test, and then compile again with pdflatex test.\n\nW~Ackermann (1896--1962).\\index{Ackermann}\n...\nAckermann function\\index{Ackermann!function}\n...\nrate of growth\\index{Ackermann!function!growth rate}\n\n\nAll three index entries will get a page number, such as \u2018Ackermann, 22\u2019. LaTeX will format the second as a subitem of the first, on the line below it and indented, and the third as a subitem of the second. Three levels deep is as far as you can nest subentries. (If you add \\index{Ackermann!function!growth rate!comparison} then makeindex says \u2018Scanning input file test.idx....done (4 entries accepted, 1 rejected)\u2019 and nothing appears in the index).\n\nIf you enter a second \\index with the same index-entry-string then you will get a single index entry with two page numbers (unless they happen to fall on the same page). Thus, adding as for Ackermann.\\index{Ackermann} later in the same document as above will give an index entry like \u2018Ackermann, 22, 151\u2019. Also, you can enter the index entries in any order, so for instance \\index{Ackermann!function} could come before \\index{Ackermann}.\n\nGet a page range in the output, like \u2018Hilbert, 23--27\u2019, as here.\n\nW~Ackermann (1896--1962).\\index{Ackermann}\n...\nD~Hilbert (1862--1943)\\index{Ackermann!Hilbert$$} ... disapproved of his marriage.\\index{Ackermann!Hilbert$$}\n\n\nIf the beginning and ending of the page range are equal then the system just gives a single page entry, not a range.\n\nIf you index subentries but not a main entry, as with \\index{Jones!program} and \\index{Jones!results}, then the output is the item \u2018Jones\u2019 with no comma or page number, followed by two subitems, like \u2018program, 50\u2019 and \u2018results, 51\u2019.\n\nGenerate a index entry that says \u2018See\u2019 by using a vertical bar character: \\index{Ackermann!function|see{P\\'eter's function}}. You can instead get \u2018See also\u2019 with seealso. (The text \u2018See\u2019 is defined by \\seename, and \u2018See also\u2019 by \\alsoname. You can redefine these either by using an internationalization package such as babel or polyglossia, or directly as with \\renewcommand{\\alsoname}[1]{Also see #1}.)\n\nThe \u2018See\u2019 feature is part of a more general functionality. After the vertical bar you can put the name of a one-input command, as in \\index{group|textit} (note the missing backslash on the \\textit command) and the system will apply that command to the page number, here giving something like \\textit{7}. You can define your own one-input commands, such as \\newcommand{\\definedpage}[1]{{\\color{blue}#1}} and then \\index{Ackermann!function|definedpage} will give a blue page number (see Color). Another, less practical, example is this,\n\n\\newcommand\\indexownpage[1]{#1, \\thepage}\n... Epimenides.\\index{self-reference|indexownpage}\n\n\nwhich creates an entry citing the page number of its own index listing.\n\nThe two functions just described combine, as here\n\n\\index{Ackermann!function|(definedpage}\n...\n\\index{Ackermann!function|)}\n\n\nwhich outputs an index entry like \u2018function, 23--27\u2019 where the page number range is in blue.\n\nConsider an index entry such as \u2018\u03b1-ring\u2019. Entering it as $\\alpha$-ring will cause it to be alphabetized according to the dollar sign. You can instead enter it using an at-sign, as \\index{alpha-ring@$\\alpha$-ring}. If you specify an entry with an at-sign separating two strings, pos@text, then pos gives the alphabetical position of the entry while text produces the text of the entry. Another example is that \\index{Saint Michael's College@SMC} produces an index entry \u2018SMC\u2019 alphabetized into a different location than its spelling would naturally give it.\n\nTo put a !, or @, or | character in an index entry, preceding it with a double quote, \". (The double quote gets deleted before alphabetization.)\n\nA number of packages on CTAN have additional functionality beyond that provided by makeidx. One is index, which allows for multiple indices and contains a command \\index*{index-entry-string} that prints the index-entry-string as well as indexing it.\n\nThe \\index command writes the indexing information to the file root-name.idx file. Specifically, it writes text of the command \\indexentry{index-entry-string}{page-num}, where page-num is the value of the \\thepage counter. On occasion, when the \\printindex command is confused, you have to delete this file to start with a fresh slate.\n\nIf you omit the closing brace of an \\index command then you get a message like this.\n\nRunaway argument? {Ackermann!function\n! Paragraph ended before \\@wrindex was complete.\n\n\nNext: , Previous: , Up: Indexes \u00a0 [Contents][Index]\n\n#### 25.2.2 makeindex\n\nSynopsis, one of:\n\nmakeindex filename\nmakeindex -s style-file filename\nmakeindex options filename0 ...\n\n\nSort, and otherwise process, the index information in the auxiliary file filename. This is a command line program. It takes one or more raw index files, filename.idx files, and produces the actual index file, the filename.ind file that is input by \\printindex (see \\printindex).\n\nThe first form of the command suffices for many uses. The second allows you to format the index by using an index style file, a .isty file. The third form is the most general; see the full documentation on CTAN.\n\nThis is a simple .isty file.\n\n% book.isty\n% $makeindex -s book.isty -p odd book.idx % creates the index as book.ind, starting on an odd page. preamble \"\\\\pagestyle{empty} \\\\small \\\\begin{theindex} \\\\thispagestyle{empty}\" postamble \"\\n \\\\end{theindex}\" The description here covers only some of the index formatting possibilities in style-file. For a full list see the documentation on CTAN. A style file consists of a list of pairs: specifier and attribute. These can appear in the file in any order. All of the attributes are strings, except where noted. Strings are surrounded with double quotes, \", and the maximum length of a string is 144 characters. The \\n is for a newline and \\t is for a tab. Backslashes are escaped with another backslash, \\\\. If a line begins with a percent sign, %, then it is a comment. preamble Preamble of the output file. Defines the context in which the index is formatted. Default: \"\\\\begin{theindex}\\n\". postamble Postamble of the output file. Default: \"\\n\\n\\\\end{theindex}\\n\". group_skip Traditionally index items are broken into groups, typically a group for entries starting with \u2018a\u2019, etc. This specifier gives what is inserted when a new group begins. Default: \"\\n\\n \\\\indexspace\\n\" (\\indexspace is a command inserting a rubber length, by default 10pt plus5pt minus3pt). lethead_flag An integer. It governs what is inserted for a new group or letter. If it is 0 (which is the default) then other than group_skip nothing will be inserted before the group. If it is positive then at a new letter the lethead_prefix and lethead_suffix will be inserted, with that letter in uppercase between them. If it is negative then what will be inserted is the letter in lowercase. The default is 0. lethead_prefix If a new group begins with a different letter then this is the prefix inserted before the new letter header. Default: \"\" lethead_suffix If a group begins with a different letter then this is the suffix inserted after the new letter header. Default: \"\". item_0 What is put between two level 0 items. Default: \"\\n \\\\item \". item_1 Put between two level 1 items. Default: \"\\n \\\\subitem \". item_2 put between two level 2 items. Default: \"\\n \\\\subsubitem \". item_01 What is put between a level 0 item and a level 1 item. Default: \"\\n \\\\subitem \". item_x1 What is put between a level 0 item and a level 1 item in the case that the level 0 item doesn\u2019t have any page numbers (as in \\index{aaa|see{bbb}}). Default: \"\\n \\\\subitem \". item_12 What is put between a level 1 item and a level 2 item. Default: \"\\n \\\\subsubitem \". item_x2 What is put between a level 1 item and a level 2 item, if the level 1 item doesn\u2019t have page numbers. Default: \"\\n \\\\subsubitem \". delim_0 Delimiter put between a level 0 key and its first page number. Default: a comma followed by a blank, \", \". delim_1 Delimiter put between a level 1 key and its first page number. Default: a comma followed by a blank, \", \". delim_2 Delimiter between a level 2 key and its first page number. Default: a comma followed by a blank, \", \". delim_n Delimiter between two page numbers for the same key (at any level). Default: a comma followed by a blank, \", \". delim_r What is put between the starting and ending page numbers of a range. Default: \"--\". line_max An integer. Maximum length of an index entry\u2019s line in the output, beyond which the line wraps. Default: 72. indent_space What is inserted at the start of a wrapped line. Default: \"\\t\\t\". indent_length A number. The length of the wrapped line indentation. The default indent_space is two tabs and each tab is eight spaces so the default here is 16. page_precedence A document may have pages numbered in different ways. For example, a book may have front matter pages numbered in lowercase roman while main matter pages are in arabic. This string specifies the order in which they will appear in the index. The makeindex command supports five different types of numerals: lowercase roman r, and numeric or arabic n, and lowercase alphabetic a, and uppercase roman R, and uppercase alphabetic A. Default: \"rnaRA\". There are a number of other programs that do the job makeindex does. One is xindy (https:\/\/ctan.org\/pkg\/xindy), which does internationalization and can process indexes for documents marked up using LaTeX and a number of other languages. It is written in Lisp, highly configurable, both in markup terms and in terms of the collating order of the text, as described in its documentation. A more recent indexing program supporting Unicode is xindex, written in Lua (https:\/\/ctan.org\/pkg\/xindex). Previous: , Up: Indexes [Contents][Index] #### 25.2.3 \\printindex Synopsis: \\printindex Place the index into the output. To get an index you must first include \\usepackage{makeidx}\\makeindex in the document preamble and compile the document, then run the system command makeindex, and then compile the document again. See Indexes, for further discussion and an example of the use of \\printindex. Previous: , Up: Front\/back matter [Contents][Index] ### 25.3 Glossaries Synopsis: \\usepackage{glossaries} \\makeglossaries ... \\newglossaryentry{label}{settings} ... \\gls{label}. ... \\printglossaries The glossaries package allows you to make glossaries, including multiple glossaries, as well as lists of acronyms. To get the output from this example, compile the document (for instance with pdflatex filename), then run the command line command makeglossaries filename, and then compile the document again. \\documentclass{...} \\usepackage{glossaries} \\makeglossaries \\newglossaryentry{tm}{% name={Turing machine}, description={A model of a machine that computes. The model is simple but can compute anything any existing device can compute. It is the standard model used in Computer Science.}, } \\begin{document} Everything begins with the definition of a \\gls{tm}. ... \\printglossaries \\end{document} That gives two things. In the main text it outputs \u2018... definition of a Turing machine\u2019. In addition, in a separate sectional unit headed \u2018Glossary\u2019 there appears a description list. In boldface it says \u2018Turing machine\u2019 and the rest of the item says in normal type \u2018A model of a machine \u2026 Computer Science\u2019. The command \\makeglossary opens the file that will contain the entry information, root-file.glo. Put the \\printglossaries command where you want the glossaries to appear in your document. The glossaries package is very powerful. For instance, besides the commands \\newglossaryentry and \\gls, there are similar commands for a list of acronyms. See the package documentations on CTAN. Next: , Up: Glossaries [Contents][Index] #### 25.3.1 \\newglossaryentry Synopsis, one of: \\newglossaryentry{label} { name={name}, description={description}, other options, ... } or \\longnewglossaryentry{label} { name={name}, other options ..., } {description} Declare a new entry for a glossary. The label must be unique for the document. The settings associated with the label are pairs: key=value. This puts the blackboard bold symbol for the real numbers \u211d, in the glossary. \\newglossaryentry{R} { name={\\ensuremath{\\mathbb{R}}}, description={the real numbers}, } Use the second command form if the description spans more than one paragraph. For a full list of keys see the package documentation on CTAN but here are a few. name (Required.) The word, phrase, or symbol that you are defining. description (Required.) The description that will appear in the glossary. If this has more than one paragraph then you must use the second command form given in the synopsis. plural The plural form of name. Refer to the plural form using \\glspl or \\Glspl (see \\gls). sort How to place this entry in the list of entries that the glossary holds. symbol A symbol, such as a mathematical symbol, besides the name. Previous: , Up: Glossaries [Contents][Index] #### 25.3.2 \\gls Synopsis, one of: \\gls{label} \\glspl{label} \\Gls{label} \\Glspl{label} Refer to a glossary entry. The entries are declared with \\newglossaryentry (see \\newglossaryentry). This \\newglossaryentry{N}{% name={the natural numbers}, description={The numbers$0$,$1$,$2$,$\\ldots\\$\\@},\nsymbol={\\ensuremath{\\mathbb{N}}},\n}\n...\nConsider \\gls{N}.\n\n\ngives the output \u2018Consider the natural numbers\u2019.\n\nThe second command form \\glspl{label} produces the plural of name (by default it tries adding an \u2018s\u2019). The third form capitalizes the first letter of name, as does the fourth form, which also takes the plural.\n\nNext: , Previous: , Up: Top \u00a0 [Contents][Index]\n\n## 26 Letters\n\nSynopsis:\n\n\\documentclass{letter}\n\\signature{sender name}\n\\begin{document}\n\\opening{salutation}\nletter body\n\\closing{closing text}\n\\end{letter}\n...\n\\end{document}\n\n\nProduce one or more letters.\n\nEach letter is in a separate letter environment, whose argument recipient address often contains multiple lines separated with a double backslash,\u00a0(\\\\). For example, you might have:\n\n \\begin{letter}{Ninon de l'Enclos \\\\\nl'h\\^otel Sagonne}\n...\n\\end{letter}\n\n\nThe start of the letter environment resets the page number to 1, and the footnote number to 1 also.\n\nThe sender address and sender name are common to all of the letters, whether there is one or more, so these are best put in the preamble. As with the recipient address, often sender address contains multiple lines separated by a double backslash\u00a0(\\\\). LaTeX will put the sender name under the closing, after a vertical space for the traditional hand-written signature.\n\nEach letter environment body begins with a required \\opening command such as \\opening{Dear Madam or Sir:}. The letter body text is ordinary LaTeX so it can contain everything from enumerated lists to displayed math, except that commands such as \\chapter that make no sense in a letter are turned off. Each letter environment body typically ends with a \\closing command such as \\closing{Yours,}.\n\nAdditional material may come after the \\closing. You can say who is receiving a copy of the letter with a command like \\cc{the Boss \\\\ the Boss's Boss}. There\u2019s a similar \\encl command for a list of enclosures. And, you can add a postscript with \\ps.\n\nLaTeX\u2019s default is to indent the sender name and the closing above it by a length of \\longindentation. By default this is 0.5\\textwidth. To make them flush left, put \\setlength{\\longindentation}{0em} in your preamble.\n\nTo set a fixed date use something like \\renewcommand{\\today}{1958-Oct-12}. If put in your preamble then it will apply to all the letters.\n\nThis example shows only one letter environment. The three lines marked as optional are typically omitted.\n\n\\documentclass{letter}\n\\signature{Sender's name \\\\ Sender's title}\n% optional: \\location{Mailbox 13}\n% optional: \\telephone{(102) 555-0101}\n\\begin{document}\n\\opening{Sir:}\n% optional: \\thispagestyle{firstpage}\nI am not interested in entering a business arrangement with you.\n\\end{letter}\n\\end{document}\n\n\nThese commands are used with the letter class.\n\nNext: , Up: Letters \u00a0 [Contents][Index]\n\n### 26.1 \\address\n\nSynopsis:\n\n\\address{senders address}\n\n\nSpecify the return address, as it appears on the letter and on the envelope. Separate multiple lines in senders address with a double backslash,\u00a0\\\\.\n\nBecause it can apply to multiple letters this declaration is often put in the preamble. However, it can go anywhere, including inside an individual letter environment.\n\nThis command is optional: if you do not use it then the letter is formatted with some blank space on top, for copying onto pre-printed letterhead paper. If you do use the \\address declaration then it is formatted as a personal letter.\n\nHere is an example.\n\n\\address{Stephen Maturin \\\\\nThe Grapes of the Savoy}\n\n\nNext: , Previous: , Up: Letters \u00a0 [Contents][Index]\n\n### 26.2 \\cc\n\nSynopsis:\n\n\\cc{name0 \\\\\n... }\n\n\nProduce a list of names to which copies of the letter were sent. This command is optional. If it appears then typically it comes after \\closing. Put the names on different lines by separating them with a double backslash, \\\\, as in:\n\n\\cc{President \\\\\nVice President}\n\n\nNext: , Previous: , Up: Letters \u00a0 [Contents][Index]\n\n### 26.3 \\closing\n\nSynopsis:\n\n\\closing{text}\n\n\nProduce the letter\u2019s closing. This is optional, but usual. It appears at the end of a letter, above a handwritten signature. For example:\n\n\\closing{Regards,}\n\n\nNext: , Previous: , Up: Letters \u00a0 [Contents][Index]\n\n### 26.4 \\encl\n\nSynopsis:\n\n\\encl{first enclosed object \\\\\n... }\n\n\nProduce a list of things included with the letter. This command is optional; when it is used, it typically is put after \\closing. Separate multiple lines with a double backslash, \\\\.\n\n\\encl{License \\\\\nPassport}\n\n\nNext: , Previous: , Up: Letters \u00a0 [Contents][Index]\n\n### 26.5 \\location\n\nSynopsis:\n\n\\location{text}\n\n\nThe text appears centered at the bottom of the page. It only appears if the page style is firstpage.\n\nNext: , Previous: , Up: Letters \u00a0 [Contents][Index]\n\n### 26.6 \\makelabels\n\nSynopsis:\n\n\\makelabels % in preamble\n\n\nOptional, for a document that contains letter environments. If you just put \\makelabels in the preamble then at the end of the document you will get a sheet with labels for all the recipients, one for each letter environment, that you can copy to a sheet of peel-off address labels.\n\nCustomize the labels by redefining the commands \\startlabels, \\mlabel, and \\returnaddress (and perhaps \\name) in the preamble. The command \\startlabels sets the width, height, number of columns, etc., of the page onto which the labels are printed. The command \\mlabel{return address}{recipient address} produces the two labels (or one, if you choose to ignore the return address) for each letter environment. The first argument, return address, is the value returned by the macro \\returnaddress. The second argument, recipient address, is the value passed in the argument to the letter environment. By default \\mlabel ignores the first argument, the return address, causing the default behavior described in the prior paragraph.\n\nThis illustrates customization. Its output includes a page with two columns having two labels each.\n\n\\documentclass{letter}\nOshkosh, Mineola 12305}\n\\newcommand*\\originalMlabel{}\n\\let\\originalMlabel\\mlabel\n\\def\\mlabel#1#2{\\originalMlabel{}{#1}\\originalMlabel{}{#2}}\n\\makelabels\n...\n\\begin{document}\n\\begin{letter}{A Einstein \\\\\n112 Mercer Street \\\\\nPrinceton, New Jersey, USA 08540}\n...\n\\end{letter}\n\\begin{letter}{K G\\\"odel \\\\\n145 Linden Lane \\\\\n`","date":"2021-09-21 20:45:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 2, \"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\": 20, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9269099235534668, \"perplexity\": 3542.1130485321814}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057227.73\/warc\/CC-MAIN-20210921191451-20210921221451-00616.warc.gz\"}"} | null | null |
Bill McGill (né le à San Angelo, Texas et mort le ) est un ancien joueur américain professionnel de basket-ball.
Biographie
Intérieur de issu de l'université d'Utah, McGill fut sélectionné par les Zephyrs de Chicago avec le premier choix de la draft 1962. Il joua trois saisons (de 1962 à 1965) dans la NBA et deux saisons (de 1968 à 1970) dans l'ABA. Lors de sa carrière ABA/NBA, il inscrivit un total de .
Notes et références
Liens externes
Joueur américain de basket-ball
Joueur des Zephyrs de Chicago
Joueur des Bullets de Baltimore
Joueur des Knicks de New York
Joueur des Hawks de Saint-Louis
Joueur des Lakers de Los Angeles
Joueur des Rockets de Denver
Joueur des Chaparrals de Dallas
Joueur des Stars de Los Angeles
Joueur des Pipers de Pittsburgh
Naissance en septembre 1939
Joueur de basket-ball des Utes de l'Utah
Décès en juillet 2014
Décès à 74 ans
Naissance à San Angelo
Décès à Salt Lake City | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,112 |
{"url":"https:\/\/www.josephkirwin.com\/2011\/11\/29\/micro-optimization-for-c-loops\/","text":"## Micro-Optimization for C# Loops\n\nI was recently trying some of the basic problems from Project Euler, which is highly recommendable for\u00a0interesting problems to solve that also teach you lessons on how to improve the way you code.\n\nI had to optimise the loop that I was doing\u00a0a for through a\u00a0large amount of items with each increment\u00a0performing a computation. To shave 10ths of a second from my computation I tried micro-optimising by using a decrementing loop.\n\nHere's a generalised version of what I did\n\nLet max be the number that you want to loop until.\n\nIncremental Loop\n\nfor (int i = 0; i < max; i++) {\n}\n\n\nDecremental Loop\n\nfor (int i = max - 1; \u00a0i\u00a0>=\u00a00;\u00a0 i--) {\n}\n\n\nThings to note\n\n\u2022 The decremented version is only faster when the calculation of max-1 is less complex than whatever is in the curly braces.\n\u2022 From a high level view, the reason that the decremental loop is faster is that most CPUs already have a decrement to zero functionality in-built so on an MSIL level it is just duplicating a section of this code. Whereas incrementing to an arbitrary maximum is not something that is fundamental.\n\u2022 This only makes the performance notably faster if there are many items to iterate through, there are nested loops and also it the code within the curly braces is relatively time consuming.\n\nGood discussions and articles on this\n\nAnyone with\u00a0better explanations of why decrementing is better, feel free to comment...","date":"2019-04-21 09:15:07","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\": 0.2520507574081421, \"perplexity\": 1498.882251524115}, \"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-2019-18\/segments\/1555578530505.30\/warc\/CC-MAIN-20190421080255-20190421102255-00473.warc.gz\"}"} | null | null |
Q: Why does 'read -p' only remember one variable? I figured out through a tutorial that using read -p would allow me to take the response as a variable. So I did this madlibs kind of thing to test it, hoping that I could repeatedly have user input translated to a variable, then back into output.
Here's an exerpt:
read -p "What is your favorite fruit?" fruit
echo Oh, I see...
sleep 2s
read -p "And what is your mama's name?" mama
echo Amazing...
sleep2s
read -p "And how many years have you been a little bitch?" years
echo Understood...
sleep 2s
echo So let me get this straight...
echo Your favorite fruit is $fruit
echo Your mama's name is $mama
echo And you have been a little bitch for $years years.
Now let's say I answered Apples, Martha, 3.
When I do this, my output comes out:
So let me get this straight...
Your favorite fruit is Apples.
Your mama's name is $mama
And you have been a little bitch for $years
Only the first variable is being output properly, the rest isn't. I have tried using different notations for the variables mama and years, changing them for mumuh and yuurs, but no alas.
Is there something obvious I am missing? Am I misunderstanding how the read command handles the created variables?
A: This has nothing to do with read. your script is simply missing some quotes for the string sequences.
so your output lines should more look like this:
echo "So let me get this straight..."
echo "Your favorite fruit is" $fruit
echo "Your mama's name is" $mama
echo "And you have been a little bitch for" $years "years."
you could even wrap the variables in the string like this:
echo "And you have been a little bitch for ${years} years."
for a bit more details, you might want to read this question.
A: The problem lies in this single line:
echo Your mama's name is $mama
because the shell finds an apostrophe (single quote) which must be matched (closed) by another, but there is none.
So, as the previous reply says, it is a problem of quoting, but only this line needs it because of the literal single quote which must be echoed literally and not interpreted.
A mean shell should complain that there is an unmatched quote; my bash says:
Your favorite fruit is Apple
-bash: test.sh: line 15: unexpected EOF while looking for matching `''
Modify that line so it reads:
echo "Your mama's name is " $mama
and you are set.
If it happens that a matching closing single quote is found, the text between the apostrophes is not expanded (no variable substitution), and it seems this happened to you: in fact, the output still contained the "$". Didn't you see an error message? Strange.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,852 |
Q: goto specific cell upper left corner In excel there is a simple function to position active cell at left upper corner of window: Application.Goto Range("H40"), True
Is there anything like that for google sheets? If not can someone help me with the code? thanks.
A: Go To Cell
In your case you would deploy it as gotoCell('H40');
function gotoCell(A1NotationString) {
SpreadsheetApp.getActiveSheet().getRange(A1NotationString).activate();
}
*
*Sheet.getRange()
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 333 |
\section{Introduction}
The Bjorken Sum Rule (BSR)~\cite{Bjorken:1966jh, Bjorken:1969mm}, which describes the polarized spin structure of nucleon, has been measured via polarized deep inelastic scattering (DIS) by various experimental collaborations~\cite{Anthony:1996mw, Abe:1994cp, Abe:1995mt, Abe:1995dc, Abe:1995rn, Abe:1997qk, Abe:1997dp, Abe:1998wq, Anthony:1999py, Anthony:1999rm, Anthony:2000fn, Anthony:2002hy, Adams:1994zd, Adams:1994id, Adams:1995ufa, Adams:1997hc, Adams:1997tq, Ackerstaff:1997ws, Ackerstaff:1998ja, Airapetian:1998wi, Airapetian:2002rw, Airapetian:2006vy, Alexakhin:2006oza, Alekseev:2010hc, Adolph:2015saz, Adolph:2016myg, Wesselmann:2006mw, Slifer:2008xu, Prok:2014ltt}. Using the operator product expansion (OPE), the BSR of the spin structure function can be calculated by separating the perturbative contribution of the matrix elements of local product operators from its non-perturbative contributions~\cite{Bjorken:1966jh, Bjorken:1969mm}, e.g.
\begin{eqnarray}
\Gamma^{p-n}_1(Q^2)&=&\int^1_0 dx[g^p_1(x, Q^2)-g^n_1(x, Q^2)]\nonumber\\
&=&\frac{g_A}{6}\left[1-E_{\rm ns}(Q^2)\right] +\sum\limits_{i=2}^{\infty}\frac{\mu_{2i}^{p-n}(Q^2)}{(Q^2)^{i-1}},
\label{gammapn}
\end{eqnarray}
where $g_1^{p,n}(x, Q^2)$ is the spin-dependent proton or neutron structure function with Bjorken scaling variable $x$, and $g_A$ is the nucleon axial charge. The BSR relates the difference of the proton and the neutron structure functions $\Gamma^p_1$ and $\Gamma^n_1$, and only the flavor non-singlet quark operators appear in perturbative part, resulting as the perturbative non-singlet leading-twist contributions $E_{\rm ns}(Q^2)$. The non-perturbative contribution is generally power suppressed in comparison to the leading-twist terms, which has been written as a power series over $1/Q^{2}$. Contributions from the high-twist terms could be sizable in low and intermediate $Q^2$ regions, and then the BSR provides a good platform for testing the perturbative and non-perturbative QCD contributions.
Analyses of $E_{\rm ns}(Q^2)$ under the $\rm{\overline{MS}}$-scheme have been given in the literature, such as Refs.\cite{Deur:2004ti, Deur:2008ej, Chen:2005tda, Deur:2014vea, Blumlein:2016xcy}. Additional treatment on extending the pQCD prediction to low $Q^2$-region has been done by using low-energy models for the strong coupling constant ($\alpha_s$) such as the analytic perturbation theory (APT), the ``massive analytic pQCD theory" (MPT), the $2\delta$- or $3\delta$-analytic QCD variants~\cite{Ayala:2018ulm, Ayala:2020scz, Pasechnik:2008th, Pasechnik:2009yc, Khandramai:2011zd, Khandramai:2013haz}. In all those treatments, there are large renormalization scale ($\mu_r$) dependence for the perturbative part due to the using of ``guessed" $\mu_r$; that is, in those analyses, the central (``optimal") value of $E_{\rm ns}(Q^2)$ is usually derived by setting $\mu_r = Q$, and then by varying it within an arbitrary range such as $[Q/2, 2Q]$ to estimate its uncertainty. Such guessing choice breaks the renormalization group invariance~\cite{Brodsky:2012ms, Wu:2014iba} and leads to conventional renormalization scale-and-scheme ambiguities due to the mismatching of the perturbative coefficients and the $\alpha_s$ at each order. In the literature, the principle of maximum conformality (PMC)~\cite{Brodsky:2011ta, Mojaza:2012mf, Brodsky:2012rj, Brodsky:2013vpa} has been suggested to eliminate such renormalization scale-and-scheme ambiguities. It is well known that the $\alpha_s$-running behavior is governed by the renormalization group equation (RGE). The existence of the $\{\beta_i\}$-terms emerged in the perturbative series is thus helpful for fixing exact $\alpha_s$-value of the pQCD approximant of a physical observable. And instead of choosing an optimal $\mu_r$, the PMC fixes the correct magnitude of $\alpha_s$ by using RGE, whose argument is called as the PMC scale, which is independent to any choice of $\mu_r$. The PMC prediction is scale-and-scheme independent, more detail and applications of the PMC can be found in the reviews~\cite{Wu:2013ei, Wu:2015rga, Wu:2019mky}.
To achieve a reliable prediction for the BSR high-twist contributions, it is important to have an accurate pQCD prediction on $E_{\rm ns}(Q^2)$. In the present paper, we shall first adopt the PMC single-scale approach~\cite{Shen:2017pdu} to deal with the perturbative part of the BSR, and then give a new determination of the non-perturbative high-twist contributions by comparing with the JLab data. The PMC singlet-scale approach follows the same idea of the original multi-scale approach~\cite{Brodsky:2011ta, Mojaza:2012mf, Brodsky:2012rj, Brodsky:2013vpa}, which determines an overall effective momentum flow of the process by using the RGE, whose magnitude corresponds to the weighted average of the multi-scales of the multi-scale approach at each order. It has also been demonstrated that the prediction under the PMC singlet-scale approach is scheme-and-scale independent up to any fixed order~\cite{Wu:2018cmb}. Though different from conventional scale ambiguity, there is residual scale dependence for fixed-order prediction due to unknown perturbative terms~\cite{Zheng:2013uja}. Such residual scale dependence can be greatly suppressed due to both $\alpha_s$-power suppression and exponential suppression. A detailed discussion on the residual scale dependence can be found in the recent review~\cite{Wu:2019mky}.
The remaining parts of the paper are organized as follows. In Sec.II, we present the calculation technology for the polarized Bjorken sum rule $\Gamma^{p-n}_1$. The PMC treatment of the pQCD contributions to the leading-twist part and the non-perturbative high-twist contributions shall be given. In Sec.III, we give the numerical results and discussions. Sec.V is reserved for a summary.
\section{Calculation technology}
In large $Q^2$-region, contributions from the leading-twist terms are dominant and those of the non-perturbative high-twist terms are generally power suppressed. In low and intermediate $Q^2$-region, contributions from the high-twist terms may have large contributions. In the following, we shall analyze the pQCD contributions to the leading-twist terms by using the PMC single-scale approach, and then give an estimation of the contributions from the non-perturbative high-twist terms. In low $Q^2$-region, the low-energy $\alpha_s$ models should be used; and for clarity, we shall adopt four low-energy $\alpha_s$ models to do our discussion.
\subsection{Perturbative series of the leading-twist terms}
The perturbative expansion over $\alpha_s$ for the hard part of the leading-twist terms $E_{\rm ns}(Q^2)$ has been calculated up to next-to-next-to-next-to leading order ($\rm{N^3LO}$), which can be written as
\begin{eqnarray}
E_{\rm ns}(Q^2,\mu_r)=\sum^4_{i=1}r_{i}(\mu_r)a^{i}(\mu_r),
\end{eqnarray}
where $a(\mu_r)=\alpha_s(\mu_r)/\pi$ and the perturbative coefficients $r_i$ are power series of the active flavor numbers $n_f$,
\begin{displaymath}
r_i =c_{i,0}+c_{i,1} n_f+\cdots +c_{i,n-1} n^{n-1}_f.
\end{displaymath}
The explicit expressions of the coefficients $c_{i,j}$ have been given in Refs.\cite{Baikov:2010je, Baikov:2012zm}. To apply the PMC, we need to use the general QCD degeneracy relations~\cite{Bi:2015wea} among different orders to make the transformation of the $n_f$-series to $\{\beta_i\}$-series, i.e. we need to rewrite $E_{\rm ns}(Q^2)$ in the following form,
\begin{eqnarray}
E_{\rm ns}(Q^2)&=&r_{1,0}a(\mu_r)+(r_{2,0}+\beta_0r_{2,1})a^2(\mu_r) \nonumber\\
&+&(r_{3,0}+\beta_1r_{2,1}+2\beta_{0}r_{3,1}+\beta_{0}^2r_{3,2})a^3(\mu_r) \nonumber\\
&+&(r_{4,0}+\beta_2r_{2,1}+2\beta_{1}r_{3,1}+\frac{5}{2}\beta_0\beta_1r_{3,2} \nonumber\\
&+&3\beta_0r_{4,1}+3\beta_0^2r_{4,2}+\beta_0^3r_{4,3})a^4(\mu_r) +\cdots,
\label{leadingtwist}
\end{eqnarray}
where the coefficients $r_{i,j}$ up to $\rm{N^3LO}$-order level are
\begin{eqnarray}
r_{1,0} &=& c_{1,0}, \\
r_{2,0} &=& c_{2,0} + {33\over2} c_{2,1}, \\
r_{2,1} &=& -6 c_{2,1}, \\
r_{3,0} &=&-{321\over8} c_{2,1}+ c_{3,0}+{33\over2} c_{3,1} + {1089\over 4} c_{3,2}, \\
r_{3,1} &=& {57\over4}c_{2,1}-3 c_{3,1}-99 c_{3,2}, \\
r_{3,2} &=& 36 c_{3,2}, \\
r_{4,0} &=& {11675\over 256}c_{2,1}-{321\over8}c_{3,1}-{10593\over8}c_{3,2}+c_{4,0}\nonumber\\
&&+{33\over2}c_{4,1}+{1089\over4}c_{4,2}+{35937\over8}c_{4,3}, \\
r_{4,1} &=& -{479\over16}c_{2,1}+{19\over2}c_{3,1}+{4113\over8}c_{3,2}-2 c_{4,1}\nonumber\\&&
-66c_{4,2}-{3267\over2}c_{4,3},\\
r_{4,2} &=& {325\over48}c_{2,1}-{285\over2}c_{3,2}+12c_{4,2}+594c_{4,3},\\
r_{4,3} &=& -216c_{4,3}.
\end{eqnarray}
Generally, the coefficients $r_{i, {j\neq0}}$ are functions of the logarithm ${\rm ln}(\mu_r^2/ Q^2)$. If setting $\mu_r=Q$, all those types of log-terms becomes zero, leading to a renormalon-free more convergent pQCD series; this explains why people usually choose $\mu_r=Q$ as the optimal scale for conventional scale-setting approach. Those coefficients can be reexpressed as
\begin{eqnarray}
r_{i,j} = \sum_{k=0}^{j} C_j^k \ln^k(\mu_r^2/Q^2) \hat{r}_{i-k,j-k},
\label{rij}
\end{eqnarray}
where the combination coefficients $C_j^k={j!}/{k!(j-k)!}$, and the coefficients $\hat{r}_{i,j}=r_{i,j}|_{\mu_r=Q}$. For convenience, we put the reduced coefficients $\hat{r}_{i,j}$ in the Appendix A.
The RGE, or the $\beta$-function, is defined as
\begin{equation} \label{RGEe}
\beta(a(\mu_r)) =-\sum_{i=0}^{\infty}\beta_{i}a^{i+2}(\mu_r).
\end{equation}
Following the decoupling theorem~\cite{Appelquist:1974tg}, the first two $\{\beta_{i\geq2}\}$-functions $\beta_{0}$ and $\beta_{1}$ are scheme-independent, and we have $\beta_0=\frac{1}{4}(11-\frac{2}{3}n_f)$ and $\beta_1=\frac{1}{4^2}(102-\frac{38}{3} n_f)$ for the ${\rm SU_{C}}(3)$-color group. The scheme dependent $\{\beta_{i\geq2}\}$-functions have been calculated up to five-loop level under the $\overline{\rm MS}$-scheme~\cite{Gross:1973id, Politzer:1973fx, Caswell:1974gg, Tarasov:1980au, Larin:1993tp, vanRitbergen:1997va, Chetyrkin:2004mf, Czakon:2004bu, Baikov:2016tgj}. A collection of all the known $\{\beta_{i}\}$-functions can be found in Ref.\cite{Wu:2013ei}.
In Eq.(\ref{leadingtwist}), the $\{\beta_i\}$-terms at each perturbative order govern the correct $\alpha_s$-running behavior, which inversely can be used to determine the effective magnitude of $\alpha_s$. Practically, by requiring all the RGE-involved non-conformal $\{\beta_i\}$-terms to be zero, one can achieve an overall effective $\alpha_s$ and hence the PMC scale $Q_\star$, and then the resultant pQCD series becomes the following scheme-independent conformal series:
\begin{equation}
E_{\rm ns}|_{\rm PMC}(Q^2) = \sum^{4}_{i\ge1} \hat{r}_{i,0} a^{i}(Q_\star),
\label{Enspmc}
\end{equation}
where the PMC scale $Q_{\star}$ can be fixed up to next-to-next-to-leading-log ($\rm NNLL$) accuracy by using the $\rm{N^3LO}$ perturbative series, e.g.
\begin{eqnarray}
\ln{\frac{Q_{\star}^2}{Q^2}}&=& T _{0} +T_{1} {\alpha_{s}(Q)\over \pi}+T_{2} {\alpha_{s}^{2}(Q)\over \pi},
\label{Enspmcscale}
\end{eqnarray}
where
\begin{widetext}
\begin{eqnarray}
T_0 &=& -{\hat{r}_{2,1}\over \hat{r}_{1,0}}, \;\;
T_1 = {2(\hat{r}_{2,0}\hat{r}_{2,1}-\hat{r}_{1,0}\hat{r}_{3,1})\over \hat{r}_{1,0}^2} +{\hat{r}_{2,1}^2-\hat{r}_{1,0}\hat{r}_{3,2}\over \hat{r}_{1,0}^2}\beta_0, \\
T_{2}&=&\frac{4 (\hat{r}_{1,0}\hat{r}_{2,0}\hat{r}_{3,1}-\hat{r}^2_{2,0}\hat{r}_{2,1})
+3(\hat{r}_{1,0}\hat{r}_{2,1}\hat{r}_{3,0}-\hat{r}^2_{1,0}\hat{r}_{4,1})}{ \hat{r}^3_{1,0}}+\frac{3(\hat{r}^2_{2,1}-\hat{r}_{1,0}\hat{r}_{3,2})}{2\hat{r}^2_{1,0}}
\beta_1\nonumber\\
&&-\frac{3\hat{r}_{2,0}\hat{r}^2_{2,1}- 4\hat{r}_{1,0}\hat{r}_{2,1} \hat{r}_{3,1}-2\hat{r}_{1,0}\hat{r}_{2,0}\hat{r}_{3,2}
+3\hat{r}^2_{1,0}\hat{r}_{4,2}}{\hat{r}^3_{1,0}}\beta_0 +\frac{2\hat{r}_{1,0}\hat{r}_{2,1}\hat{r}_{3,2} -\hat{r}^2_{1,0}\hat{r}_{4,3} -\hat{r}^3_{2,1}}{\hat{r}^3_{1,0}}\beta^2_0 .
\end{eqnarray}
\end{widetext}
Those equations show the PMC scale $Q_{\star}$ is exactly free of $\mu_r$, together with the $\mu_r$-independent conformal coefficients $\hat{r}_{i,0}$, the PMC prediction is exactly independent to any choice of $\mu_r$. Thus the conventional scale-setting ambiguity can be eliminated at any fixed-order by applying the PMC~\cite{Wu:2018cmb}. As a byproduct, due to the elimination of divergent renormalon terms in the resultant PMC perturbative series (\ref{Enspmc}), the pQCD convergence can be naturally improved. Those properties greatly improve the precision of the pQCD theory.
For a perturbative theory, it is important to have a reliable way to estimate the magnitude of the uncalculated higher-order terms. The scale-invariant and scheme-invariant PMC conformal series, which is also more convergent than the conventional series, is quite suitable for such purpose. A way of using the PMC series together with the Pad$\acute{e}$ approximation approach (PAA)~\cite{Basdevant:1972fe, Samuel:1992qg, Samuel:1995jc} has been suggested in Ref.\cite{Du:2018dma}. Some successful applications of this method can be found in Refs.\cite{Yu:2020tri, Huang:2020rtx, Yu:2019mce, Yu:2018hgw}. We shall adopt this method to estimate the magnitude of the unknown ${\cal O}(\alpha_s^5)$-terms of $E_{\rm ns}(Q^2)$, and in the following, we give a brief introduction of PAA.
The PAA offers a feasible conjecture that yields the $(n+1)_{\rm th}$-order coefficient by using a given $n_{\rm th}$-order perturbative series. For the purpose, people usually adopts a fractional function as the generating function. More explicitly, the $[N/M]$-type generating function of a pQCD approximant $\rho_n(Q)=\sum\limits_{i=1}^{n} \hat{r}_{i,0} a^{i}$ is defined as
\begin{eqnarray}
\rho^{[N/M]}_n(Q) &=& a \times \frac{b_0+b_1 a + \cdots + b_N a^N}{1 + c_1 a + \cdots + c_M a^M} \label{PAAseries0} \\
&=& \sum_{i=1}^{n} C_{i} a^{i} + C_{n+1}\; a^{n+1}+\cdots, \label{PAAseries}
\end{eqnarray}
where $M\geq 1$ and $N+M+1=n$. The perturbative coefficients $C_i$ in Eq.(\ref{PAAseries}) can be expressed by the known coefficients $b_{i\in[0,N]}$ and $c_{j\in[1,M]}$. Inversely, if we have known the coefficients $C_i$'s up to $n_{\rm th}$-order level, one can determine the coefficients $b_{i\in[0,N]}$ and $c_{j\in[1,M]}$, and then achieve a prediction for the uncalculated $(n+1)_{\rm th}$-order coefficient $C_{n+1}$.
At the present, the leading-twist term $E_{\rm ns}|_{\rm PMC}$ has been known up to $\rm{N^3LO}$-level, and the four coefficients are known, $C_{i}=\hat{r}_{i,0}$ for $i\in[1, 4]$. Then the predicted ${\rm N^4LO}$-coefficient becomes
\begin{eqnarray}
\hat{r}_{5,0} =&& \frac{\hat{r}^4_{2,0} -3\hat{r}_{1,0} \hat{r}^2_{2,0} \hat{r}_{3,0} +\hat{r}^2_{1,0} \hat{r}^2_{3,0} +2\hat{r}^2_{1,0} \hat{r}_{2,0} \hat{r}_{4,0}}{\hat{r}^3_{1,0}},
\label{PAA03}
\end{eqnarray}
where the $[0/n-1]$-type PAA generating function has been implicitly adopted, which is the preferable type for the convergent PMC series~\cite{Du:2018dma}.
\subsection{Contributions from the non-perturbative high-twist terms}
The non-perturbative contributions to the BSR can be expanded in $1/Q^{2}$-power series as Eq.(\ref{gammapn}). The $\mathcal{O}(Q^{-2})$-term $\mu_4^{p-n}$ can be written as~\cite{Ji:1993sv, Shuryak:1981pi, Kawamura:1996gg}
\begin{eqnarray}
\mu_4^{p-n}=\frac{M^2}{9}(a^{p-n}_2 +4d^{p-n}_2 +4f^{p-n}_2),
\label{twist4}
\end{eqnarray}
where $M\approx0.94~\rm GeV$ is the nucleon mass. The leading-twist target mass correction $a^{p-n}_2$ can be calculated by using the leading-twist part of $g^{p-n}_1$, which is kinematically of high-twist~\cite{Blumlein:1998nv} and its magnitude at $Q^2=1~\rm{GeV^2}$ is $0.031\pm0.010$~\cite{Deur:2014vea}. The twist-3 matrix element $d^{p-n}_2$ is given by
\begin{eqnarray}
d^{p-n}_2=\int^1_0 dx x^2(2g_1^{p-n}+3g_2^{p-n}),
\end{eqnarray}
whose magnitude at $Q^2=1~\rm{GeV^2}$ is $0.008\pm0.0036$~\cite{Deur:2014vea}. The dynamical values of the twist-2 and twist-3 contributions can be measured by polarized lepton scattering off transversely and longitudinally polarized target. The twist-2 and twist-3 contributions are calculated by the $x^2$-weighted moment of the structure function in orders of $M^2/Q^2$, thus $a^{p-n}_2$ and $d^{p-n}_2$ change logarithmically, and we shall fix their values to be the above ones at $Q^2=1~\rm{GeV^2}$. Then the remaining undetermined term in $\mu_4^{p-n}$ is $f^{p-n}_2$. The twist-4 term $f^{p-n}_2$, which is related to the color electric and magnetic polarizabilities of nucleon, plays a pivotal role in phenomenological studies of the high-twist contributions. $f^{p-n}_2$ is sensitive to $Q^2$ and its $Q^2$-evolution satisfies~\cite{Shuryak:1981pi, Kawamura:1996gg}
\begin{eqnarray}
f^{p-n}_2(Q^2)=f^{p-n}_2(1)\left(\frac{a(Q)}{a(1)} \right)^{\gamma_0/8\beta_0},
\end{eqnarray}
where $\gamma_0/8\beta_0=32/81$ with $n_f=3$. The magnitude of $f^{p-n}_2(1)$ shall be fit by comparing with the data. Moreover, it has been argued that the $\mathcal{O}(Q^{-4})$-term $\mu_6^{p-n}$ may also have sizable contribution, so we take $\mu_6/Q^4$-term into consideration to have a better fit of the data.
\subsection{The strong coupling constant $\alpha_s$}
The $\alpha_s$-running behavior in perturbative region is governed by the RGE (\ref{RGEe}). Its solution can be written as an expansion over the inverse powers of the logarithm $L=\ln{\mu^{2}_{r} / \Lambda^2}$; and up to four-loop level, we have~\cite{Chetyrkin:1997sg}
\begin{eqnarray}
\alpha_s(\mu_r)&=&{\pi\over\beta_0L}\bigg\{1-{\beta_1\over\beta^2_0}{\ln L\over L}+{1\over\beta^2_0L^2} \left[{\beta^2_1\over\beta^2_0}(\ln^2L-\ln L \right. \nonumber\\
&& \left. -1)+{\beta_2\over\beta_0} \right] +{1\over\beta^3_0L^3} \bigg[{\beta^3_1\over\beta^3_0}(-\ln^3L+{5\over2}\ln^2L\nonumber\\
&&+2\ln L-{1\over2})-3{\beta_1\beta_2\over\beta^2_0}\ln L+{\beta_3\over2\beta_0}\bigg] \bigg\},
\end{eqnarray}
where $\Lambda$ is the scheme-dependent asymptotic scale, which could be fixed by matching the measured value of $\alpha_s$ at a reference scale such as $M_Z$ or $m_\tau$ to its predicted value under a specific scheme.
In infrared region, when the scale is close to $\Lambda$ or even smaller, $\alpha_s$ becomes large whose magnitude cannot be well described by the RGE. To make the QCD prediction more reliable, we shall adopt four low-energy models for the $\alpha_s$ to do our calculation.
The first low-energy model is based on the analytical perturbation theory (APT)~\cite{Shirkov:1997wi, Shirkov:1997nx}, and we call it as the APT model. In APT model, its strong coupling constant $\alpha^{\rm{APT}}_s$ is described by applying the perturbation theory directly to the spectral function, which takes the following form,
\begin{eqnarray}
\alpha^{\rm{APT}}_s(\mu)&=&\frac{\pi}{\beta_0}\bigg(\frac{1}{\ln {\rm y}}+\frac{1}{1-{\rm y}}\bigg),
\label{asapt}
\end{eqnarray}
where $\mu$ is the energy scale, ${\rm y}={\mu^2}/{\Lambda^2}$ with
\begin{equation}
\Lambda^2=\mu^2 {\rm exp}[-\phi(\beta_0\alpha_s(\mu)/\pi)],
\label{Lambdaapt}
\end{equation}
where $\phi(z)$ satisfies $1/\phi(z)+1/(1-{\rm exp}[\phi(z)])=z$. Its freezing value is close to ${\alpha^{\rm{APT}}_s(10^{-10})/\pi}\approx0.43$.
The second low-energy model is an alteration of Eq.(\ref{asapt}), we call it as the WEB model~\cite{Webber:1998um}, which is suggested to suppress the nonperturbative power corrections of the APT model, and it takes the following form
\begin{eqnarray}
\alpha^{\rm WEB}_{s}(\mu)=\frac{\pi}{\beta_0}\bigg[\frac{1}{\ln {\rm y}}+\frac{{\rm y}+b}{(1-{\rm y})(1+b)}(\frac{1+c}{{\rm y}+c})^p\bigg],
\end{eqnarray}
where these phenomenological parameters $b=1/4$ and $p=c=4$. The obtained corresponding approximate freezing value $\sim \alpha^{\rm WEB}(10^{-10})/\pi\approx0.21$.
The third low-energy model is based on the ``massive analytic pQCD theory" (MPT)~\cite{Shirkov:1999hm, Shirkov:2012ux}, which takes the phenomenological glue-ball mass $m_{gl}=\sqrt{\xi}\Lambda$ as the infrared regulator, and we call it as the MPT model. It takes the following form
\begin{eqnarray}
\alpha^{\rm{MPT}}_s(\mu)&=&a_{cr}\bigg\{1+a_{cr}\frac{\beta_{0}}{\pi}\ln\left(1+ \frac{\mu^2}{m_{gl}^2}\right)+a_{cr}\frac{\beta_1}{\pi\beta_0}\times \nonumber\\
&&\ln\left[1+a_{cr}\frac{\beta_{0}}{\pi}\ln\left( 1+ \frac{\mu^2}{m_{gl}^2}\right)\right] +...\bigg\}^{-1},
\end{eqnarray}
whose freezing value at the origin satisfies $a_{cr}=\pi/(\beta_0 \ln\xi)$. Under the Landau gauge, we have $a_{cr}|_{\xi=10\pm2}=0.61\mp0.05$, which leads to the freezing point ${\alpha^{\rm{MPT}}_s(0)/\pi}=0.19^{-0.01}_{+0.02}$.
The fourth low-energy model is based on the continuum theory~\cite{Halzen:1992vd} and we call it as the CON model, where the exchanging gluons with effective dynamical mass $m_g$ is adopted and the non-perturbative dynamics of gluons is governed by the corresponding Schwinger-Dyson equation. It takes the following from
\begin{eqnarray}
\alpha^{\rm{CON}}_s(\mu)=\frac{\pi}{\beta_0\ln\left(\frac{4M^2_g+\mu^2}{\Lambda^2}\right)},
\end{eqnarray}
whose $M^2_g=m^2_g[\ln({\rm y}+4m^2_g/\Lambda^2)/\ln(4m^2_g/\Lambda^2)]^{-12/11}$ and $m_g=500\pm200$ MeV~\cite{Halzen:1992vd, cornwall:1982dy}, which leads to the freezing point ${\alpha^{\rm{CON}}_s(0)/\pi}=0.21^{-0.05}_{+0.19}$.
\section{Numerical results}
\begin{figure}[h]
\centering
\includegraphics[width=0.48\textwidth]{coupling.eps}
\caption{Typical $\alpha_s$-running behavior in low-energy scales for four typical low-energy models, APT, WEB, MPT, and CON, respectively. The $\alpha_s$-running behavior derived from RGE under $\overline{\rm MS}$-scheme is given as a comparison.}
\label{coupling}
\end{figure}
To do the numerical analysis, we take the nucleon axial charge ratio $g_A=1.2724\pm0.0023$~\cite{PDG:2020}. The asymptotic QCD scale $\Lambda$ can be fixed by using the $\alpha_s$-value at the reference point such as $\alpha^{\overline{\rm MS}}_s(m_\tau)=0.325\pm0.016$~\cite{PDG:2020}, which gives $\Lambda_{\overline{\rm MS}}|_{n_f=3}=0.346^{+0.028}_{-0.029}$ GeV by using the four-loop RGE. Using the relation (\ref{Lambdaapt}), we obtain $\Lambda_{\rm APT}|_{n_f=3}=0.244^{+0.033}_{-0.031}$ GeV. In Fig.~\ref{coupling}, we present the typical running behaviors of $\alpha_s/\pi$ under four low-energy models, where the parameters are set to be $\xi=10$ for MPT and $m_g=700~\rm MeV$ for CON, respectively. The $\alpha_s$-running behavior derived from the RGE under $\overline{\rm MS}$-scheme is given as a comparison. Fig.~\ref{coupling} shows the importance of the using of low-energy models in the region of small energy-scale. Using the criteria suggested in Ref.\cite{Deur:2014qfa} for the analytic matching of $\alpha_s$ in perturbative and nonperturbative regimes, we obtain the transition scales ($Q_0$) for various low-energy models, which are $\sim 1.77$ GeV, $\sim 1.78$ GeV, $\sim 1.78$ GeV, $\sim 1.19$ GeV for APT, WEB, MPT and CON models, respectively. As a subtle point, because the transition scales $Q_0$ for the cases of WEB and MPT are slightly bigger than $m_\tau$, and for self-consistency, we use the low-energy $\alpha^{\rm WEB/MPT}_s(m_\tau)=0.325\pm0.016$ to fix $\Lambda$, which is $0.206\pm{0.022}$ GeV or $0.294^{+0.033}_{-0.032}$ GeV, respecitvely.
For later convenience, in the following discussions, we simply use $\alpha_s^{\overline{\rm MS}}$ to stand for the case of using $\overline{\rm MS}$-scheme $\alpha_s$ in all $Q^2$-region, $\alpha_s^{\rm APT}$ to stand for the case of using APT model in low-energy region ($Q<Q_0$, as mentioned above, $Q_0$ is different for different low-energy model) and $\overline{\rm MS}$-scheme $\alpha_s$ in large $Q^2$-region, $\alpha_s^{\rm WEB}$ to stand for the case of using WEB model in low-energy model and $\overline{\rm MS}$-scheme $\alpha_s$ in large $Q^2$-region, $\alpha_s^{\rm MPT}$ to stand for the case of using MPT model in low-energy model and $\overline{\rm MS}$-scheme $\alpha_s$ in large $Q^2$-region, and $\alpha_s^{\rm CON}$ to stand for the case of using CON model in low-energy model and $\overline{\rm MS}$-scheme $\alpha_s$ in large $Q^2$-region.
\subsection{Perturbative contributions to the leading-twist part of BSR up to $\rm{N^4LO}$ level}
\begin{figure*}[htb]
\centering
\subfigure[Leading-twist contributions using $\alpha^{\rm APT}_s$.]{
\includegraphics[width=0.45\textwidth]{apt0.eps}
}
\quad
\subfigure[Leading-twist contributions using $\alpha^{\rm WEB}_s$.]{
\includegraphics[width=0.45\textwidth]{web0.eps}
}
\quad
\subfigure[Leading-twist contributions using $\alpha^{\rm MPT}_s$.]{
\includegraphics[width=0.45\textwidth]{mpt0.eps}
}
\quad
\subfigure[Leading-twist contributions using $\alpha^{\rm CON}_s$.]{
\includegraphics[width=0.45\textwidth]{con0.eps}
}
\caption{Perturbative leading-twist contributions to the spin structure function $\Gamma^{p-n}_1(Q^2)$ up to $\rm{N^3LO}$ versus momentum Q, under four $\alpha_s$ models: (a) the APT model; (b) The WEB model; (c) the MPT model; and (d) the CON model. The solid line is for conventional scale setting approach with $\mu_r=Q$ and the shaded band shows its scale uncertainty by varying $\mu_r\in[Q/2,2Q]$. The dot-dashed line is the prediction $\Gamma^{p-n}_1(Q^2)$ up to $\rm{N^4LO}$ for PMC scale-setting approach, which is free of renormalization scale dependence.}
\label{leadingBSR}
\end{figure*}
The perturbative contributions to the leading-twist part $E_{\rm ns}(Q^2)$ has been known up to $\rm{N^3LO}$. Under conventional scale-setting approach, the pQCD series is scale dependent, and by setting $\mu_r=Q$, we obtain
\begin{eqnarray}
E_{\rm ns}(Q^2)|_{\rm Conv.}&=& a(Q) +3.58 a^2(Q) +20.22 a^3(Q)\nonumber\\
&&+175.70 a^4(Q).
\label{EnscoefficientsC}
\end{eqnarray}
On the other hand, the pQCD series becomes scale invariant by applying the PMC, and we obtain
\begin{eqnarray}
E_{\rm ns}(Q^2)|_{\rm PMC}&=& a(Q_\star) +1.15 a^2(Q_\star) +0.14 a^3(Q_\star) \nonumber\\
&&+0.76 a^4(Q_\star)
\label{EnscoefficientsS}
\end{eqnarray}
for any choice of renormalization scale, where $Q_\star$ is of perturbative nature, which can be determined up to NNLL accuracy
\begin{eqnarray}
\ln\frac{{Q_{*}}^2}{Q^2}&=&-1.08 -1.87 a(Q) -24.06 a^2(Q).
\end{eqnarray}
One may observe that the perturbative coefficients in PMC series (\ref{EnscoefficientsS}) are much smaller than those of conventional series (\ref{EnscoefficientsC}), especially for those of high-orders, which are due to the elimination of divergent renormalon terms as $n!\beta^n_0 a_s^n$. This indicates that a much more convergent perturbative series can be achieved by applying the PMC. At the same time, the PMC scale $Q_\star$ also shows a fast convergent at high $Q$-range, e.g. the relative absolute values of the LL, the NLL and the NNLL terms are 1: 0.064 : 0.030 for $Q=100$ GeV. Thus the residual scale dependence due to unknown even higher-order terms can be greatly suppressed.
Using the convergent PMC perturbative series, one can obtain a reliable prediction of unknown $\mathcal{O}(a^5)$-term by using the PAA, e.g. by using Eq.(\ref{PAA03}), we obtain
\begin{equation}
E_{\rm ns}(Q^2)|^{\rm N^4LO}_{\rm PAA}=2.92 a^5(Q_\star).
\end{equation}
We present the predicted leading-twist part of the spin structure function $\Gamma^{p-n}_1(Q^2)$ under four low-energy models in Fig.~\ref{leadingBSR}, where the results under conventional and PMC scale-setting approaches are presented. The experimental data are from SLAC~\cite{Abe:1994cp, Abe:1995mt, Abe:1995dc, Abe:1995rn, Abe:1998wq}, DESY~\cite{Ackerstaff:1997ws, Ackerstaff:1998ja, Airapetian:1998wi, Airapetian:2002rw, Airapetian:2006vy}, CREN~\cite{Alexakhin:2006oza, Alekseev:2010hc, Adolph:2015saz} and JLab~\cite{Deur:2004ti, Deur:2008ej, Deur:2014vea}. The PMC predictions are independent to any choice of $\mu_r$, and the shaded band shows the conventional renormalization scale uncertainty by varying $\mu_r\in[Q/2, 2Q]$. Under conventional scale-setting approach, the spin structure function $\Gamma^{p-n}_1(Q^2)$ shows large scale dependence, especially in low-energy region. In low-energy region, the results by using the IR-fixed couplings are much more reliable. And since couplings behaves differently in low-energy region, the spin structure function $\Gamma^{p-n}_1(Q^2)$ behaves quite differently for $Q\to 0$. When the energy scale is large enough, such as $Q> 1.5-2.0$ GeV, the perturbative leading-twist terms could explain the experimental data well. Fig.~\ref{leadingBSR} also shows that in low-scale region, the leading-twist terms alone cannot explain the data and one must take the high-twist terms into consideration. By comparing with the data, this fact inversely provides us a good platform to achieve reliable predictions on the magnitudes of high-twist contributions.
\subsection{Analysis of high-twist contributions under various low-energy models}
\begin{table*}[htb]
\centering
\begin{tabular}{ c c c c c c c c c}
\hline
& ~$\alpha_s$ models ~ &~ & ~~~$f_2^{p-n}(1)$~~~ &~~~$\mu_6$~~~ & ~~~$\chi^2/d.o.f$ \\
\hline
&~ &$\mu_r=Q/2$ &~~$-0.176\pm{0.000}\pm{0.013}$~~ & ~~$0.004\pm{0.000}\pm{0.000}$~~ &149 \\
&$\rm{APT}|_{Conv}$ &$\mu_r=Q$ &$-0.088\pm{0.000}\pm{0.013}$ &$0.002\pm{0.000}\pm{0.000}$ &62 \\
&~ &$\mu_r=2Q$ &$-0.107\pm{0.000}\pm{0.013}$ &$0.003\pm{0.000}\pm{0.000}$ &117 \\
&$\rm{APT|_{PMC}}$ &$\mu_r\in[Q/2,2Q]$ &$-0.120\pm{0.000}\pm{0.013}$ &$0.003\pm{0.000}\pm{0.000}$ &62 \\
\hline
&~ &$\mu_r=Q/2$ &$-0.193\pm{0.000}\pm{0.013}$ &$0.005\pm{0.000}\pm{0.000}$ &193 \\
&$\rm{WEB}|_{Conv}$ &$\mu_r=Q$ &$-0.047\pm{0.000}\pm{0.013}$ &$0.001\pm{0.000}\pm{0.000}$ &168 \\
&~ &$\mu_r=2Q$ &$-0.105\pm{0.000}\pm{0.013}$ &$0.004\pm{0.000}\pm{0.000}$ &160 \\
&$\rm{WEB|_{PMC}}$ &$\mu_r\in[Q/2,2Q]$ &$-0.081\pm{0.000}\pm{0.013}$ &$0.001\pm{0.000}\pm{0.000}$ & 45 \\
\hline
&~ &$\mu_r=Q/2$ &$-0.173\pm{0.000}\pm{0.013}$ &$0.004\pm{0.000}\pm{0.000}$ &151 \\
&$\rm{MPT}|_{Conv}$ &$\mu_r=Q$ &$-0.080\pm{0.000}\pm{0.013}$ &$0.002\pm{0.000}\pm{0.000}$ &56\\
&~ &$\mu_r=2Q$ &$-0.105\pm{0.000}\pm{0.013}$ &$0.003\pm{0.000}\pm{0.000}$ &126\\
&$\rm{MPT|_{PMC}}$ &$\mu_r\in[Q/2,2Q]$ &$-0.128\pm{0.000}\pm{0.013}$ &$0.003\pm{0.000}\pm{0.000}$ &50 \\
\hline
&~ &$\mu_r=Q/2$ &$-0.175\pm{0.000}\pm{0.013}$ &$0.003\pm{0.000}\pm{0.000}$ &125 \\
&$\rm{CON}|_{Conv}$ &$\mu_r=Q$ &$-0.070\pm{0.000}\pm{0.013}$ &$0.001\pm{0.000}\pm{0.000}$ &60 \\
&~ &$\mu_r=2Q$ &$-0.102\pm{0.000}\pm{0.013}$ &$0.002\pm{0.000}\pm{0.000}$ &138 \\
&$\rm{CON|_{PMC}}$ &$\mu_r\in[Q/2,2Q]$ &$-0.139\pm{0.001}\pm{0.013}$ &$0.002\pm{0.000}\pm{0.000}$ &49 \\
\hline
\end{tabular}
\caption{The fitted parameters $f^{p-n}_2(Q^2=1~\rm GeV^2)$ and $\mu_6$ and their corresponding quality of fit $\chi^2/d.o.f$ under four $\alpha_s$ models before and after applying the PMC, where the first and the second errors are caused by the statistical and systematic errors of the data~\cite{Deur:2008ej, Deur:2014vea}. The twist-6 coefficient $\mu_6$ is almost independent to the choices of statistical and systematic errors.}
\label{fitf2mu6}
\end{table*}
Following the discussions of Sec.II.B, we need to fit two parameters, $f^{p-n}_2(1~\rm GeV^2)$ and $\mu_6$, so as to determine the high-twist contributions. We adopt the most recent data listed in Refs.\cite{Deur:2008ej, Deur:2014vea} to do the fitting, whose momentum transfer lies in the range of ${0.054}~{\rm GeV^2} \leq Q^2\leq 4.739~{\rm GeV^2}$. We adopt the APT, WEB, MPT, and the CON couplings in doing the fitting. The quality of fit is measured by the parameter of $\chi^2/ d.o.f$, e.g.
\begin{equation}
\chi^2/{d.o.f} = {1\over {N-d}}\sum\limits^{N}_{j=1}\frac{(\Gamma^{p-n}_{1, {\rm the.}}(Q^2_j) -\Gamma^{p-n}_{1,{\rm exp.}}(Q^2_j))^2}{\sigma^2_{j,{\rm stat.}}},
\end{equation}
where the symbol ``$d.o.f$" (short notation of the degree of freedom) is equal to $N-d$ with $N=31$ being the number of data points and $d=2$ being the number of fitted parameters, ``the." stands for theoretical prediction, ``exp." stands for measured value, and ``$\sigma_{j,{\rm stat.}}$" is the statistical error at each point $Q_j$. Comparing theoretical prediction $\Gamma^{p-n}_{1, {\rm the.}}(Q^2_j)$ with the measured value $\Gamma^{p-n}_{1, {\rm exp.}}(Q^2_j)$ at all the data points $Q_{j\in[1,N]}$, we can derive the preferable $f^{p-n}_2$ and $\mu_6$ by requiring them to achieve the minimum value of $\chi^2/d.o.f$. To do the fitting, we also take into account the systematic error $\sigma_{j, {\rm sys.}}$ at each point $Q_j$, which has sizable contributions to the fitted values of $f^{p-n}_2$ and $\mu_6$. For convenience, we put the detailed calculation technology in Appendix B.
Our results for the two parameters $f^{p-n}_2(1~\rm GeV^2)$ and $\mu_6$ are presented in Table.~\ref{fitf2mu6}. The right-most column shows the smallest $\chi^2/d.o.f$ for the predictions before and after applying the PMC under four $\alpha_s$ models. The magnitudes of those two parameters are small, which agree with the usual consideration that at large $Q^2$-region, the high-twist terms are power suppressed and are negligible. However in low $Q^2$-region, they will have sizable contributions; especially $f^{p-n}_2(1~\rm GeV^2)$ is important for a reliable theoretical prediction on $\Gamma^{p-n}_{1, {\rm the.}}(Q^2)$ in low $Q^2$-region. Table.~\ref{fitf2mu6} shows that the fitted parameters under conventional scale-setting approach have strong scale dependence, whose quality of fit $\chi^2/d.o.f$ varies from tens to hundreds, and the optimal fit are achieved for the case of $\mu_r\sim Q$. This, together with a better pQCD convergence due to the elimination of divergent log-terms $\ln\mu_r^2/Q^2$, in some sense explain why $\mu_r=Q$ is usually taken as the preferable renormalization scale for conventional scale-setting approach. On the other hand, the fitted parameters for the PMC scale-setting approach is independent for any choice of renormalization scale, thus a more reliable and accurate prediction is achieved.
\begin{figure}[htb]
\centering
\includegraphics[width=0.48\textwidth]{f2pnv2.eps}
\caption{The twist-4 coefficient $f_2^{p-n}(1{\rm~GeV^2})$ obtained from the PMC predictions under four $\alpha_s$ low-energy models, in which the predictions using JLab data~\cite{Deur:2004ti, Deur:2008ej, Deur:2014vea}, the QCD sum rule predictions~\cite{Stein:1995si, Balitsky:1989jb}, and the predictions using the model of the instanton-based QCD vacuum~\cite{Lee:2001ug, Sidorov:2006vu} and the Bag model prediction~\cite{Ji:1993sv} are also presented. }
\label{f2pn}
\end{figure}
At present, the twist-4 coefficient $f^{p-n}_2(1~\rm GeV^2)$ has been calculated under various approaches, such as Refs.~\cite{Stein:1995si, Balitsky:1989jb, Lee:2001ug, Sidorov:2006vu, Ji:1993sv, Deur:2004ti, Deur:2008ej, Deur:2014vea, Balla:1997hf}. We present a comparison of various predictions in Fig.~\ref{f2pn}. The results of Refs.\cite{Deur:2004ti, Deur:2008ej, Deur:2014vea} are fitted by using conventional pQCD series for the leading-twist part with fixing $\mu_r=Q$ and the JLab data within different ranges, $0.8~{\rm GeV^2}<Q^2_j<10~{\rm GeV^2}$~\cite{Deur:2004ti}, $0.66~{\rm GeV^2}<Q^2_j<10~{\rm GeV^2}$~\cite{Deur:2008ej} and $0.84~{\rm GeV^2}<Q^2_j<10~{\rm GeV^2}$~\cite{Deur:2014vea}. By using $f^{p-n}_2$, we can evaluate the color polarizability, $\chi_E^{p-n}={2\over3}(2d^{p-n}_2+f^{p-n}_2)$ and $\chi_B^{p-n}={1\over3}(4d^{p-n}_2-f^{p-n}_2)$, which describes the response of the color magnetic and electric fields to the spin of the nucleon~\cite{Ji:1995qe, Stein:1995si}. Using the PMC predictions for the hard-part of the leading-twist contributions, we obtain
\begin{eqnarray}
\chi_B^{p-n}|_{\rm{APT}}&=&0.051\pm0.009, \\
\chi_B^{p-n}|_{\rm{WEB}}&=&0.038\pm0.009, \\
\chi_B^{p-n}|_{\rm{MPT}}&=&0.053\pm0.009, \\
\chi_B^{p-n}|_{\rm{CON}}&=&0.057\pm0.009, \\
\chi_E^{p-n}|_{\rm{APT}}&=&-0.069\pm0.013,\\
\chi_E^{p-n}|_{\rm{WEB}}&=&-0.043\pm0.013,\\
\chi_E^{p-n}|_{\rm{MPT}}&=&-0.075\pm0.013,\\
\chi_E^{p-n}|_{\rm{CON}}&=&-0.082\pm0.013,
\end{eqnarray}
where the errors are squared average of those from $\Delta d^{p-n}_2=\pm0.0036$ and $\Delta f^{p-n}_2$ for the four low-energy $\alpha_s$ models (e.g. Table.~\ref{fitf2mu6}).
\begin{figure*}[htb]
\centering
\subfigure[APT model]{
\includegraphics[width=0.45\textwidth]{aptfit46.eps}
}
\quad
\subfigure[WEB model]{
\includegraphics[width=0.45\textwidth]{webfit46.eps}
}
\quad
\subfigure[MPT model]{
\includegraphics[width=0.45\textwidth]{mptfit46.eps}
}
\quad
\subfigure[CON model]{
\includegraphics[width=0.45\textwidth]{confit46.eps}
}
\caption{The spin structure function $\Gamma^{p-n}_1(Q^2)$ with both leading-twist and high-twist contributions under four $\alpha_s$ models: (a) the APT model; (b) the WEB model; (c) the MPT model; and (d) the CON model. The leading-twist perturbative contributions have been calculated up to ${\rm N^3LO}$ level and ${\rm N^4LO}$ level before and after applying the PMC scale-setting approach, respectively. The shaded band shows the prediction under conventional scale-setting approach by varying $\mu_r\in[Q/2,2Q]$. The solid line is the scale-invariant PMC prediction.}
\label{higherBSR}
\end{figure*}
We present the prediction of $\Gamma^{p-n}_1(Q^2)$ with both leading-twist and high-twist contributions in Fig.~\ref{higherBSR}. Comparing with Fig.~\ref{leadingBSR}, Fig.~\ref{higherBSR} shows that a more reasonable prediction can be achieved by including high-twist contributions. Under conventional scale-setting approach, the large scale dependence for the leading-twist prediction of $\Gamma^{p-n}_{1, {\rm Conv.}}(Q^2)$ can be greatly suppressed by including high-twist terms due to the cancellation of scale dependence among different twist-terms. Under PMC scale-setting approach, the scale-invariant $\Gamma^{p-n}_{1, {\rm PMC}}(Q^2)$ under APT, MPT and CON $\alpha_s$ models are close in shape, which as shown by Table.~\ref{fitf2mu6} also have close quality of fit $\chi^2/d.o.f$; while the PMC prediction under WEB model is slightly different from those of other $\alpha_s$ models.
As a final remark, to improve the quality of fit, as suggested by Ref.\cite{Ayala:2018ulm}, we use the JLab data points with $Q^2 > 0.268~{\rm GeV^2}$ to do fit. By using the scale-invariant PMC pQCD series, the quality of fit $\chi^2/d.o.f$ improves to be $\sim 34$ for APT model, $\sim 52$ for WEB model, $\sim 34$ for MPT model and $\sim 38$ for CON model, respectively, which correspond to the $p$-value around $95\%-99\%$~\cite{PDG:2020}.
\subsection{An analysis of high-twist contributions with massive high-twist expression}
As shown by Fig.~\ref{higherBSR}, the predictions drops down quickly in very small $Q^2$-region, and the quality of fit is greatly affected by the data within this $Q^2$-region, indicating the twist-expansion could be failed in very small $Q^2$-region. It has been suggested that by using the ``massive" high-twist expansion to do the data fitting, cf.~\cite{Ayala:2018ulm, Ayala:2020scz, Teryaev:2013qba, Khandramai:2016kbh, Gabdrakhmanov:2017dvg, Aguilar:2014tka}, one may obtain a better explanation of the data in very low $Q^2$ region. As an attempt, we take the following ``massive" high-twist expansion to do the fit~\cite{Ayala:2018ulm}
\begin{eqnarray}
\Gamma^{p-n}_1(Q^2)=\frac{g_A}{6}\left[1-E_{\rm ns}(Q^2)\right]+\frac{\mu^{p-n}_4}{Q^2+m^2}+\cdots,
\label{massivehightwist}
\end{eqnarray}
where the parameter $m$ represents a dynamical effective gluon mass, whose square satisfies
\begin{eqnarray}
m^2=\frac{m^2(1~\rm GeV^2)(1+1/\mathcal{M}^2)^{1+\rm p}}{(1+Q^2/\mathcal{M}^2)^{1+{\rm p}}}.
\end{eqnarray}
Here we have set the initial scale of the squared mass as $1$ GeV, and we shall take the parameters $\mathcal{M}^2=0.5~{\rm GeV}^2$ and $\rm p=0.1$ to do the calculation, which are within the suggested range of Ref.\cite{Aguilar:2014tka}. At present, to fit the magnitude of the ``massive" high-twist terms, the parameters $f^{p-n}_2(1~\rm GeV^2)$ and $m^2(1~\rm GeV^2)$ are used to fit with the data \cite{Deur:2008ej, Deur:2014vea}. When doing the fitting with the experiments data within the range of $0.054\rm GeV^2\le Q^2\le4.739\rm GeV^2$, we adopt four $\alpha_s$ models. The results for the two parameters $f^{p-n}_2(1~\rm GeV^2)$, $m^2(1~\rm GeV^2)$ and their corresponding quality of fit $\chi^2/d.o.f$ are presented in Table.~\ref{fitmassiveBSR}. Those two parameters are obtained by considering the systematic error $\sigma_{j,sys}$ at each data point $Q^2_j$ into the fitting; We put the details of fitting in the end of Appendix B. Comparing the smallest $\chi^2/d.o.f$ listed in Table.~\ref{fitf2mu6} and Table.~\ref{fitmassiveBSR}, one may observes that the ``massive" BSR shows a better behavior with smaller quality of fit $\chi^2/d.o.f$. The conventional predictions for twist-4 $f^{p-n}_2(1~\rm GeV^2)$ apparently depends on the choice of $\mu_r$. The quality of fit $\chi^2/d.o.f$ for conventional predictions with $\alpha_s$ under WEB model varies from tens to hundreds, while similar $\chi^2/d.o.f$ for conventional predictions with $\alpha_s$ under APT, MPT and CON models are along with different fit parameters $f^{p-n}_2(1~\rm GeV^2)$ and $m^2(1~\rm GeV^2)$, respectively. If using the PMC scale-independent series and the ``massive" high-twist term, we can obtain the corresponding color polarizability $\chi^{p-n}_E$ and $\chi^{p-n}_B$:
\begin{eqnarray}
\chi_B^{p-n}|_{\rm{APT}}&=&0.057\pm0.009, \\
\chi_B^{p-n}|_{\rm{WEB}}&=&0.038\pm0.009, \\
\chi_B^{p-n}|_{\rm{MPT}}&=&0.057\pm0.009, \\
\chi_B^{p-n}|_{\rm{CON}}&=&0.060\pm0.009, \\
\chi_E^{p-n}|_{\rm{APT}}&=&-0.083\pm0.014,\\
\chi_E^{p-n}|_{\rm{WEB}}&=&-0.043\pm0.014,\\
\chi_E^{p-n}|_{\rm{MPT}}&=&-0.082\pm0.014,\\
\chi_E^{p-n}|_{\rm{CON}}&=&-0.087\pm0.014,
\end{eqnarray}
where the errors are squared average of those from $\Delta d^{p-n}_2=\pm0.0036$ and $\Delta f^{p-n}_2$ for the four low-energy $\alpha_s$ models (e.g. Table.~\ref{fitmassiveBSR}).
\begin{table*}[htb]
\centering
\begin{tabular}{ c c c c c c c c c}
\hline
& ~$\alpha_s$ models ~ &~ & ~~~$f_2^{p-n}(1)$~~~ &~~~${m}^2(1)$~~~ & ~~~$\chi^2/d.o.f$ \\
\hline
&~ &$\mu_r=Q/2$ &~~$-0.217\pm{0.004}\pm{0.013}$~~ & ~~$0.203\pm{0.016}\pm{0.102}$~~ &48 \\
&$\rm{APT}|_{Conv}$ &$\mu_r=Q$ &$-0.113\pm{0.004}\pm{0.013}$ &$0.285\pm{0.041}\pm{0.249}$ &42 \\
&~ &$\mu_r=2Q$ &$-0.166\pm{0.005}\pm{0.013}$ &$0.505\pm{0.045}\pm{0.205}$ &43 \\
&$\rm{APT|_{PMC}}$ &$\mu_r\in[Q/2,2Q]$ &$-0.140\pm{0.004}\pm{0.013}$ &$0.162\pm{0.021}\pm{0.011}$ &28 \\
\hline
&~ &$\mu_r=Q/2$ &$-0.235\pm{0.004}\pm{0.013}$ &$0.184\pm{0.013}\pm{0.006}$ &37 \\
&$\rm{WEB}|_{Conv}$ &$\mu_r=Q$ &$-0.138\pm{0.009}\pm{0.013}$ &$2.233\pm{0.373}\pm{0.044}$ &80 \\
&~ &$\mu_r=2Q$ &$-0.183\pm{0.006}\pm{0.013}$ &$0.717\pm{0.056}\pm{0.026}$ &145 \\
&$\rm{WEB|_{PMC}}$ &$\mu_r\in[Q/2,2Q]$ &$-0.081\pm{0.003}\pm{0.013}$ &$0.038\pm{0.010}\pm{0.011}$ & 45 \\
\hline
&~ &$\mu_r=Q/2$ &$-0.220\pm{0.004}\pm{0.013}$ &$0.220\pm{0.016}\pm{0.046}$ &44 \\
&$\rm{MPT}|_{Conv}$ &$\mu_r=Q$ &$-0.099\pm{0.004}\pm{0.013}$ &$0.229\pm{0.043}\pm{0.044}$ &38\\
&~ &$\mu_r=2Q$ &$-0.168\pm{0.005}\pm{0.013}$ &$0.538\pm{0.047}\pm{0.058}$ &40\\
&$\rm{MPT|_{PMC}}$ &$\mu_r\in[Q/2,2Q]$ &$-0.139\pm{0.004}\pm{0.013}$ &$0.096\pm{0.015}\pm{0.009}$ &29 \\
\hline
&~ &$\mu_r=Q/2$ &$-0.215\pm{0.004}\pm{0.013}$ &$0.191\pm{0.016}\pm{0.007}$ &39 \\
&$\rm{CON}|_{Conv}$ &$\mu_r=Q$ &$-0.071\pm{0.003}\pm{0.013}$ &$0.045\pm{0.020}\pm{0.016}$ &60 \\
&~ &$\mu_r=2Q$ &$-0.174\pm{0.005}\pm{0.013}$ &$0.605\pm{0.052}\pm{0.019}$ &37 \\
&$\rm{CON|_{PMC}}$ &$\mu_r\in[Q/2,2Q]$ &$-0.147\pm{0.004}\pm{0.013}$ &$0.075\pm{0.011}\pm{0.008}$ &36 \\
\hline
\end{tabular}
\caption{The fitted parameters $f^{p-n}_2(Q^2=1~\rm GeV^2)$ and ${m}^2(Q^2=1~\rm GeV^2)$ and their corresponding quality of fit $\chi^2/d.o.f$ under four $\alpha_s$ models before and after applying the PMC, where the first and the second errors are caused by the statistical and systematic errors of the experiments data.}
\label{fitmassiveBSR}
\end{table*}
To compare with Fig.~\ref{higherBSR}, Fig.~\ref{massiveBSR} shows that by using the ``massive" high-twist term with the fitted parameters $f^{p-n}_2(1~\rm GeV^2)$ and $m^2(1~\rm GeV^2)$, a better prediction in agreement with the experiments data for $Q^2$ below 0.5 $\rm GeV^2$ can be achieved, which results as a smaller $\chi^2/d.o.f$ in Table.~\ref{fitmassiveBSR}. Different from the PMC predictions, the scale-dependence for conventional predictions is enhanced in small $Q^2$ region. Then, without renormalization scale dependence, the PMC predictions for the twist-4 contribution are more reliable; more explicitly, we observe that the quality of fit $\chi^2/d.o.f$ can be improved as $\sim 28$ for APT model, $\sim 45$ for WEB model, $\sim 29$ for MPT model and $\sim 36$ for CON model, respectively, all of which correspond to a $p$-value $\ge99\%$.
\begin{figure*}[htb]
\centering
\subfigure[APT model]{\includegraphics[width=0.45\textwidth]{aptr.eps}}
\quad
\subfigure[WEB model]{\includegraphics[width=0.45\textwidth]{webr.eps}}
\quad
\subfigure[MPT model]{\includegraphics[width=0.45\textwidth]{mptr.eps}}
\quad
\subfigure[CON model]{\includegraphics[width=0.45\textwidth]{conr.eps}}
\caption{The spin structure function $\Gamma^{p-n}_1(Q^2)$ with both leading-twist and the ``massive" high-twist contributions under four $\alpha_s$ models: (a) the APT model; (b) the WEB model; (c) the MPT model; and (d) the CON model. The leading-twist perturbative contributions have been calculated up to ${\rm N^3LO}$ level and ${\rm N^4LO}$ level before and after applying the PMC scale-setting approach, respectively. The shaded band shows the prediction under conventional scale-setting approach by varying $\mu_r\in[Q/2,2Q]$. The solid line is the scale-invariant PMC prediction.}
\label{massiveBSR}
\end{figure*}
\section{Summary}
In the paper, we have applied the PMC single-scale approach to deal with the perturbative series of the leading-twist part of $\Gamma^{p-n}_{1}(Q^2)$ up to ${\rm N^3LO}$ level. The pQCD series for both $\Gamma^{p-n}_{1}(Q^2)$ and the PMC scale $Q_*$ are convergent in large $Q^2$-region. We have also provided a prediction on the uncalculated ${\rm N^4LO}$ by using the more convergent and scheme-and-scale invariant PMC conformal series. Thus a more accurate pQCD prediction on $\Gamma^{p-n}_{1}(Q^2)$ can be achieved by applying the PMC.
Basing on the PMC predictions on the perturbative part, we then provide a novel determination of the high-twist contributions by using the JLab data, whose momentum transfer lies in the range of ${0.054}~{\rm GeV^2} \leq Q^2\leq 4.739~{\rm GeV^2}$. In large $Q^2$-region, the high-twist contributions to $\Gamma^{p-n}_{1}(Q^2)$ are power suppressed and negligible, which are however sizable in low and intermediate $Q^2$-region; Fig.~\ref{leadingBSR} shows that in low $Q^2$-region, the leading-twist terms alone cannot explain the JLab data. The high-twist term is necessary and it can fix this problem with two fit parameters as Fig.~\ref{massiveBSR} shows. Taking the high-twist contributions up to twist-6 accuracy, we have fixed the twist-4 coefficient $f_2^{p-n}$ and the twist-6 coefficient $\mu_6$ by using four typical $\alpha_s$-models, which give $f_2^{p-n}|_{\rm APT}=-0.120\pm0.013$ and $\mu_6|_{\rm APT}=0.003\pm{0.000}$, $f_2^{p-n}|_{\rm WEB}=-0.081\pm0.013$ and $\mu_6|_{\rm WEB}=0.001\pm{0.000}$, $f_2^{p-n}|_{\rm MPT}=-0.128\pm0.013$ and $\mu_6|_{\rm MPT}=0.003\pm{0.000}$, $f_2^{p-n}|_{\rm CON}=-0.139\pm0.013$ and $\mu_6|_{\rm CON}=0.002\pm0.000$, respectively. Here the errors are squared averages of those from the statistical and systematic errors of the measured data. As an attempt, by taking the ``massive" high-twist expansion such as Eq.(\ref{massivehightwist}) to do the fit, we have shown that a better explanation of the data in very low $Q^2$ range can be achieved.
\hspace{2cm}
\noindent {\bf Acknowledgments:} This work was supported in part by the Natural Science Foundation of China under Grant No.11625520 and No.12047564, by graduate research and innovation foundation of Chongqing, China (Grant No.CYB21045), by the Fundamental Research Funds for the Central Universities under Grant No.2020CQJQY-Z003, and by the Chongqing Graduate Research and Innovation Foundation under Grant No.ydstd1912.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,793 |
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