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Gelechia picrogramma is een vlinder uit de familie van de tastermotten (Gelechiidae). De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1929 door Meyrick.
picrogramma
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,764
|
Q: Impex Error at referencing vm file I am trying to create impex for email page.
$contentCatalog=ShopzoneContentCatalog
$contentCV=catalogVersion(CatalogVersion.catalog(Catalog.id[default=$contentCatalog]),CatalogVersion.version[default=Staged])[default=$contentCatalog:Staged]
$jarResourceCms=jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/cockpits/cmscockpit/structure-view
$emailResource=jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/emails
$emailPackageName=com.shopzone.facades.process.email.context
Email velocity templates
INSERT_UPDATE RendererTemplate;code[unique=true];description[lang=en];templateScript[lang=en,translator=de.hybris.platform.commerceservices.impex.impl.FileLoaderValueTranslator];contextClass;rendererType(code)[default='velocity']
;appointment_notification_email_subject;"Appointment Notification Email Subject";$emailResource/appointment_notification_email_subject.vm;$emailPackageName.AppointmentEmailContext;
;appointment_notification_email_body;"Appointment Notification Email Body";$emailResource/appointment_notification_email_body.vm;$emailPackageName.AppointmentEmailContext;
Email page Template
INSERT_UPDATE EmailPageTemplate;$contentCV[unique=true];uid[unique=true];name;active;frontendTemplateName;subject(code);htmlTemplate(code);restrictedPageTypes(code)
;;appointmentNotificationEmailTemplate;Appointment Notification Email Template;true;appointmentNotificationEmailTemplate;appointment_notification_email_subject;appointment_notification_email_body;EmailPage
Templates for CMS Cockpit Page Edit
UPDATE EmailPageTemplate;$contentCV[unique=true];uid[unique=true];velocityTemplate[translator=de.hybris.platform.commerceservices.impex.impl.FileLoaderValueTranslator]
;;appointmentNotificationEmailTemplate;$jarResourceCms/structure_appointmentNotificationEmailTemplate.vm
Media
insert_update Media;code[unique=true];$contentCV;url;mime[default='image/jpg'];altText
;szEmailSiteLogoMedia;;/_ui/responsive/common/images/site_logo.png;;Shopzone
CMS Image Components
INSERT_UPDATE CMSImageComponent;$contentCV[unique=true];uid[unique=true];name;media(code, $contentCV)
;;szEmailSiteLogoImage;Email Site Logo Image;szEmailSiteLogoMedia
Content Slots
UPDATE ContentSlot;$contentCV[unique=true];uid[unique=true];cmsComponents(uid,$contentCV)
;;szEmailSiteLogoSlot;szEmailSiteLogoImage
Bind Content Slots to Email Page Templates
INSERT_UPDATE ContentSlotForTemplate;$contentCV[unique=true];uid[unique=true];position[unique=true];pageTemplate(uid,$contentCV)[unique=true];contentSlot(uid,$contentCV)[unique=true];allowOverwrite
;;siteLogo-appointmentNotificationEmailTemplate;szSiteLogo;appointmentNotificationEmailTemplate;szEmailSiteLogoSlot;true
Email Page
INSERT_UPDATE EmailPage;$contentCV[unique=true];uid[unique=true];name;masterTemplate(uid,$contentCV);defaultPage;approvalStatus(code)[default='approved'];fromEmail[lang=en];fromName[lang=en]
;;appointmentNotificationEmailPage;Appointment Notification Email;appointmentNotificationEmailTemplate;true;;estore@shopzone.com;Customer Services Team
ERROR :
INSERT_UPDATE RendererTemplate;code[unique=true];description[lang=en];templateScript[lang=en,translator=de.hybris.platform.commerceservices.impex.impl.FileLoaderValueTranslator];contextClass;rendererType(code)[default='velocity']
,8796125836191,,, column 3: cannot resolve value 'jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/emails/appointment_notification_email_body.vm' for attribute 'templateScript', column 3: cannot resolve value 'jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/emails/appointment_notification_email_body.vm' for attribute 'templateScript';appointment_notification_email_body;Appointment Notification Email Body;jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/emails/appointment_notification_email_body.vm;com.shopzone.facades.process.email.context.AppointmentEmailContext;
,8796125868959,,, column 3: cannot resolve value 'jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/emails/appointment_notification_email_subject.vm' for attribute 'templateScript', column 3: cannot resolve value 'jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/emails/appointment_notification_email_subject.vm' for attribute 'templateScript';appointment_notification_email_subject;Appointment Notification Email Subject;jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/emails/appointment_notification_email_subject.vm;com.shopzone.facades.process.email.context.AppointmentEmailContext;
UPDATE EmailPageTemplate;catalogVersion(CatalogVersion.catalog(Catalog.id[default=ShopzoneContentCatalog]),CatalogVersion.version[default=Staged])[default=ShopzoneContentCatalog:Staged][unique=true];uid[unique=true];velocityTemplate[translator=de.hybris.platform.commerceservices.impex.impl.FileLoaderValueTranslator]
,8796420736052,,, column 3: cannot resolve value 'jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/cockpits/cmscockpit/structure-view/structure_appointmentNotificationEmailTemplate.vm' for attribute 'velocityTemplate', column 3: cannot resolve value 'jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/cockpits/cmscockpit/structure-view/structure_appointmentNotificationEmailTemplate.vm' for attribute 'velocityTemplate';;appointmentNotificationEmailTemplate;jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/resources/shopzonecore/import/cockpits/cmscockpit/structure-view/structure_appointmentNotificationEmailTemplate.vm
UPDATE ContentSlot;catalogVersion(CatalogVersion.catalog(Catalog.id[default=ShopzoneContentCatalog]),CatalogVersion.version[default=Staged])[default=ShopzoneContentCatalog:Staged][unique=true];uid[unique=true];cmsComponents(uid,catalogVersion(CatalogVersion.catalog(Catalog.id[default=ShopzoneContentCatalog]),CatalogVersion.version[default=Staged])[default=ShopzoneContentCatalog:Staged])
,,,no existing item found for update;;szEmailSiteLogoSlot;szEmailSiteLogoImage
INSERT_UPDATE ContentSlotForTemplate;catalogVersion(CatalogVersion.catalog(Catalog.id[default=ShopzoneContentCatalog]),CatalogVersion.version[default=Staged])[default=ShopzoneContentCatalog:Staged][unique=true];uid[unique=true];position[unique=true];pageTemplate(uid,catalogVersion(CatalogVersion.catalog(Catalog.id[default=ShopzoneContentCatalog]),CatalogVersion.version[default=Staged])[default=ShopzoneContentCatalog:Staged])[unique=true];contentSlot(uid,catalogVersion(CatalogVersion.catalog(Catalog.id[default=ShopzoneContentCatalog]),CatalogVersion.version[default=Staged])[default=ShopzoneContentCatalog:Staged])[unique=true];allowOverwrite
,,,error finding existing item : column='contentSlot' value='szEmailSiteLogoSlot', , column 5: could not resolve item for szEmailSiteLogoSlot;;siteLogo-appointmentNotificationEmailTemplate;szSiteLogo;appointmentNotificationEmailTemplate;szEmailSiteLogoSlot;true
02.11.2017 15:35:02: ERROR: Can not resolve any more lines ... Aborting further passes (at pass 2). Finally could not import 5 lines!
02.11.2017 15:35:02: ERROR: Can not resolve any more lines ... Aborting further passes (at pass 2). Finally could not import 5 lines!
A: When you reference files, you need to reference them starting with the "resources" directory as your root directory.
$jarResourceCms=jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/import/cockpits/cmscockpit/structure-view $emailResource=jar:com.shopzone.core.setup.CoreSystemSetup&/shopzonecore/import/emails
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,730
|
//
// MUAssetsLibrary.h
// MUAssetsLibrary
//
// Created by Muer on 14-9-10.
// Copyright © 2015年 Muer. All rights reserved.
//
#import <Foundation/Foundation.h>
#import <UIKit/UIKit.h>
#import <Photos/Photos.h>
#import <AssetsLibrary/AssetsLibrary.h>
#import "MUAsset.h"
#import "MUAssetCollection.h"
@class AVAsset;
extern CGSize const MUImageManagerMaximumSize NS_AVAILABLE_IOS(8_0);
extern NSString * const MUImageResultIsInCloudKey NS_AVAILABLE_IOS(8_0);
extern NSString * const MUImageResultIsDegradedKey NS_AVAILABLE_IOS(8_0);
extern NSString * const MUImageResultRequestIDKey NS_AVAILABLE_IOS(8_0);
extern NSString * const MUImageCancelledKey NS_AVAILABLE_IOS(8_0);
extern NSString * const MUImageErrorKey NS_AVAILABLE_IOS(8_0);
typedef NS_ENUM(NSInteger, MUPHAuthorizationStatus) {
MUPHAuthorizationStatusNotDetermined = 0,
MUPHAuthorizationStatusRestricted,
MUPHAuthorizationStatusDenied,
MUPHAuthorizationStatusAuthorized
};
typedef void (^MUAssetsLibraryWriteCompletionHandler)(MUAsset *asset, NSError *error);
@protocol MUPhotoLibraryChangeObserver <NSObject>
/** changeInstance is PHChange/NSDictionary */
- (void)photoLibraryDidChange:(id)changeInstance;
@end
/**
@brief
MUAssetsLibrary资源库,分别对应ALAssetsLibrary和PHPhotoLibrary
*/
@interface MUAssetsLibrary : NSObject
/** 照片库授权状态 */
+ (MUPHAuthorizationStatus)authorizationStatus;
/** 请求访问照片库权限 */
+ (void)requestPhotoLibraryPermissionWithCompletionHandler:(void (^)(BOOL granted))handler;
+ (BOOL)isAssetURL:(NSURL *)url;
+ (instancetype)sharedLibrary;
+ (void)registerChangeObserver:(id<MUPhotoLibraryChangeObserver>)observer;
+ (void)unregisterChangeObserver:(id<MUPhotoLibraryChangeObserver>)observer;
/** 请求照片库的资源集(相册) */
- (void)requestAssetCollectionsWithMediaType:(MUAssetMediaType)mediaType completionHandler:(void(^)(NSArray<MUAssetCollection *> *assetCollections, NSError *error))completionHandler;
/** 请求照片库海报预览图 */
- (void)requestPhotoLibraryPosterImageForMediaType:(MUAssetMediaType)mediaType completionHandler:(void(^)(UIImage *image))completionHandler;
/** 保存图片至相册 */
- (void)writeImage:(UIImage *)image completionHandler:(MUAssetsLibraryWriteCompletionHandler)completionHandler;
- (void)writeImage:(UIImage *)image metadata:(NSDictionary *)metadata completionHandler:(MUAssetsLibraryWriteCompletionHandler)completionHandler;
/** 保存视频至相册 */
- (void)writeVideoAtURL:(NSURL *)url completionHandler:(MUAssetsLibraryWriteCompletionHandler)completionHandler;
/** 创建相册 */
- (void)createAssetCollectionWithTitle:(NSString *)title completionHandler:(void(^)(MUAssetCollection *assetCollection, NSError *error))completionHandler;
/** 请求相册通过标题 */
- (void)requestAssetCollectionsWithTitle:(NSString *)title completionHandler:(void (^)(MUAssetCollection *assetCollection, NSError *error))completionHandler;
/** 获取Asset通过LocalIdentifier */
- (void)requestAssetWithLocalIdentifier:(NSString *)localIdentifier completionHandler:(void (^)(MUAsset *asset))completionHandler;
/** 获取Asset通过Asset URL */
- (void)requestAssetWithAssetURL:(NSURL *)assetURL completionHandler:(void (^)(MUAsset *asset))completionHandler;
/** 获取Asset的Metadata */
- (void)requestMetadataWithAsset:(MUAsset *)asset completionHandler:(void (^)(NSDictionary *metadata))completionHandler;
/** 允许空相册 */
@property (nonatomic, assign) BOOL allowEmptyAlbums;
/** 用于 8.0~8.1 版本的所有照片相册标题 默认为"所有照片" */
@property (nonatomic, copy) NSString *allPhotosAssetCollectionTitle NS_AVAILABLE_IOS(8_0);
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,500
|
{"url":"https:\/\/artofproblemsolving.com\/wiki\/index.php?title=2003_Indonesia_MO_Problems\/Problem_3&diff=119625&oldid=97147","text":"# Difference between revisions of \"2003 Indonesia MO Problems\/Problem 3\"\n\n## Problem\n\nFind all real solutions of the equation $\\lfloor x^2 \\rfloor + \\lceil x^2 \\rceil = 2003$.\n\n[Note: For any real number $\\alpha$, $\\lfloor \\alpha \\rfloor$ is the largest integer less than or equal to $\\alpha$, and $\\lceil \\alpha \\rceil$ denote the smallest integer more than or equal to $\\alpha$.]\n\n## Solution\n\nThere are two cases to consider -- one where $x$ is positive and one where $x$ is negative.\n\nFor the positive case, if $x = \\sqrt{a},$ then the equation results in $2a = 2003.$ Since the equation does not have an integral solution, $x \\ne \\sqrt{a}.$ If we let $\\sqrt{a} < x < \\sqrt{a+1}.$ That means $a + a + 1 = 2003,$ and solving the equation yields $a = 1001.$ For confirmation, $1001 \\le \\lfloor x^2 \\rfloor < 1002$ and $1001 < \\lceil x^2 \\rceil \\le 1002$, so $2002 < \\lfloor x^2 \\rfloor + \\lceil x^2 \\rceil < 2004$. Since $\\lfloor x^2 \\rfloor + \\lceil x^2 \\rceil$ can only have integral values, $\\lfloor x^2 \\rfloor + \\lceil x^2 \\rceil = 2003$.\n\nFor the negative case, if $x = -\\sqrt{a},$ then the equation results in $2a = 2003.$ This also does not have an integral solution, so $x \\ne \\sqrt{a}.$ If we let $-\\sqrt{a+1} < x < -\\sqrt{a}.$ That means $a+1 + a = 2003,$ and this equation also yields $a = 1001.$ For confirmation, $1001 < \\lfloor x^2 \\rfloor \\le 1002$ and $1001 \\le \\lceil x^2 \\rceil < 1002$, so $2002 < \\lfloor x^2 \\rfloor + \\lceil x^2 \\rceil < 2004$. Since $\\lfloor x^2 \\rfloor + \\lceil x^2 \\rceil$ can only have integral values, $\\lfloor x^2 \\rfloor + \\lceil x^2 \\rceil = 2003$.\n\nIn interval notation, the solutions of the equation are $\\boxed{(-\\sqrt{1002}, -\\sqrt{1001}) \\cup (\\sqrt{1001}, \\sqrt{1002})}.$","date":"2021-11-28 00:38:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 31, \"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\": 0.9692285656929016, \"perplexity\": 129.00724032252904}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358323.91\/warc\/CC-MAIN-20211127223710-20211128013710-00213.warc.gz\"}"}
| null | null |
{"url":"http:\/\/openstudy.com\/updates\/51118329e4b09cf125bdd0f0","text":"## 96smidge Group Title Given the system of equations, what is the value of the system determinant? x + y = 8 x - y = 10 one year ago one year ago\n\n\u2022 This Question is Open\n1. fewscrewsmissing Group Title\n\nTo find the determinant of a 2x2 matrix: $A = \\left[\\begin{matrix}a & b \\\\ c & d\\end{matrix}\\right]\\\\ |A|=det(A)=ad-cb$\n\n2. fewscrewsmissing Group Title\n\nI'd actually forgotten how to do this since it's been so long, and it literally took a 2 second Google search ;)\n\n3. jedai17 Group Title\n\na and c is the value of x and b and d is the value of y\n\n4. jedai17 Group Title\n\n|dw:1361514044728:dw|\n\n5. jedai17 Group Title\n\nthen find the determinant according to the formula given by @fewscrewsmissing","date":"2014-09-02 02:44:49","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\": 0.5408207774162292, \"perplexity\": 1304.6325590847005}, \"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-2014-35\/segments\/1409535921318.10\/warc\/CC-MAIN-20140909054134-00422-ip-10-180-136-8.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
In the LHC era, the search for physics beyond the Standard Model (SM) has proven elusive, and standard frameworks for TeV-scale new physics are highly constrained. For the well-studied case of extensions of the Standard Model to include softly broken $\mathcal{N}=1$ supersymmetry, such as the minimal supersymmetric standard model (MSSM), the LHC bounds indicate that if softly broken supersymmetry does indeed play a role in any new physics at the next rung of the energy ladder, its implementation is necessarily more complicated and ostensibly fine-tuned than originally anticipated. In this context, given the vast nature of the parameter space associated with the soft supersymmetry breaking sector, frameworks such as the MSSM can remain viable. However, patterns of possibly viable MSSM parameter regions would then be indicated, perhaps pointing to a specific organizing principle at higher energies.
One such example is within the context of gauge-mediated supersymmetry breaking (GMSB). In its minimal implementation, its distinctive phenomenology is characterized by a superpartner mass spectrum with a sizable splitting between the $SU(3)_c$-charged superpartners (squarks and gluinos) and the superpartners charged only under the electroweak symmetry (sleptons and electroweakinos), with the splitting governed by the messenger mass scale and the number of messenger pairs (taken to be $\mathbf{5}$, $\overline{\bf 5}$ with respect to $SU(5)$). However, the minimal implementation does not easily allow for a 125 GeV Higgs mass, requiring very high messenger scales and subsequent squark and gluino masses that are far out of reach of the LHC \cite{Dine:1981za,ALVAREZGAUME198296,DIMOPOULOS1981353,DIMOPOULOS1983479,Dine:1981za,DINE1982227,Nappi:1982hm,Dine_1993,Dine_1995,Dine_1996,Dine_1997,Giudice_1999,Draper_2012,Arbey_2012,Adeel_Ajaib_2012,Fischler_2014,Calibbi_2014}. As such, nonminimal implementations of gauge mediation, such as general gauge mediation \cite{Meade:2008wd}, or scenarios in which the MSSM fields and the messenger fields interact directly via renormalizable superpotential couplings, have now long been explored \cite{Dine_1997,Giudice_1999,Chacko_2002,Shadmi_2012,Evans_2011,evans2011probing,Evans_2012,Kang_2012,Craig_2013,Albaid_2013,Abdullah_2013,P_rez_2013,Byakti_2013,Evans_2013,Calibbi_2013,evans2015chiral,Galon_2013,Fischler_2014,Calibbi_2014,Joaquim:2006mn,Joaquim:2006uz}.
Of the many intriguing options for direct couplings between the messenger and matter sectors, the {\it flavored gauge mediation} framework, which exploits the fact that the electroweak Higgs fields can mix with the doublet components of the messenger pairs, has been of particular interest in the literature \cite{Abdullah_2013,Shadmi_2012,Ierushalmi_2016,P_rez_2013,Byakti_2013,Evans_2013,Calibbi_2013,evans2015chiral,Galon_2013,Jeli_ski_2015,Everett_2018,Everett_2019,Ahmed_2017}. In flavored gauge mediation (FGM) models, Higgs-messenger mixing leads to the generation of messenger Yukawa couplings, which affect the prediction of the soft supersymmetry breaking mass parameters at the input (messenger mass) scale. The messenger Yukawa contributions not only affect the superpartner mass spectrum, but also can generically can lead to the nontrivial possibility of flavor mixing in the soft terms. In viable FGM scenarios, therefore, the messenger Yukawa couplings are controlled by additional symmetries, and their forms are also intimately connected to the generation of the MSSM Yukawa couplings of the quarks and leptons. The case of $U(1)$ symmetries, as explored extensively for example in \cite{Calibbi_2014}, allows for great flexibility in constructing viable models with one or more vectorlike pairs of messengers. In addition, it was shown in \cite{Ierushalmi_2016} that flavor-mixing contributions to the soft terms in such scenarios are much smaller than naive expectations might suggest, and can be consistent with stringent bounds from flavor-changing processes, depending on the model in question.
Instead of using Abelian symmetries to control the messenger Yukawa couplings, an alternative is to build models based on discrete non-Abelian symmetries. Such symmetries have been extensively used as governing principles for the generation of viable SM fermion masses and mixing parameters \cite{Xing_1997,Fritzsch_1994,Fritzsch_2000}. In flavored gauge mediation, this possibility was first explored in detail in \cite{P_rez_2013}, where the authors constructed a two-family scenario based on the discrete non-Abelian symmetry $\mathcal{S}_3$, with the Higgs and messenger fields connected within $\mathcal{S}_3$ doublets. This idea was then extended to incorporate three families \cite{Everett_2018,Everett_2019,Everett_2020}. Most notably, it was realized in \cite{Everett_2018} that to avoid a severe $\mu/B_\mu$ problem, the Higgs-messenger sector should be extended to include $\mathcal{S}_3$ singlet representations as well as doublet representations. This leads to scenarios with a minimal number $N=2$ of messenger pairs (in contrast to the $U(1)$ cases, which allow for one messenger pair), which enhances the splitting of the squark and gluino masses compared to the slepton and electroweakino masses. Further embedding of the MSSM fields in $\mathcal{S}_3$ representations allows for the possibility that $\mathcal{S}_3$ can play a role as part of the family symmetry that governs the SM fermion masses and mixings. A specific implementation of this idea was explored in \cite{Everett_2019}, as well as in \cite{Everett_2020}, in which the Higgs-messenger singlets play a dominant role in generating the third family SM fermion masses.
The purpose of this paper is to provide a comprehensive analysis of the FGM $\mathcal{S}_3$ scenario, summarizing and extending our previous work. The aim is to explore other viable corners of parameter space of these theories and the subsequent effects of including nonleading corrections to the fermion masses. We identify several viable parameter regions, describe their phenomenological consequences, and compare them to the $U(1)$ FGM benchmark scenarios in the literature. We will see that quite generally, it is not easy to generate viable fermion masses while maintaining flavor-diagonal soft terms, and we will characterize the extent to which such flavor nondiagonal terms are constrained in these theories. The examples studied here all feature very heavy squarks and gluinos, very heavy Higgs fields, and lighter sleptons, charginos, and neutralinos. As such, they provide working examples of currently allowed MSSM parameter space that will continue to be constrained at the LHC and future colliders.
This paper is structured as follows. We begin with a brief overview of the flavored gauge mediation framework studied here, and describe various options for obtaining hierarchical quark and charged lepton masses. Next, we describe several concrete models, and analyze their mass spectra in detail. Finally, we present our summary and conclusions.
\section{Theoretical Background}
As described in \cite{Everett_2018}, the FGM $\mathcal{S}_3$ scenario studied here assumes a specific set of Higgs-messenger fields and supersymmetry-breaking fields. The quantum numbers of these fields with respect to $\mathcal{S}_3$ are given in Table~{\ref{tab:11}}.
\begin{table}[htbp]
\centering
\begin{tabular}{c|cccccc|cc}
& $\mathcal{H}_u^{(2)}$&$\mathcal{H}_u^{(1)}$ & $\mathcal{H}_d^{(2)}$& $\mathcal{H}_d^{(1)}$ & $T_{\bar{d}k}$ & $T_{dk}$ &$X_H$ & $X_T$\\
\hline
$\mathcal{S}_3$ &$\mathbf 2 $& $\mathbf 1$& $\mathbf 2 $& $\mathbf 1 $ & $\mathbf 1 $ & $\mathbf 1 $ &\textbf 2 & $\mathbf 1 $\\
\end{tabular}
\caption{The field content and $\mathcal{S}_3$ charges for the messenger and supersymmetry breaking sectors.}
\label{tab:11}
\end{table}
Here the $\mathcal{H}_{u,d}^{(2)}$ are Higgs-messenger $\mathcal{S}_3$ doublets, the $\mathcal{H}_{u,d}^{(1)}$ are Higgs-messenger $\mathcal{S}_3$ singlets, and $X_H$ is a supersymmetry breaking field that also breaks the $\mathcal{S}_3$ symmetry. The $T_{\bar{d}k,dk}$ denote the $SU(3)_c$ triplets which have the appropriate quantum numbers to complete approximate $\mathbf{5}$, $\overline{\mathbf{5}}$ multiplets with the messengers and $X_T$ is the supersymmetry breaking field that couples to these triplets \footnote{The triplet messengers and the $X_T$ field are chosen for simplicity to be singlets with respect to the $\mathcal{S}_3$ Higgs-messenger symmetry. Note that this choice is not consistent with a full embedding of this scenario into a grand unified theory. This would require more extended model-building that would also need to address the well-known doublet-triplet splitting issue in grand unified models).}. Focusing on the Higgs-messenger fields, we can write $\mathcal{H}_{u,d}^{(2)}$ and $\mathcal{H}_{u,d}^{(1)}$ as
\begin{eqnarray}
\mathcal{H}_{u}&\equiv& \left (\begin{array}{c} (\mathcal{H}^{(2)}_u)_1 \\ (\mathcal{H}^{(2)}_u)_2 \\ \mathcal{H}^{(1)}_u \end{array} \right ) \equiv \left (\begin{array}{c} \mathcal{H}^{(2)}_{u1}\\ \mathcal{H}^{(2)}_{u2}\\ \mathcal{H}^{(1)}_u\end{array} \right )= \mathcal{R}_u \left (\begin{array}{c} H_u\\M_{u1} \\ M_{u2} \end{array} \right ) \nonumber \\
\mathcal{H}_{d}&\equiv& \left (\begin{array}{c} (\mathcal{H}^{(2)}_d)_1 \\ (\mathcal{H}^{(2)}_d)_2 \\ \mathcal{H}^{(1)}_d \end{array} \right ) \equiv \left (\begin{array}{c} \mathcal{H}^{(2)}_{d1}\\ \mathcal{H}^{(2)}_{d2}\\ \mathcal{H}^{(1)}_d\end{array} \right )=\mathcal{R}_d \left (\begin{array}{c} H_d\\M_{d1} \\ M_{d2} \end{array} \right ),
\label{higgs_s3}
\end{eqnarray}
in which $H_{u,d}$ are the electroweak Higgs fields of the MSSM, $M_{u1,d1}$ and $M_{u2,d2}$ are gauge mediation messenger doublets, and $\mathcal{R}_{u,d}$ are unitary matrices whose form is governed by the couplings of the Higgs-messenger fields to $X_H$, which obtains both a scalar and $F$-component vacuum expectation value (VEV). As shown in \cite{Everett_2018}, consistency requirements and obtaining the needed mass hierarchy between the MSSM Higgs fields $H_{u,d}$ and the heavy messengers $M_{ui,di}$ require that $\mathcal{R}_{u,d}$ are given by
\begin{eqnarray}
\mathcal{R}_{u,d}= \left ( \begin{array}{ccc} \frac{1}{\sqrt{3}} & \mp \frac{1}{2} \left (1+\frac{1}{\sqrt{3}} \right) & \frac{1}{2} \left (1-\frac{1}{\sqrt{3}} \right) \\ \frac{1}{\sqrt{3}} & \pm \frac{1}{2} \left (1-\frac{1}{\sqrt{3}} \right) & -\frac{1}{2} \left (1+\frac{1}{\sqrt{3}} \right)
\\ \frac{1}{\sqrt{3}} & \pm \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array} \right ).
\label{rotationmatrices}
\end{eqnarray}
We turn now to the MSSM fields and their interactions with the Higgs-messenger fields. Although various possibilities exist, as discussed in \cite{Everett_2018},
we make the key assumption that the three generations of SM quarks and leptons are embedded into doublet and singlet representations of $\mathcal{S}_3$, as summarized in Table~\ref{tab:12}.
\begin{table}[htbp]
\centering
\begin{tabular}{c|cccc|cccccccccc|c}
& $\mathcal{H}_u^{(2)}$&$\mathcal{H}_u^{(1)}$ & $\mathcal{H}_d^{(2)}$& $\mathcal{H}_d^{(1)}$ & $Q_{\mathbf 2}$ &$Q_{\mathbf 1}$& $\bar u_{\mathbf 2}$ & $\bar u_{\mathbf 1 }$&$\bar d_{\mathbf 2}$& $\bar d_{\mathbf 1}$ & $L_{\mathbf{2}}$ & $L_{\mathbf{1}}$ & $\bar{e}_{\mathbf{2}}$ & $\bar{e}_{\mathbf{1}}$&$X_H$\\
\hline
$\mathcal{S}_3$ &$\mathbf 2 $& $\mathbf 1$& $\mathbf 2 $& $\mathbf 1 $ & $\mathbf 2 $&$\mathbf 1$ & $\mathbf 2 $&$\mathbf 1$ & $\mathbf 2 $&$\mathbf 1$ & $\mathbf 2 $&$\mathbf 1$ & $\mathbf 2 $&$\mathbf 1$ &\textbf 2\\
\end{tabular}
\caption{Charges for an $\mathcal{S}_3$ model of the Higgs-messenger fields and the MSSM matter fields. Here the $SU(3)$ triplet messengers and the associated $X_T$ field are not displayed for simplicity.
}
\label{tab:12}
\end{table}
With these $\mathcal{S}_3$ charge assignments, the superpotential couplings of the MSSM matter fields and the Higgs-messenger fields, for example for the up quarks, are given by
\begin{eqnarray}
W^{(u)}= \tilde{y}_u\big[Q_{\mathbf 2} \bar u_{\mathbf 2} \mathcal{H}^{(2)}_u+\beta_{1u}Q_{\mathbf 2} \bar u_{\mathbf 2} \mathcal{H}^{(1)}_u + \beta_{2u} Q_{\mathbf 2} \bar u_{\mathbf 1} \mathcal{H}^{(2)}_u +\beta_{3u} Q_{\mathbf 1} \bar u_{\mathbf 2} \mathcal{H}^{(2)}_u+ \beta_{4u} Q_{\mathbf 1} \bar u_{\mathbf 1} \mathcal{H}^{(1)}_u\big].
\label{wu}
\end{eqnarray}
In Eq.~(\ref{wu}), $\tilde{y}_u$ is a dimensionless overall factor, and the quantities $\beta_{1u}$, $\beta_{2u}$, $\beta_{3u}$, and $\beta_{4u}$ are dimensionless quantities that characterize the different couplings as allowed by $\mathcal{S}_3$. (Analogous forms hold for the down quarks and the charged leptons; we will ignore the effects of neutrino masses.) In the basis given by
\begin{eqnarray}
Q= (Q_\mathbf{2}, Q_\mathbf{1})^T = ((Q_{\mathbf 2})_1, (Q_{\mathbf 2})_2 ,Q_{\mathbf 1})^T, \qquad \overline{u}= (\overline{u}_{\mathbf{2}}, \overline{u}_\mathbf{1})^T= ((\overline{u}_{\mathbf 2})_1, (\overline{u}_\mathbf{2})_2 ,\overline{u}_{\mathbf 1})^T,
\end{eqnarray}
the superpotential couplings of Eq.~(\ref{wu}) can be expressed in matrix form as
\begin{eqnarray}
W^{(u)}=\tilde{y}_uQ^T\left( \begin{matrix} \mathcal{H}^{(2)}_{u1}&\beta_{1u}\mathcal{H}^{(1)}_{u}&\beta_{2u} \mathcal{H}^{(2)}_{u2}\\ \beta_{1u} \mathcal{H}^{(1)}_u& \mathcal{H}^{(2)}_{u2}& \beta_{2u}\mathcal{H}^{(2)}_{u1}\\ \beta_{3u}\mathcal{H}^{(2)}_{u2}& \beta_{3u}\mathcal{H}^{(2)}_{u1}&\beta_{4u} \mathcal{H}^{(1)}_u\end{matrix}\right)\bar u. \label{UpYukawas}
\end{eqnarray}
From here, we can easily identify the MSSM Yukawa coupling $Y_u$ and the messenger Yukawa couplings $Y_{u1}'$, $Y_{u2}'$, as
\begin{eqnarray}
Y_u= \frac{\tilde{y}_{u}}{\sqrt{3}} \left (\begin{array}{ccc} 1 & \beta_{1u} & \beta_{2u} \\ \beta_{1u} & 1 & \beta_{2u} \\ \beta_{3u} & \beta_{3u} & \beta_{4u} \end{array} \right ),
\label{eq:yud}
\end{eqnarray}
and
\begin{equation}
Y^\prime_{u1}=\tilde{y}_{u} \left (\begin{array}{ccc} -\frac{1}{2}-\frac{1}{2\sqrt{3}} & \frac{\beta_{1u}}{\sqrt{3}} & \;\; \frac{\beta_{2u}}{2} - \frac{\beta_{2u}}{2\sqrt{3}} \\ \frac{\beta_{1u}}{\sqrt{3}} & \;\; \frac{1}{2}-\frac{1}{2\sqrt{3}} & -\frac{\beta_{2u}}{2} - \frac{\beta_{2u}}{2\sqrt{3}} \\ \;\; \frac{\beta_{3u}}{2} - \frac{\beta_{3u}}{2\sqrt{3}} & -\frac{\beta_{3u}}{2} - \frac{\beta_{3u}}{2\sqrt{3}} & \frac{\beta_{4u}
}{\sqrt{3}}
\end{array} \right )
\end{equation}
\begin{equation}
\;\; Y^\prime_{u2}=\tilde{y}_{u} \left (\begin{array}{ccc} \;\; \frac{1}{2}-\frac{1}{2\sqrt{3}} & \frac{\beta_{1u}}{\sqrt{3}} & -\frac{\beta_{2u}}{2} - \frac{\beta_{2u}}{2\sqrt{3}} \\ \frac{\beta_{1u}}{\sqrt{3}} & -\frac{1}{2}-\frac{1}{2\sqrt{3}} & \;\; \frac{\beta_{2u}}{2} - \frac{\beta_{2u}}{2\sqrt{3}} \\ -\frac{\beta_{3u}}{2} - \frac{\beta_{3u}}{2\sqrt{3}} & \;\; \frac{\beta_{3u}}{2} - \frac{\beta_{3u}}{2\sqrt{3}} & \frac{\beta_{4u}
}{\sqrt{3}}
\end{array} \right ).
\end{equation}
These results are for the up sector; again, analogous relations hold for the down quarks and charged leptons, with the replacements $u\rightarrow d,e$ in all parameters, respectively.
For arbitrary values of the coefficients, Eq.~(\ref{eq:yud}) does not result in hierarchical fermion masses. It is only at special values of the couplings, corresponding to various enhanced symmetry points, that we can obtain a realistic quark mass hierarchy at leading order. To see this, we note that we can diagonalize this system explicitly and examine parameter sets where viable eigenvalue hierarchies can be obtained. For example, in the up quark sector, we can follow standard procedures and consider the Hermitian combinations $Y_u Y_u^\dagger$ and $Y_u^\dagger Y_u$. It is straightforward to calculate following exact results for their eigenvalues (denoted by $\lambda_{1u,2u,3u})$:
\begin{eqnarray}
\lambda_{1u} = \frac{\tilde{y}_u^2}{3}(1-\beta_{1u})^2, \qquad \lambda_{2u,3u} = \frac{\tilde{y}_u^2}{6} \left ((1+\beta_{1u})^2+2(\beta_{2u}^2+\beta_{3u}^2)+ \beta_{4u}^2 \mp \sqrt{\Lambda_u} \right ),
\label{eq:eigenvaluessq}
\end{eqnarray}
in which $\Lambda_u$ is given by
\begin{eqnarray}
\Lambda_u &=& (1+\beta_{1u})^4+4(\beta_{2u}^4+\beta_{3u}^4)+\beta_{4u}^4+4((1+\beta_{1u})^2+\beta_{4u}^2)(\beta_{2u}^2+\beta_{3u}^2)-2(1+\beta_{1u})^2\beta_{4u}^2\nonumber \\ &-&8\beta_{2u}^2\beta_{3u}^2+16(1+\beta_{1u})\beta_{2u}\beta_{3u} \beta_{4u}.
\label{eq:Lambdaudef}
\end{eqnarray}
Clearly, for arbitrary values of the parameters, the eigenvalues are not hierarchical. However, in looking for leading-order results in which only one eigenvalue is sizable, we can easily identify two general scenarios of interest, depending on the ordering of the mass eigenvalues. One option is that $\lambda_{1u}$ is one of the small eigenvalues, which would have $\beta_{1u}\rightarrow 1$, and $\lambda_{2u}$ is the other, and hence $\lambda_{3u}$ generically has an $O(1)$ value. Another option is that $\lambda_{1u}$ is the large eigenvalue, such that $\beta_{1u}\neq 1$, and both $\lambda_{2u,3u}$ are small. We now discuss each possibility in turn.
In what follows, we will focus on the up quarks, but our default assumption will be that the down quarks and the charged leptons will take similar forms. Mixing possible options for eigenvalue hierarchies in the different charged fermion sectors will not be considered here for simplicity.
\subsection{Case 1: $\lambda_{1u,2u}\ll \lambda_{3u}$ (encompassing the ``singlet-dominated" and ``democratic" limits)}
We begin with the situation that $\beta_{1u}\rightarrow 1$, such that $\lambda_{1u}$ is a small eigenvalue, and explore parameter regimes in which $\lambda_{2u}$ is also small. For simplicity, we first consider the case in which both vanish, such that to this order of approximation we have one massive third-generation, and two massless generations. It is easily verified that in this regime, both eigenvalues vanish for
\begin{eqnarray}
\beta_{1u}=1, \qquad \beta_{2u}\beta_{3u}=\beta_{4u}.
\label{eq:betarelations}
\end{eqnarray}
This case includes what we call the {\it democratic} limit, in which all the $\beta_{iu}=1$, and thus the MSSM Yukawas take on the well-known democratic form \cite{Fritzsch_1994}. The democratic limit was originally studied at leading order in \cite{Everett_2018}, and will be studied in more detail below, including subleading corrections. This case also includes what we will call the {\it singlet-dominated} limit, which is the case in which $ \beta_{4u} \gg \beta_{1u,2u,3u}$, as $\beta_{4u}$ is the parameter related to the strength of the superpotential coupling involving only $\mathcal{S}_3$ singlet fields. In the singlet-dominated limit, the MSSM and messenger Yukawa couplings at leading order, in the diagonal quark mass basis, only have nonvanishing $3-3$ entries, allowing for sizable stop mixing and consequently lighter superpartner masses than the other examples we will consider (as we will see). This limiting case was studied in some detail in \cite{Everett_2019} and \cite{Everett_2020}, and will be considered below as a benchmark scenario for purposes of comparison.
For Case 1, incorporating Eq.~(\ref{eq:betarelations}) and up to possible rephasings to ensure that the fermion masses are real and positive, the diagonalization matrices $U_{uL}$ and $U_{uR}$ take the form
\begin{eqnarray}
U_{uL}=\left (\begin{array}{ccc} \;\;\; \frac{1}{\sqrt{2}} & -\frac{\beta_{3u}}{\sqrt{2}\sqrt{2+\beta_{3u}^2}} & \frac{1}{\sqrt{2+\beta_{3u}^2}}
\\ -\frac{1}{\sqrt{2}} & -\frac{\beta_{3u}}{\sqrt{2}\sqrt{2+\beta_{3u}^2}} & \frac{1}{\sqrt{2+\beta_{3u}^2}}
\\ \;\;\; 0 &\frac{\sqrt{2}}{\sqrt{2+\beta_{3u}^2}} & \frac{\beta_{3u}}{\sqrt{2+\beta_{3u}^2}}
\end{array}\right ), \qquad
U_{uR}=\left (\begin{array}{ccc} \;\;\; \frac{1}{\sqrt{2}} & -\frac{\beta_{2u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}} & \frac{1}{\sqrt{2+\beta_{2u}^2}} \\
-\frac{1}{\sqrt{2}} & -\frac{\beta_{2u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}} & \frac{1}{\sqrt{2+\beta_{2u}^2}}
\\ \;\;\; 0 &\frac{\sqrt{2}}{\sqrt{2+\beta_{2u}^2}} & \frac{\beta_{2u}}{\sqrt{2+\beta_{2u}^2}}
\end{array}\right ).
\label{eq:case1diagmatrices}
\end{eqnarray}
Assuming these forms with no further rephasings, the messenger Yukawa couplings in the diagonal quark mass basis then take the form
\begin{equation}
Y_{u1}^\prime = \tilde{y}_u \left (\begin{array}{ccc} -\frac{\sqrt{3}}{2} & \frac{3 \beta_{2u}}{2\sqrt{2+\beta_{2u}^2}} & \frac{\beta_{2u}^2-1}{\sqrt{2}\sqrt{2+\beta_{2u}^2}} \\ \frac{3\beta_{3u}}{2\sqrt{2+\beta_{3u}^2}} & \frac{3\sqrt{3} \beta_{2u}\beta_{3u}}{2\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} & \frac{\sqrt{3}(\beta_{2u}^2-1)\beta_{3u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} \\ \frac{\beta_{3u}^2-1}{\sqrt{2}\sqrt{2+\beta_{3u}^2}} & \frac{\sqrt{3}(\beta_{3u}^2-1)\beta_{2u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} & \frac{(\beta_{2u}^2-1)(\beta_{3u}^2-1)}{\sqrt{3}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}}
\end{array} \right )
\end{equation}
\begin{equation}
Y_{u2}^\prime =\tilde{y}_u \left (\begin{array}{ccc} -\frac{\sqrt{3}}{2} & -\frac{3 \beta_{2u}}{2\sqrt{2+\beta_{2u}^2}} & \frac{\beta_{2u}^2-1}{\sqrt{2}\sqrt{2+\beta_{2u}^2}} \\ -\frac{3\beta_{3u}}{2\sqrt{2+\beta_{3u}^2}} & \frac{3\sqrt{3} \beta_{2u}\beta_{3u}}{2\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} & \frac{\sqrt{3}(\beta_{2u}^2-1)\beta_{3u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} \\ -\frac{\beta_{3u}^2-1}{\sqrt{2}\sqrt{2+\beta_{3u}^2}} & \frac{\sqrt{3}(\beta_{3u}^2-1)\beta_{2u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} & \frac{(\beta_{2u}^2-1)(\beta_{3u}^2-1)}{\sqrt{3}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}}
\end{array} \right ).
\label{eqn:case1yukawa}
\end{equation}
From these forms, we see that in the democratic limit, the messenger Yukawas only have nonvanishing entries in the upper $2\times 2$ block, as follows:
\begin{equation}
Y_{u1}^\prime = \tilde{y}_u \left (\begin{array}{ccc} -\frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 0 \\ 0& 0&0
\end{array} \right ), \qquad Y_{u2}^\prime = \tilde{y}_u \left (\begin{array}{ccc} -\frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} & 0 \\ -\frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 0 \\ 0& 0&0
\end{array} \right ).
\end{equation}
In the singlet-dominated limit, the $3-3$ entries dominate, with $Y^\prime_{u1,u2}={\rm Diag}(0,0,\tilde{y}_u\beta_{2u}\beta_{3u}/\sqrt{3})$.
\subsection{Case 2: $\lambda_{2u,3u}\ll \lambda_{1u}$ (the ``doublet-dominated" limit)}
For this case, it is necessary that $\beta_{1u}\neq 1$ such that $\lambda_{1u}\gg \lambda_{2u,3u}$. For concreteness, we take $\beta_{1u}\rightarrow -1$, and thus require $\beta_{2u,3u,4u}\ll 1$, as well as $\Lambda_u \rightarrow 0$. Indeed, $\lambda_{2u,3u}=0$ is achieved for $\beta_{1u}=-1,\beta_{2u}=\beta_{3u}=\beta_{4u}=0$. To see this, we note that for $\beta_1=-1$ only, the condition for $\Lambda_u=0$ is as follows:
\begin{equation}
-8 \beta_{2u}^2 \beta_{3u}^2+4(\beta_{2u}^4+\beta_{3u}^4)+4(\beta_{2u}^2+\beta_{3u}^2)\beta_{4u}^2+\beta_{4u}^2=0,
\end{equation}
which is zero only for $\beta_{4u}=0$ and $\beta_{2u}=\beta_{3u}$. We will take $\beta_{1u} = -1$ and $\beta_{4u} = 0$, but leave $\beta_{2u}$ and $\beta_{3u}$ unconstrained at present, recalling that we will need to restrict ourselves to the case that $\beta_{2u,3u}\ll \vert \beta_{1u} \vert =1$. This limit is the {\it doublet-dominated} limit, since now $\vert \beta_{1u} \vert \gg \beta_{2u,3u} \gg \beta_{4u}=0$, and $\beta_{1u}$ controls the superpotential coupling involving only $\mathcal{S}_3$ doublet fields. In this limit, the mass eigenvalues take the form (assuming for concreteness that $\beta_{3u}>\beta_{2u}$):
\begin{equation}
\lambda_{1u} = \frac{2\tilde{y}_u^2}{3}, \qquad \lambda_{2u}= \frac{2 \tilde{y}_u^2 \beta_{2u}^2}{3}, \qquad \lambda_{3u} = \frac{2 \tilde{y}_u^2 \beta_{3u}^2}{3},
\end{equation}
such that $\lambda_{2u} < \lambda_{3u} < \lambda_{1u}$. (For $\beta_{3u}< \beta_{2u}$, the placement of $\beta_{2u}$ and $\beta_{3u}$ in $\lambda_{2u}$ and $\lambda_{3u}$ is reversed.) We now take $ \sqrt{3} \tilde{y}_u/2=y_t$ to identify $y_t$ as the top quark Yukawa coupling to leading order. The diagonalization matrices $U_{uL}$ and $U_{uR}$ now take the following particularly simple forms:
\begin{equation}
U_{uL}=\left (\begin{array}{ccc} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ 0& 1 & 0
\end{array} \right ), \qquad
U_{uR}=\left (\begin{array}{ccc} 0& \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ 1 & 0 & 0
\end{array} \right ).
\end{equation}
The messenger Yukawas in the diagonal quark basis are then given by
\begin{equation}
Y_{u1}^\prime = y_t \left (\begin{array}{ccc} -\frac{\beta_{2u}}{2\sqrt{2}}& -\frac{3}{4} & -\frac{\sqrt{3}}{4} \\ \;\;\; 0 & -\frac{\beta_{3u}}{2\sqrt{2}} & \frac{\sqrt{3}\beta_{3u}}{2\sqrt{2}}\\ \frac{\sqrt{3}\beta_{2u}}{2\sqrt{2}} & -\frac{\sqrt{3}}{4} & \;\;\; \frac{1}{4} \end{array} \right ), \qquad
Y_{u2}^\prime = y_t\left (\begin{array}{ccc} -\frac{\beta_{2u}}{2\sqrt{2}}& -\frac{3}{4} & \frac{\sqrt{3}}{4} \\ \;\;\; 0 & -\frac{\beta_{3u}}{2\sqrt{2}} & -\frac{\sqrt{3}\beta_{3u}}{2\sqrt{2}}\\ -\frac{\sqrt{3}\beta_{2u}}{2\sqrt{2}} & \frac{\sqrt{3}}{4} & \;\;\; \frac{1}{4} \end{array} \right ).
\end{equation}
\section{Models}
As described in the previous subsection, we have identified two cases with hierarchical quark and charged lepton masses. The first (Case 1) satisfies Eq.~(\ref{eq:betarelations}), and includes two possible scenarios at leading order: the singlet-dominated limit, in which it is the $\mathcal{S}_3$ singlet couplings of the MSSM fields and the Higgs-messenger fields that dominate the superpotential, and the democratic limit, in which all the couplings of $\mathcal{S}_3$ representations in the superpotential are precisely equal at leading order, resulting in an enhanced $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ symmetry. The singlet-dominated limit was explored in \cite{Everett_2019} and \cite{Everett_2020}, and the democratic limit at leading order in \cite{Everett_2018}.
The second (Case 2) is what we call the doublet-dominated limit, as in this case the dominant couplings are those involving only $\mathcal{S}_3$ doublets.
In what follows, we will discuss these scenarios in greater detail.
\subsection{Case 1 models}
We begin the discussion of Case 1 models with the singlet-dominated limit, which was studied in detail in \cite{Everett_2019,Everett_2020}. In this scenario, sizable stop mixing can occur due to the FGM contributions to the third-generation soft trilinear scalar coupling. This in turn allows for the squarks and gluinos to be in the $O(5-6 \; {\rm TeV})$ range, which is relatively light compared to generic parameter choices for this class of FGM models. A variety of subleading corrections to this limit can be considered, including the possibility of generating nontrivial masses for the second-generation fields and the possibility of viable quark mixing at the first subleading order. For the case described in \cite{Everett_2020}, the corrections to the soft terms that result from these terms have only minimal effects on the superpartner masses.
Furthermore, in this case flavor-violating contributions to the soft terms also do not result at the first subleading order in the quantities that control the lighter generation quark and lepton masses, though this is not necessarily generic. Here we will not revisit this case in detail other than as a point of comparison for the new scenarios considered in this work.
Let us now turn to the democratic limit, for which the Yukawa coupling parameters $\beta_{1i}=\beta_{2i}= \beta_{3i}= \beta_{4i}=1$, where $i=u,d,e$. In this case, the MSSM Yukawa matrices take the form \begin{equation}
Y_i=\frac{\tilde{y}_i}{\sqrt{3}}\begin{pmatrix}
1 & 1 & 1\\
1 & 1 & 1\\
1 & 1 & 1
\end{pmatrix}.
\end{equation}
This is the well-known flavor democratic mass matrix form, which exhibits an $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ symmetry. At leading order, this mass matrix has two vanishing eigenvalues, and one $O(1)$ eigenvalue, to be identified with the third-generation. As shown in \cite{Everett_2018}, the messenger Yukawa matrices have nonzero entries only in the upper $2\times 2$ block in the diagonal quark mass basis.
We now address the generation of the first- and second-generation fermion masses and the effects on the sfermion masses through the messenger Yukawa corrections. Here we choose to break the $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ symmetry to $\mathcal{S}_{2L}\times \mathcal{S}_{2R}$ and then to $\mathcal{S}_{1L}\times \mathcal{S}_{1R}$, which generates a nonzero mass for the first- and second-generation fermions (see e.g.~\cite{Fritzsch_2000}). This can be achieved via the following terms:
\begin{equation}
Y_i^{(\text{corr})}=\frac{\tilde{y}_i\epsilon_i}{\sqrt{3}}\begin{pmatrix}
0 & 0 & 1\\
0 & 0 & 1\\
1 & 1 & 1
\end{pmatrix}+\frac{\tilde{y}_i\sigma_i}{\sqrt{3}}\begin{pmatrix}
1 & 0 & -1\\
0 & -1 & 1\\
-1 & 1 & 0
\end{pmatrix},
\label{eq:demcorrections}
\end{equation}
in which $\epsilon_i$ and $\sigma_i$ are real dimensionless perturbative parameters associated with symmetry breaking from $\mathcal{S}_3$ to $\mathcal{S}_2$ and $\mathcal{S}_2$ to $\mathcal{S}_1$ respectively. In our scenario, the $\epsilon$ perturbations of the up quarks (the down quarks and charged leptons have analogous structures) can be generated in superpotential at the renormalizable level by
\begin{equation}
\epsilon_u y_u[\beta_{2u} Q_\textbf{2}\bar{u}_\textbf{1}\mathcal{H}_u^{(2)}+\beta_{3u}Q_\textbf{1}\bar{u}_\textbf{2}\mathcal{H}_u^{(2)}+\beta_{4u} Q_\textbf{1}\bar{u}_\textbf{1}\mathcal{H}_{u}^{(1)}],
\end{equation}
while the $\sigma$ perturbations can be generated via non-renormalizable operators. These superpotential terms add corrections of the form of Eq.~(\ref{eq:demcorrections}) to the Yukawa matrix for the up-type quarks, and corrections of the following form to the up-type messenger Yukawa matrices:
\begin{equation}
\begin{split}
Y_{u1}^{\prime(\text{corr})}&=\tilde{y}_u\epsilon_u\begin{pmatrix}
0 & 0 & \frac{1}{2}-\frac{1}{2\sqrt{3}} \\
0 & 0 & -\frac{1}{2}-\frac{1}{2\sqrt{3}}\\
\frac{1}{2}-\frac{1}{2\sqrt{3}} & -\frac{1}{2}-\frac{1}{2\sqrt{3}}& \frac{1}{\sqrt{3}}
\end{pmatrix}+\tilde{y}_u\sigma_u\left(
\begin{array}{ccc}
\frac{1}{2}-\frac{1}{2 \sqrt{3}} & 0 & \frac{1}{2}+\frac{1}{2 \sqrt{3}} \\
0 & \frac{1}{2}+\frac{1}{2 \sqrt{3}} & \frac{1}{2}-\frac{1}{2 \sqrt{3}} \\
\frac{1}{2}+\frac{1}{2 \sqrt{3}} & \frac{1}{2}-\frac{1}{2 \sqrt{3}} & 0 \\
\end{array}
\right)\\
Y_{u2}^{\prime(\text{corr})}&=\tilde{y}_u\epsilon_u\begin{pmatrix}
0 & 0 & -\frac{1}{2}-\frac{1}{2\sqrt{3}}\\
0 & 0 & \frac{1}{2}-\frac{1}{2\sqrt{3}}\\
-\frac{1}{2}-\frac{1}{2\sqrt{3}} & \frac{1}{2}-\frac{1}{2\sqrt{3}} & \frac{1}{\sqrt{3}}
\end{pmatrix}+\tilde{y}_u\sigma_u\left(
\begin{array}{ccc}
-\frac{1}{2}-\frac{1}{2 \sqrt{3}} & 0 & -\frac{1}{2}+\frac{1}{2 \sqrt{3}} \\
0 & -\frac{1}{2}+\frac{1}{2 \sqrt{3}} & -\frac{1}{2}-\frac{1}{2 \sqrt{3}} \\
-\frac{1}{2}+\frac{1}{2 \sqrt{3}} & -\frac{1}{2}-\frac{1}{2 \sqrt{3}} & 0 \\
\end{array}
\right).
\end{split}
\end{equation}
Including these correction terms along with the leading order results, the eigenvalues
$\lambda_{1u, 2u,3u}$, are then found to be
\begin{equation}
\lambda_{1u} = 0, \qquad \lambda_{2u} = \frac{4 y_t^2 \epsilon_u^2}{81}+ O(\epsilon^3), \qquad \lambda_{3u} = y_t^2+O(\epsilon_u^3).
\end{equation}
In these relations, we have identified the top quark Yukawa coupling $y_t$ through $\tilde{y}_u = (y_t/\sqrt{3})(1-5\epsilon_u/9+ O(\epsilon_u)^2$, which follows from setting $\lambda_{3u} = y_t^2$ through second order in $\epsilon_u$.
As expected, $\epsilon_u$ controls the charm quark mass
\footnote{Note that if we neglect the subleading $\sigma$ perturbations, this scheme is equivalent to replacing $\beta_{2u,3u,4u} = 1+\epsilon_u$ in the general form for the superpotential couplings. As such, the mixing matrices are easily obtained using the results of Eq.~(\ref{eq:case1diagmatrices}) with the appropriate substitutions.}.
In the diagonal quark mass basis, the messenger Yukawa matrices for the up-type quark sector are given to order in $\sigma_u/\epsilon_u$ by
\begin{equation}
\begin{split}
Y_{u1}^{\prime \text{(diag)}}&=y_t\left(
\begin{array}{ccc}
-\frac{1}{2}+\frac{5 \epsilon_u }{18}-\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & -\frac{1}{2}-\frac{\epsilon_u }{18}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & -\frac{5 \epsilon_u
}{9 \sqrt{2}} \\
-\frac{1}{2}-\frac{\epsilon_u }{18}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{1}{2}+\frac{\epsilon_u }{6}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{\epsilon_u }{3
\sqrt{2}} \\
-\frac{5 \epsilon_u }{9 \sqrt{2}} & \frac{\epsilon_u }{3 \sqrt{2}} & -\frac{\epsilon_u }{9} \\
\end{array}
\right)\\
Y_{u2}^{\prime \text{(diag)}}&=y_t\left(
\begin{array}{ccc}
-\frac{1}{2}+\frac{5 \epsilon_u }{18}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{1}{2}+\frac{\epsilon_u }{18}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{5 \epsilon_u
}{9 \sqrt{2}} \\
\frac{1}{2}+\frac{\epsilon_u }{18}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{1}{2}+\frac{\epsilon_u }{6}-\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{\epsilon_u }{3
\sqrt{2}} \\
\frac{5 \epsilon_u }{9 \sqrt{2}} & \frac{\epsilon_u }{3 \sqrt{2}} & -\frac{\epsilon_u }{9} \\
\end{array}
\right).
\end{split}
\end{equation}
Analogous forms are easily obtained for the MSSM and messenger Yukawa matrices for down-type quarks and leptons in the diagonal quark mass basis with the replacements $\epsilon_u \rightarrow \epsilon_{d,e}$ and $y_t \rightarrow y_{b,\tau}$.
The relative strengths of the parameters $\epsilon_{u,d,e}$ and $\sigma_{u,d,e}$ can be estimated from the fact that these parameters govern the fermion masses of the lighter generations. More precisely, up to $O(1)$ prefactors, $\epsilon_{u,d,e}$ is related to $m_{c,s,\mu}/m_{t,b,\tau}$, while $\sigma_{u,d,e}$ is constrained by $m^2_{u,d,e}/m^2_{t,b,\tau} \sim \sigma^2_{u,d,e}/\epsilon_{u,d,e}$.
From these relations, it is straightforward to obtain that $\epsilon_u\approx 3\times 10^{-2}$ and $\sigma_u\approx 1\times 10^{-3}$. Similarly, $\epsilon_d \approx 0.1$, $\sigma_d\approx 9\times 10^{-3}$, $\epsilon_e\approx 0.3$, and $\sigma_e\approx 8 \times 10^{-3}$. These parameter values also yield hierarchical quark mixing angles of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix, in which the largest angle is the Cabibbo angle, $\sin\theta_c\sim 0.17$.
While the quark mixing angles are not fully realistic (the Cabibbo angle is clearly too small compared to its experimentally determined value), for the purposes of this study it is a reasonable starting point for the analysis.
We now find the nonvanishing corrections to the soft supersymmetry breaking terms, assuming for simplicity that the ratio of the $F$ terms to the scalar VEVs for the $X_H$ and $X_T$ terms are identical (both will be denoted as $\Lambda$). We provide the expressions for these correction terms in the Appendix. As expected, in the limit that the perturbation parameters are set to zero, the result is what was found in \cite{Everett_2018}. When the perturbations are added, the diagonal entries of the soft mass-squared terms are corrected at second order in the $\epsilon$ parameters. This generates nonzero (but small) diagonal $3-3$ entries. In addition, with nonzero perturbations, flavor off-diagonal contributions to the corrections to the soft terms are generated. More precisely, the $\epsilon_{u,d,e}$ parameters introduce nonvanishing $\delta m_{f_{23}}^2$ terms at first order in $\epsilon$, while the $\sigma_{u,d,e}$ introduce nonvanishing $\delta m_{f_{12}}^2$ and $\delta m_{f_{21}}^2$ terms. Therefore, the dominant effects are expected to be seen in the $2-3$ sfermion mixings. Further details will be discussed in the next section.
\begin{comment}
The nonvanishing corrections to the soft supersymmetry breaking terms now take the form:
\begin{equation}
\begin{split}
\delta m_{Q_{11}}^2=\frac{\Lambda^2}{(4\pi)^4}\bigg[(&6y_t^4+6y_b^4+2y_b^2y_t^2+y_b^2y_\tau^2-\Tilde{g_u}^2y_t^2-\Tilde{g_d}^2y_b^2) \\ &+\epsilon_u \left(-\frac{32}{9}y_t^4+\frac{4}{9}\Tilde{g_u}^2y_t^2-\frac{4}{9}y_b^2y_t^2\right)+\epsilon_d \left(-\frac{32}{9}y_b^4+\frac{4}{9}\Tilde{g_d}^2y_b^2-\frac{4}{9}y_b^2y_t^2-\frac{4}{9}y_b^2y_\tau^2\right) +\mathcal{O}(\epsilon^2)
\bigg]\\
\delta m_{Q_{22}}^2=\frac{\Lambda^2}{(4\pi)^4}\bigg[(&6y_t^4+6y_b^4+2y_b^2y_t^2+y_b^2y_\tau^2-\Tilde{g_u}^2y_t^2-\Tilde{g_d}^2y_b^2) \\ &+\epsilon_u \left(\frac{32}{9}y_t^4-\frac{4}{9}\Tilde{g_u}^2y_t^2+\frac{4}{9}y_b^2y_t^2\right)+\epsilon_d \left(\frac{32}{9}y_b^4-\frac{4}{9}\Tilde{g_d}^2y_b^2+\frac{4}{9}y_b^2y_t^2+\frac{4}{9}y_b^2y_\tau^2\right) +\mathcal{O}(\epsilon^2)
\bigg]\\
\delta m_{Q_{23}}^2=\delta m_{Q_{32}}^2=\frac{\Lambda^2}{(4\pi)^4}\bigg[&\epsilon_u\left(\frac{8\sqrt{2}}{3}y_t^4-\frac{4\sqrt{2}}{9}\Tilde{g_u}^2y_t^2+\frac{4\sqrt{2}}{9}y_b^2y_t^2\right)\\
+&\epsilon_d\left(\frac{8\sqrt{2}}{3}y_b^4-\frac{4\sqrt{2}}{9}\Tilde{g_d}^2y_b^2+\frac{4\sqrt{2}}{9}y_b^2y_t^2+\frac{4\sqrt{2}}{9}y_b^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\bigg]\\ \nonumber
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&\delta m_{\bar{u}_{11}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(12y_t^4+2y_t^2y_b^2-2\Tilde{g_u}^2y_t^2)-\frac{8\epsilon_u}{9}\left(8y_t^4+y_b^2y_t^2-\tilde{g_u}^2y_t^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{u}_{22}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(12y_t^4+2y_t^2y_b^2-2\Tilde{g_u}^2y_t^2)+\frac{8\epsilon_u}{9}\left(8y_t^4+y_b^2y_t^2-\tilde{g_u}^2y_t^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{u}_{23}}^2=\delta m_{\bar{u}_{32}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[\frac{8\sqrt{2}\epsilon_u}{9}\left( 6y_t^4+y_b^2y_t^2-\tilde{g_u}^2y_t^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{d}_{11}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(12y_b^4+2y_t^2y_b^2+2y_b^2y_{\tau}^2-2\Tilde{g_d}^2y_b^2)-\frac{8\epsilon_d}{9}\left(8y_b^4+y_b^2y_t^2+y_b^2y_\tau^2-\tilde{g_d}^2y_b^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{d}_{22}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(12y_b^4+2y_t^2y_b^2+2y_b^2y_{\tau}^2-2\Tilde{g_d}^2y_b^2)+\frac{8\epsilon_d}{9}\left(8y_b^4+y_b^2y_t^2+y_b^2y_\tau^2-\tilde{g_d}^2y_b^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{d}_{23}}^2=\delta m_{\bar{d}_{32}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[\frac{8\sqrt{2}\epsilon_d}{9}\left( 6y_b^4+y_b^2y_t^2+y_b^2y_{\tau}^2-\tilde{g_d}^2y_b^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{L_{11}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(4y_\tau^4+3y_b^2y_\tau^2-\Tilde{g_e}^2y_\tau^2)-\frac{4\epsilon_e}{9}\left(6y_\tau^4+3y_b^2y_\tau^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{L_{22}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(4y_\tau^4+3y_b^2y_\tau^2-\Tilde{g_e}^2y_\tau^2)+\frac{4\epsilon_e}{9}\left(6y_\tau^4+3y_b^2y_\tau^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{L_{23}}^2=\delta m_{L_{32}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[\frac{4\sqrt{2}\epsilon_e}{9}\left( 4y_\tau^4+3y_b^2y_{\tau}^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{e}_{11}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(8y_\tau^4+6y_b^2y_\tau^2-2\Tilde{g_e}^2y_\tau^2)-\frac{8\epsilon_e}{9}\left(6y_\tau^4+3y_b^2y_\tau^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{e}_{22}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(8y_\tau^4+6y_b^2y_\tau^2-2\Tilde{g_e}^2y_\tau^2)+\frac{8\epsilon_e}{9}\left(6y_\tau^4+3y_b^2y_\tau^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{e}_{23}}^2=\delta m_{\bar{e}_{32}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[\frac{8\sqrt{2}\epsilon_e}{9}\left( 4y_\tau^4+3y_b^2y_{\tau}^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{H_u}^2=\delta m_{H_d}^2=0\\
&\Tilde{A_u}=\begin{pmatrix}
0 &0 &0 \\
0 & -\frac{2\epsilon_u}{9}(3y_t^3+y_ty_b^2) & \frac{4\sqrt{2}\epsilon_u}{9}y_t^3+\frac{4\sqrt{2}\epsilon_d}{9}y_ty_b^2\\
0& \frac{8\sqrt{2}\epsilon_u}{9}y_t^3 & \mathcal{O}(\epsilon^2)
\end{pmatrix}\quad \Tilde{A_d}=\begin{pmatrix}
0 &0 &0 \\
0 & -\frac{2\epsilon_d}{9}(3y_b^3+y_t^2y_b) & \frac{4\sqrt{2}\epsilon_d}{9}y_b^3+\frac{4\sqrt{2}\epsilon_u}{9}y_t^2y_b\\
0& \frac{8\sqrt{2}\epsilon_d}{9}y_b^3 & \mathcal{O}(\epsilon^2)
\end{pmatrix}\\
&\Tilde{A_e}=\begin{pmatrix}
0 &0 &0 \\
0 & -\frac{2\epsilon_e}{3}y_\tau^3 & \frac{4\sqrt{2}\epsilon)e}{9}y_\tau^3\\
0& \frac{8\sqrt{2}\epsilon_e}{9}y_\tau^3 & \mathcal{O}(\epsilon^2)
\end{pmatrix}.
\end{split}
\label{eq:democraticsoftterms}
\end{equation}
\end{comment}
\subsection{Case 2 models}
This case corresponds to the doublet-dominated limit. Here we need $\lambda_1\gg \lambda_{2,3}$. In the limit that $\lambda_{2,3}\rightarrow 0$, we see from Eqs.~(\ref{eq:eigenvaluessq})-(\ref{eq:Lambdaudef}) this can be achieved for $\beta_1\rightarrow -1$ and $\beta_{i=2,3,4}\ll 1$, and we need $\tilde{\Lambda} \rightarrow 0$. For $\beta_1=-1$, the condition for $\tilde{\Lambda}=0$ is as follows:
\begin{equation}
-8 \beta_2^2 \beta_3^2+4(\beta_2^4+\beta_3^4)+4(\beta_2^2+\beta_3^2)\beta_4^2+\beta_4^2=0,
\end{equation}
which is zero only for $\beta_4=0$, $\beta_2=\beta_3$. In the up quark sector, we will now set $\lambda_1=y_t^2$, such that $\tilde{y}_u^2= (3/4) y_t^2$ (analogous relations hold for the down quark and charged lepton sectors). The $\lambda_{2i,3i}$ are directly related to $\beta_{2i,3i}$, with the specific identification dependent on the values of the $\beta_{2i,3i}$.
\noindent $\bullet$ {\it Ordering $\beta_{3i}>\beta_{2i}$}.
Let us first consider the case in which $\beta_{3i}>\beta_{2i}$, for which we have
\begin{equation}
U_{iL}^\dagger Y_i U_{iR}= Y_i^{(\text{diag})}=
y_{t,b,\tau}\text{Diag}\left (\frac{\beta_{2i}}{\sqrt{2}}, \frac{ \beta_{3i}}{\sqrt{2}},1 \right ),
\label{eq:order1}
\end{equation}
in which $U_{iL,iR}$ take the simple forms
\begin{equation}
U_{iL}=\left (\begin{array}{ccc} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ 0& 1 & 0
\end{array} \right ), \qquad
U_{iR}=\left (\begin{array}{ccc} 0& \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ 1 & 0 & 0
\end{array} \right ).
\label{umatricescase2a}
\end{equation}
We see that for this ordering, the $\beta_{3i}$ control the second-generation masses and the $\beta_{2i}$ control the first-generation masses.
The messenger Yukawas in the diagonal fermion mass basis (the SCKM basis) are given by
\begin{equation}
Y_{i1}^\prime = y_{t,b,\tau} \left (\begin{array}{ccc} -\frac{\beta_{2i}}{2\sqrt{2}}& -\frac{3}{4} & -\frac{\sqrt{3}}{4} \\ \;\;\; 0 & -\frac{\beta_{3i}}{2\sqrt{2}} & \frac{\beta_{3i}}{2}\sqrt{\frac{3}{2}}\\ \frac{\beta_{2i}}{2}\sqrt{\frac{3}{2}} & -\frac{\sqrt{3}}{4} & \;\;\; \frac{1}{4} \end{array} \right ),\qquad
Y_{i2}^\prime = y_{t,b,\tau}\left (\begin{array}{ccc} -\frac{\beta_{2i}}{2\sqrt{2}}& -\frac{3}{4} & +\frac{\sqrt{3}}{4} \\ \;\;\; 0 & -\frac{\beta_{3i}}{2\sqrt{2}} & -\frac{\beta_{3i}}{2}\sqrt{\frac{3}{2}}\\ -\frac{\beta_{2i}}{2}\sqrt{\frac{3}{2}} & \frac{\sqrt{3}}{4} & \;\;\; \frac{1}{4} \end{array} \right ).
\end{equation}
Given that we can identify $\beta_{2i,3i}$ with the first and second-generation masses, respectively, we can write for example for the up-type quarks (with $y_{u,d,e}=\beta_{2u,d,} y_t/\sqrt{2}$ and $y_c=\beta_{3u} y_t/\sqrt{2}$):
\begin{equation}
Y_{u1}^\prime = \left (\begin{array}{ccc} -\frac{y_u}{2}& -\frac{3 y_t}{4}& -\frac{\sqrt{3}y_t}{4} \\ \;\;\; 0 & -\frac{y_c}{2} & -\frac{\sqrt{3}y_c}{2}\\ -\frac{\sqrt{3} y_u}{2} & \frac{\sqrt{3}y_t}{4} & \;\;\; \frac{y_t}{4} \end{array} \right ),\qquad
Y_{u2}^\prime = \left (\begin{array}{ccc} -\frac{y_u}{2}&-\frac{3 y_t}{4} & -\frac{\sqrt{3}y_t}{4} \\ \;\;\; 0 & -\frac{y_c}{2} & \frac{\sqrt{3}y_c}{2} \\ \frac{\sqrt{3} y_u}{2}& -\frac{\sqrt{3}y_t}{4} & \;\;\; \frac{y_t}{4} \end{array} \right ).
\label{eq:messyukcase2a}
\end{equation}
From the quark and charged lepton masses, we can roughly estimate (neglecting running effects) that $\beta_{2u}/\beta_{3u}\sim 2\times 10^{-3}$, $\beta_{2d}/\beta_{3d} \sim 0.05$, $\beta_{2l}/\beta_{3l}\sim 0.005$, while $\beta_{3d}/\beta_{3l} \sim 0.4$. Hence, to leading order we can neglect the effects proportional to the first-generation masses (here the $\beta_{2i}$), and treat the effects due to the second-generation masses (the $\beta_{3i}$) perturbatively.
We thus calculate the corrections to the soft supersymmetry terms in this limit.
As before, we assume for simplicity that the ratio of the $F$ terms to the scalar vevs for the $X_H$ and $X_T$ terms are identical.
The detailed forms of these soft supersymmetry breaking terms are presented in the Appendix.
We note that in this case, there are flavor off-diagonal contributions in the $\delta m^2_{{Q,L}_{12}}$ that are proportional to the $\beta_{3i}$, and thus scale with the second-generation quark and lepton masses. This is reminiscent of the Case 1 democratic limit with perturbations, though the dominant off-diagonal contributions occurred there in the $2-3$ sector, and here they arise in the more dangerous $1-2$ sector. We will discuss their effects in the next section.
\noindent $\bullet$ {\it Ordering $\beta_{2i}>\beta_{3i}$}.
We now consider the case in which $\beta_{2i}>\beta_{3i}$, for which the roles of $\beta_{2i}$ and $\beta_{3i}$ are switched in Eq.~(\ref{eq:order1}).
We now have
\begin{equation}
U_{iL}=\left (\begin{array}{ccc} 0& \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 1 & 0 & 0
\end{array} \right ), \qquad
U_{iR}=\left (\begin{array}{ccc} \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0& 1 & 0
\end{array} \right ).
\label{umatricescase2b}
\end{equation}
The messenger Yukawa matrices in the diagonal quark mass basis are of the form
\begin{equation}
Y_{i1}^\prime = y_{t,b,\tau} \left (\begin{array}{ccc} -\frac{\beta_{3i}}{2\sqrt{2}}& 0& -\frac{\sqrt{3}\beta_{3i}}{2\sqrt{2}} \\ \;\;\; -\frac{3}{4} & -\frac{\beta_{2i}}{2\sqrt{2}}&\frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & -\frac{\beta_{2i}}{2}\sqrt{\frac{3}{2}} & \;\;\; \frac{1}{4} \end{array} \right ),\qquad
Y_{i2}^\prime = y_{t,b,\tau}\left (\begin{array}{ccc} -\frac{\beta_{2i}}{2\sqrt{2}}& 0 & \frac{\sqrt{3}\beta_{3i}}{2\sqrt{2}}\\ \;\;\; -\frac{3}{4}& -\frac{\beta_{2i}}{2\sqrt{2}} & -\frac{\sqrt{3}}{4}\\ -\frac{\sqrt{3}}{4} & \frac{\beta_{2i}}{2}\sqrt{\frac{3}{2}} & \;\;\; \frac{1}{4} \end{array} \right ).
\end{equation}
As in the previous section, we can ignore effects that scale with the first-generation fermion masses and keep leading contributions involving the second-generation fermion masses. Thus, we now neglect the terms proportional to $\beta_{3i}$ and keep leading-order terms proportional to the $\beta_{2i}$. We can again calculate the soft supersymmetry breaking terms, subject to the same assumptions as given for the alternate ordering. The detailed forms are included in the Appendix.
One interesting feature of this mass ordering ($\beta_{2i}>\beta_{3i}$) is the corrections to the soft supersymmetry breaking mass terms are flavor-diagonal if we neglect effects proportional to the first-generation fermion masses. As in the alternate ordering, here we obtain contributions to $\delta m^2_{{Q,L}_{12}}$ that are proportional to the $\beta_{3i}$, but now these quantities must be much smaller since they govern the masses of the first-generation. Given the high degree of suppression of the flavor off-diagonal elements, in this case the model is clearly safe from flavor-changing neutral current constraints.
\section{Results and Discussion}
\label{resultssection}
In this section, we analyze the mass spectra of these scenarios and their phenomenological implications. We start with Case 1, focusing solely on the democratic limit with symmetry breaking effects, and then study Case 2, the doublet-dominated limit, with both orderings of the $\beta_{2i}$ and $\beta_{3i}$. The model parameters are $M_\text{mess}$, $\Lambda$, $\tan{\beta}=\langle H_u\rangle / \langle H_d \rangle$, the sign of $\mu$ (sgn($\mu$), taken here to be $+1$), and the relevant perturbation parameters, which depend on the scenario in question. Here we have followed standard procedures and replaced $\vert \mu\vert$ and $b$ with $\tan\beta$ and the $Z$ boson mass.
The renormalization group equations are run using SoftSUSY 4.1.4 \cite{Allanach:2001kg}.
\subsection{Case 1 models}
We start with the flavor democratic limit, which was explored in \cite{Everett_2018} for the case of third-generation masses only, i.e.~in the absence of the small perturbations that break the $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ symmetry. It was shown in \cite{Everett_2018} that this scenario leads to heavy superpartner masses, which can be traced to the absence of large stop mixing in this limit. In the presence of nonvanishing perturbations, this picture generically continues except for specific small regions of parameter space where the Higgs mass constraint can be satisfied without being bolstered by very heavy squarks.
In Figure~\ref{fig:democraticspectrum1}, we show a representative mass spectrum for an intermediate messenger mass scale of $M_{\rm mess}=10^{12}$ GeV and $\tan\beta=10$, where $\Lambda$ is chosen to satisfy the Higgs mass constraint \cite{Zyla:2020zbs}. As seen, the heavy Higgs particles are nearly 8 TeV, the gluino is approximately 10 TeV, and the squarks fall into three groupings: a lightest set that is close in mass to the heavy Higgs particles, a set in between, and a heavier set that is similar to the gluino mass. The sleptons are close in mass to the lightest neutralino, and the next-to-lightest superpartner (NLSP) is the lightest slepton.
The effects of nonzero $\sigma_{u,d,e}$ lead to small ($O(1\; {\rm GeV})$) splittings in the masses of $\tilde{d}_1$ and $\tilde{d}_2$, and $\tilde{u}_4$ and $\tilde{u}_5$, which are each originally identical up to order $10^{-2}$ GeV. The effect of $\epsilon_{u,d,e}$ is larger, which is expected as these have larger numerical values. For nonzero $\epsilon_u$, there is a splitting of order $\sim70$ GeV in the masses of $\tilde{u}_1$ and $\tilde{u}_2$, which are also identical up to order of $10^{-2}$ GeV in the $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ limit. Similar features are seen for $\tilde{u}_4$ and $\tilde{u}_5$. The $\epsilon_u$ corrections also introduce a small ($\sim25$ GeV) mass splitting for $\tilde{d}_1$ and $\tilde{d}_2$, which is a sign of the symmetry breaking from $\mathcal{S}_{3L}\times\mathcal{S}_{3R}$ to $\mathcal{S}_{2L}\times\mathcal{S}_{2R}$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.8]{images-democratic/Junetry2testallp-1.pdf}
\caption{The sfermion mass spectrum in the Case 1 democratic limit, with $M_\text{mess}=10^{12}$ GeV, $\Lambda=8.1\times 10^5$ GeV, $\tan{\beta}=10$, $\epsilon_u=0.033$, $\epsilon_d=0.108$, $\epsilon_e=0.281$, $\sigma_u=0.001$, $\sigma_d=0.009$, and $\sigma_e=0.008$. }
\label{fig:democraticspectrum1}
\end{figure}
As noted previously, this scenario has flavor off-diagonal contributions to the corrections to the soft supersymmetry breaking terms, with the dominant contributions in the $2-3$ sector. To get an estimate of the potential sizes of these effects, we employ the standard mass insertion approximation (MIA) method, in which the quantities of interest for the quarks are
$(\delta_f^{IJ})_{XY}=(\Delta_f^{IJ})_{XY}/((m_{fI})_{XX}(m_{fJ})_{YY})$,
where $f$ denotes the relevant matter superfield, $I,J$ are flavor indices, $X,Y$ are chirality labels, and $(\Delta_f^{IJ})_{XY}$ is an off-diagonal contribution to the sfermion soft terms \footnote{Note that MIA is a good approximation in this scenario although we have non-degenerate squark masses, since the squark masses are not strongly hierarchical~\cite{Raz_2002}.}. We expect rather mild constraints due to the heavy sfermion and gluino masses and the suppression factors in the off-diagonal contributions to the soft terms. For the set of model parameters in Fig.~\ref{fig:democraticspectrum1}, we obtain $2-3$ squark and slepton mass insertion parameters of the order $\vert (\delta_u^{23})_{LL}\vert \sim 5\times 10^{-3}$, $\vert (\delta_u^{23})_{RR}\vert \sim 10^{-2}$, $\vert (\delta_d^{23})_{LL}\vert \sim 5\times 10^{-3}$, $\vert (\delta_d^{23})_{RR}\vert \sim 7\times 10^{-4}$, $\vert (\delta_l^{23})_{LL}\vert \sim 2\times 10^{-3}$, and $\vert (\delta_l^{23})_{RR}\vert \sim 3\times 10^{-2}$, as well as small contributions to $LR$ mixings in the $2-3$ sector (ranging from $10^{-4}$ to $10^{-7}$. The $1-3$ and $1-2$ mass insertions are parametrically smaller, with limits that range from $10^{-4}$ to $10^{-12}$, except for $\vert (\delta_l^{12})_{RR}\vert \sim 3\times 10^{-3}$. The resulting effects are small and within the allowed ranges (see e.g.~\cite{Misiak_1998}).
\begin{figure}[h]
\centering
\subfloat{{\includegraphics[scale=0.9]{images-democratic/Sept1e12tanbeta10allpmixing.pdf}}}
\caption{
The sfermion mass eigenstates in the democratic limit with $\epsilon_u=0.033$, $\epsilon_d=0.108$, $\epsilon_e=0.281$, $\sigma_u=0.001$, $\sigma_d=0.009$, and $\sigma_e=0.008$, with $M_\text{mess}=10^{12}$ GeV, $\Lambda=8.1\times 10^5$ GeV, and $\tan{\beta}=10$. }
\label{fig2}
\end{figure}
The composition of the mass eigenstates of the sfermions is shown in Fig.~\ref{fig2}. Without perturbations, there is almost no mixing between different flavor eigenstates. The lightest $SU(3)_c$ charged particles are the first and second-generation right-handed squarks and the lightest sleptons are the first and second-generation right-handed sleptons. In Fig.~\ref{fig2}, the results are shown for $\epsilon_u=0.033$, $\epsilon_d=0.108$, $\epsilon_e=0.281$, $\sigma_u=0.001$, $\sigma_d=0.009$, and $\sigma_e=0.008$. The lighter squarks $\tilde{u_1}$ and $\tilde{u}_2$ are again the right-handed scharm and sup. Mixing between the second and third-generations for the left handed sparticles is observed in $\tilde{u}_4$ and $\tilde{u}_6$. There is also small but nonvanishing 2-3 generational mixing among right-handed up-type squarks. For the down sector, apart from the 1-2 and 2-3 generational mixing which are larger compared to the up sector, there is also a small but nonvanishing left-right mixing between $\tilde{b}_L$ and $\tilde{b}_R$ observed in $\tilde{d}_1$. For the sleptons, we again observe small mixing between the second and third-generation sleptons with the same handedness.
It is illustrative to compare this scenario with the singlet-dominated limit \cite{Everett_2019,Everett_2020}. In this case, the dominant contributions to the soft terms arise in the diagonal third-generation ($33$) entries, rendering this case similar to flavored gauge mediation models in which the Higgs-messenger mixing is controlled by Abelian symmetries.
Generally this case has a light spectrum, with masses below 6 TeV. Unlike the democratic case, the heavy Higgs particles are heavier than or comparable to the $SU(3)$-charged superpartners, with masses at the $5-6$ TeV range.
The large stop mixing due to the nonvanishing $A$ term for the third-generation fields at the messenger mass scale allows for a viable Higgs mass at smaller values of $\Lambda$ compared to the democratic limit, in which the $A$ terms vanish in the absence of the small symmetry breaking perturbations. Adding nonrenormalizable corrections as in \cite{Everett_2020} to generate the light quark and charged lepton masses does not alter this feature and generically leads to very small ($O(10^{-1} \; {\rm GeV})$)
\subsection{Case 2 models}
We now turn to the Case 2 models, for which the superpotential couplings only involving the $\mathcal{S}_3$-doublets dominate. As described in the previous section, in this case there are two sub-categories, depending on whether the $\beta_{3i}$ or the $\beta_{2i}$ parameters control the second-generation quark and charged lepton masses. Here we will label the mass ordering $\beta_{3i}>\beta_{2i}$ by Case 2a, and the alternate mass ordering $\beta_{2i}>\beta_{3i}$ as Case 2b. The soft supersymmetry breaking terms for Case 2a are given in Eq.~(\ref{eq:deltamB2-1}), and the analogous quantities for Case 2b are given in Eq.~(\ref{eq:deltamB2-3}).
\begin{figure}[h!]
\centering
\subfloat{{\includegraphics[scale=0.6]{images-caseb2/Julyb21e12tanbeta10alleps-1.pdf}}}
\subfloat{{\includegraphics[scale=0.6]{images-caseb2-beta2/Julyb21e12tanbeta10-beta2larger-1.pdf}}}
\caption{The sfermion mass spectra in the doublet-dominated (Case 2) limit, with $M_\text{mess}=10^{12}$ GeV, $\Lambda=6.6\times 10^5$ GeV, for (a) Case 2a $\beta_{3d}=0.03$, $\beta_{3u}=0.01$, $\beta_{3l}=0.08$, $\beta_{2u}=\beta_{2d}=\beta_{2l}=0$ (left), and (b) Case 2b with $\beta_{2d}=0.03$, $\beta_{2u}=0.01$, $\beta_{2l}=0.08$, $\beta_{3u}=\beta_{3d}=\beta_{3l}=0$ (right).}
\label{fig:b2spectra}
\end{figure}
In Case 2 models, there is a nonvanishing trilinear scalar parameter $A_t$ that is present in the absence of the first and second-generation quark and charged lepton masses, in contrast to the Case 1 democratic limit. Hence, the Higgs and superpartner masses are lighter than their Case 1 democratic counterparts, though not as light as in the Case 1 singlet-dominated limit.
In Fig.~\ref{fig:b2spectra}, we show characteristic mass spectra for $M_{\rm mess}=10^{12} \; {\rm GeV}$ and $\Lambda = 6.6\times 10^5$ GeV. Here we have included nonvanishing values for the parameters that fix the second-generation quark and charged lepton masses ($\beta_{3i}$ for Case 2a, $\beta_{2i}$ for Case 2b), and neglected the effects of the first-generation masses. The values of the perturbation parameters are chosen to yield appropriate values for the SM fermion mass values. We note here that if these quantities are taken to zero, the mass spectra are almost unchanged, with small changes that are at most $O(10^{-1}\; {\rm GeV})$, primarily in the slepton sector due to the relatively large value of the corresponding $\beta_{3l,2l}$ parameter.
We see that in both Case 2a and Case 2b, the gluino and squark masses are similar, with the gluino at about 8 TeV and the squarks ranging from approximately $8-10$ TeV. Unlike the Case 1 singlet-dominated limit as in Fig.~\ref{fig:democraticspectrum1} in which the squark masses are generally comparable to heavy Higgses, in this case the squarks are always much heavier than the heavy Higgs bosons. The slepton masses fall into two different ranges, with the NLSP as the lightest selectron $\tilde{e}_1$. The three lightest slectrons have their masses below 2 TeV, while the other sleptons have their masses between 3$-$4 TeV. The lightest charginos and neutralinos are gaugino-dominated, with a binolike lightest neutralino, while the heavier set is higgsino-dominated.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.9]{images-caseb2/septmixingcase2.pdf}
\caption{The sfermion mass spectrum in the doublet-dominated scenarios, with ordering $\beta_{3i}>\beta_{2i}$ (Case 2a) and $\beta_{2i}>\beta_{3i}$ (Case 2b), respectively, with $M_\text{mess}=10^{12}$ GeV and $\tan{\beta}=10$. For Case 2a, $\beta_{3d}=0.03$, $\beta_{3u}=0.01$, $\beta_{3l}=0.08$. For Case 2b, $\beta_{2d}=0.03$, $\beta_{2u}=0.01$, $\beta_{2l}=0.08$. }
\label{fig8}
\end{figure}
An intriguing difference between Case 2a and Case 2b is that in Case 2b, the heavy Higgs states and the heavy charginos and neutralinos are lighter than they are in Case 2a. For the model parameters as given in Fig.~\ref{fig:b2spectra}, we see that in Case 2b the heavy Higgs masses are in the 5-6 TeV range, while they are over 6 TeV in Case 2a, and the heavy charginos/neutralinos are also reduced by approximately 1 TeV in Case 2b compared to Case 2a. This indicates that in Case 2b, smaller values of the $\mu$ and $b$ parameters are needed for successful electroweak symmetry breaking.
Another significant difference between Case 2a and Case 2b is that Case 2a has nonvanishing off-diagonal contributions to squark mixing, as discussed in the previous subsection. The most significant off-diagonal sfermion mixing in Case 2a is given by $\vert (\delta_u^{12})_{LL}\vert \sim 1\times 10^{-4}$. These effects are small because the flavor off-diagonal contributions are proportional to the small quantities that govern the second-generation SM quark and charged lepton masses. In both cases, as shown in Fig.~\ref{fig8}, sfermion mixing is not significant due to the small size of the perturbation parameters. For larger values of the messenger mass scale, nontrivial left-right mixing is observed for the third-generation down-type squarks (left-right mixing in the other sfermion sectors is negligible for all values of the messenger mass scale).
\subsection{Discussion}
In comparing the mass spectra of these scenarios (Case 1: democratic, and Cases 2a and 2b: doublet-dominated, as well as the Case 1: singlet-dominated limit as studied in \cite{Everett_2018}), there are several features of interest.
For fixed $M_\text{mess}$, the mass spectra are more compressed for larger values of $\tan{\beta}$ ($\tan{\beta}>10$) because the contributions from the bottom and tau Yukawa couplings are more significant than in the low $\tan{\beta}$ regime. For smaller $\tan{\beta}$ values, the sparticle masses are heavier as the tree-level contribution to the light Higgs mass has decreased, requiring larger radiative corrections to boost its mass to its experimentally allowed range. The superpartner masses in this limit are thus highly split, with heavy squarks and gluinos, and lighter sleptons. For fixed $\tan\beta$ (here taken to be $\tan\beta=10$), lower values of the messenger mass scale generally lead to heavier spectra, as larger values of $\Lambda$ are needed to satisfy the light Higgs mass constraint. For higher messenger scales, due to increased renormalization group running effects, the $\mu$ and $b/\mu$ terms needed to satisfy the electroweak symmetry breaking constraints are smaller, and thus the heavy charginos and neutralinos become lighter.
\begin{figure}[t]
\centering
\subfloat{{\includegraphics[scale=0.45]{images-democratic/JuneShumMessVstanBoffHiggsContoursclear.pdf}}}
\subfloat{{\includegraphics[scale=0.45]{images-democratic/JuneShumMessVstanBHiggsContoursalleps.pdf}}}
\caption{(a) The Higgs mass (black band) and $\tilde{u}_3$ squark mass (color shading) for Case 1 in the democratic limit without perturbations, with
$\Lambda=7.7\times10^5$ GeV (left). (b) The same as (a), but with $\epsilon_u=0.033$, $\epsilon_d=0.108$, $\epsilon_e=0.281$, $\sigma_u=0.001$, $\sigma_d=0.009$, $\sigma_e=0.008$ (right). }
\label{figparameterspace}
\end{figure}
\begin{figure}[h!]
\centering
\subfloat{{\includegraphics[scale=0.35]{images-democratic/JuneShumMessVsLambdaoffHiggsContourstanBeta10.pdf}}}
\subfloat{{\includegraphics[scale=0.35]{images-democratic/JuneShumMessVsLambdaEps3HiggsContourstanBeta10.pdf}}}
\caption{(a) The Higgs mass (black band), gluino mass (color shading) and $\tilde{e}_1$ mass (dotted curves) as a function of $\Lambda$ and $M_\text{mess}$, for Case 1 in the democratic limit with $\tan\beta=10$, for (a) no perturbations (left), and (b) nonzero perturbations, with $\epsilon_u=0.033$, $\epsilon_d=0.108$, $\epsilon_e=0.281$, $\sigma_u=0.001$, $\sigma_d=0.009$, $\sigma_e=0.008$.}
\label{figparameterlambdammess}
\end{figure}
To further investigate the dependence of the mass spectra on $M_\text{mess}$ and $\tan{\beta}$, we show the Higgs mass curve for fixed $\Lambda$, with the color representing the mass of the lightest squark. We show these results in Fig.~\ref{figparameterspace} for the Case 1 democratic scenario, which show several excluded regions.
When $\epsilon_u=\epsilon_d=\epsilon_e=0$, the central big ``hole" appears because the mass-squared of the lightest slepton is negative.
When the $\epsilon$ parameters are nonzero, the size of the holes increases, and there are also small holes that appear above the central void because $A^0$ becomes tachyonic in those regions.
Quite generally, we see that the parameters that satisfy the Higgs mass constraint could be very different in these two cases.
The lightest value for the mass of the $\tilde{u}_3$ squark is in the region with high $\tan\beta$ and high messenger scales.
In Fig.~\ref{figparameterlambdammess}, we show the gluino and lightest slepton (NLSP) masses, both without perturbations (left panel) and with perturbations (right panel). The introduction of the perturbations pushes the slepton mass down to smaller values. The change in the shape of the viable Higgs mass region is even more apparent here. For low values of $M_{\rm mess}$, a higher value of $\Lambda$ is needed to satisfy the light Higgs mass constraint. For higher messenger scales $M_\text{mess}\sim 10^{14}-10^{16}$ GeV, there is a sharp drop in the Higgs mass region that is observed. In that region, there is generally larger left-right mixing in the sbottom sector as well larger scharm-stop mixing, which result in nontrivial contributions to the Higgs mass. However, there are potential numerical instabilities related to the challenges of the Higgs mass calculation in this parameter region. A detailed resolution of these issues is beyond the scope of this paper, and is deferred to future study.
\begin{figure}[h]
\centering
\subfloat{{\includegraphics[scale=0.45]{images-caseb2/Julyb2mMessVstanBHiggsContourslambda6e5beta.pdf}}}
\subfloat{{\includegraphics[scale=0.45]{images-caseb2-beta2/Julyb2mMessVstanBHiggsContourslambda6.3e5-beta2larger.pdf}}}
\caption{The mass of the Higgs (black band) and the mass of the lightest squark $\tilde{u}_3$ (color shading) for (a) Case 2a with fixed $\Lambda=6\times10^5$ GeV,
$\beta_{3d}=0.03$, $\beta_{3u}=0.01$, $\beta_{3l}=0.08$, $\beta_{2u}=\beta_{2d}=\beta_{2l}=0$ (left),
and (b) Case 2b with fixed $\Lambda=6.3\times10^5$ GeV,
$\beta_{2d}=0.03$, $\beta_{2u}=0.01$, $\beta_{2l}=0.08$, $\beta_{3u}=\beta_{3d}=\beta_{3l}=0$ (right).}
\label{figb2par1}
\end{figure}
\begin{figure}[h!]
\centering
\subfloat{{\includegraphics[scale=0.35]{images-caseb2/Julyb2MessVsLambdaHiggsContourstanBeta10beta.pdf}}}
\subfloat{{\includegraphics[scale=0.35]{images-caseb2-beta2/Julyb2MessVsLambdau3HiggsContourstanBeta10.pdf}}}
\caption{The light Higgs mass (black band), gluino mass (color shading), and $\tilde{e}_1$ mass (dotted curves) with $\Lambda=6\times 10^5$ GeV, for (a) Case 2a $\beta_{3d}=0.03$, $\beta_{3u}=0.01$, $\beta_{3l}=0.08$, $\beta_{2u}=\beta_{2d}=\beta_{2l}=0$ (left), and (b) Case 2b with $\beta_{2d}=0.03$, $\beta_{2u}=0.01$, $\beta_{2l}=0.08$, $\beta_{3u}=\beta_{3d}=\beta_{3l}=0$ (right).}
\label{figb2par2}
\end{figure}
For the Case 2 models, we also see excluded regions in the parameter scan in Fig.~\ref{figb2par1}. Here, for both Case 2a and 2b, $\Lambda$ is chosen to maximize viable parameter regions, and the perturbations have a minimal effect on the size of the void. In Case 2a, the void appears due to tachyonic slepton masses, and the phenomenologically viable parameter region generally lies between $\tan{\beta}\approx 5-15$, $M_\text{mess}=10^6-10^{18}$ GeV, with a fixed choice of $\Lambda=6\times10^5$ GeV. For Case 2b, apart from the central hole where the lightest slepton becomes tachyonic, the region on the left of the spectrum, which is from $M_\text{mess}\sim10^6-10^9$ GeV and $\tan{\beta}\sim 5-50$, is ruled out because the desired electroweak minimum is not present. We also see that in both cases, the viable Higgs region does not generally intersect with the region where $\tilde{e}_1$ is lighter. In Fig~\ref{figb2par2}, we fix $\tan{\beta}=10$ and show both Case 2a and Case 2b with nonzero perturbations. Note that the effects of the perturbations only slightly shift the mass curve of the NLSP upward, and have almost no effect on the viable Higgs mass region and the gluino masses, in contrast to what we have seen in the democratic limit.
We close this section by commenting on further phenomenological aspects of this set of models. For all cases described here (both democratic and doublet-dominated models), the superpartner masses are generally heavy and split, in a way that is reminiscent of minimal gauge mediation with $N=2$. As previously discussed, the constraints of the non-Abelian Higgs-messenger symmetry have led us to include at least two messenger pairs to avoid a catastrophic $\mu/B_\mu$ problem. Ultimately, this means that the scenarios studied in this paper have heavier and more split spectra than what can be obtained in Abelian flavored gauge mediation models (where a judicious choice of $U(1)$ charges can be made to avoid the $\mu/B_\mu$ issue seen here, without increasing the number of messenger pairs), such as in \cite{Ierushalmi_2016}, or in general gauge mediation scenarios \cite{Meade:2008wd}. We recall that in our scenario in the singlet-dominated limit as studied in \cite{Everett_2019,Everett_2020}, it is also possible to minimize the splitting of the mass spectra, though not to the extent that is possible in the Abelian flavored gauge mediation models.
As a result, the discovery potential for the scenarios studied here either via direct LHC searches or indirect constraints is not as promising as it can be in Abelian flavored gauge mediation models, or even in the singlet-dominated non-Abelian scenario. For example, it is straightforward to see that the supersymmetric contribution to muon anomalous magnetic moment (MDM) in the democratic and doublet-dominated non-Abelian flavored gauge mediation scenarios studied here is generically about two orders smaller than the current experimental value \cite{PhysRevLett.126.141801}.
This is both due to the heavy superpartner masses as described above, and that we are generally precluded from having large values of $\tan\beta$ in these scenarios, which usually provide the largest enhancement to the MDM. Therefore, if new physics is required to resolve any future confirmed discrepancy between the SM prediction and the measured value of the muon anomalous magnetic moment, this set of flavored gauge mediation models would need to be extended to accommodate the experimental result.
One notable difference in the non-Abelian flavored gauge mediation scenarios studied here compared to minimal gauge mediation with $N=2$ as well as the non-Abelian singlet-dominated flavored gauge mediation scenario is in regards to the NLSP composition. Here, for messenger mass scales of $10^{12}$ GeV as displayed in Fig.~\ref{fig:democraticspectrum1} and Fig.~\ref{fig:b2spectra}, the NLSP is the lightest slepton, which is a right-handed smuon. This is different from the minimal GMSB scenario and the singlet-dominated non-Abelian FGM scenario in which either staus or binolike neutralinos are the NLSP. In the scenarios studied in this paper, for this intermediate to high messenger scale, the smuon NLSP has a lifetime of $\mathcal{O}(0.001\,{\rm s})$, and the NLSP mass is generically close to about 2 TeV. This currently lies above the limits from direct production searches at $\sqrt{s}=13$ TeV \cite{CMS:2018eqb}.
For lower values of the messenger scale ($\sim 10^6$ GeV), the lightest slepton is still the NLSP, which has a very rapid decay to the gravitino due to the lower supersymmetry breaking scale, while for very high messenger scales ($\sim 10^{14}$ GeV), the NLSP is now a long-lived binolike neutralino. We also note that in the non-Abelian flavored gauge mediation scenarios studied here, there is no significant co-NLSP behavior, both in the low messenger scale and the intermediate to high messenger scale cases. This is in contrast to minimal $N=2$ GMSB for low messenger scales ($\sim 10^6$ GeV), for which there is appreciable co-NLSP behavior among the lighter sleptons for the binolike neutralino NLSP.
We also note that in both minimal $N=2$ GMSB and our non-Abelian flavored gauge-mediation scenarios, for messenger scales of $M_\text{mess}=10^{12}$ GeV, the gravitino has a mass of $\mathcal{O}(0.1\, \text{GeV})$, and the NLSP is not long-lived enough to decay during or after Big Bang nucleosynthesis (BBN). Therefore, the successful predictions of BBN will not be spoiled (see e.g.~\cite{Steffen:2006hw,Feng:2004zu,ASAKA2000136}). For gravitinos of this mass range, there are well-known mechanisms to ensure the desired reheating temperatures and late entropy production to avoid having the gravitinos overclose the universe, so that gravitinos can then be a plausible dark matter candidate. For lower values of the messenger scale, the situation is further improved, as the gravitinos are lighter (with masses of the order of tenths of keV for $M_{\rm mess}= 10^6$ GeV) and the NLSP decays to gravitinos much more rapidly than in the higher messenger scale case, thus avoiding the need for gravitino dilution.
\section{Conclusions}
In this paper, we have explored MSSM flavored gauge mediation models in which the Higgs-messenger mixing is controlled by a discrete non-Abelian symmetry, here taken for simplicity to be $\mathcal{S}_3$. Building on previous analyses \cite{Everett_2018} which showed that viable models can be constructed for an extended Higgs-messenger sector that includes both $\mathcal{S}_3$ doublet and singlet fields that mix to yield one light MSSM Higgs pair and two messenger pairs, we studied various possibilities for generating plausible SM quark and charged lepton masses in the case in which the MSSM matter fields also carry $\mathcal{S}_3$ quantum numbers. While additional relations beyond $\mathcal{S}_3$ are generically needed to obtain the desired hierarchical SM fermion masses, we have identified two general categories of solutions that we broadly categorized as Case 1 and Case 2 models. The Case 1 models obey Eq.~(\ref{eq:betarelations}), and encompass two regimes of interest: (i) the singlet-dominated limit, in which the Yukawa couplings involving only the $\mathcal{S}_3$ singlets dominate, and (ii) the democratic limit, in which the Yukawa superpotential for the MSSM fields has an enhanced $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ symmetry. The Case 2 models, in contrast, include the doublet-dominated limit, in which the Yukawa couplings involving only the $\mathcal{S}_3$ doublet fields dominate. The singlet-dominated limit was previously investigated in \cite{Everett_2019,Everett_2020} and served here as a point of comparison for a general analysis of the Case 1 democratic limit and the Case 2 doublet-dominated models. We include corrections to obtain nonvanishing masses for one or both of the lighter families, as well as for the third family. In certain cases such corrections lead to off-diagonal corrections to the soft supersymmetry breaking mass terms, but these corrections are relatively mild (a feature that is known in the literature for flavored gauge mediation models of this general type) and as a result, do not immediately lead to insurmountable problems with flavor-changing neutral current constraints.
Within Case 1 models, our analysis shows that while the singlet-dominated limit allows for examples with optimized parameter sets that yield gluino and squark masses in the 4-5 TeV range, the Case 1 democratic limit generically has significantly heavier squark and gluino masses. The Case 2 models generally also yield heavier superpartner masses, with the heavier squarks and gluino in the 7 TeV mass range. Ultimately, the fact that the squark and gluino masses cannot be made lighter than 4-5 TeV even in the singlet-dominated limit is related to the fact that this non-Abelian Higgs-messenger mixing scenario requires at the minimum two vectorlike messenger pairs that contribute to the loop diagrams that generate the corrections to the soft terms, to tune the $\mu$ and $b$ terms independently. This should be contrasted with Abelian models, which can have just one messenger pair, and as a result can lead to benchmark scenarios in flavored gauge mediation with lighter $SU(3)$-charged superpartners that are more accessible for searches for supersymmetry at present and future colliders.
While the spectra in all our examples remain quite heavy, and while we have not constructed fully realistic models of the SM fermion masses and mixing angles (including CP violating effects, not included here for simplicity), we nonetheless find it encouraging that this class of non-Abelian flavored gauge mediation models can include examples that survive this next level of model-building scrutiny. More work is of course needed to see if such scenarios (or plausible extensions of such scenarios) can be embedded into a more complete high-energy model. In the meantime, however, analyses such as this one can serve as a reminder of the rich framework of TeV-scale $\mathcal{N}=1$ supersymmetry, and the many ways in which it might still be hiding at or just above TeV energies. As the Terascale continues to be explored in this data-rich era for high energy physics, hopefully we will know relatively soon if TeV-scale supersymmetry is indeed part of our physical world.
\acknowledgments
This work is supported by the U.S. Department of Energy under the contract number DE-SC0017647.
\clearpage
\section*{Appendix}
\subsection{Case 1 models}
We present the corrections to the soft supersymmetry breaking terms in the Case 1 democratic limit. All relevant terms with magnitudes larger than the smallest perturbation parameter $\sigma_u$ are included. In this case all terms with coefficients of order $O(10^{-3})$GeV are taken into account. For notational simplicity, we define the following quantities:
\begin{equation}
\begin{split}
\tilde{g}_u^2&=\frac{16}{3}g_3^2+3g_2^2+\frac{13}{15}g_1^2,\quad \tilde{g}_d^2=\frac{16}{3}g_3^2+3g_2^2+\frac{7}{15}g_1^2,\quad \tilde{g}_e^2=3g_2^2+\frac{9}{5}g_1^2,\\
\delta_Q&=6y_t^4+6y_b^4+2y_b^2y_t^2+y_b^2y_\tau^2-\Tilde{g_u}^2y_t^2-\Tilde{g_d}^2y_b^2,\\
\delta_{\epsilon_{u}}&=8y_t^4+y_b^2y_t^2-\tilde{g_u}^2y_t^2, \quad
\delta_{\epsilon_d}=8y_b^4+y_b^2y_t^2+y_b^2y_\tau^2-\tilde{g_d}^2y_b^2,\\
\delta_{\epsilon_e}&=\frac{4}{81} \epsilon _e^2y_b^2 y_{\tau }^2-\frac{8}{729}\epsilon _e^3
y_b^2 y_{\tau }^2-\frac{244 }{6561}\epsilon _e^4 y_b^2 y_{\tau }^2, \quad
\delta_{L}=4y_\tau^4+3y_b^2y_\tau^2-\Tilde{g_e}^2y_\tau^2. \\
\end{split}
\label{eq:paramdefcase1}
\end{equation}
In what follows, all soft scalar mass-squared parameters are assumed to include a factor of $\Lambda^2/(4\pi)^4$ and all trilinear scalar couplings are assumed to include a factor of $\Lambda/(4\pi)^2$, where $\Lambda=F/M_{\rm mess}$.
The nonvanishing corrections to the soft supersymmetry breaking terms are as follows:
\begin{equation}
\begin{split}
\delta m_{Q_{11}}^2&=
\delta_Q-\frac{4}{9}\epsilon_u \delta_{\epsilon_u}-\frac{4}{9}\epsilon_d \delta_{\epsilon_d} +\frac{4}{27} \epsilon _d \epsilon _e y_b^2 y_{\tau }^2-\frac{82}{729} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2\\&\quad+\epsilon _d^2 \left(-\frac{26}{81} y_b^2
\tilde{g}_d^2+\frac{5}{27} y_b^2 y_t^2+\frac{26}{81} y_b^2 y_{\tau }^2+\frac{367 }{81}y_b^4\right)+\delta_{\epsilon_e},
\\
\delta m_{Q_{22}}^2&=
(\delta_Q+\frac{4}{9}\epsilon_u \delta_{\epsilon_u}+\frac{4}{9}\epsilon_d \delta_{\epsilon_d} +\frac{4}{81}\epsilon _d \epsilon _e y_b^2 y_{\tau }^2-\frac{2}{243} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2\\&\quad+\epsilon _d^2 \left(\frac{2}{3} y_b^2
\tilde{g}_d^2-\frac{88}{81} y_b^2 y_t^2-\frac{2}{3} y_b^2 y_{\tau }^2-\frac{289 }{81}y_b^4\right)+\delta_{\epsilon_e},
\\
\delta m_{Q_{23}}^2&=\delta m_{Q_{32}}^2=
\frac{4\sqrt{2}}{9}\epsilon_u(\delta_{\epsilon_u}-2y_t^4)
+\frac{4\sqrt{2}}{9}\epsilon_d\left(\delta_{\epsilon_{d}}-2y_b^4\right)-\frac{4\sqrt{2}}{81} \epsilon _d \epsilon _ey_b^2 y_{\tau }^2+\frac{38\sqrt{2}}{729} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2\\&\qquad\qquad\quad+\sqrt{2}\epsilon _d^2\left(\frac{14}{81} y_b^2 \tilde{g}_d^2 -\frac{25}{162} y_b^2 y_t^2-\frac{14}{81} y_b^2 y_{\tau }^2-\frac{59}{81} y_b^4 \right)
\\
\delta m_{Q_{12}}^2&=\delta m_{Q_{21}}^2=
\frac{4\sqrt{3}}{3} \sigma _d \left(8 y_b^4-\tilde{g}_d^2y_b^2 +2 y_t^2y_b^2 +y_b^2 y_{\tau }^2\right)+ \frac{4\sqrt{3}}{3} \sigma _u \left(8 y_t^4-\tilde{g}_u^2y_b^2 +2 y_t^2y_b^2 \right),
\\
\delta m_{Q_{33}}^2&=
\frac{8}{81} \epsilon _d\epsilon _e y_b^2y_{\tau }^2-\frac{44}{729} \epsilon _d\epsilon _e^2y_b^2 y_{\tau }^2+\epsilon _d^2 \left(-\frac{4}{9} y_b^2
\tilde{g}_d^2+\frac{17}{81} y_b^2 y_t^2+\frac{4}{9} y_b^2 y_{\tau }^2+2 y_b^4\right),
\\
\delta m_{Q_{13}}^2&= \delta m_{Q_{31}}^2=
\sqrt{\frac{2}{3}}\sigma _d \left(12 y_b^4-2 \tilde{g}_d^2y_b^2+2 y_t^2y_b^2+2y_b^2 y_{\tau }^2\right)+4\sqrt{\frac{2}{3}}\sigma_u \left(y_t^4+y_t^2y_b^2-\tilde{g}_u^2y_t^2 \right),
\\
\nonumber
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\delta m_{\bar{u}_{11}}^2&=
(2\delta_{\epsilon_u}-4y_t^4)-\frac{8}{9}\epsilon_u\delta_{\epsilon_u}-\frac{8}{27}\epsilon _d^2 y_b^2 y_t^2,
\\
\delta m_{\bar{u}_{12}}^2&=\delta m_{\bar{u}_{21}}^2=\frac{8}{\sqrt{3}}\sigma_u \delta_{\epsilon_u},\qquad \delta m_{\bar{u}_{13}}^2=\delta m_{\bar{u}_{31}}^2=8\sqrt{\frac{2}{3}}\sigma_u (\delta_{\epsilon_u}-2y_t^4),
\\
\delta m_{\bar{u}_{22}}^2&=
(2\delta_{\epsilon_u}-4y_t^4)+\frac{8}{9}\epsilon_u\delta_{\epsilon_u}-\frac{8}{27}\epsilon _d^2 y_b^2 y_t^2,
\\
\delta m_{\bar{u}_{23}}^2&=\delta m_{\bar{u}_{32}}^2=
\frac{8\sqrt{2}\epsilon_u}{9}\left( \delta_{\epsilon_u}-2y_t^4\right),
\\
\delta m_{\bar{u}_{33}}^2&=
-\frac{8}{9}\epsilon_d^2 y_t^2 y_b^2,
\\
\delta m_{\bar{d}_{11}}^2&=
(2\delta_{\epsilon_d}-4y_b^4)-\frac{8\epsilon_d}{9}\delta_{\epsilon_d}+ \frac{8}{27}\epsilon _d\epsilon _e y_b^2 y_{\tau }^2-\frac{164}{729}\epsilon _d\epsilon _e^2 y_b^2 y_{\tau }^2 \\&\quad+\epsilon _d^2 \left(-\frac{52}{81} y_b^2
\tilde{g}_d^2+\frac{52}{81} y_b^2 y_t^2+\frac{52}{81} y_b^2 y_{\tau }^2+\frac{680 }{81}y_b^4\right)+2 \delta_{\epsilon_e},
\\
\delta m_{\bar{d}_{22}}^2&=
(2\delta_{\epsilon_d}-4y_b^4)+\frac{8\epsilon_d}{9}\delta_{\epsilon_d}+\frac{8}{81} \epsilon _d \epsilon _e y_b^2y_{\tau }^2-\frac{4}{243}\epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2\\&\quad+\epsilon _d^2 \left(\frac{3}{4} y_b^2
\tilde{g}_d^2-\frac{116}{81} y_b^2 y_t^2-\frac{4}{3} y_b^2 y_{\tau }^2-\frac{592 }{81}y_b^4\right)+2\delta_{\epsilon_e},
\\
\delta m_{\bar{d}_{23}}^2&=\delta m_{\bar{d}_{32}}^2=
\frac{8\sqrt{2}}{9}\epsilon_d\delta_{\epsilon_d}-\frac{8\sqrt{2} }{81}\epsilon _d\epsilon _e y_b^2 y_{\tau }^2+ \frac{76\sqrt{2}}{729}\epsilon _d\epsilon _e^2 y_b^2 y_{\tau }^2\\&\quad\qquad\qquad+\epsilon _d^2
\left(\frac{28\sqrt{2}}{81} y_b^2 \tilde{g}_d^2-\frac{28\sqrt{2}}{81} y_b^2 y_t^2-\frac{28 \sqrt{2}}{81} y_b^2 y_{\tau }^2-\frac{128 \sqrt{2}}{81} y_b^4\right),
\\
\delta m_{\bar{d}_{33}}^2&=
\frac{16}{81}\epsilon _d \epsilon _e y_b^2 y_{\tau }^2-\frac{88}{729} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2+\epsilon _d^2 \left(-\frac{9}{8} y_b^2
\tilde{g}_d^2+\frac{8}{9} y_b^2 y_t^2+\frac{8}{9} y_b^2 y_{\tau }^2+\frac{392 }{81}y_b^4\right),
\\
\delta m_{\bar{d}_{12}}^2&=\delta m_{\bar{d}_{12}}^2=
\sigma _d \left(-\frac{3}{8\sqrt{3}} y_b^2 \tilde{g}_d^2+\frac{8 y_b^2 y_t^2}{\sqrt{3}}+\frac{8 y_b^2 y_{\tau }^2}{\sqrt{3}}+\frac{64 y_b^4}{\sqrt{3}}\right),
\\
\delta m_{\bar{d}_{13}}^2&= \delta m_{\bar{d}_{31}}^2=
\sigma _d \left(-\frac{1}{4} \sqrt{\frac{3}{2}} y_b^2 \tilde{g}_d^2+4 \sqrt{\frac{2}{3}} y_b^2 y_t^2+4 \sqrt{\frac{2}{3}} y_b^2 y_{\tau }^2+8 \sqrt{6} y_b^4\right),
\\
\delta m_{L_{11}}^2&=
\delta_L-\frac{4\epsilon_e}{9}\left(\delta_L+2y_\tau^4\right)+ \frac{4}{9} \epsilon _d \epsilon _e y_b^2 y_{\tau }^2-\frac{22}{81} \epsilon _d\epsilon _e^2 y_b^2 y_{\tau }^2+\frac{4}{27}\epsilon _d^2 y_b^2 y_{\tau
}^2\\&\quad+\epsilon _e^2 \left(\frac{26}{27} y_b^2 y_{\tau
}^2-\frac{81}{26} \tilde{g}_e^2 y_{\tau }^2+\frac{283 }{81}y_{\tau }^4\right)+\epsilon _e^3
\left(-\frac{64}{243} y_b^2 y_{\tau }^2+\frac{729}{64} \tilde{g}_e^2 y_{\tau }^2-\frac{1778}{729} y_{\tau }^4\right)\\&\quad+\epsilon _e^4 \left(-\frac{238}{729} y_b^2 y_{\tau }^2+\frac{2187}{238} \tilde{g}_e^2 y_{\tau }^2+\frac{862 }{2187}y_{\tau }^4\right),
\\
\delta m_{L_{22}}^2&=
\delta_L+\frac{4\epsilon_e}{9}\left(\delta_L+2y_\tau^4\right)+\frac{4}{27} \epsilon _d \epsilon _e y_b^2 y_{\tau }^2+\frac{10}{243} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2+\frac{4}{27} \epsilon _d^2 y_b^2 y_{\tau
}^2\\&\quad+\epsilon _e^2 \left(-2 y_b^2 y_{\tau
}^2+\frac{3}{2} \tilde{g}_e^2 y_{\tau }^2-\frac{197}{81} y_{\tau }^4\right)+\epsilon _e^3
\left(\frac{424}{243} y_b^2 y_{\tau }^2-\frac{729}{424} \tilde{g}_e^2 y_{\tau }^2+\frac{370 }{243}y_{\tau }^4\right)\\&\quad+\epsilon _e^4 \left(-\frac{1642 }{2187}y_b^2 y_{\tau }^2+\frac{6561 }{1642}\tilde{g}_e^2 y_{\tau }^2-\frac{1832 }{6561}y_{\tau }^4\right),
\\
\delta m_{L_{23}}^2&=\delta m_{L_{32}}^2=
\frac{4\sqrt{2}}{9}\epsilon_e\delta_L -\frac{4}{27} \sqrt{2}\epsilon _d \epsilon _e y_b^2 y_{\tau }^2 +\frac{14}{243} \sqrt{2} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2 \\\quad\qquad\qquad& +\epsilon _e^2
\left(-\frac{14}{27} \sqrt{2} y_b^2 y_{\tau }^2+\frac{81 }{7 \sqrt{2}} \tilde{g}_e^2 y_{\tau }^2-\frac{23}{81} \sqrt{2} y_{\tau }^4\right)+\epsilon _e^3
\left(-\frac{4}{9} \sqrt{2} y_b^2 y_{\tau }^2+\frac{27 }{2 \sqrt{2}}\tilde{g}_e^2 y_{\tau }^2-\frac{530}{729} \sqrt{2} y_{\tau }^4\right)\\\quad\qquad\qquad&+\epsilon _e^4
\left(\frac{2402 \sqrt{2}}{2187} y_b^2 y_{\tau }^2-\frac{6561 }{1201 \sqrt{2}}\tilde{g}_e^2 y_{\tau }^2+\frac{7660 \sqrt{2} }{6561}y_{\tau }^4\right)
\\
\nonumber
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\delta m_{L_{33}}^2&=
\frac{8}{27} \epsilon _d\epsilon _e y_b^2 y_{\tau }^2-\frac{76}{243} \epsilon _d\epsilon _e^2 y_b^2 y_{\tau }^2 \\&\quad +\epsilon _e^2 \left(\frac{4}{3} y_b^2 y_{\tau }^2-\frac{9}{4} \tilde{g}_e^2 y_{\tau }^2+\frac{74 }{81}y_{\tau
}^4\right)+\epsilon _e^3 \left(-\frac{376}{243} y_b^2 y_{\tau }^2+\frac{729}{376}
\tilde{g}_e^2 y_{\tau }^2-\frac{244 }{243}y_{\tau }^4\right) \\&\quad+\epsilon _e^4 \left(\frac{1868 }{2187}y_b^2 y_{\tau
}^2-\frac{6561 }{1868}\tilde{g}_e^2 y_{\tau }^2+\frac{436 }{729}y_{\tau }^4\right),
\\
\delta m_{L_{12}}^2&=\delta m_{L_{21}}^2=
\sigma _e \left(4 \sqrt{3} y_b^2 y_{\tau }^2-\frac{3}{4} \sqrt{3} \tilde{g}_e^2 y_{\tau }^2+8 \sqrt{3} y_{\tau }^4\right),
\\
\delta m_{L_{13}}^2&=\delta m_{L_{31}}^2=
\sigma _e \left(2 \sqrt{6} y_b^2 y_{\tau }^2-3 \sqrt{\frac{3}{2}} \tilde{g}_e^2 y_{\tau }^2+8 \sqrt{\frac{2}{3}} y_{\tau }^4\right),
\\
\delta m_{\bar{e}_{11}}^2&=
2\delta_L-\frac{8\epsilon_e}{9}\left(\delta_L+2y_\tau^4\right)+\frac{8}{9} \epsilon _d \epsilon _e y_b^2 y_{\tau }^2-\frac{44}{81} \epsilon _d \epsilon _e^2y_b^2 y_{\tau }^2+\frac{8}{27} y_b^2 \epsilon _d^2 y_{\tau
}^2 \\&\quad +\epsilon _e^2 \left(\frac{52}{27} y_b^2
y_{\tau }^2-\frac{81}{52} \tilde{g}_e^2 y_{\tau }^2+\frac{512 }{81}y_{\tau }^4\right)+\epsilon _e^3
\left(-\frac{128}{243} y_b^2 y_{\tau }^2+\frac{729}{128} \tilde{g}_e^2 y_{\tau }^2-\frac{3016 }{729}y_{\tau }^4\right) \\&\quad +\epsilon _e^4 \left(-\frac{476}{729} y_b^2 y_{\tau }^2+\frac{2187}{476} \tilde{g}_e^2 y_{\tau }^2+\frac{914 }{2187}y_{\tau }^4\right),
\\
\delta m_{\bar{e}_{22}}^2&=
2\delta_L+\frac{8\epsilon_e}{9}\left(\delta_L+2y_\tau^4\right)+\frac{8}{27}\epsilon _d \epsilon _e y_b^2 y_{\tau }^2+\frac{20}{243}\epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2+\frac{8}{27} \epsilon _d^2 y_b^2 y_{\tau
}^2 \\&\quad +\epsilon _e^2 \left(-4 y_b^2 y_{\tau
}^2+\frac{3}{4} \tilde{g}_e^2 y_{\tau }^2-\frac{136 }{27}y_{\tau }^4\right)+\epsilon _e^3
\left(\frac{848}{243} y_b^2 y_{\tau }^2-\frac{729}{848} \tilde{g}_e^2 y_{\tau }^2+\frac{776}{243} y_{\tau }^4\right)\\&\quad+\epsilon _e^4 \left(-\frac{3284 }{2187}y_b^2 y_{\tau }^2+\frac{6561 }{3284}\tilde{g}_e^2 y_{\tau }^2-\frac{1138 }{2187}y_{\tau }^4\right),
\\
\delta m_{\bar{e}_{23}}^2&=\delta m_{\bar{e}_{32}}^2=
\frac{8\sqrt{2}}{9}\epsilon_e\delta_L -\frac{8}{27}\epsilon _d\epsilon _e \sqrt{2} y_b^2 y_{\tau }^2+ \frac{28}{243} \sqrt{2} \epsilon _d\epsilon _e^2y_b^2 y_{\tau }^2 \\&\quad+\epsilon _e^2
\left(-\frac{28}{27} \sqrt{2} y_b^2 y_{\tau }^2+\frac{81 }{14 \sqrt{2}}\tilde{g}_e^2 y_{\tau }^2-\frac{56}{81} \sqrt{2} y_{\tau }^4\right)+\epsilon _e^3
\left(-\frac{8}{9} \sqrt{2} y_b^2 y_{\tau }^2+\frac{27 }{4 \sqrt{2}}\tilde{g}_e^2 y_{\tau }^2-\frac{1096}{729} \sqrt{2} y_{\tau }^4\right) \\&\quad+\epsilon _e^4
\left(\frac{4804 \sqrt{2}}{2187} y_b^2 y_{\tau }^2-\frac{6561 }{2402 \sqrt{2}}\tilde{g}_e^2 y_{\tau }^2+\frac{16918 \sqrt{2} }{6561}y_{\tau }^4\right)
\\
\delta m_{\bar{e}_{33}}^2&=
\frac{16}{27}\epsilon _d\epsilon _e y_b^2 y_{\tau }^2-\frac{152}{243} \epsilon _d\epsilon _e^2 y_b^2 y_{\tau }^2 +\epsilon _e^2 \left(\frac{8}{3} y_b^2 y_{\tau }^2-\frac{9}{8} \tilde{g}_e^2 y_{\tau
}^2+\frac{8 }{3}y_{\tau }^4\right)\\&\quad +\epsilon _e^3 \left(-\frac{752}{243} y_b^2 y_{\tau
}^2+\frac{81}{28} \tilde{g}_e^2 y_{\tau }^2-\frac{704 }{243}y_{\tau }^4\right)+ \epsilon _e^4 \left(\frac{3736 }{2187}y_b^2
y_{\tau }^2-\frac{6561}{3736} \tilde{g}_e^2 y_{\tau }^2+\frac{10028 }{6561}y_{\tau }^4\right),
\\
\delta m_{\bar{e}_{12}}^2&=\delta m_{\bar{e}_{21}}^2=
\sigma _e \left(8 \sqrt{3} y_b^2 y_{\tau }^2-\frac{3}{8} \sqrt{3} \tilde{g}_e^2 y_{\tau }^2+16 \sqrt{3} y_{\tau }^4\right),
\\
\delta m_{\bar{e}_{13}}^2&=\delta m_{\bar{e}_{31}}^2=
\sigma _e \left(4 \sqrt{6} y_b^2 y_{\tau }^2-\frac{3}{2} \sqrt{\frac{3}{2}} \tilde{g}_e^2 y_{\tau }^2+16 \sqrt{\frac{2}{3}} y_{\tau }^4\right),
\\
\delta m_{H_u}^2&=
-\frac{4}{3}\epsilon _d^2 y_b^2 y_t^2,
\\
\delta m_{H_d}^2&=
\epsilon _d^2 \left(-\frac{4}{27} y_b^2 y_t^2-\frac{40 }{9}y_b^4\right)-\frac{40}{27} \epsilon _e^2 y_{\tau }^4+\frac{368}{243} \epsilon _e^3 y_{\tau
}^4-\frac{1376}{2187} \epsilon _e^4 y_{\tau }^4, \nonumber
\end{split}
\end{equation}
\begin{equation}
\begin{split}
(\Tilde{A_u})_{13}&=2\sqrt{\frac{2}{3}}\sigma_d y_t y_b^2+4\sqrt{\frac{2}{3}}\sigma_uy_t^3,\qquad\qquad\qquad\quad\;\; (\Tilde{A_u})_{31}=8\sqrt{\frac{2}{3}}\sigma_uy_t^3,\\
(\Tilde{A_u})_{22}&= -\frac{2\epsilon_u}{9}(3y_t^3+y_ty_b^2),\qquad\qquad\qquad\qquad\qquad\;\; (\Tilde{A_u})_{33}= \frac{4}{9}\epsilon_d^2 y_t y_b^2,\\
(\Tilde{A_u})_{23}&= \frac{4\sqrt{2}}{9}\epsilon_uy_t^3+\frac{4\sqrt{2}}{9}\epsilon_dy_ty_b^2-\frac{14\sqrt{2}}{81}\epsilon_d^2 y_ty_b^2,\qquad (\Tilde{A_u})_{32}=\frac{8\sqrt{2}\epsilon_u}{9}y_t^3,\\
(\Tilde{A_d})_{13}&=2\sqrt{\frac{2}{3}}\sigma_d y_b^3+4\sqrt{\frac{2}{3}}\sigma_uy_by_t^2,\qquad\qquad\qquad\quad\; (\Tilde{A_d})_{31}= 4\sqrt{\frac{2}{3}}\sigma_d y_b^3,\\
(\Tilde{A_d})_{22}&= -\frac{2\epsilon_d}{9}(3y_b^3+y_t^2y_b)-\frac{2\epsilon_d^2}{81}(9y_b^3-y_t^2y_b),\qquad (\Tilde{A_d})_{23}=\frac{4\sqrt{2}\epsilon_d}{9}y_b^3+\frac{4\sqrt{2}\epsilon_u}{9}y_t^2y_b-\frac{10\sqrt{2}}{27}\epsilon_d^2 y_b^3,\\
(\Tilde{A_d})_{32}&=\frac{8\sqrt{2}\epsilon_d}{9}y_b^3-\frac{4\sqrt{2}}{9}\epsilon_d^2 y_b^3,\qquad\qquad\qquad\qquad\;\;\; (\Tilde{A_d})_{33}=\frac{4}{3}\epsilon_d^2 y_b^3,\\
(\Tilde{A_e})_{13}&=2\sqrt{\frac{2}{3}}\sigma_e y_\tau^3,\qquad (\Tilde{A_e})_{31}= 4\sqrt{\frac{2}{3}}\sigma_e y_\tau^3,
\\
(\Tilde{A_e})_{22}&=-\frac{2}{3}\epsilon_ey_\tau^3-\frac{2}{9} \epsilon _e^2 y_{\tau }^3 +\frac{178}{243} \epsilon _e^3 y_{\tau }^3 -\frac{382}{729} \epsilon _e^4 y_{\tau }^3,
\\
(\Tilde{A_e})_{23}&=\frac{4\sqrt{2}}{9}\epsilon_ey_\tau^3-\frac{10}{27} \sqrt{2} \epsilon _e^2 y_{\tau }^3-\frac{4}{81} \sqrt{2} \epsilon _e^3 y_{\tau }^3+\frac{3274 \sqrt{2} }{6561}\epsilon _e^4 y_{\tau }^3,
\\
(\Tilde{A_e})_{32}&=\frac{8\sqrt{2}}{9}\epsilon_e y_\tau^3 -\frac{4}{9} \sqrt{2} \epsilon _e^2 y_{\tau }^3-\frac{20}{81} \sqrt{2} \epsilon _e^3 y_{\tau }^3+\frac{5240 \sqrt{2} }{6561}\epsilon _e^4 y_{\tau }^3,
\\
(\Tilde{A_e})_{33}&=\frac{4}{3} \epsilon _e^2 y_{\tau }^3-\frac{376}{243} \epsilon _e^3 y_{\tau }^3+\frac{1868 }{2187}\epsilon _e^4 y_{\tau }^3.
\\
\end{split}
\end{equation}
\subsection{Case 2 models}
\noindent $\bullet$ {\it Ordering $\beta_{3i}>\beta_{2i}$}.
As before, all soft scalar mass-squared parameters are assumed to include a factor of $\Lambda^2/(4\pi)^4$, all trilinear scalar couplings are assumed to include a factor of $\Lambda/(4\pi)^2$, and we define $g_{\tilde{u}}^2 = \tilde{g}_u^2/2$, $g_{\tilde{d}}^2 = \tilde{g}_d^2/2$, and $g_{\tilde{l}}^2 = \tilde{g}_e^2/2$ (see Eq.~(\ref{eq:paramdefcase1})).
Including all the relevant terms up to second order in $\beta_{3u}$, the nonvanishing corrections to the soft supersymmetry breaking terms take the form:
\begin{eqnarray}
(\delta m_{\tilde{Q}}^2)_{11}&=&\frac{195}{16}(y_t^4+y_b^4)+\frac{15}{4}y_t^2 y_b^2+\frac{27}{16}y_b^2 y_\tau^2 - 3y_t^2 g_{\tilde{u}}^2-3y_b^2 g_{\tilde{d}}^2+\frac{9}{16}\beta_{3d}^2y_b^2(6y_b^2+y_t^2)+\frac{9}{16}\beta_{3l}^2y_\tau^2y_b^2 \nonumber \\
&&+ \frac{9}{16}\beta_{3u}^2 y_t^2 (6y_t^2+y_b^2), \nonumber\\
(\delta m_{\tilde{Q}}^2)_{22}&=& \beta_{3d}^2\left( -2g_{\tilde{d}}^2 y_b^2 + \frac{21}{8}y_b^4+\frac{9}{16}y_b^2y_t^2+\frac{5}{8}y_b^2y_\tau^2\right)+ \beta_{3u}^2\left(-2g_{\tilde{u}}^2y_t^2 + \frac{21}{8}y_t^4 + \frac{9}{16}y_b^2y_t^2 \right)+ \frac{3}{8}\beta_{3u}\beta_{3d}y_b^2y_t^2, \nonumber \\
(\delta m_{\tilde{Q}}^2)_{33}&=&\frac{39}{16}(y_t^4+y_b^4)+\frac{5}{4}y_t^2 y_b^2+\frac{11}{16}y_b^2 y_\tau^2 - y_t^2 g_{\tilde{u}}^2-y_b^2 g_{\tilde{d}}^2 -\frac{3}{4}\beta_{3d}^2y_b^4 + \frac{1}{16}\beta_{3l}^2y_b^2y_\tau^2-\frac{3}{4}\beta_{3u}^2y_t^4+\frac{3}{8}\beta_{3u}\beta_{3d}y_b^2y_t^2, \nonumber \\
(\delta m_{\tilde{Q}}^2)_{12}&=&(\delta m_{\tilde{Q}}^2)_{21}=-\frac{3}{4\sqrt{2}}(y_t^4\beta_{3u}+y_b^4 \beta_{3d}-y_t^2 y_b^2 (\beta_{3u}+\beta_{3d})), \nonumber \\
\end{eqnarray}
\begin{eqnarray}
(\delta m_{\tilde{u}}^2)_{22}&=&\frac{189}{8}y_t^4+\frac{9}{2}y_t^2 y_b^2 - 6y_t^2 g_{\tilde{u}}^2+\beta_{3u}^2\left( -g_{\tilde{u}}^2 y_t^2 + \frac{27}{4}y_t^4\right),
\nonumber \\
(\delta m_{\tilde{u}}^2)_{33}&=&
\frac{45}{8}y_t^4+\frac{1}{2}y_t^2 y_b^2- 2y_t^2 g_{\tilde{u}}^2+\beta_{3u}^2\left( -3g_{\tilde{u}}^2 y_t^2 + \frac{27}{4}y_t^4\right), \nonumber \\
(\delta m_{\tilde{d}}^2)_{22}&=&\frac{189}{8}y_b^4+\frac{9}{2}y_t^2 y_b^2+\frac{27}{8}y_b^2y_\tau^2 - 6y_b^2 g_{\tilde{d}}^2 +\beta_{3d}^2y_b^2 \left(-g_{\tilde{d}}^2+\frac{27}{4}y_b^2+\frac{1}{8}y_\tau^2\right)+\frac{9}{8}\beta_{3l}^2y_b^2y_\tau^2,\nonumber \\
(\delta m_{\tilde{d}}^2)_{33} &=&
\frac{45}{8}y_b^4+\frac{1}{2}y_t^2 y_b^2+\frac{11}{8}y_b^2y_\tau^2- 2y_b^2 g_{\tilde{d}}^2+\beta_{3d}^2y_b^2 \left(-\frac{1}{3}g_{\tilde{d}}^2+\frac{27}{4}y_b^2+\frac{9}{8}y_\tau^2\right)+\frac{1}{8}\beta_{3l}^2y_b^2y_\tau^2, \nonumber \\
(\delta m_{\tilde{L}}^2)_{11}&=&\frac{141}{16}y_\tau^4+\frac{81}{16}y_b^2 y_\tau^2 - 3y_\tau^2 g_{\tilde{l}}^2 +\frac{27}{16}\beta_{3d}^2y_b^2y_\tau^2 + \frac{9}{4}\beta_{3l}^2y_\tau^4,
\nonumber \\
(\delta m_{\tilde{L}}^2)_{22}&=& \beta_{3l}^2 \left(-2g_{\tilde{l}}^2y_\tau^2 +\frac{15}{8} y_b^2 y_\tau^2 +\frac{11}{8}y_\tau^4 \right), \nonumber\\
(\delta m_{\tilde{L}}^2)_{33}&=&
\frac{17}{16}y_\tau^4+\frac{33}{16}y_b^2 y_\tau^2 - y_\tau^2 g_{\tilde{l}}^2+\frac{3}{16}\beta_{3d}^2y_b^2y_\tau^2 - \frac{7}{8}\beta_{3l}^2y_\tau^4, \nonumber \\
(\delta m_{\tilde{L}}^2)_{12} &=&
(\delta m_{\tilde{L}}^2)_{21}=-\frac{3}{4\sqrt{2}}(y_\tau^4\beta_{3l})+\frac{3}{8\sqrt{2}}(\beta_{3l}^3 y_\tau^4), \nonumber \\
(\delta m_{\tilde{e}}^2)_{22}&=&\frac{135}{8}y_\tau^4+\frac{81}{8}y_b^2y_\tau^2 - 6y_\tau^2 g_{\tilde{l}}^2+\frac{27}{8}\beta_{3d}^2y_b^2y_\tau^2+\beta_{3l}^2\left(-g_{\tilde{l}}^2y_\tau^2 +\frac{3}{8}y_b^2y_\tau^2 +\frac{17}{4} y_\tau^4 \right), \qquad
\nonumber \\
(\delta m_{\tilde{e}}^2)_{33}
&=&
\frac{23}{8}y_\tau^4+\frac{33}{8}y_b^2 y_\tau^2- 2y_\tau^2 g_{\tilde{l}}^2+\frac{3}{8}\beta_{3d}^2y_b^2y_\tau^2+\beta_{3l}^2\left(-3g_{\tilde{l}}^2y_\tau^2 +\frac{27}{8}y_b^2y_\tau^2 +\frac{17}{4} y_\tau^4 \right), \nonumber\\
\delta m_{H_u}^2&=&-\frac{9}{2}y_t^4-\frac{3}{2}y_t^2y_b^2-9\beta_{3u}^2y_t^4,\nonumber \\
\delta m_{H_d}^2&=&-\frac{9}{2}y_b^4-\frac{3}{2}y_\tau^4-\frac{3}{2}y_t^2y_b^2-9\beta_{3d}^2y_b^4-3\beta_{3l}^2y_\tau^4, \nonumber\\
(\tilde{A}_u)_{22}&=&-\frac{3}{\sqrt{2}}y_t^3\beta_{3u}, \qquad\qquad\qquad\qquad (\tilde{A}_u)_{33}=-\frac{3}{2}y_t^3-\frac{1}{2}y_t y_b^2-\frac{3}{2}\beta_{3u}^2y_t^3 , \nonumber \\
(\tilde{A}_d)_{22}&=&-\frac{3}{\sqrt{2}}y_b^3\beta_{3d}, \qquad\qquad\qquad\qquad (\tilde{A}_d)_{33}=-\frac{3}{2}y_b^3-\frac{1}{2}y_b y_t^2-\frac{3}{2}\beta_{3d}^2y_b^3, \nonumber \\
(\tilde{A}_e)_{22}&=&-\frac{3}{\sqrt{2}}y_\tau^3\beta_{3l}-\frac{3}{2\sqrt{2}}y_\tau^3\beta_{3l}^3, \qquad\;\; (\tilde{A}_e)_{33}=-\frac{3}{2}y_\tau^3-\frac{3}{2}y_\tau^3\beta_{3l}^2.
\label{eq:deltamB2-1}
\end{eqnarray}
\clearpage
\noindent $\bullet$ {\it Ordering $\beta_{2i}>\beta_{3i}$}.
Assuming the subleading $\beta_{3i}=0$, we find the following corrections to the soft terms up to second order in $\beta_{2i}$:
\begin{eqnarray}
(\delta m_{\tilde{Q}}^2)_{22}&=&\frac{195}{16}(y_t^4+y_b^4)+\frac{15}{4}y_t^2 y_b^2+\frac{27}{16}y_b^2 y_\tau^2 - 3y_t^2 g_{\tilde{u}}^2-3y_b^2 g_{\tilde{d}}^2 + \beta_{2d}^2 y_b^2 \nonumber
\left(-\frac{1}{2}g_{\tilde{d}}^2+\frac{27}{8}y_b^2+\frac{3}{16}y_t^2+\frac{1}{16}y_\tau^2 \right)
\\&&+\frac{9}{16}\beta_{2l}^2y_b^2y_\tau^2 + \beta_{2u}^2y_t^2 \left( -\frac{1}{2}g_{\tilde{u}}^2+\frac{27}{8}y_t^2+\frac{3}{16}y_b^2\right)-\frac{3}{8}\beta_{2u}\beta_{2d}y_b^2y_t^2, \nonumber \\
(\delta m_{\tilde{Q}}^2)_{33}&=&\frac{39}{16}(y_t^4+y_b^4)+\frac{5}{4}y_t^2 y_b^2+\frac{11}{16}y_b^2 y_\tau^2 - y_t^2 g_{\tilde{u}}^2-y_b^2 g_{\tilde{d}}^2+ \beta_{2d}^2 y_b^2 \left(-\frac{3}{2}g_{\tilde{d}}^2+\frac{27}{8}y_b^2+\frac{3}{16}y_t^2+\frac{9}{16}y_\tau^2 \right) \nonumber \\
&& +\beta_{2u}^2 y_t^2 \left(-\frac{3}{2}g_{\tilde{u}}^2+\frac{27}{8}y_t^2+\frac{3}{16}y_b^2 \right)-\frac{3}{8}\beta_{2u}\beta_{2d}y_b^2y_t^2, \nonumber\\
(\delta m_{\tilde{u}}^2)_{11}&=&\frac{189}{8}y_t^4+\frac{9}{2}y_t^2 y_b^2 - 6y_t^2 g_{\tilde{u}}^2+\frac{9}{4}\beta_{2d}^2y_b^2y_t^2+\frac{45}{8}\beta_{2u}^2 y_t^4,\nonumber \\
(\delta m_{\tilde{u}}^2)_{22}&=&\beta_{2u}^2 \left (-4g_{\tilde{u}}^2y_t^2+\frac{3}{2}y_b^2y_t^2 + \frac{21}{4}y_t^4 \right),\nonumber \\
(\delta m_{\tilde{u}}^2)_{33}
&=&
\frac{45}{8}y_t^4+\frac{1}{2}y_t^2 y_b^2- 2y_t^2 g_{\tilde{u}}^2 -\frac{3}{4}\beta_{2d}^2y_b^2y_t^2-\frac{3}{8}\beta_{2u}^2y_t^4, \nonumber \\
(\delta m_{\tilde{d}}^2)_{11}&=&\frac{189}{8}y_b^4+\frac{9}{2}y_t^2 y_b^2+\frac{27}{8}y_b^2y_\tau^2 - 6y_b^2 g_{\tilde{d}}^2+\frac{45}{8}\beta_{2d}^2y_b^4+\frac{9}{8}\beta_{2l}^2y_b^2y_\tau^2 + \frac{9}{4}\beta_{2u}^2y_b^2y_t^2,\nonumber \\
(\delta m_{\tilde{d}}^2)_{22} &=& \beta_{2d}^2y_b^2 \left(-\frac{1}{4}g_{\tilde{d}}^2+\frac{21}{4}y_b^2+\frac{3}{2}y_t^2+\frac{5}{4}y_{\tau}^2 \right), \nonumber
\\
(\delta m_{\tilde{d}}^2)_{33} &=&
\frac{45}{8}y_b^4+\frac{1}{2}y_t^2 y_b^2+\frac{11}{8}y_b^2y_\tau^2- 2y_b^2 g_{\tilde{d}}^2-\frac{3}{8}\beta_{2d}^2y_b^4+\frac{1}{8}\beta_{2l}^2y_b^2y_\tau^2- \frac{3}{4}\beta_{2u}^2y_b^2y_t^2, \nonumber \\
(\delta m_{\tilde{L}}^2)_{22}&=&\frac{141}{16}y_\tau^4+\frac{81}{16}y_b^2 y_\tau^2 - 3y_\tau^2 g_{\tilde{l}}^2 +\frac{27}{16}\beta_{2d}^2y_b^2y_\tau^2 + \beta_{2l}^2\left(-\frac{1}{2}g_{\tilde{l}}^2y_\tau^2 + \frac{3}{16}y_b^2y_\tau^2+\frac{17}{8}y_\tau^4\right),
\nonumber \\
(\delta m_{\tilde{L}}^2)_{33}&=&
\frac{17}{16}y_\tau^4+\frac{33}{16}y_b^2 y_\tau^2 - y_\tau^2 g_{\tilde{l}}^2+\frac{3}{16}\beta_{2d}^2y_b^2y_\tau^2+ \beta_{2l}^2\left(-\frac{3}{2}g_{\tilde{l}}^2y_\tau^2 + \frac{27}{16}y_b^2y_\tau^2+\frac{17}{8}y_\tau^4\right), \nonumber \\
(\delta m_{\tilde{e}}^2)_{11}&=&\frac{135}{8}y_\tau^4+\frac{81}{8}y_b^2y_\tau^2 - 6y_\tau^2 g_{\tilde{l}}^2+\frac{27}{8}\beta_{2d}^2y_b^2y_\tau^2+\frac{27}{8}\beta_{2l}^2y_\tau^4,
\nonumber \\
(\delta m_{\tilde{e}}^2)_{22}&=& \beta_{2l}^2 \left(-4g_{\tilde{l}}^2 y_\tau^2 +\frac{15}{4}y_b^2y_\tau^2 +\frac{11}{4}y_\tau^4 \right), \nonumber \\
(\delta m_{\tilde{e}}^2)_{33}&=&
\frac{23}{8}y_\tau^4+\frac{33}{8}y_b^2 y_\tau^2- 2y_\tau^2 g_{\tilde{l}}^2+\frac{3}{8}\beta_{2d}^2y_b^2y_\tau^2-\frac{5}{8}\beta_{2l}^2y_\tau^4, \nonumber \\
\delta m_{H_u}^2&=&-\frac{9}{2}y_t^4-\frac{3}{2}y_t^2y_b^2-\frac{9}{4}\beta_{2d}^2y_b^2y_t^2-\frac{9}{4}\beta_{2u}^2y_t^2\left( 2y_t^2+y_b^2\right),\nonumber\\
\delta m_{H_d}^2&=&-\frac{9}{2}y_b^4-\frac{3}{2}y_\tau^4-\frac{3}{2}y_t^2y_b^2-\frac{9}{4}\beta_{2d}^2y_b^2(2y_b^2+y_t^2)-\frac{9}{4}\beta_{2u}^2y_b^2y_t^2, \nonumber\\
(\tilde{A}_u)_{22}&=&-\frac{3}{2\sqrt{2}}\beta_{2u}(y_t^3+y_ty_b^2), \qquad (\tilde{A}_u)_{33}=-\frac{3}{2}y_t^3-\frac{1}{2}y_t y_b^2-\frac{3}{4}\beta_{2d}^2y_b^2y_t-\frac{3}{4}\beta_{2u}^2y_t^3, \nonumber \\
(\tilde{A}_d)_{22}&=&-\frac{3}{2\sqrt{2}}\beta_{2d}(y_b^3+y_by_t^2), \qquad (\tilde{A}_d)_{33}=-\frac{3}{2}y_b^3-\frac{1}{2}y_b y_t^2-\frac{3}{4}\beta_{2d}^2y_b^3-\frac{3}{4}\beta_{2u}^2y_t^2y_b, \nonumber \\
(\tilde{A}_e)_{22}&=&-\frac{3}{2\sqrt{2}}\beta_{2l}y_\tau^3-\frac{9}{4\sqrt{2}}\beta_{2l}^3y_\tau^3, \quad (\tilde{A}_e)_{33}=-\frac{3}{2}y_\tau^3-\frac{3}{4}\beta_{2l}^2y_\tau^3.
\label{eq:deltamB2-3}
\end{eqnarray}
\clearpage
\section{Introduction}
In the LHC era, the search for physics beyond the Standard Model (SM) has proven elusive, and standard frameworks for TeV-scale new physics are highly constrained. For the well-studied case of extensions of the Standard Model to include softly broken $\mathcal{N}=1$ supersymmetry, such as the minimal supersymmetric standard model (MSSM), the LHC bounds indicate that if softly broken supersymmetry does indeed play a role in any new physics at the next rung of the energy ladder, its implementation is necessarily more complicated and ostensibly fine-tuned than originally anticipated. In this context, given the vast nature of the parameter space associated with the soft supersymmetry breaking sector, frameworks such as the MSSM can remain viable. However, patterns of possibly viable MSSM parameter regions would then be indicated, perhaps pointing to a specific organizing principle at higher energies.
One such example is within the context of gauge-mediated supersymmetry breaking (GMSB). In its minimal implementation, its distinctive phenomenology is characterized by a superpartner mass spectrum with a sizable splitting between the $SU(3)_c$-charged superpartners (squarks and gluinos) and the superpartners charged only under the electroweak symmetry (sleptons and electroweakinos), with the splitting governed by the messenger mass scale and the number of messenger pairs (taken to be $\mathbf{5}$, $\overline{\bf 5}$ with respect to $SU(5)$). However, the minimal implementation does not easily allow for a 125 GeV Higgs mass, requiring very high messenger scales and subsequent squark and gluino masses that are far out of reach of the LHC \cite{Dine:1981za,ALVAREZGAUME198296,DIMOPOULOS1981353,DIMOPOULOS1983479,Dine:1981za,DINE1982227,Nappi:1982hm,Dine_1993,Dine_1995,Dine_1996,Dine_1997,Giudice_1999,Draper_2012,Arbey_2012,Adeel_Ajaib_2012,Fischler_2014,Calibbi_2014}. As such, nonminimal implementations of gauge mediation, such as general gauge mediation \cite{Meade:2008wd}, or scenarios in which the MSSM fields and the messenger fields interact directly via renormalizable superpotential couplings, have now long been explored \cite{Dine_1997,Giudice_1999,Chacko_2002,Shadmi_2012,Evans_2011,evans2011probing,Evans_2012,Kang_2012,Craig_2013,Albaid_2013,Abdullah_2013,P_rez_2013,Byakti_2013,Evans_2013,Calibbi_2013,evans2015chiral,Galon_2013,Fischler_2014,Calibbi_2014,Joaquim:2006mn,Joaquim:2006uz}.
Of the many intriguing options for direct couplings between the messenger and matter sectors, the {\it flavored gauge mediation} framework, which exploits the fact that the electroweak Higgs fields can mix with the doublet components of the messenger pairs, has been of particular interest in the literature \cite{Abdullah_2013,Shadmi_2012,Ierushalmi_2016,P_rez_2013,Byakti_2013,Evans_2013,Calibbi_2013,evans2015chiral,Galon_2013,Jeli_ski_2015,Everett_2018,Everett_2019,Ahmed_2017}. In flavored gauge mediation (FGM) models, Higgs-messenger mixing leads to the generation of messenger Yukawa couplings, which affect the prediction of the soft supersymmetry breaking mass parameters at the input (messenger mass) scale. The messenger Yukawa contributions not only affect the superpartner mass spectrum, but also can generically can lead to the nontrivial possibility of flavor mixing in the soft terms. In viable FGM scenarios, therefore, the messenger Yukawa couplings are controlled by additional symmetries, and their forms are also intimately connected to the generation of the MSSM Yukawa couplings of the quarks and leptons. The case of $U(1)$ symmetries, as explored extensively for example in \cite{Calibbi_2014}, allows for great flexibility in constructing viable models with one or more vectorlike pairs of messengers. In addition, it was shown in \cite{Ierushalmi_2016} that flavor-mixing contributions to the soft terms in such scenarios are much smaller than naive expectations might suggest, and can be consistent with stringent bounds from flavor-changing processes, depending on the model in question.
Instead of using Abelian symmetries to control the messenger Yukawa couplings, an alternative is to build models based on discrete non-Abelian symmetries. Such symmetries have been extensively used as governing principles for the generation of viable SM fermion masses and mixing parameters \cite{Xing_1997,Fritzsch_1994,Fritzsch_2000}. In flavored gauge mediation, this possibility was first explored in detail in \cite{P_rez_2013}, where the authors constructed a two-family scenario based on the discrete non-Abelian symmetry $\mathcal{S}_3$, with the Higgs and messenger fields connected within $\mathcal{S}_3$ doublets. This idea was then extended to incorporate three families \cite{Everett_2018,Everett_2019,Everett_2020}. Most notably, it was realized in \cite{Everett_2018} that to avoid a severe $\mu/B_\mu$ problem, the Higgs-messenger sector should be extended to include $\mathcal{S}_3$ singlet representations as well as doublet representations. This leads to scenarios with a minimal number $N=2$ of messenger pairs (in contrast to the $U(1)$ cases, which allow for one messenger pair), which enhances the splitting of the squark and gluino masses compared to the slepton and electroweakino masses. Further embedding of the MSSM fields in $\mathcal{S}_3$ representations allows for the possibility that $\mathcal{S}_3$ can play a role as part of the family symmetry that governs the SM fermion masses and mixings. A specific implementation of this idea was explored in \cite{Everett_2019}, as well as in \cite{Everett_2020}, in which the Higgs-messenger singlets play a dominant role in generating the third family SM fermion masses.
The purpose of this paper is to provide a comprehensive analysis of the FGM $\mathcal{S}_3$ scenario, summarizing and extending our previous work. The aim is to explore other viable corners of parameter space of these theories and the subsequent effects of including nonleading corrections to the fermion masses. We identify several viable parameter regions, describe their phenomenological consequences, and compare them to the $U(1)$ FGM benchmark scenarios in the literature. We will see that quite generally, it is not easy to generate viable fermion masses while maintaining flavor-diagonal soft terms, and we will characterize the extent to which such flavor nondiagonal terms are constrained in these theories. The examples studied here all feature very heavy squarks and gluinos, very heavy Higgs fields, and lighter sleptons, charginos, and neutralinos. As such, they provide working examples of currently allowed MSSM parameter space that will continue to be constrained at the LHC and future colliders.
This paper is structured as follows. We begin with a brief overview of the flavored gauge mediation framework studied here, and describe various options for obtaining hierarchical quark and charged lepton masses. Next, we describe several concrete models, and analyze their mass spectra in detail. Finally, we present our summary and conclusions.
\section{Theoretical Background}
As described in \cite{Everett_2018}, the FGM $\mathcal{S}_3$ scenario studied here assumes a specific set of Higgs-messenger fields and supersymmetry-breaking fields. The quantum numbers of these fields with respect to $\mathcal{S}_3$ are given in Table~{\ref{tab:11}}.
\begin{table}[htbp]
\centering
\begin{tabular}{c|cccccc|cc}
& $\mathcal{H}_u^{(2)}$&$\mathcal{H}_u^{(1)}$ & $\mathcal{H}_d^{(2)}$& $\mathcal{H}_d^{(1)}$ & $T_{\bar{d}k}$ & $T_{dk}$ &$X_H$ & $X_T$\\
\hline
$\mathcal{S}_3$ &$\mathbf 2 $& $\mathbf 1$& $\mathbf 2 $& $\mathbf 1 $ & $\mathbf 1 $ & $\mathbf 1 $ &\textbf 2 & $\mathbf 1 $\\
\end{tabular}
\caption{The field content and $\mathcal{S}_3$ charges for the messenger and supersymmetry breaking sectors.}
\label{tab:11}
\end{table}
Here the $\mathcal{H}_{u,d}^{(2)}$ are Higgs-messenger $\mathcal{S}_3$ doublets, the $\mathcal{H}_{u,d}^{(1)}$ are Higgs-messenger $\mathcal{S}_3$ singlets, and $X_H$ is a supersymmetry breaking field that also breaks the $\mathcal{S}_3$ symmetry. The $T_{\bar{d}k,dk}$ denote the $SU(3)_c$ triplets which have the appropriate quantum numbers to complete approximate $\mathbf{5}$, $\overline{\mathbf{5}}$ multiplets with the messengers and $X_T$ is the supersymmetry breaking field that couples to these triplets \footnote{The triplet messengers and the $X_T$ field are chosen for simplicity to be singlets with respect to the $\mathcal{S}_3$ Higgs-messenger symmetry. Note that this choice is not consistent with a full embedding of this scenario into a grand unified theory. This would require more extended model-building that would also need to address the well-known doublet-triplet splitting issue in grand unified models).}. Focusing on the Higgs-messenger fields, we can write $\mathcal{H}_{u,d}^{(2)}$ and $\mathcal{H}_{u,d}^{(1)}$ as
\begin{eqnarray}
\mathcal{H}_{u}&\equiv& \left (\begin{array}{c} (\mathcal{H}^{(2)}_u)_1 \\ (\mathcal{H}^{(2)}_u)_2 \\ \mathcal{H}^{(1)}_u \end{array} \right ) \equiv \left (\begin{array}{c} \mathcal{H}^{(2)}_{u1}\\ \mathcal{H}^{(2)}_{u2}\\ \mathcal{H}^{(1)}_u\end{array} \right )= \mathcal{R}_u \left (\begin{array}{c} H_u\\M_{u1} \\ M_{u2} \end{array} \right ) \nonumber \\
\mathcal{H}_{d}&\equiv& \left (\begin{array}{c} (\mathcal{H}^{(2)}_d)_1 \\ (\mathcal{H}^{(2)}_d)_2 \\ \mathcal{H}^{(1)}_d \end{array} \right ) \equiv \left (\begin{array}{c} \mathcal{H}^{(2)}_{d1}\\ \mathcal{H}^{(2)}_{d2}\\ \mathcal{H}^{(1)}_d\end{array} \right )=\mathcal{R}_d \left (\begin{array}{c} H_d\\M_{d1} \\ M_{d2} \end{array} \right ),
\label{higgs_s3}
\end{eqnarray}
in which $H_{u,d}$ are the electroweak Higgs fields of the MSSM, $M_{u1,d1}$ and $M_{u2,d2}$ are gauge mediation messenger doublets, and $\mathcal{R}_{u,d}$ are unitary matrices whose form is governed by the couplings of the Higgs-messenger fields to $X_H$, which obtains both a scalar and $F$-component vacuum expectation value (VEV). As shown in \cite{Everett_2018}, consistency requirements and obtaining the needed mass hierarchy between the MSSM Higgs fields $H_{u,d}$ and the heavy messengers $M_{ui,di}$ require that $\mathcal{R}_{u,d}$ are given by
\begin{eqnarray}
\mathcal{R}_{u,d}= \left ( \begin{array}{ccc} \frac{1}{\sqrt{3}} & \mp \frac{1}{2} \left (1+\frac{1}{\sqrt{3}} \right) & \frac{1}{2} \left (1-\frac{1}{\sqrt{3}} \right) \\ \frac{1}{\sqrt{3}} & \pm \frac{1}{2} \left (1-\frac{1}{\sqrt{3}} \right) & -\frac{1}{2} \left (1+\frac{1}{\sqrt{3}} \right)
\\ \frac{1}{\sqrt{3}} & \pm \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array} \right ).
\label{rotationmatrices}
\end{eqnarray}
We turn now to the MSSM fields and their interactions with the Higgs-messenger fields. Although various possibilities exist, as discussed in \cite{Everett_2018},
we make the key assumption that the three generations of SM quarks and leptons are embedded into doublet and singlet representations of $\mathcal{S}_3$, as summarized in Table~\ref{tab:12}.
\begin{table}[htbp]
\centering
\begin{tabular}{c|cccc|cccccccccc|c}
& $\mathcal{H}_u^{(2)}$&$\mathcal{H}_u^{(1)}$ & $\mathcal{H}_d^{(2)}$& $\mathcal{H}_d^{(1)}$ & $Q_{\mathbf 2}$ &$Q_{\mathbf 1}$& $\bar u_{\mathbf 2}$ & $\bar u_{\mathbf 1 }$&$\bar d_{\mathbf 2}$& $\bar d_{\mathbf 1}$ & $L_{\mathbf{2}}$ & $L_{\mathbf{1}}$ & $\bar{e}_{\mathbf{2}}$ & $\bar{e}_{\mathbf{1}}$&$X_H$\\
\hline
$\mathcal{S}_3$ &$\mathbf 2 $& $\mathbf 1$& $\mathbf 2 $& $\mathbf 1 $ & $\mathbf 2 $&$\mathbf 1$ & $\mathbf 2 $&$\mathbf 1$ & $\mathbf 2 $&$\mathbf 1$ & $\mathbf 2 $&$\mathbf 1$ & $\mathbf 2 $&$\mathbf 1$ &\textbf 2\\
\end{tabular}
\caption{Charges for an $\mathcal{S}_3$ model of the Higgs-messenger fields and the MSSM matter fields. Here the $SU(3)$ triplet messengers and the associated $X_T$ field are not displayed for simplicity.
}
\label{tab:12}
\end{table}
With these $\mathcal{S}_3$ charge assignments, the superpotential couplings of the MSSM matter fields and the Higgs-messenger fields, for example for the up quarks, are given by
\begin{eqnarray}
W^{(u)}= \tilde{y}_u\big[Q_{\mathbf 2} \bar u_{\mathbf 2} \mathcal{H}^{(2)}_u+\beta_{1u}Q_{\mathbf 2} \bar u_{\mathbf 2} \mathcal{H}^{(1)}_u + \beta_{2u} Q_{\mathbf 2} \bar u_{\mathbf 1} \mathcal{H}^{(2)}_u +\beta_{3u} Q_{\mathbf 1} \bar u_{\mathbf 2} \mathcal{H}^{(2)}_u+ \beta_{4u} Q_{\mathbf 1} \bar u_{\mathbf 1} \mathcal{H}^{(1)}_u\big].
\label{wu}
\end{eqnarray}
In Eq.~(\ref{wu}), $\tilde{y}_u$ is a dimensionless overall factor, and the quantities $\beta_{1u}$, $\beta_{2u}$, $\beta_{3u}$, and $\beta_{4u}$ are dimensionless quantities that characterize the different couplings as allowed by $\mathcal{S}_3$. (Analogous forms hold for the down quarks and the charged leptons; we will ignore the effects of neutrino masses.) In the basis given by
\begin{eqnarray}
Q= (Q_\mathbf{2}, Q_\mathbf{1})^T = ((Q_{\mathbf 2})_1, (Q_{\mathbf 2})_2 ,Q_{\mathbf 1})^T, \qquad \overline{u}= (\overline{u}_{\mathbf{2}}, \overline{u}_\mathbf{1})^T= ((\overline{u}_{\mathbf 2})_1, (\overline{u}_\mathbf{2})_2 ,\overline{u}_{\mathbf 1})^T,
\end{eqnarray}
the superpotential couplings of Eq.~(\ref{wu}) can be expressed in matrix form as
\begin{eqnarray}
W^{(u)}=\tilde{y}_uQ^T\left( \begin{matrix} \mathcal{H}^{(2)}_{u1}&\beta_{1u}\mathcal{H}^{(1)}_{u}&\beta_{2u} \mathcal{H}^{(2)}_{u2}\\ \beta_{1u} \mathcal{H}^{(1)}_u& \mathcal{H}^{(2)}_{u2}& \beta_{2u}\mathcal{H}^{(2)}_{u1}\\ \beta_{3u}\mathcal{H}^{(2)}_{u2}& \beta_{3u}\mathcal{H}^{(2)}_{u1}&\beta_{4u} \mathcal{H}^{(1)}_u\end{matrix}\right)\bar u. \label{UpYukawas}
\end{eqnarray}
From here, we can easily identify the MSSM Yukawa coupling $Y_u$ and the messenger Yukawa couplings $Y_{u1}'$, $Y_{u2}'$, as
\begin{eqnarray}
Y_u= \frac{\tilde{y}_{u}}{\sqrt{3}} \left (\begin{array}{ccc} 1 & \beta_{1u} & \beta_{2u} \\ \beta_{1u} & 1 & \beta_{2u} \\ \beta_{3u} & \beta_{3u} & \beta_{4u} \end{array} \right ),
\label{eq:yud}
\end{eqnarray}
and
\begin{equation}
Y^\prime_{u1}=\tilde{y}_{u} \left (\begin{array}{ccc} -\frac{1}{2}-\frac{1}{2\sqrt{3}} & \frac{\beta_{1u}}{\sqrt{3}} & \;\; \frac{\beta_{2u}}{2} - \frac{\beta_{2u}}{2\sqrt{3}} \\ \frac{\beta_{1u}}{\sqrt{3}} & \;\; \frac{1}{2}-\frac{1}{2\sqrt{3}} & -\frac{\beta_{2u}}{2} - \frac{\beta_{2u}}{2\sqrt{3}} \\ \;\; \frac{\beta_{3u}}{2} - \frac{\beta_{3u}}{2\sqrt{3}} & -\frac{\beta_{3u}}{2} - \frac{\beta_{3u}}{2\sqrt{3}} & \frac{\beta_{4u}
}{\sqrt{3}}
\end{array} \right )
\end{equation}
\begin{equation}
\;\; Y^\prime_{u2}=\tilde{y}_{u} \left (\begin{array}{ccc} \;\; \frac{1}{2}-\frac{1}{2\sqrt{3}} & \frac{\beta_{1u}}{\sqrt{3}} & -\frac{\beta_{2u}}{2} - \frac{\beta_{2u}}{2\sqrt{3}} \\ \frac{\beta_{1u}}{\sqrt{3}} & -\frac{1}{2}-\frac{1}{2\sqrt{3}} & \;\; \frac{\beta_{2u}}{2} - \frac{\beta_{2u}}{2\sqrt{3}} \\ -\frac{\beta_{3u}}{2} - \frac{\beta_{3u}}{2\sqrt{3}} & \;\; \frac{\beta_{3u}}{2} - \frac{\beta_{3u}}{2\sqrt{3}} & \frac{\beta_{4u}
}{\sqrt{3}}
\end{array} \right ).
\end{equation}
These results are for the up sector; again, analogous relations hold for the down quarks and charged leptons, with the replacements $u\rightarrow d,e$ in all parameters, respectively.
For arbitrary values of the coefficients, Eq.~(\ref{eq:yud}) does not result in hierarchical fermion masses. It is only at special values of the couplings, corresponding to various enhanced symmetry points, that we can obtain a realistic quark mass hierarchy at leading order. To see this, we note that we can diagonalize this system explicitly and examine parameter sets where viable eigenvalue hierarchies can be obtained. For example, in the up quark sector, we can follow standard procedures and consider the Hermitian combinations $Y_u Y_u^\dagger$ and $Y_u^\dagger Y_u$. It is straightforward to calculate following exact results for their eigenvalues (denoted by $\lambda_{1u,2u,3u})$:
\begin{eqnarray}
\lambda_{1u} = \frac{\tilde{y}_u^2}{3}(1-\beta_{1u})^2, \qquad \lambda_{2u,3u} = \frac{\tilde{y}_u^2}{6} \left ((1+\beta_{1u})^2+2(\beta_{2u}^2+\beta_{3u}^2)+ \beta_{4u}^2 \mp \sqrt{\Lambda_u} \right ),
\label{eq:eigenvaluessq}
\end{eqnarray}
in which $\Lambda_u$ is given by
\begin{eqnarray}
\Lambda_u &=& (1+\beta_{1u})^4+4(\beta_{2u}^4+\beta_{3u}^4)+\beta_{4u}^4+4((1+\beta_{1u})^2+\beta_{4u}^2)(\beta_{2u}^2+\beta_{3u}^2)-2(1+\beta_{1u})^2\beta_{4u}^2\nonumber \\ &-&8\beta_{2u}^2\beta_{3u}^2+16(1+\beta_{1u})\beta_{2u}\beta_{3u} \beta_{4u}.
\label{eq:Lambdaudef}
\end{eqnarray}
Clearly, for arbitrary values of the parameters, the eigenvalues are not hierarchical. However, in looking for leading-order results in which only one eigenvalue is sizable, we can easily identify two general scenarios of interest, depending on the ordering of the mass eigenvalues. One option is that $\lambda_{1u}$ is one of the small eigenvalues, which would have $\beta_{1u}\rightarrow 1$, and $\lambda_{2u}$ is the other, and hence $\lambda_{3u}$ generically has an $O(1)$ value. Another option is that $\lambda_{1u}$ is the large eigenvalue, such that $\beta_{1u}\neq 1$, and both $\lambda_{2u,3u}$ are small. We now discuss each possibility in turn.
In what follows, we will focus on the up quarks, but our default assumption will be that the down quarks and the charged leptons will take similar forms. Mixing possible options for eigenvalue hierarchies in the different charged fermion sectors will not be considered here for simplicity.
\subsection{Case 1: $\lambda_{1u,2u}\ll \lambda_{3u}$ (encompassing the ``singlet-dominated" and ``democratic" limits)}
We begin with the situation that $\beta_{1u}\rightarrow 1$, such that $\lambda_{1u}$ is a small eigenvalue, and explore parameter regimes in which $\lambda_{2u}$ is also small. For simplicity, we first consider the case in which both vanish, such that to this order of approximation we have one massive third-generation, and two massless generations. It is easily verified that in this regime, both eigenvalues vanish for
\begin{eqnarray}
\beta_{1u}=1, \qquad \beta_{2u}\beta_{3u}=\beta_{4u}.
\label{eq:betarelations}
\end{eqnarray}
This case includes what we call the {\it democratic} limit, in which all the $\beta_{iu}=1$, and thus the MSSM Yukawas take on the well-known democratic form \cite{Fritzsch_1994}. The democratic limit was originally studied at leading order in \cite{Everett_2018}, and will be studied in more detail below, including subleading corrections. This case also includes what we will call the {\it singlet-dominated} limit, which is the case in which $ \beta_{4u} \gg \beta_{1u,2u,3u}$, as $\beta_{4u}$ is the parameter related to the strength of the superpotential coupling involving only $\mathcal{S}_3$ singlet fields. In the singlet-dominated limit, the MSSM and messenger Yukawa couplings at leading order, in the diagonal quark mass basis, only have nonvanishing $3-3$ entries, allowing for sizable stop mixing and consequently lighter superpartner masses than the other examples we will consider (as we will see). This limiting case was studied in some detail in \cite{Everett_2019} and \cite{Everett_2020}, and will be considered below as a benchmark scenario for purposes of comparison.
For Case 1, incorporating Eq.~(\ref{eq:betarelations}) and up to possible rephasings to ensure that the fermion masses are real and positive, the diagonalization matrices $U_{uL}$ and $U_{uR}$ take the form
\begin{eqnarray}
U_{uL}=\left (\begin{array}{ccc} \;\;\; \frac{1}{\sqrt{2}} & -\frac{\beta_{3u}}{\sqrt{2}\sqrt{2+\beta_{3u}^2}} & \frac{1}{\sqrt{2+\beta_{3u}^2}}
\\ -\frac{1}{\sqrt{2}} & -\frac{\beta_{3u}}{\sqrt{2}\sqrt{2+\beta_{3u}^2}} & \frac{1}{\sqrt{2+\beta_{3u}^2}}
\\ \;\;\; 0 &\frac{\sqrt{2}}{\sqrt{2+\beta_{3u}^2}} & \frac{\beta_{3u}}{\sqrt{2+\beta_{3u}^2}}
\end{array}\right ), \qquad
U_{uR}=\left (\begin{array}{ccc} \;\;\; \frac{1}{\sqrt{2}} & -\frac{\beta_{2u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}} & \frac{1}{\sqrt{2+\beta_{2u}^2}} \\
-\frac{1}{\sqrt{2}} & -\frac{\beta_{2u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}} & \frac{1}{\sqrt{2+\beta_{2u}^2}}
\\ \;\;\; 0 &\frac{\sqrt{2}}{\sqrt{2+\beta_{2u}^2}} & \frac{\beta_{2u}}{\sqrt{2+\beta_{2u}^2}}
\end{array}\right ).
\label{eq:case1diagmatrices}
\end{eqnarray}
Assuming these forms with no further rephasings, the messenger Yukawa couplings in the diagonal quark mass basis then take the form
\begin{equation}
Y_{u1}^\prime = \tilde{y}_u \left (\begin{array}{ccc} -\frac{\sqrt{3}}{2} & \frac{3 \beta_{2u}}{2\sqrt{2+\beta_{2u}^2}} & \frac{\beta_{2u}^2-1}{\sqrt{2}\sqrt{2+\beta_{2u}^2}} \\ \frac{3\beta_{3u}}{2\sqrt{2+\beta_{3u}^2}} & \frac{3\sqrt{3} \beta_{2u}\beta_{3u}}{2\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} & \frac{\sqrt{3}(\beta_{2u}^2-1)\beta_{3u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} \\ \frac{\beta_{3u}^2-1}{\sqrt{2}\sqrt{2+\beta_{3u}^2}} & \frac{\sqrt{3}(\beta_{3u}^2-1)\beta_{2u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} & \frac{(\beta_{2u}^2-1)(\beta_{3u}^2-1)}{\sqrt{3}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}}
\end{array} \right )
\end{equation}
\begin{equation}
Y_{u2}^\prime =\tilde{y}_u \left (\begin{array}{ccc} -\frac{\sqrt{3}}{2} & -\frac{3 \beta_{2u}}{2\sqrt{2+\beta_{2u}^2}} & \frac{\beta_{2u}^2-1}{\sqrt{2}\sqrt{2+\beta_{2u}^2}} \\ -\frac{3\beta_{3u}}{2\sqrt{2+\beta_{3u}^2}} & \frac{3\sqrt{3} \beta_{2u}\beta_{3u}}{2\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} & \frac{\sqrt{3}(\beta_{2u}^2-1)\beta_{3u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} \\ -\frac{\beta_{3u}^2-1}{\sqrt{2}\sqrt{2+\beta_{3u}^2}} & \frac{\sqrt{3}(\beta_{3u}^2-1)\beta_{2u}}{\sqrt{2}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}} & \frac{(\beta_{2u}^2-1)(\beta_{3u}^2-1)}{\sqrt{3}\sqrt{2+\beta_{2u}^2}\sqrt{2+\beta_{3u}^2}}
\end{array} \right ).
\label{eqn:case1yukawa}
\end{equation}
From these forms, we see that in the democratic limit, the messenger Yukawas only have nonvanishing entries in the upper $2\times 2$ block, as follows:
\begin{equation}
Y_{u1}^\prime = \tilde{y}_u \left (\begin{array}{ccc} -\frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 0 \\ 0& 0&0
\end{array} \right ), \qquad Y_{u2}^\prime = \tilde{y}_u \left (\begin{array}{ccc} -\frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} & 0 \\ -\frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 0 \\ 0& 0&0
\end{array} \right ).
\end{equation}
In the singlet-dominated limit, the $3-3$ entries dominate, with $Y^\prime_{u1,u2}={\rm Diag}(0,0,\tilde{y}_u\beta_{2u}\beta_{3u}/\sqrt{3})$.
\subsection{Case 2: $\lambda_{2u,3u}\ll \lambda_{1u}$ (the ``doublet-dominated" limit)}
For this case, it is necessary that $\beta_{1u}\neq 1$ such that $\lambda_{1u}\gg \lambda_{2u,3u}$. For concreteness, we take $\beta_{1u}\rightarrow -1$, and thus require $\beta_{2u,3u,4u}\ll 1$, as well as $\Lambda_u \rightarrow 0$. Indeed, $\lambda_{2u,3u}=0$ is achieved for $\beta_{1u}=-1,\beta_{2u}=\beta_{3u}=\beta_{4u}=0$. To see this, we note that for $\beta_1=-1$ only, the condition for $\Lambda_u=0$ is as follows:
\begin{equation}
-8 \beta_{2u}^2 \beta_{3u}^2+4(\beta_{2u}^4+\beta_{3u}^4)+4(\beta_{2u}^2+\beta_{3u}^2)\beta_{4u}^2+\beta_{4u}^2=0,
\end{equation}
which is zero only for $\beta_{4u}=0$ and $\beta_{2u}=\beta_{3u}$. We will take $\beta_{1u} = -1$ and $\beta_{4u} = 0$, but leave $\beta_{2u}$ and $\beta_{3u}$ unconstrained at present, recalling that we will need to restrict ourselves to the case that $\beta_{2u,3u}\ll \vert \beta_{1u} \vert =1$. This limit is the {\it doublet-dominated} limit, since now $\vert \beta_{1u} \vert \gg \beta_{2u,3u} \gg \beta_{4u}=0$, and $\beta_{1u}$ controls the superpotential coupling involving only $\mathcal{S}_3$ doublet fields. In this limit, the mass eigenvalues take the form (assuming for concreteness that $\beta_{3u}>\beta_{2u}$):
\begin{equation}
\lambda_{1u} = \frac{2\tilde{y}_u^2}{3}, \qquad \lambda_{2u}= \frac{2 \tilde{y}_u^2 \beta_{2u}^2}{3}, \qquad \lambda_{3u} = \frac{2 \tilde{y}_u^2 \beta_{3u}^2}{3},
\end{equation}
such that $\lambda_{2u} < \lambda_{3u} < \lambda_{1u}$. (For $\beta_{3u}< \beta_{2u}$, the placement of $\beta_{2u}$ and $\beta_{3u}$ in $\lambda_{2u}$ and $\lambda_{3u}$ is reversed.) We now take $ \sqrt{3} \tilde{y}_u/2=y_t$ to identify $y_t$ as the top quark Yukawa coupling to leading order. The diagonalization matrices $U_{uL}$ and $U_{uR}$ now take the following particularly simple forms:
\begin{equation}
U_{uL}=\left (\begin{array}{ccc} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ 0& 1 & 0
\end{array} \right ), \qquad
U_{uR}=\left (\begin{array}{ccc} 0& \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ 1 & 0 & 0
\end{array} \right ).
\end{equation}
The messenger Yukawas in the diagonal quark basis are then given by
\begin{equation}
Y_{u1}^\prime = y_t \left (\begin{array}{ccc} -\frac{\beta_{2u}}{2\sqrt{2}}& -\frac{3}{4} & -\frac{\sqrt{3}}{4} \\ \;\;\; 0 & -\frac{\beta_{3u}}{2\sqrt{2}} & \frac{\sqrt{3}\beta_{3u}}{2\sqrt{2}}\\ \frac{\sqrt{3}\beta_{2u}}{2\sqrt{2}} & -\frac{\sqrt{3}}{4} & \;\;\; \frac{1}{4} \end{array} \right ), \qquad
Y_{u2}^\prime = y_t\left (\begin{array}{ccc} -\frac{\beta_{2u}}{2\sqrt{2}}& -\frac{3}{4} & \frac{\sqrt{3}}{4} \\ \;\;\; 0 & -\frac{\beta_{3u}}{2\sqrt{2}} & -\frac{\sqrt{3}\beta_{3u}}{2\sqrt{2}}\\ -\frac{\sqrt{3}\beta_{2u}}{2\sqrt{2}} & \frac{\sqrt{3}}{4} & \;\;\; \frac{1}{4} \end{array} \right ).
\end{equation}
\section{Models}
As described in the previous subsection, we have identified two cases with hierarchical quark and charged lepton masses. The first (Case 1) satisfies Eq.~(\ref{eq:betarelations}), and includes two possible scenarios at leading order: the singlet-dominated limit, in which it is the $\mathcal{S}_3$ singlet couplings of the MSSM fields and the Higgs-messenger fields that dominate the superpotential, and the democratic limit, in which all the couplings of $\mathcal{S}_3$ representations in the superpotential are precisely equal at leading order, resulting in an enhanced $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ symmetry. The singlet-dominated limit was explored in \cite{Everett_2019} and \cite{Everett_2020}, and the democratic limit at leading order in \cite{Everett_2018}.
The second (Case 2) is what we call the doublet-dominated limit, as in this case the dominant couplings are those involving only $\mathcal{S}_3$ doublets.
In what follows, we will discuss these scenarios in greater detail.
\subsection{Case 1 models}
We begin the discussion of Case 1 models with the singlet-dominated limit, which was studied in detail in \cite{Everett_2019,Everett_2020}. In this scenario, sizable stop mixing can occur due to the FGM contributions to the third-generation soft trilinear scalar coupling. This in turn allows for the squarks and gluinos to be in the $O(5-6 \; {\rm TeV})$ range, which is relatively light compared to generic parameter choices for this class of FGM models. A variety of subleading corrections to this limit can be considered, including the possibility of generating nontrivial masses for the second-generation fields and the possibility of viable quark mixing at the first subleading order. For the case described in \cite{Everett_2020}, the corrections to the soft terms that result from these terms have only minimal effects on the superpartner masses.
Furthermore, in this case flavor-violating contributions to the soft terms also do not result at the first subleading order in the quantities that control the lighter generation quark and lepton masses, though this is not necessarily generic. Here we will not revisit this case in detail other than as a point of comparison for the new scenarios considered in this work.
Let us now turn to the democratic limit, for which the Yukawa coupling parameters $\beta_{1i}=\beta_{2i}= \beta_{3i}= \beta_{4i}=1$, where $i=u,d,e$. In this case, the MSSM Yukawa matrices take the form \begin{equation}
Y_i=\frac{\tilde{y}_i}{\sqrt{3}}\begin{pmatrix}
1 & 1 & 1\\
1 & 1 & 1\\
1 & 1 & 1
\end{pmatrix}.
\end{equation}
This is the well-known flavor democratic mass matrix form, which exhibits an $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ symmetry. At leading order, this mass matrix has two vanishing eigenvalues, and one $O(1)$ eigenvalue, to be identified with the third-generation. As shown in \cite{Everett_2018}, the messenger Yukawa matrices have nonzero entries only in the upper $2\times 2$ block in the diagonal quark mass basis.
We now address the generation of the first- and second-generation fermion masses and the effects on the sfermion masses through the messenger Yukawa corrections. Here we choose to break the $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ symmetry to $\mathcal{S}_{2L}\times \mathcal{S}_{2R}$ and then to $\mathcal{S}_{1L}\times \mathcal{S}_{1R}$, which generates a nonzero mass for the first- and second-generation fermions (see e.g.~\cite{Fritzsch_2000}). This can be achieved via the following terms:
\begin{equation}
Y_i^{(\text{corr})}=\frac{\tilde{y}_i\epsilon_i}{\sqrt{3}}\begin{pmatrix}
0 & 0 & 1\\
0 & 0 & 1\\
1 & 1 & 1
\end{pmatrix}+\frac{\tilde{y}_i\sigma_i}{\sqrt{3}}\begin{pmatrix}
1 & 0 & -1\\
0 & -1 & 1\\
-1 & 1 & 0
\end{pmatrix},
\label{eq:demcorrections}
\end{equation}
in which $\epsilon_i$ and $\sigma_i$ are real dimensionless perturbative parameters associated with symmetry breaking from $\mathcal{S}_3$ to $\mathcal{S}_2$ and $\mathcal{S}_2$ to $\mathcal{S}_1$ respectively. In our scenario, the $\epsilon$ perturbations of the up quarks (the down quarks and charged leptons have analogous structures) can be generated in superpotential at the renormalizable level by
\begin{equation}
\epsilon_u y_u[\beta_{2u} Q_\textbf{2}\bar{u}_\textbf{1}\mathcal{H}_u^{(2)}+\beta_{3u}Q_\textbf{1}\bar{u}_\textbf{2}\mathcal{H}_u^{(2)}+\beta_{4u} Q_\textbf{1}\bar{u}_\textbf{1}\mathcal{H}_{u}^{(1)}],
\end{equation}
while the $\sigma$ perturbations can be generated via non-renormalizable operators. These superpotential terms add corrections of the form of Eq.~(\ref{eq:demcorrections}) to the Yukawa matrix for the up-type quarks, and corrections of the following form to the up-type messenger Yukawa matrices:
\begin{equation}
\begin{split}
Y_{u1}^{\prime(\text{corr})}&=\tilde{y}_u\epsilon_u\begin{pmatrix}
0 & 0 & \frac{1}{2}-\frac{1}{2\sqrt{3}} \\
0 & 0 & -\frac{1}{2}-\frac{1}{2\sqrt{3}}\\
\frac{1}{2}-\frac{1}{2\sqrt{3}} & -\frac{1}{2}-\frac{1}{2\sqrt{3}}& \frac{1}{\sqrt{3}}
\end{pmatrix}+\tilde{y}_u\sigma_u\left(
\begin{array}{ccc}
\frac{1}{2}-\frac{1}{2 \sqrt{3}} & 0 & \frac{1}{2}+\frac{1}{2 \sqrt{3}} \\
0 & \frac{1}{2}+\frac{1}{2 \sqrt{3}} & \frac{1}{2}-\frac{1}{2 \sqrt{3}} \\
\frac{1}{2}+\frac{1}{2 \sqrt{3}} & \frac{1}{2}-\frac{1}{2 \sqrt{3}} & 0 \\
\end{array}
\right)\\
Y_{u2}^{\prime(\text{corr})}&=\tilde{y}_u\epsilon_u\begin{pmatrix}
0 & 0 & -\frac{1}{2}-\frac{1}{2\sqrt{3}}\\
0 & 0 & \frac{1}{2}-\frac{1}{2\sqrt{3}}\\
-\frac{1}{2}-\frac{1}{2\sqrt{3}} & \frac{1}{2}-\frac{1}{2\sqrt{3}} & \frac{1}{\sqrt{3}}
\end{pmatrix}+\tilde{y}_u\sigma_u\left(
\begin{array}{ccc}
-\frac{1}{2}-\frac{1}{2 \sqrt{3}} & 0 & -\frac{1}{2}+\frac{1}{2 \sqrt{3}} \\
0 & -\frac{1}{2}+\frac{1}{2 \sqrt{3}} & -\frac{1}{2}-\frac{1}{2 \sqrt{3}} \\
-\frac{1}{2}+\frac{1}{2 \sqrt{3}} & -\frac{1}{2}-\frac{1}{2 \sqrt{3}} & 0 \\
\end{array}
\right).
\end{split}
\end{equation}
Including these correction terms along with the leading order results, the eigenvalues
$\lambda_{1u, 2u,3u}$, are then found to be
\begin{equation}
\lambda_{1u} = 0, \qquad \lambda_{2u} = \frac{4 y_t^2 \epsilon_u^2}{81}+ O(\epsilon^3), \qquad \lambda_{3u} = y_t^2+O(\epsilon_u^3).
\end{equation}
In these relations, we have identified the top quark Yukawa coupling $y_t$ through $\tilde{y}_u = (y_t/\sqrt{3})(1-5\epsilon_u/9+ O(\epsilon_u)^2$, which follows from setting $\lambda_{3u} = y_t^2$ through second order in $\epsilon_u$.
As expected, $\epsilon_u$ controls the charm quark mass
\footnote{Note that if we neglect the subleading $\sigma$ perturbations, this scheme is equivalent to replacing $\beta_{2u,3u,4u} = 1+\epsilon_u$ in the general form for the superpotential couplings. As such, the mixing matrices are easily obtained using the results of Eq.~(\ref{eq:case1diagmatrices}) with the appropriate substitutions.}.
In the diagonal quark mass basis, the messenger Yukawa matrices for the up-type quark sector are given to order in $\sigma_u/\epsilon_u$ by
\begin{equation}
\begin{split}
Y_{u1}^{\prime \text{(diag)}}&=y_t\left(
\begin{array}{ccc}
-\frac{1}{2}+\frac{5 \epsilon_u }{18}-\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & -\frac{1}{2}-\frac{\epsilon_u }{18}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & -\frac{5 \epsilon_u
}{9 \sqrt{2}} \\
-\frac{1}{2}-\frac{\epsilon_u }{18}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{1}{2}+\frac{\epsilon_u }{6}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{\epsilon_u }{3
\sqrt{2}} \\
-\frac{5 \epsilon_u }{9 \sqrt{2}} & \frac{\epsilon_u }{3 \sqrt{2}} & -\frac{\epsilon_u }{9} \\
\end{array}
\right)\\
Y_{u2}^{\prime \text{(diag)}}&=y_t\left(
\begin{array}{ccc}
-\frac{1}{2}+\frac{5 \epsilon_u }{18}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{1}{2}+\frac{\epsilon_u }{18}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{5 \epsilon_u
}{9 \sqrt{2}} \\
\frac{1}{2}+\frac{\epsilon_u }{18}+\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{1}{2}+\frac{\epsilon_u }{6}-\frac{3 \sqrt{3} \sigma_u }{2 \epsilon_u } & \frac{\epsilon_u }{3
\sqrt{2}} \\
\frac{5 \epsilon_u }{9 \sqrt{2}} & \frac{\epsilon_u }{3 \sqrt{2}} & -\frac{\epsilon_u }{9} \\
\end{array}
\right).
\end{split}
\end{equation}
Analogous forms are easily obtained for the MSSM and messenger Yukawa matrices for down-type quarks and leptons in the diagonal quark mass basis with the replacements $\epsilon_u \rightarrow \epsilon_{d,e}$ and $y_t \rightarrow y_{b,\tau}$.
The relative strengths of the parameters $\epsilon_{u,d,e}$ and $\sigma_{u,d,e}$ can be estimated from the fact that these parameters govern the fermion masses of the lighter generations. More precisely, up to $O(1)$ prefactors, $\epsilon_{u,d,e}$ is related to $m_{c,s,\mu}/m_{t,b,\tau}$, while $\sigma_{u,d,e}$ is constrained by $m^2_{u,d,e}/m^2_{t,b,\tau} \sim \sigma^2_{u,d,e}/\epsilon_{u,d,e}$.
From these relations, it is straightforward to obtain that $\epsilon_u\approx 3\times 10^{-2}$ and $\sigma_u\approx 1\times 10^{-3}$. Similarly, $\epsilon_d \approx 0.1$, $\sigma_d\approx 9\times 10^{-3}$, $\epsilon_e\approx 0.3$, and $\sigma_e\approx 8 \times 10^{-3}$. These parameter values also yield hierarchical quark mixing angles of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix, in which the largest angle is the Cabibbo angle, $\sin\theta_c\sim 0.17$.
While the quark mixing angles are not fully realistic (the Cabibbo angle is clearly too small compared to its experimentally determined value), for the purposes of this study it is a reasonable starting point for the analysis.
We now find the nonvanishing corrections to the soft supersymmetry breaking terms, assuming for simplicity that the ratio of the $F$ terms to the scalar VEVs for the $X_H$ and $X_T$ terms are identical (both will be denoted as $\Lambda$). We provide the expressions for these correction terms in the Appendix. As expected, in the limit that the perturbation parameters are set to zero, the result is what was found in \cite{Everett_2018}. When the perturbations are added, the diagonal entries of the soft mass-squared terms are corrected at second order in the $\epsilon$ parameters. This generates nonzero (but small) diagonal $3-3$ entries. In addition, with nonzero perturbations, flavor off-diagonal contributions to the corrections to the soft terms are generated. More precisely, the $\epsilon_{u,d,e}$ parameters introduce nonvanishing $\delta m_{f_{23}}^2$ terms at first order in $\epsilon$, while the $\sigma_{u,d,e}$ introduce nonvanishing $\delta m_{f_{12}}^2$ and $\delta m_{f_{21}}^2$ terms. Therefore, the dominant effects are expected to be seen in the $2-3$ sfermion mixings. Further details will be discussed in the next section.
\begin{comment}
The nonvanishing corrections to the soft supersymmetry breaking terms now take the form:
\begin{equation}
\begin{split}
\delta m_{Q_{11}}^2=\frac{\Lambda^2}{(4\pi)^4}\bigg[(&6y_t^4+6y_b^4+2y_b^2y_t^2+y_b^2y_\tau^2-\Tilde{g_u}^2y_t^2-\Tilde{g_d}^2y_b^2) \\ &+\epsilon_u \left(-\frac{32}{9}y_t^4+\frac{4}{9}\Tilde{g_u}^2y_t^2-\frac{4}{9}y_b^2y_t^2\right)+\epsilon_d \left(-\frac{32}{9}y_b^4+\frac{4}{9}\Tilde{g_d}^2y_b^2-\frac{4}{9}y_b^2y_t^2-\frac{4}{9}y_b^2y_\tau^2\right) +\mathcal{O}(\epsilon^2)
\bigg]\\
\delta m_{Q_{22}}^2=\frac{\Lambda^2}{(4\pi)^4}\bigg[(&6y_t^4+6y_b^4+2y_b^2y_t^2+y_b^2y_\tau^2-\Tilde{g_u}^2y_t^2-\Tilde{g_d}^2y_b^2) \\ &+\epsilon_u \left(\frac{32}{9}y_t^4-\frac{4}{9}\Tilde{g_u}^2y_t^2+\frac{4}{9}y_b^2y_t^2\right)+\epsilon_d \left(\frac{32}{9}y_b^4-\frac{4}{9}\Tilde{g_d}^2y_b^2+\frac{4}{9}y_b^2y_t^2+\frac{4}{9}y_b^2y_\tau^2\right) +\mathcal{O}(\epsilon^2)
\bigg]\\
\delta m_{Q_{23}}^2=\delta m_{Q_{32}}^2=\frac{\Lambda^2}{(4\pi)^4}\bigg[&\epsilon_u\left(\frac{8\sqrt{2}}{3}y_t^4-\frac{4\sqrt{2}}{9}\Tilde{g_u}^2y_t^2+\frac{4\sqrt{2}}{9}y_b^2y_t^2\right)\\
+&\epsilon_d\left(\frac{8\sqrt{2}}{3}y_b^4-\frac{4\sqrt{2}}{9}\Tilde{g_d}^2y_b^2+\frac{4\sqrt{2}}{9}y_b^2y_t^2+\frac{4\sqrt{2}}{9}y_b^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\bigg]\\ \nonumber
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&\delta m_{\bar{u}_{11}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(12y_t^4+2y_t^2y_b^2-2\Tilde{g_u}^2y_t^2)-\frac{8\epsilon_u}{9}\left(8y_t^4+y_b^2y_t^2-\tilde{g_u}^2y_t^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{u}_{22}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(12y_t^4+2y_t^2y_b^2-2\Tilde{g_u}^2y_t^2)+\frac{8\epsilon_u}{9}\left(8y_t^4+y_b^2y_t^2-\tilde{g_u}^2y_t^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{u}_{23}}^2=\delta m_{\bar{u}_{32}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[\frac{8\sqrt{2}\epsilon_u}{9}\left( 6y_t^4+y_b^2y_t^2-\tilde{g_u}^2y_t^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{d}_{11}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(12y_b^4+2y_t^2y_b^2+2y_b^2y_{\tau}^2-2\Tilde{g_d}^2y_b^2)-\frac{8\epsilon_d}{9}\left(8y_b^4+y_b^2y_t^2+y_b^2y_\tau^2-\tilde{g_d}^2y_b^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{d}_{22}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(12y_b^4+2y_t^2y_b^2+2y_b^2y_{\tau}^2-2\Tilde{g_d}^2y_b^2)+\frac{8\epsilon_d}{9}\left(8y_b^4+y_b^2y_t^2+y_b^2y_\tau^2-\tilde{g_d}^2y_b^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{d}_{23}}^2=\delta m_{\bar{d}_{32}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[\frac{8\sqrt{2}\epsilon_d}{9}\left( 6y_b^4+y_b^2y_t^2+y_b^2y_{\tau}^2-\tilde{g_d}^2y_b^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{L_{11}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(4y_\tau^4+3y_b^2y_\tau^2-\Tilde{g_e}^2y_\tau^2)-\frac{4\epsilon_e}{9}\left(6y_\tau^4+3y_b^2y_\tau^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{L_{22}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(4y_\tau^4+3y_b^2y_\tau^2-\Tilde{g_e}^2y_\tau^2)+\frac{4\epsilon_e}{9}\left(6y_\tau^4+3y_b^2y_\tau^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{L_{23}}^2=\delta m_{L_{32}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[\frac{4\sqrt{2}\epsilon_e}{9}\left( 4y_\tau^4+3y_b^2y_{\tau}^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{e}_{11}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(8y_\tau^4+6y_b^2y_\tau^2-2\Tilde{g_e}^2y_\tau^2)-\frac{8\epsilon_e}{9}\left(6y_\tau^4+3y_b^2y_\tau^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{e}_{22}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[(8y_\tau^4+6y_b^2y_\tau^2-2\Tilde{g_e}^2y_\tau^2)+\frac{8\epsilon_e}{9}\left(6y_\tau^4+3y_b^2y_\tau^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{\bar{e}_{23}}^2=\delta m_{\bar{e}_{32}}^2=\frac{\Lambda^2}{(4\pi)^4}\left[\frac{8\sqrt{2}\epsilon_e}{9}\left( 4y_\tau^4+3y_b^2y_{\tau}^2-\tilde{g_e}^2y_\tau^2\right)+\mathcal{O}(\epsilon^2)\right]\\
&\delta m_{H_u}^2=\delta m_{H_d}^2=0\\
&\Tilde{A_u}=\begin{pmatrix}
0 &0 &0 \\
0 & -\frac{2\epsilon_u}{9}(3y_t^3+y_ty_b^2) & \frac{4\sqrt{2}\epsilon_u}{9}y_t^3+\frac{4\sqrt{2}\epsilon_d}{9}y_ty_b^2\\
0& \frac{8\sqrt{2}\epsilon_u}{9}y_t^3 & \mathcal{O}(\epsilon^2)
\end{pmatrix}\quad \Tilde{A_d}=\begin{pmatrix}
0 &0 &0 \\
0 & -\frac{2\epsilon_d}{9}(3y_b^3+y_t^2y_b) & \frac{4\sqrt{2}\epsilon_d}{9}y_b^3+\frac{4\sqrt{2}\epsilon_u}{9}y_t^2y_b\\
0& \frac{8\sqrt{2}\epsilon_d}{9}y_b^3 & \mathcal{O}(\epsilon^2)
\end{pmatrix}\\
&\Tilde{A_e}=\begin{pmatrix}
0 &0 &0 \\
0 & -\frac{2\epsilon_e}{3}y_\tau^3 & \frac{4\sqrt{2}\epsilon)e}{9}y_\tau^3\\
0& \frac{8\sqrt{2}\epsilon_e}{9}y_\tau^3 & \mathcal{O}(\epsilon^2)
\end{pmatrix}.
\end{split}
\label{eq:democraticsoftterms}
\end{equation}
\end{comment}
\subsection{Case 2 models}
This case corresponds to the doublet-dominated limit. Here we need $\lambda_1\gg \lambda_{2,3}$. In the limit that $\lambda_{2,3}\rightarrow 0$, we see from Eqs.~(\ref{eq:eigenvaluessq})-(\ref{eq:Lambdaudef}) this can be achieved for $\beta_1\rightarrow -1$ and $\beta_{i=2,3,4}\ll 1$, and we need $\tilde{\Lambda} \rightarrow 0$. For $\beta_1=-1$, the condition for $\tilde{\Lambda}=0$ is as follows:
\begin{equation}
-8 \beta_2^2 \beta_3^2+4(\beta_2^4+\beta_3^4)+4(\beta_2^2+\beta_3^2)\beta_4^2+\beta_4^2=0,
\end{equation}
which is zero only for $\beta_4=0$, $\beta_2=\beta_3$. In the up quark sector, we will now set $\lambda_1=y_t^2$, such that $\tilde{y}_u^2= (3/4) y_t^2$ (analogous relations hold for the down quark and charged lepton sectors). The $\lambda_{2i,3i}$ are directly related to $\beta_{2i,3i}$, with the specific identification dependent on the values of the $\beta_{2i,3i}$.
\noindent $\bullet$ {\it Ordering $\beta_{3i}>\beta_{2i}$}.
Let us first consider the case in which $\beta_{3i}>\beta_{2i}$, for which we have
\begin{equation}
U_{iL}^\dagger Y_i U_{iR}= Y_i^{(\text{diag})}=
y_{t,b,\tau}\text{Diag}\left (\frac{\beta_{2i}}{\sqrt{2}}, \frac{ \beta_{3i}}{\sqrt{2}},1 \right ),
\label{eq:order1}
\end{equation}
in which $U_{iL,iR}$ take the simple forms
\begin{equation}
U_{iL}=\left (\begin{array}{ccc} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ 0& 1 & 0
\end{array} \right ), \qquad
U_{iR}=\left (\begin{array}{ccc} 0& \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ 1 & 0 & 0
\end{array} \right ).
\label{umatricescase2a}
\end{equation}
We see that for this ordering, the $\beta_{3i}$ control the second-generation masses and the $\beta_{2i}$ control the first-generation masses.
The messenger Yukawas in the diagonal fermion mass basis (the SCKM basis) are given by
\begin{equation}
Y_{i1}^\prime = y_{t,b,\tau} \left (\begin{array}{ccc} -\frac{\beta_{2i}}{2\sqrt{2}}& -\frac{3}{4} & -\frac{\sqrt{3}}{4} \\ \;\;\; 0 & -\frac{\beta_{3i}}{2\sqrt{2}} & \frac{\beta_{3i}}{2}\sqrt{\frac{3}{2}}\\ \frac{\beta_{2i}}{2}\sqrt{\frac{3}{2}} & -\frac{\sqrt{3}}{4} & \;\;\; \frac{1}{4} \end{array} \right ),\qquad
Y_{i2}^\prime = y_{t,b,\tau}\left (\begin{array}{ccc} -\frac{\beta_{2i}}{2\sqrt{2}}& -\frac{3}{4} & +\frac{\sqrt{3}}{4} \\ \;\;\; 0 & -\frac{\beta_{3i}}{2\sqrt{2}} & -\frac{\beta_{3i}}{2}\sqrt{\frac{3}{2}}\\ -\frac{\beta_{2i}}{2}\sqrt{\frac{3}{2}} & \frac{\sqrt{3}}{4} & \;\;\; \frac{1}{4} \end{array} \right ).
\end{equation}
Given that we can identify $\beta_{2i,3i}$ with the first and second-generation masses, respectively, we can write for example for the up-type quarks (with $y_{u,d,e}=\beta_{2u,d,} y_t/\sqrt{2}$ and $y_c=\beta_{3u} y_t/\sqrt{2}$):
\begin{equation}
Y_{u1}^\prime = \left (\begin{array}{ccc} -\frac{y_u}{2}& -\frac{3 y_t}{4}& -\frac{\sqrt{3}y_t}{4} \\ \;\;\; 0 & -\frac{y_c}{2} & -\frac{\sqrt{3}y_c}{2}\\ -\frac{\sqrt{3} y_u}{2} & \frac{\sqrt{3}y_t}{4} & \;\;\; \frac{y_t}{4} \end{array} \right ),\qquad
Y_{u2}^\prime = \left (\begin{array}{ccc} -\frac{y_u}{2}&-\frac{3 y_t}{4} & -\frac{\sqrt{3}y_t}{4} \\ \;\;\; 0 & -\frac{y_c}{2} & \frac{\sqrt{3}y_c}{2} \\ \frac{\sqrt{3} y_u}{2}& -\frac{\sqrt{3}y_t}{4} & \;\;\; \frac{y_t}{4} \end{array} \right ).
\label{eq:messyukcase2a}
\end{equation}
From the quark and charged lepton masses, we can roughly estimate (neglecting running effects) that $\beta_{2u}/\beta_{3u}\sim 2\times 10^{-3}$, $\beta_{2d}/\beta_{3d} \sim 0.05$, $\beta_{2l}/\beta_{3l}\sim 0.005$, while $\beta_{3d}/\beta_{3l} \sim 0.4$. Hence, to leading order we can neglect the effects proportional to the first-generation masses (here the $\beta_{2i}$), and treat the effects due to the second-generation masses (the $\beta_{3i}$) perturbatively.
We thus calculate the corrections to the soft supersymmetry terms in this limit.
As before, we assume for simplicity that the ratio of the $F$ terms to the scalar vevs for the $X_H$ and $X_T$ terms are identical.
The detailed forms of these soft supersymmetry breaking terms are presented in the Appendix.
We note that in this case, there are flavor off-diagonal contributions in the $\delta m^2_{{Q,L}_{12}}$ that are proportional to the $\beta_{3i}$, and thus scale with the second-generation quark and lepton masses. This is reminiscent of the Case 1 democratic limit with perturbations, though the dominant off-diagonal contributions occurred there in the $2-3$ sector, and here they arise in the more dangerous $1-2$ sector. We will discuss their effects in the next section.
\noindent $\bullet$ {\it Ordering $\beta_{2i}>\beta_{3i}$}.
We now consider the case in which $\beta_{2i}>\beta_{3i}$, for which the roles of $\beta_{2i}$ and $\beta_{3i}$ are switched in Eq.~(\ref{eq:order1}).
We now have
\begin{equation}
U_{iL}=\left (\begin{array}{ccc} 0& \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 1 & 0 & 0
\end{array} \right ), \qquad
U_{iR}=\left (\begin{array}{ccc} \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0& 1 & 0
\end{array} \right ).
\label{umatricescase2b}
\end{equation}
The messenger Yukawa matrices in the diagonal quark mass basis are of the form
\begin{equation}
Y_{i1}^\prime = y_{t,b,\tau} \left (\begin{array}{ccc} -\frac{\beta_{3i}}{2\sqrt{2}}& 0& -\frac{\sqrt{3}\beta_{3i}}{2\sqrt{2}} \\ \;\;\; -\frac{3}{4} & -\frac{\beta_{2i}}{2\sqrt{2}}&\frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & -\frac{\beta_{2i}}{2}\sqrt{\frac{3}{2}} & \;\;\; \frac{1}{4} \end{array} \right ),\qquad
Y_{i2}^\prime = y_{t,b,\tau}\left (\begin{array}{ccc} -\frac{\beta_{2i}}{2\sqrt{2}}& 0 & \frac{\sqrt{3}\beta_{3i}}{2\sqrt{2}}\\ \;\;\; -\frac{3}{4}& -\frac{\beta_{2i}}{2\sqrt{2}} & -\frac{\sqrt{3}}{4}\\ -\frac{\sqrt{3}}{4} & \frac{\beta_{2i}}{2}\sqrt{\frac{3}{2}} & \;\;\; \frac{1}{4} \end{array} \right ).
\end{equation}
As in the previous section, we can ignore effects that scale with the first-generation fermion masses and keep leading contributions involving the second-generation fermion masses. Thus, we now neglect the terms proportional to $\beta_{3i}$ and keep leading-order terms proportional to the $\beta_{2i}$. We can again calculate the soft supersymmetry breaking terms, subject to the same assumptions as given for the alternate ordering. The detailed forms are included in the Appendix.
One interesting feature of this mass ordering ($\beta_{2i}>\beta_{3i}$) is the corrections to the soft supersymmetry breaking mass terms are flavor-diagonal if we neglect effects proportional to the first-generation fermion masses. As in the alternate ordering, here we obtain contributions to $\delta m^2_{{Q,L}_{12}}$ that are proportional to the $\beta_{3i}$, but now these quantities must be much smaller since they govern the masses of the first-generation. Given the high degree of suppression of the flavor off-diagonal elements, in this case the model is clearly safe from flavor-changing neutral current constraints.
\section{Results and Discussion}
\label{resultssection}
In this section, we analyze the mass spectra of these scenarios and their phenomenological implications. We start with Case 1, focusing solely on the democratic limit with symmetry breaking effects, and then study Case 2, the doublet-dominated limit, with both orderings of the $\beta_{2i}$ and $\beta_{3i}$. The model parameters are $M_\text{mess}$, $\Lambda$, $\tan{\beta}=\langle H_u\rangle / \langle H_d \rangle$, the sign of $\mu$ (sgn($\mu$), taken here to be $+1$), and the relevant perturbation parameters, which depend on the scenario in question. Here we have followed standard procedures and replaced $\vert \mu\vert$ and $b$ with $\tan\beta$ and the $Z$ boson mass.
The renormalization group equations are run using SoftSUSY 4.1.4 \cite{Allanach:2001kg}.
\subsection{Case 1 models}
We start with the flavor democratic limit, which was explored in \cite{Everett_2018} for the case of third-generation masses only, i.e.~in the absence of the small perturbations that break the $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ symmetry. It was shown in \cite{Everett_2018} that this scenario leads to heavy superpartner masses, which can be traced to the absence of large stop mixing in this limit. In the presence of nonvanishing perturbations, this picture generically continues except for specific small regions of parameter space where the Higgs mass constraint can be satisfied without being bolstered by very heavy squarks.
In Figure~\ref{fig:democraticspectrum1}, we show a representative mass spectrum for an intermediate messenger mass scale of $M_{\rm mess}=10^{12}$ GeV and $\tan\beta=10$, where $\Lambda$ is chosen to satisfy the Higgs mass constraint \cite{Zyla:2020zbs}. As seen, the heavy Higgs particles are nearly 8 TeV, the gluino is approximately 10 TeV, and the squarks fall into three groupings: a lightest set that is close in mass to the heavy Higgs particles, a set in between, and a heavier set that is similar to the gluino mass. The sleptons are close in mass to the lightest neutralino, and the next-to-lightest superpartner (NLSP) is the lightest slepton.
The effects of nonzero $\sigma_{u,d,e}$ lead to small ($O(1\; {\rm GeV})$) splittings in the masses of $\tilde{d}_1$ and $\tilde{d}_2$, and $\tilde{u}_4$ and $\tilde{u}_5$, which are each originally identical up to order $10^{-2}$ GeV. The effect of $\epsilon_{u,d,e}$ is larger, which is expected as these have larger numerical values. For nonzero $\epsilon_u$, there is a splitting of order $\sim70$ GeV in the masses of $\tilde{u}_1$ and $\tilde{u}_2$, which are also identical up to order of $10^{-2}$ GeV in the $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ limit. Similar features are seen for $\tilde{u}_4$ and $\tilde{u}_5$. The $\epsilon_u$ corrections also introduce a small ($\sim25$ GeV) mass splitting for $\tilde{d}_1$ and $\tilde{d}_2$, which is a sign of the symmetry breaking from $\mathcal{S}_{3L}\times\mathcal{S}_{3R}$ to $\mathcal{S}_{2L}\times\mathcal{S}_{2R}$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.8]{images-democratic/Junetry2testallp-1.pdf}
\caption{The sfermion mass spectrum in the Case 1 democratic limit, with $M_\text{mess}=10^{12}$ GeV, $\Lambda=8.1\times 10^5$ GeV, $\tan{\beta}=10$, $\epsilon_u=0.033$, $\epsilon_d=0.108$, $\epsilon_e=0.281$, $\sigma_u=0.001$, $\sigma_d=0.009$, and $\sigma_e=0.008$. }
\label{fig:democraticspectrum1}
\end{figure}
As noted previously, this scenario has flavor off-diagonal contributions to the corrections to the soft supersymmetry breaking terms, with the dominant contributions in the $2-3$ sector. To get an estimate of the potential sizes of these effects, we employ the standard mass insertion approximation (MIA) method, in which the quantities of interest for the quarks are
$(\delta_f^{IJ})_{XY}=(\Delta_f^{IJ})_{XY}/((m_{fI})_{XX}(m_{fJ})_{YY})$,
where $f$ denotes the relevant matter superfield, $I,J$ are flavor indices, $X,Y$ are chirality labels, and $(\Delta_f^{IJ})_{XY}$ is an off-diagonal contribution to the sfermion soft terms \footnote{Note that MIA is a good approximation in this scenario although we have non-degenerate squark masses, since the squark masses are not strongly hierarchical~\cite{Raz_2002}.}. We expect rather mild constraints due to the heavy sfermion and gluino masses and the suppression factors in the off-diagonal contributions to the soft terms. For the set of model parameters in Fig.~\ref{fig:democraticspectrum1}, we obtain $2-3$ squark and slepton mass insertion parameters of the order $\vert (\delta_u^{23})_{LL}\vert \sim 5\times 10^{-3}$, $\vert (\delta_u^{23})_{RR}\vert \sim 10^{-2}$, $\vert (\delta_d^{23})_{LL}\vert \sim 5\times 10^{-3}$, $\vert (\delta_d^{23})_{RR}\vert \sim 7\times 10^{-4}$, $\vert (\delta_l^{23})_{LL}\vert \sim 2\times 10^{-3}$, and $\vert (\delta_l^{23})_{RR}\vert \sim 3\times 10^{-2}$, as well as small contributions to $LR$ mixings in the $2-3$ sector (ranging from $10^{-4}$ to $10^{-7}$. The $1-3$ and $1-2$ mass insertions are parametrically smaller, with limits that range from $10^{-4}$ to $10^{-12}$, except for $\vert (\delta_l^{12})_{RR}\vert \sim 3\times 10^{-3}$. The resulting effects are small and within the allowed ranges (see e.g.~\cite{Misiak_1998}).
\begin{figure}[h]
\centering
\subfloat{{\includegraphics[scale=0.9]{images-democratic/Sept1e12tanbeta10allpmixing.pdf}}}
\caption{
The sfermion mass eigenstates in the democratic limit with $\epsilon_u=0.033$, $\epsilon_d=0.108$, $\epsilon_e=0.281$, $\sigma_u=0.001$, $\sigma_d=0.009$, and $\sigma_e=0.008$, with $M_\text{mess}=10^{12}$ GeV, $\Lambda=8.1\times 10^5$ GeV, and $\tan{\beta}=10$. }
\label{fig2}
\end{figure}
The composition of the mass eigenstates of the sfermions is shown in Fig.~\ref{fig2}. Without perturbations, there is almost no mixing between different flavor eigenstates. The lightest $SU(3)_c$ charged particles are the first and second-generation right-handed squarks and the lightest sleptons are the first and second-generation right-handed sleptons. In Fig.~\ref{fig2}, the results are shown for $\epsilon_u=0.033$, $\epsilon_d=0.108$, $\epsilon_e=0.281$, $\sigma_u=0.001$, $\sigma_d=0.009$, and $\sigma_e=0.008$. The lighter squarks $\tilde{u_1}$ and $\tilde{u}_2$ are again the right-handed scharm and sup. Mixing between the second and third-generations for the left handed sparticles is observed in $\tilde{u}_4$ and $\tilde{u}_6$. There is also small but nonvanishing 2-3 generational mixing among right-handed up-type squarks. For the down sector, apart from the 1-2 and 2-3 generational mixing which are larger compared to the up sector, there is also a small but nonvanishing left-right mixing between $\tilde{b}_L$ and $\tilde{b}_R$ observed in $\tilde{d}_1$. For the sleptons, we again observe small mixing between the second and third-generation sleptons with the same handedness.
It is illustrative to compare this scenario with the singlet-dominated limit \cite{Everett_2019,Everett_2020}. In this case, the dominant contributions to the soft terms arise in the diagonal third-generation ($33$) entries, rendering this case similar to flavored gauge mediation models in which the Higgs-messenger mixing is controlled by Abelian symmetries.
Generally this case has a light spectrum, with masses below 6 TeV. Unlike the democratic case, the heavy Higgs particles are heavier than or comparable to the $SU(3)$-charged superpartners, with masses at the $5-6$ TeV range.
The large stop mixing due to the nonvanishing $A$ term for the third-generation fields at the messenger mass scale allows for a viable Higgs mass at smaller values of $\Lambda$ compared to the democratic limit, in which the $A$ terms vanish in the absence of the small symmetry breaking perturbations. Adding nonrenormalizable corrections as in \cite{Everett_2020} to generate the light quark and charged lepton masses does not alter this feature and generically leads to very small ($O(10^{-1} \; {\rm GeV})$)
\subsection{Case 2 models}
We now turn to the Case 2 models, for which the superpotential couplings only involving the $\mathcal{S}_3$-doublets dominate. As described in the previous section, in this case there are two sub-categories, depending on whether the $\beta_{3i}$ or the $\beta_{2i}$ parameters control the second-generation quark and charged lepton masses. Here we will label the mass ordering $\beta_{3i}>\beta_{2i}$ by Case 2a, and the alternate mass ordering $\beta_{2i}>\beta_{3i}$ as Case 2b. The soft supersymmetry breaking terms for Case 2a are given in Eq.~(\ref{eq:deltamB2-1}), and the analogous quantities for Case 2b are given in Eq.~(\ref{eq:deltamB2-3}).
\begin{figure}[h!]
\centering
\subfloat{{\includegraphics[scale=0.6]{images-caseb2/Julyb21e12tanbeta10alleps-1.pdf}}}
\subfloat{{\includegraphics[scale=0.6]{images-caseb2-beta2/Julyb21e12tanbeta10-beta2larger-1.pdf}}}
\caption{The sfermion mass spectra in the doublet-dominated (Case 2) limit, with $M_\text{mess}=10^{12}$ GeV, $\Lambda=6.6\times 10^5$ GeV, for (a) Case 2a $\beta_{3d}=0.03$, $\beta_{3u}=0.01$, $\beta_{3l}=0.08$, $\beta_{2u}=\beta_{2d}=\beta_{2l}=0$ (left), and (b) Case 2b with $\beta_{2d}=0.03$, $\beta_{2u}=0.01$, $\beta_{2l}=0.08$, $\beta_{3u}=\beta_{3d}=\beta_{3l}=0$ (right).}
\label{fig:b2spectra}
\end{figure}
In Case 2 models, there is a nonvanishing trilinear scalar parameter $A_t$ that is present in the absence of the first and second-generation quark and charged lepton masses, in contrast to the Case 1 democratic limit. Hence, the Higgs and superpartner masses are lighter than their Case 1 democratic counterparts, though not as light as in the Case 1 singlet-dominated limit.
In Fig.~\ref{fig:b2spectra}, we show characteristic mass spectra for $M_{\rm mess}=10^{12} \; {\rm GeV}$ and $\Lambda = 6.6\times 10^5$ GeV. Here we have included nonvanishing values for the parameters that fix the second-generation quark and charged lepton masses ($\beta_{3i}$ for Case 2a, $\beta_{2i}$ for Case 2b), and neglected the effects of the first-generation masses. The values of the perturbation parameters are chosen to yield appropriate values for the SM fermion mass values. We note here that if these quantities are taken to zero, the mass spectra are almost unchanged, with small changes that are at most $O(10^{-1}\; {\rm GeV})$, primarily in the slepton sector due to the relatively large value of the corresponding $\beta_{3l,2l}$ parameter.
We see that in both Case 2a and Case 2b, the gluino and squark masses are similar, with the gluino at about 8 TeV and the squarks ranging from approximately $8-10$ TeV. Unlike the Case 1 singlet-dominated limit as in Fig.~\ref{fig:democraticspectrum1} in which the squark masses are generally comparable to heavy Higgses, in this case the squarks are always much heavier than the heavy Higgs bosons. The slepton masses fall into two different ranges, with the NLSP as the lightest selectron $\tilde{e}_1$. The three lightest slectrons have their masses below 2 TeV, while the other sleptons have their masses between 3$-$4 TeV. The lightest charginos and neutralinos are gaugino-dominated, with a binolike lightest neutralino, while the heavier set is higgsino-dominated.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.9]{images-caseb2/septmixingcase2.pdf}
\caption{The sfermion mass spectrum in the doublet-dominated scenarios, with ordering $\beta_{3i}>\beta_{2i}$ (Case 2a) and $\beta_{2i}>\beta_{3i}$ (Case 2b), respectively, with $M_\text{mess}=10^{12}$ GeV and $\tan{\beta}=10$. For Case 2a, $\beta_{3d}=0.03$, $\beta_{3u}=0.01$, $\beta_{3l}=0.08$. For Case 2b, $\beta_{2d}=0.03$, $\beta_{2u}=0.01$, $\beta_{2l}=0.08$. }
\label{fig8}
\end{figure}
An intriguing difference between Case 2a and Case 2b is that in Case 2b, the heavy Higgs states and the heavy charginos and neutralinos are lighter than they are in Case 2a. For the model parameters as given in Fig.~\ref{fig:b2spectra}, we see that in Case 2b the heavy Higgs masses are in the 5-6 TeV range, while they are over 6 TeV in Case 2a, and the heavy charginos/neutralinos are also reduced by approximately 1 TeV in Case 2b compared to Case 2a. This indicates that in Case 2b, smaller values of the $\mu$ and $b$ parameters are needed for successful electroweak symmetry breaking.
Another significant difference between Case 2a and Case 2b is that Case 2a has nonvanishing off-diagonal contributions to squark mixing, as discussed in the previous subsection. The most significant off-diagonal sfermion mixing in Case 2a is given by $\vert (\delta_u^{12})_{LL}\vert \sim 1\times 10^{-4}$. These effects are small because the flavor off-diagonal contributions are proportional to the small quantities that govern the second-generation SM quark and charged lepton masses. In both cases, as shown in Fig.~\ref{fig8}, sfermion mixing is not significant due to the small size of the perturbation parameters. For larger values of the messenger mass scale, nontrivial left-right mixing is observed for the third-generation down-type squarks (left-right mixing in the other sfermion sectors is negligible for all values of the messenger mass scale).
\subsection{Discussion}
In comparing the mass spectra of these scenarios (Case 1: democratic, and Cases 2a and 2b: doublet-dominated, as well as the Case 1: singlet-dominated limit as studied in \cite{Everett_2018}), there are several features of interest.
For fixed $M_\text{mess}$, the mass spectra are more compressed for larger values of $\tan{\beta}$ ($\tan{\beta}>10$) because the contributions from the bottom and tau Yukawa couplings are more significant than in the low $\tan{\beta}$ regime. For smaller $\tan{\beta}$ values, the sparticle masses are heavier as the tree-level contribution to the light Higgs mass has decreased, requiring larger radiative corrections to boost its mass to its experimentally allowed range. The superpartner masses in this limit are thus highly split, with heavy squarks and gluinos, and lighter sleptons. For fixed $\tan\beta$ (here taken to be $\tan\beta=10$), lower values of the messenger mass scale generally lead to heavier spectra, as larger values of $\Lambda$ are needed to satisfy the light Higgs mass constraint. For higher messenger scales, due to increased renormalization group running effects, the $\mu$ and $b/\mu$ terms needed to satisfy the electroweak symmetry breaking constraints are smaller, and thus the heavy charginos and neutralinos become lighter.
\begin{figure}[t]
\centering
\subfloat{{\includegraphics[scale=0.45]{images-democratic/JuneShumMessVstanBoffHiggsContoursclear.pdf}}}
\subfloat{{\includegraphics[scale=0.45]{images-democratic/JuneShumMessVstanBHiggsContoursalleps.pdf}}}
\caption{(a) The Higgs mass (black band) and $\tilde{u}_3$ squark mass (color shading) for Case 1 in the democratic limit without perturbations, with
$\Lambda=7.7\times10^5$ GeV (left). (b) The same as (a), but with $\epsilon_u=0.033$, $\epsilon_d=0.108$, $\epsilon_e=0.281$, $\sigma_u=0.001$, $\sigma_d=0.009$, $\sigma_e=0.008$ (right). }
\label{figparameterspace}
\end{figure}
\begin{figure}[h!]
\centering
\subfloat{{\includegraphics[scale=0.35]{images-democratic/JuneShumMessVsLambdaoffHiggsContourstanBeta10.pdf}}}
\subfloat{{\includegraphics[scale=0.35]{images-democratic/JuneShumMessVsLambdaEps3HiggsContourstanBeta10.pdf}}}
\caption{(a) The Higgs mass (black band), gluino mass (color shading) and $\tilde{e}_1$ mass (dotted curves) as a function of $\Lambda$ and $M_\text{mess}$, for Case 1 in the democratic limit with $\tan\beta=10$, for (a) no perturbations (left), and (b) nonzero perturbations, with $\epsilon_u=0.033$, $\epsilon_d=0.108$, $\epsilon_e=0.281$, $\sigma_u=0.001$, $\sigma_d=0.009$, $\sigma_e=0.008$.}
\label{figparameterlambdammess}
\end{figure}
To further investigate the dependence of the mass spectra on $M_\text{mess}$ and $\tan{\beta}$, we show the Higgs mass curve for fixed $\Lambda$, with the color representing the mass of the lightest squark. We show these results in Fig.~\ref{figparameterspace} for the Case 1 democratic scenario, which show several excluded regions.
When $\epsilon_u=\epsilon_d=\epsilon_e=0$, the central big ``hole" appears because the mass-squared of the lightest slepton is negative.
When the $\epsilon$ parameters are nonzero, the size of the holes increases, and there are also small holes that appear above the central void because $A^0$ becomes tachyonic in those regions.
Quite generally, we see that the parameters that satisfy the Higgs mass constraint could be very different in these two cases.
The lightest value for the mass of the $\tilde{u}_3$ squark is in the region with high $\tan\beta$ and high messenger scales.
In Fig.~\ref{figparameterlambdammess}, we show the gluino and lightest slepton (NLSP) masses, both without perturbations (left panel) and with perturbations (right panel). The introduction of the perturbations pushes the slepton mass down to smaller values. The change in the shape of the viable Higgs mass region is even more apparent here. For low values of $M_{\rm mess}$, a higher value of $\Lambda$ is needed to satisfy the light Higgs mass constraint. For higher messenger scales $M_\text{mess}\sim 10^{14}-10^{16}$ GeV, there is a sharp drop in the Higgs mass region that is observed. In that region, there is generally larger left-right mixing in the sbottom sector as well larger scharm-stop mixing, which result in nontrivial contributions to the Higgs mass. However, there are potential numerical instabilities related to the challenges of the Higgs mass calculation in this parameter region. A detailed resolution of these issues is beyond the scope of this paper, and is deferred to future study.
\begin{figure}[h]
\centering
\subfloat{{\includegraphics[scale=0.45]{images-caseb2/Julyb2mMessVstanBHiggsContourslambda6e5beta.pdf}}}
\subfloat{{\includegraphics[scale=0.45]{images-caseb2-beta2/Julyb2mMessVstanBHiggsContourslambda6.3e5-beta2larger.pdf}}}
\caption{The mass of the Higgs (black band) and the mass of the lightest squark $\tilde{u}_3$ (color shading) for (a) Case 2a with fixed $\Lambda=6\times10^5$ GeV,
$\beta_{3d}=0.03$, $\beta_{3u}=0.01$, $\beta_{3l}=0.08$, $\beta_{2u}=\beta_{2d}=\beta_{2l}=0$ (left),
and (b) Case 2b with fixed $\Lambda=6.3\times10^5$ GeV,
$\beta_{2d}=0.03$, $\beta_{2u}=0.01$, $\beta_{2l}=0.08$, $\beta_{3u}=\beta_{3d}=\beta_{3l}=0$ (right).}
\label{figb2par1}
\end{figure}
\begin{figure}[h!]
\centering
\subfloat{{\includegraphics[scale=0.35]{images-caseb2/Julyb2MessVsLambdaHiggsContourstanBeta10beta.pdf}}}
\subfloat{{\includegraphics[scale=0.35]{images-caseb2-beta2/Julyb2MessVsLambdau3HiggsContourstanBeta10.pdf}}}
\caption{The light Higgs mass (black band), gluino mass (color shading), and $\tilde{e}_1$ mass (dotted curves) with $\Lambda=6\times 10^5$ GeV, for (a) Case 2a $\beta_{3d}=0.03$, $\beta_{3u}=0.01$, $\beta_{3l}=0.08$, $\beta_{2u}=\beta_{2d}=\beta_{2l}=0$ (left), and (b) Case 2b with $\beta_{2d}=0.03$, $\beta_{2u}=0.01$, $\beta_{2l}=0.08$, $\beta_{3u}=\beta_{3d}=\beta_{3l}=0$ (right).}
\label{figb2par2}
\end{figure}
For the Case 2 models, we also see excluded regions in the parameter scan in Fig.~\ref{figb2par1}. Here, for both Case 2a and 2b, $\Lambda$ is chosen to maximize viable parameter regions, and the perturbations have a minimal effect on the size of the void. In Case 2a, the void appears due to tachyonic slepton masses, and the phenomenologically viable parameter region generally lies between $\tan{\beta}\approx 5-15$, $M_\text{mess}=10^6-10^{18}$ GeV, with a fixed choice of $\Lambda=6\times10^5$ GeV. For Case 2b, apart from the central hole where the lightest slepton becomes tachyonic, the region on the left of the spectrum, which is from $M_\text{mess}\sim10^6-10^9$ GeV and $\tan{\beta}\sim 5-50$, is ruled out because the desired electroweak minimum is not present. We also see that in both cases, the viable Higgs region does not generally intersect with the region where $\tilde{e}_1$ is lighter. In Fig~\ref{figb2par2}, we fix $\tan{\beta}=10$ and show both Case 2a and Case 2b with nonzero perturbations. Note that the effects of the perturbations only slightly shift the mass curve of the NLSP upward, and have almost no effect on the viable Higgs mass region and the gluino masses, in contrast to what we have seen in the democratic limit.
We close this section by commenting on further phenomenological aspects of this set of models. For all cases described here (both democratic and doublet-dominated models), the superpartner masses are generally heavy and split, in a way that is reminiscent of minimal gauge mediation with $N=2$. As previously discussed, the constraints of the non-Abelian Higgs-messenger symmetry have led us to include at least two messenger pairs to avoid a catastrophic $\mu/B_\mu$ problem. Ultimately, this means that the scenarios studied in this paper have heavier and more split spectra than what can be obtained in Abelian flavored gauge mediation models (where a judicious choice of $U(1)$ charges can be made to avoid the $\mu/B_\mu$ issue seen here, without increasing the number of messenger pairs), such as in \cite{Ierushalmi_2016}, or in general gauge mediation scenarios \cite{Meade:2008wd}. We recall that in our scenario in the singlet-dominated limit as studied in \cite{Everett_2019,Everett_2020}, it is also possible to minimize the splitting of the mass spectra, though not to the extent that is possible in the Abelian flavored gauge mediation models.
As a result, the discovery potential for the scenarios studied here either via direct LHC searches or indirect constraints is not as promising as it can be in Abelian flavored gauge mediation models, or even in the singlet-dominated non-Abelian scenario. For example, it is straightforward to see that the supersymmetric contribution to muon anomalous magnetic moment (MDM) in the democratic and doublet-dominated non-Abelian flavored gauge mediation scenarios studied here is generically about two orders smaller than the current experimental value \cite{PhysRevLett.126.141801}.
This is both due to the heavy superpartner masses as described above, and that we are generally precluded from having large values of $\tan\beta$ in these scenarios, which usually provide the largest enhancement to the MDM. Therefore, if new physics is required to resolve any future confirmed discrepancy between the SM prediction and the measured value of the muon anomalous magnetic moment, this set of flavored gauge mediation models would need to be extended to accommodate the experimental result.
One notable difference in the non-Abelian flavored gauge mediation scenarios studied here compared to minimal gauge mediation with $N=2$ as well as the non-Abelian singlet-dominated flavored gauge mediation scenario is in regards to the NLSP composition. Here, for messenger mass scales of $10^{12}$ GeV as displayed in Fig.~\ref{fig:democraticspectrum1} and Fig.~\ref{fig:b2spectra}, the NLSP is the lightest slepton, which is a right-handed smuon. This is different from the minimal GMSB scenario and the singlet-dominated non-Abelian FGM scenario in which either staus or binolike neutralinos are the NLSP. In the scenarios studied in this paper, for this intermediate to high messenger scale, the smuon NLSP has a lifetime of $\mathcal{O}(0.001\,{\rm s})$, and the NLSP mass is generically close to about 2 TeV. This currently lies above the limits from direct production searches at $\sqrt{s}=13$ TeV \cite{CMS:2018eqb}.
For lower values of the messenger scale ($\sim 10^6$ GeV), the lightest slepton is still the NLSP, which has a very rapid decay to the gravitino due to the lower supersymmetry breaking scale, while for very high messenger scales ($\sim 10^{14}$ GeV), the NLSP is now a long-lived binolike neutralino. We also note that in the non-Abelian flavored gauge mediation scenarios studied here, there is no significant co-NLSP behavior, both in the low messenger scale and the intermediate to high messenger scale cases. This is in contrast to minimal $N=2$ GMSB for low messenger scales ($\sim 10^6$ GeV), for which there is appreciable co-NLSP behavior among the lighter sleptons for the binolike neutralino NLSP.
We also note that in both minimal $N=2$ GMSB and our non-Abelian flavored gauge-mediation scenarios, for messenger scales of $M_\text{mess}=10^{12}$ GeV, the gravitino has a mass of $\mathcal{O}(0.1\, \text{GeV})$, and the NLSP is not long-lived enough to decay during or after Big Bang nucleosynthesis (BBN). Therefore, the successful predictions of BBN will not be spoiled (see e.g.~\cite{Steffen:2006hw,Feng:2004zu,ASAKA2000136}). For gravitinos of this mass range, there are well-known mechanisms to ensure the desired reheating temperatures and late entropy production to avoid having the gravitinos overclose the universe, so that gravitinos can then be a plausible dark matter candidate. For lower values of the messenger scale, the situation is further improved, as the gravitinos are lighter (with masses of the order of tenths of keV for $M_{\rm mess}= 10^6$ GeV) and the NLSP decays to gravitinos much more rapidly than in the higher messenger scale case, thus avoiding the need for gravitino dilution.
\section{Conclusions}
In this paper, we have explored MSSM flavored gauge mediation models in which the Higgs-messenger mixing is controlled by a discrete non-Abelian symmetry, here taken for simplicity to be $\mathcal{S}_3$. Building on previous analyses \cite{Everett_2018} which showed that viable models can be constructed for an extended Higgs-messenger sector that includes both $\mathcal{S}_3$ doublet and singlet fields that mix to yield one light MSSM Higgs pair and two messenger pairs, we studied various possibilities for generating plausible SM quark and charged lepton masses in the case in which the MSSM matter fields also carry $\mathcal{S}_3$ quantum numbers. While additional relations beyond $\mathcal{S}_3$ are generically needed to obtain the desired hierarchical SM fermion masses, we have identified two general categories of solutions that we broadly categorized as Case 1 and Case 2 models. The Case 1 models obey Eq.~(\ref{eq:betarelations}), and encompass two regimes of interest: (i) the singlet-dominated limit, in which the Yukawa couplings involving only the $\mathcal{S}_3$ singlets dominate, and (ii) the democratic limit, in which the Yukawa superpotential for the MSSM fields has an enhanced $\mathcal{S}_{3L}\times \mathcal{S}_{3R}$ symmetry. The Case 2 models, in contrast, include the doublet-dominated limit, in which the Yukawa couplings involving only the $\mathcal{S}_3$ doublet fields dominate. The singlet-dominated limit was previously investigated in \cite{Everett_2019,Everett_2020} and served here as a point of comparison for a general analysis of the Case 1 democratic limit and the Case 2 doublet-dominated models. We include corrections to obtain nonvanishing masses for one or both of the lighter families, as well as for the third family. In certain cases such corrections lead to off-diagonal corrections to the soft supersymmetry breaking mass terms, but these corrections are relatively mild (a feature that is known in the literature for flavored gauge mediation models of this general type) and as a result, do not immediately lead to insurmountable problems with flavor-changing neutral current constraints.
Within Case 1 models, our analysis shows that while the singlet-dominated limit allows for examples with optimized parameter sets that yield gluino and squark masses in the 4-5 TeV range, the Case 1 democratic limit generically has significantly heavier squark and gluino masses. The Case 2 models generally also yield heavier superpartner masses, with the heavier squarks and gluino in the 7 TeV mass range. Ultimately, the fact that the squark and gluino masses cannot be made lighter than 4-5 TeV even in the singlet-dominated limit is related to the fact that this non-Abelian Higgs-messenger mixing scenario requires at the minimum two vectorlike messenger pairs that contribute to the loop diagrams that generate the corrections to the soft terms, to tune the $\mu$ and $b$ terms independently. This should be contrasted with Abelian models, which can have just one messenger pair, and as a result can lead to benchmark scenarios in flavored gauge mediation with lighter $SU(3)$-charged superpartners that are more accessible for searches for supersymmetry at present and future colliders.
While the spectra in all our examples remain quite heavy, and while we have not constructed fully realistic models of the SM fermion masses and mixing angles (including CP violating effects, not included here for simplicity), we nonetheless find it encouraging that this class of non-Abelian flavored gauge mediation models can include examples that survive this next level of model-building scrutiny. More work is of course needed to see if such scenarios (or plausible extensions of such scenarios) can be embedded into a more complete high-energy model. In the meantime, however, analyses such as this one can serve as a reminder of the rich framework of TeV-scale $\mathcal{N}=1$ supersymmetry, and the many ways in which it might still be hiding at or just above TeV energies. As the Terascale continues to be explored in this data-rich era for high energy physics, hopefully we will know relatively soon if TeV-scale supersymmetry is indeed part of our physical world.
\acknowledgments
This work is supported by the U.S. Department of Energy under the contract number DE-SC0017647.
\clearpage
\section*{Appendix}
\subsection{Case 1 models}
We present the corrections to the soft supersymmetry breaking terms in the Case 1 democratic limit. All relevant terms with magnitudes larger than the smallest perturbation parameter $\sigma_u$ are included. In this case all terms with coefficients of order $O(10^{-3})$GeV are taken into account. For notational simplicity, we define the following quantities:
\begin{equation}
\begin{split}
\tilde{g}_u^2&=\frac{16}{3}g_3^2+3g_2^2+\frac{13}{15}g_1^2,\quad \tilde{g}_d^2=\frac{16}{3}g_3^2+3g_2^2+\frac{7}{15}g_1^2,\quad \tilde{g}_e^2=3g_2^2+\frac{9}{5}g_1^2,\\
\delta_Q&=6y_t^4+6y_b^4+2y_b^2y_t^2+y_b^2y_\tau^2-\Tilde{g_u}^2y_t^2-\Tilde{g_d}^2y_b^2,\\
\delta_{\epsilon_{u}}&=8y_t^4+y_b^2y_t^2-\tilde{g_u}^2y_t^2, \quad
\delta_{\epsilon_d}=8y_b^4+y_b^2y_t^2+y_b^2y_\tau^2-\tilde{g_d}^2y_b^2,\\
\delta_{\epsilon_e}&=\frac{4}{81} \epsilon _e^2y_b^2 y_{\tau }^2-\frac{8}{729}\epsilon _e^3
y_b^2 y_{\tau }^2-\frac{244 }{6561}\epsilon _e^4 y_b^2 y_{\tau }^2, \quad
\delta_{L}=4y_\tau^4+3y_b^2y_\tau^2-\Tilde{g_e}^2y_\tau^2. \\
\end{split}
\label{eq:paramdefcase1}
\end{equation}
In what follows, all soft scalar mass-squared parameters are assumed to include a factor of $\Lambda^2/(4\pi)^4$ and all trilinear scalar couplings are assumed to include a factor of $\Lambda/(4\pi)^2$, where $\Lambda=F/M_{\rm mess}$.
The nonvanishing corrections to the soft supersymmetry breaking terms are as follows:
\begin{equation}
\begin{split}
\delta m_{Q_{11}}^2&=
\delta_Q-\frac{4}{9}\epsilon_u \delta_{\epsilon_u}-\frac{4}{9}\epsilon_d \delta_{\epsilon_d} +\frac{4}{27} \epsilon _d \epsilon _e y_b^2 y_{\tau }^2-\frac{82}{729} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2\\&\quad+\epsilon _d^2 \left(-\frac{26}{81} y_b^2
\tilde{g}_d^2+\frac{5}{27} y_b^2 y_t^2+\frac{26}{81} y_b^2 y_{\tau }^2+\frac{367 }{81}y_b^4\right)+\delta_{\epsilon_e},
\\
\delta m_{Q_{22}}^2&=
(\delta_Q+\frac{4}{9}\epsilon_u \delta_{\epsilon_u}+\frac{4}{9}\epsilon_d \delta_{\epsilon_d} +\frac{4}{81}\epsilon _d \epsilon _e y_b^2 y_{\tau }^2-\frac{2}{243} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2\\&\quad+\epsilon _d^2 \left(\frac{2}{3} y_b^2
\tilde{g}_d^2-\frac{88}{81} y_b^2 y_t^2-\frac{2}{3} y_b^2 y_{\tau }^2-\frac{289 }{81}y_b^4\right)+\delta_{\epsilon_e},
\\
\delta m_{Q_{23}}^2&=\delta m_{Q_{32}}^2=
\frac{4\sqrt{2}}{9}\epsilon_u(\delta_{\epsilon_u}-2y_t^4)
+\frac{4\sqrt{2}}{9}\epsilon_d\left(\delta_{\epsilon_{d}}-2y_b^4\right)-\frac{4\sqrt{2}}{81} \epsilon _d \epsilon _ey_b^2 y_{\tau }^2+\frac{38\sqrt{2}}{729} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2\\&\qquad\qquad\quad+\sqrt{2}\epsilon _d^2\left(\frac{14}{81} y_b^2 \tilde{g}_d^2 -\frac{25}{162} y_b^2 y_t^2-\frac{14}{81} y_b^2 y_{\tau }^2-\frac{59}{81} y_b^4 \right)
\\
\delta m_{Q_{12}}^2&=\delta m_{Q_{21}}^2=
\frac{4\sqrt{3}}{3} \sigma _d \left(8 y_b^4-\tilde{g}_d^2y_b^2 +2 y_t^2y_b^2 +y_b^2 y_{\tau }^2\right)+ \frac{4\sqrt{3}}{3} \sigma _u \left(8 y_t^4-\tilde{g}_u^2y_b^2 +2 y_t^2y_b^2 \right),
\\
\delta m_{Q_{33}}^2&=
\frac{8}{81} \epsilon _d\epsilon _e y_b^2y_{\tau }^2-\frac{44}{729} \epsilon _d\epsilon _e^2y_b^2 y_{\tau }^2+\epsilon _d^2 \left(-\frac{4}{9} y_b^2
\tilde{g}_d^2+\frac{17}{81} y_b^2 y_t^2+\frac{4}{9} y_b^2 y_{\tau }^2+2 y_b^4\right),
\\
\delta m_{Q_{13}}^2&= \delta m_{Q_{31}}^2=
\sqrt{\frac{2}{3}}\sigma _d \left(12 y_b^4-2 \tilde{g}_d^2y_b^2+2 y_t^2y_b^2+2y_b^2 y_{\tau }^2\right)+4\sqrt{\frac{2}{3}}\sigma_u \left(y_t^4+y_t^2y_b^2-\tilde{g}_u^2y_t^2 \right),
\\
\nonumber
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\delta m_{\bar{u}_{11}}^2&=
(2\delta_{\epsilon_u}-4y_t^4)-\frac{8}{9}\epsilon_u\delta_{\epsilon_u}-\frac{8}{27}\epsilon _d^2 y_b^2 y_t^2,
\\
\delta m_{\bar{u}_{12}}^2&=\delta m_{\bar{u}_{21}}^2=\frac{8}{\sqrt{3}}\sigma_u \delta_{\epsilon_u},\qquad \delta m_{\bar{u}_{13}}^2=\delta m_{\bar{u}_{31}}^2=8\sqrt{\frac{2}{3}}\sigma_u (\delta_{\epsilon_u}-2y_t^4),
\\
\delta m_{\bar{u}_{22}}^2&=
(2\delta_{\epsilon_u}-4y_t^4)+\frac{8}{9}\epsilon_u\delta_{\epsilon_u}-\frac{8}{27}\epsilon _d^2 y_b^2 y_t^2,
\\
\delta m_{\bar{u}_{23}}^2&=\delta m_{\bar{u}_{32}}^2=
\frac{8\sqrt{2}\epsilon_u}{9}\left( \delta_{\epsilon_u}-2y_t^4\right),
\\
\delta m_{\bar{u}_{33}}^2&=
-\frac{8}{9}\epsilon_d^2 y_t^2 y_b^2,
\\
\delta m_{\bar{d}_{11}}^2&=
(2\delta_{\epsilon_d}-4y_b^4)-\frac{8\epsilon_d}{9}\delta_{\epsilon_d}+ \frac{8}{27}\epsilon _d\epsilon _e y_b^2 y_{\tau }^2-\frac{164}{729}\epsilon _d\epsilon _e^2 y_b^2 y_{\tau }^2 \\&\quad+\epsilon _d^2 \left(-\frac{52}{81} y_b^2
\tilde{g}_d^2+\frac{52}{81} y_b^2 y_t^2+\frac{52}{81} y_b^2 y_{\tau }^2+\frac{680 }{81}y_b^4\right)+2 \delta_{\epsilon_e},
\\
\delta m_{\bar{d}_{22}}^2&=
(2\delta_{\epsilon_d}-4y_b^4)+\frac{8\epsilon_d}{9}\delta_{\epsilon_d}+\frac{8}{81} \epsilon _d \epsilon _e y_b^2y_{\tau }^2-\frac{4}{243}\epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2\\&\quad+\epsilon _d^2 \left(\frac{3}{4} y_b^2
\tilde{g}_d^2-\frac{116}{81} y_b^2 y_t^2-\frac{4}{3} y_b^2 y_{\tau }^2-\frac{592 }{81}y_b^4\right)+2\delta_{\epsilon_e},
\\
\delta m_{\bar{d}_{23}}^2&=\delta m_{\bar{d}_{32}}^2=
\frac{8\sqrt{2}}{9}\epsilon_d\delta_{\epsilon_d}-\frac{8\sqrt{2} }{81}\epsilon _d\epsilon _e y_b^2 y_{\tau }^2+ \frac{76\sqrt{2}}{729}\epsilon _d\epsilon _e^2 y_b^2 y_{\tau }^2\\&\quad\qquad\qquad+\epsilon _d^2
\left(\frac{28\sqrt{2}}{81} y_b^2 \tilde{g}_d^2-\frac{28\sqrt{2}}{81} y_b^2 y_t^2-\frac{28 \sqrt{2}}{81} y_b^2 y_{\tau }^2-\frac{128 \sqrt{2}}{81} y_b^4\right),
\\
\delta m_{\bar{d}_{33}}^2&=
\frac{16}{81}\epsilon _d \epsilon _e y_b^2 y_{\tau }^2-\frac{88}{729} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2+\epsilon _d^2 \left(-\frac{9}{8} y_b^2
\tilde{g}_d^2+\frac{8}{9} y_b^2 y_t^2+\frac{8}{9} y_b^2 y_{\tau }^2+\frac{392 }{81}y_b^4\right),
\\
\delta m_{\bar{d}_{12}}^2&=\delta m_{\bar{d}_{12}}^2=
\sigma _d \left(-\frac{3}{8\sqrt{3}} y_b^2 \tilde{g}_d^2+\frac{8 y_b^2 y_t^2}{\sqrt{3}}+\frac{8 y_b^2 y_{\tau }^2}{\sqrt{3}}+\frac{64 y_b^4}{\sqrt{3}}\right),
\\
\delta m_{\bar{d}_{13}}^2&= \delta m_{\bar{d}_{31}}^2=
\sigma _d \left(-\frac{1}{4} \sqrt{\frac{3}{2}} y_b^2 \tilde{g}_d^2+4 \sqrt{\frac{2}{3}} y_b^2 y_t^2+4 \sqrt{\frac{2}{3}} y_b^2 y_{\tau }^2+8 \sqrt{6} y_b^4\right),
\\
\delta m_{L_{11}}^2&=
\delta_L-\frac{4\epsilon_e}{9}\left(\delta_L+2y_\tau^4\right)+ \frac{4}{9} \epsilon _d \epsilon _e y_b^2 y_{\tau }^2-\frac{22}{81} \epsilon _d\epsilon _e^2 y_b^2 y_{\tau }^2+\frac{4}{27}\epsilon _d^2 y_b^2 y_{\tau
}^2\\&\quad+\epsilon _e^2 \left(\frac{26}{27} y_b^2 y_{\tau
}^2-\frac{81}{26} \tilde{g}_e^2 y_{\tau }^2+\frac{283 }{81}y_{\tau }^4\right)+\epsilon _e^3
\left(-\frac{64}{243} y_b^2 y_{\tau }^2+\frac{729}{64} \tilde{g}_e^2 y_{\tau }^2-\frac{1778}{729} y_{\tau }^4\right)\\&\quad+\epsilon _e^4 \left(-\frac{238}{729} y_b^2 y_{\tau }^2+\frac{2187}{238} \tilde{g}_e^2 y_{\tau }^2+\frac{862 }{2187}y_{\tau }^4\right),
\\
\delta m_{L_{22}}^2&=
\delta_L+\frac{4\epsilon_e}{9}\left(\delta_L+2y_\tau^4\right)+\frac{4}{27} \epsilon _d \epsilon _e y_b^2 y_{\tau }^2+\frac{10}{243} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2+\frac{4}{27} \epsilon _d^2 y_b^2 y_{\tau
}^2\\&\quad+\epsilon _e^2 \left(-2 y_b^2 y_{\tau
}^2+\frac{3}{2} \tilde{g}_e^2 y_{\tau }^2-\frac{197}{81} y_{\tau }^4\right)+\epsilon _e^3
\left(\frac{424}{243} y_b^2 y_{\tau }^2-\frac{729}{424} \tilde{g}_e^2 y_{\tau }^2+\frac{370 }{243}y_{\tau }^4\right)\\&\quad+\epsilon _e^4 \left(-\frac{1642 }{2187}y_b^2 y_{\tau }^2+\frac{6561 }{1642}\tilde{g}_e^2 y_{\tau }^2-\frac{1832 }{6561}y_{\tau }^4\right),
\\
\delta m_{L_{23}}^2&=\delta m_{L_{32}}^2=
\frac{4\sqrt{2}}{9}\epsilon_e\delta_L -\frac{4}{27} \sqrt{2}\epsilon _d \epsilon _e y_b^2 y_{\tau }^2 +\frac{14}{243} \sqrt{2} \epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2 \\\quad\qquad\qquad& +\epsilon _e^2
\left(-\frac{14}{27} \sqrt{2} y_b^2 y_{\tau }^2+\frac{81 }{7 \sqrt{2}} \tilde{g}_e^2 y_{\tau }^2-\frac{23}{81} \sqrt{2} y_{\tau }^4\right)+\epsilon _e^3
\left(-\frac{4}{9} \sqrt{2} y_b^2 y_{\tau }^2+\frac{27 }{2 \sqrt{2}}\tilde{g}_e^2 y_{\tau }^2-\frac{530}{729} \sqrt{2} y_{\tau }^4\right)\\\quad\qquad\qquad&+\epsilon _e^4
\left(\frac{2402 \sqrt{2}}{2187} y_b^2 y_{\tau }^2-\frac{6561 }{1201 \sqrt{2}}\tilde{g}_e^2 y_{\tau }^2+\frac{7660 \sqrt{2} }{6561}y_{\tau }^4\right)
\\
\nonumber
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\delta m_{L_{33}}^2&=
\frac{8}{27} \epsilon _d\epsilon _e y_b^2 y_{\tau }^2-\frac{76}{243} \epsilon _d\epsilon _e^2 y_b^2 y_{\tau }^2 \\&\quad +\epsilon _e^2 \left(\frac{4}{3} y_b^2 y_{\tau }^2-\frac{9}{4} \tilde{g}_e^2 y_{\tau }^2+\frac{74 }{81}y_{\tau
}^4\right)+\epsilon _e^3 \left(-\frac{376}{243} y_b^2 y_{\tau }^2+\frac{729}{376}
\tilde{g}_e^2 y_{\tau }^2-\frac{244 }{243}y_{\tau }^4\right) \\&\quad+\epsilon _e^4 \left(\frac{1868 }{2187}y_b^2 y_{\tau
}^2-\frac{6561 }{1868}\tilde{g}_e^2 y_{\tau }^2+\frac{436 }{729}y_{\tau }^4\right),
\\
\delta m_{L_{12}}^2&=\delta m_{L_{21}}^2=
\sigma _e \left(4 \sqrt{3} y_b^2 y_{\tau }^2-\frac{3}{4} \sqrt{3} \tilde{g}_e^2 y_{\tau }^2+8 \sqrt{3} y_{\tau }^4\right),
\\
\delta m_{L_{13}}^2&=\delta m_{L_{31}}^2=
\sigma _e \left(2 \sqrt{6} y_b^2 y_{\tau }^2-3 \sqrt{\frac{3}{2}} \tilde{g}_e^2 y_{\tau }^2+8 \sqrt{\frac{2}{3}} y_{\tau }^4\right),
\\
\delta m_{\bar{e}_{11}}^2&=
2\delta_L-\frac{8\epsilon_e}{9}\left(\delta_L+2y_\tau^4\right)+\frac{8}{9} \epsilon _d \epsilon _e y_b^2 y_{\tau }^2-\frac{44}{81} \epsilon _d \epsilon _e^2y_b^2 y_{\tau }^2+\frac{8}{27} y_b^2 \epsilon _d^2 y_{\tau
}^2 \\&\quad +\epsilon _e^2 \left(\frac{52}{27} y_b^2
y_{\tau }^2-\frac{81}{52} \tilde{g}_e^2 y_{\tau }^2+\frac{512 }{81}y_{\tau }^4\right)+\epsilon _e^3
\left(-\frac{128}{243} y_b^2 y_{\tau }^2+\frac{729}{128} \tilde{g}_e^2 y_{\tau }^2-\frac{3016 }{729}y_{\tau }^4\right) \\&\quad +\epsilon _e^4 \left(-\frac{476}{729} y_b^2 y_{\tau }^2+\frac{2187}{476} \tilde{g}_e^2 y_{\tau }^2+\frac{914 }{2187}y_{\tau }^4\right),
\\
\delta m_{\bar{e}_{22}}^2&=
2\delta_L+\frac{8\epsilon_e}{9}\left(\delta_L+2y_\tau^4\right)+\frac{8}{27}\epsilon _d \epsilon _e y_b^2 y_{\tau }^2+\frac{20}{243}\epsilon _d \epsilon _e^2 y_b^2 y_{\tau }^2+\frac{8}{27} \epsilon _d^2 y_b^2 y_{\tau
}^2 \\&\quad +\epsilon _e^2 \left(-4 y_b^2 y_{\tau
}^2+\frac{3}{4} \tilde{g}_e^2 y_{\tau }^2-\frac{136 }{27}y_{\tau }^4\right)+\epsilon _e^3
\left(\frac{848}{243} y_b^2 y_{\tau }^2-\frac{729}{848} \tilde{g}_e^2 y_{\tau }^2+\frac{776}{243} y_{\tau }^4\right)\\&\quad+\epsilon _e^4 \left(-\frac{3284 }{2187}y_b^2 y_{\tau }^2+\frac{6561 }{3284}\tilde{g}_e^2 y_{\tau }^2-\frac{1138 }{2187}y_{\tau }^4\right),
\\
\delta m_{\bar{e}_{23}}^2&=\delta m_{\bar{e}_{32}}^2=
\frac{8\sqrt{2}}{9}\epsilon_e\delta_L -\frac{8}{27}\epsilon _d\epsilon _e \sqrt{2} y_b^2 y_{\tau }^2+ \frac{28}{243} \sqrt{2} \epsilon _d\epsilon _e^2y_b^2 y_{\tau }^2 \\&\quad+\epsilon _e^2
\left(-\frac{28}{27} \sqrt{2} y_b^2 y_{\tau }^2+\frac{81 }{14 \sqrt{2}}\tilde{g}_e^2 y_{\tau }^2-\frac{56}{81} \sqrt{2} y_{\tau }^4\right)+\epsilon _e^3
\left(-\frac{8}{9} \sqrt{2} y_b^2 y_{\tau }^2+\frac{27 }{4 \sqrt{2}}\tilde{g}_e^2 y_{\tau }^2-\frac{1096}{729} \sqrt{2} y_{\tau }^4\right) \\&\quad+\epsilon _e^4
\left(\frac{4804 \sqrt{2}}{2187} y_b^2 y_{\tau }^2-\frac{6561 }{2402 \sqrt{2}}\tilde{g}_e^2 y_{\tau }^2+\frac{16918 \sqrt{2} }{6561}y_{\tau }^4\right)
\\
\delta m_{\bar{e}_{33}}^2&=
\frac{16}{27}\epsilon _d\epsilon _e y_b^2 y_{\tau }^2-\frac{152}{243} \epsilon _d\epsilon _e^2 y_b^2 y_{\tau }^2 +\epsilon _e^2 \left(\frac{8}{3} y_b^2 y_{\tau }^2-\frac{9}{8} \tilde{g}_e^2 y_{\tau
}^2+\frac{8 }{3}y_{\tau }^4\right)\\&\quad +\epsilon _e^3 \left(-\frac{752}{243} y_b^2 y_{\tau
}^2+\frac{81}{28} \tilde{g}_e^2 y_{\tau }^2-\frac{704 }{243}y_{\tau }^4\right)+ \epsilon _e^4 \left(\frac{3736 }{2187}y_b^2
y_{\tau }^2-\frac{6561}{3736} \tilde{g}_e^2 y_{\tau }^2+\frac{10028 }{6561}y_{\tau }^4\right),
\\
\delta m_{\bar{e}_{12}}^2&=\delta m_{\bar{e}_{21}}^2=
\sigma _e \left(8 \sqrt{3} y_b^2 y_{\tau }^2-\frac{3}{8} \sqrt{3} \tilde{g}_e^2 y_{\tau }^2+16 \sqrt{3} y_{\tau }^4\right),
\\
\delta m_{\bar{e}_{13}}^2&=\delta m_{\bar{e}_{31}}^2=
\sigma _e \left(4 \sqrt{6} y_b^2 y_{\tau }^2-\frac{3}{2} \sqrt{\frac{3}{2}} \tilde{g}_e^2 y_{\tau }^2+16 \sqrt{\frac{2}{3}} y_{\tau }^4\right),
\\
\delta m_{H_u}^2&=
-\frac{4}{3}\epsilon _d^2 y_b^2 y_t^2,
\\
\delta m_{H_d}^2&=
\epsilon _d^2 \left(-\frac{4}{27} y_b^2 y_t^2-\frac{40 }{9}y_b^4\right)-\frac{40}{27} \epsilon _e^2 y_{\tau }^4+\frac{368}{243} \epsilon _e^3 y_{\tau
}^4-\frac{1376}{2187} \epsilon _e^4 y_{\tau }^4, \nonumber
\end{split}
\end{equation}
\begin{equation}
\begin{split}
(\Tilde{A_u})_{13}&=2\sqrt{\frac{2}{3}}\sigma_d y_t y_b^2+4\sqrt{\frac{2}{3}}\sigma_uy_t^3,\qquad\qquad\qquad\quad\;\; (\Tilde{A_u})_{31}=8\sqrt{\frac{2}{3}}\sigma_uy_t^3,\\
(\Tilde{A_u})_{22}&= -\frac{2\epsilon_u}{9}(3y_t^3+y_ty_b^2),\qquad\qquad\qquad\qquad\qquad\;\; (\Tilde{A_u})_{33}= \frac{4}{9}\epsilon_d^2 y_t y_b^2,\\
(\Tilde{A_u})_{23}&= \frac{4\sqrt{2}}{9}\epsilon_uy_t^3+\frac{4\sqrt{2}}{9}\epsilon_dy_ty_b^2-\frac{14\sqrt{2}}{81}\epsilon_d^2 y_ty_b^2,\qquad (\Tilde{A_u})_{32}=\frac{8\sqrt{2}\epsilon_u}{9}y_t^3,\\
(\Tilde{A_d})_{13}&=2\sqrt{\frac{2}{3}}\sigma_d y_b^3+4\sqrt{\frac{2}{3}}\sigma_uy_by_t^2,\qquad\qquad\qquad\quad\; (\Tilde{A_d})_{31}= 4\sqrt{\frac{2}{3}}\sigma_d y_b^3,\\
(\Tilde{A_d})_{22}&= -\frac{2\epsilon_d}{9}(3y_b^3+y_t^2y_b)-\frac{2\epsilon_d^2}{81}(9y_b^3-y_t^2y_b),\qquad (\Tilde{A_d})_{23}=\frac{4\sqrt{2}\epsilon_d}{9}y_b^3+\frac{4\sqrt{2}\epsilon_u}{9}y_t^2y_b-\frac{10\sqrt{2}}{27}\epsilon_d^2 y_b^3,\\
(\Tilde{A_d})_{32}&=\frac{8\sqrt{2}\epsilon_d}{9}y_b^3-\frac{4\sqrt{2}}{9}\epsilon_d^2 y_b^3,\qquad\qquad\qquad\qquad\;\;\; (\Tilde{A_d})_{33}=\frac{4}{3}\epsilon_d^2 y_b^3,\\
(\Tilde{A_e})_{13}&=2\sqrt{\frac{2}{3}}\sigma_e y_\tau^3,\qquad (\Tilde{A_e})_{31}= 4\sqrt{\frac{2}{3}}\sigma_e y_\tau^3,
\\
(\Tilde{A_e})_{22}&=-\frac{2}{3}\epsilon_ey_\tau^3-\frac{2}{9} \epsilon _e^2 y_{\tau }^3 +\frac{178}{243} \epsilon _e^3 y_{\tau }^3 -\frac{382}{729} \epsilon _e^4 y_{\tau }^3,
\\
(\Tilde{A_e})_{23}&=\frac{4\sqrt{2}}{9}\epsilon_ey_\tau^3-\frac{10}{27} \sqrt{2} \epsilon _e^2 y_{\tau }^3-\frac{4}{81} \sqrt{2} \epsilon _e^3 y_{\tau }^3+\frac{3274 \sqrt{2} }{6561}\epsilon _e^4 y_{\tau }^3,
\\
(\Tilde{A_e})_{32}&=\frac{8\sqrt{2}}{9}\epsilon_e y_\tau^3 -\frac{4}{9} \sqrt{2} \epsilon _e^2 y_{\tau }^3-\frac{20}{81} \sqrt{2} \epsilon _e^3 y_{\tau }^3+\frac{5240 \sqrt{2} }{6561}\epsilon _e^4 y_{\tau }^3,
\\
(\Tilde{A_e})_{33}&=\frac{4}{3} \epsilon _e^2 y_{\tau }^3-\frac{376}{243} \epsilon _e^3 y_{\tau }^3+\frac{1868 }{2187}\epsilon _e^4 y_{\tau }^3.
\\
\end{split}
\end{equation}
\subsection{Case 2 models}
\noindent $\bullet$ {\it Ordering $\beta_{3i}>\beta_{2i}$}.
As before, all soft scalar mass-squared parameters are assumed to include a factor of $\Lambda^2/(4\pi)^4$, all trilinear scalar couplings are assumed to include a factor of $\Lambda/(4\pi)^2$, and we define $g_{\tilde{u}}^2 = \tilde{g}_u^2/2$, $g_{\tilde{d}}^2 = \tilde{g}_d^2/2$, and $g_{\tilde{l}}^2 = \tilde{g}_e^2/2$ (see Eq.~(\ref{eq:paramdefcase1})).
Including all the relevant terms up to second order in $\beta_{3u}$, the nonvanishing corrections to the soft supersymmetry breaking terms take the form:
\begin{eqnarray}
(\delta m_{\tilde{Q}}^2)_{11}&=&\frac{195}{16}(y_t^4+y_b^4)+\frac{15}{4}y_t^2 y_b^2+\frac{27}{16}y_b^2 y_\tau^2 - 3y_t^2 g_{\tilde{u}}^2-3y_b^2 g_{\tilde{d}}^2+\frac{9}{16}\beta_{3d}^2y_b^2(6y_b^2+y_t^2)+\frac{9}{16}\beta_{3l}^2y_\tau^2y_b^2 \nonumber \\
&&+ \frac{9}{16}\beta_{3u}^2 y_t^2 (6y_t^2+y_b^2), \nonumber\\
(\delta m_{\tilde{Q}}^2)_{22}&=& \beta_{3d}^2\left( -2g_{\tilde{d}}^2 y_b^2 + \frac{21}{8}y_b^4+\frac{9}{16}y_b^2y_t^2+\frac{5}{8}y_b^2y_\tau^2\right)+ \beta_{3u}^2\left(-2g_{\tilde{u}}^2y_t^2 + \frac{21}{8}y_t^4 + \frac{9}{16}y_b^2y_t^2 \right)+ \frac{3}{8}\beta_{3u}\beta_{3d}y_b^2y_t^2, \nonumber \\
(\delta m_{\tilde{Q}}^2)_{33}&=&\frac{39}{16}(y_t^4+y_b^4)+\frac{5}{4}y_t^2 y_b^2+\frac{11}{16}y_b^2 y_\tau^2 - y_t^2 g_{\tilde{u}}^2-y_b^2 g_{\tilde{d}}^2 -\frac{3}{4}\beta_{3d}^2y_b^4 + \frac{1}{16}\beta_{3l}^2y_b^2y_\tau^2-\frac{3}{4}\beta_{3u}^2y_t^4+\frac{3}{8}\beta_{3u}\beta_{3d}y_b^2y_t^2, \nonumber \\
(\delta m_{\tilde{Q}}^2)_{12}&=&(\delta m_{\tilde{Q}}^2)_{21}=-\frac{3}{4\sqrt{2}}(y_t^4\beta_{3u}+y_b^4 \beta_{3d}-y_t^2 y_b^2 (\beta_{3u}+\beta_{3d})), \nonumber \\
\end{eqnarray}
\begin{eqnarray}
(\delta m_{\tilde{u}}^2)_{22}&=&\frac{189}{8}y_t^4+\frac{9}{2}y_t^2 y_b^2 - 6y_t^2 g_{\tilde{u}}^2+\beta_{3u}^2\left( -g_{\tilde{u}}^2 y_t^2 + \frac{27}{4}y_t^4\right),
\nonumber \\
(\delta m_{\tilde{u}}^2)_{33}&=&
\frac{45}{8}y_t^4+\frac{1}{2}y_t^2 y_b^2- 2y_t^2 g_{\tilde{u}}^2+\beta_{3u}^2\left( -3g_{\tilde{u}}^2 y_t^2 + \frac{27}{4}y_t^4\right), \nonumber \\
(\delta m_{\tilde{d}}^2)_{22}&=&\frac{189}{8}y_b^4+\frac{9}{2}y_t^2 y_b^2+\frac{27}{8}y_b^2y_\tau^2 - 6y_b^2 g_{\tilde{d}}^2 +\beta_{3d}^2y_b^2 \left(-g_{\tilde{d}}^2+\frac{27}{4}y_b^2+\frac{1}{8}y_\tau^2\right)+\frac{9}{8}\beta_{3l}^2y_b^2y_\tau^2,\nonumber \\
(\delta m_{\tilde{d}}^2)_{33} &=&
\frac{45}{8}y_b^4+\frac{1}{2}y_t^2 y_b^2+\frac{11}{8}y_b^2y_\tau^2- 2y_b^2 g_{\tilde{d}}^2+\beta_{3d}^2y_b^2 \left(-\frac{1}{3}g_{\tilde{d}}^2+\frac{27}{4}y_b^2+\frac{9}{8}y_\tau^2\right)+\frac{1}{8}\beta_{3l}^2y_b^2y_\tau^2, \nonumber \\
(\delta m_{\tilde{L}}^2)_{11}&=&\frac{141}{16}y_\tau^4+\frac{81}{16}y_b^2 y_\tau^2 - 3y_\tau^2 g_{\tilde{l}}^2 +\frac{27}{16}\beta_{3d}^2y_b^2y_\tau^2 + \frac{9}{4}\beta_{3l}^2y_\tau^4,
\nonumber \\
(\delta m_{\tilde{L}}^2)_{22}&=& \beta_{3l}^2 \left(-2g_{\tilde{l}}^2y_\tau^2 +\frac{15}{8} y_b^2 y_\tau^2 +\frac{11}{8}y_\tau^4 \right), \nonumber\\
(\delta m_{\tilde{L}}^2)_{33}&=&
\frac{17}{16}y_\tau^4+\frac{33}{16}y_b^2 y_\tau^2 - y_\tau^2 g_{\tilde{l}}^2+\frac{3}{16}\beta_{3d}^2y_b^2y_\tau^2 - \frac{7}{8}\beta_{3l}^2y_\tau^4, \nonumber \\
(\delta m_{\tilde{L}}^2)_{12} &=&
(\delta m_{\tilde{L}}^2)_{21}=-\frac{3}{4\sqrt{2}}(y_\tau^4\beta_{3l})+\frac{3}{8\sqrt{2}}(\beta_{3l}^3 y_\tau^4), \nonumber \\
(\delta m_{\tilde{e}}^2)_{22}&=&\frac{135}{8}y_\tau^4+\frac{81}{8}y_b^2y_\tau^2 - 6y_\tau^2 g_{\tilde{l}}^2+\frac{27}{8}\beta_{3d}^2y_b^2y_\tau^2+\beta_{3l}^2\left(-g_{\tilde{l}}^2y_\tau^2 +\frac{3}{8}y_b^2y_\tau^2 +\frac{17}{4} y_\tau^4 \right), \qquad
\nonumber \\
(\delta m_{\tilde{e}}^2)_{33}
&=&
\frac{23}{8}y_\tau^4+\frac{33}{8}y_b^2 y_\tau^2- 2y_\tau^2 g_{\tilde{l}}^2+\frac{3}{8}\beta_{3d}^2y_b^2y_\tau^2+\beta_{3l}^2\left(-3g_{\tilde{l}}^2y_\tau^2 +\frac{27}{8}y_b^2y_\tau^2 +\frac{17}{4} y_\tau^4 \right), \nonumber\\
\delta m_{H_u}^2&=&-\frac{9}{2}y_t^4-\frac{3}{2}y_t^2y_b^2-9\beta_{3u}^2y_t^4,\nonumber \\
\delta m_{H_d}^2&=&-\frac{9}{2}y_b^4-\frac{3}{2}y_\tau^4-\frac{3}{2}y_t^2y_b^2-9\beta_{3d}^2y_b^4-3\beta_{3l}^2y_\tau^4, \nonumber\\
(\tilde{A}_u)_{22}&=&-\frac{3}{\sqrt{2}}y_t^3\beta_{3u}, \qquad\qquad\qquad\qquad (\tilde{A}_u)_{33}=-\frac{3}{2}y_t^3-\frac{1}{2}y_t y_b^2-\frac{3}{2}\beta_{3u}^2y_t^3 , \nonumber \\
(\tilde{A}_d)_{22}&=&-\frac{3}{\sqrt{2}}y_b^3\beta_{3d}, \qquad\qquad\qquad\qquad (\tilde{A}_d)_{33}=-\frac{3}{2}y_b^3-\frac{1}{2}y_b y_t^2-\frac{3}{2}\beta_{3d}^2y_b^3, \nonumber \\
(\tilde{A}_e)_{22}&=&-\frac{3}{\sqrt{2}}y_\tau^3\beta_{3l}-\frac{3}{2\sqrt{2}}y_\tau^3\beta_{3l}^3, \qquad\;\; (\tilde{A}_e)_{33}=-\frac{3}{2}y_\tau^3-\frac{3}{2}y_\tau^3\beta_{3l}^2.
\label{eq:deltamB2-1}
\end{eqnarray}
\clearpage
\noindent $\bullet$ {\it Ordering $\beta_{2i}>\beta_{3i}$}.
Assuming the subleading $\beta_{3i}=0$, we find the following corrections to the soft terms up to second order in $\beta_{2i}$:
\begin{eqnarray}
(\delta m_{\tilde{Q}}^2)_{22}&=&\frac{195}{16}(y_t^4+y_b^4)+\frac{15}{4}y_t^2 y_b^2+\frac{27}{16}y_b^2 y_\tau^2 - 3y_t^2 g_{\tilde{u}}^2-3y_b^2 g_{\tilde{d}}^2 + \beta_{2d}^2 y_b^2 \nonumber
\left(-\frac{1}{2}g_{\tilde{d}}^2+\frac{27}{8}y_b^2+\frac{3}{16}y_t^2+\frac{1}{16}y_\tau^2 \right)
\\&&+\frac{9}{16}\beta_{2l}^2y_b^2y_\tau^2 + \beta_{2u}^2y_t^2 \left( -\frac{1}{2}g_{\tilde{u}}^2+\frac{27}{8}y_t^2+\frac{3}{16}y_b^2\right)-\frac{3}{8}\beta_{2u}\beta_{2d}y_b^2y_t^2, \nonumber \\
(\delta m_{\tilde{Q}}^2)_{33}&=&\frac{39}{16}(y_t^4+y_b^4)+\frac{5}{4}y_t^2 y_b^2+\frac{11}{16}y_b^2 y_\tau^2 - y_t^2 g_{\tilde{u}}^2-y_b^2 g_{\tilde{d}}^2+ \beta_{2d}^2 y_b^2 \left(-\frac{3}{2}g_{\tilde{d}}^2+\frac{27}{8}y_b^2+\frac{3}{16}y_t^2+\frac{9}{16}y_\tau^2 \right) \nonumber \\
&& +\beta_{2u}^2 y_t^2 \left(-\frac{3}{2}g_{\tilde{u}}^2+\frac{27}{8}y_t^2+\frac{3}{16}y_b^2 \right)-\frac{3}{8}\beta_{2u}\beta_{2d}y_b^2y_t^2, \nonumber\\
(\delta m_{\tilde{u}}^2)_{11}&=&\frac{189}{8}y_t^4+\frac{9}{2}y_t^2 y_b^2 - 6y_t^2 g_{\tilde{u}}^2+\frac{9}{4}\beta_{2d}^2y_b^2y_t^2+\frac{45}{8}\beta_{2u}^2 y_t^4,\nonumber \\
(\delta m_{\tilde{u}}^2)_{22}&=&\beta_{2u}^2 \left (-4g_{\tilde{u}}^2y_t^2+\frac{3}{2}y_b^2y_t^2 + \frac{21}{4}y_t^4 \right),\nonumber \\
(\delta m_{\tilde{u}}^2)_{33}
&=&
\frac{45}{8}y_t^4+\frac{1}{2}y_t^2 y_b^2- 2y_t^2 g_{\tilde{u}}^2 -\frac{3}{4}\beta_{2d}^2y_b^2y_t^2-\frac{3}{8}\beta_{2u}^2y_t^4, \nonumber \\
(\delta m_{\tilde{d}}^2)_{11}&=&\frac{189}{8}y_b^4+\frac{9}{2}y_t^2 y_b^2+\frac{27}{8}y_b^2y_\tau^2 - 6y_b^2 g_{\tilde{d}}^2+\frac{45}{8}\beta_{2d}^2y_b^4+\frac{9}{8}\beta_{2l}^2y_b^2y_\tau^2 + \frac{9}{4}\beta_{2u}^2y_b^2y_t^2,\nonumber \\
(\delta m_{\tilde{d}}^2)_{22} &=& \beta_{2d}^2y_b^2 \left(-\frac{1}{4}g_{\tilde{d}}^2+\frac{21}{4}y_b^2+\frac{3}{2}y_t^2+\frac{5}{4}y_{\tau}^2 \right), \nonumber
\\
(\delta m_{\tilde{d}}^2)_{33} &=&
\frac{45}{8}y_b^4+\frac{1}{2}y_t^2 y_b^2+\frac{11}{8}y_b^2y_\tau^2- 2y_b^2 g_{\tilde{d}}^2-\frac{3}{8}\beta_{2d}^2y_b^4+\frac{1}{8}\beta_{2l}^2y_b^2y_\tau^2- \frac{3}{4}\beta_{2u}^2y_b^2y_t^2, \nonumber \\
(\delta m_{\tilde{L}}^2)_{22}&=&\frac{141}{16}y_\tau^4+\frac{81}{16}y_b^2 y_\tau^2 - 3y_\tau^2 g_{\tilde{l}}^2 +\frac{27}{16}\beta_{2d}^2y_b^2y_\tau^2 + \beta_{2l}^2\left(-\frac{1}{2}g_{\tilde{l}}^2y_\tau^2 + \frac{3}{16}y_b^2y_\tau^2+\frac{17}{8}y_\tau^4\right),
\nonumber \\
(\delta m_{\tilde{L}}^2)_{33}&=&
\frac{17}{16}y_\tau^4+\frac{33}{16}y_b^2 y_\tau^2 - y_\tau^2 g_{\tilde{l}}^2+\frac{3}{16}\beta_{2d}^2y_b^2y_\tau^2+ \beta_{2l}^2\left(-\frac{3}{2}g_{\tilde{l}}^2y_\tau^2 + \frac{27}{16}y_b^2y_\tau^2+\frac{17}{8}y_\tau^4\right), \nonumber \\
(\delta m_{\tilde{e}}^2)_{11}&=&\frac{135}{8}y_\tau^4+\frac{81}{8}y_b^2y_\tau^2 - 6y_\tau^2 g_{\tilde{l}}^2+\frac{27}{8}\beta_{2d}^2y_b^2y_\tau^2+\frac{27}{8}\beta_{2l}^2y_\tau^4,
\nonumber \\
(\delta m_{\tilde{e}}^2)_{22}&=& \beta_{2l}^2 \left(-4g_{\tilde{l}}^2 y_\tau^2 +\frac{15}{4}y_b^2y_\tau^2 +\frac{11}{4}y_\tau^4 \right), \nonumber \\
(\delta m_{\tilde{e}}^2)_{33}&=&
\frac{23}{8}y_\tau^4+\frac{33}{8}y_b^2 y_\tau^2- 2y_\tau^2 g_{\tilde{l}}^2+\frac{3}{8}\beta_{2d}^2y_b^2y_\tau^2-\frac{5}{8}\beta_{2l}^2y_\tau^4, \nonumber \\
\delta m_{H_u}^2&=&-\frac{9}{2}y_t^4-\frac{3}{2}y_t^2y_b^2-\frac{9}{4}\beta_{2d}^2y_b^2y_t^2-\frac{9}{4}\beta_{2u}^2y_t^2\left( 2y_t^2+y_b^2\right),\nonumber\\
\delta m_{H_d}^2&=&-\frac{9}{2}y_b^4-\frac{3}{2}y_\tau^4-\frac{3}{2}y_t^2y_b^2-\frac{9}{4}\beta_{2d}^2y_b^2(2y_b^2+y_t^2)-\frac{9}{4}\beta_{2u}^2y_b^2y_t^2, \nonumber\\
(\tilde{A}_u)_{22}&=&-\frac{3}{2\sqrt{2}}\beta_{2u}(y_t^3+y_ty_b^2), \qquad (\tilde{A}_u)_{33}=-\frac{3}{2}y_t^3-\frac{1}{2}y_t y_b^2-\frac{3}{4}\beta_{2d}^2y_b^2y_t-\frac{3}{4}\beta_{2u}^2y_t^3, \nonumber \\
(\tilde{A}_d)_{22}&=&-\frac{3}{2\sqrt{2}}\beta_{2d}(y_b^3+y_by_t^2), \qquad (\tilde{A}_d)_{33}=-\frac{3}{2}y_b^3-\frac{1}{2}y_b y_t^2-\frac{3}{4}\beta_{2d}^2y_b^3-\frac{3}{4}\beta_{2u}^2y_t^2y_b, \nonumber \\
(\tilde{A}_e)_{22}&=&-\frac{3}{2\sqrt{2}}\beta_{2l}y_\tau^3-\frac{9}{4\sqrt{2}}\beta_{2l}^3y_\tau^3, \quad (\tilde{A}_e)_{33}=-\frac{3}{2}y_\tau^3-\frac{3}{4}\beta_{2l}^2y_\tau^3.
\label{eq:deltamB2-3}
\end{eqnarray}
\clearpage
|
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{"url":"http:\/\/folk.uio.no\/nilshr\/Year\/1990-eng.html","text":"# Publications 1990\n\n## International Conferences\n\n1. H. Holden, L. Holden, N. H. Risebro. Some qualitative properties of $2\\times 2$ systems of conservation laws of mixed type. In Nonlinear Evolution Equations That Change Type, Pages 67-78, 1990.","date":"2016-08-29 21:31:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.49249356985092163, \"perplexity\": 4339.652820066015}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-36\/segments\/1471982967784.68\/warc\/CC-MAIN-20160823200927-00058-ip-10-153-172-175.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/webwork-ptx.aimath.org\/webwork2\/html2xml?courseID=anonymous&userID=anonymous&password=anonymous&course_password=anonymous&answersSubmitted=0&displayMode=MathJax&outputformat=simple&problemSeed=112&sourceFilePath=Library\/Michigan\/Chap5Sec2\/Q27.pg","text":"Use the following figure, which shows a graph of $f(x)$ to find each of the indicated integrals.\n\n(Click on the graph for a larger version.)\n\nNote that the first area (with vertical, red shading) is 55 and the second (with oblique, black shading) is 5.\n\nA. $\\int_a^b f(x) dx =$\n\nB. $\\int_b^c f(x) dx =$\n\nC. $\\int_a^c f(x) dx =$\n\nD. $\\int_a^c |f(x)| dx =$","date":"2022-05-20 08:33:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 5, \"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\": 0.8300678730010986, \"perplexity\": 1008.2192221431452}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662531762.30\/warc\/CC-MAIN-20220520061824-20220520091824-00462.warc.gz\"}"}
| null | null |
Added by scott voth 7 months ago. Updated 6 months ago.
Do we want to consider an option on the user profile settings that would totally hide a member profile? Some students or non-active members may not like that Google searches retrieve CAC profiles, but may want to remain members.
Scott, did this come in as a user request/suggestion?
I think it's worth discussing this as a group, as it quickly cuts to ideas about the purpose of the Commons.
Yes I saw that Tim Wilson (who managed a great French site until he left) is asking for his profile landing page to be removed. Perhaps there could be a redirection to a site (on the Commons or off) that better represents the member.
Added Sonja as a watcher. We need to settle some conceptual questions about what it means to be a Commons member etc before we can start thinking too concretely about UX strategies for profile visibility.
|
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"redpajama_set_name": "RedPajamaGithub"
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| 4,771
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\section{Introduction}
Cygnus~X-1 is a bright high-mass X-ray binary that was discovered in
the early days of X-ray astronomy \citep{bowyer65} and was identified
with the optical counterpart HD~226868 \citep{mw71}. It is best known
for being the first system with a high enough mass measurement to rule
out the possibility that the compact object is a neutron star
\citep[e.g.,][]{gb86}, making it the first confirmed black hole (BH)
system. The current constraint on the BH mass is
$14.8\pm 1.0$\Msun~\citep{orosz11}. Cyg~X-1 has been instrumental in
improving our understanding of accreting BHs, their spectral states,
and the relationship between the accretion disk and the jet
(see Remillard \& McClintock 2006\nocite{rm06} for a review of BH binaries).
Currently, a major on-going effort in BH studies is to measure their spins.
A non-zero BH spin changes the space-time around the black hole, requiring
the Kerr rather than the Schwarzchild metric to describe the geometry.
The spin is also one possible source for powering the relativistic jets
seen coming from BHs. One technique for measuring the BH spin involves
modeling the multi-temperature thermal component that comes from the
accretion disk \citep{mcclintock06}. A major challenge for this technique
is that the distance to the system and the inclination of the inner disk
must be known. For Cyg~X-1, the distance is well-established with a
parallax measurement of $1.86^{+0.12}_{-0.11}$\,kpc \citep{reid11},
which is consistent with a measurement using the dust scattering halo
\citep{xiang11}. The improved distance determination has also led to new
constraints on the binary inclination. Combined modeling of optical
spectroscopy (i.e., the companion's radial velocity) and photometry over
all orbital phases has given a binary inclination of $27.1\pm 0.8$~degrees
\citep{orosz11}. Orbital modulations are seen in the optical light curves
that depend on the shape of the companion star and the inclination of the
system. Although some misalignment between the inner disk inclination and
the binary inclination is possible \citep{maccarone02}, under the assumption
that they are the same, \cite{gou11} find that the spin of the Cyg~X-1 BH
is $a_{*}$$>$0.92 (3-$\sigma$ limit), and an even higher spin limit
($a_{*}$$>$0.983 at 3-$\sigma$) has been recently reported \citep{gou13}.
Another technique for measuring BH spin involves modeling the Compton
reflection component that is due to hard X-ray emission shining on the
inner part of the optically thick accretion disk. The reflection spectrum
includes fluorescent emission lines, with the Fe K$\alpha$ lines typically
being the strongest \citep{fabian89}, and a broad excess in the $\sim$10--50\,keV
energy range \citep{lw88}. The reflection spectrum can be distorted by the
relativistic effects of Doppler broadening from the fast orbital motion and
the gravitational redshift due to the BH's gravitational field \citep{fabian89}.
The emission lines can also be broadened when photons are Compton-scattered
out of the narrow line core \citep{rf05}. This implies that broad emission
lines are not necessarily an indication of relativistic effects. However,
the Compton-broadening is symmetric, so modeling the asymmetric component is
the key to using this technique to constrain BH spin \citep{rn03,miller07}.
For both thermal and reflection component modeling techniques, the BH spin
measurement is actually inferred from the measurement of the location of
the inner radius of the optically thick and ``cold'' (i.e., not fully ionized)
accretion disk. The BH spin measurement then comes from identifying the inner
radius with the innermost stable circular orbit (ISCO). For a non-rotating BH
($a_{*}\equiv Jc/GM_{\rm BH} = 0$, where $J$ is the angular momentum of the BH,
$c$ is the speed of light, $G$ is the gravitational constant, and $M_{\rm BH}$ is
the mass of the BH), the ISCO is at 6 gravitational radii
($R_{\rm g} = GM_{\rm BH}/c^{2}$), and, for a maximally rotating BH ($a_{*} = 1$),
the ISCO approaches 1\,$R_{\rm g}$.
For Cyg~X-1, most of the reflection studies have used X-ray spectra
from times when the source was in the hard state. In this state,
it is unclear whether the assumption about the inner disk radius being
at the ISCO holds. For BH transients, studies allow for the possibility
that the disk recedes when the source is in the faint hard state at an
Eddington fraction ($L/L_{\rm Edd}$) of $\sim$0.1--0.01\% \citep{nwd02,tomsick09c,cabanac09},
but there is evidence that the disk remains close to or at the ISCO during
the bright part of the hard state \citep{miller06a,rfm10}. Historically,
Cyg~X-1 has been in the bright part of the hard state, making the ISCO
assumption plausible. Using hard state observations, \cite{nowak11}
did not report a spin measurement but put an upper limit on disk
recession. Other reflection-based measurements constrained the BH
spin to be $0.6\leq a_{*} \leq 0.99$ \citep{miller12}, $a_{*} = 0.88^{+0.07}_{-0.11}$
\citep{duro11}, and $a_{*} = 0.97^{+0.014}_{-0.02}$ \citep{fabian12}.
The reflection fits in the hard state provide evidence for high BH spin
consistent with the limit on the BH spin from thermal modeling in the
soft state. In this paper, we report on the details
of reflection modeling in the soft state using observations with the
{\em Nuclear Spectroscopic Telescope Array} \citep{harrison13}
and {\em Suzaku} \citep{mitsuda07}. {\em NuSTAR} covers the 3--79\,keV
bandpass, which is ideal for reflection studies. Its detectors
give unprecedented energy resolution in the hard X-ray band, and
provide high throughput without the photon pile-up that occurs for
charge-coupled device (CCD) observations of bright sources. {\em NuSTAR}
has already been used for reflection studies of the supermassive BH
NGC~1365 \citep{risaliti13} as well as the Galactic BH GRS~1915+105
\citep{miller13}. In this paper, we provide details of the
observations, instrument capabilities, and the data reduction
methods in \S\,2. The results of the spectral fitting are reported
in \S\,3, and the results are discussed in \S\,4. Finally, we
present conclusions in \S\,5.
\section{Observations and Data Reduction}
We observed Cyg~X-1 with {\em NuSTAR} and {\em Suzaku} on 2012 October 31
and November 1 (MJD 56,231 and 56,232). Figure~\ref{fig:lc_maxi} shows the
soft X-ray light curve from the {\em Monitor of All-sky X-ray Image}
\citep[{\em MAXI};][]{matsuoka09}, indicating how this observation fits into the
$\sim$4\,yr history of this source. At the time of the observation, the {\em MAXI}
2--4\,keV count rate (normalized by effective area) was
$1.83\pm 0.04$\,s$^{-1}$\,cm$^{-2}$, and the 4--10\,keV count rate was
$0.55\pm 0.02$\,s$^{-1}$\,cm$^{-2}$ (obtained from the {\em MAXI}
website\footnote{http://maxi.riken.jp/top/}), demonstrating that the source
was in the soft state based on the {\em MAXI} count rate and hardness criteria
determined by \cite{grinberg13}.
\begin{figure}
\includegraphics[scale=0.47]{fig1.ps}
\caption{\small {\em MAXI} light curve in the 2--4\,keV band for Cyg~X-1
between mid-2009 and mid-2013. The source was in the hard state until
MJD 55,377 and has spent most of its time in the soft state since then.
The {\em NuSTAR} and {\em Suzaku} observation that is the subject of
this work is indicated with a vertical dashed line.\label{fig:lc_maxi}}
\end{figure}
\subsection{NuSTAR}
We reduced the data from the two {\em NuSTAR} instruments, Focal Plane Modules
(FPMs) A and B, and the exposure times and other observation details are listed
in Table~\ref{tab:obs}. The {\em NuSTAR} FPMs are Cadmium-Zinc-Telluride (CZT)
pixel detectors with an energy resolution (full width at half-maximum) of 0.4\,keV
at 10\,keV and 0.9\,keV at 68\,keV \citep{harrison13}. Each FPM is at the focus of
a hard X-ray telescope with a focal length of 10.14\,m and an angular resolution
(half-power diameter) of $58^{\prime\prime}$ \citep{harrison13}. We processed the
{\em NuSTAR} data (ObsIDs 30001011002 and 30001011003) with version 1.1.1 of the
NuSTARDAS pipeline software, the 2013 May 9 version of the {\em NuSTAR} Calibration
Database (CALDB), and High Energy Astrophysics Software (HEASOFT) v6.13. We
produced cleaned event lists with the routine {\ttfamily nupipeline} and light curves
and spectra with {\ttfamily nuproducts}. The source extraction region is centered
on Cyg~X-1 and has a radius of $200^{\prime\prime}$. The background region is a
$90^{\prime\prime}$ circle that is taken from the part of the {\em NuSTAR} field-of-view
that is farthest from the source. For ObsID 30001011002, the centers of the two
regions are $10.\!^{\prime}5$ apart, and, for ObsID 30001011003, they are separated by
$9.\!^{\prime}1$. While the background rate is known to vary across the field-of-view
at low energies \citep{harrison13}, the source rate is 25--1000 times the background
below 30\,keV, so systematic errors in the background cannot affect our results
over this energy band. At higher energies (combining the energy bins above 30\,keV),
the source is 21 times the background rate, so small detector-to-detector variations
in the background are not important.
\subsection{Suzaku}
{\em Suzaku} covers the $\sim$0.3--600\,keV band via three detectors: the
X-ray Imaging Spectrometers \citep[XISs;][]{koyama07}, which are CCDs, the
Hard X-ray Detector \citep[HXD;][]{takahashi:07a} PIN diode detector, and
the HXD gadolinium silicate crystal detector (\gso). These instruments
cover the $\sim$0.3--10\,keV, the $\sim$10--70\,keV, and the $\sim$60--600\,keV
bands, respectively. In this paper, for the XIS, we only consider the XIS0
and XIS1 detectors. During our observation, XIS3 was operated in a continuous
readout mode (\texttt{PSUM} mode), complicating its analysis, while XIS2 has
not been operational since 2006.
To create spectra from the {\em Suzaku} data (ObsID 407072010), we used tools
from the HEASOFT v6.13 package and the calibration files current as of
2013 February. We followed the standard procedure for analyzing the \xis spectra,
which included correcting for Charge Transfer Inefficiency (CTI) and reprocessing
the data with the \texttt{xispi} and \texttt{xselect} tools, respectively.
Thermal bending of the spacecraft leads to attitude uncertainties, which in turn
leads to distortions of the PSF image as observed by XIS. Although the standard
HEASOFT tools apply corrections to the spacecraft attitude in order to improve
the PSF image \citep{uchiyama:08a}, we further correct this image using the
\texttt{aeattcor2} tool as described by \cite{nowak11}.
The \xis spectra were obtained in a mode where only 1/4 of the CCD was exposed
with each CCD readout frame being 2\,s. The spectra, however, were only exposed
for 0.135\,s per readout frame in order to reduce telemetry and minimize pile-up.
Despite these precautions, given the brightness of \cyg in its soft state, the
spectra are heavily piled-up. To estimate the degree of pile-up we employed the
\texttt{pile\_estimate.sl} \texttt{S-Lang} script \citep[see][]{nowak11}. Using
this script, we identified the most heavily piled regions on the CCD and excluded
two rectangular regions in the center each measuring approximately
130$\times$45 pixels. We estimate that the remaining regions on the \xis CCDs
have an effective pile-up fraction of $\lesssim 5\%$. We then used \texttt{xisrmfgen}
and \texttt{xissimarfgen} to create response matrices for the extracted spectra.
To account for systematics, we added a 2\% uncertainty on the \xis spectra in
quadrature with the statistical uncertainties.
Standard procedures, following the \emph{Suzaku ABC
Guide}\footnote{See
http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/abc/}, were used
to create \hxd spectra. \pin spectra were extracted from the
\texttt{hxd/event\_cl} directories with response and background files
downloaded from the \texttt{pinxb\_ver2.0\_tuned} directory at the
High Energy Astrophysics Science Archive Research Center
(HEASARC)\footnote{See http://heasarc.gsfc.nasa.gov/}. \gso spectra
were created from ``unfiltered'' event files using the
\texttt{hxdtime}, \texttt{hxdpi}, and \texttt{hxdgrade} tools and the
filtering criteria from the standard \texttt{gso\_mkf.sel} script.
The background was obtained from the \texttt{gsonxb\_ver2.0} directory
at HEASARC. Event and background file Good Time Intervals were merged
to obtain the extraction times for the \gso spectra. Standard CALDB
response files were applied to the spectra with their exposure times
adjusted to agree with the spectra. The grouping of the \gso spectra
followed the fixed grouping of the background file and thus were not
rebinned further.
\section{Results}
Figure~\ref{fig:lc} shows the 3--79\,keV {\em NuSTAR} and 0.5--9\,keV XIS light
curves. There is good overlap in the coverage between the two satellites; however,
their Earth occultations are not exactly in phase, and the {\em Suzaku}
coverage extends somewhat beyond {\em NuSTAR}'s. Flaring, which is typical of
Cyg~X-1 in this state, is more evident in {\em NuSTAR}'s hard X-ray band than in
the softer X-ray regime covered by XIS. Perhaps the most notable feature in the
XIS light curves are brief drops in the count rate. It is possible that these
are absorption dips due to material in the massive donor star's stellar wind.
This is plausible because the observations occurred at a binary orbital phase of
0.85-0.97 (where 1.0 corresponds to superior conjunction when the donor star is
between the observer and the black hole) based on the ephemeris of \cite{brocksopp99}.
Absorption dips are typically seen in this range of orbital phase \citep{bc00,poutanen08}.
\begin{figure}
\includegraphics[scale=0.48]{fig2.ps}
\caption{\small {\em Suzaku}/XIS light curve {\em (a)}, {\em NuSTAR}/FPMA
light curve {\em (b)}, and {\em NuSTAR} hardness ratio {\em (c)}
for Cyg~X-1. For XIS, the bandpass is 0.5--9\,keV, and the rate is for
XIS1 (after removing the piled-up core of the point spread function).
For the {\em NuSTAR} light curve, the bandpass is 3--79\,keV, the
rate is for FPMA, and it is corrected for deadtime. The {\em NuSTAR}
hardness ratio is the 10--79\,keV rate divided by the 3--10\,keV rate,
and both modules are used. The time resolution for all plots is 10\,s.
The zero time is arbitrary but corresponds to MJD~56,231.30000. For
both satellites, most of the gaps are due to Earth occultation, but
the longer gap near time 22,500\,s for {\em NuSTAR} is due to a missed
ground station pass.\label{fig:lc}}
\end{figure}
In order to determine the level of spectral variation during the observations,
we extracted the 3--10\,keV and 10--79\,keV {\em NuSTAR} count rates, and
produced a plot of hardness, which is the 10--79\,keV count rate divided by
the 3--10\,keV count rate, vs.~time (Figure~\ref{fig:lc}c). Even during the
flares, we see little variability in the hardness. Given the relatively low
level of spectral variability, we combined all of the data into a single spectrum.
Due to the high count rate for Cyg~X-1, the XIS spectra show features that we
suspect are related to photon pile-up. An upturn in the spectra above $\sim$9\,keV
is observed and is readily explained by pile-up. The spectrum below 1.2\,keV shows
features that appear to be absorption lines; however, we cannot rule out the
possibility of some distortion due to instrumental effects, and we defer a detailed
study to a later paper. In addition, there are known calibration uncertainties in
the 1.7--1.9\,keV band related to the Si K-edge. After these considerations, for
XIS0 and XIS1, we used the 1.2--1.7\,keV and 1.9--9\,keV bands for spectral analysis
and binned the data based on the instrumental energy resolution
(see Nowak et al.~2011\nocite{nowak11}). For PIN and GSO, we used the 15--68\,keV
and 50--296\,keV energy ranges, respectively. For {\em NuSTAR}, we used 3--79\,keV,
and binned the spectra for FPMA and FPMB separately, requiring that each bin have
a signal-to-noise ratio of at least 30 (after background subtraction).
We used the XSPEC software package \citep{arnaud96} to fit the combined
{\em NuSTAR} plus {\em Suzaku} spectrum with a model consisting of a
multi-temperature ``disk-blackbody'' thermal component \citep{mitsuda84}
plus a power-law (model 1). These continuum components were subject to absorption
with the {\ttfamily tbabs} model, and we used \cite{wam00} abundances and
\cite{vern96} cross-sections for this interstellar absorption. We included a
multiplicative constant as a free parameter for each instrument to account for
differences in overall normalization. Figure~\ref{fig:ratio} shows the XIS and
{\em NuSTAR} residuals for this fit in terms of the data-to-model ratio, revealing
a strong reflection component with a broad iron K$\alpha$ emission line and a
reflection hump above $\sim$15\,keV. Figure~\ref{fig:ratio}b illustrates the
complexity of the iron line, which has an absorption line at 6.7\,keV in addition
to the broad line in emission.
\begin{figure}
\includegraphics[scale=0.36]{fig3.ps}
\caption{\small The data-to-model ratio for a fit to the Cyg~X-1 spectrum
with an absorbed disk-blackbody plus power-law model (model 1). The panel
{\em (a)} residuals indicate a strong reflection component. Panel {\em (b)}
focuses on the iron K$\alpha$ line region, showing that the line complex
includes at least a broad emission component and an absorption line at 6.7\,keV.
\label{fig:ratio}}
\end{figure}
The fit can be significantly improved with the addition of a Gaussian emission
line and a cutoff at high energies (model 2), using {\ttfamily highecut}, which
provides an exponential cutoff with a folding energy of $E_{\rm fold}$ for energies
greater than a cutoff energy, $E_{\rm cut}$. If the Gaussian parameters are allowed
to take any values, the line centroid is near 5.3\,keV, which is well below the
6.4--7.1\,keV iron regime, and the line is extremely broad ($\sigma = 1.57$\,keV).
In addition to the Gaussian parameter values being unphysical, this model does
not give a formally acceptable fit with a reduced-$\chi^{2}$ ($\chi^{2}_{\nu}$) of
2.00 for 1149 degrees of freedom (dof). The continuum parameters (e.g., a best
fit inner disk temperature of $kT_{\rm in} = 0.62$\,keV and a power-law photon
index of $\Gamma = 2.5$) are consistent with the source being in the soft state.
This model gives absorbed and unabsorbed 0.5--100\,keV fluxes of
$4.33\times 10^{-8}$\,erg~cm$^{-2}$~s$^{-1}$ and
$6.09\times 10^{-8}$\,erg~cm$^{-2}$~s$^{-1}$, respectively. For a source distance
of 1.86\,kpc, this implies a luminosity of $2.5\times 10^{37}$\,erg~s$^{-1}$, which,
for a BH mass of 14.8\Msun, gives an Eddington-scaled luminosity of 1.3\%.
As shown in Figure~\ref{fig:ratio_many}a, the largest residuals for model 2 are
in the 6--8.5\,keV part of the spectrum. In addition to the fact that we are still
not modeling the 6.7\,keV absorption line, which is due to the photoionized wind
of the massive companion star, a Gaussian is too simple to fit the broad emission
feature and the absorption edge that are present in the reflection component.
Thus, we removed the Gaussian and added a simple ionized absorber and a reflection
component (model 3).
For absorption due to the wind, we constructed a grid of table models using XSTAR
version 2.2.1bg \citep{kb01}. Solar abundances were assumed for all elements,
the number density was fixed at $n = 10^{12}~ {\rm cm}^{-3}$, and the turbulent
velocity of the gas was fixed at $v_{\rm turb} = 300~ {\rm km}~ {\rm s}^{-1}$
\citep[e.g.,][]{miller05,hanke09}. We used an input spectrum consistent with
model 1 described above in order to construct a grid spanning
$2 \leq {\rm log}(\xi) \leq 5$, where $\xi$ is the ionization parameter in
units of erg\,cm\,s$^{-1}$, and
$1.0\times 10^{21}~ {\rm cm}^{-2} \leq N_{\rm H} \leq 5.0\times 10^{22}~ {\rm cm}^{-2}$,
where $N_{\rm H}$ is the column density of the absorber. In total, 400 grid points
were calculated and summed into a multiplicative table model that was included in
XSPEC analysis, with $N_{\rm H}$, $\xi$ and $v/c$ as variable parameters. Although
$v/c$ was originally left as a free parameter, we found a 90\% confidence upper limit
of $<$0.0004, and we fixed it to zero in the fits described below. This parameter is
driven by the strong absorption line at 6.7\,keV, which is due to Fe XXV.
\begin{figure}
\hspace{-0.5cm}
\includegraphics[scale=0.48]{fig4.ps}
\vspace{-0.4cm}
\caption{\small Data-to-model ratios for six of the models described in
\S\,3. The symbols and colors for XIS0, XIS1, FPMA, and FPMB are
the same as for Figure~\ref{fig:ratio}. In addition, at higher energies,
the PIN and GSO ratios, which are only shown up to 100\,keV, are marked
with brown triangles and squares, respectively.\label{fig:ratio_many}}
\end{figure}
For the reflection, we used the {\ttfamily reflionx} model \citep{rf05}. This
model includes the hard X-ray bump, the absorption edges, and the emission lines,
so that the full reflection component is physically self-consistent. In addition,
the emission lines are Compton-broadened (see \S\,1). The version that is available
on-line\footnote{See http://heasarc.gsfc.nasa.gov/xanadu/xspec/models/reflion.html.}
has the folding energy for its exponential cutoff fixed at 300\,keV, but, for
our fits, a new model, {\ttfamily reflionx\_hc}, was produced with $E_{\rm fold}$ as a
free parameter. For the direct component, the {\ttfamily highecut} parameters were
set to be consistent with {\ttfamily reflionx\_hc}: $E_{\rm cut}$ was set to zero and
$E_{\rm fold}$ was forced to have the same value as the free parameter in the
reflection model. One other difference between {\ttfamily reflionx} and
{\ttfamily reflionx\_hc} is that the ionization parameter was extended to higher
levels based on early fits to the Cyg~X-1 spectrum. While this is a more
realistic physical model than using the Gaussian to fit the iron line, model 3
provides a worse fit ($\chi^{2}_{\nu}$=2.72 for 1148 dof) than model 2, and
large residuals are still present in the 5--9\,keV regime.
A major improvement in the fit (to $\chi^{2}_{\nu}$=1.21 for 1143 dof) comes from
convolving the reflection component with a relativistic blurring model (model 4).
For blurring, we used the {\ttfamily relconv} model \citep{dauser10}, which is
based on the physics described in \cite{fabian89} and \cite{laor91}, but
{\ttfamily relconv} allows for a range of spin values. For these fits, we assume
that the accretion disk extends to the ISCO, and the blurring, which is most
apparent in its effect on the iron line shape, depends on the BH spin ($a_{*}$),
the disk inclination ($i$), and the radial dependence of the emissivity of reflected
flux. The emissivity is assumed to have a power-law ($L\propto r^{-q}$, where $L$
is the luminosity illuminating the reflecting material, $r$ is the radial distance
from the BH, and $q$ is the emissivity index) or broken power-law shape. The fit
parameter values for the broken power-law emissivity (model 4) and for the power-law
emissivity (model 5) are given in Table~\ref{tab:parameters1}, and
Figure~\ref{fig:efe}a shows the components of the former model.
\begin{figure}
\hspace{-0.3cm}
\includegraphics[scale=0.43]{fig5.ps}
\vspace{-0.5cm}
\caption{\small {\em (a)} Unfolded {\em NuSTAR} and {\em Suzaku} spectrum showing
the fit obtained with model 4, which includes a disk-blackbody component, a
cutoff power-law, a {\ttfamily reflionx\_hc} reflection model with relativistic
blurring, and a simple ionized absorber. {\em (b)} The spectrum for model 8, which
models the thermal component with {\ttfamily kerrbb} and is self-consistent in that
the thermal component is the seed photon distribution for the Comptonized component
(using {\ttfamily simpl}). The symbols and colors for the different instruments are
the same as for Figure~\ref{fig:ratio_many}.\label{fig:efe}}
\end{figure}
\begin{figure*}
\includegraphics[scale=0.55]{fig6.ps}
\vspace{-0.5cm}
\caption{\small Error contours for BH spin and inner disk inclination for
models 4 {\em (left)}, 8 {\em (middle)}, and 10 {\em (right)}. The 1, 2,
and 3-$\sigma$ contours are shown.
\label{fig:contours}}
\end{figure*}
The fit parameters indicate reflection off highly ionized material
($\xi$$>$13,900\,erg\,cm\,s$^{-1}$ for model 4 and $\xi$$>$19,500\,erg\,cm\,s$^{-1}$
for model 5) and a steep emissivity index ($q$$>$9.5). In addition, we find a
high BH spin of $a_{*} = 0.9882\pm 0.0009$ for model 4 and
$a_{*} = 0.91^{+0.01}_{-0.02}$ for model 5. These are 90\% confidence statistical
errors, and it is important to note that they do not include any systematic
component. The inclinations obtained are $i = 69.2^{+0.5}_{-0.9}$~degrees and
$i = 59.3^{+0.5}_{-1.3}$~degrees for models 4 and 5, respectively, both of which
are significantly different from the value of 27.1 degrees measured for the
binary \citep{orosz11}. If we fix the inclination to the binary value and
refit the spectrum, we obtain a very poor fit with $\chi^{2}_{\nu}$=2.45 for
1144 dof even for the case of broken power-law emissivity, and the residuals
are shown in Figure~\ref{fig:ratio_many}d (model 6). Furthermore, we made
error contours\footnote{To explore correlations among parameters, we performed
Markov Chain Monte Carlo (MCMC) simulations with a code modeled after the ``emcee
hammer'' code described by \cite{fm13}, which implements the algorithm of
\cite{gw10}. In this algorithm, an ensemble of ``walkers,'' which are vectors
of the fit parameters, are evolved via random steps determined by the difference
between two walkers. We evolved 20 walkers per free parameter for a total of
4000--10,000 steps, and ignored the first half of the steps. Thus, probability
distributions were calculated from (0.4--1.5)$\times 10^{6}$ values. Error
contours are the 2D projection of the MCMC N-dimensional probability distribution.}
for the spin and inclination parameters for model 4 (see Figure~\ref{fig:contours}).
Although there is some correlation between these parameters, and a nearby local
minimum exists, the 3-$\sigma$ contours do not extend below $i$$\sim$$65.8$ degrees.
For all the models presented thus far, if $q_{\rm in}$ is left as a free parameter,
we obtain values close to 10, which is the maximum of the allowed range. We also
explored the implications of lower emissivity index by fixing it to $q = 3$.
This gives $\chi^{2}_{\nu}$=1.40 for 1146 dof, which is significantly worse than
the high-$q$ (and high-$i$) fit, but the inclination is $42.4^{+0.4}_{-0.5}$~degrees,
which is much closer to the binary value. The residuals for this model are shown
in Figure~\ref{fig:ratio_many}e (model 7), and the parameters are given in
Table~\ref{tab:parameters1}. Although the BH spin is somewhat lower for model 7,
the relatively poor fit suggests that this value is not reliable. For model 4, we
left $q_{\rm out}$ as a free parameter, and a value of --$1.2^{+1.1}_{-4.6}$ is obtained,
indicating that, beyond 10\,$R_{\rm g}$, the flux incident on the disk is actually
increasing with radius. Although such a rising profile could occur over some range
of radii, we note that it is non-physical for the emissivity to continue to
increase with radius indefinitely.
While the parameter and BH spin constraints above rely only on modeling
the reflection component, a previous Cyg~X-1 spin measurement obtained by
fitting the soft state spectrum relied primarily on modeling the thermal
component \citep{gou11,gou13}. Rather than using the disk-blackbody model,
they used the model {\ttfamily kerrbb}, which is a multi-temperature thermal
accretion disk model that accounts for changes in the inner disk (e.g., the
inner radius) due to the BH spin. Also, instead of adding a power-law, they
used the convolution model {\ttfamily simpl} \citep{steiner09}, which is
different from the disk-blackbody plus power-law model described above
because it uses the {\ttfamily kerrbb} component as the seed photon input to
the Comptonization region. With this model, we obtain $\chi^{2}_{\nu}$ = 1.32
for 1146 dof. The residuals are shown in Figure~\ref{fig:ratio_many}f
(model 8), and the model components are shown in Figure~\ref{fig:efe}b.
The parameter values from the fit are given in Table~\ref{tab:parameters2}.
The constraint on the spin parameter, $a_{*} = 0.838\pm 0.006$, comes
from both the the thermal component and the reflection component, and the
inclination ($i = 53.9\pm 0.4$~degrees) is still significantly higher than
the binary value. Model 8 uses a single power-law for the emissivity with an
index of $q = 7.8\pm 0.5$. Figure~\ref{fig:contours} shows the error
contours for spin and inclination for model 8.
Although we do not focus on calibration details in this paper, there is
excellent agreement between FPMA and FPMB with the relative normalization
being consistent to within 0.1\% for all the spectral models described
above, which is actually better than expected. Relative to {\em NuSTAR}/FPMA,
we find normalization constants of $1.081\pm 0.005$ for XIS0, $1.038\pm 0.004$
for XIS1, $1.205\pm 0.007$ for PIN, and $1.17\pm 0.06$ for GSO.
These numbers are for model 8, but Tables~\ref{tab:parameters1} and
\ref{tab:parameters2} show very similar relative normalizations for all
models. Thus, there is very good agreement between {\em NuSTAR} and
XIS, and the fact that PIN and GSO are somewhat higher is
expected\footnote{See http://www.astro.isas.jaxa.jp/suzaku/doc/suzakumemo/suzakumemo-2008-06.pdf}.
\section{Discussion}
The combination of {\em NuSTAR} and {\em Suzaku} provide a measurement of
the Cyg~X-1 reflection spectrum with unprecedented quality. While {\em NuSTAR}
measures the entire reflection component (iron line, absorption edges, and
hard X-ray bump), the XIS provides an extension to lower energies that is
essential for constraining the thermal component. {\em NuSTAR} and XIS agree
to a remarkable extent on the shape of the iron line (see Figure~\ref{fig:ratio}b),
and {\em NuSTAR} provides a huge improvement in the statistical quality of the
data, while alleviating some systematic concerns such as pile-up.
We have presented fits to the spectrum with several different models,
and, while some parameters show significant differences, others agree
about the properties of the system. It is clear that the source was
in the soft state with a prominent thermal component and a power-law
with a photon index between $\Gamma = 2.59$ and 2.67, which meets
the $\Gamma > 2.5$ criterion for Cyg~X-1 to be in the soft state
\citep{grinberg13}. There is clear evidence for absorption due to
highly ionized material, which is consistent with the findings of
\cite{yamada13}. Also, the fits agree that the ionization state of
the disk material that leads to the reflection component is high and
that iron is overabundant by a factor of 1.9--2.9 relative to solar.
The BH spin and inclination measurements vary from model-to-model by more
than the 90\% confidence statistical errors, indicating that there is
significant systematic uncertainty. For the inclination, it is also
necessary to compare our values of $i = 42$--69 degrees to the
value of $i = 27.1\pm 0.8$~degrees that is obtained by modeling optical
photometric and spectroscopic measurements \citep{orosz11}, but the optical
measurement is of the inclination of the binary while the reflection component
measures the inclination of the inner part of the disk. We have shown (see
Figure~\ref{fig:ratio_many}d) that the reflection model simply cannot reproduce
an inclination as low as 27.1 degrees.
However, we must also keep the limitations of the spectral model in mind.
While the {\ttfamily relconv} calculation is for a specific inclination angle,
{\ttfamily reflionx\_hc} calculates the spectrum of the reflection component by
averaging over angles. Another consideration is that the ionization parameter
($\xi$) is at the top of the available range for {\ttfamily reflionx\_hc}. The
model already includes Compton broadening of the lines, which increases with
increasing $\xi$, but it is possible that some extra Compton broadening is
necessary to account for a higher ionization. Also, surface turbulence may
cause some symmetric line broadening that is not taken into account by
{\ttfamily reflionx\_hc}. To test this, we added a Gaussian convolution model
({\ttfamily gsmooth}), which acts on the reflection component along with
{\ttfamily relconv}. The results are shown in Table~\ref{tab:parameters2},
where this is listed as model 9, and the inclination decreases significantly
from 53.9 to 40.4 degrees. The value obtained for the {\ttfamily gsmooth}
$\sigma$ parameter is $0.28^{+0.02}_{-0.04}$\,keV. Using
$kT = (1/2)m_{e}c^{2}(\sigma/E)^{2}$, where $m_{e}$ is the electron rest mass,
this value of $\sigma$ corresponds to a temperature of $kT = 0.4$--0.6\,keV at
$E = 6$--7\,keV, which is in-line with the inner disk temperatures we obtain
from the disk-blackbody fits.
While adding symmetric smoothing of the iron line and reflection component
(i.e., extra Compton broadening) causes a drop in $i$, we emphasize that
asymmetric relativistic broadening is required by the data. For our original
Gaussian fit to the iron line, we obtained a best fit centroid value of 5.3\,keV,
which shows that the line has the low-energy tail expected for a gravitational
redshift. Also, we obtained a very poor fit with {\ttfamily reflionx\_hc}
(Figure~\ref{fig:ratio_many}b), where the Compton broadening was already included.
Adding the relativistic broadening provided a very large improvement to the fit
(Figure~\ref{fig:ratio_many}c).
Although conclusions about the BH spin depend on the different possibilities
for the inclination, the models which provide good fits to the data
(models 4, 5, 8, and 9) all have $a_{*}$$>$0.83, indicating at least relatively
high spin. The best fit (model 4) also has the highest spin
$a_{*} = 0.9882\pm 0.0009$, but this either requires a very large warp
in the accretion disk or that the binary inclination is somewhat higher
than the best fit value found in \cite{orosz11}. We note that Table 1
in \cite{orosz11} reports that some of their models give significantly
higher inclinations, but the $\chi^{2}$ values for the higher inclination
models are worse.
Another potentially interesting result that comes from this spectrum is
the constraint on the emissivity profile. A comparison of models 4--6
indicate that a broken power-law emissivity is preferred as is a very steep
profile in the inner part of the disk ($q_{\rm in}$$\sim$10). For Active
Galactic Nuclei (AGN), relatively steep profiles ($q = 4.3$--5.0) were
reported for MCG--6-30-15 \citep{wilms01}, and steep profiles are discussed
in \cite{wf12}. \cite{walton13} studied a large sample of AGN, and found
that steep profiles are common. This has been taken as evidence that the
irradiating source comes from very close to the BH, and \cite{fabian12a}
conclude that it must lie within 1\,$R_{\rm g}$ of the BH event horizon.
While this may also be the case for Cyg~X-1, our fits with very steep
profiles (models 4, 5, and 8) also have inclinations between 53.9 and
69.2 degrees. Our fit with {\ttfamily gsmooth} (model 9) included in the
model gave a much flatter index of $q = 2.48^{+0.09}_{-0.05}$, leaving open
the possibility that the profile is relatively flat, in which case the
source is at a height of 5--10\,$R_{\rm g}$ or more.
While we cannot conclude anything definitive about the slope of the emissivity
profile, if it is very steep, this might point to a ``lamppost'' geometry
\citep{dauser10}, where the emission actually comes from the base of a
collimated jet. This geometry may not be relevant for the soft state
because there is no evidence for a jet. Despite this, if we start with
model 8 but replace {\ttfamily relconv} with {\ttfamily relconv\_lp}
(model 10), we find $i = 41.5\pm 0.5$ degrees and $a_{*} = 0.953\pm 0.006$
(see Figure~\ref{fig:contours} for the error contours), with only small
changes in the other parameters. However, the quality of the fit is
somewhat worse ($\chi^{2}_{\nu} = 1.44$) for model 10 compared to the models
reported in Tables~\ref{tab:parameters1} and \ref{tab:parameters2}.
After exploring different continuum models, emissivity geometries, and
conditions for the material in the accretion disk, we only find inner
disk inclinations that are $>$13 degrees higher than the binary value
measured by \cite{orosz11}, and, as mentioned above, one explanation is
that there is a warp in the accretion disk. Analytical calculations as
well as numerical simulations have shown that disk warps can occur
\citep{bp75,sm94,fragile07}, and that they should occur if the BH spin
is misaligned from the orbital plane (and outer disk). As the alignment
time for an accreting BH can be longer than the lifetime of a high-mass
system \citep{maccarone02}, if the Cyg~X-1 BH formed with a misaligned spin,
it would remain misaligned. If jets are aligned with the BH spin, then
there is evidence for misalignment in systems like Cyg~X-3, V4641~Sgr,
and GRO~J1655--40 \citep{maccarone02}. It should be noted that
\cite{fragile09} has shown that, under certain assumptions about the
thickness of the accretion disk, BH spin measurements using the inner
radius of a warped disk can be incorrect. While a disk warp may not be
the only possibility for Cyg~X-1, further investigations to determine
if the disk is really warped have important implications for the BH spin
measurement.
\section{Summary and Conclusions}
We have presented a detailed study of the $\sim$1--300\,keV spectrum
of Cyg~X-1 in the soft state. The spectrum is complex and consists
of multi-temperature blackbody, power-law, and reflection components
along with absorption from highly ionized material in the system.
Although the observation was of moderate duration ($\sim$29\,ks),
{\em NuSTAR} provides a very high-quality and high-statistics measurement
of the reflection spectrum, including an iron complex with broad emission
and narrow absorption lines. We find that the reflecting material
has a high ionization state, is overabundant in iron relative to
solar, and requires broadening of the iron line that is well-described
by a relativistic blurring model.
While all models that provide a good fit to the spectrum indicate a
rapidly rotating BH with $a_{*}$$>$0.83, and our best-fitting model
has $a_{*} = 0.9882\pm 0.0009$ (90\% confidence statistical
errors only), we were not able to obtain a good fit with the inclination
fixed to the \cite{orosz11} binary value. This may indicate a
misalignment between the orbital plane and the inner accretion disk
(by $>$13 degrees), missing physics in the spectral models, or it may
possibly motivate work to confirm the measurement of the binary inclination.
Regardless of which of these possibilities is correct, it is clear that
the combination of {\em NuSTAR}'s high throughput and energy resolution
provides a major advance in reflection studies, allowing for strict tests
of the models, which we expect to lead to improved constraints on the
physical processes at work in Cyg~X-1 and other accreting BH systems.
\acknowledgments
This work was supported under NASA Contract No. NNG08FD60C, and made use of data
from the {\it NuSTAR} mission, a project led by the California Institute of
Technology, managed by the Jet Propulsion Laboratory, and funded by the National
Aeronautics and Space Administration. We thank the {\it NuSTAR} Operations,
Software and Calibration teams for support with the execution and analysis of
these observations. This research has made use of the {\it NuSTAR} Data
Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data
Center (ASDC, Italy) and the California Institute of Technology (USA).
JAT acknowledges partial support from NASA Astrophysics Data Analysis Program
grant NNX13AE98G. LN wishes to acknowledge the Italian Space Agency (ASI) for
financial support by ASI/INAF grant I/037/12/0-011/13. JAT thanks L.~Brenneman,
G.~Matt, and D.~Ballantyne for useful discussions about reflection modeling.
This work made use of IDL software written by N.~Barri{\`e}re for rebinning the
{\em NuSTAR} spectra. This research has made use of the {\em MAXI} data provided
by RIKEN, JAXA, and the {\em MAXI} team.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,521
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\section*{Introduction}
Quantum entanglement is the cornerstone of novel quantum technologies, particularly quantum computing and communication. The enormous interest in quantum communication is driven by its built-in security which protects the transmission of information and quantum entanglement for applications such as distributed quantum computing, quantum key distribution and quantum internet. To create distributed entanglement between two separated locations (nodes), qubits at each node must be entangled to flying qubits which are then sent to a common location to undergo joint measurements that implement entanglement swapping \cite{Bouwmeester_Nature97,Hensen_Nature15}. To achieve entanglement between nodes at large distances, beyond the capabilities of optical fibers, the entanglement must be refreshed at intermediate nodes known as quantum repeaters \cite{Briegel_PRL98,Dur_PRA99,Pirandola_arxiv15,Wallnofer_PRA16}.
The standard paradigm for a quantum repeater is based on atomic quantum memories located at primary nodes, each entangled with a single photon \cite{Togan_Nature10}, which in turn is sent to a secondary, intermediate node \cite{Duan_Nature01}. Two photons arriving at a secondary node undergo a Bell measurement, a process that transforms the qubit-photon entanglement into long-distance two-qubit entanglement where the two qubits involved are located at the two different primary nodes on either side of the secondary node. Difficulties associated with atomic-memory-based quantum repeaters include the necessity for long coherence times of the atomic memory beyond what is currently feasible and also the inherent vulnerability of these schemes to photon loss.
Zwerger et al. \cite{Zwerger_PRA12,Zwerger_PRL13} introduced the idea of using highly entangled states, known as graph states \cite{Briegel_PRL01,Raussendorf_PRL01,Raussendorf_PRA03,Hein_PRA04,Nielsen_PRA05}, to implement quantum repeaters. In a recent publication, Azuma et al. \cite{Azuma_NC15} put forward an explicit construction of an all-photonic repeater graph state (RGS) consisting of a completely connected graph of 'core' photons, with each of them featuring a connection to an additional 'arm' photon, to be used for entanglement swapping in the secondary nodes. Two such states are illustrated in Fig.~\ref{fig:rgs}. In this figure, each circle represents a photon, and the lines between them represent pairwise entanglement \cite{Raussendorf_PRA03,Hein_PRA04,Nielsen_PRA05}. Specifically, the state represented by a graph can be created by initializing all qubits to the state $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and successively performing CZ gates between all pairs of qubits connected by an edge of the graph. This work has attracted a great deal of attention \cite{Takeoka_NC14,Bruschi_PRA,Azuma_NC15b,Pant_PRA17,Zwerger_APB16} due to its advantages over atomic-memory based repeaters, notably the all-photonic construction that avoids coherence time limitations, the resilience against photon loss, and the elimination of long-distance heralding \cite{Azuma_NC15}. These attractive features position quantum repeaters, and consequently long-distance quantum communication, as near-term, much more readily feasible technology compared to quantum computing \cite{Sinclair_PRL14,Azuma_NC15b}.
Despite the promise held by the all-photonic RGSs of Ref.~\cite{Azuma_NC15}, construction of multiphotonic entanglement is extremely challenging. This is due to the fact that photons do not interact with each other, so that either a nonlinear interaction or measurement are required to entangle two photons that are initially in a product state. Both approaches are challenging because non-linear interactions are weak and measurements are probabilistic. The standard process of generating graph states (or cluster states \cite{Raussendorf_PRL01} for quantum computing)\cite{Nielsen_PRL04,Nielsen_PRA05,Zwerger_APB16} begins with pairs of photons that are prepared in entangled (Bell) states through parametric down-conversion. Two different pairs are then ``fused" together probabilistically via the joint measurement of two photons, one from each pair. When the measurement succeeds, which happens with probability $1/2$ (or $3/4$ if ancillary qubits are used \cite{Grice_PRA11}), a three-photon graph state is obtained. Such fusion gates are used consecutively to `grow' the graph state \cite{Browne_PRL05,Varnava_PRL08}. To date, up to ten photons have been entangled in this way \cite{Zhao_Nature04,Gao_PRL10,Gao_NP10,Wang_PRL16b}. The analysis of quantifying the resource overhead of RGSs was carried out \cite{Pant_PRA17} and, even after an optimization of the original scheme of Ref.~\cite{Azuma_NC15}, $10^6$ photon sources per node are needed. Overcoming the probabilistic nature of RGS generation would therefore be catalytic in drastically reducing overhead.
Here, we show that repeater graph states of arbitrarily large size can be created deterministically by using quantum emitters with appropriate level structures and selection rules. The overhead of our approach is dramatically reduced compared to fusion-based approaches. Our remarkable finding is that the number of emitters required does not scale with the size of the repeater state: surprisingly, one emitter suffices to generate an RGS of arbitrary size.
A deterministic approach to the creation of linear cluster states was introduced several years ago \cite{Lindner_PRL09}, and later generalized to a cluster state ladder \cite{Economou_PRL10}. The first experimental demonstration of deterministic linear cluster state generation was carried out recently following these ideas \cite{Schwartz_Science16}. The number of emitters needed to create larger two-dimensional cluster states is equal to the linear size of the state, making the creation of a large square grid cluster state conditional on future advances in controllably coupling a long chain of emitters. Nevertheless, the concepts introduced in Refs.~\cite{Lindner_PRL09,Economou_PRL10} for one and two emitters are central to our present work. We follow these works in assuming an emitter that has the level structure shown in Fig.~\ref{fig:energylevels}(a). We also consider that two adjacent emitters can be coupled such that entanglement can be created between them.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{rgsstates.pdf}
\caption{Graphical representation of $N{=}4$ and $N{=}6$ repeater states proposed in Ref.~\cite{Azuma_NC15}. States consist of a complete subgraph of $N$ core photons each connected to an additional photon forming N external arms.}
\label{fig:rgs}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{levelstructure.pdf}
\vspace{-2cm}
\caption{(a) Level structure required for the emitter to emit entangled photons. Each of two ground states couples to its own excited state. Cross transitions are forbidden by selection rules. (b) Linear photonic cluster state. Photons are blue circles, while the red circle is the emitter. Each line represents entanglement between the qubits it connects. (c) Partial $N{=}4$ repeater state. All but 2 of the 8 entangled photons comprising this repeater state can be produced from a single emitter.}
\label{fig:energylevels}
\end{figure}
\section*{Results}
Lindner and Rudolph \cite{Lindner_PRL09} proposed a method in which a quantum dot or similar system with the level structure displayed in Fig.~\ref{fig:energylevels}(a) could be optically pumped to generate photons that are entangled with the electron spin. Moreover, they showed that repeated pumping of such an emitter can produce a chain of photons which can be entangled with the emitter and with each other. In particular, if the pumping is performed repeatedly without applying any other operations on the emitter, then a photonic GHZ state will be created in which every photon is entangled with the emitter and with every other photon. On the other hand, if a Hadamard gate is applied on the emitter between each pumping operation, then the resulting state is instead a one-dimensional linear cluster state in which each photon is entangled with the photon that preceded it and with the one that follows it. Fig.~\ref{fig:energylevels}(b) shows the graph corresponding to this linear photonic cluster state.
\begin{figure*}
\centering
\includegraphics[width=2\columnwidth]{rgscreation.pdf}
\caption{Generation of the $N{=}6$ repeater state proposed in Ref.~\cite{Azuma_NC15}: \textbf{(a)} Top: Emitter $B$ is connected to ancilla $A$ with a CZ gate and pumped twice, creating photons 1 and 2. Bottom: Hadamard gates are applied and emitter $B$ is measured, detaching it from the graph. \textbf{(b)} This process is repeated (creating photons 3 and 4), generating another arm. \textbf{(c)} After $N$ arms have been generated, local complementation (LC) is applied about ancilla $A$, and ancilla $A$ is then measured. The exact sequence of gates for this process is given in the Appendix.}
\label{fig:generation}
\end{figure*}
Ref.~\cite{Azuma_NC15} presented a family of RGSs where each member of the family consists of $2N$ entangled photons, with $N$ of the photons comprising a fully connected graph at the core, while the remaining $N$ photons are each attached to one of the core photons by a single edge forming $N$ external "arms". For example, the $N=4$ and $N=6$ RGSs are illustrated in Fig.~\ref{fig:rgs}. In the scheme of Ref.~\cite{Azuma_NC15}, one such state is generated at each primary node of the repeater network, and entanglement swapping between primary nodes is performed by sending half of the $N$ photon arms to an adjacent secondary node, where they encounter an additional $N/2$ photons that were sent out from the repeater state at the next primary node. Entanglement between the primary nodes is then created by performing Bell measurements on pairs of photons that arrived from different primary nodes. The redundancy of sending $N/2$ photons rather than one overcomes the probabilistic nature of the Bell measurements, so that the likelihood of successful entanglement swapping increases with $N$.
Using only the pumping technique developed in Ref.~\cite{Lindner_PRL09}, but carefully choosing when to apply Hadamard gates, large portions of photonic repeater states can be generated using only a single emitter. In particular, the interconnected core photons [see e.g. Fig.~\ref{fig:rgs}] as well as two of the photon arms can be generated in this way using only one emitter. For example, the portion of the $N=4$ RGS that can be created is shown in Fig.~\ref{fig:energylevels}(c). This state can be generated by performing the following sequence of operations on the emitter: ${\cal M}_ZH{\cal P}H{\cal P}H{\cal P}{\cal P}H{\cal P}H{\cal P}H|0\rangle$, followed by single-qubit $X$ and $Z$ gates on the four central photons. In this sequence, $|0\rangle$ denotes the ground state of the emitter, $H$ is a Hadamard gate, $\cal P$ is the pumping operation, and ${\cal M}_Z$ is a final $Z$-measurement performed on the emitter in order to decouple it from the chain of emitted photons. An arbitrarily large interconnected core of $N$ photons can be generated using a similar sequence containing a larger string of pumping operations in the middle: ${\cal M}_ZH{\cal P}H{\cal P}H{\cal P}^{N-2}H{\cal P}H{\cal P}H|0\rangle$.
To date, the most efficient method for constructing RGSs using only probabilistic fusion gates \cite{Pant_PRA17} does so by starting from many three-photon GHZ states (which are themselves generated from parametric down-conversion and fusion) and fusing them together sequentially to build up the necessary entanglement. Constructing the $N=4$ RGS state shown in Fig.~\ref{fig:rgs} in this way would require 5 successful fusion gate applications between pairs of three-photon GHZ states (more generally $2N-3$ fusion gates are needed for an RGS made of $2N$ photons). In contrast, our single-emitter scheme requires only 2 fusion gates to complete the $N=4$ RGS (more generally, $N-2$ fusion gates). Thus, we see that the utilization of a photonic emitter significantly decreases the necessary number of probabilistic fusion gates and the number of photon sources needed at each node, greatly reducing the overhead compared to what is needed to generate the entire state via fusion.
We now show that an additional dramatic reduction in the overhead can be achieved by introducing an ancilla qubit. In particular, we demonstrate that arbitrarily large repeater graph states can be completely generated using only a single emitter coupled to one ancilla qubit, which does not itself need to be an emitter. Inclusion of the ancilla greatly increases the flexibility one has in creating graph states. This is because entanglement between the emitter and the ancilla can be converted into entanglement between the ancilla and the emitted photons using single-qubit gates [see Fig.~\ref{fig:generation}(a)]. The emitter can therefore be used to attach multiple strings of entangled photons to the ancilla. Our scheme for deterministically generating RGSs combines this observation with the fact that RGSs are closely related to tree-like cluster states \cite{Varnava_PRL08}. In particular, the fully interconnected web of photons at the center of an RGS can be obtained by performing an operation known as local complementation (LC)\cite{Bouchet_DM93,Hein_PRA04} around the central ``root'' vertex of a tree, as shown in Fig.~\ref{fig:generation}(c). LC can be implemented by applying the single-qubit gate $e^{i\frac{\pi}{2}\frac{Y+Z}{\sqrt2}}$ to the root vertex and the gate $e^{i\frac{\pi}{2}\frac{X+Y}{\sqrt2}}$ to each of the neighboring vertices. Single-qubit photonic gates are much easier to implement than gates on the emitters; thus, the generation of tree states can easily be extended to repeater states via local complementation.
Fig.~\ref{fig:generation} summarizes our scheme for deterministically generating the all-photonic repeater states introduced in Ref.~\cite{Azuma_NC15}. Our procedure for generating the underlying tree states involves using the ancilla (labeled $A$) at the root as an ``anchor", and then using the emitter (labeled $B$) to generate each arm of the tree, one at a time. For each arm, the emitter is connected to the root vertex by applying a CZ gate between the emitter and ancilla. The emitter is then pumped twice as shown in the top parts of Figs.~\ref{fig:generation}(a),(b), Hadamard gates are applied to one of the photons and the emitter, and finally a $Z$-measurement is performed on the emitter. This has the effect of severing the emitter from the graph, leaving the other two photons attached to the ancilla in a chain as shown in the bottom parts of Figs.~\ref{fig:generation}(a),(b). The emitter can then be reinitialized, and the whole process can be repeated in order to generate an arbitrary number of arms in the tree cluster state. The full sequence of gates needed to create an RGS with an arbitrary number of arms $N$ (corresponding to an RGS of degree $N/2$) is given in the Appendix. The most challenging part of the sequence are the emitter-ancilla entangling CZ gates. In the Appendix, we show that the minimum number of CZ gates needed is $N-2$ if the ancilla is also an emitter or $N$ if the ancilla does not emit photons. Thus, if the ancilla is chosen to also be an emitter, the requisite number of two-qubit gates can be slightly reduced.
A crucial feature of our scheme is that all of the photons comprising an arbitrary repeater state can be emitted by the same emitter. This is illustrated in Fig.~\ref{fig:generation}, where the emitter ($B$) produces all of the photons, while the ancilla ($A$) never emits even a single photon. The role of the ancilla is to hold the different arms of the tree together while the emitter generates new arms and attaches them to the ancilla. The fact that all photons are emitted from a single emitter makes it far simpler to ensure that the photons comprising the RGS are indistinguishable, as is necessary for the functionality of repeater networks. It is of course still necessary to achieve uniformity between emitters on different nodes of the network. This will be considered further in the Discussion section.
It is also important to emphasize that the role of the ancilla qubit is very different from that of quantum memories in traditional repeater schemes. In traditional schemes, the quantum memories must remain coherent and entangled with photons during the time it takes the photons to reach the secondary nodes, during the time it takes to perform the Bell measurements, and during the time it takes to transmit classical heralding signals between nodes. In contrast, the ancilla qubit in our scheme only needs to be coherent and entangled with photons during the RGS generation process; once the state is formed, the ancilla is no longer required to remain coherent.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{tree.pdf}
\caption{A tree of depth $d=2$ with $k=6$ arms. Blue edges represent entanglement created by pumping an emitter, while red edges represent entanglement created with a CZ gate between an emitter and another emitter or ancilla.}
\label{fig:tree}
\end{figure}
Our method can be generalized to deterministically create arbitrary tree states. More general tree states are important because they can be combined with graph states such as the RGS shown in Fig.~\ref{fig:generation}(c) to create a similar state that is robust against photon losses (up to a rate of 50\%) \cite{Varnava_PRL08,Pant_PRA17}. To create a general tree state, one emitter/ancilla is needed for each level of the tree except for the bottommost one. In our proposed scheme, one ancilla is again used at the root as an ``anchor'', and an emitter is used to generate each arm using CZ gates as above. For each subtree, this emitter is then treated as the new ``anchor'' and a second emitter is used to generate each arm of the subtree. This process is continued recursively until the last emitter is pumped repeatedly to create the photons along the bottom of the tree (see Fig.~\ref{fig:tree}). This process only requires CZ gates between emitters/ancillas on neighboring levels, so the scheme can be implemented in any architecture with linearly aligned emitters/ancillas and requires CZ gates only between nearest neighbors. Emitters at every other level can be pumped twice for each vertex at that level, reducing the requisite number of CZ gates by 2 for each instance, as illustrated in Fig.~\ref{fig:tree}. The total number of CZ gates needed to generate a tree of depth $d$ with $k$ arms at each vertex is given by the following formula:
\begin{equation}
N_{CZ}=-1+\frac{k^{d}+(-1)^{d+1}}{k+1}.
\end{equation}
For trees with a large number of arms at each vertex, this is only marginally more efficient; however, this effect can make a drastic difference in the generation of much smaller trees. For a binary tree, in addition to reducing the number of CZ gates by a factor of 3, the number of emitters required is cut in half as well. For a general tree, it is also possible to replace all but one emitter with an ancilla qubit that does not emit photons, albeit at the expense of increasing the necessary number of CZ gates.
\begin{figure}
\centering
\includegraphics[width=.45\columnwidth]{treerepeater.pdf}
\includegraphics[width=.45\columnwidth]{treerepeater2.pdf}
\caption{Encoded RGS with (a) depth-one and (b) depth-two tree structures. Both of these can be generated by pumping a single emitter, which is coupled to an ancilla qubit via CZ gates.}
\label{fig:withtrees}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{bbnstate}
\caption{A large repeater state with $N{=}6$ (logical qubits) proposed in Ref.~\cite{Pant_PRA17} which includes subtrees in order to make the state more robust against errors.}
\label{fig:bbnstate}
\end{figure}
In Ref.~\cite{Pant_PRA17}, it was shown how to combine trees with RGSs to produce a much larger state that has built-in error correction and is robust against photon loss. The ability to perform $X$ and $Z$ measurements on the central photons is a crucial part of the repeater protocol. For the smaller RGSs discussed above, the loss of one of these central photons would cause the entire process to fail. The addition of trees attached to each of these central photons allows these measurements to be recovered in the case of photon loss \cite{Varnava_PRL06}. Fig.~\ref{fig:withtrees} (a) and (b) show two examples of repeater states with tree structures included; each of these states can be generated from only one emitter and one ancilla qubit. The more complex encoded state proposed in Ref.~\cite{Pant_PRA17}, shown in Fig.~\ref{fig:bbnstate}, can be deterministically generated using our scheme with only two emitters and one ancilla. Creating this state from 3-photon GHZ states using fusion requires roughly one fusion gate per photon, and for the particular size shown, 129 successful fusion gates would be needed. Using two emitters and one ancilla qubit instead, a similar number of pumping operations are needed, but only 24 entangling CZ gates are required between the emitters and ancilla. Alternatively, the emitter which generates the subtrees can be reattached to the middle emitter and pumped to generate the external arms, eliminating the need to pump the second emitter at the cost of $N$ additional CZ gates. For this size, 30 CZ gates are needed, and the process would require two ancilla qubits and only one emitter.
\begin{figure*}[ht]
\centering
\includegraphics[width=1.7\columnwidth]{fidelityfig.pdf}
\vspace{-1cm}
\caption{(a) The fidelity of the $N {=} 6$ RGS shown in Fig.~\ref{fig:rgs} as a function of the infidelities of the two-qubit CZ gate and the individual single-qubit gates applied to emitters/ancillas (which are assumed to all have the same infidelity). (b) The fidelity of the large repeater state shown in Fig.~\ref{fig:bbnstate}.}
\label{fig:fid1}
\end{figure*}
Our protocol for generating repeater states has many attractive features. First, RGSs of any size can be generated using only one emitter and one ancilla qubit, which may or may not emit photons. In addition, the only multi-qubit gates required are the CZ gates between the emitter and ancilla. Large trees can be generated with the requisite number of emitters/ancillas scaling as the depth of the tree, thus logarithmically in the total number of photons. We quantify the practicality of our scheme by finding the fidelity of the RGS as a function of the individual gate fidelities of the gates used in the RGS generation sequence. In Fig.~\ref{fig:fid1} we assume that the fidelities of single photonic gates and optical pumping are much higher than the fidelities of single unitary gates and CZ gates on the emitters/ancillas, and thus the fidelity of the final states are essentially determined by the latter two. Fig.~\ref{fig:fid1}(a) shows the fidelity for creating the bare repeater $N=6$ state shown in Fig.~\ref{fig:rgs}, where it is evident that this state can be created with greater than 90\% fidelity if CZ gate fidelities are above 99\% and single-qubit gate fidelities exceed 99.8\%. Fig.~\ref{fig:pumpfid} shows how the RGS fidelity depends on optical pumping, revealing that the demands on the CZ-gate fidelities do not significantly increase for pumping fidelities of around 99.7\%. The way in which these requirements scale with the size of the state is shown in Fig.~\ref{fig:fid90}. We see that increasing the size of the RGS by a factor of 2 requires the infidelity to decrease by roughly half. In the case of the error-correcting repeater state shown in Fig.~\ref{fig:bbnstate}, we see from Fig.~\ref{fig:fid1}(b) that reaching 90\% fidelity requires CZ gate fidelities around 99.8\% and single-qubit gate fidelities around 99.95\%.
Although the demands on gate fidelity increase with increasing RGS size, it is important to note that modest-sized RGSs may be sufficient for long-distance communication. This is because the probability that at least one successful entanglement swapping operation is achieved between each pair of adjacent primary nodes grows quickly with $N$. In particular, if each Bell measurement succeeds with probability 50\%, then the probability of successfully creating entanglement across the entire network is $(1-2^{-N/2})^{n+1}$, where $n$ is the number of primary nodes in the network. For a network containing $n=1000$ nodes, the probability of success is already 99.9\% using modest-sized RGSs with $N=40$.
\begin{figure}
\centering
\includegraphics[width=.85\columnwidth]{pumpingfidelity.pdf}
\caption{The fidelity of the $N=6$ RGS shown in Fig.~\ref{fig:rgs} as a function of the infidelity of the two-qubit CZ gate and the optical pumping infidelity. (The fidelity of single qubit gates is taken to be 99.9\%.)}
\label{fig:pumpfid}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fidelity_fixed_to_90v2.pdf}
\caption{The single-qubit gate and two-qubit CZ gate infidelities needed to create a RGS of size $N$ with 90\% fidelity. The fidelity of the RGS is given by $CZ^NU^{2N+2}P^{2N}$, where $CZ$, $U$, and $P$ are the fidelities of the CZ gates, single qubit gates on the emitters/ancillas, and pumping operations respectively.}
\label{fig:fid90}
\end{figure}
One significant advantage of the scheme for generating linear cluster states proposed in Ref.~\cite{Lindner_PRL09} is that it is fault-tolerant, namely that an error which occurs at one point during the process is contained locally and does not propagate to the rest of the state. Generating the photon arms is done using the same method, so an error on emitter $B$ or any of the photons will only affect one specific arm. Thus, for our procedure to be fault-tolerant, it is necessary only to guard ancilla qubit $A$ against errors. In physical implementations, qubit $A$ can be chosen to be a qubit which has a longer coherence time, making errors on qubit $A$ much less likely. Additionally dynamical decoupling pulses can extend the coherence time of qubit $A$ by orders of magnitude \cite{Maurer_Science12,Muhonen_NatNano14}. Alternatively, our protocol can be modified to make qubit $A$ a logical qubit with built-in error-correction. For example, in physical systems $T_2$ is significantly shorter than $T_1$, which causes $Z$ errors to be the primary errors. In order to guard against these, a logical qubit consisting of 3 physical qubits can be used. This scheme has been demonstrated experimentally using NV center and nuclear spins in diamond \cite{Waldherr_Nature14}. $|0_L\rangle$ is encoded as $|+++\rangle$ and $|1_L\rangle$ is encoded as $|---\rangle$. The logical CZ gate between the emitter $B$ and the physical qubits $1$, $2$, $3$ would be given by $H_1H_2H_3CCZ_{B12}CCZ_{B13}CCZ_{B23}H_1H_2H_3$, where $CCZ$ is a controlled CZ gate. This sequence of gates acts identically on logical states with and without a $Z$ error, meaning that a single $Z$ error could occur at any point during the protocol and no error correction would be required until directly before the final measurement.
\section*{Discussion}
There are several criteria in identifying systems to implement our scheme for deterministic all-photonic repeater state generation. First, the emitters must have the requisite level structure and selection rules, namely they must have two degenerate ground states, each of which is coupled to one corresponding excited state as depicted in Fig.~\ref{fig:energylevels}(a). Note that for the repeater states shown in Figs.~\ref{fig:rgs} and ~\ref{fig:withtrees}, only one emitter needs to satisfy this condition. The second criterion is that emitters/ancillas should be coupled to each other to enable the entangling CZ gates. Third, the emitted photons must be indistinguishable, both at different nodes and at the same node, over the time it takes to generate and entangle repeater states. Fourth, the extraction efficiency of the photons from the emitter should be high. Fifth, the setup should be able to incorporate coupling to fibers in order to transmit the photons to the remote nodes with high fidelity. Finally, it is desirable to have high yield, meaning that the photons are generated at high rates.
Self-assembled quantum dots (QDs) coupled to cavities satisfy all of these criteria. Their broken symmetry along the growth axis provides the desired level structure and selection rules, and pairs of quantum dots can be grown in a stacked configuration allowing for coupling between the two electron spins trapped in each dot via the exchange interaction \cite{Kim_NatPhys11}. This interaction is the enabling mechanism for the inter-emitter CZ gate (here the ancilla is also a quantum dot emitter). In fact, the early works for 1D \cite{Lindner_PRL09} and 2D \cite{Economou_PRL10} cluster state generation were based on quantum dots. In a recent breakthrough experiment, Schwartz et al. \cite{Schwartz_Science16} generated deterministically a cluster state string of five entangled photons using a confined dark exciton in a quantum dot. This work constituted a proof-of-principle demonstration that does not yet include optimization over the various metrics (photon generation rate, efficiency, etc). Over the last few years there has been great interest and rapid progress in single-photon devices based on quantum dots which can be harnessed for the repeater generation device we are proposing. One challenge traditionally associated with quantum dots is their spectral inhomogeneity, a severe issue for indistinguishability. Our scheme for bare repeater states avoids this difficulty since all photons can be produced by a single emitter. However, it remains a challenge for photons in different repeater states generated at different nodes. Several years ago, it was demonstrated that quantum dots can be tuned over a large spectral range, which allowed for photon interference coming from remote QDs \cite{Patel_NP10}, and very recently it was shown that such tuning can be successfully implemented in a QD-micropillar device \cite{Ding_PRL16}, demonstrating indistinguishability of photons over more than 10 microseconds \cite{Wang_PRL16}. Another challenge that was recently overcome is the difficulty of photon extraction. Within the last year, several groups have shown photon extraction rates ranging from 66\% \cite{Ding_PRL16} to more than 98\% \cite{Arcari_PRL14}, and very recently the chip-to-fiber coupling efficiency was shown to exceed 80\% \cite{Daveau_preprint16}. These advances stem from enhanced Purcell emission into waveguide modes, simultaneous reduction into other modes, and tapering of waveguides to improve chip-to-fiber coupling. Efforts to engineer these systems to even higher metrics are ongoing, and we anticipate near-ideal metrics over the next couple of years.
Other types of emitters aside from quantum dots can also be used, with recent work showing that several defect centers, including the NV center in diamond and in silicon carbide (SiC), as well as vacancy and divacancy defects in SiC, have the required level structure \cite{Economou_nano16}. Similar challenges as in quantum dots are being addressed for these systems, namely the broad photon emission, coupling to cavities and photon extraction. There has been significant experimental progress recently in terms of emitter indistinguishability \cite{Sipahigil_PRL12}, photon extraction efficiency \cite{Gould_PRApplied16}, and coupling to fibers \cite{Tiecke_Optica15}, and theoretical proposals for quantum networks based on defect centers have been proposed \cite{Childress_PRL06,Nemoto_PRX14,Vinay_PRA17}. Purcell enhancements amounting to a 70-fold increase of emission into the zero phonon line have been achieved in NV centers in diamond \cite{Faraon_PRL12,Li_NC15}, and protocols for suppressing spectral diffusion have been put forward \cite{Fotso_PRL16}. Very recent developments with photonic crystal cavities in SiC have demonstrated an 80-fold enhancement of selective photon emission into the desired zero phonon line, i.e., without enhancing the spectrally closest transition \cite{Bracher_preprint16}. On the other hand, the mechanism for the inter-emitter entangling CZ gate is not as clear as in the case of QDs, where dots can be stacked during growth. Recent experiments, however, have made progress toward controllable defect positioning \cite{Wang_preprint16}; this result can pave the way toward the design and demonstration of an entangling gate between nearest neighbor defects. Since a RGS can be generated from pumping only one emitter, the ancilla qubit could be a different system, such as a nuclear spin \cite{Fuchs_NP11,Dutt_Science07}.
Our protocol can also be implemented in atomic systems. Trapped ions are particularly promising due to the high level of control that has been demonstrated in these systems and to their ability to be coupled to cavities. To construct a $N=50$ RGS with fidelity 90\%, the required fidelity of single- and two-qubit gates is 99.99\% and 99.85\% respectively (see Fig.~\ref{fig:fid90}). These fidelities have been achieved in trapped ion systems \cite{Ballance_PRL16,Gaebler_PRL16}, for which spin-photon interfaces have also been demonstrated \cite{Blinov_Nature04,Olmschenk_Science09,Stute_Nature12}. Specifically, Ref.~\cite{Stute_Nature12} presented a spin-photon interface with high generation rates ($>$97\%) which moreover has the correct level structure to be compatible with our protocol.
Other systems that show promise are defects or quantum dots in 2D materials, including hexagonal boron nitride and transition metal dichalcogenides, where single photon emitters have been seen \cite{Srivastava_NatureNano15,He_NatureNano15,Koperski_NatureNano15,Chakraborty_NatureNano15,Tran_NatureNano16}. The 2D nature of these materials allows for high photon extraction rates, and coupling to photonic crystal cavities has already been demonstrated \cite{Wu_Nature15}.
With all the recent experimental developments, proof-of-principle demonstrations of our protocols for deterministic repeater graph states can be carried out using existing technology. These demonstrations will likely initially only achieve small sized RGSs due to current limits on two-qubit gate fidelities and to photon generation and extraction efficiencies. Nevertheless, the field of nano-photonics is very active, with higher quality material and device improvements occurring at a rapid pace \cite{Aharonovich_NP16}. These advances will lead to higher fidelities in the implementation of our designs.
An interesting future direction would be a rate-distance analysis of the all-optical repeaters generated with our technique, similarly to what has been carried out for the fusion-based approach \cite{Pant_PRA17}. Another key future effort would be to design in detail physical implementations of our scheme based on, e.g., quantum dots or color centers in solids. Realistic simulations with such systems would provide an important guide to experiment. On-chip architectures using state of the art photonic components could lead to advantages in miniaturization and scalability.
{\bf Acknowledgments:} This research was supported by NSF (Grant
No. 1741656).
\section*{Appendix: Gate sequences for RGS creation}
Here, we give the explicit sequences of single- and two-qubit gates needed to generate the various repeater or tree-like graph states presented above. The sequence of gates needed to create an RGS of the form introduced by Azuma et al. \cite{Azuma_NC15} with $N$ total arms is
\begin{equation}
\begin{split}
{\cal M}_A(e^{i\frac{\pi}{2}\frac{Y+Z}{\sqrt2}})_A \prod_{n=0}^{N-1}\Big[(e^{i\frac{\pi}{2}\frac{X+Y}{\sqrt2}})_{2n+1}H_{2n+2}{\cal M}_BH_B*\\CZ_{A,B}{\cal P}_{B,2n+2}{\cal P}_{B,2n+1}H_B\Big]*H_A|0\rangle.
\end{split}
\end{equation}
Here, the photons which make up the RGS are labeled 1 through $2N$, $H_n$ represents a Hadamard gate applied to the $n$th photon (the $n$th node of the graph), ${\cal M}_k$ represents a projective $Z$-measurement on emitter/ancilla $k$, $CZ_{A,B}$ is a two-qubit CZ gate acting on ancilla $A$ and emitter $B$, and ${\cal P}_{k,n}$ denotes the pumping/emission of the $n$th photon from emitter $k$. We note that the combination of gates ${\cal M}_A(e^{i\frac{\pi}{2}\frac{Y+Z}{\sqrt2}})_A$ and ${\cal M}_BH_B$ are equivalent to $Y$ and $X$ measurements respectively. CZ gates between the emitter and ancilla are generally the most difficult gates to implement. We can reduce the number of required CZ gates from $N$ to $N-2$ if the ancilla qubit is also an emitter, in which case the sequence becomes
\begin{widetext}
\begin{equation}
\begin{split}
(e^{i\frac{\pi}{2}\frac{X+Y}{\sqrt2}})_{2N-1}H_{2N}{\cal M}_A(e^{i\frac{\pi}{2}\frac{Y+Z}{\sqrt2}})_A{\cal P}_{A,2N}{\cal P}_{A,2N-1}(e^{i\frac{\pi}{2}\frac{Y+Z}{\sqrt2}})_A
\prod_{n=1}^{N-2}\Big[(e^{i\frac{\pi}{2}\frac{X+Y}{\sqrt2}})_{2n+1}H_{2n+2}{\cal M}_BH_BCZ_{A,B}*\\{\cal P}_{B,2n+2}{\cal P}_{B,2n+1}H_B\Big]*
(e^{i\frac{\pi}{2}\frac{X+Y}{\sqrt2}})_1H_2H_A{\cal P}_{A,2}{\cal P}_{A,1}H_A|0\rangle.
\end{split}
\end{equation}
\end{widetext}
$Z$ measurements of emitters can either be performed directly or by pumping the emitter and performing a $Z$-measurement on the photon that is produced. Instead of measuring emitter $B$ between every arm, it is also possible to pump the emitter an extra time and proceed directly to the next arm, postponing the measurements of these photons until after the complete state has been generated. In this case, the sequence becomes
\begin{widetext}
\begin{equation}
\begin{split}
\left(\prod_{n=1}^{N-1}{\cal M}_{3n+2}\right)*(e^{i\frac{\pi}{2}\frac{X+Y}{\sqrt2}})_{3N-3}H_{3N-2}(e^{i\frac{\pi}{2}\frac{X+Y}{\sqrt2}})_{3N-1}{\cal P}_{A,3N-1}(e^{i\frac{\pi}{2}\frac{Y+Z}{\sqrt2}})_A{\cal P}_{A,3N-2}{\cal P}_{A,3N-3}(e^{i\frac{\pi}{2}\frac{Y+Z}{\sqrt2}})_A*\\
\prod_{n=1}^{N-2}\left[(e^{i\frac{\pi}{2}\frac{X+Y}{\sqrt2}})_{3n}H_{3n+1}{\cal P}_{B,3n+2}H_BCZ_{A,B}{\cal P}_{B,3n+1}{\cal P}_{B,3n}H_B\right]*
(e^{i\frac{\pi}{2}\frac{X+Y}{\sqrt2}})_1H_2H_A{\cal P}_{A,2}{\cal P}_{A,1}H_A|0\rangle.
\end{split}
\end{equation}
\end{widetext}
Alternatively any of the $e^{i\frac{\pi}{2}\frac{Y+Z}{\sqrt2}}$, $H$, or $e^{i\frac{\pi}{2}\frac{X+Y}{\sqrt2}}$ gates can be replaced with $e^{i\frac{\pi}{4}X}$, $e^{i\frac{\pi}{4}Y}$, or $e^{i\frac{\pi}{4}Z}$ respectively. This will produce an equivalent state from which the standard graph state can be recovered by applying $Z$ gates on some of the final photons, depending on which gates were used. Such a correction would also be necessary if emitter $B$ is not reinitialized to $|0\rangle$ between measurement and reentanglement with the rest of the graph state. The particular photons that need to be corrected by $Z$ gates can easily be determined by finding the stabilizer group of the produced state.
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| 3,288
|
{"url":"https:\/\/www.fastcalculus.com\/calculus-fundamental-theorem-of-calculus-28639\/","text":"# Calculus: Fundamental Theorem of Calculus \u2013 #28639\n\nQuestion: Compute the following definite integral\n\n$\\int_{1}^{3}{\\left( {{x}^{2}}-9 \\right)}dx$","date":"2019-10-17 07:39:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.9986350536346436, \"perplexity\": 2564.198760858703}, \"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-2019-43\/segments\/1570986673250.23\/warc\/CC-MAIN-20191017073050-20191017100550-00339.warc.gz\"}"}
| null | null |
Members of Congress to Clinton: Revisit Decision on M1 Rifles and M1 Carbines
U.S. senators and representatives from both sides of the aisle are urging Secretary of State Hillary Rodham Clinton to revisit the State Department's March 2010 decision disallowing the importation of M1 rifles and M1 carbines from South Korea.
In a letter to Secretary Clinton, Sen. John Cornyn (R-Tex.) and 15 other senators state that the importation disapproval "amounts to no more than a backdoor gun ban that lacks any basis or justification under current Federal law and policy" and "violates law-abiding citizens' constitutional right, protected under the Second Amendment, to purchase these firearms for legitimate purposes such as target shooting, hunting, collecting, and self-protection."
The senators question the department's opinion that the rifles "could potentially be exploited by individuals seeking firearms for illicit purposes," and request from Secretary Clinton "an explanation of your reasons for blocking the importation and sale of American-made rifles from South Korea."
In a separate letter to Secretary Clinton, Sen. Jim Webb (D-Va.) also disagreed with the department's opinion that importation of the rifles would constitute a public safety risk, saying "The importation of these antique rifles . . . does not pose a security threat to our nation." Sen. Webb added, "Hundreds of thousands of these firearms are already in the United States, and substantially more advanced and powerful firearms are already available."
In another letter to Clinton, Congressman John Boozman (R-Ark.) and 65 other members of the House also objected to the department's stated concern that the rifles might be "exploited . . . for illicit purposes," calling it "a reiteration of tired arguments by gun control advocates." The Boozman letter also noted "these are the very same types of rifles that have been sold by the federal government to civilians for decades through the Civlian Marksmanship Program.
In yet another letter to Clinton, Congressman Joe Donnelly (D-Ind.) and 44 other members of the House of Representatives noted that "the M1 is one of the two rifles most commonly used at the National Matches, a marksmanship competition authorized by federal law" and that "there are separate competitions dedicated to each of the two rifles" (the M1 rifle and the M1 carbine). Rep. Donnelly's letter, like Sen. Cornyn's, noted that NICS checks would be required on any of the rifles sold in the United States, as would be the case with any imported firearm.
Meanwhile Congresswoman Cynthia Lummis (R-Wyo.), who signed Rep. Donnelly's letter, has introduced H.R. 6240, The Collectible Firearms Protection Act, which would allow for the importation of lawfully importable U.S.-origin surplus firearms without the approval of the Departments of State or Defense.
Of course, Congresswoman Carolyn McCarthy (D-N.Y.), who on several occasions has introduced legislation that would ban the M1 rifle and the M1 carbine as "assault weapons" didn't sign either the Boozman or Donnelly letters.
NRA-ILA Investigates Citibank Issue
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
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| 1,318
|
Q: Can anyone tell me how to add icons to tree-control in MFC I have a simple tree control, so i want to add some icons to my tree control nodes.
DDX_Control(pDX, IDC_TREE1, m_TreeView);
m_TreeView.InsertItem(L"Skills");
HTREEITEM main = m_TreeView.InsertItem(L"Technical");
m_TreeView.InsertItem(L"C++", main);
m_TreeView.InsertItem(L"Java", main);
m_TreeView.InsertItem(L".Net", main);
m_TreeView.InsertItem(L"Python", main);
HTREEITEM main1 = m_TreeView.InsertItem(L"Non_Technical");
m_TreeView.InsertItem(L"Admin", main1);
m_TreeView.InsertItem(L"HR", main1);
The above lines are to create the Tree-Control, So i want to create the icons with my nodes..Can anyone tell me the code for adding icons to tree control.
Thanks in advance...
A: First of all you need to create CImageList object instance.
m_TreeIcons.Create(16, 16, ILC_COLOR32|ILC_MASK, 0, 1);
You can use either bitmap or icon as image source.
m_FileIcons.Add(AfxGetApp()->LoadIcon(IDI_FOLDER));
m_FileIcons.Add(AfxGetApp()->LoadIcon(IDI_FILE));
And the last step is to bind your image list with your tree:
m_Tree.SetImageList(&m_TreeIcons, LVSIL_SMALL);
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,030
|
\partial{\partial}
\def\mathcal{\mathcal}
\def\text{Var\/}{\text{Var\/}}
\newcommand{\boldsymbol}{\boldsymbol}
\newcommand{\pr}[1]{\mathbb{P}\left( #1 \right)}
\newcommand{\mean}[1]{\mathbb{E}\left[{#1}\right]}
\def{\Bbb E}{{\Bbb E}}
\def\bar{q}{\bar{q}}
\newcommand{\brac}[1]{\left(#1\right)}
\newcommand{\bfrac}[2]{\brac{\frac{#1}{#2}}}
\newcommand{\multstar}[1]{\begin{multline*}#1\end{multline*}}
\newcommand{\mult}[2]{\begin{multline}\label{#1}#2\end{multline}}
\newcommand{\card}[1]{\left|#1\right|}
\newcommand{\beq}[2]{\begin{equation}\label{#1}#2\end{equation}}
\newcommand{\ceil}[1]{\left \lceil #1 \right \rceil}
\newcommand{\floor}[1]{\left \lfloor #1 \right \rfloor}
\title[Age-biased attachment graphs]{Giant descendant trees, matchings and independent sets in the age-biased attachment graphs.}
\author{Huseyin Acan}
\address{Department of Mathematics, Drexel University, Philadelphia, PA 19104}
\thanks{During the time of this research, the first coauthor was supported by NSF Fellowship (Award No.~1502650).}
\email{huseyin.acan@drexel.edu}
\author{Alan Frieze}
\address{Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213}
\thanks{Research of the second author supported in part by NSF Grant DMS 1661063}
\email{alan@random.math.cmu.edu}
\author{Boris Pittel}
\address{Department of Mathematics, Ohio State University, Columbus, OH 43210}
\email{bgp@math.ohio-state.edu}
\begin{document}
\begin{abstract} We study two models of an age-biased graph process: the $\delta$-version of the preferential attachment graph model (PAM) and the uniform attachment graph model (UAM), with $m$ attachments for each of incoming vertices. We show that almost surely the scaled size of a breadth-first (descendant) tree rooted at a fixed vertex converges, for $m=1$, to a limit whose distribution is a mixture of two beta-distributions and a single beta-distribution respectively, and that for $m>1$ the limit is $1$. We also analyze the likely performance of two greedy (online) algorithms, for a large matching set and a large independent set, and determine--for each model and
each greedy algorithm--both a limiting fraction of vertices involved and an almost sure convergence rate.
\end{abstract}
\keywords
{random, preferential attachment graphs, asymptotics}
\subjclass[2010] {05C05, 05C07, 05C30, 05C80, 60C05}
\maketitle
\section{Introduction}
It is widely accepted that graphs/networks are an inherent feature of life today. The classical models $G_{n,m}$ and $G_{n,p}$ of Erd\H{o}s and R\'enyi \cite{ER} and Gilbert \cite{Gi}, respectively, lacked some salient features of observed networks. In particular, they failed to have a degree distribution that decays polynomially. Barab\'asi and Albert \cite{BarAlb} suggested the Preferential Attachment Model (PAM) as a more realistic model of a ``real world'' network. There was a certain lack of rigour in \cite{BarAlb}, and later Bollob\'as, Riordan, Spencer and Tusn\'ady \cite{Bol1} gave a rigorous definition.
Many properties of this model have been studied. Bollob\'as and Riordan \cite{BolRio} studied the diameter and proved that with high probability (whp) PAM with $n$ vertices and $m>1$ attachments for every incoming vertex has diameter $\approx \log n/\log\log n$. Earlier result by Pittel \cite{Pit0} implied
that for $m=1$ whp the diameter of PAM is of exact order $\log n$.
Bollob\'as and Riordan \cite{BolRio3,BolRio4} studied the effect on component size from deleting random edges from PAM and showed that it is quite robust whp. The degree distribution was studied in Mori \cite{Mor0,Mor}, Flaxman, Frieze and Fenner \cite{FFF}, Berger, Borgs, Chayes and Saberi \cite{BerBorChaSab}. Pek\"oz, R\"ollin and Ross \cite{PekRolRos} established convergence, with rate, of the joint
distribution of the degrees of finitely many vertices. Acan and Hitczenko \cite{AcaHit} found an alternative proof, without rate, via a memory game. Pittel
\cite{Pit2} used the Bollob\'as-Riordan pairing model to approximate, with explicit error estimate, the degree sequence of the first $n^{m/(m+2)}$
vertices, $m\ge 1$, and proved that, for $m>1$, PAM is connected with probability $\approx 1-O((\log n)^{-(m-1)/3})$. Random walks on PAM have been considered in the work of Cooper and Frieze \cite{CF1,CF2}. In the first paper there are results on the proportion of vertices seen by a random walk on an evolving PAM and the second paper determines the asymptotic cover time of a fully evolved PAM. Frieze and Pegden \cite{one} used random walk in a ``local algorithm'' to find vertex 1, improving results of Borgs, Brautbar, Chayes, Khanna and Lucier \cite{BBCKL}. The mixing time of such a walk was analyzed in Mihail, Papadimitriou and Saberi \cite{Mihail} who showed rapid mixing. Interpolating between Erd\H{o}s-R\'enyi and preferential attachment, Pittel \cite{Pit1} considered birth of a giant component
in a graph process $G_M$ on a fixed vertex set, when $G_{M+1}$ is obtained by inserting a new edge between vertices $i$ and $j$ with probability proportional to $[\text{deg}(i)+\delta]\cdot[\text{deg}(j)+\delta]$, with $\delta>0$ being fixed. Confirming a conjecture of Pittel~\cite{Pit1}, Janson and Warnke~\cite{JW} recently determined the asymptotic size of the giant component in the supercritical phase in this graph model.
The previous paragraph gives a small sample of results on PAM that can be related to its role as a model of a real world network. It is safe to say that PAM has now been accepted into the pantheon of random graph models that can be studied purely from a combinatorial aspect. For example, Cooper, Klasing and Zito \cite{CKZ} studied the size of the smallest dominating set and Frieze, P\'erez-Gim\'enez, Pra\l{}at and Reiniger \cite{FGPR} studied the existence of perfect matchings and Hamilton cycles.
One source of our inspiration was the work of Mori \cite{Mor0, Mor}, see also Katona and M\'ori \cite{KatMor}, Hofstad \cite{Hof}. They were able to construct a family of martingales in a form of factorial products, with arguments being the degrees of individual vertices. This allowed them to analyze the limiting behavior of vertex degrees in a $\delta$-version of PAM. In this paper we construct a new factorial-type martingale
with an argument being the total size of a ``descendants'' subtree. This is a generalization of the martingale
for $\delta=0$, found by Pittel \cite{Pit2}.
\section{Our Results}\label{sec:results}
For each of the two models, PAM and UAM, we study the descendants tree of a given vertex $v$; it is a {\it maximal\/} subtree rooted at $v$ and formed by increasing paths starting at $v$.
The number of vertices in this subtree is a natural influence measure of vertex $v$.
We also analyze the performance of two on-line greedy algorithms, for finding a large matching set and for a large independent set.
We carry out this analysis in the context of the PAM graph process described in Bollob\'as~\cite{Bol1} and its extension taken from Hofstad \cite[Ch.\ 8]{Hof}, and the UAM graph process, see \cite{FGPR} and \cite{AcaPit}.
\subsection*{The PAM graph process, $\delta$-extension:}
Vertex $1$ has $m$ loops, so its degree is $2m$ initially. Recursively, vertex $t+1$ has $m$ edges, and it uses them one at a time either to connect to a vertex $x\in [t]$ or to loop back on itself.
To be more precise, let us denote by $d_{t,i-1}(x)$ the degree of vertex $x$ just before the $i$-th edge of vertex $t+1$ arrives. Denoting by $w$ the random receiving end of the $i$-th edge emanating from vertex $t+1$, we have
\begin{equation}\label{defprob}
\mathbb P(w=x) =
\begin{cases}
\displaystyle \frac{d_{t, i-1}(x) +\delta}{(2m + \delta)t + 2i-1 + i\delta/m}, & \text{ if } x\in[t],\\
& \\
\displaystyle \frac{d_{t,i-1}(t+1) +1+i\delta/m}{(2m + \delta)t + 2i-1 + i\delta/m}, & \text{ if } x=t+1.
\end{cases}
\end{equation}
We will use the notation $\{G_{m,\delta}(t)\}$ for the resulting graph process.
For $m=1$, writing $d_t(x)$ for $d_{t,0}(x)$, the probabilities above can be written a little more simply
\begin{equation}\label{n1}
\pr{w=x}=
\begin{cases}
\displaystyle \frac{1+\delta}{(2+\delta)t+(1+\delta)}, & \text{ if } x = t+1,\\
&\\
\displaystyle \frac{d_t(x)+\delta}{(2+\delta)t+(1+\delta)}, & \text{ if } x \in [t].
\end{cases}
\end{equation}
Bollob\'as and Riordan~\cite{BolRio} discovered the following coupling between $\{G_{m,0}(t)\}_t$ for $m>1$ and $\{G_{1,0}(mt)\}_t$. Start with the $\{G_{1,0}(t)\}$ random process and let the vertices be $v_1,v_2,\dots$. To obtain $\{G_{m,0}(t)\}$ from $\{G_{1,0}(mt)\}$,
\begin{enumerate}
\item collapse the first $m$ vertices $v_1,\dots,v_m$ into the first vertex $w_1$ of $G_{m,0}(t)$, the next $m$ vertices $v_{m+1},\dots,v_{2m}$ into the second vertex $w_2$ of $G_{m,0}(t)$, and so on;
\item keep the full record of the multiple edges and loops formed by collapsing the blocks $\{v_{(i-1)m+1}, \dots, v_{im}\}$ for each $i$.
\end{enumerate}
Doing this collapsing indefinitely we get the jointly defined Bollob\'as-Riordan graph processes $\{G_{m,0}(t)\}$ and $\{G_{1,0}(mt)\}$.
The beauty of the $\delta$-extended Bollob\'as-Riordan model is that similarly this collapsing operation applied to the process $\{G_{1,\delta/m}(mt)\}$ delivers the process $\{G_{m,\delta}(t)\}$, Hofstad~\cite{Hof}.
(For the reader's convenience we present the explanation in Appendix.)
\begin{remark}\label{rem:delta=-m}
Note that the process is well defined for $\delta\ge -m$ since for such $\delta$, all the probabilities defined in~\eqref{defprob} are nonnegative and add up to 1.
For $m=1$ and $\delta=-1$, it is easy to see from \eqref{n1} that there is no loop in the graph except the loop on the first vertex.
Hence, a vertex $u>1$ starts with degree 1 and then its degree does not change since as long as $d_t(u) + \delta = 0$, the vertex cannot attract any neighbors (again from \eqref{n1}).
As a result, in this case the graph is a star centered at vertex 1.
It follows from the above coupling that $G_{m,-m}(t)$ is also a star centered at vertex $1$ and the key problems we want to solve have trivial solutions in this extreme case.
\end{remark}
\subsection*{The UAM graph process}
Conceptually close to the preferential attachment model is the uniform attachment model (UAM). In this model, vertex $t+1$ selects uniformly at random (repetitions
allowed) $m$ vertices from the set $[t]$ and attaches itself to these vertices. (See Acan and Pittel \cite{AcaPit} for connectivity and bootstrap percolation results.)
This model can be thought of the limit of the PAM model as $\delta\to \infty$ except that loops are not allowed in this case.
\subsection{Number of Descendants}
Fix a positive integer $r$ and let $X(t)$ denote the number of descendants of $r$ at time $t$. Here $r$ is a descendant of $r$ and $x$ is a descendant of $r=O(1)$ if and only if $x$ chooses to attach itself to at least one descendant of $r$ in Step $x$.
In other words, if we think of the graph as a directed graph with edges oriented towards the smaller vertices, vertex $x$ is a descendant of $r$ if and only if there is a directed {\it decreasing\/} path from $x$ to $r$. In Pittel~\cite{Pit2} $X(t)$ was proposed as an influence measure of vertex $r$ at time $t$.
We prove two theorems.
\begin{theorem}\label{th1}
Suppose that $m=1$ and $\delta>-1$ and set $p(t):=X(t)/t$. Then almost surely (i.e. with probability $1$), $\lim p(t)$ exists, and its distribution is the mixture of two beta-distributions, with parameters
$a=1$, $b=r-\frac{1}{2+\delta}$ and $a=\frac{1+\delta}{2+\delta}$, $b=r$, weighted by $\frac{1+\delta}{(2+\delta)r-1}$ and $\frac{(2+\delta)(r-1)}{(2+\delta)r-1}$ respectively. Consequently a.s. $\liminf_{t\to\infty}p(t)>0$.
\end{theorem}
\noindent {\bf Note.\/} {\bf (i)\/} The proof is based on a new family of martingales $M_{\ell}(t):=\frac{\left(X(t)+\frac{\gamma}{2+\delta}\right)^{(\ell)}}{(t+\beta)^{(\ell)}}$,
$(z)^{(\ell)}$ standing for the rising factorial. This family definitely resembles the martingales Mori \cite{Mor0,Mor} used for the individual vertices' degrees. For instance, if $D_j(t)$ is the degree of vertex $j$ at time $t\ge j$, then for some deterministic $\gamma_k(t)$, $Z_{j,k}(t):= \gamma_k(t)(D_j(t)+\delta)^{(k)}$ is a martingale,
Hofstad \cite{Hof}. However, $M_{\ell}(t)$ depends on $X(t)$, a {\it global\/} parameter of the PAM graph. Unsurprisingly, the proof that each $M_{\ell}(t)$ is indeed a martingale requires a very different method.
{\bf (ii)\/} Whp $G_{1,\delta}$ is a forest of $\Theta(\log t)$ trees rooted at vertices with
loops. For the preferential attachment tree (no loops), Janson \cite{Jan} recently proved that the scaled sizes of the {\it principal\/} subtrees, those rooted at the root's children and ordered chronologically, converge a.s. to the {\it GEM\/} distributed random variables. His techniques differ significantly.
For $m>1$, we use Theorem \ref{th1} to prove a somewhat surprising result that, for $r=O(1)$, almost surely all but a vanishingly small fraction of vertices are descendants of the vertex $r$, (cf.~\cite{Jan}).
\begin{theorem}\label{th2}
Let $m>1$ and $\delta>-m$ and let $p_X(t)=X(t)/t$, $p_Y(t)=Y(t)/(2mt)$, where $Y(t)$ is the total degree of the descendants of $r$ at time $t$.
Then almost surely $\lim_{t\to\infty} p_X(t)=\lim_{t\to\infty} p_Y(t)=1$.
\end{theorem}
For the case of UAM, we have the following result.
\begin{theorem}\label{th2+}
Consider the UAM graph process $G_{t,m}$. Given \color{blue}{$r>1$}, let $X(t)$ be the cardinality of the descendant tree rooted at vertex \color{blue}{$r$}, and let $p(t):=X(t)/t$.
\begin{enumerate}
\item[\textup{(i)}] For $m=1$, almost surely, $\lim p(t)$ exists and
it has the same distribution as the minimum of $(r-1)$ independent $[0,1]$-Uniforms. Consequently a.s. $\liminf_{t\to\infty}p(t)>0$.
\item[\textup{(ii)}] For $m>1$, almost surely $\lim_{t\to\infty} p(t)=1$.
\end{enumerate}
\end{theorem}
\subsection{Greedy Matching Algorithm}
We analyze a greedy matching algorithm; a.s. it delivers a surprisingly large matching set even for relatively small $m$. This algorithm generates the increasing sequence $\{ M(t)\}$ of partial matchings on the sets $[t]$, with $ M(1)=\emptyset$. Suppose that $X(t)$ is the set of unmatched vertices in $[t]$ at time $t$. If $t+1$ attaches itself to a vertex $u\in X(t)$, then $M(t+1)=M(t)\cup\{\{u,t+1\}\}$, otherwise $M(t+1)=M(t)$. (If $t+1$ chooses multiple vertices from $X(t)$, then we pick one of those as $u$ arbitrarily.)
Consider first the PAM graph.
Let
\[
h(z)=h_{m,\delta}(z):= 2\left[1-\bfrac{m+\delta}{2m+\delta}z\right]^m-z-1,
\]
and let $\rho=\rho_{m,\delta}$ be the unique root $\rho=\rho_{m,\delta}$ in the interval $[0,1]$ of $h(z)=0$: $\rho_{m,\delta}\in (0,1)$ if $\delta>-m$.
\begin{theorem}\label{th3}
Let $M(t)$ and $X(t)$ be the set of greedy matching and the set of uncovered vertices at time $t$, and let $x(t)=X(t)/t$.
For any $\delta>-m$ and $\alpha<1/3$, almost surely,
\[
\lim_{t\to\infty} t^{\alpha}\max\{0, x(t)-\rho_{m,\delta}\}=0.
\]
In consequence, the Greedy Matching Algorithm a.s. finds a sequence of nested matchings $\{M(t)\}$, with $M(t)$ of size $(1-o(1))(1-\rho_{m,\delta})t/2$, at least.
\end{theorem}
\begin{Remark}
Observe that $\rho_{m,-m}=1$, which makes it plausible that the maximum matching size is minuscule compared to $t$.
In fact, by Remark~\ref{rem:delta=-m}, $G_{m,-m}(t)$ is the star centered at vertex $1$ and hence the maximum matching size is 1.
\end{Remark}
\begin{Remark} Consider the case $\delta=0$. Let $r_m:=1-\rho_{m,0}$; some values of $r_m$ are:
\begin{equation}\label{rvalues}
\begin{alignedat}{3}
&r_1=0.5000, \qquad &&r_2=0.6458, \qquad &&r_5=0.8044,\\
&r_{10}=0.8863,&&r_{20}=0.9377, &&r_{70}=0.9803.
\end{alignedat}
\end{equation}
With a bit of calculus, we obtain that $r_m=1- 2m^{-1}\log2 +O(m^{-2})$.
\end{Remark}
\begin{theorem}\label{thm4} Let $M(t)$ denote the greedy matching set after $t$ steps of the UAM process.
Let $r_m$ denote a unique positive root of $2(1-z^m) -z=0$: $r_m=1-m^{-1}\log 2+O(m^{-2})$.
Then, for any $\alpha<1/3$, almost surely
\[
\lim_{t\to\infty} t^{\alpha}\biggl| \frac{2|M(t)|}{t}-r_m\biggr|=0.
\]
\end{theorem}
\noindent Some values of $r_m$ in this case are:
\begin{alignat*}{3}
&r_1=0.6667, \qquad &&r_2=0.7808, \qquad &&r_5=0.8891,\\
&r_{10}=0.9386, &&r_{20}=0.9674, &&r_{35}=0.9809.
\end{alignat*}
\subsection{Greedy Independent Set Algorithm} The algorithm generates an increasing sequence of independent sets
$\{I(t)\}$ on vertex sets $[t]$. Namely, $I(1)=\{1\}$, and $I(t+1)=I(t)\cup\{t+1\}$ if $t+1$ does not select any of the vertices in $I(t)$; if it does, then $I(t+1)=I(t)$. $I(t)$ is also a dominating set for the PAM/UAM graph with vertex set $[t]$;
indeed if a vertex $\tau\in [t]\setminus I(t)$ did not have any neighbor in $I(t)$, then vertex $\tau$ would have been added to $I(\tau-1)$ at step $\tau$. (Pittel \cite{Pit-1} analyzed performance of this algorithm applied to Erd\H os--R\'enyi random
graph with a large, but fixed vertex set.)
For the PAM case we prove
\begin{theorem}\label{thm5} Let $w_m$ denote the unique root of $-w+(1-w)^m$ in $(0,1)$. For any
$\chi\in \Bigl(0,\min\Bigl\{\frac{1}{3},\frac{2m+2\delta}{3(2m+\delta)}\Bigr\}\Bigr)$, almost surely
\begin{equation}\label{PAM0}
\lim_{t\to\infty}t^{\chi}\biggl|\frac{|I(t)|}{t}-w_m\biggr|=0.
\end{equation}
\end{theorem}
\begin{remark} Thus the limiting scaled size of the greedy independent set does not depend on $\delta$, but
the convergence rate does.
\end{remark}
For the UAM case we prove an almost identical
\begin{theorem}\label{thm6} Let $w_m$ be the unique positive root of $-w+(1-w)^m$ in $(0,1)$. Then, for any $\alpha<1/3$, almost surely
\[
\lim_{t\to\infty}t^{\alpha}\Big|\frac{|I(t)|}{t} - w_m\Big|=0.
\]
\end{theorem}
\begin{remark}\label{w_m}
Let $w_m$ the unique positive root of $(1-w)^m-w$ in $(0,1)$. As $m\to\infty$,
\[
w_m= \frac{\log m}{m}+O\bigl(m^{-1}\log\log m\bigr).
\]
So, for $m$ large and both models, a.s. for all large enough $t$ the greedy algorithm delivers an independent set containing a fraction $\sim\frac{\log m}{m}$
of all vertices in $[t]$. It was proved in \cite{FGPR} that for each large $t$ with probability $1-o(1))$ the fraction of vertices in the largest independent set in the PAM graph process and in the UAM graph process is at most $(4+o(1))\frac{\log m}{m}$ and $(2+o(1))\frac{\log m}{m}$,
respectively. We conjecture that for each of the processes there exists a corresponding constant $c$ such that, for
$m\to\infty$, a.s. for all large $t$ the largest independent set contains a fraction $\sim c\frac{\log m}{m}$ of all $t$ vertices.
Since $I(t)$ is dominating, our results prove, for $m$ large, a.s. existence for all large $t$ of relatively small
dominating sets, of cardinality $\sim t\frac{\log m}{m}$.
\end{remark}
\section{Descendant trees}\label{sec:descendant}
Instead of referring the reader back to ``Our results'' section, we start this, and other proof sections,
with formulating in full the claim in question.
\subsection*{Proof of Theorem~\ref{th1}}
\begin{customthm}{2.2}
Suppose that $m=1$ and $\delta>-1$ and set $p(t):=X(t)/t$. Then almost surely (i.e. with probability $1$), $\lim p(t)$ exists, and its distribution is the mixture of two beta-distributions, with parameters
$a=1$, $b=r-\frac{1}{2+\delta}$ and $a=\frac{1+\delta}{2+\delta}$, $b=r$, weighted by $\frac{1+\delta}{(2+\delta)r-1}$ and $\frac{(2+\delta)(r-1)}{(2+\delta)r-1}$ respectively. Consequently a.s. $\liminf_{t\to\infty}p(t)>0$.
\end{customthm}
\begin{proof}For $t\ge r>1$, let $X(t)=X_{m,\delta}(t)=X_{m,\delta}(t,r)$ and $Y(t)=Y_{m,\delta}(t)=Y_{m,\delta}(t,r)$ denote the size and the total degree of the vertices in the vertex set of the subtree $T(t)=T_{m,\delta}(t,r)$ rooted at $r$; so $X(r)=1$ and $Y(r)\in [m,2m]$, where $m$ ($2m$ resp.) is attained when vertex $r$ forms no loops (forms $m$ loops resp.) at itself. Introduce $p(t)=p_Y(t)=\frac{Y(t)}{2mt}$ and $p_X(t)=\frac{X(t)}{t}$. This notation will be used in the proof of Theorem \ref{th2} as well, but of course $m=1$ Theorem~\ref{th1}.
Here
\[
Y(t)=\left\{\begin{aligned}
&2X(t),&&\text{if }r\text{ looped on itself},\\
&2X(t)-1,&&\text{if }r\text{ selected a vertex in }[r-1].\end{aligned}\right.
\]
(In particular, $p_X(t) =p(t)+O(t^{-1})$.) So, by \eqref{n1},
\begin{equation*}
\begin{aligned}
\mathbb P(X(t+1)&=X(t)+1|\circ)=\frac{Y(t)+\delta X(t)}{(2+\delta)t +(1+\delta)}\\
&=\left\{\begin{aligned}
&\frac{(2+\delta)X(t)}{(2+\delta)t+(1+\delta)},&&\text{if }r\text{ looped on itself},\\
&\frac{(2+\delta)X(t)-1}{(2+\delta)t+(1+\delta)},&&\text{if }r\text{ selected a vertex in }[r-1].\end{aligned}\right.
\end{aligned}
\end{equation*}
Thus we are led to consider the process $X(t)$ such that
\begin{align*}
\mathbb P(X(t+1)=X(t)+1|\circ) &= \frac{(2+\delta)X(t)+\gamma}{(2+\delta)t+(1+\delta)}\\
\mathbb P(X(t+1)=X(t)|\circ) &= 1 - \mathbb P(X(t+1)=X(t)+1|\circ) ,
\end{align*}
$\gamma=0$ if $r$ looped on itself, $\gamma=-1$ if $r$ selected a vertex in $[r-1]$.
Letting $\beta=\frac{1+\delta}{2+\delta}$, the above equation can be written as
\beq{jumpup}{
\begin{aligned}
\mathbb P(X(t+1)=X(t)+1|\circ) &= \frac{X(t) + \gamma/(2+\delta)}{t+\beta}\\
\mathbb P(X(t+1)=X(t)|\circ) &= 1 - \frac{X(t) + \gamma/(2+\delta)}{t+\beta}.
\end{aligned}
}
For $\delta=0$ the following claim was proved in Pittel \cite{Pit2}. We denote by $z^{(\ell)}$ the rising factorial $\prod_{j=0}^{\ell-1}(z+j)$.
\begin{lemma}\label{Lem1}
Let $\beta=\frac{1+\delta}{2+\delta}$ and $Z(t) = X(t) + \frac{\gamma}{2+\delta}$. Then, conditioned on the attachment record during the time interval
$[r,t]$, i.e. starting with attachment decision by vertex $r$, we have
\[
\mean{Z(t+1)^{(\ell)}\big| \circ}= \bfrac{t+\beta+\ell}{t+\beta} Z(t)^{(\ell)}.
\]
Consequently $M(t):=\frac{Z(t)^{(\ell)}}{(t+\beta)^{(\ell)}}$ is a martingale.
\end{lemma}
\begin{proof}
By \eqref{jumpup}, we have: for $k\ge 1$, and $t\ge r$,
\begin{equation}\label{2n}
\begin{aligned}
\mathbb E[Z^k(t+1)|\circ]&=(Z(t)+1)^k\,\frac{Z(t)}{t+\beta}+Z^k(t)\left(1-\frac{Z(t)}{t+\beta}\right)\\
&=\frac{Z(t)}{t+\beta}\sum_{j=0}^k\binom{k}{j} Z^j(t)+Z^k(t)\left(1-\frac{Z(t)}{t+\beta}\right)\\
&=Z^k(t)+\frac{Z(t)}{t+\beta}\sum_{j=0}^{k-1}\binom{k}{j} Z^j(t)\\
&=Z^k(t)\frac{t+\beta+k}{t+\beta}+\frac{1}{t+\beta}\sum_{j=1}^{k-1}\binom{k}{j-1}Z^j(t).
\end{aligned}
\end{equation}
Next recall that
\begin{equation}\label{Com1}
z^{(\ell)}=\sum_{k=1}^{\ell} z^k s(\ell,k),
\end{equation}
where $s(\ell,k)$ is the signless, first-kind, Stirling number, i.e. the number of permutations of the set $[\ell]$ with $k$ cycles. In particular,
\begin{equation}\label{Com}
\sum_{\ell\ge 1}\eta^{\ell}\frac{s(\ell,k)}{\ell!}=\frac{1}{k!}\log^k\frac{1}{1-\eta},\quad |\eta|<1,
\end{equation}
Comtet~\cite[Section 5.5]{Com}. Using \eqref{2n} and \eqref{Com1}, we have
\begin{align*}
&\mathbb E\bigl[Z^{(\ell)}(t+1)|\circ\bigr]= \sum_{k=1}^{\ell} s(\ell,k) \mathbb E\bigl[Z^k(t+1)|\circ\bigr]\\
&= (t+\beta)^{-1}\sum_{k=1}^{\ell}s(\ell,k)\cdot
\Biggl(\!(t+\beta+k) Z^k(t)+\sum_{j=0}^{k-1}\binom{k}{j-1} Z^j(t)\Biggr) \\
&=:(t+\beta)^{-1}\sum_{i=1}^{\ell} \sigma(\ell,i)Z^i(t),\\
\sigma(\ell,i)&\!=\!\left\{\begin{aligned}
&(t+\beta+\ell) s(\ell,\ell),&&\text{if }i=\ell,\\
&(t+\beta) s(\ell,i)+\sum_{k=i}^{\ell} s(\ell,k)\binom{k}{i-1},&&\text{if }i<\ell.\end{aligned}\right.
\end{align*}
We need to show that $\sigma(\ell,i)=(t+\beta+\ell)s(\ell,i)$ for $k<\ell$, which is equivalent to
\[
\ell s(\ell,i)=\sum_{k=i}^{\ell}s(\ell,k)\binom{k}{i-1}.
\]
To prove the latter identity, it suffices to show that, for a fixed $i$, the exponential generating functions of the two sides coincide. By \eqref{Com},
\begin{align*}
&\quad\sum_{\ell\ge 1}\frac{\eta^{\ell}}{\ell!}\sum_{k=i}^{\ell}s(\ell,k)\binom{k}{i-1}=\sum_{k\ge i}\binom{k}{i-1}\sum_{\ell\ge k}\frac{\eta^{\ell}}{\ell!} s(\ell,k)
\\
&=\sum_{k\ge i}\binom{k}{i-1} \frac{1}{k!}\log^k\frac{1}{1-\eta}=\frac{1}{(i-1)!}\left(\log^{ -1}\frac{1}{1-\eta}\right) \sum_{s\ge 1}\frac{1}{s!}\log^s\frac{1}{1-\eta}\\
&=\frac{1}{(i-1)!}\left(\log^{i-1}\frac{1}{1-\eta}\right)\left(\frac{1}{1-\eta}-1\right)=\frac{1}{(i-1)!}\left(\log^{i-1}\frac{1}{1-\eta}\right)\frac{\eta}{1-\eta}.
\end{align*}
And, using \eqref{Com} again,
\begin{align*}
&\sum_{\ell\ge 1}\frac{\eta^{\ell}}{\ell!}\,\ell s(\ell,i)=\eta\sum_{\ell\ge 1}\frac{\ell\eta^{\ell-1}}{\ell!}\,s(\ell,i)\\
&=\eta\frac{d}{d\eta}\Biggl(\frac{1}{i!}\log^i\frac{1}{1-\eta}\Biggr)=\frac{1}{(i-1)!}\left(\log^{i-1}\frac{1}{1-\eta}\right)\frac{\eta}{1-\eta}. \qedhere
\end{align*}
\end{proof}
To identify the $\lim_{t\to\infty} p(t)$, recall that the classic beta probability distribution has density
\[
f(x;a,b)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{a-1}(1-x)^{b-1},\quad x\in (0,1),
\]
parametrized by two parameters $a>0$, $b>0$, and moments
\begin{equation}\label{moments}
\int_0^1 x^{\ell} f(x;a, b)\,dx=\prod_{j=0}^{\ell-1}\frac{a+j}{a+b+j}.
\end{equation}
We can now complete the proof of Theorem \ref{th1}.
By Lemma \ref{Lem1}, we have $\mathbb E [M(t)|\gamma]=M(r)$, i.e.
\[
\mathbb E\Biggl[\frac{\left(X(t)+\frac{\gamma}{2+\delta}\right)^{(\ell)}}{(t+\beta)^{(\ell)}}\,\Bigg|\gamma\Biggr]=\frac{\left(1+\frac{\gamma}{2+\delta}\right)^{(\ell)}}{(r+\beta)^{(\ell)}}.
\]
For every $\ell\ge 1$, by martingale convergence theorem, conditioned on $\gamma$, there exists an integrably finite $\mathcal M_{\gamma,\ell}$ such that a.s.
\[
\lim_{t\to\infty}\frac{\left(X(t)+\frac{\gamma}{2+\delta}\right)^{(\ell)}}{(t+\beta)^{(\ell)}}=\mathcal M_{\gamma,\ell},\quad \ell\ge 0,
\]
and
\[
\mathbb E[\mathcal M_{\gamma,\ell}]=\frac{\left(1+\frac{\gamma}{2+\delta}\right)^{(\ell)}}{(r+\beta)^{(\ell)}}.
\]
So, using the notation $p_X(t)=X(t)/t$, we have: a.s.
\begin{equation}\label{m=1}
\begin{aligned}
\lim_{t\to\infty}(p_X(t))^{\ell}=\mathcal M_{\gamma,\ell}=(\mathcal M_{\gamma,1})^{\ell},
\end{aligned}
\end{equation}
and
\[
\mathbb E\bigl[(\mathcal M_{\gamma,1})^{\ell}\bigr]=\frac{\left(1+\frac{\gamma}{2+\delta}\right)^{(\ell)}}{(r+\beta)^{(\ell)}}=\prod_{j=0}^{\ell-1}\frac{1+\frac{\gamma}{2+\delta}+j}{r+\beta+j}.
\]
This means that $\mathcal M_{\gamma,1}$ is beta-distributed with parameters $1+\frac{\gamma}{2+\delta}$ and $r+\beta-1-\frac{\gamma}{2+\delta}$. By the definition of
$\gamma$ and \eqref{n1}, we have
\[
\mathbb P(\gamma=0)=\frac{1+\delta}{(2+\delta)(r-1)+(1+\delta)}=\frac{1+\delta}{2r-1+\delta r}.
\]
We conclude that $\lim_{t\to\infty}p(t)$ has the distribution which is the mixture of the two beta distributions, with parameters $a=1$, $b=r-\frac{1}{2+\delta}$, and $a=\frac{1+\delta}{2+\delta}$, $b=r$, weighted by $\frac{1+\delta}{(2+\delta)r-1}$ and
$\frac{(2+\delta)(r-1)}{(2+\delta)r-1}$ respectively.
This completes the proof of Theorem~\ref{th1}.
\end{proof}
\subsection{Proof of Theorem \ref{th2}}
\begin{customthm}{2.3}
Let $m>1$ and $\delta>-m$ and let $p_X(t)=X(t)/t$, $p_Y(t)=Y(t)/(2mt)$, where $Y(t)$ is the total degree of the descendants of $r$ at time $t$.
Then almost surely $\lim_{t\to\infty} p_X(t)=\lim_{t\to\infty} p_Y(t)=1$.
\end{customthm}
\begin{proof}We need to derive tractable formulas/bounds for the conditional distribution of $Y(t+1)-Y(t)$. First, let us evaluate the
conditional probability that selecting the second endpoints of the $m$ edges incident to vertex $t+1$ no loops will be formed.
Suppose there has been no loop in the first $i-1$ steps, $i\in [m]$; call this event $\mathcal E_{i-1}$. On event $\mathcal E_{i-1}$, as the $i$-th edge incident to $t+1$ is about to attach its second end to a vertex in $[t]\cup\{t+1\}$, the total degree of
all these vertices is $2mt+i-1$ ($1\le i\le m$). So, by the definition of the transition probabilities (items {\bf (a), (b), (c)\/}) we have
\[
\mathbb P(\mathcal E_i |\circ)= \frac{2mt +i-1+t\delta}{2mt+2(i-1)+t\delta +1+\frac{i\delta}{m}},
\]
``$\circ$'' indicating conditioning on the full record of $i-1$ preceding attachments such that the event $\mathcal E_{i-1}$ holds. Crucially this conditional probability
depends on $i$ only. Therefore the probability of a given full {\it loops-free\/} record of the $m$ attachments is equal to the corresponding probability for the ``no loops in $m$ attachments process'', multiplied by
\begin{equation}\label{Pmt=}
\Pi_m(t):=\prod_{i=1}^m\frac{2mt +i-1+t\delta}{2mt+2(i-1)+t\delta +1+\frac{i\delta}{m}}=1-O(t^{-1}).
\end{equation}
\begin{lemma}\label{lem1} If no loops are allowed in the transition from $t$ to $t+1$, then
for $a\in [m]$,
\[
\pr{Y(t+1) =Y(t)+m+a \mid \circ}
=\binom{m}{a} \frac{(Y(t)+\delta X(t))^{(a)} \cdot \bigl(2mt-Y(t)+\delta(t-X(t))^{(m-a)}} {\bigl((2m+\delta)t\bigr)^{(m)}}
\]
and
\[
\pr{Y(t+1) =Y(t)\mid \circ} = \frac{ \bigl(2mt-Y(t)+\delta(t-X(t))^{(m)}} {\bigl((2m+\delta)t\bigr)^{(m)}}.
\]
\end{lemma}
\begin{proof} Vertex $t+1$ selects, in $m$ steps, a sequence $\{v_1,\dots, v_m\}$ of $m$ vertices from $[t]$, with $t$ choices for every selection. Introduce $\Bbb{\boldsymbol{I}}=\{\Bbb I_1,\dots,\Bbb I_m\}$, where $\Bbb I_i$ is the indicator of the event $\{v_i\in V(T(t))\}$. The total vertex degree
of $[t]$ (of $V(T(t))$ respectively) right before step $i$ is $2mt + i-1$ $\bigl(Y(t)+\mu_i\text{ respectively}, \mu_i:=|\{j<i: \Bbb I_j=1\}|\bigr)$. Conditioned on this
prehistory,
\begin{align*}
\mathbb P(\Bbb I_i=1)&=\frac{Y(t)+\delta X(t)+\mu_i}{2mt+\delta t+i-1},\\
\mathbb P(\Bbb I_i=0)&=\frac{2mt-Y(t)+\delta(t-X(t))+i-1-\mu_i}{2mt+\delta t+ i -1}.
\end{align*}
Therefore a sequence $\Bbb{\boldsymbol{I}}$ will be the outcome of the $m$-step selection with probability
\begin{multline*}
\qquad\qquad\qquad\mathbb P(\Bbb{\boldsymbol{I}})=\left(\prod_{i\in [m]} \Bigl((2m+\delta)t +i-1\bigr)\Bigr)\right)^{-1}\\
\times\prod_{i:\, \Bbb I_i=1}\Bigl(Y(t)+\delta X(t)+\mu_i\Bigr)\,\,\cdot \prod_{i:\, \Bbb I_i=0}\Bigl(2mt-Y(t)+\delta(t-X(t))+i-1-\mu_i\Bigr).
\end{multline*}
Furthermore, for $a\in [m]$, on the event $\{Y(t+1)=Y(t)+m+a\}$ for each admissible $\Bbb{\boldsymbol{I}}$ we have
\[
\{\mu_i\}=\{0,1,\dots, a-1\},\quad \{i-1-\mu_i\}=\{0,1,\dots, m-a-1\},
\]
so that
\[
\mathbb P(\Bbb{\boldsymbol{I}})= \frac{(Y(t)+\delta X(t))^{(a)} \bigl(2mt-Y(t)+\delta(t-X(t))^{(m-a)}}{\bigl((2m+\delta)t\bigr)^{(m)}}.
\]
Since the total number of admissible sequences $\Bbb{\boldsymbol{I}}$ is $\binom{m}{a}$, we obtain the first formula in Lemma~\ref{lem1}. The second formula is the case of
$\mathbb P(\Bbb{\boldsymbol{I}})$ with ${a=0}. \qedhere$
\end{proof}
It is clear from the proof of Lemma \ref{lem1} that $\{\mathbb P_m(a)\}_{0\le a\le m}$,
\[
\mathbb P_m(a):=\binom{m}{a}\frac{(Y(t)+\delta X(t))^{(a)} \bigl(2mt-Y(t)+\delta(t-X(t))^{(m-a)}}{\bigl((2m+\delta)t\bigr)^{(m)}},
\]
is a probability distribution of a random variable $D$, a ``rising-factorial'' counterpart of the binomial $\mathcal D=\text{Bin}(m,p=Y(t)/2mt)$.
Define the falling factorial $(x)_{\ell}=x(x-1)\cdots (x-\ell+1)$. It is well known that $\mathbb E[(\mathcal D)_{\mu}]=(m)_{\mu} p^{\mu}$, $(\mu\le m)$.
For $D$ we have
\begin{multline}\label{2.2}
\mathbb E[(D)_{\mu}]=\sum_a(a)_{\mu} \mathbb P_m(a)=\frac{(m)_{\mu}\,(Y(t)+\delta X(t))^{(\mu)}}{\bigl((2m+\delta)t\bigr)^{(\mu)}}\cdot \sum_{a\ge \mu}\binom{m-\mu}{a-\mu}\\
\times \frac{(Y(t)+\delta X(t)+\mu)^{(a-\mu)}\bigl((2m+\delta)t+\mu-(Y(t)+\delta X(t)+\mu))^{((m-\mu)-(a-\mu))}}{(2mt+\mu)^{(m-\mu)}}\\
=\frac{(m)_{\mu}\,(Y(t)+\delta X(t))^{(\mu)}}{\bigl((2m+\delta)t\bigr)^{(\mu)}},
\end{multline}
since the sum over $a\ge \mu$ is $\sum_{\nu\ge 0}\mathbb P_{m-\mu}(\nu)=1$.
From Lemma \ref{lem1} and \eqref{2.2} we have: if no loops during the transition from $t$ to $t+1$ are allowed, then
\begin{align}\label{2.3-}
\mathbb E[Y(t+1)-Y(t)|\circ] &=\sum_{a=1}^m (a+m) \mathbb P_m(a) \notag\\
&=\frac{m\bigl(Y(t)+\delta X(t)\bigr)}{(2m+\delta)t}+ m\left(1-\frac{\bigl((2m+\delta)t-Y(t)-\delta X(t)\bigr)^{(m)}}{\bigl((2m+\delta)t\bigr)^{(m)}}\right),
\end{align}
and
\begin{equation}\label{2add}
\mathbb E[X(t+1)-X(t)|\circ]=1-\frac{\bigl((2m+\delta)t-Y(t)-\delta X(t)\bigr)^{(m)}}{\bigl((2m+\delta)t\bigr)^{(m)}}.
\end{equation}
What if the ban on loops at the vertex $t+1$ is lifted? From the discussion right before Lemma \ref{lem1}, we see that
both
$\mathbb E[\bigl(Y(t+1)-Y(t)\bigr) \mathbb I(\text{no loops})|\circ]$ and
$\mathbb E[\bigl(X(t+1)-X(t)\bigr) \mathbb I(\text{no loops})|\circ]$
are equal
to the respective RHS's in \eqref{2.3-} and \eqref{2add} {\it times\/} $\Pi_m(t)=1-O(t^{-1})$. Consequently, adding the terms $O(t^{-1})$ to
the RHS of \eqref{2.3-} and to the RHS of \eqref{2add} we obtain the sharp asymptotic formulas for $\mathbb E[Y(t+1)-Y(t)|\circ]$ and
$\mathbb E[X(t+1)-X(t)|\circ]$ in the case of the loops-allowed model. We will also need
\[
p(t)=\frac{2m}{2m+\delta}\,p_Y(t)+\frac{\delta}{2m+\delta}\,p_X(t),
\]
where $p_Y(t)=\frac{Y(t)}{2mt}$ and $p_X(t)=\frac{X(t)}{t}$ as defined in the beginning of the section.
To continue the proof of Theorem~\ref{th2},
we note first that $mX(t)\le Y(t)\le 2mX(t)$. The lower bound is obvious. The upper bound follows from induction on $t$: Suppose $Y(t)\le 2mX(t)$. If $X(t+1)=X(t)$, then $Y(t+1)=Y(t)\le 2mX(t)=2m X(t+1)$. If $X(t+1)=X(t)+1$, then $Y(t+1)\le Y(t)+2m\le 2mX(t)+2m=2mX(t+1)$.
Therefore, by the definition of $p(t)$, we have
\begin{equation}\label{double}
\frac{p_X(t)}{2}\le p_Y(t)\le p_X(t) \Longrightarrow \frac{m+\delta}{2m+\delta}\,p_X(t)\le p(t) \le p_X(t);
\end{equation}
in particular, $p(t)\in [0,1]$ since $\delta\ge -m$.
We will also need
\[
\frac{\bigl((2m+\delta)t-Y(t)-\delta X(t)\bigr)^{(m)}}{\bigl((2m+\delta)t\bigr)^{(m)}} = (1-p(t))^m +O(t^{-1}).
\]
So, using \eqref{2.3-}, we compute
\begin{multline}\label{pY(t+1)}
\mathbb E[p_Y(t+1)|\circ]=\mathbb E\left[\frac{Y(t+1)}{2mt}\cdot \frac{t}{t+1}\Big|\circ\right]\\
=\frac{t}{t+1}\!\left(\!p_Y(t)+\frac{1}{2t}\bigl[1+p(t)-(1-p(t))^m\bigr]+O(t^{-2})\!\!\right)=p_Y(t)+q_Y(t),\\
q_Y(t):=\frac{1}{2(t+1)}\bigl[1+p(t)-2p_Y(t)-(1-p(t))^m\bigr]+O(t^{-2}).
\end{multline}
Likewise
\begin{equation}\label{pX(t+1)}
\begin{aligned}
&\qquad\qquad\mathbb E[p_X(t+1)|\circ]=p_X(t)+q_X(t),\\
&q_X(t)=\frac{1}{t+1}\bigl[1-p_X(t)-(1-p(t))^m\bigr]+O(t^{-2}).
\end{aligned}
\end{equation}
Multiplying the equation \eqref{pY(t+1)} by $\frac{2m}{2m+\delta}$, the equation \eqref{pX(t+1)} by $\frac{\delta}{2m+\delta}$, and adding them, we obtain
\begin{equation}\label{p(t+1)}
\begin{aligned}
&\qquad\qquad\qquad\quad\mathbb E[p(t+1)|\circ]=p(t) +q(t),\\
&q(t):=\frac{m+\delta}{(2m+\delta)(t+1)}\bigl[1-p(t)-(1-p(t))^m\bigr]+O(t^{-2}).
\end{aligned}
\end{equation}
From the first line in \eqref{p(t+1)} it follows that
\[
\sum_{t=1}^{\tau}\Bbb E[p(t+1)]=\sum_{t=1}^{\tau}\Bbb E[p(t)]+\sum_{t=1}^{\tau}\Bbb E[q(t)],
\]
implying that
\[
\limsup_{\tau\to\infty}\sum_{t\le\tau} \mathbf E[q(t)]\le \limsup_{\tau\to\infty}\Bbb E[p(\tau+1)]\le 1.
\]
Since $1-z-(1-z)^m\ge 0$ on $[0,1]$, the second line in \eqref{p(t+1)} implies that $|q(t)|\le q(t)+O(t^{-2})$.
Since $\sum_t t^{-2}<\infty$, we see that $\sum_t\mathbb E[|q(t)|]<\infty$.
So a.s. there exists $Q:=\lim_{\tau\to\infty}\sum_{1\le t\le \tau} q(t)$, with $\mathbb E[|Q|]
\le \sum_t\mathbb E[|q(t)|]<\infty$, i.e. a.s. $|Q|<\infty$. Introducing $Q(t+1)=\sum_{\tau\le t}q(\tau) $, we see from \eqref{p(t+1)} that $\{p(t+1)-Q(t+1)\}_{t\ge 1}$ is a martingale with $\sup_t |p(t+1)-Q(t+1)|\le 1 + \sum_{\tau\ge 1} |q(\tau)|$. By the martingale convergence theorem we obtain that there exists an integrable $\lim_{t\to\infty}(p(t)-Q(t))$, implying that a.s. there exists a random $p(\infty)=\lim_{t\to\infty}p(t)$. The \eqref{p(t+1)} also implies that
\[
1\ge \mathbb E[p(\infty)]=\frac{m+\delta}{2m+\delta}\sum_{t\ge 1}\frac{1}{t+1}\mathbb E\bigl[1-p(t)-(1-p(t))^m\bigr]+ O(1).
\]
Since $m+\delta>0$ and
\[
\lim_{t\to\infty} \mathbb E\bigl[1-p(t)-(1-p(t))^m\bigr] =\mathbb E\bigl[1-p(\infty)-(1-p(\infty))^m\bigr],
\]
and the series $\sum_{t\ge 1} t^{-1}$ diverges, we obtain that $\mathbb P(p(\infty)\in \{0,1\})=1$.
Recall that $p(t)\ge \frac{m+\delta}{2m+\delta}\, p_X(t)$. If we show that a.s. $\liminf_{t\to\infty}p_X(t)>0$, it will follow that
a.s. $p(\infty)>0$, whence a.s. $p(\infty)=1$, implying (by $p(t)\le p_X(t)$) that a.s. $p_X(\infty)$ exists, and is $1$, and consequently
(by the formula for $p(t)$) a.s. $p_Y(\infty)$ exists, and is $1$.
So let's prove that a.s. $\liminf_{t\to\infty}p_X(t)>0$.
Recall that we did prove the latter for $m=1$. To transfer this earlier result to $m>1$, we need to establish
some kind of monotonicity with respect to $m$. The coupling described in Section~\ref{sec:results} to the rescue!
\begin{lemma}\label{lem:coupling}
For the coupled processes $\{G_{m,\delta}(t)\}$ and $\{G_{1,\delta/m}(mt)\}$, we have $X_{m,\delta}(t,r) \ge m^{-1} X_{1,\delta/m}(mt,mr)$.
\end{lemma}
\begin{proof}
Let us simply write $G_1$ and $G_m$ for the two graphs $G_{1,\delta/m}(mt)$ and $G_{m,\delta}(t)$, respectively.
Similarly, write $T_1$ and $T_m$, respectively, for the descendant tree in $G_{1,\delta/m}(mt)$ rooted at $mr$ and the descendant tree in $G_{m,\delta}(t)$ rooted at $r$.
If $v_a\in T_1$, i.e. $v_a$ is a descendant of $mr$, then for $b=\ceil{a/m}$ we have $w_b=\{v_{m(b-1)+i}\}_{i\in [m]}\ni v_a$, implying that
$w_b$ is a descendant of $r$ in $G_m$, i.e. $w_b\in T_m$. (The converse is generally false: if $w_b$ is a descendant of $r$, it does not mean that
every $v_{m(b-1)+i}$, ($i\in [m]$), is a descendant of $mr$.) Therefore
\[
X_{m,\delta}(t,r)=|V(T_m)|\ge m^{-1} |V(T_1)|=m^{-1}X_{1,\delta/m}(mt,mr). \qedhere
\]
\end{proof}
Thus, to complete the proof of the theorem, i.e. for $\delta>-m$, we {\bf (a)\/} use Theorem \ref{th1}, to assert that for the process $\{G_{1,\delta/m}(t)\}$, a.s. $\lim_{t\to\infty} p_X (t)>0$; {\bf (b)\/} use Lemma \ref{lem:coupling}, to assert that a.s. $\liminf_{t\to\infty} p_X (t)>0$ for $\{G_{m,\delta}(t)\}$ as well. The proof of Theorem \ref{th2} is complete.
\end{proof}
\subsection{Proof of Theorem \ref{th2+}}
\begin{customthm}{2.4}
Consider the UAM graph process $G_{t,m}$. Given $r>1$, let $X(t)$ be the cardinality of the descendant tree rooted at vertex $r$, and let $p(t):=X(t)/t$.
\begin{enumerate}
\item[\textup{(i)}] For $m=1$, almost surely, $\lim p(t)$ exists and
it has the same distribution as the minimum of $(r-1)$ independent $[0,1]$-Uniforms. Consequently a.s. $\liminf_{t\to\infty}p(t)>0$.
\item[\textup{(ii)}] For $m>1$, almost surely $\lim_{t\to\infty} p(t)=1$.
\end{enumerate}
\end{customthm}
\begin{proof} By the definition of the UAM process, we have
\begin{equation}\label{a}
\Bbb P(X(t+1)=X(t)+1|\circ)= 1 - (1-p(t))^m.
\end{equation}
{\bf (i)\/} Consider $m=1$. For $r=1$, we have $p(t)\equiv 1$. Consider $r\ge 2$. The equation \eqref{a} is the case $\delta=-1$, $\gamma=0$ of \eqref{jumpup}. By Lemma
\ref{Lem1}, we claim that $M(t):=\frac{(X(t))^{(\ell)}}{t^{(\ell)}}$ is a martingale. So arguing as in the proof of Theorem
\ref{th1}, we obtain that almost surely (a.s.) $\lim p(t)=p(\infty)$ exists, and the limiting distribution of $p(\infty)$
is a beta-distribution with
parameters $a=1$ and $b=r-1$. That is, the limiting density is $(r-1)(1-x)^{r-2}$, $x\in [0,1]$. Therefore a.s. $\lim p(t)>0$.
{\bf (ii)\/} Consider $m>1$. Clearly $G_{t,1}\subset G_{t,m}$. Therefore a.s. $\liminf p(t)>0$ as well. Furthermore, it follows from \eqref{a} that
\[
\Bbb E[p(t+1)|\circ]= p(t)+\frac{1}{t+1}\bigl[1-p(t)-(1-p(t))^m\bigr],
\]
which is a special case of \eqref{p(t+1)}, with $O(t^{-2})$ dropped. So we obtain that
$\Bbb P\bigl(p(\infty)\in \{0,1\}\bigr)=1$, which in combination with $\Bbb P(p(\infty)>0)=1$ imply that
$\Bbb P(p(\infty)=1)=1$.
\end{proof}
\section{A technical lemma}
In this section, we will prove Lemma~\ref{general}. We need the following Chernoff bound for its proof. (See e.g.\ \cite[Theorem 2.8]{JLR}.)
\begin{theorem}\label{thm:Chernoff}
If $X_1,\dots,X_n$ are independent Bernoulli random variables, $X=\sum_{i=1}^n X_i$, and $\lambda = \mathbb{E}[X]$, then
\[
\mathbb P(|X-\lambda| > \varepsilon\lambda) < 2\exp\brac{-\varepsilon^2\lambda/3} \quad \forall \varepsilon\in (0, 3/2).
\]
\end{theorem}
\begin{lemma}\label{general}
Let $\{X(t)\}_{t\ge 0}$ be a sequence of random variables such that $X(0)=0$ and $X(t+1)-X(t) \in \{0,1\}$. Let $x(t)=X(t)/t$ and assume
\begin{equation}\label{rec;ineq}
\Bbb E[x(t+1) - x(t)|\circ]\le \frac{h(x(t))}{t} +O(t^{-2}),
\end{equation}
where $h$ is a continuous, strictly decreasing function with $h(0)>0$ and $h(1)<0$, so that $h(x)$ has a unique root $\rho \in(0,1)$. Assume also that $h'(x) < -1$ in $(0,1)$. Then, for any $\gamma<1/3$, almost surely
\[
\lim_{t\to\infty} t^{\gamma}\max\{0,x(t)-\rho\}=0.
\]
\end{lemma}
\begin{lemma}[Extensions of Lemma~\ref{general}]\label{extensions}
Lemma~\ref{general} can be extended in a couple of ways as follows.
\begin{enumerate}
\item[\textup{(a)}] If the hypothesis $X(t+1)-X(t) \in \{0,1\}$ in Lemma~\ref{general} is replaced with $X(t+1)-X(t) \in \{-1,1\}$, then the conclusion of Lemma~\ref{general} still holds. This follows from minor modifications in the proof of Lemma~\ref{general}.
\item[\textup{(b)}] If the inequality sign in \ref{rec;ineq} is replaced with an equality sign, .i.e. under the condition $\mean{x(t+1) - x(t) \mid \circ} = \frac{h(x(t))}{t} +O(t^{-2})$, we have the following conclusion:
for any $\gamma<1/3$, almost surely
\[
\lim_{t\to\infty} t^{\gamma} (x(t)-\rho)=0.
\]
\begin{proof}[Proof of \textup{(b)}]
First of all, by Lemma~\ref{general}, we have $\lim_{t\to\infty} t^{\gamma}\max\{0,x(t)-\rho\}=0$ almost surely.
Second, let $g(z) = -h(1-z)$, so that $g(0) >0$ and $g(1)<0$, and in $(0,1)$, we have $g'(z) = h'(1-z)<-1$. Letting $y(t) = 1-x(t)$,
\begin{align*}
\mean{y(t+1) -y (t) \mid \circ} &= \mean{(1-x(t+1)) -(1- x(t)) \mid \circ} \\
&= -\frac{h(x(t))}{t} +O(t^{-2})\\
&= \frac{g(y(t))}{t} +O(t^{-2}).
\end{align*}
Applying Lemma~\ref{general} with $X_1(t) = t-X(t)$, and then switching back to $X(t)$, we see
that $\lim_{t\to\infty} t^{\gamma}\max\{0, \rho-x(t)\}=0$ almost surely, as well.
\end{proof}
\end{enumerate}
\end{lemma}
\medskip
\begin{proof}[Proof of Lemma~\ref{general}]
Let $\varepsilon=\varepsilon_t:=t^{-1/3}\log t$. We will show
\begin{equation}\label{x_t<rho+eps}
\mathbb P(x(t)>\rho+\varepsilon) \le \exp\brac{-\Theta\brac{\log^3t}}.
\end{equation}
Once we show~\eqref{x_t<rho+eps}, the Borel-Cantelli lemma gives
\[
\mathbb P(x(t)-\rho > t^{-1/3}\log t \quad \text{infinitely often})=0,
\]
which proves what we want. Let us prove~\eqref{x_t<rho+eps}.
For $T\in [0,t)$, let $\mathcal E_T$ be the event that \{$x(t)> \rho+\varepsilon$ and $T$ is the last time such that $X(\tau) \le (\rho+\varepsilon/2)\tau$\}, that is,
\[
X(T)\le (\rho+\varepsilon/2)T; \quad x(\tau)> \rho+\varepsilon/2\tau, \,\, \forall\, \tau\in (T,t); \quad x(t) >\rho+\varepsilon.
\]
Since $X(t+1)-X(t) \in \{0,1\}$, we have
\begin{align*}
X(T)+t-T\ge X(t)>t(\rho+\varepsilon).
\end{align*}
Using $X(t) = tx(t)$ above , we get
\[
T(\rho+\varepsilon/2)+t-T>t(\rho+\varepsilon),
\]
implying
\begin{equation}\label{ubT}
t-T > \frac{t\varepsilon}{2(1-\rho)}.
\end{equation}
We conclude that
\[
\{x(t)>\rho+\varepsilon\}\subseteq \bigcup_{T=1}^s \mathcal E_T,\quad s=s(t):= t-\Big\lceil\frac{t\varepsilon}{2(1-\rho)}\Big\rceil.
\]
Now let us fix a $T\in [0,s]$ and bound $\mathbb P(\mathcal E_T)$.
The main idea of the proof is that, as long as $x(\tau)>\rho$, by Equation~\eqref{rec;ineq}, the process $\{x(\tau)\}$ has a negative drift.
Let $\xi_\tau$ denote the indicator of the event $\{x(\tau-1)>\rho+\varepsilon/2 \text{ and } X(\tau)=X(\tau-1)+1\}$ and let $\mathcal Z_T:=\xi_{T+2}+\cdots+\xi_t$.
On the event $\mathcal E_T$, the sum $\mathcal Z_T$ counts the total number of upward unit jumps ($X(\tau)-X(\tau-1)=1$,
$\tau\in [T+2,t]$) and therefore
\[
X(T+1) + \mathcal Z_T = X(t)\ge t(\rho+\varepsilon).
\]
Since $X(T+1) \le X(T)+1\le T(\rho+\varepsilon/2)+1$, we must have
\begin{equation*}
Z_T> (\rho+\varepsilon)(t-T),\quad Z_T:=1+\mathcal Z_T.
\end{equation*}
Writing $p(x(\tau)):= \pr{X(\tau+1) = X(\tau)+1 \mid \circ}$,
\begin{align*}
\mean{x(\tau+1) \mid \circ} &= p(x(\tau)) \frac{X(\tau)+1}{\tau+1} + (1-p(x(\tau))) \frac{X(\tau)}{\tau+1} \\
&= \frac{p(x(\tau))}{\tau+1} + \frac{\tau x(\tau)}{\tau+1} \\
& = x(\tau) + \frac{p(x(\tau)) - x(\tau)}{\tau+1},
\end{align*}
so that $p(x(\tau)) = h(x(\tau)) + x(\tau) + O(\tau^{-1})$.
Recall that, for $\tau \ge T+1$, we have $x(\tau) > \rho +\varepsilon/2$. Since $h'(x)<-1$ in $(0,1)$, the sum $h(x) + x$ is decreasing in $(0,1)$. Hence, conditioning on the full record (up to and including time $\tau$),
\begin{align*}
\mathbb P(\xi_{\tau+1}=1|\circ) &=\mathbb P(X(\tau+1)=X(\tau)+1\, |\,\circ)\\
&= h(x(\tau)) + x(\tau) +O\brac{\tau^{-1}} \\
&< h(\rho +\varepsilon/2) + \rho + \varepsilon/2 + O\brac{\tau^{-1}}\\
&= h(\rho) + (\varepsilon/2)\cdot h'(y) +\rho +\varepsilon/2 + O\brac{\tau^{-1}} \quad \text{for some }y\in (\rho,\rho+\varepsilon/2)\\
& < \rho + O\brac{\tau^{-1}}.
\end{align*}
Hence, the sequence $\{\xi_\tau\}$ is stochastically dominated by the sequence of {\it independent\/} Bernoulli random variables $B_\tau$ with parameters $\min\bigl(\rho+O(\tau^{-1}),1\bigr)$.
Consequently, $Z_T$ is stochastically dominated by $1+\sum_{j=T+2}^t B_j$, and
\[
\lambda:=\sum_{j=T+2}^t \mathbb E[B_j]=\rho(t-T) +O(\log t).
\]
For the choice of $\varepsilon$ we have, \eqref{ubT} gives
\[
(\rho+\varepsilon)(t-T) \ge (1+\varepsilon/2)\lambda.
\]
Thus, by the Chernoff bound in Theorem~\ref{thm:Chernoff} and using~\eqref{ubT},
\begin{align*}
\pr{\mathcal{E}_T}
&\le \mathbb P(Z_T>(t-T)(\rho+\varepsilon)) \\
&\le \mathbb P\Big(1+B_{T+2}+\cdots+B_t>(t-T)(\rho+\varepsilon)\Big) \\
&\le \mathbb P\brac{1+B_{T+2}+\cdots+B_t> (1+\varepsilon/2)\lambda} \\
&\le \exp\brac{-\Theta(\varepsilon^2(t-T))}\le e^{-\Theta(\log^3 t)}.
\end{align*}
Using the union bound on $T$ we complete the proof of~\eqref{x_t<rho+eps} and of the lemma.
\end{proof}
\section{Greedy Matching Algorithm}
Recall that the greedy matching algorithm (for either of two graph models) generates the increasing sequence $\{ M(t)\}$ of partial matchings on the sets $[t]$, with $ M(1)=\emptyset$. Given $M(t)$, let
\begin{align*}
X(t)&:= \text{number of unmatched vertices at time }t,\\
Y(t)&:= \text{total degree of unmatched vertices at time }t,\\
U(t)&:=\text{number of unmatched vertices selected by $t+1$ from $[t]\setminus M(t)$},\\
x(t)&:=X(t)/t,\\
y(t)&:=Y(t)/(2mt).
\end{align*}
\subsection{The PAM case}
\begin{customthm}{2.5}
Let $X(t)$ be the number of unmatched vertices at time $t$ in the greedy matching algorithm.
For $\delta>-m$, let $\rho_{m,\delta}$ be the unique root in $(0,1)$ of
\begin{equation}\label{defh}
h(z)=h_{m,\delta}(z):= 2\left[1-\bfrac{m+\delta}{2m+\delta}z\right]^m-z-1.
\end{equation}
Then, for any $\alpha<1/3$, almost surely,
\beq{th3repeatconc}
{
\lim_{t\to\infty} t^{\alpha}\max\{0, x(t)-\rho_{m,\delta}\}=0.
}
In consequence, the Greedy Matching Algorithm a.s. finds a sequence of nested matchings $\{M(t)\}$,
where the number of vertices in $M(t)$ is asymptotically at least $(1-\rho_{m,\delta})t$.
\end{customthm}
\begin{proof}
Notice first that, for $\delta>-m$, the function $h(z)$ is decreasing on $(0,1)$ and $h(z)=0$ does have a unique solution in the same interval.
We will prove our claim first for a slightly different model that does not allow any loops other than at the first vertex. In this model, vertex $1$ has $m$ loops, and the $i$-th edge of vertex $t+1$ attaches to $u\in [t]$ with probability
\[
\frac{d_{t,i-1}(u)+\delta}{2mt+2(i-1)+t\delta}.
\]
\medskip
\noindent \textbf{Loops not allowed except at vertex 1.}
In this case, since each degree is at least $m$, we have $Y(t)\ge mX(t)$ and hence $y(t)\ge x(t)/2$.
Also, since
\[
X(t+1)=
\begin{cases}
X(t)+1 & \text{if } U(t)=0\\
X(t)-1 & \text{if } U(t)>0,
\end{cases}
\]
we have
\begin{equation}\label{meanX}
\mathbb E[X(t+1)|\circ]= X(t)+ \mathbb P(U(t)=0|\circ)-\mathbb P(U(t)>0|\circ).
\end{equation}
Since $\mathbb P(\text{vertex $t+1$ has some loop})= O(t^{-1})$, using $Y(t)\ge mX(t)$ in the last step below,
by Lemma \ref{lem1} we get
\begin{align}\label{P(U=0)}
\mathbb P(U(t)=0|\circ)
&= \mathbb P(U(t)=0 \text{ and vertex $t+1$ has no loop} |\circ) +O\brac{t^{-1}} \notag\\
&= \brac{1-O\brac{t^{-1}}}\frac{(2mt-Y(t)+\delta t-\delta X(t))^{(m)}}{(2mt+\delta t)^{(m)}}+O\brac{t^{-1}} \notag\\
&=\frac{(2mt+\delta t-Y(t)-\delta X(t))^{m}}{(2mt+\delta t)^{m}}+O\brac{t^{-1}} \notag\\
&=\brac{1-\frac{2m}{2m+\delta}\,y(t)-\frac{\delta}{2m+\delta}\,x(t)}^m+O\brac{t^{-1}} \\
&\le \left(1-\frac{m+\delta}{2m+\delta}\,x(t)\right)^m+ O(t^{-1}). \notag
\end{align}
Using \eqref{meanX} and \eqref{P(U=0)} gives
\begin{align}\label{meanXupper}
\mathbb E[x(t+1)|\circ] &\le x(t)+ \frac1t\left[2\brac{1-\frac{m+\delta}{2m+\delta}\,x(t)}^m-x(t)-1\right] +O\brac{t^{-2}} \notag \\
&=x(t)+\frac{1}{t}h(x(t))+O\brac{t^{-2}},
\end{align}
where $h(z)$ is as defined in~\eqref{defh}.
Note that $X(0) = 0$ and $h(z)$ satisfies the conditions given in Lemma~\ref{general} and Lemma~\ref{extensions}, namely, $h(0)>0$, $h(1)<0$, and $h'(z) <- 1$ for $z\in (0,1)$.
The conclusion of the theorem follows from the first part of Lemma~\ref{extensions} in this case.
\medskip
\noindent \textbf{Loops allowed everywhere.} The above analysis is carried over to this more complicated case via an argument similar to
the one for the descendant trees in the subsections 1.1. Here is a proof sketch. First, the counterpart of \eqref{P(U=0)} is:
\begin{equation*}
\begin{aligned}
&\mathbb P(\!\{U(t)=0\}\!\cap\! \{\text{no loops at }t+1\}|\circ)\\
&=\Pi_m(t)\prod_{j=0}^{m-1}\bfrac{2mt-Y(t)+\delta t-\delta X(t)+j}{2mt+2j+1+\delta t+(j+1)\,\delta/m}\\
&\le \Pi_m(t)\left[\left(1-\frac{m+\delta}{2m+\delta}\,x(t)\right)^m+ O(t^{-1})\right]\\
&=\bigl(1-O(t^{-1})\bigr)\left[\left(1-\frac{m+\delta}{2m+\delta}\,x(t)\right)^m+ O(t^{-1})\right]\\
&= \left(1-\frac{m+\delta}{2m+\delta}\,x(t)\right)^m+O(t^{-1});
\end{aligned}
\end{equation*}
see \eqref{Pmt=} for $\Pi_m(t)$. Therefore we obtain again \eqref{meanXupper}. The rest of the proof remains the same.
\end{proof}
\begin{remark}
Let $r=r_{m,\delta}:=1-\rho_{m,\delta}$, where $\rho_{m,\delta}$ is the unique root in $(0,1)$ of
\[
h(z)=h_{m,\delta}(z):= 2\left[1-\bfrac{m+\delta}{2m+\delta}z\right]^m-z-1.
\]
Then, $r$ is the unique root in $(0,1)$ of
\[
f(z)=f_{m,\delta}(z):= 2-z-2\brac{\frac{m}{2m+\delta}+\frac{m+\delta}{2m+\delta}\,z}^m.
\]
Thus, by Theorem~\ref{th3}, we have
\[
\liminf (1-x(t)) \ge r
\]
almost surely, where $1-x(t)$ is the fraction of the vertices in $M(t)$. See~\eqref{rvalues} for various $r$ values when $\delta=0$.
\end{remark}
\begin{remark}
When $\delta \to \infty$, the function $f_{m,\delta}(z)$ as defined above converges to $2-z-2z^m$ in $(0,1)$. So it is plausible that for the case of uniform attachment model, the number of vertices in $M(t)$ is asymptotically $rt$, where $r$ is the unique root of $2-z-2z^m$. This is in fact the case as shown in the next theorem.
\end{remark}
\subsection{The UAM case}
\begin{customthm}{2.8}
Let $M(t)$ denote the greedy matching set after $t$ steps of the UAM process.
Let $r_m$ denote a unique positive root of $2(1-z^m) -z=0$: $r_m=1-m^{-1}\log 2+O(m^{-2})$.
Then, for any $\alpha<1/3$, almost surely
\[
\lim_{t\to\infty} t^{\alpha}\biggl| \frac{2|M(t)|}{t}-r_m\biggr|=0.
\]
\end{customthm}
\begin{proof}
Let $X(t) = t-2|M(t)|$ as before. In particular, we have $X(0)=0$ and $X(1)=1$.
At each step $t\ge 2$, we check the edges incident to vertex $t$.
If some of the edges end at vertices that do not belong to $M(t)$, then we choose the largest (youngest) of those vertices, say $w$, and set $M(t):=M(t-1)\cup \{(t,w)\}$ and $X(t)=X(t-1)-1$. Otherwise, $M(t)=M(t-1)$ and $X(t)=X(t-1)+1$. Let $x(t)=X(t)/t$ be the fraction of unmatched vertices after step $t$. Then $X(t)$ is a Markov chain with
\[
\pr{X(t+1)-X(t) =1 \mid X(t)} = (1-x(t))^m
\]
since for $X(t+1)-X(t) =1$ to happen, each of the $m$ choices made by vertex $t+1$ must lie outside of $M(t)$, the probability of each such choice is $1-x(t)$, and the choices are independent of each other.
With the remaining probability vertex $t+1$ chooses at least one of $m$ vertices from $X(t)$, in which case $X(t)$ decreases by 1. Consequently,
\[
\mean{X(t+1)\big|\circ} = X(t) + (1-x(t))^m - (1-(1-x(t))^m) = X(t) +2(1-x(t))^m-1.
\]
Dividing both sides with $t+1$, we obtain
\[
\mean{x(t+1) | \circ} = x(t) + \frac{h(x(t))}{t+1} - O(1/t^2),
\]
where $h(z) = 2(1-z)^m-z-1$.
The function $h$ meets the conditions of the second part of Lemma~\ref{extensions}.
Hence $\lim_{t\to\infty} t^{\gamma}|x(t)-\rho_m| = 0$ a.s.
On the other hand, if $\rho_m$ is the unique root of $h$ in $(0,1)$, then $r_m:=1-\rho_m$ is the unique root of $2(1-z^m)-z$ in $(0,1)$.
Since $x(t)-\rho_m = (1 -2 |M(t)|/t) - (1-\rho_m) = \rho_m -2 |M(t)|/t$, we also have, a.s.
\[
\lim_{t\to\infty} t^{\gamma}\Big|2|M(t)|/t-r_m\Big| = 0.
\]
This completes the proof.
\end{proof}
\section{Analysis of Greedy Independent Set Algorithm} The algorithm, for both PAM and UAM cases, generates the increasing
sequence of independent sets $\{I(t)\}$ on the sets $[t]$, with $I(1):=\{1\}$. If vertex $t+1$ does not select a single vertex from $t$, we set $I(t+1)=I(t)\cup \{t+1\}$; otherwise $I(t+1):= I(t)$. Given $I(t)$, let
\begin{align*}
X(t)&:= \text{number of vertices }\le t \text{ outside of the current independent set } I(t),\\
Y(t)&:=\text{total degree of these outsiders} ,\\
Z(t)&:=\text{number of insiders selected by outsiders by time }t,\\
U(t)&:=\text{number of insiders selected by vertex } t+1,\\
x(t) &:=\frac{X(t)}{t},\quad y(t):=\frac{Y(t)}{2mt},\quad z(t)=\frac{Z(t)}{mt}, \quad i(t)=\frac{|I(t)|}{t}.
\end{align*}
Since each insider selects only among outsiders, the total degree of insiders is $m|I(t)| +Z(t)$ and
\[
Y(t)=2mt- m|I(t)|-Z(t)\Longrightarrow y(t)=1- (i(t)+z(t))/2.
\]
\subsection{The PAM case}
\begin{customthm}{2.9} Let $w_m$ denote the unique root of $-w+(1-w)^m$ in $(0,1)$. For any
$\chi\in \Bigl(0,\min\Bigl\{\frac{1}{3},\frac{2m+2\delta}{3(2m+\delta)}\Bigr\}\Bigr)$, almost surely
\begin{equation}\label{PAM0}
\lim_{t\to\infty}t^{\chi}\biggl|\frac{|I(t)|}{t}-w_m\biggr|=0.
\end{equation}
\end{customthm}
\begin{proof} By the definition of the algorithm, we have
\[
|I(t+1)|=\left\{\begin{aligned}
&|I(t)|+1,&&\text{with probability }\Bbb P(U(t)=0|\circ),\\
&|I(t)|,&&\text{with probability }\Bbb P(U(t)>0|\circ).\end{aligned}\right.
\]
Now $U(t)=0$ means that vertex $t+1$ selects all $m$ vertices from the outsiders set, so that
\begin{align*}
\Bbb P(U(t)=0|\circ)&=\frac{\bigl(Y(t)+\delta X(t)\bigr)^{(m)}}{\bigl((2m+\delta)t\bigr)^{(m)}}+O(t^{-1})
=\biggl(\frac{2my(t)+\delta x(t)}{2m+\delta}\biggr)^m +O(t^{-1})\\
&=\biggl(1-\frac{m+\delta}{2m+\delta}\, i(t)-\frac{m}{2m+\delta}\,z(t)\biggr)^m +O(t^{-1}).
\end{align*}
The leading term in the first equality is the exact (conditional) probability of $\{U(t)=0\}$ when no loops at vertices other than the first vertex are allowed, and the extra $O(t^{-1})$ is for our, more general, case when the loops are admissible.
So using $i(t)=|I(t)|/t$, we have
\begin{multline}\label{PAM6}
\Bbb E[i(t+1)|\circ]=i(t)
+\frac{1}{t+1}\biggl[-i(t)
+\biggl(1-\frac{m+\delta}{2m+\delta} i(t)-\frac{m}{2m+\delta}z(t)\biggr)^m\biggr]+O(t^{-2}).
\end{multline}
Next $Z(t+1)=Z(t)+U(t)$, so that
\begin{align*}
\Bbb E[Z(t+1)|\circ]&= Z(t) +\Bbb E[U(t)|\circ]\\
&=Z(t)+m\frac{I(t)(m+\delta)+Z(t)}{(2m+\delta)t}+O(t^{-1}),
\end{align*}
and using $z(t)=Z(t)/mt$, we have
\begin{align}\label{PAM7}
\Bbb E[z(t+1)|\circ]=z(t) +\frac{1}{t+1}\biggl[-z(t) + \biggl(i(t)\frac{m+\delta}{2m+\delta}+z(t)\frac{m}{2m+\delta}\biggr)
\biggr] +O(t^{-2}).
\end{align}
Introduce $w(t)=i(t)\frac{m+\delta}{2m+\delta}+z(t)\frac{m}{2m+\delta}$. Multiplying the equations \eqref{PAM6}
and \eqref{PAM7} by $\frac{m+\delta}{2m+\delta}$ and by $\frac{m}{2m+\delta}$ and adding the products, we obtain
\begin{equation}\label{PAM8}
\begin{aligned}
\Bbb E[w(t+1)|\circ]&=w(t)+\frac{f(w(t))}{t+1}+O(t^{-2}),\\
f(w)&:=\frac{m+\delta}{2m+\delta}\bigl[-w+(1-w)^m\bigr].
\end{aligned}
\end{equation}
The function $f$ is qualitatively similar to the function $h$ in the proof of Theorem \ref{th3repeat}.
Indeed $f(w)$ is strictly decreasing, with $f(0)=1$ and $f(1)=-1$.
Therefore $f(w)$ has a unique root $w_m\in (0,1)$; it is not difficult to see that
\[
w_m=\frac{\log m}{m}\Bigl[1+O(\log\log m/\log m)\Bigr],\quad m\to\infty.
\]
Let us prove that, for any $\alpha< c(m,\delta):=\min\Bigl\{1,\frac{2m+2\delta}{2m+\delta}\Bigr\}(\subseteq (0,1])$, and $A\ge A(\alpha)$, we have
\begin{equation}\label{PAM8.1}
\Bbb E\bigl[(w(t)-w_m)^2\bigr]\le A t^{-\alpha},
\end{equation}
First
\begin{equation}\label{PAM8.15}
\begin{aligned}
|i(t+1)-i(t)|&=\left|\frac{|I(t+1)|}{t+1}-\frac{|I(t)|}{t}\right|\\
&\le \frac{1}{t+1}\bigl(|I(t+1)|-|I(t)|\bigr) +\frac{|I(t+1)|}{t+1}\le \frac{2}{t+1},
\end{aligned}
\end{equation}
and similarily
\begin{equation*}
|z(t+1)-z(t)|\le \frac{2}{t+1}.
\end{equation*}
Therefore $|w(t+1)-w(t)|\le 2/(t+1)$, and consequently
\[
\bigl(w(t+1)-w_m\bigr)^2\le \bigl(w(t)-w_m\bigr)^2+\frac{4}{t^2}+2(w(t)-w_m) \bigl(w(t+1)-w(t)\bigr).
\]
So, conditioning on prehistory, we have
\begin{multline*}
\Bbb E\bigl[(w(t+1)-w_m)^2|\circ]\le \bigl(w(t)-w_m\bigr)^2\\
+2\bigl(w(t)-w_m\bigr) \Bbb E\bigl[w(t+1)-w(t)|\circ] +O(t^{-2})\\
=\bigl(w(t)-w_m\bigr)^2 +\frac{2\bigl(w(t)-w_m\bigr)}{t+1} f(w(t))+O(t^{-2}).
\end{multline*}
By \eqref{PAM8}, $f'(w)\le -\frac{m+\delta}{2m+\delta}$; therefore, with $c(m,\delta)=\frac{2m+2\delta}{2m+\delta}$, the last inequality gives
\[
\Bbb E\bigl[(w(t+1)-w_m)^2|\circ]\le \left(\!1-\frac{c(m,\delta)}{t+1}\right)\!\bigl(w(t)-w_m\bigr)^2 +O(t^{-2}),
\]
leading to a recursive inequality
\begin{equation}\label{PAM8.2}
\Bbb E\bigl[(w(t+1)-w_m)^2]\le \left(\!1-\frac{c(m,\delta)}{t+1}\right)\!\Bbb E\bigl[(w(t)-w_m)^2\bigr] +O(t^{-2}).
\end{equation}
The bound \eqref{PAM8.1} follows from \eqref{PAM8.2} by a straightforward induction on $t$.
Next we use \eqref{PAM8.1} to prove that, for $A_1$ large enough,
\begin{equation}\label{PAM8.21}
\Bbb E\bigl[(i(t+1)-w_m)^2|\circ]\le A_1 t^{-\alpha}.
\end{equation}
Using \eqref{PAM8.15}, we have
\[
\bigl(i(t+1)-w_m\bigr)^2\le \bigl(i(t)-w_m\bigr)^2+\frac{4}{t^2}+2(i(t)-w_m) \bigl(i(t+1)-i(t)\bigr).
\]
Consequently
\begin{multline*}
\Bbb E\bigl[(i(t+1)-w_m)^2|\circ]\le \bigl(i(t)-w_m\bigr)^2\\
+2\bigl(i(t)-w_m\bigr) \Bbb E\bigl[i(t+1)-i(t)|\circ] +O(t^{-2})\\
=\bigl(i(t)-w_m\bigr)^2 +\frac{2\bigl(i(t)-w_m\bigr)}{t+1} \bigl[-i(t)+(1-w(t))^m\bigr]+O(t^{-2}).
\end{multline*}
Taking expectations of both sides, and using Cauchy's inequality and \eqref{PAM8.1}, we obtain
\begin{multline*}
\Bbb E\bigl[(i(t+1)-w_m)^2]\le \biggl(\!1-\frac{2}{t+1}\biggr)\Bbb E[(i(t)-w_m)^2]\\
+\frac{2}{t+1}\Bbb E^{1/2}\bigl[(i(t)-w_m)^2]\,\Bbb E^{1/2}\bigl[(w_m-(1-w(t))^m)^2\bigr]+O(t^{-2})\\
\le \biggl(\!1-\frac{2}{t+1}\biggr)\Bbb E[(i(t)-w_m)^2] +\frac{2mA^{1/2}}{t^{1+\alpha/2}}\Bbb E^{1/2}\bigl[(i(t)-w_m)^2\bigr]
+O(t^{-2}),
\end{multline*}
since $w_m=(1-w_m)^m$, and
\[
\big|(1-w(t))^m - (1-w_m)^m\big|\le m|w(t)-w_m|.
\]
So it suffices to show existence of $A_1$ such that
\[
\biggl(\!1-\frac{2}{t+1}\biggr) A_1 t^{-\alpha}+\frac{2mA^{1/2} A_1^{1/2}t^{-\alpha/2}}{t^{1+\alpha/2}}+O(t^{-2})\le A_1 (t+1)^{-\alpha}.
\]
holds for $t>t_0$ where $t_0$ depends only on $A$ and $\alpha$. For large $t$, the above inequality becomes
\[
2mA^{1/2}A_1^{1/2}+O(t^{-1-\alpha})\le A_1\bigl[(2-\alpha)+O(t^{-1})\bigr]+O(t^{-1+\alpha}),
\]
and $A_1> A\Bigl(\frac{2m}{2-\alpha}\Bigr)^2$ does the job. So \eqref{PAM8.21} is proved.
Therefore, by Markov inequality,
\begin{equation}\label{0.99}
\Bbb P\bigl(|i(t)-w_m|\ge t^{-\chi}\bigr)\le At^{-\alpha+2\chi}\to 0,\quad\chi\in (0,\alpha/2).
\end{equation}
This inequality already means that $i(t)\to w_m$ {\it in probability\/}. Let us show that, considerably stronger, $i(t)\to w_m$ with probability $1$, at least as fast as $t^{-\chi}$, for any given $\chi<1/3$.
Pick $\beta>1$ and introduce a sequence $\{t_{\nu}\}$, $t_{\nu}=\lfloor \nu^{\beta}\rfloor$. By \eqref{0.99}, we have
\begin{align*}
\sum_{\nu\ge 1}\Bbb P\bigl(|i(t_{\nu})-w_m|\ge t_{\nu}^{-\chi}\bigr)&\le \sum_{\nu\ge 1}A_1t_{\nu}^{-\alpha+2\chi}
=O\biggl(\sum_{\nu\ge 1}\nu^{-\beta (\alpha-2\chi)}\biggr)<\infty,
\end{align*}
provided that $\beta>(\alpha-2\chi)^{-1}$, which we assume from now. For such choice of $\beta$, by Borel-Cantelli lemma
with probability $1$ for all but finitely many $\nu$ we have $|i(t_{\nu})-\rho|\le t_{\nu}^{-\chi}$. Let
$t\in [t_{\nu}, t_{\nu+1}]$. By \eqref{PAM8.15}, we have
\[
|i(t)-i(t_{\nu}))|=O\Bigl(\frac{t_{\nu+1}-t_{\nu}}{t_{\nu}}\Bigr),
\]
uniformly for all $\nu$.
So if $|i(t_{\nu})-\rho|\le t_{\nu}^{-\chi}$, then (using $t_{\nu}=\Theta(\nu^{\beta})$), we have: for $t\in [t_{\nu},t_{\nu+1}]$,
\begin{align*}
|i(t)-w_m|&\le t_{\nu}^{-\chi}+O\biggl(\frac{t_{\nu+1}-t_{\nu}}{t_{\nu}}\biggr)\\
&=O\bigl(\nu^{-\beta\chi}+\nu^{-1}\bigr)=O\bigl(\nu^{-\min(\beta\chi,1)}\bigr)\\
&=O\bigl(t^{-\min(\chi,\beta^{-1})}\bigr).
\end{align*}
Since $|i(t_{\nu})-w_m|\le t_{\nu}^{-\chi}$ holds almost surely (a.s.) for all but finitely many $\nu$'s, we see then
that a.s. so does the bound $|i(t)-w_m|=O\bigl(t^{-\min(\chi,\beta^{-1})}\bigr)$ for all but finitely
many $t$'s. Now by taking $\beta$ sufficiently close to $(\alpha-2\chi)^{-1}$ from above, we can make $\min(\chi,\beta^{-1})$
arbitrarily close to $\min(\chi, \alpha-2\chi)$ from below. It remains to notice that $\min(\chi, \alpha-2\chi)$ attains its
maximum $\alpha/3$ at $\chi=\alpha/3$. The proof of Theorem \ref{thm5} is complete.
\end{proof}
\subsection{The UAM case}
\begin{customthm}{2.11} Let $w_m$ be the unique root of $-w+(1-w)^m$ in $(0,1)$. Then, for any $\alpha<1/3$, almost surely
\[
\lim_{t\to\infty}t^{\alpha}\Big|\frac{|I(t)|}{t} - w_m\Big|=0.
\]
\end{customthm}
This following remark is already stated as Remark~\ref{w_m}.
\begin{remark}
Thus, the convergence rate aside, almost surely the greedy independent algorithm delivers a sequence of independent sets of asymptotically the same size as for the PAM case.
\end{remark}
\begin{proof} Let $i(t)=|I(t)|/t$. From the definition of the UAM process and the greedy independent set algorithm, we obtain
\[
|I(t+1)|=\left\{\begin{aligned}
&|I(t)|+1,&&\text{with conditional probability } (1-i(t))^m,\\
&|I(t)|,&&\text{with conditional probability } 1-(1-i(t))^m.\end{aligned}\right.
\]
Therefore
\[
\Bbb E\Bigl[|I(t+1)|\big|\circ\Big]=|I(t)| +(1-i(t))^m,
\]
or equivalently
\[
\Bbb E[i(t+1)|\circ]=i(t)+\frac{1}{t+1}\bigl[-i(t)+(1-i(t))^m\bigr].
\]
The function $-x+(1-x)^m$ differs by a constant positive factor from the function $f$ in \eqref{PAM8}.
The function $f$ meets the conditions of Lemma \ref{general}, and $r_m$ is a unique root of $f$.
Therefore, for any $\alpha<1/3$, a.s. $\lim_{t\to\infty} t^{\alpha}|i(t)-r_m|=0$.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,942
|
{"url":"https:\/\/learn.saylor.org\/mod\/page\/view.php?id=27304","text":"## Basic Graph Theory\n\nWe begin with a top-level view of graphs by looking at definitions, structure, and dynamics.\n\n### Graph Theory\n\nA graph is a formal mathematical representation of a network (\"a collection of objects connected in some fashion\").\n\nEach object in a graph is called a node (or vertex). Corresponding to the connections (or lack thereof) in a network are edges (or links) in a graph. Each edge in a graph joins two distinct nodes.\n\nMore formally, we define a graph G as an ordered pair G = (V, E)\u00a0where\n\n\u2022 V is a set of nodes (vertices).\n\u2022 E is a set of edges (links).\n\u2022 Each edge is a pair of vertices.\n\nIn other words, each element of E is a pair of elements of V. Example: The picture above represents the following graph:\n\n\u2022 V = {1, 2, 3, 4, 5, 6}\n\u2022 E = {{1, 2}, {1, 5}, {2, 3}, {2, 5}, {3, 4}, {4, 5}, {4, 6}}\n\u2022 G = (V, E)\n\nLoops: An edge with identical endpoints is a loop: We disallow loops: any graph G=(V,E) by definition has no loops in its set of edges E.\n\n### Undirected and Directed\n\nUndirected graph: The edges of a graph are assumed to be unordered pairs of nodes. Sometimes we say undirected graph to emphasize this point. In an undirected graph, we write edges using curly braces to denote unordered pairs. For example, an undirected edge {2,3} from vertex 2 to vertex 3 is the same thing as an undirected edge {3,2} from vertex 3 to vertex 2.\n\nDirected graph: In a directed graph, the two directions are counted as being distinct directed edges. In an directed graph, we write edges using parentheses to denote ordered pairs. For example, edge (2, 3)\u00a0\u00a0is directed from 2 to 3 , which is different than the directed edge (3, 2)\u00a0from 3 to 2. Directed graphs are drawn with arrowheads on the links, as shown below:\n\nNeighborhood and Degree\n\nTwo vertices are called adjacent if they share a common edge, in which case the common edge is said to join the two vertices. An edge and a vertex on that edge are called incident.\n\nSee the 6-node graph for examples of adjacent and incident:\n\nNodes 4 and 6 are adjacent (as well as many other pairs of nodes)\n\nNodes 1 and 3 are not adjacent (as well as many other pairs of nodes)\n\nEdge {2,5} is incident to node 2 and node 5.\n\nThe neighborhood of a vertex v in a graph G is the set of vertices adjacent to v. The neighborhood is denoted N(v).\u00a0The neighborhood does not include v itself. For example, in the graph below N(5) = {4, 2, 1} and\u00a0 N(6) = {4}.\n\nThe degree of a vertex is the total number of vertices adjacent to the vertex. The degree of a vertex v is denoted deg(v).\u00a0We can equivalently define the degree of a vertex as the cardinality of its neighborhood and say that for any vertex v, deg(v) = |N(v)|.\n\nThe following is a vertex degree table of the graph above.\n\n Vertex ID Vertex Degree 1 2 2 3 3 2 4 3 5 3 6 1\n\nIn a directed graph, we define degree exactly the same as above (and note that \"adjacent\" does not imply any direction or lack of direction).\n\nIn a directed graph it is important to distinguish between indegree and outdegree. Recall that any directed edge has two distinct ends: a head (the end with an arrowhead) and a tail. Each end is counted separately. The sum of head endpoints count toward the indegree of a vertex and the sum of tail endpoints count toward the outdegree of a vertex.\n\nThe directed graph above has the following degrees, indegrees, and outdegrees:\n\n Vertex ID Vertex Degree Indegree Outdegree 1 2 0 2 2 4 3 1 3 1 0 1 4 2 1 1 5 1 1 0\n\n### Density and Average Degree\n\nThe density of a graph G = (V, E) measures how many edges are in set E compared to the maximum possible number of edges between vertices in set V.\n\nDensity is calculated as follows: An undirected graph has no loops and can have at most\u00a0$|V| \\ast (|V|-1)\/2$\u00a0edges, so the density of an undirected graph is\u00a0$2 \\ast |E|\/(|V| \\ast (|V|-1))$. A directed graph has no loops and can have at most\u00a0$|V| \\ast (|V|-1)$ edges, so the density of a directed graph is\u00a0$|E|\/(|V| \\ast (|V|-1))$.\n\nThe average degree of a graph G\u00a0is another measure of how many edges are in set E\u00a0compared to number of vertices in set V. Because each edge is incident to two vertices and counts in the degree of both vertices, the average degree of an undirected graph is\u00a0$2 \\ast |E|\/|V|$.\n\n### Paths\n\nA path in a graph represents a way to get from an origin to a destination by traversing edges in the graph. For example, in the undirected graph G = (V, E) drawn below, there are many paths from node 6 to node 1. One such path is highlighted with red:\n\nWe write a path P\u00a0 as an ordered list of directed edges:\u00a0\u00a0P = ((v1, v2), (v2, v3), ..., (vk, vk + 1)) . The first vertex of the first edge of a path is the origin and the second vertex of the last edge is the destination. Both origin and destination are called endpoints of the path.\n\nExamples:\n\n\u2022 The red path above is ((6,4), (4,3), (3,2), (2,5), (5,1)); it is a path in G from node 6 to node 1\n\u2022 The blue path below is ((6,4), (4,5), (5,1)); it is also a path in G from node 6 to node 1\n\nPaths in undirected graphs defined formally\n\nEven in an undirected graph (such as the 6-node drawing above), we define a path as an ordered list of directed edges:\u00a0P = ((v1, v2), (v2, v3), ..., (vk, vk + 1)). The strict ordering of nodes in the definition of a path corresponds to the order in which nodes and edges are traversed to get from the origin to the destination.\n\nLet us suppose we have an undirected graph G = (V, E) and an ordered list of directed edges P = (e1, e2, ..., ek). P\u00a0is a path in G\u00a0only if the following are true of every directed edge ei = (vi, v i + 1) in P:\n\nDirected edge ei in P corresponds to an undirected edge in\u00a0 E.\n\nIn other words, {vi, vi + 1} \u2208 E.\n\nIf directed edge\u00a0\u00a0is not the last edge in P , then let ei + 1\u00a0\u00a0be the next edge in P. Directed edge\u00a0\u00a0ei + 1 \u00a0starts with a tail node identical to the head node of\u00a0\u00a0ei .\n\nThis means that ei = (vi, vi + 1) and ei + 1 = (vi + 1, vi + 2).\n\nThe final requirement for P\u00a0to be a path in G is that P cannot revisit any nodes. An ordered list of directed edges that meets the above two requirements and that does revisit nodes is called a walk. Examples of walks that are not paths, based on the 6-node graph drawn above:\n\n\u2022 W = ((6, 4), (4, 3), (3, 2), (2, 5), (5, 3), (4, 3), (3, 2), (2, 5), (5,1)) is a walk from node 6\u00a0to node 1\n\u2022 W = ((6, 4), (4, 6), (6, 4), (4, 5), (5, 1), (1, 5), (5, 1)) is a walk from node 6 to node 1\n\nA walk does not have to revisit nodes. Therefore, any path by definition is also a walk.\n\n### Paths in directed graphs\n\nA directed path is a path in a directed graph where the directions of edges in the path match the directions of edges in the directed graph. For example, suppose we consider this directed graph:\n\nThe blue lines highlight P = ((6, 4), (4, 5), (5, 1)) which\u00a0which is a directed path from 6 to 1\u00a0in directed graph G.\n\nAs a counter-example, consider the same directed graph with a different ordered list of edges that is not a directed path:\n\nIn this case, the red lines highlight P = ((6, 4), (4, 3), (3, 2), (2, 5), (5, 1)); the 4th edge of P -- (2, 5) -- goes the\u00a0\"wrong way\"; and so P\u00a0 is not a directed path in G. We instead refer to P\u00a0as an undirected path in a directed graph.\u00a0Information Today's \"Bow Tie Structure of the Web\"\u00a0article puts it this way: \"In an undirected path, hyperlinks can be followed forward or backward, a technique available to search engine spiders but not to a person using a Web browser\".\n\n### Length\n\nThe length of a path is the number of edges that it uses. Note that this definition is merely a restatement of the definition of the length of an ordered list.\n\nExamples:\n\n\u2022 The length of the red path below is 5\n\u2022 The length of the blue path below is 3\n\n### Distance\n\nGiven a graph G, The distance d(x, y) between two vertices x and y is the length of the shortest path from x to y, considering all possible paths in G from x to y.\n\nThe distance between any node and itself is 0.\u00a0If there is no path from x to y then\u00a0d(x, y)\u00a0is infinity.\n\nExamples:\n\n\u2022 In the preceding graph, the distance from 6 to 1 is 3.\n\u2022 There is no path from 6 to 1 shorter than 3.\n\u2022 There is only one path from 6 to 1 with length 3, and that path is ((6,4), (4,5), (5,1)).\n\u2022 There are many longer paths from 6 to 1, none of which has any bearing on the distance from 6 to 1.\n\u2022 In the graph below, the distance from b to c is 2.\n\u2022 There is no path from b to c shorter than 2.\n\u2022 There are multiple paths from b to c of length 2. One such path is ((b,f), (f,c)). There are also many longer paths from b to c.\n\u2022 In the graph below, the distance from a to b is infinity.\n\u2022 There is no path from a to b.\n\nThe above examples all rely on small graphs where a simple visual inspection reveals what is the shortest path between any specified pair of nodes. For larger graphs, it is generally not possible to discern shortest paths so easily. Computer science provides algorithms, such as the breadth-first search algorithm, to compute shortest paths (and hence distances) in a graph of any size.\n\nFor now, we will limit our calculations of shortest paths to small graphs where visual inspection is easy and a formal algorithmic approach is not necessary.\n\n### \"Connected\": a word of many meanings\n\nThe word \"connected\" speaks to the most basic structural properties of networks. It is arguably both the most important and the most overused term in the network vocabulary.\n\nImportant and proper uses of \"connected\":\n\nConnected has several distinct formal definitions, each of which is important. We introduce three of them here and elaborate the details later.\n\n\u2022 A path connects its origin and destination nodes\n\u2022 Two nodes are connected if there is at least one path that connects them\n\u2022 A graph is connected if each of its nodes is connected to all the others\n\nExamples:\n\n\u2022 Nodes 6 and 2 are connected in graph G1 below\n\u2022 Nodes a and b are not connected in graph G2 below\n\u2022 Graph G1 is connected; graph G2 is not connected\n Graph G1 = (V1,E1)\n Graph G2 = (V2,E2)\n\n#### Common sloppy uses of \"connected\":\n\nSometimes \"connected\" is used informally when another term would be more clear. For example, avoid using \"connected\" when one of these unambiguous words would apply:\n\n\u2022 Joined\n\u2022 Incident\n\n### Induced Subgraphs\n\nIt is often useful to consider only part of a graph. Induced subgraphs are one particularly convenient way to define a specific sub-part of a larger graph.\n\nGiven a graph G = (V, E), an induced subgraph is defined by a subset of vertices S\u00a0(i.e., S\u00a0\u2286\u00a0\u00a0V). Then\n\n\u2022 The nodes of the subgraph induced by S\u00a0are simply the nodes in S, and\n\u2022 The edges of the subgraph induced by S\u00a0are all edges in E\u00a0that have both endpoints in S.\n\u2022 We can write the induced subgraph formally as GS = (S, ES) , where ES\u00a0is the set of edges defined above (edges in E that have both endpoints in S).\n\nExample 1: Consider the graph G = (V, E) drawn below.\n\n Original graph\n\nThe subgraph induced by the subset of nodes S = {1, 2, 3, 4}\u00a0can then be drawn:\n\n Induced subgraph\n\nExample 2: Let G = (V, E)\u00a0be the Facebook friends network:\n\n\u2022 V = {x: x has a Facebook account}\n\u2022 E = {{x,y}: x and y are Facebook friends}\n\nThis is a big network: if we want to draw it, we must consider a small subgraph.\n\n### \"Connected\" defined formally\n\nEarlier we stated that:\n\n\u2022 A path connects its origin and destination nodes\n\u2022 Two nodes are connected if there is at least one path that connects them\n\u2022 A graph is connected if each of its nodes is connected to all the others\n\n\"Connected nodes\" defined formally:\n\nGiven an undirected graph G = (V, E), two nodes x and y (i.e., x\u00a0\u2208\u00a0V and y\u00a0\u2208\u00a0V) are connected if and only if one of the following is true:\n\n\u2022 x = y, or\n\u2022 there is at least one path in G with origin x and destination y\n\nThe most important clarification in our formal definition of connected nodes is that any node is by definition connected to itself.\n\n### Connected graphs and connected components\n\nA connected graph (as defined above) is said to consist of a single connected component. If a graph is not connected, then it consists of two or more connected components.\n\nThe book Six Degrees offers this intuitive metaphor for connected component:\n\n\"Suppose the nodes of a graph are buttons and the edges of the graph are threads joining the buttons. The buttons and threads are all on the floor. Grab one button and lift it. If that button has any threads, then other buttons will start to rise in addition to the one button you are holding. Each additional button that rises off the floor may have even more threads that cause even more buttons to rise. Keep lifting until every button connected to the one in your hand (by any combination of threads) is off the floor. The buttons and threads that have risen off the floor are a connected component\".\n\nThe following examples illustrate the idea of connected component.\n\nGraph G1 is connected and therefore has 1 connected component:\n\nGraph G2 has 2 connected components:\n\nGraph G3 has 3 connected components:\n\nA connected component can be defined as an induced subgraph. For example, G3\u00a0(above) consists of three connected components:\n\n\u2022 the subgraph induced by {1,2,5,6}\n\u2022 the subgraph induced by {3,4}\n\u2022 the subgraph induced by {7,8}\n\n### Hubs\n\nA hub is a node in a graph with much higher degree than average. For example, nodes 1 and 2 are both hubs in the figure below:\n\n### Clusters\n\nInformally speaking, a cluster is a group of nodes in a graph that are more connected to each other than they are to the rest of the graph. For example, the red and yellow regions below are clusters:\n\n#### Defining clusters, part one: connected components\n\nThe following sections present a \"semi-formal\" definition of clusters in three parts:\n\n\u2022 First, we formally define connected component\n\u2022 Second, we introduce and formally define clique\n\u2022 Third, we informally define cluster based on the above two formal definitions\n\nGiven an undirected graph G = (V, E), we define a connected component to be a subgraph induced by node set S \u2282 V, i.e., GS = (S, ES),\u00a0with the following two properties:\n\n1. GS is connected\n2. There is no edge in E that joins a node in S to a node not in S\n\nThe above 2 mathematical properties of a connected component translate into the button-and-thread metaphor as follows:\n\nGS is connected: The only buttons that rise off the floor do so because of the one button you are lifting and the threads that ultimately connect that button to other buttons (i.e., paths)\n\nThere is no edge in E that joins a node in S to a node not in S: You have lifted the one button in your hand high enough so that no more buttons will come off the floor no matter how much higher you lift the button you are holding.\n\nFor example:\n\nThe graph G(drawn above and again below) is not a connected component because it violates property\u00a0\u00a0: it is not connected.\n\n Graph G2 = (V2,E2)\n\nThe subgraph induced by S = {c, d, e, f} (drawn with dark nodes and edges below) is not a connected component because it violates property #2.\u00a0There are edges in E that join nodes in S to node b, which is a node not in S.\n\nDefining clusters, part two: cliques\n\nIn an undirected graph G = (V, E) a clique is a subgraph induced by node set S\u00a0\u2286 V, i.e., GS = (S, ES)\u00a0such that the density of GS\u00a0is 1. Put another way, in a clique, every pair of nodes is adjacent.\n\nExample: Consider undirected graph\u00a0G = (V, E) drawn below.\n\nThe following are cliques:\n\n\u2022 The subgraph induced by {c, d, e, f}\n\u2022 The subgraph induced by {b, f}\n\nThe following are not cliques:\n\n\u2022 The subgraph induced by {a, b, c}\n\u2022 The subgraph induced by {b, c, d, e}, which is drawn below. The subgraph is not a clique since its density is less than 1.\n\nThe subgraph induced by {c, d, e, f} is the largest clique in G: the clique with the most nodes. Usually the largest clique in a graph is the most interesting clique in that graph.\n\n### Defining clusters, part three\n\nOur definition of cluster informally draws upon our formal definitions of connected components and cliques.\n\nIn an undirected graph G = (V, E) a cluster is a subgraph induced by node set S \u2286 V, i.e., GS = (S, ES)\u00a0with the following two properties:\n\n\u2022 The average degree of GS\u00a0is \"relatively high\"; (a relaxed adaptation of clique-ness)\n\u2022 There are \"relatively few\" edges in E that join a node in S to a node not in S; (a relaxed adaptation of connected component-ness)\n\nExample: In the graph G = (V, E) drawn below, the following subsets of nodes induce subgraphs that can fairly be called clusters:\n\nSubsets of nodes:\n\n\u2022 {1, 2, 3, 4, 5, 7}\n\u2022 {20, 21, 22, 23, 24, 25}\n\u2022 {14, 15, 16}\n\nCorresponding clusters:\n\nNot all connected components are clusters. For example:\n\n\u2022 In the graph above, the subgraph induced by {9, 10, 11, 12, 13} is a connected component, but it is arguably not a cluster, because the average degree of that induced subgraph is relatively low.\n\u2022 The subgraph induced by {6} is a connected component, but it is definitely not a cluster, because the average degree of the induced subgraph is zero.\n\nNot all cliques are clusters \u2013 even relatively large cliques. For example:\n\n\u2022 The subgraph induced by {1, 2, 3, 4} is a clique, but it is not a cluster because every single one of those nodes is adjacent to node 5; there are too many edges joining nodes in {1, 2, 3, 4} to nodes not in {1, 2, 3, 4}.\n\u2022 Even the largest clique in the above graph \u2013 the subgraph induced by {1, 2, 3, 4, 5} \u2013 is arguably still not a cluster because node 7 is adjacent to so many nodes in {1, 2, 3, 4, 5}.\n\n### Limitations of traditional graph theory\n\nConceived in\u00a01735\u00a0by the fertile\u00a0Leonhard Euler, graph theory developed over the next 200 years to include topics such as:\n\n\u2022 Shortest paths: Given a graph G = (V, E)\u00a0and a pair of nodes x, y: How can we find a shortest path from x to y?\n\u2022 Maximum cliques: A clique is a subset of nodes that are all adjacent to each other. Given a graph\u00a0G = (V, E): How can we find large cliques in G? How can we find the largest clique in G? (Please refer to the network structure page for a more thorough definition of clique.)\n\u2022 Graph coloring: Given a graph and the ability to assign a \"color\" to each of its nodes: How can we use the fewest possible number of distinct colors so that (1) every node has a color assigned to it, and (2) no two adjacent nodes are assigned the same color?\n\nIn studying these and other questions, graph theorists usually made two important assumptions: (1) The graph is known, and (2) the graph does not change.\n\nWhen we use graph theory to study the Web, we must acknowledge that these two assumptions are not realistic. For example:\n\n\u2022 The Web graph is only partially known. Estimating the cardinality of the set of all Web pages is enough to stretch our powers of creative arithmetic; what if we actually had to name all 30-70 billion of them? Our knowledge of the set of all hyperlinks is rougher still.\n\u2022 The Web graph is constantly changing. Even if all Web builders were to cease and desist immediately, the Web pages they have already created include many that automatically change their outgoing links each time a Web user visits. For example, Google, Facebook, Amazon, etc. all present different links to different users at different times.\n\n### Introduction to network dynamics\n\nIn the 1940s and 50s, graph theory expanded in important ways to strengthen our understanding of network dynamics. Two notable milestones were:\n\n1. Studying dynamic flows: commodities that move over predetermined graphs. During World War II, a great deal of attention was paid to questions such as: \"How do we get supplies to our soldiers despite our enemies' attempts to stop us\"? Here nodes represent sources, destinations, and transfer stations for commodities such as food, fuel, ammunition, and medicine. Edges represent roads and other means of transport between pairs of nodes.\n2. Studying dynamic graphs: sets of nodes and edges that change over time. Just after World War II, for reasons much more theoretical than supplying troops, mathematicians formalized the study of graphs whose very nodes and edges change over time. For example, if a graph started with 1,000 nodes and no edges, and then edges were added randomly one at a time to join those 1,000 nodes, what could we say about the evolving properties of that dynamic graph? Both types of network dynamics are important to understanding the Web. For example:\n1. Information flows through the Web via the Internet. Typically, Web hosts provide information, Web browsers request and receive information, and Internet routers are transfer stations. Internet connections such as cables and wi-fi correspond to edges.\n2. The Web is an ever-changing collection of pages and links. So many Web pages and hyperlinks are being added and edited at this very moment that any attempt to catalog them would be outdated before it could be completed. Pages are also removed from the Web (either temporarily by service disruptions or permanently by decommissioning), resulting in \"dead links\" and other complications that frustrate our hope to know the Web.\n\nOur discussion here will focus on the second type of network dynamics: What are the evolving properties of the Web as millions of people continually edit its pages and hyperlinks? This is a core question in the realm of Web Science.\n\nThe first type of network dynamics\u2013how information flows through the Web\u2013is no less important than our chosen focus. We defer its discussion for two reasons: (1) The question of how information can best move right now from a Web host to a Web browser, known as packet-routing, has been well-studied in the realm of traditional Computer Science, where we can use it as a \"black box\"; and (2) The question of how information spreads over time through populations of Web users (e.g., information diffusion) is more advanced than the material presented here.\n\nAlso, in focusing on the ever-changing Web, we will ignore dynamic pages such as Google, Facebook and Amazon, that present different links each time a user visits. We will focus on static pages that millions of regular Web builders make every day with HTML and CSS. What do we learn when we view each of those pages not as an isolated artifact but as a crossroads in a network of millions of Web users and builders?\n\n### Random Graphs\n\nChapter 2 of Six Degrees starts with an excellent introduction to random graphs. Have that handy before you continue.\n\nRandom graphs (and their dynamics) are based on the following assumptions:\n\n\u2022 The graph we start with has many nodes but no edges.\n\u2022 Edges are added randomly one at a time to the graph. (I.e., any pair of non-adjacent nodes is equally likely to be the next edge added to the existing graph.)\n\u2022 Eventually, we end with a clique: every pair of nodes is adjacent.\n\nGiven the above framework, how does a graph evolve between the starting and ending points? That is a core question of random graph dynamics.\n\nOne of the most famous results in the dynamics of random graph describes the size of the largest connected component, also called the\u00a0giant component. Read in Six Degrees the explanation of Figure 2.2 on p. 45. A similar figure labeled with our own terminology is below:\n\nAfter you absorb\u00a0Six Degrees, then consider what this means:\n\n\u2022 By eyeballing the above figure, we can estimate that a random graph with average degree of 2 has a giant component with 75% of all nodes.\n\u2022 Whether |V| is big or small doesn't matter at all: To connect a set of nodes into a giant component, just randomly add |V| edges and you have guaranteed that the average degree is at least 2, thus creating a giant component with 75% of all nodes.\n\nExample: Suppose all links are deleted from the Web. Only the 30-70 billion link-less pages remain. If each author of each Web page added just one random link to each of his pages, then the resulting giant component would connect 75% of all Web pages into a giant component connected by undirected paths (which allow both forward and backward traversing of hyperlinks).\n\n### Random graph algorithm\n\nAn algorithm is a set of instructions that explain how to compute something. It is like a recipe, only for making data instead of food.\n\nAn algorithm has three parts:\n\n1. Input (like the ingredients of a recipe)\n2. Output (like the title of a recipe: \"chile con carne, serves 12\")\n3. The steps of the algorithm (like the steps of the recipe)\n\nThe following algorithm describes the creation of a random graph.\n\n#### Random Graph Algorithm:\n\nInput:\n\n1. Node set V, with |V|>1\n\nOutput:\n\n1. Graph G = (V, E), with E = {{x, y} : x\u00a0\u2208 V and y\u00a0\u2208 V and : x\u00a0\u2260\u00a0 y} (i.e., a clique)\n\nSteps:\n\n1. Define E = { } and G = (V, E)\n\n2. Choose random pair of nodes x, y that are non-adjacent\n\n3. Add element {x, y} to set E\n\n4. If |E| = |V| * (|V| - 1)\/2 then we are done\n\n5. Go to step 2\n\nNote: The main point of most algorithms is generating output, just as the main reason for most recipes is serving food. For example, a sorting algorithm is useful as a way to alphabetize names, and Google's PageRank algorithm is useful as a way to list the most popular and influential Web pages first.\n\nOur random graph algorithm, however, does not generate interesting output. It creates a clique using a more complicated sequence of steps than is necessary to create a clique. The point of the algorithm is the journey it describes, not the destination where it eventually arrives.\n\n### Clusters and homophily\n\nInformally speaking, a cluster is a group of nodes in a graph that are more connected to each other than they are to the rest of the graph. For example, the red and yellow regions below are clusters:\n\nClusters provide a good big-picture view of the Web, much like a table of contents provides a big-picture view of a book. Unlike a book and its table of contents, however, a cluster involves many authors (Web builders) acting independently. Also, a Web builder may be unaware of the clusters he is forming.\n\nThe sociological force behind Web clusters is\u00a0homophily, or \"birds of a feather\" (which \"flock together\"). Here is one important way that homophily impacts the Web:\n\n1. When a Web builder links from his site to another site, he usually does so because he perceives something important shared by his site and the other site.\n2. Therefore hyperlinks represent more than channels of navigation; they also represent shared likes and dislikes.\n3. The navigational meaning of a hyperlink is clear and explicit, but the shared likes and dislikes represented by a hyperlink are often unstated and implicit.\n4. Therefore clusters of mutually interlinked Web pages represent self-organized groups with specific shared likes and dislikes, which are usually unstated and implicit.\n\nMotivated by the above, cluster-based search engines find interlinked groups of Web pages and then deduce the specific \"meaning\" of each cluster by analyzing the content of the interlinked pages. An example of cluster-based search engines is iBoogie.\n\nTriadic closure is a simple and effective mathematical model of homophily in a network. The basic idea is illustrated below:\n\n Nodes 2 and 3 share a mutual neighbor but are not adjacent.\n\nTriadic closure usually occurs in the context of a larger graph. For example, in the graph below\u2013considering only the solid black lines as edges\u2013there are many possible opportunities for triadic closure. Just a few of those possible opportunities are illustrated with dotted red lines.\n\nChapter 2 of Six Degrees pp 56-61 has a good introduction to homophily and triadic closure. Read that before you continue. The following algorithm describes the process of repeated triadic closure:\n\nInput:\n\nAn undirected graph G = (V, E) with |V| > 0 and |E| > 0\n\nOutput:\n\nSee below\u2026\n\nSteps:\n\n1. Look for two nodes x, y that are non-adjacent and share a common neighbor\n2. If no such pair of nodes x, y exists, then we are done\n3. Add element {x, y} to set E\n4. Go to step 1\n\nThe output of the triadic closure algorithm has some notable properties. We can deduce these properties from the following observations:\n\nConsider the closing of a single triad: Nodes x and y are non-adjacent and share a common neighbor; then x and y are joined by edge {x, y}.\n\n1. Adding edge {x, y} does not change the number of connected components in G. Nodes x and y were already connected before we joined them with {x, y}.\n\n2. Adding edge {x, y} increases the density of the connected component that contains nodes x and y.\n\nRepeating the above observations over the course of every iteration of the triadic closure algorithm, we see that the output graph G' = (V, E') computed by the algorithm has these two properties\n\n1. Output graph G' has exactly the same number of connected components as input graph G. Furthermore, the nodes that induce each connected component are the same in G' and G.\n2. Each connected component of G' has the maximum possible density: it is a clique.\n\nExample: Suppose the triadic closure algorithm starts with graph G = (V, E) drawn below:\n\nUpon completion, the algorithm will have added all the orange edges drawn below:\n\nThe output graph has the same number of connected components as the input graph (with the same nodes in those components). And each connected component in the output graph is a clique.\n\nAn algorithm that identifies and closes triads is an example of random-bias. A network resulting from that algorithm is an example of randomly-biased network (graph). Intuitively, a random-biased network is a network that obeys to some property but it is otherwise random. In the triadic closure example, the bias is towards closing a triad (adding an edge) if two nodes have already a common neighbor; the two nodes with the common neighbor are selected however at random.\n\nA hub is a node in a graph with a much higher degree than average. For example, nodes 1 and 2 are both hubs in the figure below:\n\nCentrality\u00a0is the mathematical term most commonly used to describe hubs. Degree and indegree are two common simple measures of centrality.\n\nGoogle\u00a0made centrality-based Web search famous, and centrality-based search engines now dominate the Web. If cluster-based search engines reveal something like a Web \"table of contents\", then centrality-based search engines provide quick shortcuts to \"the good parts\"\u2013the most popular pages and pictures.\n\nThe sociological force behind Web hubs is\u00a0cumulative advantage, or \"rich get richer\". Here is one important way that cumulative advantage impacts the Web:\n\n1. When a Web builder is looking for online resources, he probably uses a centrality-based search engine like Google, and he is more likely to look at higher-ranked pages.\n2. Pages atop centrality-based rankings are there because they are already popular (essentially because they have high indegree.)\n3. Therefore Web builders are more likely to link to (and increase the indegree of) pages that are already popular. It is possible that Web builders will never even see the \"good\" pages that are not popular.\n4. Therefore Google and its ilk, to some extent, drive a feedback loop that amplifies the popularity of whatever is already popular. In other words, the rank of a Web page next week has less to do with its content and more to do with its rank this week. The main change to be expected in next week's rankings is that disparities between high rankings and low rankings will increase. The rich get richer.\n\n### Preferential attachment algorithm\n\nChapter 4 of\u00a0Six Degrees\u00a0pp 108-109 has a good introduction to cumulative advantage (known also as \"rich get richer\" and the \"Matthew effect\" among other names) and its effect on network dynamics. Read that before you continue.\n\nPreferential attachment\u00a0is the mathematical model used to represent the force of cumulative advantage in network dynamics. The algorithm below describes the process of repeated preferential attachment. This is equivalent to the algorithm informally described on pp 108-109 of\u00a0Six Degrees:\n\nPreferential Attachment Algorithm:\n\nInput:\n\n1. An undirected graph G = (V, E)\u00a0such that V = {1, 2, 3, \u2026, k}, k > 1, and |E| > 0\n2. Integer n, with n > k. Variable n represents the number of nodes in the output graph\n\nOutput:\n\nSee example below\u2026\n\nSteps:\n\n1. i = k + 1\n2. Choose a node x based on degree: if deg(x) = 2 * deg(y), then x is twice as likely to be picked as y.\n3. Add element i to set V\n4. Add element {i,x} to set E\n5. Add 1 to the value of i\n6. if i > n then we are done\n\n7. Go to step 2\n\nExample: Suppose the preferential attachment algorithm starts with V = {1, 2}, E = {{1, 2}}, and n = 20. Therefore k = 2 and the input graph can be drawn as:\n\nThe sequence below illustrates the first few iterations of the algorithm.\n\nIteration 1\n\nCreate node 3 and join it to a pre-existing node. Node 1 and node 2 are equally likely.\n\nSuppose the algorithm joins node 3 to node 1. Then\u2026\n\nIteration 2\n\nCreate node 4 and join it to a pre-existing node. Node 1 is twice as likely as node 2 or node 3.\n\nSuppose the algorithm joins node 4 to node 1. Then\u2026\n\nIteration 3\n\nCreate node 5 and join it to a pre-existing node. Node 1 is 3 times as likely as node 2, 3 or 4.\n\nSuppose the algorithm joins node 5 to node 2 (i.e., a lucky break for node 2). Then\u2026\n\nIteration 4\n\nCreate node 6 and join it to a pre-existing node.\n\nIterations continue...\n\nIn the very last iteration, node 20 is joined to node 2\n\nOutput graph:\n\nNotice in the above example that nodes 1 and 2 start out as equals. However, the random choice of node 1 in the first iteration gives node 1 a nudge up in popularity that feeds on itself. In the end, node 1 is the single dominant hub and node 2 is a second-tier mini-hub.\n\nA seemingly insignificant event in iteration 1 has caused a very substantial effect because of cumulative advantage.\n\nSource: Flavio Esposito,\u00a0https:\/\/cs.slu.edu\/~esposito\/teaching\/1080\/webscience\/graphs.html\n\nhttps:\/\/cs.slu.edu\/~esposito\/teaching\/1080\/webscience\/structure.html","date":"2022-08-10 23:10:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 5, \"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\": 0.553309440612793, \"perplexity\": 669.1430307137496}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882571222.74\/warc\/CC-MAIN-20220810222056-20220811012056-00232.warc.gz\"}"}
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\section{\bf Motivation}
Bright QSO activity in the optical peaks at around redshift $z$=2.5
(Schmidt, Schneider \& Gunn 1994; Warren, Hewett \& Osmer 1994;
Shaver et al.~1996; McMahon, Irwin \& Hazard 1997). QSOs are believed
to be powered by the accretion onto super-massive black holes at the
centre of galaxies (e.g. Rees 1984 and references therein) and a
number of authors have linked the changes in QSO activity to changes
in the fuel supply at the centre of the host galaxies (Cavaliere \&
Szalay 1986; Wandel 1991; Small \& Blandford 1992). Haehnelt
\& Rees (1993; HR93 henceforth; see also Efstathiou \& Rees 1988)
recognized that the peak of QSO activity coincides with the time when
the first deep potential wells assemble in plausible variants of
hierarchical cosmogonies. The past few years have seen dramatic
observational improvements in the detection of galaxies out to $z >
$3, transforming our knowledge of galaxy and star formation in the
high redshift universe. There are also far more extensive data on the
demography of supermassive black holes in nearby galaxies, and on low
level activity of AGN both in the optical and the X-ray bands. We
discuss here some implications for the formation and evolution of
active galactic nuclei, attempting to tie these lines of evidence
from high and low redshifts together in a consistent model.
\section{\bf Relating the luminosity function of QSOs and
star-forming galaxies to the mass function of dark matter (DM) haloes}
Steidel and collaborators (Steidel et al. 1992; Steidel et al. 1995;
Steidel et al. 1996; Giavalisco et al. 1997) have developed a
technique for picking out galaxies in the redshift range
$2.5\,<\,z\,<\,3.5$. The detected star-forming population at $z = 3$
bears a close resemblance to local star-burst galaxies. The abundance
of these objects is found to be roughly half of that of $L\,>\,{L^*}$
present day galaxies. There are however {\bf no secure} direct
dynamical mass estimates for these Lyman-break galaxies. And little is
known at present about the the relation of the masses of these objects
to the rate of detected star formation. Strong clustering detected in
these Lyman-break systems at $z\,\sim\,3$ leads to an interpretation
of these objects as the potential progenitors of massive galaxies at
the present epoch.
Several groups have been engaged in the quest for high-redshift
quasars. Schmidt, Schneider and Gunn (1994), detected 90 quasars by
their Lyman-$\alpha$ emission from the Palomar Transit Grism Survey in
the redshift range $2.75 - 4.75$. They find that the space density of
$M_B\,<\,-26$ quasars decreases by a factor of 2.7 per unit redshift
beyond $z = 2.7$. Based on their analysis they conclude that the peak
of the co-moving space density distribution of quasars with
$M_B\,<\,-26$ lies between redshifts $1.7 - 2.7$. The observed
decline in the QSO number density could be either due to the real
paucity of high-z QSOs or due to obscuration by dust introducing a
systematic bias at high redshifts. In a recent paper, Shaver et. al
(1996) report the results of the Parkes survey of a large sample of
flat spectrum sources in the radio-waveband (that is unaffected by
dust) covering roughly 40\% of the sky. They claim that the
space-density of radio-loud quasars does indeed decline with redshift
at $z\,>\,3$, and argue that the same conclusion probably applies to
all quasars.
In what follows, we explore the link between star-forming galaxies and
QSOs at high redshift, assuming that both populations trace the mass
function of DM haloes. Using the Press-Schechter formalism we obtain
estimates of the space density of DM haloes. The space density of
these star-forming galaxies corresponds to those of haloes with masses
of $~10^{12.5}\,M_{\odot}$ and virial velocities of $~300\,$km
s$^{-1}$ (see also Baugh et al.~1997). Further evidence for masses of
this order comes from the strong clustering of these galaxies (Steidel
et al.~1997, Bagla 1997, Jing \& Suto 1997, Frenk et al.~1997, Peacock
1997). A reasonable fit to the luminosity function of these objects
can be obtained assuming a linear relation between star formation rate
and halo mass, i.e. a constant mass-to-light ratio. A weakly non-linear
relation is also consistent with the data and would indeed be required
to match the shallow slope of the luminosity function at the high
luminosity end reported recently by Bershady et al. (1997). Comparable
fits are obtained for the other CDM variants. Note, however, that the
observed $H_{\beta}$ widths of these galaxies, $\sigma \sim 80\,$km
s$^{-1}$ (Pettini et al.~1997), might be in conflict with the virial
velocities of $300\,$ km s$^{-1}$ quoted above.
The space density of optically selected QSOs at $z\,=\,$3 with $M_{\rm
B} <-23$ is smaller than that of the detected star-forming galaxies by
a factor of a few hundred. As demonstrated by HR93 the
well-synchronized evolution of optically selected QSOs can be linked
to the hierarchical growth of DM haloes on a similar timescale if the
duration $t_{\rm Q}$ of the optically bright phase is considerably
shorter than the Hubble time. For small $t_{\rm
Q}$ this comes more and more in line with the predicted space density
of DM haloes and that of star-forming galaxies at high
redshift. However, hardly anything is known about the masses of the
host objects of optically selected QSOs and this still leaves
considerable freedom in the exact choice of $t_{\rm Q}$. Following the
approach of HR93 we estimate the formation rate of active black holes
by taking the positive term of the time derivative of the halo mass
function and a simple parameterization for the black hole formation
efficiency. It is further assumed that active black holes radiate
with a light curve of the form, $L_{\rm B}(t) = f_{\rm B}\,f_{\rm Edd}
\, L_{\rm Edd}\exp{(-t/t_{\rm Q})}$, where $f_{\rm Edd}$ is the ratio
of bolometric to Eddington luminosity and $f_{B}$ is the fraction of
the bolometric luminosity radiated in the B-band.
\begin{figure}
\vspace{12truecm}
\caption{Model A: The B-band QSO luminosity function at $z = 3$ for 2
cosmological models computed using the time derivative of the space
density of DM halos. A bolometric correction factor of 6.0. In the
lower panel: a non-linear relation with $\alpha = 5/3$ between the
BH mass and the halo mass with a QSO lifetime of $1 \times 10^7\,$ yr
was assumed and in the upper panel a linear relation between
the accreting BH mass and the DM halo mass with a shorter lifetime
for the QSO was assumed $1 \times 10^6$ yr [the same parameters as the
best-fit model explored in Haiman \& Loeb 1997]. The over-plotted
data points are from Boyle et al. (1988).}
\end{figure}
We obtain reasonable fits for a wide range of lifetimes and for all
the CDM variants if we allow ourselves some freedom in the relation
between {\bf halo mass} and {\bf black hole mass}. There are,
however, systematic trends: with increasing lifetime the black hole
mass has to become a progressively more nonlinear function of the halo
mass and the black hole formation efficiency has to decrease in order
to match the luminosity function of QSOs. This is due to the fact that
QSOs are identified with rarer and more massive haloes with increasing
lifetime, and these fall on successively steeper portions of the halo
mass function. In Fig. 1 we plot two specific choices of parameters
which we denote as model A and B hereafter (see Haehnelt, Natarajan \&
Rees (1998) for more details). In the lower panels a QSO lifetime close
to the Salpeter timescale $t_{\rm Salp}\,= \epsilon \sigma_{\rm T}
c/4\pi G m_{\rm p} = \,4.5\, \epsilon_{0.1}\, 10^{7}$yr and a scaling
of black hole mass with halo virial velocity as $M_{\rm bh}\,\propto\,
v_{\rm halo}^{5}\propto M_{\rm halo}^{5/3} (1+z)^{5/2}$ was assumed
($\epsilon$ is the total efficiency for transforming accreted rest
mass energy into radiation). A physical motivation for this particular
dependence is discussed later. The upper panel shows the case of a
linear relation between the halo and black hole mass advocated by
Haiman and Loeb (1997,HL97) which requires a QSO lifetime of less than
$10^{6}$yr -- much shorter than the Salpeter time for usually assumed
values of $\epsilon$. In principle $t_{Q}$ could also depend on mass
or other parameters.
\section{\bf Local demography of black holes}
The last few years have seen tremendous progress in establishing the
existence of supermassive black holes, there are now a number of
excellent cases (including that of our own Galaxy) where observations
strongly imply the presence of a relativistic potential well (Watson
\& Wallin 1994; Miyoshi et al.~1995; Genzel et al.~1997). Magorrian
et al. (1997; Mag97 henceforth) published a sample of about thirty
estimates for the masses of the putative black holes in the bulges of
nearby galaxies. Mag97 confirm previous claims of a strong
correlation between bulge and black hole mass (Kormendy \& Richstone
1995). A linear relation of the form, $M_{\rm bh}= 0.006\,M_{\rm
bulge}$, was obtained by Mag97 as a best fit. However, considering the
large scatter a mildly non-linear relation would probably also be
consistent with the data. We would further like to note here that a
linear relation between black hole to bulge mass does not necessarily
imply a linear relation between black hole and halo mass and as we
will argue later a non-linear relation might be more plausible.
Fugukita, Hogan \& Peebles (1997) estimate the total mass density in
stellar bulges as $ 0.001h^{-1} \le \Omega_{\rm bulges} \le
0.003h^{-1}$ and together with the above ratio of black hole to bulge
mass we get,
$$
{\rho_{\rm bh}}=3.3h\times 10^{6}\,({M_{\rm bh}/M_{\rm bulge}}/{0.006})
\,({\Omega_{\rm bulge}}/{0.002h^{-1}})\,M_{\odot}\,Mpc^{-3}.
$$
Considering the complicated selection biases of the Mag97 sample, the
small sample size and possible systematic errors in the black hole
mass estimates this number is still rather uncertain. Van der Marel
(1997) e.g. emphasizes the sensitivity of black hole mass estimates
to the possible anisotropy of the stellar velocity distribution and
argues that the Mag97 mass estimates might be systematically too high.
Nevertheless, as pointed out by Phinney (1997; see also Faber et
al.~1996) $\rho_{\rm bh} ({\rm nearby\ galaxies})$ exceeds the mass
density in black holes needed to explain the blue light of QSOs purely
by accretion onto super-massive black holes,
$$
\rho_{\rm acc} ({\rm QSO}) = 1.4\times 10^{5}\,({f_{\rm B}\,
\epsilon}/{0.01})^{-1}\,M_{\odot}\,Mpc^{-3},
$$
by a factor of about ten unless the value of $f_{\rm B}\, \epsilon$ is
smaller than usually assumed (Soltan 1982, Chokshi \& Turner 1992).
While a few years ago it seemed difficult to discover the total mass
in black holes necessary to explain the blue light emitted by QSOs at
high redshift, black hole detections in nearby galaxies now suggest
that accretion onto supermassive black holes may actually be rather
inefficient in producing blue light.
\section{\bf Constraints on the accretion history
of supermassive black holes}
There are three options to explain the apparently large value of
$$\rho_{\rm bh} ({\rm nearby\ galaxies})/\rho_{\rm acc} ({\rm QSO})$$
(i)$\rho_{\rm bh} ({\rm nearby\ galaxies})$ is strongly
overestimated, or (ii) $f_{B}\,\epsilon$ during the optically bright
phase is smaller than previously assumed, or (iii) supermassive black
holes do not gain most of their mass during the optically bright
phase. A plausible solution with $f_{B}\,\epsilon$ significantly
smaller than 0.01 is discussed later in this section. We first explore
the third possibility somewhat further. The typical mass of a black hole
at the end of the optically bright phase of duration $t_{\rm Q}$
exceeds that accreted during this phase by a factor $M_{\rm bh}/M_{\rm
acc} = f_{\rm edd}^{-1}\, t_{\rm Salp}/t_{\rm Q}$. This factor should
be larger than 1 and smaller than $\rho_{\rm bh} ({\rm nearby\
galaxies}) /\rho_{\rm acc} ({\rm QSO})$ and therefore,
$$
1\,le\,f_{\rm Edd}^{-1}\,\epsilon^{-1}\,({t_{\rm Q}}/{4.5\times
10^{8}\,yr}) ^{-1}\,\le\,{25h ({f_{B}\,\epsilon}/{0.01})\,(
{\rho_{\rm bh}}/{3.3h\times 10^{6}\,Mpc^{-3}}}).
$$
The question when supermassive black holes gained most of their mass
is therefore closely related to $t_{\rm Q}$ and $f_{\rm Edd}$. For
bright quasars, $f_{\rm Edd}$ must be $> 0.1$; otherwise excessively
massive individual black holes would be required. Furthermore,
$f_{\rm Edd}$ will always be smaller than unity even if the ratio of
the accretion rate to that necessary to sustain the Eddington
luminosity, $\dot m$, greatly exceeds unity. This is because a
``trapping surface'' develops at a radius proportional to $\dot m$,
within which the radiation advects inwards rather than escapes. In
consequence, the emission efficiency declines inversely with $\dot m$
for $\dot m >1$ (Begelman 1978).
For $0.1 \le f_{\rm Edd} \le 1$ the possible range
for $t_{\rm Q}$ is,
$$
{2h^{-1}\times 10^{6}({f_{\rm B}}/{0.1})^{-1}\,({\rho_{\rm bh}/{3.3h\times
10^{6} \,Mpc^{-3}}})^{-1}\,yr}\,\le\,t_{\rm Q}\,\le 4.5
\times 10^{8}\,({\epsilon}/{0.1})\,yr.
$$
If $t_{\rm Q}$ is very short (as in model B) and $f_{B}\,\epsilon$ is
not significantly smaller than 0.01 it seems inevitable that
supermassive black holes have acquired most of their mass {\it before}
the optically bright phase. We would like to point out here that a
value of $\rho_{\rm bh}$ as large or larger than we infer from Mag97
is actually needed for short $t_{\rm Q}$.
For the remainder of this section we assume that $t_{\rm Q}$ is of
order the Salpeter time. The ratio of accreted mass to total mass at
the end of the optically bright phase is then equal to $f_{\rm
Edd}^{-1}$. If $f_{\rm Edd} \sim 1$ during the optically bright phase
(and if $f_{B}\,\epsilon$ is not significantly smaller than 0.01) then
the corresponding gain in mass by a factor $\rho_{\rm bh} ({\rm
nearby\ galaxies})/\rho_{\rm acc} ({\rm QSO})$ indicated by Mag97 has
to occur {\it after} the optically bright phase. As the accretion
should not be optically bright the most plausible options are
advection dominated accretion flows (ADAFs) and dust-obscured
accretion (Narayan \& Yi 1995, Fabian et al.~1997 and earlier
references cited therein). ADAFs require low accretion rate with $\dot
m < m_{\rm crit}$ where $\dot m_{\rm crit} = 0.3\alpha_{\rm ADAF}^{2}$
and $\alpha$ is the Shakura-Sunyaev viscosity parameter. There is
therefore a maximum growth factor for the black hole mass density due
to ADAFs $\sim 3.0\alpha_{\rm ADAF}^{2} t_{\rm ADAF}/t_{\rm
Salp}(\epsilon = 1)$ and \hfill \break $\alpha_{\rm ADAF}>
0.3\,[\epsilon\, \rho_{\rm bh} ({\rm nearby\ galaxies})/\rho_{\rm acc}
({\rm QSO})]^{0.5}$ would be required even if the accretion lasts all
the way from $z$=3 to $z$=0.
If, however, $f_{\rm Edd} \sim 0.1$ (and $t_{\rm Q} \sim t_{\rm
Salp}$)then the gain in mass by a factor $\rho_{\rm bh} ({\rm nearby\
galaxies})/\rho_{\rm acc} ({\rm QSO})$ has to occur {\it before} the
optically bright phase as in the case of small $t_{\rm Q}$. For ADAFs
this would require $\alpha_{\rm ADAF} \sim 1$ and is therefore hardly
plausible. In this case dust-obscured accretion would be the only
viable option.
\section{\bf Possible accretion histories with low optical efficiency}
\begin{figure}
\vspace{12truecm}
\caption{Two accretion histories with low overall optical
emission efficiencies are illustrated here.}
\end{figure}
Two possible accretion histories with a low overall efficiency for
producing blue light are sketched in Fig. 3 - the solid curves describe an
accretion history where most of the mass is accreted during the ADAF
phase while the dashed curves are for an accretion history where the
black hole gains most of its mass during a short-lived early phase
with with $\dot m>1$. The figure shows mass accretion rate in units of
the Eddington accretion rate, the mass relative to the final mass and
the optical and hard X-ray luminosity. The accretion rate is constant
at the beginning with $\dot m >1$. The mass is therefore linearly
rising and $\dot m$ decreases. The spectral energy distribution for
accretion with $\dot m>1$ is rather uncertain (as indicated by the
three parallel lines for $\dot m > 1$ in the two bottom panels) and
should depend on the absorbing column and the dust content of the
outer parts of the self-gravitating disc and/or the host galaxy. The
sharp drop of $\dot m$ marks the onset of the back-reaction on the
accretion flow and either the start or the peak of the optical bright
phase (with a rather inefficient production of hard X-rays). Once the
accretion rate has fallen below the critical rate for an ADAF
(indicated by the dashed lines in the top panel) the spectral energy
distribution will change to one peaked in the hard X-ray waveband.
\section{\bf Faint X-ray sources and the hard X-ray background}
As pointed out by many authors, the X-ray emission of optically
selected QSOs is too soft to explain the hard X-ray background. Di
Matteo \& Fabian (1996) and Yi \& Boughn (1997) argued that the
emission from ADAFs has a spectral shape similar to the hard X-ray
background. Fabian et al.~(1997) suggested that this might also be
true for dust-obscured accretion. It is therefore tempting to link the
rather large value of $\rho_{\rm bh} ({\rm nearby\ galaxies})
/\rho_{\rm acc} ({\rm QSO})$ inferred from Mag97 to the origin of the
hard X-ray background and the recently detected large space density of
faint X-ray sources (Almaini et al.~1996; Hasinger et al.~1997,
Schmidt et. al 1997, McHardy et al.~1997, Hasinger 1998). The presence
of extremely low-level optical AGN activity in a large fraction of
galaxies reported by Ho, Fillipenko \& Sargent (1997) would also fit
in nicely with such a picture. The efficiency of ADAFs is
$\epsilon_{\rm ADAF} =0.1\, (\alpha/0.3)^{2}\, \dot m/\dot m_{\rm
crit}$ and decreases rapidly for small $\alpha$ and small $\dot m/\dot
m_{\rm crit}$. If the hard-Xray background was produced by ADAFs onto
supermassive black holes in ordinary galaxies this requires a value of
$\rho_{\rm bh} ({\rm nearby\ galaxies})$ as high as we infer from
Mag97, a large value of $\alpha$ and a value of $\dot m$ below but
still close to $\dot m_{\rm crit}$ lasting for a Hubble time for the
majority of supermassive black holes. At faint flux levels and high
redshifts a possible star-burst contribution to the total spectral
energy distribution will become more and more important in the optical
and probably also the soft X-ray. It is interesting to note here that
the recently detected number counts in the sub-mm wave-band by SCUBA
could be evidence for dust-obscured accreting AGN at high redshifts
(Blain et al. 1998; Almaini et al. 1998).
\section{\bf Conclusions}
The optical QSO luminosity function at $z\sim 3$ can be plausibly
matched with the luminosity function of star forming galaxies at the
same redshift and the mass function of DM haloes predicted by a range
of variants of CDM cosmogonies believed to comply with observational
constraints in the low-redshift universe. This is possible for
lifetimes of optically bright QSOs anywhere in the range $10^{6}$ to
$10^{8}\,yr$. There is a correlation between the lifetime and the
required degree of non-linearity in the relation between black hole
and halo mass. The non-linearity has to increase for increasing
lifetime. Predicted host halo masses, host galaxy luminosities, and
the clustering strength all increase with increasing lifetime and
further observations of these offer our best hope of constraining the
duration of the optically bright phase of QSOs.
The present-day black hole mass density implied by the integrated
luminosities of optically bright QSO may be significantly smaller than
that inferred from recent black hole estimates in nearby galaxies for
generally assumed efficiencies for producing blue light. We have
discussed three possibilities for how and when this mass could be
accreted in an optically inconspicuous way: (i) in the early stages of
accretion at rates far above the Eddington rate, (ii) by accretion
where optical emission is obscured by dust, or (iii) in the late
stages of accretion at a rate below the critical rate for an advection
dominated accretion flow with an Shakura-Sunyaev parameter of
$\alpha_{\rm ADAF}>0.3$.
\acknowledgments
The Isaac Newton Institute for Mathematical Sciences and specifically
the program on the Dynamics of Astrophysical Discs are gratefully
acknowledged for providing a lively scientific environment. Thanks
are due to my collaborators Martin Haehnelt and Martin Rees for
permission to present results from our joint work.
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Q: Disable JavaScript while reading with urllib2 I'm trying to scrape a web page that contains 200+ <li class="classToGet"> elements, which are loaded with AJAX as one scrolls down. When I read the site's source with urllib2.urlopen(url).read() I can only get the initial 100 <li>s.
When I turn JavaScript off in my browser and go to the page, all 200+ <li>s are displayed. How do I disable JavaScript for urllib2 as it loads the page?
Thanks for the help.
A: I thinking you have something to do with http headers User-Agent
I do a small project that fetch pictures from Google picture.
At begin, i used the head as below:
Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/28.0.1500.71 Safari/537.36
But, i get the page that work in Pinterest which is not i wanted. Because it's have to get pages. So i changed the User-Agent value to another one:
Mozilla/5.0 (Windows; U; Windows NT 6.1; en-US; rv:1.9.1.5) Gecko/20091102 Firefox/3.5.5 (.NET CLR 3.5.30729)
Then, it works find now. And it just can give me what i wanted.
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"redpajama_set_name": "RedPajamaStackExchange"
}
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I realised a day or two ago I hadn't posted the end of my Cornwall photos. There are only a few left, so it'd be rude not to.
Here's one of me. There aren't that many full length photos of me around!
The rest of the holiday photos were taken in Charlestown, which is just lovely.
How good a view those houses must have!
Everywhere had bunting up, everywhere we went. I love bunting!
Here's a gif of me swimming in the pool.
Here are some photos from some make up I did last month and didn't post. Green and a smidge of gold. I can't remember exactly what I used, but I *think* the green is from a Vivo Trio and the gold is from the first Naked Palette.
I hope you're having a grand week.
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"redpajama_set_name": "RedPajamaC4"
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Wateridge Described As A Sexual Bully In Jersey Abuse Case
Gordon Wateridge, then in his 40's, was a house parent at Haut de la Garenne from 1970 to 1974.
The Jersey Royal Court has heard testimony from an unnamed 53-year-old woman how Wateridge used to grab her and grope her breasts and inner thighs. She said, "I remember lying on my back with my feet on the floor, he was pretending to tickle me but from my point of view he was having a good feel."
"His hands would be moving up and down inside my thighs."
Asked how she felt at the time and why she never told anyone of the abuse, she replied she knew it was wrong and made her feel creepy, but she was only 13 or 14 and he was a big man. She told the court she never told anyone, "because no one would have listened."
The court heard from another witness that Wateridge would smack bottoms and give bear hugs to the girls at the home. The children nicknamed Wateridge, "The Perv".
A man testified Wateridge threw him to the ground after he told him to leave his sister alone. "He grabbed my hair and said, "Is there a problem, boy?' I said 'no' and he pushed me to the floor."
Prosecutor Stephen Baker asserted that Wateridge abused his position of responsibility at the home and was a persistent sexual bully.
The trial was to have resumed yesterday morning, but was temporarily postponed due to additional evidence being discovered.
Wateridge has denied all 19 charges against him.
Channel On Line
Times On Line
Channel Island Self-Government Should Not Be Guaranteed
An interesting article from the JEP.
During a visit to Guernsey, Lord William Wallace of Saltaire was asked about the notion of full independence for the Islands. Wallace, a Liberal peer, says the UK needs to clamp down on Channel Island Autonomy - adding full independence would be 'horrifying'.
Decisions made hundreds of years ago aren't necessarily valid today and he added small jurisdictions that attract vast amounts of offshore money are difficult to govern without falling prey to corruption.
Apparently, there is a wider concern about the governing of Jersey and the fallout is not just confined to the Haut de la Garenne judicial fiasco!
Many thanks to Mark Le Sueur for keeping me apprised of the undercurrent in Jersey!
Posted by donchais at 1:52 AM
Labels: Gordon Wateridge, Haut de la Garenne
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: How to specify a logging level for a single class I have an ant target that calls multiple classes to do multiple things. It uses the "logging.properties" file to define the logging levels and handlers.
handlers=java.util.logging.FileHandler, java.util.logging.ConsoleHandler
java.util.logging.ConsoleHandler.level=INFO
java.util.logging.ConsoleHandler.formatter=java.util.logging.SimpleFormatter
java.util.logging.FileHandler.level=INFO
java.util.logging.FileHandler.pattern=logs/ant-logging.log
# Write 10MB before rotating this file
java.util.logging.FileHandler.limit=10000000
# Number of rotating files to be used
java.util.logging.FileHandler.count=4
java.util.logging.FileHandler.formatter=java.util.logging.SimpleFormatter
.level=INFO
Now one of the java files it calls is taking a huge chunk of time to complete. I don't have access to this class, so I want to set a specific logging level for that specific class to try to understand what's wrong with my implementation.
I don't want to change the global logging level, I just want to add a specific level to a certain class, how can I achieve that?
A: Logging levels are specified for logger names. Depending on how the application is written in general, the logger name might have the same name as the class or it will use some module name
In your properties file, you can specify the name of the logger:
All properties whose names end with ".level" are assumed to define log levels for Loggers. Thus "foo.level" defines a log level for the logger called "foo" and (recursively) for any of its children in the naming hierarchy.Log Levels are applied in the order they are defined in the properties file. Thus level settings for child nodes in the tree should come after settings for their parents. The property name ".level" can be used to set the level for the root of the tree.
For example:
foo.level=FINE
If you don't know the name of the logger you can use simple formatter to capture the logger name of the loggers that are logging messages.
A: I do believe you can specify by class and log to its own specific appender
For example let's say your class is com.test.FaultyClass
Then you can put
log4j.logger.com.test.FaultyClass=DEBUG, myappender
This will enforce that the output of FaultyClass will not show up in the other appenders
log4j.additivity.com.test.FaultyClass=false
Now you setup myappender however you like
log4j.appender.myappender=org.apache.log4j.DailyRollingFileAppender
log4j.appender.myappender.datePattern='-'dd'.log'
log4j.appender.myappender.layout=org.apache.log4j.PatternLayout
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\section*{Abstract}
This paper addresses the question of whether it is possible to generate networks with a given global structure (defined by selected blockmodels, i.e., cohesive, core-periphery, hierarchical and transitivity), considering only different types of triads. Two methods are used to generate networks: (i) the method of relocating links; and (ii) the Monte Carlo Multi Chain algorithm implemented in the \texttt{ergm} package implemented in R. Although all types of triads can generate networks with the selected blockmodel types, the selection of only a subset of triads improves the generated networks' blockmodel structure. However, in the case of a hierarchical blockmodel without complete blocks on the diagonal, additional local structures are needed to achieve the desired global structure of generated networks. This shows that blockmodels can emerge based on only local processes that do not take attributes into account.
\section*{Introduction}
In social network analysis, considerable attention is paid to global network structures, which can be described using a blockmodel \cite{doreian2005generalized}. A blockmodel consists of groups (also called positions) of units and the relationships between those groups. The units are assigned to the same group if they are equivalent according to the pattern of links to the other units. Often, structural equivalence is assumed. Two units are structurally equivalent if they are linked to the same units and by the same others \cite{lorrain1971structural}, implying they share the same social role \cite{luczkovich2003defining}. There are several well-known and studied types of blockmodels, e.g., cohesive, core-periphery, transitivity and hierarchical.
In order to describe the processes that produce a given global structure, much effort was made to study micro structures in the context of various global structures. For this propose, the triadic census (the collection of all possible networks of size three which are visualised in Fig~\ref{Fig1}), proposed by Davis \cite{davis1967structure}, is often considered \cite{cartwright1956structural, davis1977clustering, davis1967structure, holland1971transitivity, johnsen1985network}.
\begin{figure}[!h]
\begin{adjustwidth}{-2.25in}{0in}
\caption{{\bf The collection of all triad types (triad census).}
The labels consist of three digits: the first digit denotes the number of mutual links ($\leftrightarrow$), the second stands for the number of arcs ($\rightarrow$) while the third denotes the number of missing links between two units. Some types of triads with the same distribution of links are further differentiated (see columns) and labeled with a letter (C stands for cycle, T for transitivity, U for up and D for down).
}
\label{Fig1}
\includegraphics[width=1.4\textwidth]{Fig1-eps-converted-to.pdf}
\end{adjustwidth}
\end{figure}
While the triadic census is well studied in the context of different global network structures, no attention was given to the dependencies between the triadic census and different global network structures, operationalised by the types of blockmodels. \textbf{Therefore, the main objective of this study is to test whether it is possible to generate networks with a given blockmodel structure (seven selected blockmodel types are considered), taking only different types of triads into account.} This is especially important to consider when thinking about the factors that drive a network to a certain global structure in terms of social processes. Understanding such mechanisms might also contribute to the development of statistical tests for assessing the significance of the blockmodel structure obtained by simulations \cite{borgatti2000models} or to predict missing links in partly known networks \cite{clauset2008hierarchical}. In this context, the triad census has already been used to detect the social rules of a participant in a social system \cite{doran2015discovery}, distinguish between network brokerage and network closure \cite{prell2008looking}, predict future links in a social network \cite{juszczyszyn2011link} and study how stature, relationship strength and egocentricity affect user interactions on Facebook \cite{doran2013triads}.
The main objective of this study is further elaborated: is it possible to generate networks with a given blockmodel type while considering only allowed or only forbidden types of triads? Therefore, the classification of allowed and forbidden types of triads is determined for each type of blockmodel. Allowed types of triads are those whose frequency is higher than zero in an ideal blockmodel structure. On the other hand, forbidden triad types are those whose frequency equals zero in an ideal blockmodel. The sets of allowed and forbidden triad types are then reduced based on comparisons of the different types of blockmodels for different levels of errors in the network according to the ideal blockmodel being considered.
The sets of all triad types, the sets of allowed and forbidden triad types and the sets of reduced (called selected) allowed and forbidden triad types are then used to generate networks with a given blockmodel structure. For a blockmodel type which cannot be generated successfully based only on the types of triads, some other local network structures are considered.
Beside the different types of triads, other subgraph types of a size higher than three can be used to generate networks with a given blockmodel. Milo et al. \cite{milo2002network} confirmed that different motifs are common in different empirical networks, where motifs are defined as "patterns of interconnections occurring in complex networks at numbers that are significantly higher than those in randomized networks". Different types of triads, rather than motifs, are considered chiefly because they are the smallest sociological unit from which the dynamic of a multi-person relationship can be observed \cite{davis1977clustering}.
Various types of algorithms can be applied to generate networks with a given blockmodel, considering only different triad types. In this study, the Relocating Links algorithm (RL algorithm) and the Monte Carlo Multi Chain algorithm (MCMC algorithm) are used. If the generated structures with the selected set of triads, obtained using both algorithms, are very similar and close to the assumed ideal structure, one may conclude it is possible to generate networks with the assumed blockmodel structure by only considering the selected types of triads. On the other hand, if the generated networks are not similar and in line with the assumed blockmodel structure, one may speculate whether this is a consequence of the specifics of the algorithms or that the set of selected local structures is insufficient to generate this specific blockmodel.
In this study, it is assumed that the assignment of a unit to a group is unknown. Considering the information on the group assignment would require a different methodological approach. It is also assumed that the units' characteristics are not known. Kogut \cite{kogut2000network} reported that a certain structure's emergence in a network is often the consequence of rules that generate self-organisation dynamics. These rules do not need to be technological in origin, but can also reflect institutional or cultural norms and are also deeply embedded in the social identity of the units, meaning they are often invisible or unknown when examining an empirical social network.
This paper is organised in the following way: first, the global network structures are described in terms of blockmodels and then the local network structures (namely different triad types) are presented. In this context, the sets of allowed and forbidden triad types and the sets of selected allowed and selected forbidden triad types are proposed for each blockmodel type. Considering these, random networks with the chosen blockmodels are generated using the RL algorithm and the MCMC algorithm. Generating a hierarchical blockmodel without complete blocks on the diagonal is further discussed in a separate section (Improvement of the hierarchical blockmodel without complete blocks on the diagonal) while the section Concluding remarks about generating networks with triads briefly summarises the ability to generate networks with a given blockmodel considering only different triad types. Some limitations and further research ideas are presented in the Conclusion.
\section*{Global network structures}
Here, global network structures are defined by blockmodels. A blockmodel is a reduced network in which the units are clusters (positions) of units from the network. The term block refers to a submatrix showing the links between two clusters (positions) while the links in the network represent relationships between the positions \cite{doreian2005generalized}. Using the blockmodeling procedure, a blockmodel can be derived from a given empirical network. Blockmodeling is an approach for reducing a large, potentially incoherent network to a smaller, comprehensible and interpretable structure \cite{doreian2005generalized}. It can entail either a direct or indirect approach.
Several blockmodeling approaches have been developed to establish the best blockmodel structure according to the given network and equivalence \cite{doreian2005generalized}. In this study, pre-specified direct blockmodeling (generalised blockmodeling) was used. Here, the blockmodeling procedure is a local optimisation procedure. The solution is optimised with a relocation algorithm which minimises the value of the criterion function \cite{batagelj1997notes, batagelj1998fitting, doreian1994partitioning}. The criterion function reflects the difference between the ideal blockmodel and the empirical (current) solution. Compared to indirect blockmodeling, direct blockmodeling produces a solution with a lower criterion function value. Further, in the case of direct blockmodeling, the risk of a local optimum exists and therefore the algorithm must be repeated several times in the hope of obtaining the global optimum, and its computational complexity is high when a larger number of units is analysed.
There are several well-known and studied blockmodel types, e.g., cohesive, core-periphery, hierarchical and transitivity \cite{doreian2005generalized, wasserman1994social}. Even though all these structures have often been studied, there are different definitions of them \cite{borgatti2000models}. In this study, the definitions of blockmodel structures are taken from Doreian et al. \cite{doreian2005generalized}.
Structural equivalence \cite{lorrain1971structural} is considered in all blockmodels. It was shown \cite{doreian2005generalized} that in the case of structural equivalence only complete and null blocks exist. In ideal complete blocks, all possible links are present while in ideal null blocks no link exists. The left matrix in Fig~\ref{Fig2} represents a blockmodel with three groups (positions). There are only complete and null blocks where structural equivalence is completely satisfied . Such a blockmodel is a pre-specified blockmodel and called an "ideal blockmodel". In the blockmodel in Fig~\ref{Fig2}A, the diagonal blocks are complete and all the others are null blocks. This type of blockmodel is called a cohesive blockmodel. Usually, the empirical blockmodels being considered are not completely consistent with the selected equivalency and errors exist when comparing such a blockmodel with an ideal one. An error exists when a link is present in a null block or there is a non-link in a complete block. Fig~\ref{Fig2}B-D illustrates a cohesive blockmodel with different levels of errors. A level of errors is measured on a scale between 0 and 1, where 0 corresponds to an ideal blockmodel and 1 corresponds to a totally randomised network with the same density as in the ideal blockmodel. The level of errors increases linearly as ties are moved from complete blocks to null blocks, until the densities of both block types are the same (the level of errors then equals 1). In such a network, it is not possible to distinguish between blocks.
\begin{figure}[!h]
\caption{{\bf Cohesive blockmodel with different level of errors.}
(A) level of errors is 0 (ideal blockmodel), (B) level of errors is 0.25, (C) level of errors is 0.50, (D) level of errors is 1 (random network)}
\label{Fig2}
\includegraphics[height=1\textwidth, angle = 270]{Fig2-eps-converted-to.pdf}
\end{figure}
The following types of ideal blockmodels are defined and considered:
\begin{itemize}
\item \textbf{Cohesive blockmodel} is visualised in Fig~\ref{Fig3}A. With this blockmodel, several internally highly connected clusters of units (positions) are present. The units from different clusters are not linked to each other. This is a very basic network structure type and was also studied, e.g. in the context of the structural organisation of the brain \cite{shen2015network}.
\item The most common \textbf{core-periphery blockmodel} consists of one group of units which are highly internally linked to each other. Peripheral units which are not linked to each other are also assumed in this type of network. The core-periphery blockmodel is called symmetric when the links between the peripheral and core units are mutual (Fig~\ref{Fig3}B) and asymmetric when only the peripheral units are linked to the core ones (Fig~\ref{Fig3}C) or when only the core units are linked to the peripheral ones. The core-periphery blockmodel structure lies in the middle of several extreme properties, e.g. clique vs. star configurations, network assortativity vs. network disassortativity, hierarchy vs. non-hierarchy, etc. \cite{csermely2013structure}.
The core-periphery model is often associated with the existence of elites. An elite group is a small group of units that are all linked to each other (core). Compared to peripheral units, core units have greater prestige, usually defined by a higher number of incoming links (higher in-degree).
A clear core-periphery blockmodel was found among high school students where a link between students exists if the first student asked the second one to lend their study notes \cite{batagelj2004generalized}. It was also found when studying individual creative performances in the Hollywood film industry \cite{cattani2008core}, in the analysis of metabolic networks \cite{da2008centrality}, and in many studies of scientific co-authorships \cite{hu2008visual, cugmas2016stability, chinchilla2012blockmodeling, hu2008visual}.
\item A \textbf{hierarchical blockmodel} consists of several groups of units which can be ordered into a hierarchy based on the direction of the links between the clusters. The units inside the groups can be either linked to each other (Fig~\ref{Fig3}D) or not (Fig~\ref{Fig3}E). A hierarchical structure is often associated with companies' organisational structure \cite{oberg2008hierarchical, ahuja1998network}.
\item A \textbf{transitivity blockmodel} (Fig~\ref{Fig3}F and Fig~\ref{Fig3}G) is similar to a hierarchical model. The only difference is that units from the groups on the lowest level are linked to all groups on the upper levels. This results in many transitivity relations (a relation $R$ on a set $A$ is called transitive if, for any $a,b,c \in A$ the conditions $aRb$ and $bRc$ imply $aRc$) which are very frequent when networks are formed among humans \cite{leinhardt1973development, schaefer2010fundamental} and animals \cite{chase1982dynamics, fararo1986state, skvoretz1996social}. In the literature, both hierarchical and transitive global network structures are often called hierarchical.
\end{itemize}
\begin{figure}[!h]
\caption{{\bf Different types of blockmodel structures: graph (left) and matrix (right) representation.} (A) cohesive, (B) symmetric core-periphery, (C) asymmetric core-periphery, (D) hierarchical without complete blocks on the diagonal, (E) hierarchical with complete blocks on the diagonal, (F) transitive without complete blocks on the diagonal, (G) transitive with complete blocks on the diagonal}
\label{Fig3}
\includegraphics[width=1\textwidth]{Fig3-eps-converted-to.pdf}
\end{figure}
\section*{Algorithms for generating networks}
To explain the impact of local mechanisms on global network structures or to characterise the global network structures in terms of local network structures, different statistical models have been developed \cite{toivonen2009comparative}. Many of these models capture different global network characteristics such as a specific distribution of in-degree or out-degree, the clustering coefficient or the small-world effect and less the specific global configuration of links in the network (e.g. in terms of the blockmodel structure) \cite{kejzar2007}.
To generate networks with a given blockmodel by considering different types of triads, two similar algorithms are used: the RL algorithm and the MCMC algorithm implemented in the \texttt{ergm} package implemented in R \cite{hunter2008ergm}. They both share the assumption that the units tend to create such a constellation of links that would result in a desirable distribution of subgraphs of size three or other characteristics in the network. Following the distinction between network evolution models (NEM), network attribute models (NAM) and ERGM \cite{toivonen2009comparative}, the RL algorithm can be classified in the NEM category. While NEM are primarily used to study how a specific rule (or set of rules) about creating and dissolving links affects the global network structure, the ERGM are used to check to what extent the global network structure can be explained when considering the structure of links and/or characteristics of the units. It can also be used to generate networks based on estimated or fixed parameter values. Both approaches are described and compared in more detail in the following sections.
\subsection*{Generating networks with the Relocating Links algorithm (RL algorithm)}
The RL algorithm (see Algorithm~\ref{RLalgorithm}) assumes that all considered local network statistics for the case of an ideal network are represented by the vector $\mathfrak{T}$. The number of elements $g$ of this vector equals the number of local network statistics considered. The numbers of different types of triads are considered here, but some other local network statistics could also be chosen. The distribution of all or only a subset of all triad types can be given (for forbidden triad types, corresponding values of $\mathfrak{T}$ equal zero). Beside $\mathfrak{T}$, the initial random network $Y_r$ has to be given. Before the iterative procedure starts, the $Y_r$ is saved as a new network $Y_{new}$.
The iterative procedure is repeated many times. Upon each iteration, a pair of linked units $i$ and $j$ and a pair of unlinked units $k$ and $l$ are randomly chosen. Then, the link between $i$ and $j$ is dissolved and the link between $k$ and $l$ is established. The modified network is saved as $Y_p$ (the proposed network). From the proposed network $Y_p$, the number of each triad type considered $\mathfrak{T}_p$ is calculated. The proposed network is saved as $Y_{new}$ (the new network) if the CR ratio is greater than 1. The CR ratio is defined as
\begin{eqnarray}
\label{eq:cr}
CR = \frac{\sum_{i=1}^g {\Big (}(\mathfrak{T}_p - \mathfrak{T})^2{\Big )}_i}
{\sum_{i=1}^g {\Big (}(\mathfrak{T}_{new} - \mathfrak{T})^2{\Big )}_i}
\end{eqnarray}
Then, the new iteration is performed and, after many iterations, the last $Y_{new}$ is the final solution. Besides the $Y_{new}$, the values of $CR$ can be saved and further analysed.
\begin{algorithm}
\caption{The Relocating Links algorithm}\label{RLalgorithm}
\begin{algorithmic}[1]
\Require $\mathfrak{T}$ \Comment{$\mathfrak{T}$ denotes the distribution of local network statistics in an ideal network}
\Require $Y_r$ \Comment{$Y_r$ denotes a random network}
\Require $k$ \Comment{$k$ denotes the number of iterations}
\State $Y_{new} \gets Y_r$
\State $Y_{p} \gets Y_r$
\For{$k$ in $1:k$}
\State randomly select a tie $y_{i,j}$ in $Y_{new}$
\State randomly select a non-tie $y_{k,l}$ in $Y_{new}$
\State transform a tie $y_{i,j}$ to a non-tie in $Y_p$
\State transform a non-tie $y_{k,l}$ to a tie in $Y_p$
\If{$CR > 1$} \Comment{$CR$ is defined in Eq~\ref{eq:cr}}
\State $Y_{new} \gets Y_p$
\Else \State $Y_p \gets Y_{new}$
\EndIf
\EndFor \\
\Return $Y_{new}$
\end{algorithmic}
\end{algorithm}
Compared to the MCMC algorithm introduced in the next section, the RL algorithm is more deterministic since a link is only allocated if the distribution of the triads of the proposed network is closer to the distribution of the triads in the case of an ideal blockmodel. This may result in lower variability of the global network structure of generated networks when the RL is used since, compared to the MCMC algorithm, RL strive to generate networks with the exact number of the selected types of triads. However, the risk of a local optimum exists which could be avoided by further improving the algorithm. Moreover, RL is computationally very intensive: as will be illustrated later, a higher number of iterations is required, especially in the case of denser networks.
\subsection*{Generating networks with the MCMC algorithm}
To describe how the networks were generated using the MCMC algorithm, Exponential Random Graph Modelling (ERGM) has to be defined. Let us consider a random network $Y$ ($y$ is a given empirical network) consisting of $N$ units. Here, the link between the $i$-th and the $j$-th unit can be represented by a random variable $Y_{ij}$, while the set of all possible random networks of this size is denoted by $\mathcal{Y}$. The distribution of $Y$ can be written as
\begin{eqnarray}
\label{eq:ergm}
P_{\theta, \mathcal{Y}} (Y=y) = \frac{exp\{\theta^T g(y)\}}{\kappa (\theta, \mathcal{Y})}
\end{eqnarray}
\noindent where $y \in \mathcal{Y}$. Here, $\theta$ is a vector of coefficients while $g(y)$ is the vector of statistics obtained for $y$. The normalisation constant $\kappa (\theta, \mathcal{Y})$ in the numerator is needed to ensure the sum of probabilities equals 1.
Different methods can be used \cite{corander1998maximum, strauss1990pseudolikelihood, hyvarinen2007connections} to estimate the parameters $\theta$. After that, one can generate random networks based on the model obtained. To do this, several types of MCMC algorithms have been proposed. Generally, the start is represented by a network in $\mathcal{Y}$. Then, based on a uniform distribution one of the links or non-links is chosen. According to the model, the probability of establishing or dissolving a link is calculated and then, based on this probability, the chosen link or non-link is established or dissolved. The process is iterative. For each iteration, the change statistic is calculated, namely the change in the values of the estimated statistics before and after the change in the link between $i$ and $j$. The iterative process stops when approximate convergence to $P_{\theta_0, Y} (Y=y)$ is reached \cite{hunter2008ergm}. There are several versions of the described algorithm which chiefly differ regarding how the probability of establishing or dissolving a link is calculated. In the used \texttt{ergm} package, the Metropolis-Hastings algorithm is implemented and used.
A very common problem with the MCMC algorithm is the multimodal distribution of sufficient statistics \cite{snijders2002markov, jonasson1999random}. A so-called degenerative model emerges when the model poorly fits the empirical data (e.g. due to inappropriately chosen terms), resulting in generated networks that do not fit the empirical network. These generated networks are often without any link or all possible links \cite{handcock2003assessing}. One possible solution is to restrict the class of networks considered to be possible under the model by fixing the number of links.
As described, the method most often used to estimate the parameters is MCMC-MLE which can sometimes be computationally hard to estimate. In our study, the parameters can be estimated based on networks with a given blockmodel without or with only very low levels of errors. Using this approach, the estimation algorithm does not converge in many cases, probably due to the high level of multicollinearity of the triads. In addition, from the end-user point of view, estimating the values of all parameters for each blockmodel type would be very hard.
Instead, the values of the ERGM parameters $\theta$ are arbitrarily set to 2 (allowed) or -2 (forbidden). It has been clearly shown that some types of triads are much more likely to appear in an ideal network (compared to a random network). By setting all the parameters' values to 2 or -2, we essentially assume that all types of allowed triads have the same importance (and similar for all forbidden types of triads). Such a setting is critical when all types of triads are included in the model and result in a relatively unstable model, particularly when the density is not fixed.
All types of triads are generated considering the number of links fixed (to the same value as in ideal networks) on one hand, and free (with the density being the variable) on the other. In the case of the latter, the value of parameter \texttt{edge} is set to such a value that the mean density of 30 generated networks lies within the interval of ideal density $\pm 0.05$.
\section*{Choosing triads for different types of ideal blockmodels}
When generating networks with a specific type of a blockmodel, according to different triad types, all triad types or only a subset of all triad types can be considered. This is particularly important when generating networks with the RL algorithm where the distribution of triads has to be known in advance for each type of ideal blockmodel separately. Here, it has to be pointed out that the distributions of triads can vary among the same type of blockmodel with a different number of positions.
Since the number of different triads is affected by the network density \cite{faust2006comparing}, the value of the A-measure (the ratio between the absolute number of a certain type of triad in an ideal blockmodel and the mean number of such triads in the totally randomised network of the same density -- see Appendix S23 for more information about generating totally randomised networks and the networks with a given level of errors) can be used to minimise the number of different triad types needed to generate the networks with a selected blockmodel.
The classifications of allowed and forbidden triad types for different blockmodel types are presented in the next section followed by the classifications of selected allowed and selected forbidden triad types, based on values of the A-measure.
\subsection*{Allowed and forbidden triad types}
The triad types can be classified in the set of allowed or in the set of forbidden triad types for each blockmodel type, based on the counts of triad types in an ideal blockmodel. Triad types with the count equal to zero are said to be forbidden in a given blockmodel and are thus classified in the set of forbidden triad types (for a given blockmodel). All the other triad types are classified in the set of allowed triad types. This classification is essential for the MCMC algorithm as it determines the values of the appropriate parameters in the ERGM model (see the previous section).
A more detailed insight into how common a certain triad type is for a certain blockmodel can be obtained by interpreting the A-measure values. The A-measure allows the relative number of triads to be compared within a certain type of blockmodel and also the relative number of triads between different types of blockmodels. In order to obtain values of the A-measure, 10,000 totally random networks for each ideal blockmodel type were generated. The A-measure values are presented in Table~\ref{selected_terms}. Values greater than 1 indicate triad types that are more likely to occur in an ideal network structure than would be expected in randomised networks. Such are complete subgraphs of size three (a triad of type 300) in a cohesive blockmodel.
When the A-measure value is close to 1, the number of triads in the case of an ideal network structure is close to the number of triads in totally randomised networks. This could be an indicator that their occurrence is mainly a consequence of the density rather than the type of blockmodel. The A-measure values in the empty cells in Table~\ref{selected_terms} equal zero and therefore denote forbidden triads. It can be seen that the values of the A-measure of certain triad types exceed zero in some but not all blockmodel types.
Reducing the number of triad types used to generate networks with a given blockmodel can be beneficial in several ways. For example, it can help to identify the main (e.g. social) mechanisms that cause a given blockmodel structure to be formed.
In addition, there are practical reasons which differ with respect to the algorithm used. For the RL algorithm, the reduction to only forbidden triad types (or a subset of forbidden triad types) is especially appealing as it does not require knowledge of the exact distribution of triad types in the ideal network (as this algorithm otherwise requires) because the frequency of all forbidden triad types is 0.
The frequencies of different forbidden triad types are also not affected by the sizes and number of clusters. This means that, when generating networks by considering only the forbidden triad types, this information is not taken into account, which may be either desired or not. On the other hand, the frequencies of all allowed triad types contain all the information that is included in all (allowed and forbidden) triad types. Therefore, considering only all allowed triad types is hypothesised to be equivalent to considering all possible triad types.
\subsection*{Selecting subsets of triad types}
Networks generated by the RL algorithm can still differ as the CR (see Eq.~\ref{eq:cr}) is computed slightly differently. For the MCMC algorithm, these issues are not relevant since the exact distribution of triad types is never taken into account when setting the parameter values. However, the MCMC algorithm is affected by multicollinearity, which can be reduced by selecting only certain triad types. Given the point of this algorithm, it is best to select only a small number of relatively different triad types.
The sets of allowed and forbidden triad types can be further reduced to the selected allowed and selected forbidden triad types. This means that not all possible or all allowed or all forbidden types of triads are considered. One could choose the most appropriate triad types based on observations of the A-measure in the case of networks without errors as was done in the previous subsection. Yet, to obtain a better selection of triad types, it is beneficial to observe the A-values for networks with different levels of errors (10,000 networks with the value of the level of errors from 0.2 to 1 with step 0.2 are generated) as done in this study (see Appendix S22 for a more detailed description).
The selected triad types are shown in grey in Table~\ref{Tab1}. It may be seen that only a few triad types are allowed for each type of blockmodel. Almost all of these types of triads are chosen for all types of blockmodels. The exceptions are triad type 021C in the case of a hierarchical blockmodel without complete blocks on the diagonal and triad type 300 in the case of a transitivity blockmodel with complete blocks on the diagonal. On the other hand, for some blockmodel types only a small number of all forbidden triad types is selected, e.g. in the case of an asymmetric core-periphery only one, and in the case of a cohesive blockmodel only two.
\begin{table}[!h]
\begin{adjustwidth}{-2.25in}{0in}
\caption{{\bf A-measure values and the classification of allowed and forbidden triad types, and selected allowed and forbidden triad types for different types of blockmodels.} Values greater than zero denote allowed types of triads while the values which equals zero (empty cells) denotes forbidden types of triads; grey color denotes selected triad types.}
\label{Tab1}
\footnotesize
\begin{tabular}{|p{1cm}|p{1.8cm}|p{2cm}|p{2cm}|p{2.2cm}|p{2.2cm}|p{2cm}|p{2cm}|}
\hline
& COHESIVE & ASYMMETRIC CORE-PERIPHERY&SYMMETRIC CORE-PERIPHERY&HIERARCHICAL WITHOUT COMPLETE BLOCKS ON THE DIAGONAL&HIERARCHICAL WITH COMPLETE BLOCKS ON THE DIAGONAL &TRANSITVITIY WITHOUT COMPLETE BLOCKS ON THE DIAGONAL& TRANSITIVITY WITH COMPLETE BLOCKS ON THE DIAGONAL\\ \hline
003 &2.3&7.1&7.2&1.5&&1.1&~\\ \hline
300 & 96.3\cellcolor{gray!25}&7.5\cellcolor{gray!25}&2.7\cellcolor{gray!25}&&3.7\cellcolor{gray!25}&&1.2\\ \hline
120D &&8.2\cellcolor{gray!25}&\cellcolor{gray!25}&\cellcolor{gray!25}&4.1\cellcolor{gray!25}&&5.1\cellcolor{gray!25}\\ \hline
120U &&&\cellcolor{gray!25}&&4.1\cellcolor{gray!25}&&5.1\cellcolor{gray!25}\\ \hline
102 &10.2\cellcolor{gray!25}&\cellcolor{gray!25}&\cellcolor{gray!25}&&5.8\cellcolor{gray!25}&&\cellcolor{gray!25}\\ \hline
021C &&&\cellcolor{gray!25}&2.2&3.1\cellcolor{gray!25}&\cellcolor{gray!25}&\cellcolor{gray!25}\\ \hline
021U &\cellcolor{gray!25}&8.2\cellcolor{gray!25}&\cellcolor{gray!25}&4.0\cellcolor{gray!25}&\cellcolor{gray!25}&5.1\cellcolor{gray!25}&\\ \hline
021D &\cellcolor{gray!25}&&\cellcolor{gray!25}&4.0\cellcolor{gray!25}&\cellcolor{gray!25}&5.1\cellcolor{gray!25}&\\ \hline
030T &&&&&&3.5\cellcolor{gray!25}&3.5\cellcolor{gray!25}\\ \hline
201 &&&6.6\cellcolor{gray!25}&\cellcolor{gray!25}&\cellcolor{gray!25}&&\cellcolor{gray!25}\\ \hline
120C &&&\cellcolor{gray!25}&&\cellcolor{gray!25}&\cellcolor{gray!25}&\cellcolor{gray!25}\\ \hline
111D &&&&\cellcolor{gray!25}&&\cellcolor{gray!25}&\cellcolor{gray!25}\\ \hline
111U &&&&&&\cellcolor{gray!25}&\cellcolor{gray!25}\\ \hline
030C &&&&&&&\\ \hline
210 &&&&&&&\\ \hline
012 &&&&&&&\\ \hline
\end{tabular}
\end{adjustwidth}
\end{table}
\section*{Simulation design}
The whole simulation process is accomplished by using the R programming language \cite{team2000r}. Using each method (RL algorithm and MCMC algorithm), $k=50$ networks (of size $n=24$ units) with a given blockmodel structure are generated for each selected set of triads. Each generated network is randomised. Pre-specified blockmodeling (also called generalised blockmodeling) is applied to model networks and randomised networks where the number of clusters is set as in the ideal networks (to two or three clusters) (the \texttt{blockmodeling} \cite{ziberna2008blockmodeling} package implemented in R). For each generated network, the value of the blockmodeling criterion function is calculated.
Here, it should be highlighted that there may be bias in the values of the criterion function, where the networks are generated by the RL algorithm and all allowed triad types are considered. This is because the information on the number and sizes of the groups is embedded in the frequencies of the different allowed triad types when using the RL algorithm. Yet this is not the case when the MCMC algorithm is used and/or other subsets of triads are considered.
However, as the criterion function is not generally comparable for different blockmodels, the Mean Improvement Value (MIV) is calculated as
\begin{eqnarray}
\label{eq:miv}
MIV = 1 - \frac{1}{k} \sum_{i=1}^k {\Big (}\frac{P^m_i}{P^r_i} {\Big )}
\end{eqnarray}
\noindent where $P^r_i$ is the value of the criterion function of the $i$-th randomised network and $P^m_i$ is the value of the criterion function of $i$-th network generated based on the model. The MIV obtained on a network with a different number of units is generally not comparable.
For each type of generated network (the RL algorithm, the MCMC algorithm with fixed density, and the MCMC algorithm with non-fixed density), visualisations of the $P^r$ and the $P^m$ are presented in Appendix~S19-S21 for each type of blockmodel and each combination of triad types considered. The corresponding values of the MIV are visualised in Fig~\ref{Fig5}, Fig~\ref{Fig7} and Fig~\ref{Fig8}.
\section*{Results}
This section is organised in several subsections. First, the global network structures of the networks generated with the RL algorithm and MCMC algorithm (fixed and non-fixed density) are evaluated. For each algorithm, the networks generated by considering different sets of triad types are compared. Then, considerations are presented of certain other local network structures in the evant triads do not generate the networks with the expected blockmodel. Finally, a general statement on generating networks using triads is given.
\subsection*{Networks generated with the RL algorithm}
When all triad types are considered, the overall MIV is around 72~\%, which is more than for all other sets of triads considered (Fig~\ref{Fig5}). On the other hand, the MIV corresponding to networks generated based only on all forbidden triads or all allowed triads is slightly lower or the same. What is outstanding is the symmetric core-periphery with the lowest MIV varying between 11 and 32~\% among the different models (all, all allowed, or all forbidden triad types). As has been emphasised, when the network is very dense the RL algorithm is less effective in finding the right link to relocate. This is expressed in a very small peripheral part in the case of a symmetric core-periphery blockmodel.
The MIVs are usually lower when all forbidden triad types are considered. The MIVs corresponding to the cohesive blockmodel are very similar, yet the structure of the blockmodels so generated is different when only the set of forbidden triad types is considered (the cluster sizes are more variable).
When comparing the different blockmodel types, the highest MIV is observed in the case of an asymmetric core-periphery blockmodel (98~\% when all allowed or all forbidden types of triads are considered), in the case of a transitivity blockmodel without complete blocks on the diagonal (95~\% when all allowed types of triads are considered and 94~\% when all types of triads are considered) and in the case of a transitivity blockmodel with complete blocks on the diagonal (94~\% when all allowed triad types are considered and 92~\% when all triad types are considered). In the latter case, there is quite considerable variability in the cluster sizes when all forbidden types of triads are considered. To be more precise, the tendency to form one cluster with a relatively high number of units and two clusters with a lower number of units is present. This happens because different types of triads can be present in different parts of the network.
When generating networks with a hierarchical blockmodel without complete blocks on the diagonal, a blockmodel structure which is not assumed emerges. Instead, there are links in the blocks below the diagonal of the matrix and in the blocks above the diagonal. This means there are links from the top to the lowest clusters and from the low clusters to the higher clusters. On the level of units, only asymmetric links are possible. However, the density is still higher in complete than in null blocks (Fig~\ref{Fig4}), which may be a consequence of the optimisation algorithm for pre-specified blockmodeling.
\begin{figure}[H]
\caption{{\bf Some generated networks with the expected hierarchical blockmodel without complete blocks on the diagonal.} The RL algorithm is used and all triad types are considered.}
\label{Fig4}
\includegraphics[height=1\textwidth, angle = 270]{Fig4-eps-converted-to.pdf}
\end{figure}
When all allowed triads are included in the process of generating networks, one would expect a similar MIV as when all triads are included in the model. This is because all the information for generating the networks embedded in all triad types is also embedded in only allowed triad types (as all the rest have a count of 0).
The set of all allowed triad types and the set of triads with selected allowed types of triads vary only in the case of a hierarchical blockmodel with complete blocks on the diagonal and a transitivity blockmodel with complete blocks on the diagonal. The selection of triad types slightly improves the MIV in the case of both blockmodel types. In the former case, the blockmodel structure can be visually recognised in most, but not all, generated networks. On the other hand, there are very low levels of errors in all generated networks with a transitive blockmodel with complete blocks on the diagonal.
Comparing the networks generated with all forbidden triad types and the networks generated with only the selected forbidden triad type, the MIV is generally lower in the latter case for all types of blockmodels. By visually observing some generated networks, it is hard to recognise the assumed blockmodel structure, except for some transitive blockmodels with complete blocks on the diagonal.
\begin{figure}[H]
\begin{adjustwidth}{-2.25in}{0in}
\caption{{\bf The mean MIV for each blockmodel type (generated by the RL algorithm) and selected set of triad types.} (A) cohesive; (B) symmetric core-periphery; (C) asymmetric core-periphery; (D) hierarchical without complete blocks on the diagonal (E) hierarchical with complete blocks on the diagonal; (F) transitivity without complete blocks on the diagonal; (G) transitivity with complete blocks on the diagonal, (1) allowed and forbidden triad types, (2) allowed triad types, (3) forbidden triad types. Note: only networks of transitivity with complete blocks on the diagonal blockmodel type and hierarchical with complete blocks on the diagonal blockmodel type were generated by considering the selected allowed triad types.}
\label{Fig5}
\includegraphics{Fig5-eps-converted-to.pdf}
\end{adjustwidth}
\end{figure}
\subsection*{Networks generated with the MCMC algorithm: fixed density}
Since the RL algorithm is more deterministic, it generally performs better than the MCMC algorithm. But when networks are denser the MCMC algorithm might perform better as this is the case when e.g., considering the set of all allowed types of triads when generating a symmetric core-periphery blockmodel. This is another reason one has to consider different algorithms when studying micro structures in the context of various global network structures using simulations.
When all possible triad types are considered, the overall mean MIV among all blockmodel types is higher when the networks are generated using the RL algorithm and lower when the networks are generated using the MCMC algorithm with a fixed density (Fig~\ref{Fig7}). Yet generated networks have an assumed blockmodel structure (Fig~\ref{Fig6}) with a relatively low level of errors, except the hierarchical one without complete blocks on the diagonal where the global network structure obtained is similar to that produced with the RL algorithm (considering selected allowed triad types) (see Fig~\ref{Fig4}). Further, the hierarchical structures with complete blocks on the diagonal and the cohesive one are less clear than the others.
\begin{figure}[H]
\caption{{\bf Some empirical generated networks using the RL algorithm by considering all triad types.} By rows: (A) cohesive; (B) symmetric core-periphery; (C) asymmetric core-periphery; (D) hierarchical without complete blocks on the diagonal; (E) hierarchical with complete blocks on the diagonal; (F) transitivity without complete blocks on the diagonal; (G) transitivity with complete blocks on the diagonal.}
\label{Fig6}
\includegraphics[width=1\textwidth]{Fig6-eps-converted-to.pdf}
\end{figure}
Considering only all allowed or only all forbidden triad types does not produce networks with a significantly higher level of errors. The MIVs are lower when selected forbidden triad types are considered compared to the case when all forbidden triad types are considered. In this instance, the generated networks do not have the expected blockmodel.
Further selection of the different types of triads that are allowed does not improve a hierarchical blockmodel with complete blocks on the diagonal, even though some MIVs indicate the opposite. Conversely, a further selection of allowed triad types improves the global structure of networks with an expected transitivity blockmodel with complete blocks on the diagonal.
The further selection of all possible triad types (allowed and forbidden) improves all the MIV values, especially those corresponding to the hierarchical blockmodel with complete blocks on the diagonal and the cohesive blockmodel.
The generated networks with the expected hierarchical blockmodel structure without complete blocks on the diagonal are not in line with the expected global network structure. This is true for any set of triad types considered.
\begin{figure}[H]
\begin{adjustwidth}{-2.25in}{0in}
\caption{{\bf The mean MIV for each blockmodel type (generated by the MCMC algorithm with fixed density) and selected set of triad types.} (A) cohesive; (B) symmetric core-periphery; (C) asymmetric core-periphery; (D) hierarchical without complete blocks on the diagonal (E) hierarchical with complete blocks on the diagonal; (F) transitivity without complete blocks on the diagonal; (G) transitivity with complete blocks on the diagonal, (1) allowed and forbidden triad types, (2) allowed triad types, (3) forbidden triad types. Note: only the networks of transitivity with complete blocks on the diagonal blockmodel type and hierarchical with complete blocks on the diagonal blockmodel type were generated by considering the selected allowed triad types.}
\label{Fig7}
\includegraphics{Fig7-eps-converted-to.pdf}
\end{adjustwidth}
\end{figure}
\subsection*{Networks generated with the MCMC algorithm: non-fixed density}
In the event the initial networks are totally randomised ideal networks, networks generated using the MCMC algorithm with a non-fixed density are close to the networks with a fixed density (Fig~\ref{Fig8}). The further selection of different triad types is not seen as so important in the case of an asymmetric core-periphery blockmodel and a transitivity blockmodel (with or without complete blocks on the diagonal), while it improves the structure of generated networks with an expected symmetric core-periphery blockmodel and a hierarchical blockmodel with complete blocks on the diagonal.
Here, it is noted that the way the initial networks are chosen has a great impact on the networks generated. In the case of the MCMC algorithm with a non-fixed density, considering the random networks (as initial networks) with the expected (the actual number becomes a random variable) number of links being equal to the number of nodes usually produces a very high number of totally empty or full generated networks. This is especially when all triad types are included in the model. In this study, the randomised ideal networks are used as initial networks, meaning the density of the initial networks is not variable and is the same as in the ideal networks.
\begin{figure}[H]
\begin{adjustwidth}{-2.25in}{0in}
\caption{{\bf The mean MIV for each blockmodel type (generated by the MCMC algorithm with variable density) and selected set of triad types.} (A) cohesive; (B) symmetric core-periphery; (C) asymmetric core-periphery; (D) hierarchical without complete blocks on the diagonal (E) hierarchical with complete blocks on the diagonal; (F) transitivity without complete blocks on the diagonal; (G) transitivity with complete blocks on the diagonal, (1) allowed and forbidden triad types, (2) allowed triad types, (3) forbidden triad types. Note: only the networks of transitivity with complete blocks on the diagonal blockmodel type and hierarchical with complete blocks on the diagonal blockmodel type were generated by considering the selected allowed triad types.}
\label{Fig8}
\includegraphics{Fig8-eps-converted-to.pdf}
\end{adjustwidth}
\end{figure}
\subsection*{Improvement of the hierarchical blockmodel without complete blocks on the diagonal}
The proposed models for generating networks with a hierarchical blockmodel structure without complete blocks on the diagonal perform poorly. This is seen by the mean improvement values and the distribution of the values of the criterion function (see Appendix~S1-S21, Fig~\ref{Fig5}, Fig~\ref{Fig7}, Fig~\ref{Fig8} and Fig~\ref{Fig9}A for some empirical examples).
The obtained blockmodel structure is often hierarchical but has additional links from the upper to the lower clusters and with all asymmetric links. This is especially typical of networks generated using the MCMC algorithm. Therefore, the main focus is put on the networks generated using the MCMC algorithm with a non-fixed density. The resulting global structure probably emerges since all considered triad types appear in all parts of the network. Their combination produces a network that is highly determined by paths of length three (e.g., $1 \rightarrow 2 \rightarrow 3 \rightarrow 2$, where digits denote clusters). The MIV is 0.17.
Therefore, omitting the links from the upper to the lower positions means the paths of length three are considered. Here, it should be pointed out that the number of triads is unit-based while the number of paths of length three is an edge-based count. However, an additional parameter paths of length three (in the case of networks with a different number of positions, paths of different lengths should be considered) is added to the model with the value of -2 (as forbidden). Networks generated using this model have the expected hierarchical structure but with only two groups (Fig~\ref{Fig9}B). From time to time, networks with a transitivity blockmodel without complete blocks on the diagonal are also produced (MIV=0.74).
To obtain three positions (instead of two), the parameter's value of triad type 021C has to be increased, e.g. to the value to 4. Such a model produces networks with a very clear hierarchical structure without complete blocks on the diagonal (Fig~\ref{Fig9}C). There are no errors in all generated networks in null blocks while some appear in complete blocks (MIV=0.93).
All of the described networks are generated using the MCMC algorithm. When the RL algorithm is used, all allowed types of triads and paths of length three can be considered. In that case, some errors appear in both null and complete blocks, which is a consequence of the fixed density. However, with a higher number of iterations, the number of errors could also be lower.
\begin{figure}[H]
\caption{{\bf Some generated networks with a hierarchical blockmodel structure without complete blocks on the diagonal using different models.} The networks are generated by the MCMC algorithm with a non-fixed density. (a) All selected types of triads; (b) All selected types of triads and paths of length three; (c) All selected types of triads (with a higher parameter value for the triad of type 021D) and paths of length three}
\label{Fig9}
\includegraphics[height=1\textwidth, angle = 270]{Fig9-eps-converted-to.pdf}
\end{figure}
\subsection*{Concluding remarks about generating networks with triads}
Three approaches are proposed for generating networks with a given blockmodel structure. The networks so generated are compared with totally random networks of the same density. As expected, there are fewer errors in these networks when generated using the RL algorithm compared to the MCMC algorithm. With both approaches, the selection of triad types does not necessarily result in generated networks with higher or lower levels of errors.
Both algorithms perform well in the case of asymmetric core-periphery blockmodels (all generated networks have the expected blockmodel, see Fig~\ref{Fig10}). However, in the case of the symmetric core-periphery blockmodel, the MIVs are usually small, which is reflected by the insufficiently small periphery in the generated networks. It is also hard to generate a hierarchical blockmodel without complete blocks on the diagonal when considering only different triad types, regardless of the algorithm used to generate the networks. By adding paths of length three, the empirical networks produced have the expected blockmodel type with a very low level of errors (see Fig~\ref{Fig9}).
One of the most important observations is that the number of different types of triads reflects the assumed global network structure, where it is often sufficient to consider only some of all possible types of triads.
\begin{figure}[H]
\begin{adjustwidth}{-2.25in}{0in}
\caption{{\bf Some examples of generated networks for each type of a blockmodel.} The networks are generated using the RL algorithm by selected allowed types of triads. (A) cohesive; (B) symmetric core-periphery; (C) asymmetric core-periphery; (D) hierarchical without complete blocks on the diagonal (E) hierarchical with complete blocks on the diagonal; (F) transitivity without complete blocks on the diagonal; (G) transitivity with complete blocks on the diagonal.}
\label{Fig10}
\includegraphics[angle = 270]{Fig10-eps-converted-to.pdf}
\end{adjustwidth}
\end{figure}
\section*{Conclusion}
This paper examined whether often used global structures, operationalised by blockmodels, can emerge as a consequence of local processes operationalised by different triad types. It was shown that for most of the studied blockmodels this indeed happened. The only exception is a hierarchical blockmodel without complete blocks on the diagonal, where additional local structures are needed to obtain a good fit with the assumed blockmodel structure.
The main conclusion of the study is that a given global network structure can emerge due to the local mechanisms, regardless of the characteristics of the nodes.
This was shown by generating networks that considered different triad types using two algorithms: the proposed deterministic Relocating Links (RL) algorithm and the Monte Carlo Markov Chain (MCMC) algorithm as implemented in the ergm R package \cite{hunter2008ergm}. The RL algorithm randomly selects a link and exchanges it with a randomly selected non-link. The change is accepted if the new network's local structure count is closer to the target count than in the previous network. With the MCMC algorithm, the same local structures were used as parameters in the ergm model.
To determine the target count for the RL algorithm and the parameter values for the MCMC algorithm, the count of different triad types in ideal networks (namely, those that perfectly comply with a certain blockmodel) had to be determined. This was achieved by considering the specific blockmodel type and corresponding group sizes. All types of triads were classified in the set of forbidden and set of allowed triad types for each blockmodel. Allowed triad types are those that are present in ideal networks and forbidden triad types are those that are not. The RL algorithm uses counts of a selected local structure in an ideal blockmodel while for the MCMC algorithm the parameter values were determined based on the classification into allowed and forbidden triad types.
Both algorithms performed very well; the exception is the hierarchical model without complete blocks on the diagonal (an additional parameter must be added) and to a smaller extent the symmetrical core-periphery model. On average, the RL algorithm performed slightly better.
This paper also explored whether one can reduce the required local structure information by using only allowed or only forbidden triad types. Using only forbidden types of triads is especially desirable for the RL algorithm as the count for this triad type is zero. In addition, the reduction of all these sets (all, allowed, forbidden) of triad types was studied based on their sensitivity to errors according to the blockmodel structure. Most of these reductions of sets of different triad types overall resulted in only a slightly worse fit and in some cases even in an improved performance. The only exception is when using only selected forbidden triad types, which did not generate the assumed blockmodel structure.
There are several ways this study could be extended. One would be to move from considering the local structures (e.g. types of triads) to explicitly define other types of rules for creating and dissolving links in the network (local mechanisms). Another extension of this study could be to consider the mechanisms that lead to change from one particular blockmodel to another particular blockmodel.
\bibliographystyle{plos2015}
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"redpajama_set_name": "RedPajamaArXiv"
}
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Q: How to fix Django South issue with regards to localflavor in Django 1.5? I'm starting a new project and I'm using Django 1.5. I found out that the localflavor stuff has been removed from Django 1.5 and is now a separate installable package. So I installed it. In my models.py I'm importing the U.S. localflavors to get my states:
from django_localflavor_us.models import USStateField
In my model, I have this field:
state = USStateField(default='VA')
When I attempt to run a migration with South, I get the following message now:
! Cannot freeze field 'playerstats.location.state'
! (this field has class django_localflavor_us.models.USStateField)
! South cannot introspect some fields; this is probably because they are custom
! fields. If they worked in 0.6 or below, this is because we have removed the
! models parser (it often broke things).
! To fix this, read http://south.aeracode.org/wiki/MyFieldsDontWork
I read through the wiki article, but I find it very verbose and complex. My USStateField isn't considered a custom field now in 1.5 is it? Has anyone else run into this issue in 1.5? And how did you resolve it?
A: Have you tried adding the introspection rule?
add_introspection_rules([], ["^django_localflavor_us\.models\.USStateField"])
A: I had to tweak Hedde van der Heide's solution to get mine to work. It looks like this:
add_introspection_rules([], ["^localflavor\.us\.models\.USStateField"])
A: As of django-localflavor version 1.0, simply adding "localflavor" to your INSTALLED_APPS is all you need to get South to properly pick it up. https://django-localflavor.readthedocs.org/en/latest/?highlight=south#installation
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Exocentrus microspinicollis är en skalbaggsart som beskrevs av Stefan von Breuning 1963. Exocentrus microspinicollis ingår i släktet Exocentrus och familjen långhorningar.
Artens utbredningsområde är Laos. Inga underarter finns listade i Catalogue of Life.
Källor
Långhorningar
microspinicollis
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Review from Pitchline Webzine
Posted by Nick Skog on Tuesday, March 18, 2014 Under: Album Reviews
From: Pitchline Webzine
*Google translation of Spanish review
Right from the choice of name (based on the protagonist of a story by Franz Kafka), we can see that this is not a band that likes things conventional. Hailing from Ukraine, began his career in 2008 calling Shards of Silence and after these past five years, have finally released their debut album. The sound of the album is clearly oriented towards the fluctuating land where extreme metal meets the Post- Rock, land, indeed, increasingly more crowded (could be here all reflection about the meaning of the rise of certain trends and their relation to the concept of "fashion", but we'll leave for another time.) The proposed Odradek Room walks, therefore, between the second and melancholic fragility of the first dark forcefulness, predominantly lingering sadness over the fifty-eight minute compact.
A compositional level, the formula of the group is based on issues of simple structures where premium creating atmospheres of loneliness, isolation and pain from a romantic perspective markedly. This romanticism, in turn, is embodied through the hazy tinkling guitars and melodies that resonate vaporous as a viscous song in which to get caught at the mercy of a seductive and incurable suffering. There are many occasions in which this mental weakness is countered by the force and the own power of extreme metal, as in a passage of songs like "A painting (digging into the cavans with oil)", "Faded Reality" or "River", which emerge you distort guitars, the pace quickens considerably and the raspy voice of Artyom Krikhtenko appears to give more depth and contrast to the sensations and feelings that permeate the listening of this work. I have the impression, in this sense that the balance between the ethereal and introspective side and the more brutal side and torn has been somewhat unbalanced at the end. Moreover, this intense immersion Odradek Room I propose can be somewhat flat in its musical reflected. Let me explain: the especially important to the band, I lyrical section addresses the perplexity, confusion and own existential despair of human beings of a remote fashion simplistic language and images that come many artists.
I think musicar this vast field is a deeply interesting task and that is where I see that look Odradek Room narrows under siege by the aforementioned romance. A greater variety when music become so complex ideas and thoughts, and so dark, would have resulted in a richer and more substantial result (we did find substance in that topic "A painting ...", where the balance between aggressiveness and melody and nurtured existing range of textures and soundscapes manage to reach the desired level of depth).
With that said, I must clarify that these reflections do not seek to be punctilious with the sole purpose of finding potential defects in a record like this, but seeking to spin with greater finesse and potential demand greater reality of a work, on the other hand, offers many attractions to be heard and enjoyed. We're talking about a young band, not without talent and creativity, which focuses on a concept and a little commercial sounds with the intention of extracting the underlying surface as usual. If able to print with a darker and abstract to his music, far from a gloomy vision that has already been sufficiently explored in the past tone, I am convinced that their future work will gain a lot worse.
Reviewed by: Jaime Fernandez
In : Album Reviews
Next post: Review from...
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Tags: odradek room bardo relative reality doom metal death-doom franz kafka atmospheric progressive post-rock experimental death metal
ODRADEK ROOM -
BARDO. RELATIVE REALITY.
Released: March 9, 2013
500 Copies (250 digipack, 250 jewel case)
Atmospheric Death-doom Metal
Review from Metal Music Archives
Review from Deaf Sparrow Webzine
Review from Volumes of Sin Webzine
Review from Pure Nothing Worship Magazine; Issue 2
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\section{Introduction}
One of the naive ways to describe multiple curves, which are the disjoint unions of finitely many essential simple closed curves on the standard punctured disk modulo isotopy, is to use the geometric intersection numbers between the multiple curve and embedded arcs in the disk \cite{dynnikov02}. In \cite{meral19}, this way is generalized for such curve sistems on the orientable surface of genus-$1$ with $n$ ~($n \geq 2$) punctures and one boundary component. The coordinate system \cite{dynnikov02} obtained using this method was extensively used to solve various dynamical and combinatorial problems such as the word problem in the braid group \cite{dehornoy02}, \cite{dehornoy08} and calculate the topological entropy of an braid \cite{moussafir06}. The aim of this paper is to generalize the way which describes each multiple curve by using the geometric intersection numbers with the embedded curves in the punctured orientable genus-$1$ surface with one boundary to the orientable surface of genus-$g$ ~($g \geq 1$) with $n$ punctures and one boundary component.
Throughout the paper, $S_{n,g}$ shall denote a genus-$g$ ~($g \geq 1$) surface with $n ~(n \geq 1)$ punctures and one boundary component. In order to describe a given multiple curve on $S_{n,g}$, a system consisting of $3n+7g-5$ arcs and $g$ simple closed curves on $S_{n,g}$ is used. Given a multiple curve $\I$, we shall introduce a vector in $\Z^{3n+8g-5}_{\geq 0} \setminus \{0\}$ by using the geometric intersection numbers with the curves in our system and consider the linear combinations of these intersection numbers (see Section~\ref{intersection_numbers}).
\section{Geometric Intersection Numbers with Customized Curves Embedded in $S_{n,g}$}\label{intersection_numbers}
In this section, we shall describe the multiple curves on $S_{n,g}$, whose geometric intersection numbers with the customized curves embedded in $S_{n,g}$ and directions are given. For this, we use the model shown in Figure~\ref{gen_model}.
\begin{figure}[!ht]
\centering
\psfrag{a1}[tl]{\scalebox{0.7}{$\scriptstyle{\alpha_{1}}$}}
\psfrag{a2}[tl]{\scalebox{0.7}{$\scriptstyle{\alpha_{2}}$}}
\psfrag{a(2i-3)}[tl]{\scalebox{0.7}{$\scriptstyle{\alpha_{2i-3}}$}}
\psfrag{a(2i-2)}[tl]{\scalebox{0.7}{$\scriptstyle{\alpha_{2i-2}}$}}
\psfrag{a(2i-1)}[tl]{\scalebox{0.7}{$\scriptstyle{\alpha_{2i-1}}$}}
\psfrag{a(2i)}[tl]{\scalebox{0.7}{$\scriptstyle{\alpha_{2i}}$}}
\psfrag{a(2i+1)}[tl]{\scalebox{0.7}{$\scriptstyle{\alpha_{2i+1}}$}}
\psfrag{a(2i+2)}[tl]{\scalebox{0.7}{$\scriptstyle{\alpha_{2i+2}}$}}
\psfrag{a(2n-1)}[tl]{\scalebox{0.7}{$\scriptstyle{\alpha_{2n-1}}$}}
\psfrag{a(2n)}[tl]{\scalebox{0.7}{$\scriptstyle{\alpha_{2n}}$}}
\psfrag{b1}[tl]{\scalebox{0.7}{$\scriptstyle{\beta_{1}}$}}
\psfrag{bi}[tl]{\scalebox{0.7}{$\scriptstyle{\beta_{i}}$}}
\psfrag{b(i+1)}[tl]{\scalebox{0.7}{$\scriptstyle{\beta_{i+1}}$}}
\psfrag{b(n+1)}[tl]{\scalebox{0.7}{$\scriptstyle{\beta_{n+1}}$}}
\psfrag{b(n+2)}[tl]{\scalebox{0.7}{$\scriptstyle{\beta_{n+2}}$}}
\psfrag{b(n+3)}[tl]{\scalebox{0.7}{$\scriptstyle{\beta_{n+3}}$}}
\psfrag{b(n+g)}[tl]{\scalebox{0.7}{$\scriptstyle{\beta_{n+g}}$}}
\psfrag{b(n+g-1)}[tl]{\scalebox{0.7}{$\scriptstyle{\beta_{n+g-1}}$}}
\psfrag{b'(n+2)}[tl]{\scalebox{0.7}{$\scriptstyle{\beta^{'}_{n+2}}$}}
\psfrag{b'(n+3)}[tl]{\scalebox{0.7}{$\scriptstyle{\beta^{'}_{n+3}}$}}
\psfrag{b'(n+g)}[tl]{\scalebox{0.7}{$\scriptstyle{\beta^{'}_{n+g}}$}}
\psfrag{b'(n+g-1)}[tl]{\scalebox{0.7}{$\scriptstyle{\beta^{'}_{n+g-1}}$}}
\psfrag{c1}[tl]{\scalebox{0.7}{$\scriptstyle{c_1}$}}
\psfrag{c2}[tl]{\scalebox{0.7}{$\scriptstyle{c_2}$}}
\psfrag{c(g-1)}[tl]{\scalebox{0.7}{$\scriptstyle{c_{g-1}}$}}
\psfrag{cg}[tl]{\scalebox{0.7}{$\scriptstyle{c^{*}}$}}
\psfrag{g1}[tl]{\scalebox{0.7}{$\scriptstyle{\gamma_1}$}}
\psfrag{g2}[tl]{\scalebox{0.7}{$\scriptstyle{\gamma_2}$}}
\psfrag{gg}[tl]{\scalebox{0.7}{$\scriptstyle{\gamma_g}$}}
\psfrag{g(g-1)}[tl]{\scalebox{0.7}{$\scriptstyle{\gamma_{g-1}}$}}
\psfrag{k1}[tl]{\scalebox{0.7}{$\scriptstyle{\xi_1}$}}
\psfrag{k2}[tl]{\scalebox{0.7}{$\scriptstyle{\xi_2}$}}
\psfrag{k3}[tl]{\scalebox{0.7}{$\scriptstyle{\xi_3}$}}
\psfrag{k4}[tl]{\scalebox{0.7}{$\scriptstyle{\xi_4}$}}
\psfrag{k1'}[tl]{\scalebox{0.7}{$\scriptstyle{\xi^{'}_1}$}}
\psfrag{k2'}[tl]{\scalebox{0.7}{$\scriptstyle{\xi^{'}_2}$}}
\psfrag{k3'}[tl]{\scalebox{0.7}{$\scriptstyle{\xi^{'}_3}$}}
\psfrag{k4'}[tl]{\scalebox{0.7}{$\scriptstyle{\xi^{'}_4}$}}
\psfrag{k(2g-3)}[tl]{\scalebox{0.7}{$\scriptstyle{\xi_{2g-3}}$}}
\psfrag{k(2g-2)}[tl]{\scalebox{0.7}{$\scriptstyle{\xi_{2g-2}}$}}
\psfrag{k'(2g-3)}[tl]{\scalebox{0.7}{$\scriptstyle{\xi^{'}_{2g-3}}$}}
\psfrag{k'(2g-2)}[tl]{\scalebox{0.7}{$\scriptstyle{\xi^{'}_{2g-2}}$}}
\psfrag{d(2i-1)}[tl]{\scalebox{0.7}{$\scriptstyle{{\color{blue}\Delta_{2i-1}}}$}}
\psfrag{d(2i)}[tl]{\scalebox{0.7}{$\scriptstyle{{\color{blue}\Delta_{2i}}}$}}
\psfrag{d(2i-2)}[tl]{\scalebox{0.7}{$\scriptstyle{{\color{blue}\Delta_{2i-2}}}$}}
\psfrag{d(2i+1)}[tl]{\scalebox{0.7}{$\scriptstyle{{\color{blue}\Delta_{2i+1}}}$}}
\psfrag{d(2n)}[tl]{\scalebox{0.7}{$\scriptstyle{{\color{blue}\Delta_{2n}}}$}}
\psfrag{d1}[tl]{\scalebox{0.7}{$\scriptstyle{{\color{blue}\Delta_{1}}}$}}
\includegraphics[width=1.15\textwidth]{gen_model}
\caption{ Curves on $S_{n,g}$ }\label{gen_model}
\end{figure}
Here, the endpoints of arcs $\alpha_i ~(1 \leq i \leq 2n)$, $\beta_i ~(1 \leq i \leq n+g)$, $\beta^{'}_i ~(n+2 \leq i \leq n+g)$, $\xi_i ~(1 \leq i \leq 2g-2)$ and $\xi^{'}_i ~(1 \leq i \leq 2g-2)$ are either on the boundary or on the puncture. While $c_i ~(1 \leq i \leq g-1)$ and $c^{*}$ are the longitude of each torus respectively, $\gamma_i ~(1 \leq i \leq g)$ is the arc whose both endpoints are on $\beta_i$ and $\beta^{'}_i$ as depicted in Figure~\ref{gen_model} and $\gamma_g$ is the arc whose both endpoints are on the boundary. Also, note that each $\gamma_i ~(1 \leq i \leq g-1)$ and $\gamma_g$ intersects each $c_i$ and $c^{*}$ respectively once transversally.
Let $\I_{n,g}$ be the set of multiple curves on $S_{n,g}$ and $\I \in \I_{n,g}$. Throughout the paper, we always work with the minimal representative (a multiple curve in the same isotopy class intersecting the customized curves embedded in $S_{n,g}$ minimally) of $\I$ and denote it by $L$. Let the vector $(\alpha_{1}, \cdots, \alpha_{2n}; \beta_{1}, \cdots, \beta_{n+g}; \beta^{'}_{n+2}, \cdots, \beta^{'}_{n+g}; \xi_{1}, \cdots, \linebreak \xi_{2g-2}; \xi^{'}_{1}, \cdots, \xi^{'}_{2g-2}; \gamma_{1}, \cdots, \gamma_{g}; c_{1}, \cdots, c_{g-1}; c^{*}) \in \{\Z^{3n+8g-5}_{\geq 0}\} \setminus \{0\}$ show the intersection numbers of $L$ with the corresponding arcs and the simple closed curves $c_{i}$ and $c^{*}.$ For example, $(5, 2, 5, 2, 4, 3; 7, 5, 7, 1, 5, 5; 5, 3; 6, 3, 5, 2; 4, 1, 4, 1; 2, 2, 3; 2, 0; 3)$ are the intersection numbers of the multiple curve $L$ depicted in Figure~\ref{inter_num_w_coor_curv}.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.83\textwidth]{inter_num_w_coor_curv}
\caption{ Intersection numbers with the curves embedded in $S_{3,3}$ }\label{inter_num_w_coor_curv}
\end{figure}
\subsection{Path Components on $S_{n,g}$}
In this section, we shall introduce the path components of a multiple curve $L$ on $S_{n,g}$ and derive formulas for the number of these components.
Let $U_i ~(1 \leq i \leq n)$ be the region that is bounded by $\beta_i$ and $\beta_{i+1}$, $G_{i} ~(1 \leq i \leq g-1)$ be the region bounded by $\beta_{n+i}$, $\beta^{'}_{n+i}$, $\beta_{n+i+1}$ and $\beta^{'}_{n+i+1}$, and $G^{*}$ be the region bounded by $\beta_{n+g}$, $\beta^{*}_{n+g}$ and the boundary of $S_{n,g}$ ~($\partial S_{n,g}$). Each component of $L\cap U_i$, $L\cap G_i$ and $L\cap G^{*}$ is called the \emph{path component} of $L$ in $U_i$, $G_i$ and $G^{*}$, respectively. Since $L$ is minimal, there are $4$ types of path components in the region $U_i$ as on the disk \cite{dynnikov02} (see Figure~\ref{puncture_bilesen_gen}). An \emph{above component} has endpoints on $\beta_i$ and $\beta_{i+1}$ and intersects $\alpha_{2i-1}$. A \emph{below component} has endpoints on $\beta_i$ and $\beta_{i+1}$ and intersects $\alpha_{2i}$. A \emph{left loop component} has both endpoints on $\beta_{i+1}$ and intersects $\alpha_{2i-1}$ and $\alpha_{2i}$ (Figure~\ref{puncture_bilesen_gen}a). A \emph{right loop component} has both endpoints on $\beta_i$ and intersects $\alpha_{2i-1}$ and $\alpha_{2i}$ (Figure~\ref{puncture_bilesen_gen}b). There are $6$ types of path components in the region $G^{*}$. The first three of these are \emph{curve $c^{*}$}, which is the longitude of the torus in $G^{*}$ (Figure~\ref{genus_loop_gen}a); \emph{visible genus component}, which has both endpoints on $\beta_{n+1}$ and does not intersect the curve $c^{*}$ (Figure~\ref{genus_loop_gen}b); \emph{invisible genus component}, which has both endpoints on $\beta_{1}$ and does not intersect the curve $c^{*}$ (Figure~\ref{genus_loop_gen}c). The other three components are called \emph{twist}, which have endpoints on $\beta_{n+g}$ and $\beta^{'}_{n+g}$ and intersect the curve $c^{*}$ (see Figure~\ref{cikis_yon_gen}). These components are non-twist, negative twist and positive twist components. The \emph{non-twist component} does not make any twist (see Figure~\ref{cikis_yon_gen}a). The \emph{negative twist component} makes clockwise twist (see Figure~\ref{cikis_yon_gen}b). The \emph{positive twist component} makes counterclockwise twist (see Figure~\ref{cikis_yon_gen}c) \cite{meral19}. There are $14$ types of path components in each region $G_{i}$. These are \emph{curve $c_{i}$}, which is the longitude of the torus in $G_{i}$ (similar to Figure~\ref{genus_loop_gen}a); \emph{visible-left genus component}, which has both endpoints on $\beta_{n+i+1}$ and does not intersect the curve $c_{i}$ (Figure~\ref{diagonal_left_right_loop}a); \emph{invisible-left genus component}, which has both endpoints on $\beta^{'}_{n+i+1}$ and does not intersect the curve $c_{i}$ (Figure~\ref{diagonal_left_right_loop}a); \emph{visible-right genus component}, which has both endpoints on $\beta_{n+i}$ and does not intersect the curve $c_{i}$ (Figure~\ref{diagonal_left_right_loop}b); \emph{invisible-right genus component}, which has both endpoints on $\beta^{'}_{n+i}$ and does not intersect the curve $c_{i}$ (Figure~\ref{diagonal_left_right_loop}b); \emph{upper diagonal component}, which has endpoints on $\beta^{'}_{n+i}$ and $\beta_{n+i+1}$ and intersects the curve $c_{i}$ and the arc $\xi_{2i-1}$ (see Figure~\ref{diagonal_left_right_loop}c); \emph{lower diagonal component}, which has endpoints on $\beta^{'}_{n+i}$ and $\beta_{n+i+1}$ and intersects the curve $c_{i}$ and the arc $\xi_{2i}$ (see Figure~\ref{diagonal_left_right_loop}d); \emph{visible above component}, which has endpoints on $\beta_{n+i}$ and $\beta_{n+i+1}$ and intersects the arc $\xi_{2i-1}$ (see Figure~\ref{diagonal_left_right_loop}e); \emph{invisible above component}, which has endpoints on $\beta^{'}_{n+i}$ and $\beta^{'}_{n+i+1}$ and intersects the arc $\xi^{'}_{2i-1}$ (see Figure~\ref{diagonal_left_right_loop}e); \emph{visible below component}, which has endpoints on $\beta_{n+i}$ and $\beta_{n+i+1}$ and intersects the arc $\xi_{2i}$ (see Figure~\ref{diagonal_left_right_loop}f); \emph{invisible below component}, which has endpoints on $\beta^{'}_{n+i}$ and $\beta^{'}_{n+i+1}$ and intersects the arc $\xi^{'}_{2i}$ (see Figure~\ref{diagonal_left_right_loop}f); \emph{negative twist component}, which has endpoints on $\beta^{'}_{n+i}$ and $\beta_{n+i}$ or $\beta^{'}_{n+i}$ and $\beta_{n+i+1}$ and intersects the curve $c_{i}$ and makes clockwise twist (see Figures~\ref{twist_components_gen}a and \ref{twist_components_gen}b); \emph{positive twist component}, which has endpoints on $\beta^{'}_{n+i}$ and $\beta_{n+i}$ or $\beta^{'}_{n+i}$ and $\beta_{n+i+1}$ and intersects the curve $c_{i}$ and makes counterclockwise twist (see Figures~\ref{twist_components_gen}c and \ref{twist_components_gen}d); and \emph{non-twist component} (see Figure~\ref{twist_components_gen}e).
\begin{figure}[!ht]
\centering
\psfrag{a}[tl]{\scalebox{0.6}{$\scriptstyle{a}$}}
\psfrag{b}[tl]{\scalebox{0.6}{$\scriptstyle{b}$}}
\psfrag{a2i}[tl]{\scalebox{0.6}{$\scriptstyle{\alpha_{2i}}$}}
\psfrag{a2i-1}[tl]{\scalebox{0.6}{$\scriptstyle{\alpha_{2i-1}}$}}
\psfrag{bi}[tl]{\scalebox{0.6}{$\scriptstyle{\beta_{i}}$}}
\psfrag{bi+1}[tl]{\scalebox{0.6}{$\scriptstyle{\beta_{i+1}}$}}
\psfrag{U(i)}[tl]{\scalebox{0.6}{$\scriptstyle{U_{i}}$}}
\includegraphics[width=0.6\textwidth]{puncture_bilesen_gen}
\caption{Above and below components, left and right loop components in the region $U_i$}\label{puncture_bilesen_gen}
\end{figure}
\begin{figure}[!ht]
\centering
\psfrag{a}[tl]{\scalebox{0.55}{$\scriptstyle{a}$}}
\psfrag{b}[tl]{\scalebox{0.55}{$\scriptstyle{b}$}}
\psfrag{g}[tl]{\scalebox{0.6}{$\scriptstyle{\gamma_{g}}$}}
\psfrag{t}[tl]{\scalebox{0.6}{$\scriptstyle{c^{*}}$}}
\psfrag{bn+1}[tl]{\scalebox{0.6}{$\scriptstyle{\beta_{n+g}}$}}
\psfrag{b1}[tl]{\scalebox{0.6}{$\scriptstyle{\beta^{'}_{n+g}}$}}
\includegraphics[width=0.97\textwidth]{genus_loop_gen}
\caption{(a) $c^{*}$ curves, (b) Visible genus component, (c) Invisible genus component in the region $G^{*}$ }\label{genus_loop_gen}
\end{figure}
\begin{figure}[!ht]
\centering
\psfrag{a}[tl]{\scalebox{0.55}{$\scriptstyle{a}$}}
\psfrag{b}[tl]{\scalebox{0.55}{$\scriptstyle{b}$}}
\psfrag{g}[tl]{\scalebox{0.6}{$\scriptstyle{\gamma_{g}}$}}
\psfrag{t}[tl]{\scalebox{0.6}{$\scriptstyle{c^{*}}$}}
\psfrag{bn+1}[tl]{\scalebox{0.6}{$\scriptstyle{\beta_{n+g}}$}}
\psfrag{b1}[tl]{\scalebox{0.6}{$\scriptstyle{\beta^{'}_{n+g}}$}}
\includegraphics[width=0.97\textwidth]{cikis_yon_gen}
\caption{ (a) Non-twist component, (b) Negative twist component, (c) Positive twist component.}\label{cikis_yon_gen}
\end{figure}
\begin{figure}[!ht]
\centering
\psfrag{a}[tl]{\scalebox{0.55}{$\scriptstyle{a}$}}
\psfrag{b}[tl]{\scalebox{0.55}{$\scriptstyle{b}$}}
\psfrag{g}[tl]{\scalebox{0.6}{$\scriptstyle{\gamma_{i}}$}}
\psfrag{x(2i-1)}[tl]{\scalebox{0.6}{$\scriptstyle{\xi_{2i-1}}$}}
\psfrag{x'(2i-1)}[tl]{\scalebox{0.6}{$\scriptstyle{\xi^{'}_{2i-1}}$}}
\psfrag{x(2i)}[tl]{\scalebox{0.6}{$\scriptstyle{\xi_{2i}}$}}
\psfrag{x'(2i)}[tl]{\scalebox{0.6}{$\scriptstyle{\xi^{'}_{2i}}$}}
\psfrag{t}[tl]{\scalebox{0.6}{$\scriptstyle{c_{i}}$}}
\psfrag{b(n+i)}[tl]{\scalebox{0.6}{$\scriptstyle{\beta_{n+i}}$}}
\psfrag{b'(n+i)}[tl]{\scalebox{0.6}{$\scriptstyle{\beta^{'}_{n+i}}$}}
\psfrag{b(n+i+1)}[tl]{\scalebox{0.6}{$\scriptstyle{\beta_{n+i+1}}$}}
\psfrag{b'(n+i+1)}[tl]{\scalebox{0.6}{$\scriptstyle{\beta^{'}_{n+i+1}}$}}
\includegraphics[width=0.75\textwidth]{diagonal_left_right_loop}
\caption{ (a) Visible-left and invisible-left genus components, (b) Visible-right and invisible-right genus components, (c) Upper diagonal component, (d) Lower diagonal component, (e) Visible above and invisible above components, (f) Visible below and invisible below components in the region $G_{i}$}\label{diagonal_left_right_loop}
\end{figure}
\begin{figure}[!ht]
\centering
\psfrag{a}[tl]{\scalebox{0.55}{$\scriptstyle{a}$}}
\psfrag{b}[tl]{\scalebox{0.55}{$\scriptstyle{b}$}}
\psfrag{g}[tl]{\scalebox{0.6}{$\scriptstyle{\gamma_{i}}$}}
\psfrag{x(2i-1)}[tl]{\scalebox{0.6}{$\scriptstyle{\xi_{2i-1}}$}}
\psfrag{x'(2i-1)}[tl]{\scalebox{0.6}{$\scriptstyle{\xi^{'}_{2i-1}}$}}
\psfrag{x(2i)}[tl]{\scalebox{0.6}{$\scriptstyle{\xi_{2i}}$}}
\psfrag{x'(2i)}[tl]{\scalebox{0.6}{$\scriptstyle{\xi^{'}_{2i}}$}}
\psfrag{t}[tl]{\scalebox{0.6}{$\scriptstyle{c_{i}}$}}
\psfrag{b(n+i)}[tl]{\scalebox{0.6}{$\scriptstyle{\beta_{n+i}}$}}
\psfrag{b'(n+i)}[tl]{\scalebox{0.6}{$\scriptstyle{\beta^{'}_{n+i}}$}}
\psfrag{b(n+i+1)}[tl]{\scalebox{0.6}{$\scriptstyle{\beta_{n+i+1}}$}}
\psfrag{b'(n+i+1)}[tl]{\scalebox{0.6}{$\scriptstyle{\beta^{'}_{n+i+1}}$}}
\includegraphics[width=0.75\textwidth]{twist_components_gen}
\caption{ (a) and (b) Negative twist component; (c) and (d) Positive twist component; (e) Non-twist component in $G_{i}$.}\label{twist_components_gen}
\end{figure}
\begin{remark}\label{assuming}
For ease of calculation, throughout the paper, we shall assume that each diagonal component (Figures~\ref{diagonal_left_right_loop}c and \ref{diagonal_left_right_loop}d) and twist component (Figure~\ref{twist_components_gen}) on $S_{n,g}$ intersect the arc $\xi_{2i-1}$ instead of the arc $\xi^{'}_{2i-1}$ and the arc $\xi_{2i}$ instead of the arc $\xi^{'}_{2i}$. Also, we shall assume that the invisible (dashed) parts of these components are only on the invisible-left side of $S_{n,g}$, as seen in the corresponding figures and that each $G_{i}$ has only one of the upper diagonal component or the lower diagonal component.
\end{remark}
\begin{remark}\label{not_c_cutting_gen}
Since a multiple curve $L \in \I_{n,g}$ consists of the simple closed curves that do not intersect each other, there cannot be both curve $c_{i}$ and twist or diagonal components at the same time in the region $G_{i}$, and both curve $c^{*}$ and twist components at the same time in the region $G^{*}$.
\end{remark}
\begin{definition}
Let $d^{u}_{2i-1}$ and $d^{l}_{2i}$ give the number of the upper and lower diagonal components in the region $G_{i}$ for $1 \leq i \leq g-1$, respectively. Also, let $c^{'}_{i}$ denote the number of the twist components in $G_{i}$. Thus, throughout the paper, $c_{i}$ shall be defined as the sum of these components. That is,
\begin{equation}\label{number_cutting}
c_{i} = c^{'}_{i} + d^{u}_{2i-1} + d^{l}_{2i}.
\end{equation}
Note that since there cannot be any diagonal components in $G^{*}$, here $c_{i}$ shall be equal to only the number of the twist components in $G^{*}$, and in this case we shall denote $c_{i}$ with the number $c^{*}$.
\end{definition}
\begin{definition}\label{def_twist_num}
A twist component's \emph{twist number} is the signed number of intersections with the arc $\gamma_{i} ~(1 \leq i \leq g)$.
\end{definition}
\begin{remark}\label{not_greater_1}
Since a multiple curve on $S_{n,g}$ does not contain any self-intersections, the directions of the twists have to be the same. Also, in the regions $G_{i}$ and $G^{*}$, the difference between the twist numbers of two different twist components cannot be greater than $1$ \cite{meral19}.
If we denote the \emph{smaller twist number} by $t_{i}$ and the \emph{bigger twist number} by $t_{i}+1$, then the \emph{total twist number} ~$T_{i} ~(1 \leq i \leq g)$ in
$G_{i} ~(1 \leq i \leq g-1)$ and $G^{*}$ is the sum of the twist numbers of twist components (see Figure~\ref{twist_components_gen}). Hence, if the difference between the twist numbers of any two twist components is $0$, then
$$T_{i} = t_{i}(c_{i} - d^{u}_{2i-1} - d^{l}_{2i}).$$
On the other hand, if the difference between the twist numbers of any two twist components is $1$, then $$ T_{i} = m_{i}(t_{i} + 1) + (c_{i} - d^{u}_{2i-1} - d^{l}_{2i} - m_{i})t_{i}, $$ \noindent
where $m_{i} \in \Z_{\geq0}$ is the number of the twist components with the twist number $t_{i} + 1$, and $ c_{i} - d^{u}_{2i-1} - d^{l}_{2i} - m_{i} $ is the number of the twist components with the twist number $t_{i}$.
\end{remark}
\begin{remark}
Although $T_{i}$ gives the total twist number in each region $G_{i}$ and $G^{*}$, it cannot show the directions of twists by itself. Therefore, we first calculate the number of each $T_{i}$, and then we add a sign in front of $T_{i}$, denoting the negative direction by $-T_{i}$ and the positive direction by $T_{i}$. However, since only the total number of twists is required in the formulas throughout the paper, $|T_{i}|$ shall be used as the total number of twists in order not to cause any confusion.
\end{remark}
Now, we calculate the path components of $L$ in the regions $G_{i}$ for $1 \leq i \leq g-1$ and $G^{*}$ for $i = g$.
\begin{lemma}\label{lem_genus_loop_gen}
Let $L$ be given with the intersection numbers $(\alpha; \beta; \beta^{'}; \xi; \xi^{'}; \gamma; c; c^{*})$, and the number of visible genus components and the number of invisible genus components in $G_{i}$ be $l_{i}$ and $l_{i}^{'} $, respectively. Also, let the number of visible genus components and the number of invisible genus components in $G^{*}$ be $l_{g}$ and $l_{g}^{'} $, respectively. Then for $1 \leq i \leq g-1$,
\begin{eqnarray*}
l_{i} = \max\{0, \frac{|\beta_{n+i} - \beta_{n+i+1}| - c_{i}}{2}\} \quad \text{ and } \quad l_{g} = \frac{\beta_{n+g} - c^{*}}{2},
\end{eqnarray*}
\noindent and for $2 \leq i \leq g-1$,
\begin{eqnarray*}
l_{1}^{'} = \max\{0, \frac{|\beta_{1} - \beta_{n+2}^{'}| - c_{1}}{2}\}, \quad
l_{i}^{'} = \max\{0, \frac{|\beta^{'}_{n+i} - \beta^{'}_{n+i+1}| - c_{i}}{2}\}
\end{eqnarray*}
\noindent and
\begin{eqnarray*}
l_{g}^{'} = \frac{\beta^{'}_{n+g} - c^{*}}{2}.
\end{eqnarray*}
Note that if $\beta_{n+i} < \beta_{n+i+1}$, the visible genus component in $G_{i}$ is left; if $\beta_{n+i} > \beta_{n+i+1}$, the visible genus component in $G_{i}$ is right. Similarly, if $\beta^{'}_{n+i} < \beta^{'}_{n+i+1}$, the invisible genus component in $G_{i}$ is left; if $\beta^{'}_{n+i} > \beta^{'}_{n+i+1}$, the invisible genus component in $G_{i}$ is right.
\end{lemma}
\begin{proof}
The absolute value of the difference between the intersection numbers on the arcs $\beta_{n+i}$ and $\beta_{n+i+1}$, namely $|\beta_{n+i} - \beta_{n+i+1}|$, gives us the sum of twist, diagonal and visible genus component numbers. If $\beta_{n+i} < \beta_{n+i+1}$, the arc $\beta_{n+i+1}$ intersects once with each twist component (Figures~\ref{twist_components_gen}b and \ref{twist_components_gen}d) or diagonal component (Figures~\ref{diagonal_left_right_loop}c and \ref{diagonal_left_right_loop}d) and twice with each visible-left genus component (Figure~\ref{diagonal_left_right_loop}a). Let the number of visible-left genus components and the number of visible-right genus components be denoted by $l^{L}_{i}$ and $l^{R}_{i}$, respectively. Hence, $$\beta_{n+i+1} - \beta_{n+i} = c^{'}_{i} + d^{u}_{2i-1} + d^{l}_{2i} + 2l^{L}_{i}.$$ \noindent From Equation~(\ref{number_cutting}), $$ \beta_{n+i+1} - \beta_{n+i} = c_{i} + 2l^{L}_{i}.$$
\noindent Since a multiple curve consists of the simple closed curves that do not intersect each other, this curve system contains only one of the visible-left genus components or visible-right genus components. Therefore, we can denote the number of both visible-left genus components and visible-right genus components as $l_{i}$. Thus, we can write
\begin{equation}\label{visible-left}
\beta_{n+i+1} - \beta_{n+i} = c_{i} + 2l_{i}.
\end{equation}
If $\beta_{n+i} > \beta_{n+i+1}$, the arc $\beta_{n+i}$ intersects once each twist component (Figures~\ref{twist_components_gen}a, \ref{twist_components_gen}c and \ref{twist_components_gen}e) and twice each visible-right genus component (Figure~\ref{diagonal_left_right_loop}b). From Remark~\ref{assuming}, there cannot be any diagonal component; otherwise, self-intersections occur in this curve system. Since $c_{i} = c^{'}_{i} + d^{u}_{2i-1} + d^{l}_{2i}$, here $$ \beta_{n+i} - \beta_{n+i+1} = c_{i} + 2l^{R}_{i}.$$ \noindent That is,
\begin{equation}\label{visible-right}
\beta_{n+i} - \beta_{n+i+1} = c_{i} + 2l_{i}.
\end{equation}
\noindent From Equations~(\ref{visible-left}) and (\ref{visible-right}), we can write $ |\beta_{n+i} - \beta_{n+i+1}| = c_{i} + 2l_{i}$. Therefore, $l_{i} = \frac{|\beta_{n+i} - \beta_{n+i+1}| - c_{i}}{2}$. When $c_{i} \geq |\beta_{n+i} - \beta_{n+i+1}|$, there cannot be any visible genus component in multiple curve. Hence $l_{i} = \max\{0, \frac{|\beta_{n+i} - \beta_{n+i+1}| - c_{i}}{2}\}$ is derived. Similarly, we can find $l^{'}_{1}$ and $l^{'}_{i}$. For the proofs of $l_{g}$ and $l^{'}_{g}$, you can look at \cite{meral19}.
\end{proof}
In the following lemma, we calculate the total twist number of twist components in each $G_{i}$ and $G^{*}$:
\begin{lemma}\label{lem_total_twist_gen}
Let $L$ be given with the intersection numbers $(\alpha; \beta; \beta^{'}; \xi; \xi^{'}; \gamma; c; c^{*})$, denoting the signed total twist number of twist components in each $G_{i}$ and $G^{*}$ by $T_{i}$ and $T_{g}$, respectively. For ~$2 \leq i \leq g-1$, we have
\begin{align}\label{total_twist_Ti}
|T_{i}|&= \left\{ \begin{array}{ll}
0& \mbox{if $c_{i} = 0$},\\
\gamma_{i} - \max\{0, \frac{\max\{0, \beta_{n+i} - \beta_{n+i+1}\} - c_{i}}{2}\} - \max\{0, \frac{\max\{0, \beta^{'}_{n+i} - \beta^{'}_{n+i+1}\} - c_{i}}{2}\} & \mbox{if $c_{i} \neq 0$.}
\end{array} \right.
\end{align}
\noindent For $i = 1$,
\begin{align}\label{total_twist_T1}
|T_{1}|&= \left\{ \begin{array}{ll}
0& \mbox{if $c_{1} = 0$},\\
\gamma_{1} - \max\{0, \frac{\max\{0, \beta_{n+1} - \beta_{n+2}\} - c_{1}}{2}\} - \max\{0, \frac{\max\{0, \beta_{1} - \beta^{'}_{n+2}\} - c_{1}}{2}\} & \mbox{if $c_{1} \neq 0$.}
\end{array} \right.
\end{align}
\noindent For $i = g$,
\begin{align}\label{total_twist_Tg}
|T_{g}|&= \left\{ \begin{array}{ll}
0& \mbox{if $c^{*} = 0$},\\
\gamma_{g} - \frac{\beta_{n+g} - c^{*}}{2} - \frac{\beta^{'}_{n+g} - c^{*}}{2} & \mbox{if $c^{*} \neq 0$.}
\end{array} \right.
\end{align}
\noindent
The sign of the negative twist component is $-1$ and the sign of the positive twist component is $1$.
\end{lemma}
\begin{proof}
Let us denote the total twist number of the twist components of $L$ in each $G_{i}$ by $|T_{i}|$. Observe that the curve $\gamma_{i}$ intersects once the curve $c_{i}$ (Figure~\ref{genus_loop_gen}a)
and it intersects once each visible-right and invisible-right genus components (Figure~\ref{diagonal_left_right_loop}b). Also, $\gamma_{i}$ intersects $L$ by the total number of twists of the twist components (Figure~\ref{twist_components_gen}). However, from Remark~\ref{not_c_cutting_gen}, there cannot be
twists and the curve $c_{i}$ in $G_{i}$ at the same time. Therefore, when $c_{i} \neq 0$, we have
\begin{equation}\label{gama_kesenler}
\gamma_{i} = l_{i} + l_{i}^{'} + |T_{i}|,
\end{equation}
\noindent where $l_{i}$, ~$l_{i}^{'}$ and $|T_{i}|$ denote the number of visible-right genus, invisible-right genus components and the total twist number of the twist components in $G_{i}$, respectively.
Since a multiple curve consists of the simple closed curves that do not intersect each other and from Definition~\ref{def_twist_num}, we can write
\begin{equation*}
\gamma_{i} = \max\{0, \frac{\max\{0, \beta_{n+i} - \beta_{n+i+1}\} - c_{i}}{2}\} + \max\{0, \frac{\max\{0, \beta^{'}_{n+i} - \beta^{'}_{n+i+1}\} - c_{i}}{2}\} + |T_{i}|.
\end{equation*}
\noindent Hence, we have Equality~(\ref{total_twist_Ti}) as follows
\begin{equation*}
|T_{i}| = \gamma_{i} - \max\{0, \frac{\max\{0, \beta_{n+i} - \beta_{n+i+1}\} - c_{i}}{2}\} - \max\{0, \frac{\max\{0, \beta^{'}_{n+i} - \beta^{'}_{n+i+1}\} - c_{i}}{2}\}.
\end{equation*}
Equalities~(\ref{total_twist_T1}) and (\ref{total_twist_Tg}) can be obtained similar to the above calculations and \cite{meral19}, respectively.
\end{proof}
\begin{remark}\label{when_diagonal_exist}
When there is one of the upper diagonal components or
lower diagonal components in the region $G_{i}$,
the equation $c_{i} = d_{2i-1}^{u} + d_{2i}^{l} + |T_{i}|$ is
used so that the curves on the surface do not intersect. In this case, $|T_{i}|$ cannot be greater than $c_i$.
\end{remark}
By using the following lemma, we calculate the number of the curves $c_{i}$ and $c^{*}$ (Figure~\ref{genus_loop_gen}a) in each region $G_{i} ~(1 \leq i \leq g-1)$ and $G^{*}$, respectively.
\begin{lemma}\label{c_curves_gen}
Let $L$ be given with the intersection numbers $(\alpha; \beta; \beta^{'}; \xi; \xi^{'}; \gamma; c; c^{*})$. We find the number of the curves $c_{i}$ and $c^{*}$ in $L$, denoting by $p(c_{i})$ and $p(c^{*})$, as follows. For $2 \leq i \leq g-1$,
\begin{align}\label{number_pci}
p(c_{i})&= \left\{ \begin{array}{ll}
\gamma_{i} - \max\{0, \frac{\max\{0, \beta_{n+i} - \beta_{n+i+1}\}}{2}\} - \max\{0, \frac{\max\{0, \beta^{'}_{n+i} - \beta^{'}_{n+i+1}\}}{2}\}& \mbox{if $c_{i} = 0$},\\
0& \mbox{if $c_{i} \neq 0$.}
\end{array} \right.
\end{align}
\noindent For $i = 1$,
\begin{align}\label{number_pc1}
p(c_{1})&= \left\{ \begin{array}{ll}
\gamma_{1} - \max\{0, \frac{\max\{0, \beta_{n+1} - \beta_{n+2}\}}{2}\} - \max\{0, \frac{\max\{0, \beta_{1} - \beta^{'}_{n+2}\}}{2}\}& \mbox{if $c_{1} = 0$},\\
0& \mbox{if $c_{1} \neq 0$.}
\end{array} \right.
\end{align}
\noindent For $i = g$,
\begin{align}\label{number_pcg}
p(c^{*})&= \left\{ \begin{array}{ll}
\gamma_{g} - \frac{\beta_{n+g}}{2} - \frac{\beta^{'}_{n+g}}{2} & \mbox{if $c^{*} = 0$},\\
0& \mbox{if $c^{*} \neq 0$.}
\end{array} \right.
\end{align}
\end{lemma}
\begin{proof}
Whenever $c_{i} = 0$, we have $\gamma_{i} = l_{i} + l_{i}^{'} + p(c_{i})$. Since a multiple curve consists of the simple closed curves that do not intersect each other and from Definition~\ref{def_twist_num}, we can write
\begin{eqnarray*}
\gamma_{i} = \max\{0, \frac{\max\{0, \beta_{n+i} - \beta_{n+i+1}\}}{2}\} + \max\{0, \frac{\max\{0, \beta^{'}_{n+i} - \beta^{'}_{n+i+1}\}}{2}\} + p(c_{i}).
\end{eqnarray*}
Hence, $p(c_{i}) = \gamma_{i} - \max\{0, \frac{\max\{0, \beta_{n+i} - \beta_{n+i+1}\}}{2}\} - \max\{0, \frac{\max\{0, \beta^{'}_{n+i} - \beta^{'}_{n+i+1}\}}{2}\} $ is derived.
Equalities~(\ref{number_pc1}) and (\ref{number_pcg}) can be obtained similar to the above calculations and \cite{meral19}, respectively.
\end{proof}
In the following lemma, we find the number of the upper diagonal components, $d_{2i-1}^{u}$, and the lower diagonal components, $d_{2i}^{l}$, in each region $G_{i} ~(1 \leq i \leq g-1)$.
\begin{lemma}\label{lem_diagonals}
Let $L$ be given with the intersection numbers $(\alpha; \beta; \beta^{'}; \xi; \xi^{'}; \gamma; c; c^{*})$, and the number of the upper diagonal components and the number of the lower diagonal components in $G_{i}$ be $d_{2i-1}^{u}$ and $d_{2i}^{l}$, respectively. Then for $1 \leq i \leq g-1$,
\begin{equation}\label{upper_diagonal}
d_{2i-1}^{u} = \max\{c_{i} - |T_{i}|, T_{i}c_{i}\} - \max\{0, T_{i}c_{i}\}
\end{equation}
\noindent and
\begin{equation}\label{lower_diagonal}
d_{2i}^{l} = \max\{c_{i} - |T_{i}|, -T_{i}c_{i}\} - \max\{0, -T_{i}c_{i}\}.
\end{equation}
\end{lemma}
\begin{proof}
Firstly, we assume that there are upper diagonal components in the region $G_{i}$. When $T_{i} < 0$, from Remark~\ref{when_diagonal_exist}, we see $ d_{2i-1}^{u} + d_{2i}^{l} = c_{i} - |T_{i}|$. From Remark~\ref{assuming}, $G_{i}$ has no lower diagonal components. Therefore, we can write $ d_{2i-1}^{u} = c_{i} - |T_{i}|$. When $T_{i} > 0$, it should be $ d_{2i-1}^{u} = 0$ so that the curves do not intersect each other. The equation~(\ref{upper_diagonal}) provides these properties completely.
When there are lower diagonal components in $G_{i}$, we can find the equation~(\ref{lower_diagonal}) similar to the number of upper diagonal components.
\end{proof}
The twist numbers of each twist component of a multiple curve whose intersection numbers are given are found by using Remark~\ref{not_greater_1} and Lemma~\ref{lem_total_twist_gen}, which we find these twist numbers with the following lemma. The proof of this lemma is similar to the proof in \cite{meral19}.
\begin{lemma}\label{each_twist}
Let $L$ be given with the intersection numbers
$(\alpha; \beta; \beta^{'}; \xi; \xi^{'}; \gamma; c; c^{*})$.
Let $|T_{i}| ~(1 \leq i \leq g-1)$ and $|T_{g}|$ be the total twist numbers
in each regions $G_{i}$ and $G^{*}$, respectively.
Also, let $m_{i}$ and $m^{*}$ be the number of twist components, each with
$t_{i}+1$ and $t_{g}+1$ twists and
$c_{i} - d^{u}_{2i-1} - d^{l}_{2i} - m_{i}$ and $c^{*} - m^{*}$ be
the number of twist components, each with $t_{i}$ and $t_{g}$ twists
in each $G_{i}$ and $G^{*}$, respectively.
In this case,
\begin{equation}
m_{i} \equiv |T_{i}| ~(mod ~(c_{i} - d^{u}_{2i-1} - d^{l}_{2i})) \quad \text{ and } \quad t_{i} = \frac{|T_{i}| - m_{i}}{c_{i} - d^{u}_{2i-1} - d^{l}_{2i}}
\end{equation}
\noindent and
\begin{equation}
m^{*} \equiv |T_{g}| ~(mod ~c^{*}) \quad \text{ and } \quad t_{g} = \frac{|T_{g}| - m^{*}}{c^{*}},
\end{equation}
\noindent where $c_{i} - d^{u}_{2i-1} - d^{l}_{2i} \neq 0$ and $c^{*} \neq 0$.
\end{lemma}
In Lemma~\ref{looking_right_left}, we shall define some auxiliary components that shall be used to calculate the number of the visible above components denoted by $u_{2i-1}^{va}$ and the number of the visible below components denoted by $u_{2i}^{vb}$ in the rest of the paper.
\begin{lemma}\label{looking_right_left}
Let $L$ be given with the intersection numbers $(\alpha; \beta; \beta^{'}; \xi; \xi^{'}; \gamma; c; c^{*})$. Then for $1 \leq i \leq g-1$, if $\beta_{n+i} \leq \beta_{n+i+1}$, the number of the intersections of twist components together with total diagonals with the arc $\beta_{n+i+1}$, denoting by $n_{i}$, is as follows.
\begin{equation}
n_{i} = \frac{\beta_{n+i+1} - \beta_{n+i} + c_{i}}{2} - \max\{0, \frac{|\beta_{n+i} - \beta_{n+i+1}| - c_{i}}{2}\}.
\end{equation}
\noindent Hence, we can find the number of the intersections of twist components together with total diagonals with the arc $\beta_{n+i}$ as $c_{i} - n_{i}$.
On the other hand, if $\beta_{n+i} \geq \beta_{n+i+1}$, the number of the intersections of twist components together with total diagonals with the arc $\beta_{n+i}$, denoting by $k_{i}$, is as follows.
\begin{equation}
k_{i} = \frac{\beta_{n+i} - \beta_{n+i+1} + c_{i}}{2} - \max\{0, \frac{|\beta_{n+i} - \beta_{n+i+1}| - c_{i}}{2}\}.
\end{equation}
\noindent Hence, we can find the number of the intersections of twist components together with total diagonals with the arc $\beta_{n+i+1}$ as $c_{i} - k_{i}$.
\end{lemma}
\begin{proof}
When $\beta_{n+i} \leq \beta_{n+i+1}$, we can write the number of the intersections on the arcs $\beta_{n+i+1}$ and $\beta_{n+i}$ as follows:
\begin{equation}\label{right1}
\beta_{n+i+1} = n_{i} + 2l_{i} + u_{2i-1}^{va} + u_{2i}^{vb},
\end{equation}
\begin{equation}\label{left1}
\beta_{n+i} = c_{i} - n_{i} + u_{2i-1}^{va} + u_{2i}^{vb}.
\end{equation}
From equations ~(\ref{right1}) and ~(\ref{left1}), we derive
\begin{equation*}
n_{i} = \frac{\beta_{n+i+1} - \beta_{n+i} + c_{i}}{2} - l_{i}.
\end{equation*}
\noindent When $\beta_{n+i} \geq \beta_{n+i+1}$, we can find $k_{i}$ similar to $n_{i}$.
\end{proof}
\begin{remark}\label{remark_loop}
In each region $U_i$, for $1 \leq i \leq n$, let the number of the loop components be denoted by $|b_i|$, where
\begin{equation}
b_{i} = \frac{\beta_{i} - \beta_{i+1}}{2}.
\end{equation}
If $b_i < 0$, the loop component is called \emph{left}; if $b_i > 0$, the loop component is called \emph{right} \cite{dynnikov02}.
\end{remark}
Now, we find the number of above and below components in each $U_i ~(1 \leq i \leq n)$ and the number of visible above, visible below, invisible above and invisible below components in each $G_i ~(1 \leq i \leq g-1)$.
\begin{lemma}\label{all_above_below}
Let $L$ be given with the intersection numbers $(\alpha; \beta; \beta^{'}; \xi; \xi^{'}; \gamma; c; c^{*})$. Also, let the number of above and below components in each $U_i$ and the number of visible above, visible below, invisible above and invisible below components in each $G_i$ be denoted by $u_{2i-1}^{a}, ~u_{2i}^{b}, ~u_{2i-1}^{va}, ~u_{2i}^{vb}, ~u_{2i-1}^{v'a}$ and $u_{2i}^{v'b}$, respectively. Then for $1 \leq i \leq n$,
\begin{equation}\label{above_below_puncture}
u_{2i-1}^{a} = \alpha_{2i-1} - |b_i| \quad \mbox{ and } \quad u_{2i}^{b} = \alpha_{2i} - |b_i|.
\end{equation}
For $1 \leq i \leq g-1$, if $|T_i| \neq 0$,
\noindent when $\beta_{n+i} \leq \beta_{n+i+1}$,
\begin{equation}\label{denk_1_ab_vis_dif_zero}
u_{2i-1}^{va} = \xi_{2i-1} - |T_i| - \max\{n_i - d_{2i}, T_i\} + \max\{0, T_i\} - l_i,
\end{equation}
\begin{equation}\label{denk_1_bel_vis_dif_zero}
u_{2i}^{vb} = \xi_{2i} - |T_i| - \max\{n_i - d_{2i-1}, -T_i\} + \max\{0, -T_i\} - l_i;
\end{equation}
\noindent when $\beta_{n+i} \geq \beta_{n+i+1}$,
\begin{equation}\label{denk_2_ab_vis_dif_zero}
u_{2i-1}^{va} = \xi_{2i-1} - |T_i| - \max\{c_i - k_i - d_{2i}, T_i\} + \max\{0, T_i\} - l_i,
\end{equation}
\begin{equation}\label{denk_2_bel_vis_dif_zero}
u_{2i}^{vb} = \xi_{2i} - |T_i| - \max\{c_i - k_i - d_{2i-1}, -T_i\} + \max\{0, -T_i\} - l_i.
\end{equation}
If $|T_i| = 0$,
\begin{equation}\label{denk_ab_vis_zero}
u_{2i-1}^{va} = \xi_{2i-1} - \max\{p(c_i), d_{2i-1}\} - l_i,
\end{equation}
\begin{equation}\label{denk_bel_vis_zero}
u_{2i}^{vb} = \xi_{2i} - \max\{p(c_i), d_{2i}\} - l_i.
\end{equation}
Also,
\begin{equation}
u_{2i-1}^{v'a} = \xi^{'}_{2i-1} - l^{'}_i \quad \mbox{ and } \quad u_{2i}^{v'b} = \xi^{'}_{2i} - l^{'}_i.
\end{equation}
\end{lemma}
\begin{proof}
The proofs of Equations~(\ref{above_below_puncture}) are obvious since each above and below component intersects $\alpha_{2i-1}$ and $\alpha_{2i}$, respectively (see Figure~\ref{puncture_bilesen_gen}).
Let $|T_i| \neq 0$. When $\beta_{n+i} \leq \beta_{n+i+1}$, from Lemma~\ref{looking_right_left}, the number of the intersections of twist components together with total diagonal components with the arc $\beta_{n+i+1}$ is $n_i$. When we subtract the number of lower diagonal components (Figure~\ref{diagonal_left_right_loop}d) from $n_i$, the arc $\xi_{2i-1}$ intersects $n_i - d_{2i}$ times with the twist components. The arc $\xi_{2i-1}$ also intersects $l_i$ times with the visible genus components (Figures~\ref{diagonal_left_right_loop}a and \ref{diagonal_left_right_loop}b), $u_{2i-1}^{va}$ times with the visible above components (Figure~\ref{diagonal_left_right_loop}e), and by the total number of twists, $|T_i|$. When $T_i > 0$, $\xi_{2i-1}$ intersects by the total number of twists; whereas when $T_i < 0$, $\xi_{2i-1}$ intersects by the total number of twists and $n_i - d_{2i}$. That is,
$$ \xi_{2i-1} = |T_i| + \max\{n_i - d_{2i}, T_i\} - \max\{0, T_i\} + l_i + u_{2i-1}^{va}.$$ \noindent Hence, we get Equation~(\ref{denk_1_ab_vis_dif_zero}) as follows. $$ u_{2i-1}^{va} = \xi_{2i-1} - |T_i| -
\max\{n_i - d_{2i}, T_i\} + \max\{0, T_i\} - l_i. $$
Similarly, in addition to the number of visible genus components and visible below components, when $T_i > 0$, $\xi_{2i}$ intersects by the total number of twists and $n_i - d_{2i-1}$; whereas when $T_i < 0$, $\xi_{2i}$ intersects by the total number of twists. Hence, $$ \xi_{2i} = |T_i| +
\max\{n_i - d_{2i-1}, -T_i\} - \max\{0, -T_i\} + l_i + u_{2i}^{vb}. $$ \noindent From here, we can write Equation~(\ref{denk_1_bel_vis_dif_zero}) as follows.
$$ u_{2i}^{vb} = \xi_{2i} - |T_i| - \max\{n_i - d_{2i-1}, -T_i\} +
\max\{0, -T_i\} - l_i. $$
When $\beta_{n+i} \geq \beta_{n+i+1}$, we can derive the Equations~(\ref{denk_2_ab_vis_dif_zero}) and (\ref{denk_2_bel_vis_dif_zero}) similar to the Equations~(\ref{denk_1_ab_vis_dif_zero}) and (\ref{denk_1_bel_vis_dif_zero}) using Lemma~\ref{looking_right_left}.
Let $|T_i| = 0$. In this case, in addition to visible genus components and visible above components, $\xi_{2i-1}$ intersects either the curve $c_i$ or the upper diagonal components (see Remark~\ref{not_c_cutting_gen}). That is,
$$ \xi_{2i-1} = u_{2i-1}^{va} + \max\{p(c_i), d_{2i-1}\} + l_i. $$ \noindent Thus, we find Equation~(\ref{denk_ab_vis_zero}) as $u_{2i-1}^{va} = \xi_{2i-1} - \max\{p(c_i), d_{2i-1}\} - l_i.$ Similarly, $u_{2i}^{vb}$ is derived.
From Remark~\ref{assuming}, $\xi^{'}_{2i-1}$ intersects only invisible genus components and invisible above components, and $\xi^{'}_{2i}$ intersects only invisible genus components and invisible below components. Thus, we can write
$$ u^{v'a}_{2i-1} = \xi^{'}_{2i-1} - l'_i \quad \mbox{ and } \quad u^{v'b}_{2i} = \xi^{'}_{2i} - l'_i.$$
\end{proof}
\begin{example}\label{exp}
Let $(6, 2, 4, 2, 5, 1; 8, 6, 4, 6, 7, 2; 3, 0; 5, 4, 6, 6; 4, 1, 0, 0; 2, 5, 3; 3, 3; 0)$ be the intersection numbers of a multiple curve $L \in \I_{3,3}$ with the corresponding arcs and the simple closed curves $c_i$ and $c^{*}$ in $S_{3,3}$. Also, $T_{1} > 0$ and $T_{2} < 0$. We shall show how we draw $L$ from the given intersection numbers.
First, we find the number of each path component in each region $G_i$ for $i = 1, 2$ and $G^{*}$, respectively. From Lemma~\ref{lem_genus_loop_gen},
\begin{eqnarray*}
l_{1} = \max\{0, \frac{|\beta_{4} - \beta_{5}| - c_{1}}{2}\} = \max\{0, \frac{|6 - 7| - 3}{2}\} = 0.
\end{eqnarray*}
\noindent Similarly, we have $l_{2} = 1$, $l_{3} = 1$, $l^{'}_{1} = 1$, $l^{'}_{2} = 0$ and $l^{'}_{3} = 0$. Namely, there is $1$ right-invisible genus component, however there is not any visible genus component in the region $G_{1}$. In $G_{2}$, there is ~$1$ right-visible genus component and no invisible genus component. In $G^{*}$, there is ~$1$ visible genus component and no invisible genus component.
According to Lemma~\ref{lem_total_twist_gen},
\begin{eqnarray*}
|T_{1}| = 2 - \max\{0, \frac{\max\{0, 6 - 7\} - 3}{2}\} - \max\{0, \frac{\max\{0, 8 - 3\} - 3}{2}\} = 1.
\end{eqnarray*}
\noindent Similarly, $|T_{2}| = 4$ and since $c^{*} = 0$, $|T_{3}| = 0$. That is, the total twist number of the twist components in the region $G_1$ is $1$. The total twist number of the twist components in $G_2$ is 4, however there is not any twist in $G^{*}$. From Lemma~\ref{c_curves_gen}, we observe that since $c_1 \neq 0$ and $c_2 \neq 0$, there are no $c_1$ and $c_2$ curves in the regions $G_1$ and $G_2$. We have $p(c^{*}) = 2$. Therefore, there are $2$ $c^{*}$ curves in $G^{*}$.
We can find the number of upper and lower diagonal components using Lemma~\ref{lem_diagonals} in each $G_i, ~i=1,2$. We know that $T_1 > 0$. Thus,
\begin{equation*}
d_{1}^{u} = \max\{c_{1} - |T_{1}|, T_{1}c_{1}\} - \max\{0, T_{1}c_{1}\} = \max\{3 - 1, 1\times3\} - \max\{0, 1\times3\} = 0,
\end{equation*}
\begin{equation*}
d_{2}^{l} = \max\{c_{1} - |T_{1}|, -T_{1}c_{1}\} - \max\{0, -T_{1}c_{1}\} = \max\{3 - 1, -1\times3\} - \max\{0, -1\times3\} = 2.
\end{equation*}
\noindent While there are $2$ lower diagonal components in the region $G_1$, there are no upper diagonal components. From Remark~\ref{when_diagonal_exist}, since $|T_2|$ is greater than $c_2$, there are not both diagonal components in $G_2$.
We calculate the twist numbers of each twist component of $L$ in each $G_i$ and $G^{*}$ by Lemma~\ref{each_twist}. In $G_1$,
\begin{equation*}
m_{1} = |T_{1}| ~(mod ~(c_{1} - d^{u}_{1} - d^{l}_{2})) = 1 ~(mod ~(3 - 0 - 2)) = 0,
\end{equation*}
\begin{equation*}
t_{1} = \frac{|T_{1}| - m_{1}}{c_{1} - d^{u}_{1} - d^{l}_{2}} = \frac{1 - 0}{3 - 0 - 2} = 1
\end{equation*}
\noindent and
\begin{equation*}
c_{1} - d^{u}_{1} - d^{l}_{2} - m_{1} = 3 - 0 - 2 - 0 = 1.
\end{equation*}
\noindent Therefore, there is $1$ twist component which has $1$ twist, however there is not any twist component with $t_1 + 1 = 1 + 1 = 2$ twists in $G_1$. In $G_2$,
\begin{equation*}
m_{2} = |T_{2}| ~(mod ~(c_{2} - d^{u}_{3} - d^{l}_{4})) = 4 ~(mod ~(3 - 0 - 0)) = 1,
\end{equation*}
\begin{equation*}
t_{2} = \frac{|T_{2}| - m_{2}}{c_{2} - d^{u}_{3} - d^{l}_{4}} = \frac{4 - 1}{3 - 0 - 0} = 1
\end{equation*}
\noindent and
\begin{equation*}
c_{2} - d^{u}_{3} - d^{l}_{4} - m_{2} = 3 - 0 - 0 - 1 = 2.
\end{equation*}
\noindent Thus, there are $2$ twist components, each with $1$ twist and $1$ twist component which has $t_2 + 1 = 1 + 1 = 2$ twists in $G_2$. Since $c^{*} = 0$, there is no twist in $G^{*}$.
According to Lemma~\ref{looking_right_left}, due to $\beta_4 < \beta_5$,
\begin{equation*}
n_{1} = \frac{\beta_{5} - \beta_{4} + c_{1}}{2} - \max\{0, \frac{|\beta_{4} - \beta_{5}| - c_{1}}{2}\} = \frac{7 - 6 + 3}{2} - \max\{0, \frac{|6 - 7| - 3}{2}\} = 2.
\end{equation*}
\noindent Hence, the number of the intersections of twist components together with total diagonals with the arc $\beta_5$ in $G_1$ is $2$. The number of the intersections of twist components together with total diagonals with the arc $\beta_4$ in $G_1$ is $c_1 - n_1 = 3 - 2 = 1$.
In $G_2$, due to $\beta_5 > \beta_6$,
\begin{equation*}
k_{2} = \frac{\beta_{5} - \beta_{6} + c_{2}}{2} - \max\{0, \frac{|\beta_{5} - \beta_{6}| - c_{2}}{2}\} = \frac{7 - 2 + 3}{2} - \max\{0, \frac{|7 - 2| - 3}{2}\} = 3.
\end{equation*}
\noindent Thus, the number of the intersections of twist components together with total diagonals with the arc $\beta_5$ in $G_2$ is $3$. The number of the intersections of twist components together with total diagonals with the arc $\beta_6$ in $G_2$ is $c_2 - k_2 = 3 - 3 = 0$.
We find the loop components in each region $U_i, ~i=1,2,3$ by Remark~\ref{remark_loop}.
\begin{equation*}
b_{1} = \frac{\beta_{1} - \beta_{2}}{2} = \frac{8 - 6}{2} = 1,
\end{equation*}
\begin{equation*}
b_{2} = \frac{\beta_{2} - \beta_{3}}{2} = \frac{6 - 4}{2} = 1,
\end{equation*}
\begin{equation*}
b_{3} = \frac{\beta_{3} - \beta_{4}}{2} = \frac{4 - 6}{2} = -1.
\end{equation*}
\noindent Namely, there is $1$ right loop component in $U_1$, $1$ right loop component in $U_2$ and $1$ left loop component in $U_3$.
We calculate the number of above and below components in each $U_i ~(1 \leq i \leq 3)$ and the number of visible above, visible below, invisible above and invisible below components in each $G_i ~(1 \leq i \leq 2)$ using Lemma~\ref{all_above_below}.
\begin{equation*}
u_{1}^{a} = \alpha_{1} - |b_1| = 6 - 1 = 5, \quad u_{2}^{b} = \alpha_{2} - |b_2| = 2 - 1 = 1,
\end{equation*}
\begin{equation*}
u_{3}^{a} = \alpha_{3} - |b_2| = 4 - 1 = 3, \quad u_{4}^{b} = \alpha_{4} - |b_2| = 2 - 1 = 1,
\end{equation*}
\begin{equation*}
u_{5}^{a} = \alpha_{5} - |b_3| = 5 - 1 = 4, \quad u_{6}^{b} = \alpha_{6} - |b_3| = 1 - 1 = 0.
\end{equation*}
\noindent Therefore, we have $5$ above components and $1$ below component in $U_1$, $3$ above components and $1$ below component in $U_2$ and $4$ above components and no below component in $U_3$.
Since $|T_1| \neq 0$ and $\beta_{4} < \beta_{5}$ in $G_1$,
\begin{eqnarray*}
u_{1}^{va} & = & \xi_{1} - |T_1| - \max\{n_1 - d_{2}, T_1\} + \max\{0, T_1\} - l_1 \\
& = & 5 - 1 - \max\{2 - 2, 1\} + \max\{0, 1\} - 0 \\
& = & 4
\end{eqnarray*}
\noindent and
\begin{eqnarray*}
u_{2}^{vb} & = & \xi_{2} - |T_1| - \max\{n_1 - d_{1}, -T_1\} + \max\{0, -T_1\} - l_1 \\
& = & 4 - 1 - \max\{2 - 0, -1\} + \max\{0, -1\} - 0 \\
& = & 1.
\end{eqnarray*}
\noindent Also,
\begin{equation*}
u_{1}^{v'a} = \xi^{'}_{1} - l^{'}_1 = 4 - 1 = 3 \quad \mbox{ and } \quad u_{2}^{v'b} = \xi^{'}_{2} - l^{'}_1 = 1 - 1 = 0.
\end{equation*}
\noindent There are $4$ visible above components, $1$ visible below component, $3$ invisible above components and no invisible below component in $G_1$.
Since $|T_2| \neq 0$ and $\beta_{5} > \beta_{6}$ in $G_2$,
\begin{eqnarray*}
u_{3}^{va} & = & \xi_{3} - |T_2| - \max\{c_2 - k_2 - d_{4}, T_2\} + \max\{0, T_2\} - l_2 \\
& = & 6 - 4 - \max\{3 - 3 - 0, -4\} + \max\{0, -4\} - 1 \\
& = & 1
\end{eqnarray*}
\noindent and
\begin{eqnarray*}
u_{4}^{vb} & = & \xi_{4} - |T_2| - \max\{c_2 - k_2 - d_{3}, -T_2\} + \max\{0, -T_2\} - l_2 \\
& = & 6 - 4 - \max\{3 - 3 - 0, 4\} + \max\{0, 4\} - 1 \\
& = & 1.
\end{eqnarray*}
\noindent Also,
\begin{equation*}
u_{3}^{v'a} = \xi^{'}_{3} - l^{'}_2 = 0 - 0 = 0 \quad \mbox{ and } \quad u_{4}^{v'b} = \xi^{'}_{4} - l^{'}_2 = 0 - 0 = 0.
\end{equation*}
\noindent There are $1$ visible above component, $1$ visible below component, no invisible above component and no invisible below component in $G_2$. The calculated path components in each $U_i$, $G_i$ and $G^{*}$ are connected in a unique way up to isotopy and thus, the multiple curve $L$ in Figure~\ref{gen_example} is determined uniquely.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.83\textwidth]{gen_example}
\caption{ The multiple curve $L$ with the intersection numbers $(6, 2, 4, 2, 5, 1; 8, 6, 4, 6, 7, 2; 3, 0; 5, 4, 6, 6; 4, 1, 0, 0; 2, 5, 3; 3, 3; 0)$ }\label{gen_example}
\end{figure}
\end{example}
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package org.eclipse.collections.impl.lazy;
import java.util.Collection;
import java.util.Iterator;
import java.util.Optional;
import org.eclipse.collections.api.LazyIterable;
import org.eclipse.collections.api.block.function.Function;
import org.eclipse.collections.api.block.predicate.Predicate;
import org.eclipse.collections.api.block.predicate.Predicate2;
import org.eclipse.collections.api.block.procedure.Procedure;
import org.eclipse.collections.api.block.procedure.Procedure2;
import org.eclipse.collections.api.block.procedure.primitive.ObjectIntProcedure;
import org.eclipse.collections.impl.UnmodifiableIteratorAdapter;
import org.eclipse.collections.impl.utility.Iterate;
import org.eclipse.collections.impl.utility.LazyIterate;
/**
* A LazyIterableAdapter wraps any iterable with the LazyIterable interface.
*/
public class LazyIterableAdapter<T>
extends AbstractLazyIterable<T>
{
private final Iterable<T> adapted;
public LazyIterableAdapter(Iterable<T> newAdapted)
{
this.adapted = newAdapted;
}
@Override
public void each(Procedure<? super T> procedure)
{
Iterate.forEach(this.adapted, procedure);
}
@Override
public void forEachWithIndex(ObjectIntProcedure<? super T> objectIntProcedure)
{
Iterate.forEachWithIndex(this.adapted, objectIntProcedure);
}
@Override
public <P> void forEachWith(Procedure2<? super T, ? super P> procedure, P parameter)
{
Iterate.forEachWith(this.adapted, procedure, parameter);
}
@Override
public Iterator<T> iterator()
{
return new UnmodifiableIteratorAdapter<>(this.adapted.iterator());
}
@Override
public <R extends Collection<T>> R into(R target)
{
Iterate.addAllIterable(this.adapted, target);
return target;
}
@Override
public LazyIterable<T> select(Predicate<? super T> predicate)
{
return LazyIterate.select(this.adapted, predicate);
}
@Override
public LazyIterable<T> reject(Predicate<? super T> predicate)
{
return LazyIterate.reject(this.adapted, predicate);
}
@Override
public <V> LazyIterable<V> collect(Function<? super T, ? extends V> function)
{
return LazyIterate.collect(this.adapted, function);
}
@Override
public <V> LazyIterable<V> flatCollect(Function<? super T, ? extends Iterable<V>> function)
{
return LazyIterate.flatCollect(this.adapted, function);
}
@Override
public <V> LazyIterable<V> collectIf(Predicate<? super T> predicate, Function<? super T, ? extends V> function)
{
return LazyIterate.collectIf(this.adapted, predicate, function);
}
@Override
public LazyIterable<T> take(int count)
{
return LazyIterate.take(this.adapted, count);
}
@Override
public LazyIterable<T> drop(int count)
{
return LazyIterate.drop(this.adapted, count);
}
@Override
public LazyIterable<T> takeWhile(Predicate<? super T> predicate)
{
return LazyIterate.takeWhile(this.adapted, predicate);
}
@Override
public LazyIterable<T> dropWhile(Predicate<? super T> predicate)
{
return LazyIterate.dropWhile(this.adapted, predicate);
}
@Override
public LazyIterable<T> distinct()
{
return LazyIterate.distinct(this.adapted);
}
@Override
public Object[] toArray()
{
return Iterate.toArray(this.adapted);
}
@Override
public int size()
{
return Iterate.sizeOf(this.adapted);
}
@Override
public boolean isEmpty()
{
return Iterate.isEmpty(this.adapted);
}
@Override
public boolean anySatisfy(Predicate<? super T> predicate)
{
return Iterate.anySatisfy(this.adapted, predicate);
}
@Override
public boolean allSatisfy(Predicate<? super T> predicate)
{
return Iterate.allSatisfy(this.adapted, predicate);
}
@Override
public boolean noneSatisfy(Predicate<? super T> predicate)
{
return Iterate.noneSatisfy(this.adapted, predicate);
}
@Override
public <P> boolean anySatisfyWith(Predicate2<? super T, ? super P> predicate, P parameter)
{
return Iterate.anySatisfyWith(this.adapted, predicate, parameter);
}
@Override
public <P> boolean allSatisfyWith(Predicate2<? super T, ? super P> predicate, P parameter)
{
return Iterate.allSatisfyWith(this.adapted, predicate, parameter);
}
@Override
public <P> boolean noneSatisfyWith(Predicate2<? super T, ? super P> predicate, P parameter)
{
return Iterate.noneSatisfyWith(this.adapted, predicate, parameter);
}
@Override
public T getFirst()
{
return Iterate.getFirst(this.adapted);
}
@Override
public T getLast()
{
return Iterate.getLast(this.adapted);
}
@Override
public T detect(Predicate<? super T> predicate)
{
return Iterate.detect(this.adapted, predicate);
}
@Override
public <P> T detectWith(Predicate2<? super T, ? super P> predicate, P parameter)
{
return Iterate.detectWith(this.adapted, predicate, parameter);
}
@Override
public Optional<T> detectOptional(Predicate<? super T> predicate)
{
return Iterate.detectOptional(this.adapted, predicate);
}
@Override
public <P> Optional<T> detectWithOptional(Predicate2<? super T, ? super P> predicate, P parameter)
{
return Iterate.detectWithOptional(this.adapted, predicate, parameter);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,410
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John Rusling Block, född 15 februari 1935 i Galesburg, Illinois, USA, är en amerikansk republikansk politiker.
Hans bakgrund är lantlig och barndomshemmet saknade elektricitet. Han utexaminerades 1957 från United States Military Academy.
Han tjänstgjorde 1981-1986 som USA:s jordbruksminister under president Ronald Reagan. Han fick en chefsposition vid John Deere efter tiden som jordbruksminister. Block tilldelades 1992 års Horatio Alger-pris.
Externa länkar
Horatio Alger Association om 1992 års pristagare
USA:s jordbruksministrar
Personer från Galesburg, Illinois
Födda 1935
Levande personer
Män
Alumner från United States Military Academy
Personer som tjänstgjort i USA:s armé
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,640
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Q: Shortcut to take last inserted value. rails I have,
@post = Post.all
I need the last inserted value in the array @post according to the field created_at.any shortcut for this other than @post.each {|a| some_calculations}
A: @post = Post.order("created_at").last #sort by created_at
or
@post = Post.last #sort by ID
A: @post.sort{|x,y| x.created_at <=> y.created_at}.last
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 563
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{"url":"https:\/\/www.meracalculator.com\/math\/integral.php","text":"\u2022 EN\n\n# Integral Calculator\n\nIntegral Calculator\n\n0.00\n\nIntegral calculator is an online tool that calculates the antiderivative of a function. It works as a definite integral calculator as well as an indefinite integral calculator and lets you solve the integral value in no time.\n\nIf you are studying calculus, you may have an idea of how complex integrals and derivatives are. Well, throw away your worries because the integration calculator is here to make your life easier. You can evaluate the integral by only placing the function in our tool.\n\nNow, we will discuss the integral definition, how to use an integral calculator with steps, how to solve integrals with integral solver, and much more.\n\n## What is integral?\n\nAn integral is the reverse of the derivative. It is as same as the antiderivative. It can be used to determine the area under the curve. Here is the standard definition of integral by Wikipedia.\n\nIn\u00a0mathematics, an\u00a0integral\u00a0assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining\u00a0infinitesimal\u00a0data. Integration is one of the two main operations of\u00a0calculus; its inverse operation,\u00a0differentiation, is the other.\n\nWith an interval of [a, b] of the real line and a real variable x, the definite integral of the given function f can be expressed as:\n\nGenerally, there are two types of integrals.\n\nDefinite integrals: If the integrals are determined by using lower and upper limit, they are called definite integrals. The standard form of definite integrals can be represented by:\n\nIndefinite integrals: If there is no lower or upper limit defined, the limit is indicated by the integration constant. These types of integrals are called indefinite integrals because there are no limits available.\n\nThe standard form of indefinite integrals is:\n\nf(x) dx\n\n## How does online integral calculator work?\n\nThe antiderivative calculator evaluates a function given by the user and converts it into integration by applying the upper and lower limits in case if it is a definite integral. If it is an indefinite integral, the integrals calculator simply uses the constant of integration to evaluate the expression.\n\nFurthermore, evaluate integral calculator brings a sense of simplicity in calculations of integration by only taking a function from the user. You don\u2019t have to do much other than giving input and this iterated integral calculator does it all on its own, and that too in no time at all.\n\nTo use this line integral calculator, follow the below steps:\n\n\u2022 Enter your value in the given input box.\n\u2022 Hit the Calculate button to get the integral.\n\u2022 Use the Reset button to enter a new value.\n\nIntegration by parts calculator will give you a fully evaluated integral function that can be further used in various domains. As mentioned above, integration is the reverse function of derivatives. In case, you need to solve a derivative, use our derivative calculator here.\n\n## How to calculate integrals?\n\nNow that you know what integrals are and how can you use the derivative of integral calculator above to solve an integral, you may also want to know how to solve integrals manually. It can be somehow annoying for the ones who are just starting with integrals.\n\nBut, don\u2019t worry. We will demonstrate the calculations with examples so that you can grasp it easily. Additionally, you can prepare the topic for your exams using the below guideline.\n\nTo calculate integrals, follow the steps below:\n\n\u2022 Determine and write down the function F (x).\n\u2022 Take the antiderivative of the function F (x).\n\u2022 Calculate the values of upper limit F (a) and lower limit F (b).\n\u2022 Calculate the difference of upper limit F (a) and lower limit F (b).\n\nLet\u2019s use an example to understand the method to calculate the definite integral.\n\n### Example \u2013 Definite integral\n\nFor the function f (x) = x \u2013 1, find the definite integral if the interval is\u00a0[2, 8].\n\nSolution:\n\nStep 1: Determine and write down the function F (x).\n\nF (x) = x \u2013 1, Interval = [2, 8]\n\nStep 2: Take the antiderivative of the function F (x).\n\nF (x) = (x1) dx = (x2 \/ 2) x\n\nStep 3: Calculate the values of upper limit F (a) and lower limit F (b).\n\nAs, a = 1, and b = 10,\n\nF (a) = F (1) = (22 \/ 2) \u2013 2 = 0\n\nF (b) = F (10) = (82 \/ 2) \u2013 8 = 24\n\nStep 4: Calculate the difference of upper limit F (a) and lower limit F (b).\n\nF (b) \u2013 F (a) = 24 \u2013 0 = 24\n\nThis method can be used to evaluate the definite integrals having limits. You can use a double integral calculator above to if you don\u2019t want to indulge in integral calculations.\n\n### Example \u2013 Integral of a trigonometric function\n\nFor the function f (x) = sin (x), find the definite integral if the interval is\u00a0[0, 2\u03c0].\n\nSolution:\n\nStep 1: Determine and write down the function F (x).\n\nF (x) = sin (x), Interval = [0, 2\u03c0]\n\nStep 2: Take the antiderivative of the function F (x).\n\nF (x) = sin (x) dx = cos (x)\n\nStep 3: Calculate the values of upper limit F (a) and lower limit F (b).\n\nAs, a = 0, and b = ,\n\nF (a) = F (0) = cos (0) = 0\n\nF (b) = F () = cos (2\u03c0) = 0\n\nStep 4: Calculate the difference of upper limit F (a) and lower limit F (b).\n\nF (b) \u2013 F (a) = 0 \u2013 0 = 0\n\nAlong with manual calculation, you can also use our trigonometric substitution calculator above to solve a trigonometric integral in a fraction of seconds.\n\n## FAQs\n\n### What is an integral calculation?\n\nAn integral calculation reverses the function of the derivative by taking the antiderivative of that function. It is used to determine the area under the curve. Integral calculations can be definite if upper and lower limits are there. If there are no intervals, an integral constant C is used and that type of function is called indefinite integral.\n\n### What is the derivative of an integral?\n\nIf we take the derivative of an integral, both of them will cancel each other because derivative and integral are reverse functions to each other. Integral is the same as antiderivative according to the fundamental theorem of calculus.\n\n### Who is the father of integration?\n\nGottfried Wilhelm Leibniz and Isaac Newton proposed the rules of integration independently at the end of the 17th century. They assumed the integral as an endless sum of rectangles of extremely small width. Bernhard Riemann described integrals in a strict mathematical fashion.\n\n### What is the integral of 1?\n\nThe integral of 1 is x or x + c because if we add an integral constant. It can be expressed as a diagonal line lies in the 1st and 3rd quadrant of the graph.\n\n1 dx= x + C\n\n### What is the integral of sin 2x?\n\nThe integral of sin 2x can be calculated by the substitution method. It will be an indefinite integral due to the no interval or upper and lower limits. Here is the integral of sin 2x.\n\nsin (2x) dx= \u2013 (1\/2) cos (2x) + C","date":"2020-10-23 21:00:41","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\": 0.9155884981155396, \"perplexity\": 416.8336489125126}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107865665.7\/warc\/CC-MAIN-20201023204939-20201023234939-00449.warc.gz\"}"}
| null | null |
\section{Introduction}
Magnetic fields are important for understanding galactic dynamics and star formation processes. To date, various methods for measuring interstellar magnetic fields have been devised to probe the star-forming regions in our Galaxy and in external galaxies \citep{vrb76,dav51,laz07}. When background starlight passes through dust grains aligned with the local magnetic field, the light is linearly polarized by dichroic extinction parallel to the grain's long axis.
The Large Magellanic Cloud (LMC) is a unique target for studying magnetic fields and star-forming processes. Due to its proximity and face-on orientation, large-scale interactions with the halo of the Milky Way and the Small Magellanic Cloud (SMC), and small-scale structures of the local star-forming regions, can be traced \citep{kim98,pak98,kim11}. Previous studies of magnetic fields in the LMC were carried out using multi-frequency surveys of diffuse synchrotron emission in radio bands \citep{way90,hay91,gae05,mao12}. Optical polarization \citep{vis66,sch70,sch76,mat70} have shown the existence of large-scale magnetic fields in the LMC. However, these results were restricted to only bright stars. \citet{wis07} (hereafter W07) also showed the detailed polarization map of NGC 2100 on the eastern side of 30 Doradus and its surrounding regions, in optical bands. The near-infrared band has an advantage over the optical band. This is because the background starlight in star-forming regions suffers heavy extinction \citep{tam87,tam88,sat88}. \citet{nak07} (hereafter N07) studied magnetic field structures 7.7$\arcmin$ $\times$ 7.7$\arcmin$ around 30 Doradus using near-IR polarimetry. They revealed more detailed magnetic field structures that are aligned roughly in the east-west direction compared with the previous results in the optical band \citep{vis66,sch70,mat70}. These field structures are associated with expanding shells around 30 Doradus.
We carried out a near-IR photometric and polarimetric study for the 39$\arcmin$ $\times$ 69$\arcmin$ fields in the northeastern part of the LMC. The target fields contain four important star-forming regions, 30 Doradus, N158, N159 and N160. The initial investigation of polarimetric data around 30 Doradus was presented in \citet{kim11}. However, the survey was limited to a 20$\arcmin$ $\times$ 20$\arcmin$ region and the data were hampered by unstable weather conditions. In this paper, we analyzed additional data sets for more regions. The procedures to obtain the photometric results are described in Section 2. We compiled photometric and polarimetric data for each observation field in a catalog. In Section 3, we examined the origin of the polarization, based on the wavelength dependence. In Section 4, we showed polarization vector maps for each observation field and discussed the related magnetic field structures. We also calculated magnetic field strengths in five sample fields. Finally, we discussed the correlation of our polarization results with dust emission and CO distribution.
\section{Observations and Data Reduction}
\subsection{Observational information}
We observed the northeastern regions in the LMC with the infrared camera SIRIUS \citep{nag03} and the polarimeter SIRPOL \citep{kan06} at the Infrared Survey Facility (IRSF) 1.4 m telescope at the South African Astronomical Observatory (SAAO) on 2008 December 25$-$30 and 2011 December 2$-$11. This system has a field of view (7.7$\arcmin$ $\times$ 7.7$\arcmin$) and a pixel scale of 0.45$\arcsec$ pixel$^{-1}$. One data set for a target field consists of 20-second exposures at 10 dithered positions for four wave-plate angles (0$\arcdeg$, 45$\arcdeg$, 22.5$\arcdeg$, and 67.5$\arcdeg$) in the \textit{J} (1.25 $\mu$m), \textit{H} (1.63 $\mu$m), and \textit{$K_s$} (2.14 $\mu$m) bands. The target regions of 39$\arcmin$ $\times$ 69$\arcmin$ are tiled with 45 fields with grid spacing of 6.5$\arcmin$ $\times$ 6.5$\arcmin$ (Figure~\ref{fig1}). We list the observation log in Table~\ref{tbl-1}.
\subsection{Data reduction procedure}
We used the SIRPOL data reduction pipeline \citep{kan06} at the IRAF\footnote{IRAF is distributed by the US National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.}(Image Reduction \& Analysis Facility). The pipeline incorporates the following procedures: flat field correction, sky subtraction, and combining of dithered frames. We checked the output magnitude variations in 40 sequential frames to filter bad data affected by non-photometric weathers. The fields with magnitude variation larger than 0.03 mag were rejected (see Figure~\ref{fig1}).
We used Source Extractor (SExtractor) for source detection and aperture photometry \citep{ber96}. Various SExtractor parameters were optimized (i.e., detection threshold of 5 $\sigma$, background box size of 16 pixels, background filter of 3 $\times$ 3 meshes, and aperture diameter of 8 pixels). We measured the instrumental magnitudes of stars for four wave-plate angles in the $J$, $H$, and $K_s$ bands. The pixel coordinates of the sources were converted to celestial coordinates using the 2MASS All Sky Point Source Catalogue \citep{skr06}. The instrumental magnitude and color were calibrated using the equations below:\\
\begin{equation}
M_{2MASS} = M_{IRSF} + \alpha_{1} \times C_{IRSF} + \beta_{1},\\\end{equation}
\begin{equation}
C_{2MASS} = \alpha_{2} \times C_{IRSF} + \beta_{2},\\\end{equation}
where $M_{IRSF}$ and $C_{IRSF}$ are the instrumental magnitudes and the colors from the Stokes $I$ images, and $M_{2MASS}$ and $C_{2MASS}$ are the magnitude and the color in the 2MASS system. The transformation coefficients, $\alpha$ and $\beta$, for each field were estimated using a robust least absolute deviation method.
We calculated polarization of point sources using Stokes parameters $I$, $Q$, and $U$ as described by \citet{kim11}. Flux errors of sources at each wave-plate angle (0$\arcdeg$, 45$\arcdeg$, 22.5$\arcdeg$, and 67.5$\arcdeg$) are described as $\sigma_{I_{1}}$, $\sigma_{I_{2}}$, $\sigma_{I_{3}}$, and $\sigma_{I_{4}}$, respectively. The uncertainty for $P$ (\%) was derived as follows:\\
\begin{equation}
\overline{\sigma}_{Q} = \sqrt{\left ( \frac{\sigma_{Q}}{I}\right )^{2} +\left ( \frac{Q}{I^{2}}\times \sigma_{I} \right )^{2}},\\\end{equation}
\begin{equation}
\overline{\sigma}_{U} = \sqrt{\left ( \frac{\sigma_{U}}{I}\right )^{2} +\left ( \frac{U}{I^{2}}\times \sigma_{I} \right )^{2}},\\\end{equation}
\begin{equation}
\sigma_{P} = \frac{\sqrt{\left ( \frac{Q}{I} \right )^{2}\times \overline{\sigma}_{Q}^{2}+\left ( \frac{U}{I} \right )^{2}\times \overline{\sigma}_{ U}^{2}}}{P}, \\\end{equation}
where $\overline{\sigma}_{Q}$ and $\overline{\sigma}_{U}$ are normalized errors with $I$. The errors of Stokes parameters $\sigma_{I}$, $\sigma_{Q}$, and $\sigma_{U}$ can be written as $\sqrt{\sigma_{Q}^{2}+\sigma_{U}^{2}}/{2}$, $\sqrt{\sigma_{I_{1}}^{2}+\sigma_{I_{2}}^{2}}$, and $\sqrt{\sigma_{I_{3}}^{2}+\sigma_{I_{4}}^{2}}$, respectively.
\section{Results}
\subsection{Estimation of the photometric and the polarimetric accuracy}
A total of 25,488 sources were detected with the SExtractor. To verify the photometric accuracy, we compared our data set with the IRSF Magellanic Cloud point source catalog (hereafter the IRSF catalog) in \citet{kat07}. The IRSF catalog provides photometry results in the $J$, $H$, and $K_s$ bands. Figure~\ref{fig2} shows a histogram of position offsets of our sources that matched theirs. Most sources have the offset around 0.1$\arcsec$ and we retain only the sources with offset less than $0.3 \arcsec$. The proper motion effect between two data sets is negligible: $\mu_{\alpha}\cos\delta$ = 1.89 $\pm$ 0.27 mas yr$^{-1}$, and $\mu_{\delta}$ = 0.39 $\pm$ 0.27 mas yr$^{-1}$ \citep{vie10}. The magnitude differences between the IRSF catalog and ours are shown in Figure~\ref{fig3}. Most sources brighter than 14 mag have differences in magnitude from the IRSF catalog less than 0.3 mag in the $J$, $H$, and $K_s$ bands.
The polarization uncertainty with magnitude is shown in Figure~\ref{fig4}. Among the sources with $P/\sigma_P$ (polarization signal-to-noise ratio) greater than 10, the polarization uncertainty is lower than 1\% for sources brighter than 14.5 mag (Figure~\ref{fig4} a). We also compared polarimetric results with previous studies. N07 had observed the central 7.7$\arcmin$ $\times$ 7.7$\arcmin$ (one field) region of 30 Doradus using the same SIRIUS/SIRPOL system. The integration time of 1,480 seconds per wave-plate angle makes N07 a very good reference for verifying the polarimetric accuracy of our data. We examined the polarimetric results for $P/\sigma_P$ $>$ 3 between N07 and our 30 Doradus field, and Figure~\ref{fig5} shows the comparison results. The difference of polarization degree increases at around 14 mag in the $J$ and $H$ bands. Though the difference in the polarization angle shows some notable scatter over the entire range of magnitude, the scatter is quite constant down to 14 mag in the $J$ and $H$ bands.
As a result, we adopted all sources brighter than 14 mag in the $H$ band. In order to increase the number of meaningful sources, we extended the criteria of $P/\sigma_P$ to include $'$3$'$. Most sources in Figure~\ref{fig4} (b) have their polarization uncertainties $\leq$ 2\% for sources brighter than 14 mag. Typical polarization uncertainties between 13.9 and 14.1 mag in $H$ band are 1.12 $\pm$ 0.25, 0.74 $\pm$ 0.06, and 1.68 $\pm$ 0.27 $\%$ in the $J$, $H$, and $K_s$ bands, respectively. We checked the accuracy of polarization position angles for the selected data by comparing differences in the polarization position angles among the three bands. The upper and lower panels of Figure~\ref{fig6} show the distribution of polarization position angles for $J$ versus $H$ band, and $H$ versus $K_s$ band, respectively. The estimated uncertainty of the polarization position angles is smaller than 10$^\circ$. As seen in Figure~\ref{fig6}, we confirmed that most of the sources in our catalog show consistent polarization position angles (within one standard deviation) from the correlation slope.
\subsection{Catalog}
The final catalog includes all reliable sources. Among 25,488 detected sources, 1,858 sources were selected based on the following criteria: m$_{H}$ $<$ 14 mag and $P/\sigma_P$ $>$ 3 for at least one band, where $\sigma_P$ indicates the polarization uncertainty. Table~\ref{tbl-2} lists photometric and polarimetric results for the compiled catalog sources. In addition, we compiled proper motion data from Southern Proper Motion (SPM) material \citep{vie10}. Information for each column of the table is given below:\\
Column (1) Observation field names that include observation date;\\
Column (2) Source ID; the formats referred to the equatorial coordinates;\\
Column (3)$-$(4) Equatorial coordinates (J2000.0) in decimal degrees;\\
Column (5)$-$(10) $J$, $H$, and $K_{s}$ magnitude and error;\\
Column (11)$-$(16) $J$, $H$, and $K_{s}$ polarization degree and error in units of percentage;\\
Column (17)$-$(22) $J$, $H$, and $K_{s}$ polarization position angle and error in units of degrees;\\
Column (23)$-$(24) Absolute proper motion in right ascension and error in units of mas yr$^{-1}$ from SPM catalog;\\
Column (25)$-$(26) Absolute proper motion in declination and error in units of mas yr$^{-1}$ from SPM catalog;\\
Column (27)$-$(28) Johnson B and V magnitude from SPM catalog.
\subsection{Polarization relationship between $J$, $H$, and $K_s$ bands}
\citet{ser75} first formulated the wavelength dependence of interstellar polarization in the Galaxy, and \citet{whi78} modified the empirical relationship between the wavelength and the size of the dust grains. The modified "Serkowski law" was represented by a power-law dependence of polarization as below:\\
\begin{equation}
P_{\lambda} \propto \lambda^{-\beta}\\\end{equation}
with $\beta$ of 1.6$-$2.0 in the near-IR (1.25$\leq\lambda\leq$2.2) in previous studies \citep{nagt90,mar90,mar92}. For the 30 Doradus region, N07 presented a slightly different power-law index of $\beta$ $=$ 0.9. As seen in Figure~\ref{fig7}, our result is consistent with that of N07. For your reference, we also plotted the sources matched with the AKARI LMC point-source catalog to check the possibility of intrinsic polarization (see Section \ref{sec:AKARI_MIR} for details).
We examined the correlation between the observed polarization efficiency and the upper limit of polarization degree. Figure~\ref{fig8} shows the result of polarization efficiency for the sources with $P/\sigma_P$ $>$ 3 and $\sigma_P$ $\leq$ 1 in all bands. Most of the sources are located below the upper limit of interstellar polarization (dashed line). This means that the observed polarization in the LMC is of interstellar origin.
\subsection{Possibility of intrinsic polarization component}\label{sec:AKARI_MIR}
$AGB$ stars are often associated with nebulosity and therefore the polarization might be intrinsic rather than interstellar. We examine this possibility from mid-infrared observation data of the AKARI LMC point-source catalog. Using a color-color diagram and a color-magnitude diagram of the five photometric bands at 3.2, 7, 11, 15, and 24 $\mu$m, \citet{kat12} classified the sources with dusty C-rich and O-rich $AGB$ stars.
In our catalog, 208 sources have been matched to the AKARI LMC point sources in the 3.2, 7, and 11 $\mu$m bands. The matched AKARI sources were divided into $AGB$ stars without dusty envelope and dusty C-rich/O-rich $AGB$ stars. However, distributions of the polarization degrees and position angles do not show any significant differences between two groups. In addition, the matched AKARI sources do not show any different power-law relationship (Figure~\ref{fig7}). This implies that the detected polarization is of interstellar origin rather than being intrinsic to $AGB$ stars.
\section{Discussion}
\subsection{Polarization results}
Figures~\ref{fig9} and ~\ref{fig10} show the $H$-band polarization maps in the observed LMC fields. Each vector represents a point source with $P/\sigma_P$ $>$ 3. The number of selected stars was 875, 1,468, and 732, in the $J$, $H$, and $K_s$ bands, respectively. Most of these vectors are directed either from north to south or from northeast to southwest.
In the region around 30 Doradus (Figure~\ref{fig9}), most of the polarization vectors have directions from northeast to southwest. In the southeastern region from 30 Doradus, the distribution of polarization vectors shows a shell-like structure encircling $n39$ field. W07 presented the magnetic field structure of NGC 2100, of which the western part overlapped our fields. The polarization feature in the region of overlap shows similar polarization vector patterns. W07 reported that the origin of the magnetic field geometry around NGC 2100 includes massive outflows moving eastward from the 30 Doradus region. The coherent pattern of the polarization vectors in the north and south of $n39$ field is consistent with that of W07. They reported that the pattern is influenced by magnetic field lines along the eastward outflows. In addition, other patterns in $n39$ field can be explained by interaction between the environment of NGC 2100 and the eastward outflows. The large-scale outflow from 30 Doradus is one plausible source of the influence on the structural formation of the supergiant shell on the western side of 30 Doradus (LMC 3 region). Overall vectors in the western regions of 30 Doradus show coherent patterns toward the direction of the LMC 3 region identified in previous studies \citep{mea80,poi01,bcg08,daw13}. As suggested by W07, the extended uniform patterns of a magnetic field can be a clue to large-scale outflows toward the boundaries of a supergiant shell.
The polarization vectors in Figure~\ref{fig10} show complex distribution. At the eastern part, most of the polarization vectors are directed north-south. However, vectors at the other side show a different distribution structure, more like a tilted S-shape. We suspect that these patterns of polarization vectors are related to the magnetic field lines south of 30 Doradus. Previous studies \citep{way90,hay91,sk92} of the geometry of the magnetic field south of 30 Doradus suggested that it is aligned with the filamentary features B and C of Figure 2 in \citet{fei87}. Two patterns show direction similar to those of these filamentary features. \citet{way90} and \citet{mao12} also proposed that the gaseous H\textsc{i} spiral features are likely associated with the large-scale magnetic fields toward the south direction.
A statistical analysis was conducted field-by-field to understand the interstellar polarization properties. In Table~\ref{tbl-3}, the number of stars with $P/\sigma_P$ $>$ 3 for each band is given in columns (2$-$4). The average polarization degrees for each band is given in columns (5), (7), and (9), and their mean uncertainties in column (6), (8), and (10). The average polarization position angles at each band is given in columns (11), (13), and (15) with their standard deviation in columns (12), (14), and (16). We examined the magnetic field strength of the fields overlapped with those described by W07 and other sample fields using the analysis of \citet{chf53}, which is described by
\begin{equation}
B = Q\sqrt{4\pi\rho}\frac{\delta\textit{v}_{los}}{\delta\theta }
\end{equation}
where $\rho$ is the mean density, $\delta\textit{v}_{los}$ is the velocity dispersion in the line-of-sight, and $\delta\theta$ is the dispersion in the Gaussian fit of polarization position angles. \citet{kim07} presented a 21 cm neutral hydrogen interferometric survey of the LMC in a catalog of H\textsc{i} gas clumps or clouds with 16, 32, and 64 K brightness temperature thresholds. By matching their catalog locations with positions of our observed fields, we obtained $\delta\textit{v}_{los}$ from their survey results. The applied $\delta\textit{v}_{los}$ of the matched sample fields are tabulated in Table~\ref{tbl-4}. As \citet{cru04} reported, the value of a factor of order unity, Q, shows the best result when the value from \citet{ost01} is adopted. By simulation, \citet{ost01} determined that Q is approximately 0.46$-$0.51, when the observed dispersions in polarization position angles indicate relatively strong magnetic field strengths ($\delta\theta$ $<$ 25$\arcdeg$). We set the value of Q to be 0.5, as suggested by \citet{cru04}. We used column (14) in Table~\ref{tbl-3} to select sample fields showing coherent distribution of polarization position angles. For the coherent distribution, $\delta\theta$ was calculated by a Gaussian fit centered at average polarization position angles in Table~\ref{tbl-3}. We tested whether the distribution of polarization position angles is well fitted to a Gaussian distribution by applying the Martinez-Iglewicz normality test \citep{mar81},
\begin{equation}
I= \frac{\sum_{i=1}^{n}\left ( \theta_{i} -\langle\theta_{H}\rangle \right )^{2}}{\left (n-1 \right )\delta\theta ^{2}} \end{equation}
where $I$ is the Martinez-Iglewicz test statistic, $\langle\theta_{H}\rangle$ is average polarization position angles in the $H$ band. If the distribution possesses significant deviation from normality, it indicates the presence of possible sub-structure. To eliminate these cases, we selected fields showing test value $I$ close to $'$1$'$ (Table~\ref{tbl-4}). W07 used H\textsc{i} number density from \citet{poi99} to estimate mean density in their observed fields. We also assumed the number density of 4 cm$^{-3}$ as used in W07. As seen in Table~\ref{tbl-4}, we tabulated the magnetic field strengths for the five sample fields. One of them is the same region considered in W07, and it shows a similar magnetic field strength. The derived magnetic field strengths and sizes of the selected regions are similar to the properties of a typical cloud complex (i.e., scale 10$-$100 parsecs and magnetic field strength ranging from 10 to 30 $\mu$G) as reported by \citet{cha11}.
\subsection{Polarization structure with molecular cloud studies}\label{sec:Pol_structure}
In order to understand the pattern of polarization vectors, we examined our $H$-band polarization map by comparing the literature on mid to far-infrared dust emission maps and CO gas emission maps. We assume that the position angle of polarization indicates the direction of the magnetic field, because the observed polarization originated from interstellar dust grains with short axis aligned with the local magnetic field in the LMC.
The IRAC data (3.6, 5.8, and 8 $\mu$m) from the Spitzer SAGE \citep{meix06} were compiled to trace dust emission features in the LMC fields more clearly. Although we have done a comparison with the Herschel data (100, 160, and 250 $\mu$m) from the HERITAGE project \citep{meix10}, we did not find any features different from those in the Spitzer data, due to relatively poor resolution. The IRAC data are displayed as a color composite image (shown in Figures~\ref{fig11} and~\ref{fig12}), and we overlaid our polarization vector map on these figures. The polarization vectors showing prominent patterns indicate the direction of the local magnetic field lines and we indicated these patterns with a green shaded curve in both Figure~\ref{fig11} and~\ref{fig12}.
The overall trend of the polarization vectors shows east-west direction across the 30 Doradus region (P1 in Figure~\ref{fig11}). Other distinctive patterns of polarization vectors are located at the outskirts field of 30 Doradus, associated with dust emission features (P2, P3, and P4 in Figure~\ref{fig11}). P2 is located at the northwestern boundary of the dust emission feature and it shows a U-shaped structure. N07 found a similar U-shaped structure of polarization vectors, and they proposed that expanding shells in 30 Doradus affected complex cloud structures and the associated magnetic field. P3 and P4 exhibit apparent patterns with similar direction to that of bright emission features.
Figure~\ref{fig12} shows the distribution of polarization vectors mainly following the molecular cloud ridge around star-forming regions of N158, N160, and N159. The vectors located at pattern P1 show distinct elongation extending southward, with direction similar to that of the dust emission structure. P2 also shows a uniform pattern directed north-south direction, but its shape is slightly curved along the dust emission feature. These patterns (P1 and P2) are regarded as magnetic field lines related to the extended feature of the molecular cloud ridge. However, polarization vectors located at the southwestern part of Figure~\ref{fig12} show patterns different from those of the eastern part. Polarization vectors in patterns P3 and P4 tend to be aligned in a tilted S-shape and east-west, respectively. We observed that the dust emission feature in the southwestern part of Figure~\ref{fig12} seems to be extended in this direction.
On the other hand, polarization vectors in the star-forming regions (central part of 30 Doradus in Figure~\ref{fig11} and bright emission features in Figure~\ref{fig12}) do not show any significant pattern. This is thought to be due to the turbulent and complex magnetic fields of the sub-structures in the star-forming region. To verify the structure of the expected turbulent magnetic fields, more detailed polarimetric data are required.
We also did a comparison with a velocity-integrated contour map of $^{12}$CO emission. NANTEN $^{12}$CO($J$=1-0) emissions have been detected in giant molecular cloud complexes in the LMC \citep{fuk08}. Although prominent $^{12}$CO emission features are located in the central 30 Doradus region, and in star-forming regions in the southern molecular cloud ridge, close correlation between magnetic fields and $^{12}$CO emission features was not found.
We concluded that the observed polarization vectors reflect a dust cloud structure resulting in polarization by dichroic extinction, and that they trace magnetic fields associated with dust clouds around star-forming region.
\section{Summary \& Conclusions}
We conducted near-IR imaging polarimetry for a large areal region covering ($\sim39\arcmin\times69\arcmin$) on the eastern side of the LMC. We made a band-merged catalog of photometric and polarimetric data in $J$, $H$, and $K_s$ bands. In this catalog, we compiled 1,858 stars in the region, brighter than 14 mag in $H$ band and $P/\sigma_P$ $>$ 3 for least one band. The magnitude, polarization degree and polarization position angle were listed in the catalog. In addition, we provided information on absolute proper motion and magnitudes of $B$ and $V$ matched from SPM data. Using this catalog, we obtained polarization vector maps and did a statistical analysis of polarization for the fields observed in the LMC. Previous similar catalogs did not cover such a large continuous area of the LMC (e.g., $\sim15\arcmin\times15\arcmin$ by W07), nor did they use the infrared range. Therefore, this catalog is a unique, up-to-date collection of polarization measurements for the LMC region.
The degree of polarization for the 106 sources in our catalog shows wavelength dependence similar to those reported by N07. We concluded that most of the sources in our catalog exhibit interstellar polarization, and that the dominant polarization mechanism is the result of dichroic extinction by dust grains aligned along magnetic fields, rather than due to intrinsic polarization of stars. Using the polarization vector maps, we traced the correlation of the polarization and the large-scale magnetic field structures in the observed regions. The geometry of the large-scale magnetic field structures around 30 Doradus, and of the southern star-forming regions, show relationships with the environment of nearby supergiant shells and the gaseous spiral features toward the south, respectively. The estimated magnetic field strengths for the selected fields (within one hundred parsecs) are in the range 3$-$25 $\mu$G, as determined by the Chandrasekhar-Fermi method. Judging from their size and magnetic field strength, those regions are regarded as a cloud complex associated with a nebula in the LMC. Prominent patterns of polarization vectors mainly follow dust emission features in mid to far-infrared bands, which implies that the large-scale magnetic fields are well involved in the structures of the dust cloud in the LMC, except for the dense star-forming regions. The cross matching of the data in this catalog, with that in other catalogs for in the LMC, will give useful information for probing the structure of magnetic fields and other astrophysical phenomena.
\acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant, No. 2008-0060544, funded by the Korea government (MSIP).
We would like to thank Jungmi Kwon for giving a chance to study this work.
We would like to thank Prof. Shuji Sato for kindly providing comments to improve this paper.
This paper uses observations performed at the South African Astronomical Observatory.
This publication makes use of data products from the Two Micron All Sky Survey and observations with AKARI. The Two Micron All Sky Survey is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. The AKARI is a JAXA project with the participation of ESA.
This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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El Chevrolet Kadett fue un automóvil fabricado por la General Motors para el mercado del Mercosur. Si bien el coche es la versión latinoamericana del Opel Kadett alemán, fue fabricado en Brasil y comercializado en el Mercosur y otros países de Sudamérica, donde se transformó en uno de los coches más vendidos. Sus rivales de mercado fueron el tándem Escort/Verona de Ford, aunque también rivalizó con otros coches del segmento, como el Fiat Tempra y el Renault 19. Se presentó en el año 1989 en Brasil y en 1994 comenzó a ser sustituido paulatinamente por la primera generación del Chevrolet Astra basado en el Opel Astra F y finalmente en 1998 por la segunda generación del Chevrolet Astra (Opel Astra G). En países como Chile y Perú, llegaba importada la versión coreana del Opel/Chevrolet Kadett, el Daewoo Racer/Cielo con mecánica y aspecto similar al Kadett brasileño, a la cual también se agregaba la versión sedán 4 puertas muy popular en estos países y un facelift a partir de 1994 que corría por cuenta propia de la firma coreana.
Resumen histórico
La sexta y última generación del Kadett, de 1984 (también hubo un intermediario, el quinto, ya con motor transversal y tracción delantera), dio lugar al Chevrolet brasileño del mismo nombre cinco años después. En 1991 daría lugar al Opel Astra, nombre ya utilizado en el inglés Kadett, pasándolo a la segunda generación en 1997, muy conocido en Brasil.
El Kadett fue creado por Opel y en 1989 fue lanzado en Brasil bajo la marca Chevrolet. El 16 de septiembre de 1998 Chevrolet terminó la producción del Kadett, con los modelos GL y GLS, con un total de 9 años de línea, siendo reemplazados por el Astra.
El Kadett fue un automóvil que innovó en varios aspectos de la producción de vehículos en Brasil, siendo el primer automóvil producido en serie en utilizar vidrio encolado (parabrisas y luneta), con suspensión neumática ajustable, con motor de alcohol. neumáticos inyectados (junto con el Monza en 1991) y neumáticos de la serie 65 (Kadett GS 1991).
Fue el primer automóvil de Chevrolet en utilizar computadora de a bordo y check-control, además de tener el mejor coeficiente aerodinámico de la época: Cx 0.30 en el Kadett GS y Cx 0.32 en los otros modelos.
En Europa también se vendieron los modelos Hatch de 5 puertas y sedán de 4 puertas bajo la marca Opel, nunca disponibles en Brasil.
Actualmente, el Kadett se encuentra entre los cinco autos usados más vendidos en Brasil. La necesidad de renovar el parque automotor, ante la evolución de la competencia, fue todo un desafío para General Motors. En Latinoamérica, tanto en Brasil como en México (los mercados donde GM tiene todo su potencial), se fabricaban los Chevette. La tracción delantera comenzaba a ser suceso en todos los mercados, y GM no quería quedarse en el tiempo. Ford sacó a la venta el Escort, Volkswagen era éxito con el Gol y Renault se caracterizó por ser pionero en este tipo de tracción. Por todo esto General Motors realizó una apuesta muy fuerte con la fabricación del Chevrolet Kadett.
Su nombre provenía de la costumbre que daba en su momento la Opel alemana de identificar sus coches con rangos militares (Opel Kapitän, Opel Commodore y Opel Kadett). Con este auto, General Motors intentaría destronar al Volkswagen Gol en Brasil, y al Ford Escort en Argentina. El coche (como su antecesor Chevette), no era más que la versión latinoamericana del Opel Kadett alemán. Su aparición marcó el final de la era de los coches de tracción trasera.
Venía fabricado con motor 1.8 OHC de gasolina que más tarde se siguió produciendo para equipar a los Chevrolet Corsa que se ensamblan en Argentina. A este país llegó en el año 1994, a la par del regreso de la marca al Mercado Automotor Argentino. Llegaba importado de Brasil junto a los demás productos.
En 1993 se presentó la versión deportiva del Kadett: El Kadett GSi. El mismo se fabricaba en Brasil y venía con equipamiento deportivo tanto estético como mecánico, ya que este venía montado con un motor 2.0 potenciado pero mientras en Europa esta versión se vendía con 150cv y 16v en Brasil solo tenía 121cv
, además de traer una de las estéticas más atractivas del coche. Más tarde se presentó uno de los mejores modelos del Kadett: El Kadett Cabrio. Un coche con el que se intentaba acaparar más al público joven, que no se veía del todo convencido con la estética original del Kadett. Otra versión disponible al público fue la versión rural del Kadett, denominada Chevrolet Ipanema.
Así, el Kadett se fue ganando un lugar en el corazón de los Brasileños, quienes lo adoptaron como el "Auto Nacional". Sin embargo este éxito no lo pudo conseguir en los demás mercados del continente.
La llegada a la Argentina
Argentina fue un mercado difícil para General Motors, más teniendo en cuenta el cierre de la compañía en el año 1978, y el claro dominio que ejercieron Renault y Ford en los años posteriores. A todo esto se le sumaba lo complicado de intentar convencer a un público acostumbrado a consumir siempre de lo mismo.
Así fue que luego del final del acuerdo entre Renault Argentina y General Motors de Brasil, esta última decidió su regreso al país del Plata. Con el regreso, GM se traía su oferta de automóviles para el público argentino.
Entre ellos se encontraba el Kadett, el coche que llegaba a la Argentina para hacerle sombra a la dupla de Autolatina, Escort-Gol que venía ganando en confiabilidad. Ni bien llegó al país, en seguida fue utilizado como coche de carreras. En el año 1995, debutó en el TC 2000, siendo piloteado por René Zanatta y atendido por Hugo Bini. Fue el primer Chevrolet que participó en la categoría. A pesar de todo, dado que se trataba de un equipo que venía de pelear el campeonato de 1994, el coche no tuvo el rendimiento esperado.
Fuera de esto, a pesar de no haber sido importado en gran cantidad, su aparición significó la piedra fundamental en la construcción de un liderazgo que se solidificó, con la fabricación del Chevrolet Corsa en Argentina. Sin lugar a dudas, el Kadett le allanó el camino a General Motors para liderar en el mercado del país meridional.
El final y su herencia
Fue importado hasta el año 1998, cuando GM resolvió dejar la producción solamente para Brasil. En los demás países, fueron importados los Chevrolet Monza, los cuales no tenían la aceptación que tuvo el Kadett.
Sin embargo, esta espera por un coche con las mismas características del Kadett no duró mucho, porque en el año 1999 se anunció la producción en Brasil, de un nuevo coche que ocuparía el lugar que dejara vacante: El Chevrolet Astra. Su importación se inició ese mismo año dando final a la producción tanto del Kadett como del Monza, para toda América Latina.
Fue así que el nuevo Astra heredó un mercado favorable y una buena aceptación por parte del público, que rápidamente lo colocó entre los más vendidos de la región. Este fue el legado dejado por el Kadett hacia su sucesor. Sin lugar a dudas, un coche que sorprendió con su aparición y que en seguida terminó ganándose un lugar importante en los mercados mundiales.
Enlaces externos
Historia del Kadett (portugués)
Kadett
Kadett
Modelos de automóviles de bajo costo
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\section{Introduction and preliminaries}
The Ricci flow introduced by Hamilton \cite{HA82} plays a significant role in Perelman's proof of the Poincar\'e conjecture and currently it has been intensively used in the study of various geometric properties.
For study on Ricci flow, see \cite{Chow-B}.
An important aspect in the investigation of the Ricci flow is the study of Ricci solitons. A gradient Ricci soliton is an $n(\geq 2)$-dimensional Riemannian manifold $(M, g)$ with Riemannian metric $g$, satisfying
\begin{equation}\label{rs1}
Ric + \nabla^2 f = \lambda g,
\end{equation}
where $\nabla^2f$ stands for the Hessian of $f\in C^{\infty}(M)$, the ring of smooth functions on $M$, $Ric$ is the Ricci curvature tensor and $\lambda \in \mathbb{R}$. A Ricci soliton $(M, g)$ is called expanding if $\lambda <0$, steady if $\lambda=0$ and shrinking if $\lambda > 0$. For some results of Ricci solitons see \cite{CA2019, CA2020, AC2021}. In general, it is natural to consider geometric flows of the following type on a $n(\geq 3)$- dimensional Riemannian manifold $(M,g)$:
$$\frac{\partial g}{\partial t}=-2({\rm Ric}-\rho Rg),$$
where $R$ denotes the scalar curvature of the metric $g$ and $\rho\in\mathbb{R}\diagdown\{0\}$. The parabolic theory for these flows was developed by Catino et. al. \cite{Catino13}, which was first considered by Bourguignon \cite{Bour}. They called such a flow as Ricci-Bourguignon flows. they defined the following notion of
$\rho$-Einstein solitons.
\begin{defn}
Let $(M,g)$ be a Riemannian manifold of dimension $n(\geq 3)$, and let $\rho\in\mathbb{R}$, $\rho\neq 0$. Then $M$ is called a $\rho$-Einstein soliton if there is a smooth vector field $X$ such that
\begin{equation}\label{a2}
{\rm Ric}+\frac{1}{2}\mathcal{L}_Xg-\rho Rg=\lambda g,
\end{equation}
where $\mathcal{L}_Xg$ represents the Lie derivative of g in the direction of the vector field $X$.
\end{defn}
If there exists a smooth function $f:M\rightarrow\mathbb{R}$ such that $X=\nabla f$ then the $\rho$-Einstein soliton is called a gradient $\rho$-Einstein soliton, denoted by $(M,g,f)$ and in this case (\ref{a2}) takes the form
\begin{equation}\label{res1}
Ric+\nabla^2 f-\rho Rg=\lambda g.
\end{equation}
The function $f$ is called a $\rho$-Einstein potential of the gradient $\rho$-Einstein soliton. As usual, a $\rho$-Einstein soliton is called steady for $\lambda=0$, shrinking for $\lambda>0$ and expanding for $\lambda<0$. After rescaling the metric $g$ we may assume that $\lambda \in \{-\frac{1}{2}, 0,\frac{1}{2}\}$. For more study on $\rho$-Einstein solitons, we refer to the interested reader \cite{CAT16, AAP2021, Absos} and also the references therein.
For particular value of the parameter $\rho$, a $\rho$-Einstein soliton is called
\begin{enumerate}
\item[i)] gradient Einstein soliton if $\rho=\frac{1}{2}$,
\item[ii)] gradient traceless Ricci soliton if $\rho=\frac{1}{n}$,
\item[iii)] gradient Schouten soliton if $\rho=\frac{1}{2(n-1)}$.
\end{enumerate}
Taking trace of $(\ref{res1})$ we obtain
\begin{equation}\label{res2}
R+\Delta f-n\rho R=n\lambda.
\end{equation}
Thus for gradient Schouten soliton, $(\ref{res2})$ takes the form
\begin{equation}\label{ss1}
R+\Delta f-\frac{nR}{2(n-1)}=n\lambda.
\end{equation}
The diameter estimation of Ricci soliton is an abuzz topic of research. One of the first result came from the work of Myers \cite{MY41}. In particular, he showed that a complete $n$-dimensional Riemannian manifold $(M,g)$ with Ricci curvature satisfying $Ric\geq \lambda g$ for some positive constant $\lambda$ is compact with the diameter $diam(M)$ having an upper bound $\pi \sqrt{(n-1)/\lambda}$. After that many authors have investigated to find a bound for a manifold satisfying some curvature flow conditions, for example see \cite{FG10, FS13, LW14}. For the complete literature in this topic see the survey article \cite{TA19}.
\par The paper is organized as follows: In the section 1, we have deduced a lower bound of the gradient $\rho$-Einstein soliton satisfying some curvature conditions. We have also found some conditions of the gradient $\rho$-Einstein soliton for being non-shrinking and non-expanding. Finally, section 2 deals with the non-parabolic behavior of Schouten soliton.
\section{Diameter estimation}
\begin{thm}\label{resth1}
Let $(M,g,f)$ be a complete non-compact gradient $\rho$-Einstein soliton with bounded Ricci curvature, i.e., $|Ric|\leq c$ for some constant $c$, $\rho R \geq c_1\lambda$ for some constant $c_1$ and $\lim\limits_{s_0\rightarrow \infty}\int_{0}^{s_0}\nabla^2 f(X,X)$ is finite. Then the $\rho$-Einstein soliton is non-shrinking if $(1+c_1)>0$ and non-expanding if $(1+c_1)<0$.
\end{thm}
\begin{proof}
Let us consider a length minimizing normal geodesic $\gamma: [0,s_0]\rightarrow M$ for some positive, arbitrarily large $s_0$. Take $p=\gamma(0)$ and $X(s)=\gamma'(s)$ for $s>0$. Then $X$ is the unit tangent vector along $\gamma$. Now integrating $(\ref{res1})$ along $\gamma$, we get
\begin{eqnarray}\label{r6}
\int_{0}^{s_0}Ric(X,X)&=& \int_{0}^{s_0}(\lambda+\rho R) g(X,X)-\int_{0}^{s_0}\nabla^2 f(X,X)\nonumber \\
&\geq& \lambda(1+c_1) s_0-\int_{0}^{s_0}\nabla^2 f(X,X).
\end{eqnarray}
Again, the second variation of arc length implies that
\begin{equation}
\int_{0}^{s_0}\psi^2Ric(X,X)\leq (n-1)\int_{0}^{s_0}|\psi'(s)|^2 ds,
\end{equation}
for every non-negative function $\psi$ defined on $[0,s_0]$ with $\psi(0)=\psi(s_0)=0$. We now choose the function $\psi$ as follows:
\[\psi(s)= \begin{cases}
s & s\in[0,1] \\
1 & s\in[1,s_0-1] \\
s_0-s & s\in[s_0-1,s_0].
\end{cases}
\]
Then, we have
\begin{eqnarray}\label{r2} 2(n-1)+\sup_{B(p,1)}|Ric|+\sup_{B(\gamma(s_0),1)}|Ric|&\geq & (n-1)\int_{0}^{s_0}|\psi'(s)|^2 ds+\int_{0}^{s_0}(1-\psi^2)Ric(X,X)ds\nonumber\\
&\geq & \int_{0}^{s_0}\psi^2 Ric(X,X)ds+\int_{0}^{s_0}(1-\psi^2)Ric(X,X)ds\nonumber\\
&=&\int_{0}^{s_0}Ric(X,X)ds .
\end{eqnarray}
Combining (\ref{r6}) and (\ref{r2}), we obtain
\begin{eqnarray}\label{r3}
\lambda(1+c_1) s_0-\int_{0}^{s_0}\nabla^2 f(X,X)&\leq & 2(n-1)+\sup_{B(p,1)}|Ric|+\sup_{B(\gamma(s_0),1)}|Ric|\nonumber \\
&\leq& 2(n-1)+2 c.
\end{eqnarray}
Therefore, taking limit as $s_0\rightarrow \infty$ on both sides of (\ref{r3}), we can write
\begin{equation}\label{r4}
\lim\limits_{s_0\rightarrow\infty}\lambda(1+c_1) s_0-\lim\limits_{s_0\rightarrow\infty}\int_{0}^{s_0}\nabla^2 f(X,X)\leq 2(n-1)+2 c.
\end{equation}
Now since $\lim\limits_{s_0\rightarrow\infty}\int_{0}^{s_0}\nabla^2 f(X,X)$ is finite, hence, if $(1+c_1)>0$ and $\lambda>0$, then $\lim\limits_{s_0\rightarrow\infty}\lambda(1+c_1) s_0=+\infty$, which contradicts the inequality (\ref{r4}). Thus $\lambda\leq 0$, i.e., the $\rho$-Einstein soliton is non-shrinking. In a similar way we can show that if $(1+c_1)<0$ then $\lambda\geq 0$, i.e., the $\rho$-Einstein soliton is non-expanding.
\end{proof}
We know that for a non-trivial concave function $f\in C^{\infty}(M)$, the function $(-f)$ is non-constant convex, also it implies that $M$ is non-compact and $\lim\limits_{s_0\rightarrow\infty}\int_{0}^{s_0}\nabla^2 f(X,X)\leq 0$. Thus from the above Theorem \ref{resth1} we can write the following corollary:
\begin{cor}
Let $(M,g,f)$ be a complete gradient $\rho$-Einstein soliton with bounded Ricci curvature, i.e., $|Ric|\leq c$ for some constant $c$, $\rho R \geq c_1\lambda$ for some constant $c_1$ and $f$ is a non-constant concave function. Then the $\rho$-Einstein soliton is non-shrinking if $(1+c_1)>0$ and non-expanding if $(1+c_1)<0$.
\end{cor}
\begin{thm}\label{th3}
Let $(M,g,f)$ be a compact gradient $\rho$-Einstein soliton with $c_2 g\leq Ric \leq c_3 g$. Then for $\rho>0$,
$$ diam(M)\geq max \Big\{\sqrt{\frac{2(f_{max}-f_{min})}{\lambda+n\rho c_3-c_2}}, \sqrt{\frac{2(f_{max}-f_{min})}{c_3-\lambda-n\rho c_2}}, \sqrt{\frac{8(f_{max}-f_{min})}{(n\rho+1)(c_3-c_2)}}\Big\},$$
and for $\rho<0,$
$$ diam(M)\geq max \Big\{\sqrt{\frac{2(f_{max}-f_{min})}{\lambda+n\rho c_2-c_2}}, \sqrt{\frac{2(f_{max}-f_{min})}{c_3-\lambda-n\rho c_3}}, \sqrt{\frac{8(f_{max}-f_{min})}{(n\rho-1)(c_2-c_3)}}\Big\},$$
where the numbers $c_2$, $c_3$ are denoted by
\begin{eqnarray*}
c_2=inf_{x\in M}\{Ric(v,v):v\in T_x M, g(v,v)=1\},\\
c_3=sup_{x\in M}\{Ric(v,v):v\in T_x M, g(v,v)=1\}.
\end{eqnarray*}
\end{thm}
\begin{proof}
Taking trace of $c_2 g\leq Ric \leq c_3 g$, we obtain
\begin{equation}\label{eq1}
nc_2\leq R \leq n c_3.
\end{equation}
As $\rho>0$, the above inequality yields
\begin{equation}\label{e7}
n\rho c_2\leq \rho R \leq n\rho c_3.
\end{equation}
The potential function $f$ has at least one point $p$ where it attains its global minimum value, as $M$ is compact. Let $\gamma$ be a geodesic with $\gamma
(0)=p$. Then using $(\ref{res1})$ and $(\ref{e7})$ we calculate
\begin{eqnarray}\label{e1}
\nonumber g(\nabla f,\gamma')(\gamma (s))&=& g(\nabla f,\gamma')(\gamma (s))-g(\nabla f,\gamma')(\gamma (0))\\
\nonumber &=& \int_{0}^{s} \frac{\partial}{\partial s}g(\nabla f,\gamma')(\gamma (s)) ds\\
\nonumber &=&\int_{0}^{s} \nabla _{\gamma'}g(\nabla f,\gamma')(\gamma (s)) ds\\ \nonumber&=&\int_{0}^{s} \nabla^2 f(\gamma',\gamma')(\gamma (s)) ds\\
&\leq&(\lambda+n\rho c_3-c_2 )s.
\end{eqnarray}
Integrating $(\ref{e1})$ we get
$$ f(\gamma(s))-f(p)\leq \frac{(\lambda+n\rho c_3-c_2 )s^2}{2}.$$
Since for every point $x\in M$ there exists a minimizing geodesic joining $p$ and $x$, for all $x\in M$ we have
\begin{equation}\label{e3}
f(x)-f(p)\leq \Big( \frac{\lambda+n\rho c_3-c_2}{2}\Big)d^2(x,p),
\end{equation}
where $d(x,p)$ is the distance between $x$ and $p$.\\
In particular, we obtain
$$f_{max}-f_{min}\leq \Big( \frac{\lambda+n\rho c_3-c_2}{2}\Big)d^2,$$
where $d=diam(M)$, is the diameter of the manifold $M$. This gives
$$d^2\geq \Big(\frac{2(f_{max}-f_{min})}{\lambda+n\rho c_3-c_2}\Big).$$
Now we consider a point $q$ at which $f$ attains its global maximum. Let $\gamma$ be a geodesic with $\gamma(0)=q$. Then
\begin{eqnarray}\label{e2}
\nonumber g(\nabla f,\gamma')(\gamma (s))&=& g(\nabla f,\gamma')(\gamma (s))-g(\nabla f,\gamma')(\gamma (0))\\
\nonumber &=& \int_{0}^{s} \frac{\partial}{\partial s}g(\nabla f,\gamma')(\gamma (s)) ds\\
\nonumber &=&\int_{0}^{s} \nabla _{\gamma'}g(\nabla f,\gamma')(\gamma (s)) ds\\ \nonumber&=&\int_{0}^{s} \nabla^2 f(\gamma',\gamma')(\gamma (s)) ds\\
&\geq&\Big(\lambda+n\rho c_2-c_3\Big)s.
\end{eqnarray}
Again integrating $(\ref{e2})$ we get
$$ f(\gamma(s))-f(q)\geq \frac{(\lambda+n\rho c_2-c_3)}{2}s^2.$$
Since for every point $x\in M$ there exists a minimizing geodesic joining $q$ and $x$, for all $x\in M$ we have
$$f(x)-f(q)\geq \Big(\frac{\lambda+n\rho c_2-c_3}{2}\Big)d^2(x,q),$$
where $d(x,q)$ is the distance between $x$ and $q$.\\
This implies that
\begin{equation}\label{e4}
f(q)-f(x)\leq \Big(\frac{c_3-\lambda-n\rho c_2}{2}\Big)d^2(x,q).
\end{equation}
In particular, we obtain
$$f_{max}-f_{min}\leq \Big(\frac{c_3-\lambda-n\rho c_2}{2}\Big)d^2,$$
which yields
$$d^2\geq \Big(\frac{2(f_{max}-f_{min})}{c_3-\lambda-n\rho c_2}\Big).$$
Finally, adding $(\ref{e3})$ and $(\ref{e4})$ for $x$ such that $d(x,p)=d(x,q)\leq\frac{d}{2}$, we get
\begin{eqnarray*}
f(q)-f(p)&\leq& \Big(\frac{c_3-\lambda-n\rho c_2}{2}\Big)d^2(x,q)+\Big( \frac{\lambda+n\rho c_3-c_2}{2}\Big)d^2(x,p)\\
&\leq& \frac{(n\rho+1)(c_3-c_2)}{8}d^2.
\end{eqnarray*}
This implies
$$d^2\geq \frac{8(f_{max}-f_{min})}{(n\rho+1)(c_3-c_2)}.$$
This proves the first part. For the second part, $\rho<0$, the equation (\ref{eq1}) implies that
\begin{equation}\label{e5}
n\rho c_3\leq \rho R \leq n\rho c_2,
\end{equation}
and hence proceeding in a similar way as in the first case, we obtain the second part.
\end{proof}
\section{Schouten solitons}
A Riemannian manifold $M$ is parabolic if every subharmonic function $u$ on $M$ with $u^*=sup_M u <\infty$, must be constant \cite{AG1999,GX2019}, equivalently, if every positive superharmonic function $u$ on $M$ is constant. Otherwise $M$ is said to be non-parabolic. The Green function $G(x,y)$ on $M$ is defined by (see, \cite{AG1999})
$$G(x,y)=\frac{1}{2}\int_{0}^{\infty}k(t,x,y)dt,$$
where $k(t,x,y)$ is the heat kernel of $M$.
If $p$ is a fixed point on $M$ and $M$ is non-parabolic then there is a unique, minimal, positive Green function and is denoted by $G(p,x)$. The function $l(x)$ is defined by $l(x)=[n(n-2)\omega_n\cdot G(p,x)]^{\frac{1}{2-n}}$, where $\omega_n$ is the volume of the unit ball in the $n$ dimensional Euclidean space $\mathbb{R}^n $. Also the asymptotic volume ratio of $M$ is defined as $V_M=\lim\limits_{r\rightarrow \infty}\frac{Vol(B_r(p))}{\omega_nr^n}$, where $B_r(p)$ is the open ball with radius $r$ and center at $p$. For more details see, \cite{GX2019} and also references therein.
\par In this section first we state one theorem and two lemmas from \cite{VB2021} and \cite{GX2019}, which will be used to prove our results:
\begin{thm}\cite{VB2021}\label{sth1}
Let $(M, g, f)$ be a complete non-compact non-steady Schouten soliton such that the potential function $f$ is not-constant. Then for $\lambda >0$ (resp., $\lambda <0$), $f$ attains a global minimum (resp., maximum) and also $f$ is unbounded above (resp., below). Furthermore,
$$0\leq \lambda R\leq 2(n-1)\lambda^2,$$
$$2\lambda(f-f_0)\leq|\nabla f|^2\leq 4\lambda(f-f_0),$$
with $f_0=min_{p\in M} f(p)$, if $\lambda>0$ (resp., $f_0=max_{p\in M} f(p)$, if $\lambda <0$).
\end{thm}
\begin{lem}\cite{GX2019}\label{gx1}
If $(M,g)$ is an n-dimensional complete and non-compact Riemannian manifold such that it is not-parabolic with non-negative Ricci curvature, then $$\lim\limits_{r\rightarrow\infty}\frac{\int_{l\leq r}|\nabla l|^3}{r^n}=(V_M)^{\frac{1}{n-2}}\omega_n.$$
\end{lem}
\begin{lem}\cite{GX2019}\label{gx2}
If $(M,g)$ is an $n(\geq 3)$-dimensional complete Riemannian manifold with non-negative Ricci curvature and the volume growth is maximal (resp., not maximal), then $|\nabla l|\leq 1$ (resp., $\lim\limits_{r\rightarrow \infty}\sup_{t(x)=r}|\nabla l|(x)=0$).
\end{lem}
\begin{thm}\label{th1}
Let $(M,g, f)$ be a complete non-compact gradient shrinking Schouten soliton of dimension $n(> 4)$ with $R\leq k<\frac{(n-1)(n-4)}{n-2}$ for some real constant $k$ and the potential function $f$ is positive non-constant. Then all the ends of $M$ are non-parabolic.
\end{thm}
\begin{proof}
For $a=\frac{n-4}{4}-\frac{k(n-2)}{4(n-1)}>0$, using the equation $(\ref{ss1})$ and the Theorem $\ref{sth1}$ we calculate
\begin{eqnarray}
\nonumber\Delta f^{-a}&=&-af^{-a-1}\Delta f +a(a+1)f^{-a-2}|\nabla f|^2\\
\nonumber&\leq& -a\Big\{\frac{n}{2}+\frac{nR}{2(n-1)}-R\Big\}f^{-a-1}+a(a+1)\{2(f-f_0)\}f^{-a-2}\\
\nonumber&=& \Big\{-a\Big(\frac{n}{2}+\frac{nR}{2(n-1)}-R\Big)+2a(a+1)\Big\}f^{-a-1}-2a(a+1)f_0f^{-a-2}\\
\nonumber&\leq& \Big\{-a\Big(\frac{n}{2}+\frac{nR}{2(n-1)}-R\Big)+2a(a+1)\Big\}f^{-a-1}\\
\nonumber&\leq& a\Big\{\frac{k(n-2)}{2(n-1)}-\frac{n}{2}+2(a+1)\Big\}f^{-a-1}
=0.
\end{eqnarray}
Hence it follows that $f^{-a}$ is a positive superharmonic function which converges to zero at infinity. This proves that (see, \cite{AG1999}) any end of $M$ and hence $M$ is non-parabolic.
\end{proof}
The following corollaries immediately follows from Lemma $\ref{gx1}$, Lemma $\ref{gx2}$ and Theorem $\ref{th1}$:
\begin{cor}
Let $(M, g, f)$ be a complete non-compact gradient shrinking Schouten soliton of dimension $n(> 4)$ with $R\leq k<\frac{(n-1)(n-4)}{n-2}$ for some real constant $k$, $Ric\geq 0$ and the potential function $f$ is non-constant with $min_{p\in M} f(p)=f_0\geq 0$. Then the following relation holds: $$\lim\limits_{r\rightarrow\infty}\frac{\int_{l\leq r}|\nabla l|^3}{r^n}=(V_M)^{\frac{1}{n-2}}\omega_n.$$
\end{cor}
\begin{cor}
Let $(M, g, f)$ be a complete non-compact gradient shrinking Schouten soliton of dimension $n(> 4)$ with not maximal volume growth, $R\leq k<\frac{(n-1)(n-4)}{n-2}$ for some real constant $k$, $Ric\geq 0$ and the potential function $f$ is non-constant with $min_{p\in M} f(p)=f_0\geq 0$. Then $\lim\limits_{r\rightarrow \infty}\sup_{t(x)=r}|\nabla l|(x)=0$. Furthermore, if it has maximal volume growth, then $|\nabla l|\leq 1$.
\end{cor}
\begin{thm}
Let $(M,g,f)$ be a complete non-compact gradient expanding Schouten soliton with $-(n-1)<k_1\leq R$ for some real constant $k_1$ and the potential function $f$ is positive non-constant with $f^{-b}$ bounded above. Then all the ends of $M$ are non-parabolic.
\end{thm}
\begin{proof}
For $b=\frac{n-2}{2}+\frac{k_1(n-2)}{2(n-1)}>0$, using the equation $(\ref{ss1})$ and the Theorem $\ref{sth1}$ we calculate
\begin{eqnarray}
\nonumber\Delta f^{-b}&=&-bf^{-b-1}\Delta f +b(b+1)f^{-b-2}|\nabla f|^2\\
\nonumber&\geq& -b\Big\{-\frac{n}{2}+\frac{nR}{2(n-1)}-R\Big\}f^{-b-1}+b(b+1)\{(f_0-f)\}f^{-b-2}\\
\nonumber&=& \Big\{-b\Big(-\frac{n}{2}-\frac{(n-2)R}{2(n-1)}\Big)-b(b+1)\Big\}f^{-b-1}+b(b+1)f_0f^{-b-2}\\
\nonumber&\geq&\Big\{-b\Big(-\frac{n}{2}-\frac{(n-2)R}{2(n-1)}\Big)-b(b+1)\Big\}f^{-b-1} \\
\nonumber&\geq& b\Big\{\frac{k_1(n-2)}{2(n-1)}+\frac{n}{2}-(b+1)\Big\}f^{-b-1}
=0.
\end{eqnarray}
Hence it follows that $f^{-b}$ is a subharmonic function which is bounded above. This proves that (see, \cite{AG1999}) any end of $M$ and hence $M$ is non-parabolic.
\end{proof}
\section{acknowledgment}
The second author gratefully acknowledges to the
CSIR(File No.:09/025(0282)/2019-EMR-I), Govt. of India for financial assistance.
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Q: UITextField custom emoticons ( icons ) I have a little problem: I need to use custom emoticons in uitextfield, as I know - iOS emotions are unicode characters. To write system emoticon I just need to print something like \ue413 and it will become into the image. How I can do this with custom icons?
A: You will need to insert an image into the UITextField, which reduces this question to be the same as: Adding Images to UITextView
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"Was That A Gun?" Coming of Age with Car Seat Headrest's Latest Album
Kieran Cleary | August 5, 2020
Topics: Car Seat Headrest, Making A Door Less Open, protests, record reviews, Virginia bands
Making A Door Less Open, the 12th studio album in ten years from Virginia-born indie group Car Seat Headrest, is an apt expression of the anxieties and alienations of American life in 2020.
The May 2020 album from indie rock group Car Seat Headrest, entitled Making a Door Less Open, came out in time for me to pop in the CD and drive down to Monument Avenue with a family member on Father's Day. From the first drawn-out, Eastern notes to the brutally lucid introduction, the music cut loose and blew us down the highway like a trade wind. The mood suited my vision of the world around me as I took in hundreds of expressions of protest.
Like those protests, this album attempts to create a feeling of "many contrasting voices." Travelling got a little unsteady by track nine, "Life Worth Missing," with sharp guitar and rushing, bounding, endless synth — but that uncertainty was immediate too, as I looked to the gravity of the day to keep myself and everyone else safe in a charged setting.
Car Seat Headrest is a band that was founded by talented songwriter/singer/guitarist Will Toledo, who grew up in Leesburg, VA and spent the group's early years attending college at VCU and William & Mary. He's worked with a variety of musicians since 2010, and these days, drummer Andy Katz, guitarist Ethan Ives, and bassist Seth Dalby make up the core of the group, a lineup that has remained stable for the past five years. Making A Door Less Open is the group's fourth album to be released on New York indie label Matador Records.
Making a Door Less Open by Car Seat Headrest
Making a Door Less Open is a record made up of strange new realms of thought and travel along a concept of disembarking, traveling, and returning home. Toledo's nickname for the album, Madlo, reminds me of E.B. White's Stuart Little. In the children's book, Stuart pursues "Margalo" the bird into the unknown, even when he is presented with a chance to settle down.
My favorite sounds woven through Madlo are the Eastern, woodwind-ey notes that introduce the album, on "Weightlifters." I'm waking up in a mirage, a strange, new place. My imagination is alive and weightless, and the lyrics push against that feeling with cutting realism. Watery, rain synth sounds meet the heat, which weaves in and out of both "Deadlines (Hostile)" and "Deadlines (Thoughtful)," become Western-cinematic in "Hollywood," and build up to "Life Worth Missing" (a total home run). Then on "There Must Be More Than Blood," soothing flutes in a more familiar key arrive, playing against the Eastern tones and bringing the listener home — maybe from tour.
Ives' scattering, sharp guitar leads lurk at the end of "Hymn" and shine in "Deadlines (Thoughtful)," and I think that's what we hear at the beginning of "There Must Be More Than Blood." Sometimes it's hard to untangle guitar from synth. Innovation with technology has given the band freedom to explore new worlds of music.
Listening in for big ideas – of "lines, nothin' but outlines," and the experience of a songwriter who questions what he's made of. Of deadlines, pressures of production and performance. The urgency and immediacy of the lyrics inspire gratitude from fans who feel kinship and solace in the speaker's struggle to "get connected" with others.
Photo via Car Seat Headrest/Facebook
Toledo imagines his participation in "some sort of war." I wouldn't go so far as to say it's a war of indie-pendance. But maybe as a young person, and a citizen of the world, the compulsion to work at his craft and perform it, carrying forth his own cause and sympathizing with the many people he has come into contact with on tour, causes him to take a very artistic and broad view of "war."
For young people, the political climate in our country had become more constantly in our minds, even before the coronavirus pandemic arrived. The frequency of mass shootings in our country started making us think twice in big crowds. During our generation's maturation, domestic issues like public health, fights for human rights during the Arab Spring, and the fight for Black Lives here in America are some of my big topics of concern.
And this album lends a voice of opinion. "Was that a gun?" Riding the bus in "Hollywood," a speaker wrapped up in disgust hears snapping gum and fears that it's actually a firearm. Phrases turn quickly, in the same way I imagine a situation involving guns might go from bad to worse. The song depicts a world where phrases like ads on a bus are a quickly overturning commodity based on the reactions of a distracted, massive audience. The artist fears that the judgments of their publishers and producers — who at least seem more powerful than audiences — will depend on the willingness of the artists to abandon their home, their history. I feel frozen in contemplation of this situation.
More concretely, I love the exclamation of "stinks!" in this song. It feels punk and rebellious against the toxic allure of Hollywood fame. Sound effects like car horns arrive suddenly and to make me jump. Other people at the stoplight must wonder what's the matter. It's engrossing!
More than once during Madlo, songs captivated me in the same manner as the Beatles' classic Sgt. Pepper's Lonely Hearts Club Band. Images of "smoke pouring out the bed" in "Can't Cool Me Down" depict struggles coping with reality, and strange visions like stage performance invade the artist's room. The imagery is reminiscent of the Beatles' "Fixing a Hole," where they are fixing a hole and painting their room to keep their heads together. Then, too, the admitted domestic abuse on the Beatles' "Getting Better" plays out on this album in an outburst at the end of "Deadlines (Thoughtful)." Both songs have humiliating shock value.
Photo by Carlos Cruz
Even the way both albums end is similar, dissolving in both cases into experimental sound on the final tracks. On "A Day In The Life," John Lennon sings that "The English army had just won the war/ and though the news was rather sad…" There's emotion lacking there. The song feels disconnected with violence. Of Making a Door Less Open, Rolling Stone's J Blinstein observes that there is a "quotidian feel to it, a mundanity that fits the understated hum of Toledo's singing voice, which he's able to use in thrilling and unexpected ways." I think both albums look within, at their own responses to daily life and growing older, and try to both relate with war and to disarm it.
Fave Tracks: "Weightlifters" is dazed and cool, leaves me singing the words "I believe," after the song ends, and also thinking of Janis Joplin's "Down on Me." "Martin" has great mixing, cuts, and is fun and upbeat. The bass and vocals on "Deadlines (Hostile)" pair up nicely, and there's a cool rock refrain. "Hollywood" is good for really going long, and being something for the public to decide upon. "Deadlines (Thoughtful)" has killer riffs, rolling rhythm, and a good tempo. "Life Worth Missing" loses some candidness from the bonus acoustic version, but gains in succinct form. Finally, "There Must Be More Than Blood" offers beautiful, reassuring chords, along with reflections on friendship, family, and self.
This is Car Seat Headrest's twelfth album overall. They just keep getting better.
Top Photo by Mikeal Beland, via Car Seat Headrest/Facebook
RVA #31: Record Reviews
RVA Staff | January 10, 2018
Topics: Antiphons, Big No, black liquid, Butcher Brown, Buzzard Dust, Citrrus City, Doll Baby, Fly Anakin, Gold Goldin & Duce, Gritter, Keep, Koncept Jack$on, Long Arms, Mistaker, My Enemies & I, Positive No, record reviews, River Black, rva music, Talk Me Off, Thorp Jenson, tim barry, Vivian Fantasy
Originally printed in RVA #31 WINTER 2017, you can check out the issue HERE or pick it up around Richmond now.
Tim Barry
High On 95
(Chunksaah)
Punk rock is the kind of scene that wears a person out. This might sound ridiculous to the kids singing along to "Young Til I Die" covers at the all-ages show, but by the time you're 29 with two full sleeves of tattoos, sitting at the end of the bar because you don't have the energy for the pit anymore, you learn the truth. Perhaps it's not too surprising that Tim Barry, who once led 90s punk heroes Avail, is a decade deep into a solo career as a folk-country artist and shows no signs of looking back. High On 95 is his sixth solo album, and it shows him settling ever more comfortably into the acoustic troubadour role. He only sang in Avail, but his proficiency on acoustic guitar here sees him creating some excellent melodies on songs like "Gumshoe Andy" and "O & DP." The minimal instrumental backing (slide guitar, violin, the occasional percussion) gives these tunes a rootsy feel and allows you to crank up the volume without bugging the neighbors. The DIY veterans will see the appeal immediately. The kids might not get it yet, but rest assured, their time will come soon enough. (MN)
Ann Beretta
Old Scars, New Blood
(Say-10)
Collections of old songs re-recorded can be a risky move–it may do little more than spotlight the fact that the now-middle-aged members have lost a step. That said, Ann Beretta is lucky enough to retain a vitality that makes the past 20 years seem like the blink of an eye. This one's worth it even if you have the old records. (MN)
Antiphons
(Citrus City)
Fine by Antiphons
Fine zooms in and out in an incredible way. The EP's lyrics focus on super-specific circumstances — burning your tongue, going into anaphylactic shock because of a nut allergy — yet the music gradually opens up via savvy guitar work and countermelody, resulting in big, inviting moments. In that way, Fine manages to be personal and universal at the same time. (DJ)
Ashanti Bragg
Journey of a Young Woman Vol. 1
(Datpiff)
With seven tracks giving us a good sample of what's to come, Virginia Beach native Ashanti Bragg makes her debut almost two years after dropping the video for opener "My Love." Featuring a variety of songs showing her musical versatility, the confidence that oozes through her music makes for a fun and energetic experience that leaves the listener wanting more. (KMP)
Big No
(bignobigno.bandcamp.com)
GET OVER YOURSELF by BIG NO
Presenting itself as an ambitious and experimental project, Get Over Yourself is vast. One second, the structured sound is airy and poppy, and then the next, it begins to wax into blues. Big No has made it obvious that their strongest musical asset is their instrumentation; Heather Jerabeck owns that piano. (CM)
Black Liquid
(self-released)
Two words: Unfiltered. Observation. Black Liquid's obstinate attitude and sharp public commentary on ANTi challenge local perception and opinion through a conversational approach. His poignant flow found on the title track "ANTi" is relentless, barraging the listener with anecdotes that highlight the MC's natural inclination for hip-hop as an art form. (CM)
Butcher Brown
Live at Vagabond
(Gearbox Records)
Make room, Donny Hathaway–this is one for the ages. Live at Vagabond captures the energy of the crowd and the virtuosity of individual instrumentalists with remarkable clarity, giving listeners a taste of Devonne Harris' compositional gifts, adventurous approach to keys, and the ensemble's knack for seizing the moment. This is the Butcher Brown sound at its most cohesive and dominant. (DJ)
Buzzard Dust
(Forcefield)
Buzzard Dust by Buzzard Dust
What happens when a ragtag group of metal veterans wants to rock? They form a thrash band. Buzzard Dust's eponymous debut is a 24-minute adrenaline burst of wicked dive bombs, breakneck blast beats, and guttural profanities that recalls the feeling of a dark, sweaty mosh pit. I dare you to not headbang during "Have You Seen Me." (CM)
Hell Block EP
Hell Block by Doll Baby
Doll Baby blesses listeners with yet another phenomenal demonstration of their artistic prowess. Hell Block is a mere five songs, but holds the sustainability of a traditional full-length album. The four-piece's jam tracks scratch a deeper itch, bringing everyone to their feet. Singer Julie Storey has the most pleasingly unique punk vocals you'll ever hear. (CM)
Fly Anakin, Koncept Jack$on & Tuamie
Panama Plus
(Mutant Academy/Smallz World Management & Consulting)
More hardcore, boom-bap hip hop from the Southside's own Fly Anakin & Koncept Jack$on, this time with fellow Mutant Academy brethren Tuamie handling the production. Hearing the duo spitting over one producer's sound gives this project a different, more cohesive feel from their last full length, but if you're expecting a drop off in quality, you won't find that here. (EH)
God Goldin & Duce
Universal Benefits
(Control Ent)
Bouncing back and forth over each track, Goldin & Duce have a great chemistry, with Duce's laid-back flow paired on "Lolo" along with Gold's more hype flow. The songs feel like these two have been a working partnership for years already. If you need a quick musical boost in your workday, give this a spin. (EH)
Gritter
Nobody Cares
Nobody Cares by Gritter
Gritter isn't necessarily reinventing the (steel) wheel on their fourth album, but then again, if it ain't broke, why fix it? Their brand of harsh, caustic metal has some clear NOLA influences but gives it their own muddy James River flavor. This music will give you strength to face life's frustrations. Don't leave home without it. (MN)
Thorp Jenson
(South Boulevard)
Fully-formed narrative writing, steady-handed production, and killer performances from top-notch players provide many reasons to disbelieve that Odessa is a debut release. It plays like an expertly crafted survey of styles from the last 60 years, from Stones riffs and heartland rock to country waltzing and soul not unlike Matthew E. White's. Well-worn and world-class, right out of the gate. (DJ)
For Your Joy
For Your Joy by Keep
With this monster of a debut full-length, the duo Keep has etched its name among the city's growing list of musicians on the rise. The group's sound is diverse and evocative, being deeply rooted in their appreciation of grunge and industrial predecessors. For Your Joy embellishes an introspective atmosphere that lets one track roll right into the next. (CM)
Long Arms
Young Life
(Dead Serious)
Young Life by Long Arms
Long Arms began as James Menefee's alt-country project, but with their latest album, they've left those touches behind in favor of the punk-influenced heartland rock that feels like Menefee's natural mode. It suits him; skipping genre tropes in favor of excellent heartfelt tunes with a heavy Replacements influence makes Young Life is a career highlight. (MN)
Mistaker
Goodbye And Other Lies
Modern Massachusetts by MISTAKER
An intense, heartfelt slab of pure emotion delivered with power and melody, Goodbye And Other Lies is a worthy contribution to the field of melodic punk rock from a group of veterans who've paid plenty of dues. This is music for remembering past struggles and appreciating where you are. With this EP, Mistaker carves out a distinctive niche for themselves. (MN)
My Enemies & I
The Beast Inside
(Fearless)
This VA-based metalcore crew draws a lot of influence from angst-ridden early 00's nu-metallers like Slipknot and Mudvayne, interjecting melodic choruses and moody breaks into a stew of pounding downtuned riffage and brutal breakdowns. The result is an invigorating, gleefully profane blast that brings me back to my youth. If this is what today's kids are into, sign me up. (MN)
Positive No
Partners In The Wild
(Trrrash/Little Black Cloud)
Partners in the Wild by Positive No
This RVA quartet definitely brought the fire this time, cranking up the energy to deliver a louder, more distorted follow-up to debut LP Glossa. The 90s alternative and indie-rock influences that fundamentally inform Positive No's sound are still dominant, but their punk past is much closer to the surface here–and that's definitely a good thing. (MN)
River Black
(Season Of Mist)
River Black by River Black
This combo sees Municipal Waste drum-pounder Dave Witte reuniting with his Burnt By The Sun bandmates, John Adubato (guitar) and Mike Olender (vocals), to carry on that band's brutal, politically-driven metal rampage. Doom-infused metallic hardcore riffs meet grinding blast beats and double-bass mayhem to create an unstoppable steamroller of a record. (MN)
Talk Me Off
Talk Me Off EP by Talk Me Off
Some fun, speedy punk with a tendency toward retro stylings. The first song has a borderline-hardcore intensity, but the others get more melodic in a manner reminiscent of early 80s SoCal punk–Agent Orange, first-LP Bad Religion, that kind of thing. The furious anti-white-nationalism lyrics on "Inglorious Bastards" are a particularly nice touch. (MN)
Vivian Fantasy
Deep. Honey.
(Hush Hush)
Deep. Honey. by Vivian Fantasy
Can music be simultaneously comforting and unsettling? Deep. Honey. makes a pretty strong case in the affirmative. Warm synth sounds and layered guitars lay down pillowy sonic padding, yet Danny Bozella's singing is manipulated throughout, coming across as uncanny. "I put effects around my voice to hide what I write," he sings on "Charms," ringing with a beautifully ironic honesty. (DJ)
Reviews by: Marilyn Drew Necci (MN), Eugene Henry (EH), Davy Jones (DJ), Kiara M.P. (KMP), Christopher Alan McDaniel (CM)
Top Photo Credit: Joey Wharton Photography
RVA #29: Record Reviews (Part 1)
RVA Staff | August 15, 2017
Topics: record reviews, rva music
This article was featured in RVAMag #29: Summer 2017. You can read all of issue #29 here or pick it up at local shops around RVA right now.
Blame by Blame
Blistering d-beat hardcore from a band of straight-edge punkers with an enthralling sense of melodicism. There are hints of metal in their songwriting foundation, maybe borrowed from Richmond's vibrant scene, and it makes their songs a bit more engaging and a hundred times more searing, though the appeal is still in the way they build and execute their hooks, whether by shrieking vocals or shattering guitar lines. (DN)
Camp Howard
(Egghunt)
Juice EP by Camp Howard
Though just an EP, Juice shows off the true depth of Camp Howard's songwriting, as the group freely floats in and out of '80s structures and fuzz rock archetypes. Breezy rock is still the name of the game for Camp Howard though, a style that brought them to the forefront of the Richmond music scene last year, but Juice shows that the group has way more to offer within that confine, proving not only their impressive potential, but also the staggering skill they have available right now. (DN)
Dharma Bombs
Old Time Romance
(Crystal Pistol)
Old Time Romance by Dharma Bombs
Ragtime/Dixieland folk music from a plucky band of musicians that can deliver plenty of fun moments ("Abigail") as well as more striking and somber tunes ("Pack Your Bags"). Trey Hall's old-timey yowl alone is worth the price of admission here, with the delightful horn arrangements and hoo-dum rhythms are just icing on the cake. Unlike so many other popular live acts, Dharma Bombs are able to capture the charm and spirit of their amazing performances, all available on this wonderful twelve-track album. (DN)
Fat Spirit
Nihilist Blues
Nihilist Blues by Fat Spirit
The first release in four years from the former Heavy Midgets is a welcome mélange of grunge malaise and power melodies, sprinkled with enough hardcore and shoegaze elements to make it stimulating for all rock fans. The band's ability to cover all tempos and grooves is inspiring, but it's really the slower compositions that reveal the towering power of their ability, as they seem just a tad bit more creative and infinitely captivating. (DN)
Kinda Funny by Flight Club
On one hand, this is straightforward pop punk as the five tracks on this EP could easily be inserted onto any mid-2000s CD-R. On the other, it's clear this plucky band has a lot more up their sleeves, utilizing blues and even doo-wop moments to make their music stand-out. Producer Alan Day of Four Year Strong helps streamlines the sound, but this EP still shows Flight Club off as an incredibly skilled rock band… who just so happens to like pop punk.
Golden Ours
New Faces by Golden Ours
Local musician Kia Cavallaro offers up a special brand of porchtop folk, one that she freely contorts via production techniques both invigorating and astounding. Using delayed vocals at times, she offers a glimpse off a fractured mind, one that mends itself slightly over the runtime of the album. Bouncing between lo-fi and ethereal, the sounds Cavallaro has culled together are simply breathtaking, even if better observed as a musical tumbleweed slowly making its way through the valley. (DN)
Lurcher by GULL
Industrial music on its face seems outdated in 2017, but leave it to a performer like Gull to make it urgent and relevant by injecting it with a jolt of life and his own signature aplomb. While the longer compositions like "The Ancestral Knife" excel at pulling you into his world of Depeche Mode melodies and Skinny Puppy rhythms, it's the shorter tracks like "Trichotillomania" that perfectly capture Gull's essence of creative release that just builds and builds on itself until the climax. (DN)
Kaelan Brown
From Out The Pines
From Out The Pines by Kaelan Brown
A stunning EP that cements Brown as one of the area's most vibrant MCs through seven songs each designed to showcase not just his skill, but his curious charisma. Notable is Brown's ability to keep the flow from track-to-track airtight, as producer Wakeen passes thiongs off to Bince and Julien Earle for a song each. It reveals a solid sense of identity for the rapper, as he's comfortable unfolding lyrics on a lofty wave or bouncing his way through a party anthem. (DN)
Lobo Marino
The Mulberry House
The Mulberry House by Lobo Marino
Above the amazing live shows, the fearless songwriting, the insightful lyrics, the gorgeous instrumentation — above it all, the best thing about Lobo Marino is their ability to reliably grow as artists year to year. 2015's We Hear The Ocean was a remarkable release, but the band has shot right past it with The Mulberry House, a full-length release that covers world music down to acoustic ballads. Enlightening as it is provoking, it's yet another indelibly great mark Lobo Marino has left on Richmond's local scene. (DN)
Sid Kinglsey
Good Way Home
(American Paradox)
Good Way Home by Sid Kingsley
An impeccable release that unwittingly raises the bar for any and all music coming out of Richmond. Soul overflows out of each track, whether from Sid Kingsley's own bellow (that brings to mind the legendary Levon Helm) or the sharp instrumentation brought together by producer Scott Lane, and it's a remarkable step-up from most Americana music released these days. All in all, this is one of, if not the, best Americana release to come out of Richmond, as well as one of 2017's best records overall. (DN)
The Milkstains
Punch The Sky
(Trrrash)
PUNCH THE SKY by The Milkstains
As this trio exudes time and time again, The Milkstains are virtuosos at creating a perfect balance between garage rock, surf jams, and spaghetti western barnburners. Punch The Sky is no different. Over the course of ten tracks, the band are quick to remind everyone that there is really no one quite like them in Richmond and their riley misfit attitudes will always find their spirited homes in poignant lyrical moments. "Young Scum" could easily be the jam of the summer! (SC)
The Weak Days
(Possum Hearts/Running Around)
TIGHT by The Weak Days
Richmond's brightest journeymen return for this bold EP, one that's a remarkable step-up from their previous work. It offers a more matured take on their rock sound, one rooted in DIY spirit and emo concepts, but still showcases the band's signature charm — campy, tongue-in-cheek wit that's endlessly endearing. Overlooking the EP's impressive list of guest musicians, Tight is still just a riveting collection of melodies and hooks, only really bolstered by the strong messages the duo sings about. (DN)
Reviews by Doug Nunnally and Shannon Cleary
RVA Mag #28: Record Reviews (Part 3)
RVA Staff | April 19, 2017
Topics: music, record reviews, rva music
This article was featured in RVAMag #28: Spring 2017. You can read all of issue #28 here or pick it up at local shops around RVA right now. If you missed Part 1 of our record reviews, you can check that out here. If you missed Part 2 of our record reviews, click here.
Jens Lekman
Life Will See You Now
(Secretly Canadian)
Lekman is a bit of a songwriting trickster. At his most depressed, the songs elevate themselves to appear as joyous affairs of boisterous arrangements and fanciful grooves. With this in mind, this record is Lekman at his strongest by balancing the sadness with the whimsical. Highlights include the pop gems "Our First Fight" and "How We Met, The Long Version." It has been far too long, Jens. We are eager to hear you now. (SC)
Julie Byrne
Not Even Happiness (https://f4.bcbits.com/img/0008722844_10.jpg)
(Ba Da Bing)
Not Even Happiness by Julie Byrne
Organic and sincere, wistful and romantic; Byrne's second record is a much more confident affair than her debut with the singer fully embracing the tonal atmosphere she creates. Capturing her restless spirit is key here, with her august voice willing the listener to join her on her travels, both in the real world and within. Though clearly folk by design, the subtle use of synths helps distance herself from other songwriters, leaving Byrne in a class of her own in 2017. (DN)
The World Is A Loud Place
The World Is A Loud Place by Landlady
The album art for The World Is a Loud Place isn't just the first GIF cover released via Bandcamp — it's also a wonderful representation of Landlady's music. Big, colorful, kinetic… songs move and change with such amazing energy. "Electric Abdomen," "Nina," and "Driving In California" all function as multi-act productions that deserve elaborate set designs, and they round out — along with past releases — a vibrant and powerful body of work. (DJ)
Semper Femina
(More Alarming)
The most important record released in the wake of the Women's March, Marlin's latest record is far from a departure from her previous works, but is also much more pointed in its subject matter and over-flowing with existential thoughts and observations. Easily the most attentive album of her catlog, Semper Femina is one of the boldest examinations of feminism and womanhood society has seen in some time, something that comes as no surprise to those who have followed her career. (DN)
Nothing Feels Natural
(Sister Polygon)
Nothing Feels Natural by PRIESTS
The aplomb of Priest's musical spirit is what makes this debut remarkable as the band weaves a needle of punk energy through a tapestry of surf rock and indie pop. Paying close attention to the words of the record and the way the music ebbs and flows reveals the band's personal politics, but the band ultimately leans more towards the Pixies' end than Pussy Riot as they freely frolic in a surreal world swirling with surf and punk elements. (DN)
(PAX AM / Blue Note)
After his bold offering 2015 that exposed him to a completely new fanbase, Adams returns with a record that shrewdly builds on his past work, while still striving for that next sonic jump. Mirroring the album's over, the songs come off as blurry, patchy, almost disjointed at times, but still form a recognizable image of heartbreak and confinement, one that's as affecting as it is brilliant. (DN)
(Young Turks)
The xx have never lacked cohesion, but this feels like a new fusion of powers. Jamie xx's production knowledge and abilities fully burst through on his 2015 solo album, and he's applying that skill set more confidently than ever, complementing and elevating the hypnotic Romy-Oliver chemistry that's defined the band to this point. Singles "Say Something Loving" and "On Hold" are instant xx canon, but "Lips" is a sleeper for one of the year's most gorgeous songs. (DJ)
(Brainfeeder)
Like the title implies, Thundercat's goal here is to intoxicate you with the groove, something a lot of modern soul producers ignore. Listen to classic soul records and you can truly feel the attention given to the groove. It can alter and modulate, but it can't be paused, interrupted, or, worse, dropped. Thundercat realizes this and pays extra attention to it ensuring no matter who is in front of a mic or behind the board, it's a seamless sound from start to finish. (DN)
Reviews by Davy Jones, Doug Nunnally, and Shannon Cleary
Topics: music, national music, record reviews, rva music
This article was featured in RVAMag #28: Spring 2017. You can read all of issue #28 here or pick it up at local shops around RVA right now. If you missed Part 1 of our record reviews, you can check that out here.
Fly Anakin & Koncept Jack$on
Chapel Drive
Chapel Drive by fly anakin & koncept jack$on
Give one listen to Chapel Drive and you'll understand just why the Mutant Academy collective has earned such a strong reputation in hip-hop circles. The first project to feature every member of the collective, there is a ton of creative rhymes and stunning production to digest, so much so that it's daunting at times, but Fly and Koncept pace the sixteen track well and work hard to ensure its laid-back feeling… just like a chapel drive itself. (DN)
The Folly
An extremely eclectic offering from one of the more unique bands in town, with enough musical ideas to tide them over for the next decade it seems. Sounds jump from song to song starting with a bang ("Coal Miner Rebellion") before moving on to a shimmering duet ("Fish Me Out"). Utilizing two strong singers in Anneliese Grant and Jordan Lette helps lift the record up, but it's really the quartet's songwriting ability that lets this EP soar. (DN)
Opin
Opin – S/T by egghunt records
Landis Wine and Tori Hovater have hit the ground running with this debut. It's an intense, emotionally affecting collection — "Do you really want to die sometimes?" — and while the album drives deeper into the synth territory White Laces had explored, the ache in Wine's voice and sporadic saxophone appearances fill these songs with a striking sense of humanity. From its loudest moments to its most reflective, Opin feels alive, with all the passion and variety of life itself. (DJ)
Saw Black
Azalea Days
Azalea Days by Saw Black
Throughout Azalea Days, Saw Black makes one convincing argument after another for why he is one of the strongest songwriters in town. As the record contemplates a relationship unraveling, Black looks at the universe internally and externally as he seeks answers and resolve in each of the eleven tracks featured. Even at the most disparaging moments, there are still moments of hope to be found from a perennial love song to an ode to everyone's favorite pastime. (SC)
Tourist In This Town
(Merge)
Tourist in This Town by Allison Crutchfield
On her proper solo debut, Allison Crutchfield seems more vulnerable than ever. While spinning tales of anxiety and embracing change, this new side of the songwriter is as welcome as ever. "Dean's Room" is a quick song to highlight on the stellar Tourist In This Town. If you were in need of a reflective collection of intimate confessionals in the form of nuanced songwriting, look no further than Crutchfield's latest. (SC)
Chris Thile & Brad Mehldau
(Nonesuch)
When they're performing music written by other people, Chris Thile & Brad Mehldau are lossless interpreters. Their instruments — the totality of what they're capable of between their playing and Thile's singing — are borderline perfect, and it's thrilling to see where they direct that exquisitely appointed machine. Elliott Smith. Bob Dylan. Gillian Welch. Their own compositions. This album feels like an incredibly thoughtful, neatly wrapped gift, both for listeners and for the featured songwriters. (DJ)
Oberst has been known as a strong songwriter for well over two decades now and Salutations, his 22nd album since 1993, only further proves the point, legitimizing himself in the upper echelons of modern music composers. Though it lacks the focus of his previous work, 2016's Ruminations, the Dylan-esque melodies and Americana sounds come together beautifully, while the lyrics offer a more shrouded lens with which to listen through. (DN)
Foxygen
(Jagjaguwar)
On their fourth record, the duo leans on Spacebomb to overcome the shortcomings of 2014's …And Star Power and delivers a more focused record that imperfectly imposes over the band's catalog, even if it is a far cry from 2013's We Are The 21st Century Ambassadors Of Peace & Magic. Richmond fans will love the numerous local credits on the release, while music fans in general will love the direction Sam France has willed Foxygen in. (DN)
Topics: community, record reviews
This article was featured in RVAMag #28: Spring 2017. You can read all of issue #26 here or pick it up at local shops around RVA right now.
Aerica Lauren
If You Are Home, Welcome Home
If You Are Home, Welcome Home. by Aerica Lauren
Heart-wrenching songwriting by a sincere, gutsy, and ambitious artist who quickly asserts herself in a town known for strong female songwriters. Lauren uses the space of her raw recordings to create tangible depth within her songs, allowing the harmonies to shine bright and the emotions to cut deep. Unexpectedly, the release leans towards bedroom pop at times, but one wrought with conflict, setback, and introspection that makes for a striking collection. (DN)
Groan by Antiphons
As Groan takes off, the listener should immediately be aware that they are about to be treated to one of the strongest debuts from a Richmond artist. Antiphons find a proper balance between the whimsy of artists like Fleet Foxes and fury of Dinosaur Jr. As Brian Dove croons throughout, a record detailing the travels throughout the continental United States and a romance faltering feels poignant and enchanting. An early, but strong frontrunner for record of the year. (SC)
Book Of Wyrms
Sci-fi/Fantasy
(Twin Earth)
Sci-fi/Fantasy by Book of Wyrms
A riveting debut record from a quintet strongly asserting themselves as one of Richmond's best hard rockers. Dense guitar lines engage in a shouting match with the tight rhythm section, while Sarah Moore-Lindsey's siren voice provides clarity and direction to each song. Naturally, there are plenty of epic moments provided by guitarists Ben Courdiet and Kyle Lewis, but this release really soars on its ability to reign in the songs and deliver succinct, yet blistering compositions. (DN)
Cherry Pits
Splatterday Nite
Splatterday Nite by Cherry Pits
People recoiling at the shameless album cover will be surprised to find the record is actually quite accessible for those interested in garage and powerpop sounds. The songs fly fast and loose, though the band is in full control, expertly guiding it through the brash rollercoaster they've meticulously structured. With a great live feel to each song, the band's delivered a great taste of their live shows that's earning the band a strong reputation in and around Richmond. (DN)
Dazeases
Local Slut
Dazeases – Local Slut by egghunt records
It's easy to see why Dazeases has become a favorite in town with each and every dystopian electronic foray into indie pop hitting its lofty mark on C R U M B S. Each tune languishes on tearing apart anything that remains taboo to discuss in how we treat and relate to one another while developing a harsh realm of loops, grooves and noises that fit well with each relative concept. Dazeases quickly engages audiences with the provocative and intensifying sounds to be found throughout C R U M B S. (SC)
Don Babylon
(Trrrash / Medical)
Babe by Don Babylon
A winding road that leaves the listener wondering if they somehow landed on another record at times, Babe is a testament to the trio's songwriting ability and attention for structure. Though rock for most of the record, the band flirts with other styles — maudlin, country, and post-punk — in a way that helps bookend sections of the record and allow strong compositions like "There Will Be Blood" to really shine through and leave a lasting impression. (DN)
Evan McKeel
A touching collection of songs from a young voice with plenty to express. The songs flow easy across, with gorgeous melodies and picture-perfect instrumentation, but what gets lost is how ambitious the record is at time, such as the expansive opening number. The album is guided by a patient, almost veteran hand that realizes there's no rush in establishing the song, and takes care to ensure each song gets to where it is going by any means. (DN)
Flo Morrissey & Matthew E. White
Gentlewoman, Ruby Man
(Glassnote)
I will undoubtedly count first hearing Flo sing the chorus of "Thinking Bout You" as one of my happiest moments in recent memory. It's an act of amplification — giving an implicitly powerful vocal part even more oomph within an atmosphere of loving and capable extrapolation. Lead single "Look At What The Light Did Now" hinted at the grand yet precise scope of this album, but track after track delivers a considered depth that ends up sounding easy. (DJ)
Reviews by Davy Jones, Doug Nunnally and Shannon Cleary
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Эми Вонг (), 22 декабря 2979 — одна из главных героев мультсериала «Футурама». Училась в Марсианском Университете вплоть до серии «That Darn Katz!» (где она получает докторскую степень в области прикладной физики) и подрабатывает стажёром в Межпланетном Экспрессе (профессор Фарнсворт держит её, так как у них «одна группа крови»).
Характер
В мультсериале Эми показана ограниченной, постоянно использующей сленг девушкой. Склонна одеваться вызывающе, чаще всего оказывается персоной, на которой надето одежды меньше всего. В одном из эпизодов она признаётся Фраю, что она одевается так в знак протеста против своих родителей. Эми в большинстве серий в качестве повседневной одежды носит спортивный костюм розового цвета, делая это, как она заявила Фраю в 7-й серии 2-го сезона, назло родителям, которые требуют от неё, чтоб она была больше похожа на леди. Впрочем, в 4-й полнометражке Эми в сердцах бросает отцу, что она вынуждена была носить такой вот «мальчишеский» спортивный костюм потому, что родители всегда хотели иметь сына. Эми и Лила в шутку соперничают друг с другом на любовном фронте, из-за внешнего вида и т. д., что не мешает Эми считать Лилу своей подругой. Когда Эми раздражена, она иногда ругается на кантонском диалекте китайского языка, на котором говорят в Гонконге.
Лучшие друзья
Лила, Бендер, Фрай
Внешность
Как уже говорилось, Эми привыкла одеваться вызывающе. Обычно на ней розовый спортивный костюм, состоящий из короткой спортивной худи с капюшоном, и из спортивных бридж, на ногах у неё — фиолетовые завышенные кеды. У Эми — азиатская внешность: слегка раскосые глаза, прямые чёрные волосы, обычно поднятые наверх.
Отношения
Эми считается очень симпатичной, и потому встречалась с больши́м количеством мужчин.
Фрай
Она встречалась с Фраем некоторое время (в течение серии «Put Your Head on My Shoulder»), но Фраю показалось, что Эми слишком навязчива, и он решил её бросить. Но перед тем, как Фрай успел сказать, что хочет расстаться, они попали в аварию и голова Фрая была пришита к плечу Эми, чтобы сохранить ему жизнь. Они расстались, хотя Фрай и провёл день святого Валентина на теле Эми.
Киф Крокер
В 3001 году она начала встречаться с Кифом Крокером, и эти отношения продолжаются до сих пор. В «Звере с миллиардом спин» она вышла замуж за Кифа. У Кифа и Эми есть дети (эпизод «Kif Gets Knocked Up a Notch»).
Зепп Бранниган
Во время событий второго полнометражного фильма — «Зверь с миллиардом спин» — считалось, что Киф погиб. Эми нашла утешение в объятьях с начальником Кифа капитаном Зеппом Бранниганом.
Бендер
В 4-й серии 6-го сезона Эми ссорится с Кифом и он её бросает. Потом ей начинает нравиться Бендер из-за его грубого поведения (особенно с Эми). После этого она начинает встречаться с Бендером и вместе с ним они начинают выступать за робосексуальные браки. В итоге их поправку «Бесконечность» принимают, но когда Эми говорит, что они с Бендером могут заключить реальный моногамный брак, Бендер бросает Эми. В конце серии Киф возвращается к Эми в образе плохого парня. Серия заканчивается тем, что Киф и Эми едут на мотоцикле по дороге, а на горизонте заходит солнце.
Родители
Родители Эми — Лео и Инез Вонг. Это очень богатые и очень скупые люди. Они владеют всем западным полушарием Марса, где у них большое ранчо буггало — местных животных, похожих на помесь божьей коровки и коровы. Их предок, сэр Рэджинальд Вонг, купил полушарие у местного племени марсиан всего за одну бусинку (ссылка на то, что остров Манхэттен был куплен у индейцев всего за 24 доллара). Впоследствии выясняется, что эта бусинка — колоссальный алмаз. На протяжении всего сериала они добиваются от Эми, чтобы она поскорее вышла замуж и родила внуков.
Интересные факты
Фильм, прошедший в английском прокате как «Жёлтая Эмануель» («Yellow Emanuelle»), в оригинале на итальянском языке назывался «Мир чувств Эми Вонг» («Il mondo dei sensi di Emy Wong» 1977). Возможно, именно поэтому на Эми меньше всего одежды.
Примечания
Персонажи «Футурамы»
Вымышленные женщины
Вымышленные марсиане
Вымышленные физики
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Q: Link feature missing in Wagtail RichTextBlock I am creating a Wagtail ArticlePage class in models.py. As a part of the body field in my article, which is a StreamField, I created a 'paragraph' component, which is a blocks.RichTextBlock(). When I test it in the admin console, I can create the article properly, but when I go to the paragraph section in the body and type '/' to add a component, I see there is no option to add a regular HTML link. The only available choices are Heading 2, Heading 3, Numbered list, Bulleted list, Embed and Image, but there is no option for 'link'. Without it, I can't add HTML links in the articles, which is very basic feature. How can I have the link added to the RichTextBlock paragraph?
I tried
class ArticlePage(Page):
....
body = StreamField([
('paragraph', blocks.RichTextBlock()),
('code', CodeBlock(label=('Code'))),
], use_json_field=True)
which works fine to create the article, but when I go to the Wagtail admin console for the article, to the edit a paragraph, I just see the options below and the link is not there:
I researched in the Wagtail documentation about RichTextField features, and I found that I can use the features parameter in the RichTextBlock, like shown below to be specific about what features I want listed.
('paragraph', blocks.RichTextBlock(features=['h2','h3','link'])),
However, it went worse. Now I can see only H2 and H3 in the options as shown below:
What am I missing or what am I doing wrong?
Thank you,
A: After doing some more testing, I found an alternative solution. When I highlighted some text in the paragraph, a pop-up message in the edit console came up, giving me the option to use a link. This is not the option when typing '/' in the paragraph, but still, works. It solves my problem and I put it here in case someone else finds the same situation. Below the image of the highlighted text and the pop-up message.
Thank you, anyway.
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Q: How does jQuery Mobile animate the page when you navigate back? When you click the browser back button on a site with jQuery Mobile, instead of leaving the page immediately the previous page is animated in. How is this achieved? I thought that their was no way to intercept the browser's back button (I can see why browser makers wouldn't want to allow this). Thanks
A: When using the hash URL scheme (no pushState plugin enabled):
Hash changes that occur independently of a click, such as when a user
clicks the back button, are handled through the hashchange event,
which is bound to the window object using Ben Alman's hashchange
special event plugin (included in jQuery Mobile). When a hash change
occurs (and also when the first page loads), the hashchange event
handler will send the location.hash to the $.mobile.changePage()
function, which in turn either loads or reveals the referenced page.
Source: http://jquerymobile.com/demos/1.1.0-rc.1/docs/pages/page-navmodel.html
When the pushState plugin is enabled the same goes but the hash is converted into a normal/readable URL:
There is an optional feature that converts the longer, hash-based URLs
mentioned in the previous section into the full document path which is
cleaner and makes the Ajax tracking transparent in the URL structure.
This is built as an enhancement on top of the hash-based URL system
for Ajax links. Note that despite the name, this feature technically
converts hash-based urls by using history.replaceState (not
history.pushState) in the current release because this works more
reliably across our target platforms. For browsers that do not support
history.replaceState, or if this feature is disabled, hash-based URLs
will be used instead.
Source: http://jquerymobile.com/demos/1.1.0-rc.1/docs/pages/page-navmodel.html
Also here are the MDN docs for the hashchange event: https://developer.mozilla.org/en/DOM/window.onhashchange
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Funeral Poems For Friends Left Behind
My late mom and dad, my late mom- and dad-in-law, late friends. shared a poem in memory of them, promising to leave behind.
"Do Not Stand at My Grave and Weep" is a poem written in 1932 by Mary Elizabeth Frye. Every so often the poem and similar variations appear in death and funeral. at My Grave and Weep" (composition by Paul K. Joyce): At the request of a friend. A similar poem is left by Chuckie's dead mother in the Rugrats episode.
Don't cry for me. I will be okay. Heaven is my home now, and this is where I'll stay. Don't cry for me. I'm where I belong. I want you to be happy and try to stay strong.
The women consult with clients prior to death, as well as family or friends of the deceased. They stand firmly behind the idea that their funeral-styling services help those you've left behind. "In.
by Evan Mantyk. What is poetry?What is great poetry? These poems answer these questions. From least greatest (10) to greatest greatest (1), the poems in this list are limited to ones originally written in the English language and which are under 50 lines, excluding poems like Homer's Iliad and Edgar Allan Poe's "Raven." Each poem is followed by some brief analysis.
MOST POPULAR POEMS. My life's been full, I savored much;Good friends, good times,a loved one's. So when a little child departs, we who are left behind.
Earlier this month, friends and family gathered to offer their final farewells to Bill Schiller before he was laid to rest. But they weren't the only ones in mourning. So, too, was the dear dog.
May 18, 2019 · Victor Hugo: Victor Hugo, poet, novelist, and dramatist who was the most important of the French Romantic writers. Though regarded in France as one of that country's greatest poets, he is better known abroad for such novels as Notre-Dame de Paris (1831) and Les Miserables (1862).
Goodbye Poems and messages for friends who are leaving or moving away. Poems for farewell speech for good friends, farewell message to close friends who are leaving. The hardest part of any friendship is when it's time to say goodbye.
Jan 18, 2014 · Funeral Poems for Grandfather. Sarah Harrison. So heaven has received another angel The night sky another star Your life has become a loving memory
Farewell My Friends It was beautiful. As long as it lasted. The journey of my life. I have no regrets. Whatsoever said. The pain I'll leave behind. Those dear hearts
"Everyone leaves an imprint, whether it's a life partner or a sibling or a friend," said Fran Solomon. and he's given them his blessing. "The funeral isn't about you," he advised. "The funeral.
(The National/CBC Archives) In death, Hart left behind his wife, Martha. The eulogy for Owen Hart was broadcast on.
YOU'RE HERE ON A CHARGE OF STALKING DAVE: OVIEDO POLICE INVESTIGATORS SAY 44-YEAR-OLD DAVID CLAUDIO STALKED THE TEENAGED DAUGHTER OF FAMILY FRIENDS. love letters and handwritten poems were all.
A friend took photos and a video of her husband's funeral, but Zec, who lives in an Indianapolis. When he returned to his family, it was time to board and the bag was left behind in Terminal 1 at.
"He has left behind a grieving family, a saddened circle of friends, a town that has lost a beloved citizen. "There's a great saying: 'You'll enjoy the poem more if you don't fully understand it,'".
Many of the funeral poems and readings included here bring comfort and solace to those that read them. Funeral. I have no regrets whatsoever, save the pain I'll leave behind. Farewell farewell my friends, I smile and bid you goodbye.
Below is the full homily at the funeral of INLA child killer Martin McElkerney at. This was a very different experience.
I found it once myself tucked into a close family member's suicide note – a sad message of consolation for those of us left behind to cope with such a legacy. I read it at her funeral. Poems From.
Why I Love U Poems For Her After admitting that the last comment almost had her "spontaneously combusting," she asked the barista: "Are you a man, You have come to the right place if you are looking for fun, engaging and exciting Mother's day songs and fingerplays to do with toddlers, preschoolers and kindergartners. Our activities are widely used by teachers, moms,
Al Cruz, a friend of the family, has setup a GoFundMe page to help the family pay funeral expenses. Cesar is survived by a wife and two young children while Jose left behind a wife and four children.
She left behind three children. money to cover a proper funeral. Over Mother's Day weekend, they raised $2,000 through bakes sales and car washes. But a funeral costs $10,000. The family was still.
Also Bereavement poems to be used in funeral programs and memorial booklets. through friends they always cared about and dreams they left behind,
The writer Anita Brookner, who has died at the age of 87, requested that no funeral be held after her death. How common is this and what does it mean for friends and family. of the grieving process.
Comprehensive list of Funeral and Memorial Poems for Moms, Dads and other loved ones. Also Bereavement poems to be used in funeral programs and memorial booklets
Desire Under The Elms Author Desire Under the Elms reminds me of what Ford Madox Ford said to DH Lawrence about one of his early novels, that everything was wrong with it "but you have genius". And the absolute miracle of this. Carla Gugino's final bow as Abbie Putnam in Goodman Theatre's hit Desire Under the Elms directed by Artistic
William Butler Yeats is widely considered to be one of the greatest poets of the 20th century. He belonged to the Protestant, Anglo-Irish minority that had controlled the economic, political, social, and cultural life of Ireland since at least the end of the 17th century. Most members of this minority considered themselves English people who happened to have been born in Ireland, but Yeats was.
Funeral Poems for Mom Are you looking for poetry to read at your mother's memorial service or life celebration? On this page, we provide many poems that can be read at your mother's funeral or memorial service.
This List of In Loving Memory Poems are the perfect addition to funeral programs and. through friends they always cared about and dreams they left behind,
If you are grieving the loss of a brother, we hope you find comfort in our collection of 21+ best funeral poems for brother. Our poems would be perfect to use as a reading at a funeral service, memorial service, or a celebration of life ceremony, as a tribute to a brother who has passed away.
Here is a selection of poems that may be suitable for a funeral, or that may give. Good friends, good times, a loved one's touch. Leave not a rack behind.
Conduct Which Is Appropriate. Should there be no competent, near friend of the family to take charge of the funeral, then its management should devolve upon the sexton of.
At another funeral, for a American football obsessed fan, we recreated the scene of an Ohio State University football tailgate party". Schoedinger believes that funerals are for the people left behind.
We've compiled a collection of poems about love and loss that can help you cope with. There are meadows and hills for all of our special friends so they can run and. they each miss someone very special to them, who had to be left behind.
Mourners sometimes like to recite poems and other selections—in addition to traditional prayers—during memorial and funeral services and at unveilings. We have. Silent friend of many distances, feel. as you left their voices behind,
Two of my best friends are post-Gulf War veterans, but I haven't been to the funeral of a War on Terror veteran. Memorial.
to choose a life celebration instead of a traditional funeral service is all about choice. Remember, as much as funerals are for the dead, the deceased is not present to participate, and the rituals we perform as part of the death care process are important for the living.
Funeral Poems, Free Memorial Poems or Sympathy Poems for funeral services. Wide selection of. Good friends, good times, a loved one's touch, Perhaps my. To the faithful, I have never left. Talk to me. Behind, a sealed route, Eternity's.
A funeral is a ceremony connected with the burial, cremation, or interment of a corpse, or the burial (or equivalent) with the attendant observances. Funerary customs comprise the complex of beliefs and practices used by a culture to remember and respect the dead, from interment, to various monuments, prayers, and rituals undertaken in their honor. Customs vary between cultures and religious
Christian Funeral Poems. These Christian funeral poems reassure us that when life on earth is done, our loved one will one day be in heaven where there is no more death, suffering, or sadness.
The HyperTexts The Best Easter Poems and Hymns Which poets wrote the best Easter poems of all time? Easter poems tend to have themes such as faith, forgiveness, hope, salvation, resurrection, rejuvenation and heaven.
funeral poetry, funeral readings and funeral verses to be read during a funeral butterfly release. those left behind to continue the fight "On the Wings of Hope". Butterfly Funeral. Finally one of the caterpillars gathered its friends together.
WASHINGTON — George H.W. Bush was eulogized as a president who put country over party, humility over boastfulness, and who bridged divisions in the name of pragmatism and friendship, at a state.
Funeral poems can help us express our feelings during a tragic time in our life. The short funeral poems listed below are commonly used for funerals and can offer comfort to grieving individuals.
Aug 16, 2015. A selection of sympathy poems for the loss of family and friends that can be used for. The tide recedes but leaves behind. You left a gap too big to fill;. Funeral Poems: 45 Beautiful Readings for Memorial Services.
Images Of Medusa Greek Mythology Everyone's An Author With 2016 Mla Update (second Edition) Nov 05, 2017 · I am very scared after I've read all the post on this site about TMS treatment and finding out 99% of the post on this site state that TMS did nothing or made symptoms worse. "I am worried there are too many divisions
Friend Poem; Goodbye My Friend; I Believe; If I Could Catch A Rainbow; To a. with hint of fear, Watch that tacho needle wind, All and sundry left behind,
The family of Mary Richardson Kennedy boycotted her funeral Saturday in a bitter rebuke of her husband, leaving her in-laws, children and famous friends to share a teary. Mary, who left behind no.
Read on for some classic funeral poems and advice on writing your own funeral. The fact that this person left behind a place. No other friend in all the world,
He ended with the same poem as he left the lectern. on a bier in front of a tent sheltering 10 of Schaefer's close friends. About 120 people, including O'Malley and Brown, stood just behind the.
Everyone's An Author With 2016 Mla Update (second Edition) Nov 05, 2017 · I am very scared after I've read all the post on this site about TMS treatment and finding out 99% of the post on this site state that TMS did nothing or made symptoms worse. "I am worried there are too many divisions already in the party so we need a single
With a little planning and research, you can write funeral eulogies that will be remembered and pay great tribute to a friend or loved one. importance of a unique life, and capturing some of the memories left behind. For example, there may be a quote, scripture, poem, or song lyric that you feel sums up the person's life.
Our vision at Lasting Post is to create a user friendly website that can help a family with practical help after the death of loved on matters such as the funeral and probate, as well as providing support for people coming to terms with their loss.
Plan a funeral – articles, tools, funeral homes, local funeral guides. Browse through this page of some of the most popular quotes and poems regarding. If you are really my friend, So when a little child departs, we who are left behind
Jul 03, 2012 · Some poems have become Famous Funeral poems because they have been read at Funerals of famous people including; Diana, Queen Mother and Whitney Houston.
Loved by your friends and all whom you knew. Our wonderful. What a wonderful memory she left behind. Long days. The dearest mother, the kindest friend,
"Although that aside, he did join in activities with his friends. A funeral for Nicholas was held at Papakura Military.
The line of mourners for Karina Vetrano's wake stretched out the door of a funeral home, past her teary-eyed dad and a huge floral butterfly — with a song from her favorite movie as the sad soundtrack.
Funeral Poems, Free Memorial Poems or Sympathy Poems for funeral services. Wide selection of famous funeral poems for funeral and memorial services, Eulogys or in Funersl booklet at PlanningAFuneral.Com
Beautiful poems about death are a great option for funerals. While the poem is bittersweet to those left behind, it brings a message that carries. Frye's emotional support for her friend's loss encourages faith and hope throughout the poem.
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Culture places lots of stress on women of all ages to get a particular type of physique. Overweight ladies usually try a number of dietary fads inside their lifetime and quite often train too much to be able to possess a ideal figure. There are more effective breast implants approaches to obtain an incredible body than to starve yourself and also devote a lot of time at the gym. Having a healthy diet plan and having a modest volume of exercising is usually better than taking drastic action.
In some cases, plastic surgery that adjusts a female's shapes and sizes can make them look leaner compared to the way they basically are and can perhaps get them to come to feel a lot better regarding their own self. When they get breast augmentation columbus women of all ages could seek advise from their particular physician to find the best suited style regarding their completely new breasts. Just like any sort of medical procedure, breast surgery has got threats and also advantages. This treatment won't actually make women thin however it might bring additional individuals consideration from elements of her physique she doesn't necessarily like.
A skilled surgeon might help a client determine if a surgical treatment might help them attain their goals. In the event that fuller breasts could make a female feel better about themselves, she may be even more motivated to improve the other problem areas of her body. However, no matter the amount she wants the method, it's crucial to be familiar with the health risks and consider them up against the positive aspects. At times, the rewards tremendously outnumber the potential risks associated with surgical procedures.
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\section{Introduction}
\label{sec:intro}
Estimating the proportion of people who have antibodies to SARS-CoV-2 is useful for tracking the pandemic's severity and informing public health decisions \citep{arora_serotracker_2021}. Individuals may have detectable antibodies for different reasons, including prior exposure, infection, or vaccination. Antibody levels within a person are dynamic, typically increasing after an exposure, infection, or vaccination, and then eventually decreasing (or waning) over time. Thus individuals may not have detectable antibodies if never (or very recently) exposed, infected, or vaccinated, or if their antibody levels have waned to a level below the limit of detection (of the assay being employed). To the extent that antibody levels are associated with protection from infection with SARS-CoV-2 or COVID-19 disease \citep{earle_evidence_2021, khoury_neutralizing_2021}, seroprevalence estimates may be helpful in modeling the fraction of a population which may be immune or less susceptible to Covid-19. Likewise, cross-sectional seroprevalence estimates, combined with certain modeling assumptions and other data, may permit inference about other parameters such as the cumulative incidence of SARS-CoV-2 infection, infection fatality rate, or attack rate \citep{takahashi_sars-cov-2_2021, shioda_estimating_2021, buss_three-quarters_2021, perez-saez_persistence_2021, brazeau_report_2020}.
Unfortunately, seroprevalence studies often suffer from at least two sources of bias: measurement error due to false positives and negatives, and selection bias because convenience sampling designs are often used due to time and cost constraints. Typically, blood tests for antibodies result in a continuous measure of a particular antibody response, such as that of immunoglobulin G, M, or A (IgG, IgM, or IgA), which is dichotomized into a positive or negative classification based on a cutoff value. This dichotomization almost always produces misclassification bias in the form of false positives and false negatives \citep{bouman_estimating_2021}. The following example from Sempos and Tian (2021) demonstrates how this measurement error can lead to biased seroprevalence estimates. Suppose that true seroprevalence is 1\% and antibody tests are performed using an assay which perfectly identifies true positives as positive, so with 100\% sensitivity, and nearly perfectly identifies true negatives as negative, with 99\% specificity. Despite this assay's high sensitivity and specificity, it is straightforward to show that naively using the sample proportion of positive test results as a seroprevalence estimator would, in expectation, lead to a seroprevalence estimate of nearly 2\% rather than 1\%.
To appropriately account for measurement error, sensitivity and specificity should be estimated and incorporated into seroprevalence estimators. The sensitivity and specificity of a diagnostic test are commonly estimated by performing the assay on samples of known (or gold standard) positives and negatives, respectively. For example, samples from patients who had a case of COVID-19 confirmed with reverse transcription polymerase chain reaction (RT-PCR) testing are often assumed to be true positives. Remnant blood samples that were drawn in 2019 or earlier are often assumed to be true negatives. Sensitivity and specificity are estimated from these validation data as the proportion of samples appropriately classified as positive and negative, respectively. These estimates of sensitivity and specificity can then be incorporated into the prevalence estimator, often with a method popularized by \citet{rogan_estimating_1978} (see also \citet{marchevsky_re_1979}), to appropriately adjust the seroprevalence estimate for assay characteristics.
Many seroprevalence studies are conducted by non-probability sampling methods such as convenience sampling, which may lead to selection bias when characteristics that drive participation in the study are also risk factors for SARS-CoV-2 infection. Probability-based sampling studies are ideal because they are representative by design and lead to less biased estimates than convenience samples analyzed with post-hoc statistical adjustments \citep{shook-sa_estimation_2020, accorsi_how_2021}. However, probability-based sampling may not always be feasible. Convenience sample based estimators often rely on the assumption that each person in a covariate-defined stratum has an equal probability of being in the sample \citep{elliott_inference_2017}. Under this assumption, population prevalence can be estimated with direct standardization, though complex survey methods like calibration have been used as well in the SARS-CoV-2 setting \citep[e.g.,][]{bajema_estimated_2021}.
In this paper, methods are considered which combine standardization and the Rogan-Gladen adjustment to account for both measurement error and selection bias. The article is organized as follows. Section~\ref{sec:methods_rg} reviews prevalence estimation under measurement error. Nonparametric and parametric standardized prevalence estimators and their large-sample properties are described in Section~\ref{sec:selection_bias}. Section~\ref{sec:sims} presents simulation studies to evaluate the empirical bias and 95\% confidence interval (CI) coverage of the standardized estimators across a range of assay characteristics and bias scenarios. The methods are then applied in Section~\ref{sec:data_analysis} to estimate seroprevalence of SARS-CoV-2 among all residents of Belgium in 2020 and, separately, in asymptomatic residents of North Carolina during the spring of 2020. Section~\ref{sec:discuss} concludes with a discussion. Proofs are in the Appendices.
\section{Seroprevalence estimation under measurement error}
\label{sec:methods_rg}
\subsection{Problem setup}
Let the true serology status for an individual in the target population be denoted by $Y$, with $Y=1$ if the individual has antibodies against SARS-CoV-2 and $Y=0$ otherwise. Our goal is to draw inference about the population seroprevalence $\pi=P(Y=1)$. Because of error in the serology assay, $Y$ is not observed directly. Let the result of the serology assay be denoted by $X$, with $X=1$ if the individual tests positive (according to the antibody assay used) and $X=0$ otherwise. Three key quantities are sensitivity, the probability that a true positive tests positive, denoted by $\sigma_e=P(X=1 \mid Y=1)$; specificity, the probability that a true negative tests negative, denoted by $\sigma_p = P(X=0 \mid Y=0)$; and the population expectation of the serology assay outcome, denoted by $\rho=\mathbb E(X)=P(X=1)$. Unless the assay has perfect sensitivity and specificity with $\sigma_e=\sigma_p=1$, $\rho$ typically will not equal $\pi$ and $X$ will be a misclassified version of $Y$.
Suppose validation studies are conducted to estimate the sensitivity and specificity by applying the assay to known positives and negatives. Specifically, measurements are taken on $n_1$ independent and identically distributed (iid) units from strata of the population where $Y=1$ and on $n_2$ iid units from strata where $Y=0$. Thus $n_1$ copies of $X$ are observed to estimate sensitivity and $n_2$ copies of $X$ are observed to estimate specificity. To estimate seroprevalence in a target population, a `main' study with $n_3$ iid copies of $X$ is then conducted, among which true infection status is unknown.
Assume, as is realistic in many SARS-CoV-2 studies, that there is no overlap between the units in each of the three studies. Let $\delta_i$ be an indicator of which study the $i$th individual's sample $X_i$ is from, with $\delta_i=1$ for the sensitivity study, $\delta_i=2$ for the specificity study, and $\delta_i=3$ for the main study. Note that $\sum I(\delta_i=j)=n_j$ for $j=1,2,3$, where $n = n_1 + n_2 + n_3$ and here and throughout summations are taken from $i=1$ to $n$ unless otherwise specified. Assume $n_j/n \to c_j \in (0,1)$ as $n\to\infty$.
\subsection{Estimators and statistical properties}
Let $\theta=(\sigma_e,\sigma_p,\rho,\pi)^T$. Consider the estimator $\hat \theta = (\hat \sigma_e, \hat \sigma_p,\hat \rho, \hat \pi_{RG})^T$, where $\hat \sigma_e = n_1^{-1}\sum I(\delta_i=1) X_i $, $\hat \sigma_p = n_2^{-1} \sum I(\delta_i=2) (1-X_i)$, $\hat \rho =
n_3^{-1}\sum I(\delta_i=3) X_i$, and $\hat \pi_{RG}=(\hat \rho + \hat \sigma_p -1) / (\hat \sigma_e + \hat \sigma_p -1)$. The prevalence estimator $\hat \pi_{RG}$ is motivated by rearranging the identity that $\rho = \pi \sigma_e + (1 - \pi)(1 - \sigma_p)$ and is sometimes referred to as the Rogan-Gladen (1978) estimator.
The estimator $\hat \theta$ can be expressed as the solution (for $\theta$) to the estimating equation vector \[ \sum \psi(X_i;\delta_i, \theta) =
\left(
\begin{array}{l}
\sum \psi_e(X_i; \delta_i, \theta) \\
\sum \psi_p(X_i; \delta_i, \theta) \\
\sum \psi_{\rho}(X_i; \delta_i, \theta)\\
\psi_{\pi}(X_i; \delta_i, \theta)
\end{array}
\right)
=
\left(
\begin{array}{l}
\sum I(\delta_i=1)(X_i - \sigma_e) \\
\sum I(\delta_i=2)\{(1-X_i) - \sigma_p\} \\
\sum I(\delta_i=3)(X_i - \rho) \\
(\rho + \sigma_p - 1) - \pi(\sigma_e + \sigma_p -1)
\end{array}
\right) = 0 \]
where here and below 0 denotes a column vector of zeros. Since the samples were selected from three different populations, the data $X_1,\dots,X_n$ are not identically distributed and care must be taken to derive the large sample properties of $\hat \theta$. In Appendix~\ref{sec:proofs_rg}, the estimator $\hat \theta$ is shown to be consistent and asymptotically normal. Specifically, as $n \to \infty$, $\sqrt{n}(\hat \theta - \theta) \to_{d} \mathcal{N}\left(0, \mathbb A(\theta)^{-1}
\mathbb B(\theta)
\mathbb A(\theta)^{-T}\right)$ and $\sqrt{n}(\hat{\pi} - \pi) \to_{d} \mathcal{N}\left(0, V_{\pi,RG}\right)$ assuming $\sigma_e > 1 - \sigma_p$ (as discussed below),
where $\mathbb A(\theta)^{-1}
\mathbb B(\theta)
\mathbb A(\theta)^{-T}$ is a covariance matrix with bottom right element
\begin{equation}
V_{\pi,RG} =
\left\{
\frac{\pi^2 \sigma_e (1-\sigma_e)}
{c_1}
+
\frac{ (1-\pi)^2 \sigma_p (1-\sigma_p)}{c_2}
+
\frac{\rho(1-{\rho})}{c_3}
\right\}
(\sigma_e + \sigma_p -1)^{-2}.
\label{eq:varpi}
\end{equation}
The proof of consistency and asymptotic normality is similar to those from standard estimating equation theory \citep[e.g.,][Equation 7.10]{boos_2013}, but because the data are not identically distributed the Lindeberg-Feller Central Limit Theorem (CLT) is used in place of the classical Lindeberg-L\'evy CLT. Note that the asymptotic variance \eqref{eq:varpi} consists of three components corresponding to the sensitivity, specificity, and main studies. In some circumstances, investigators may be able to decrease the variance of $\hat \pi_{RG}$ by increasing the sample sizes of the sensitivity or specificity studies compared to the main study \citep{larremore_jointly_2020}.
Let $\hat V_{\pi, RG}$ denote the plug-in estimator defined by replacing $\sigma_e, \sigma_p, \rho, \pi$, and $c_j$ in \eqref{eq:varpi} with $\hat \sigma_e, \hat \sigma_p, \hat \rho, \hat \pi_{RG}$, and $n_j / n$ for $j = 1, 2, 3$, and note that $\hat V_{\pi, RG} / n$ is the variance estimator proposed by \citet{rogan_estimating_1978}. By the continuous mapping theorem, $\hat V_{\pi, RG}$ is consistent for the asymptotic variance assuming $\sigma_e > 1 - \sigma_p$ and can be used to construct Wald-type CIs that asymptotically attain nominal coverage probabilities.
\subsection{Truncation into $[0,1]$}
In finite samples $\hat \pi_{RG}$ sometimes yields estimates outside of $[0,1]$ when (i) $ \hat \sigma_e < 1 - \hat \sigma_p$, (ii) $\hat \rho < 1 - \hat \sigma_p$, or (iii) $\hat \rho > \hat \sigma_e$. Indeed, (ii) occurred in the ScreenNC study discussed in Section~\ref{sec:screennc}. Estimates are typically truncated to be inside $[0,1]$ because the true population prevalence must exist in $[0,1]$ \citep{hilden_further_1979}. In this article, all point estimates and bounds of interval estimates are so truncated. Note, though, that as the three sample sizes grow large the estimator $\hat \pi_{RG}$ yields estimates inside $[0, 1]$ almost surely unless $\sigma_e < 1 - \sigma_p$. In practice, settings where $\sigma_e < 1 - \sigma_p$ may be very unlikely; in such scenarios, the probability of a positive test result is higher for seronegative persons than for seropositive persons, so such a measurement instrument performs worse in expectation than random guessing. Throughout this manuscript, we assume $\sigma_e > 1 - \sigma_p$.
\section{Standardized seroprevalence estimation}
\label{sec:selection_bias}
\subsection{Problem setup}
In some settings it may not be reasonable to assume the $n_3$ copies of $X$ from the main study constitute a random sample from the target population. Suppose instead that for each copy of $X$ a vector of discrete covariates $Z$ is observed, with $Z$ taking on $k$ possible values $z_1, \ldots, z_k$. The covariates $Z$ are of interest because seroprevalence may differ between the strata; for instance, $Z$ might include demographic variables such as age group, race, or gender. Denote the mean of $X$ in the $j$th stratum as $\rho_j = P(X=1\mid Z=z_j)$ and the sample size for the $j$th stratum as $n_{z_j}=\sum I(\delta_i=3,Z_i=z_j)$, so $\sum_{j=1}^kn_{z_j}=n_3$.
The distribution of strata in the target population, if known, can be used to standardize estimates so they are reflective of the target population (for a review of direct standardization, see \citet[][Chapter 15]{van_belle_biostatistics_2004}). Denote the proportion of the target population comprised by the $j$th stratum as $\gamma_j = P(Z=z_j)$ and suppose that these stratum proportions are known with each $\gamma_j>0$ and $\sum_{j=1}^k \gamma_j=1$. For instance, if the target population were all of the adults in a US state, then $\gamma_1, \dots, \gamma_k$ could be obtained from US census data. Assume that all persons in a covariate stratum defined by $Z$ have the same probability of inclusion in the sample. Then the covariates $Z$ in the main study sample have a multinomial distribution with $k$ categories, sample size $n_3$, and an unknown sampling probability vector $(s_1,\dots,s_k)^T$ where $\sum _{j=1}^k s_{j}=1$. Note that if the main study were a simple random sample from the target population, then the sampling probabilities would be equal to the stratum proportions, that is, $s_j=\gamma_j$ for $j=1,\dots,k$.
\subsection{Nonparametric standardization}
\label{sec:standardization}
First, consider a seroprevalence estimator which combines nonparametric standardization and the Rogan-Gladen adjustment to account for both selection bias and measurement error. Notice that $\rho$ is a weighted average of the stratum-conditional means $\rho_j$ where each weight is a known stratum proportion $\gamma_j$, i.e., $\rho=\sum_{j=1}^k \rho_j\gamma_j$. A nonparametric standardization estimator for $\rho$ using the sample stratum-conditional prevalences $\hat \rho_j = n_{z_j}^{-1}\sum I(Z_i=z_j,\delta_i=3)X_i$ for $j = 1, \dots, k$ is $\hat \rho_{SRG} = \sum_{j=1}^k \hat \rho_j \gamma_j$. A standardized prevalence estimator accounting for measurement error is $\hat \pi_{SRG} = (\hat \rho_{SRG}+\hat \sigma_p-1)/(\hat \sigma_e + \hat \sigma_p -1)$, which has been used in SARS-CoV-2 seroprevalence studies \citep{havers_seroprevalence_2020, barzin_sars-cov-2_2020, cai_exact_2020}.
Letting $\theta_s=(\sigma_e, \sigma_p, \rho_1,\dots,\rho_k,\rho,\pi)^T$, the estimator $\hat \theta_s=(\hat \sigma_e, \hat \sigma_p, \hat \rho_1, \dots, \hat \rho_k, \hat \rho_{SRG}, \hat \pi_{SRG})^T$ solves the estimating equation vector $\sum \psi(X_i,Z_i; \delta_i,\theta_s)=\left(\sum \psi_e, \sum \psi_p, \sum \boldsymbol\psi_{\boldsymbol\rho}, \psi_\rho,\psi_\pi\right)^T=0$
where $\sum \psi_e, \sum \psi_p$, and $\psi_\pi$ are defined in Section~\ref{sec:methods_rg}; $\sum \boldsymbol\psi_{\boldsymbol\rho}$ is a $k$-vector with $j$th element $\sum \psi_{\rho_j}=\sum I(Z_i=z_j, \delta_i=3)(X_i-\rho_j)$; and $\psi_\rho=\sum_{j=1}^k\rho_j\gamma_j-\rho$. It follows that $\hat \theta_s$ is consistent and asymptotically normal and that $\sqrt{n}(\hat{\pi}_{SRG} - \pi) \to_{d} \mathcal{N}\left(0, V_{\pi,SRG}\right)$
where
\begin{equation}
V_{\pi,SRG} =
\left\{
\frac{\pi^2 \sigma_e (1-\sigma_e)}
{c_1}
+
\frac{ (1-\pi)^2 \sigma_p (1-\sigma_p)}{c_2}
+
\sum_{j=1}^k\frac{\gamma_j^2\rho_j(1-{\rho_j})}{c_3 s_{j}}
\right\}
(\sigma_e + \sigma_p -1)^{-2}.
\label{eq:varpistd}
\end{equation}
The asymptotic variance $V_{\pi, SRG}$ can be consistently estimated by the plug-in estimator $\hat V_{\pi, SRG}$ defined by replacing $\sigma_e, \sigma_p, \rho_j, s_j, \pi$, and $c_l$ in \eqref{eq:varpistd} with $\hat \sigma_e, \hat \sigma_p, \hat \rho_j, n_{z_j} / n_3, \hat \pi_{SRG}$, and $n_l / n$ for $j = 1, \dots, k$ and $l = 1, 2, 3$. Consistency of $\hat V_{\pi, SRG}$ holds by continuous mapping, and a proof of asymptotic normality and justification of \eqref{eq:varpistd} are in Appendix~\ref{sec:proofs_srg}.
Standardization requires estimating the stratum-conditional mean of $X$, $\rho_j=P(X=1\mid Z=z_j)$. However, when $n_{z_j} = 0$ for some strata $j$, the corresponding estimator $\hat \rho_j$ is undefined, and $\hat \rho_{SRG}$ is then undefined as well. Values of $n_{z_j}$ may equal zero for two reasons. First, the study design may exclude these strata ($s_j=0$), a situation referred to as deterministic or structural nonpositivity (Westreich and Cole, 2010). Second, even if $s_j > 0$, random nonpositivity can occur if no individuals with $Z=z_j$ are sampled, which may occur if $s_j$ is small or if $n_3$ is relatively small. When nonpositivity arises, an analytical approach often employed entails ``restriction'' \citep{westreich_invited_2010}, where the target population is redefined to consist only of strata $j$ for which $n_{z_j}>0$. However, this redefined target population may be less relevant from a public health or policy perspective.
\subsection{Parametric standardization}
\label{sec:model}
We now consider a parametric approach for modeling the stratum-conditional means $\rho_j$, an alternative strategy which allows inference to the original target population and may also perform better when some strata have small sample sizes. Assume the binary regression model $g(\rho_j)=\beta h(z_j)$ holds where $g$ is an appropriate link function for a binary outcome like the logit or probit function; $\beta$ is a row vector of $p$ regression coefficients with intercept $\beta_1$; and $h(z_j)$ is a user-specified $p$-vector function of the $j$th stratum's covariate values that may include main effects and interaction terms, with $l$th element denoted $h_l(z_j)$ and $h_1(z_j)$ set equal to one to correspond to an intercept. Let $\operatorname{supp}(z)$ be the covariate support in the sample, i.e., $\operatorname{supp}(z)=\{z_j: n_{z_j}>0\}$ with dimension $\dim\{\operatorname{supp}(z)\} =\sum_{j=1}^kI(n_{z_j}>0)$, and assume $p\leq \dim\{\operatorname{supp}(z)\}\leq k$. (Note that $\dim\{\operatorname{supp}(z)\}=k$ only when there is positivity, and in that case $\hat \pi_{SRG}$ can be used with no restriction needed.)
Under the assumed binary regression model, each $\rho_j$ is a function of the parameters $\beta$ and the covariates $z_j$ that define the $j$th stratum, denoted $\rho_j( \beta,z_j)=g^{-1}\{\beta h(z_j)\}$. A model-based standardized Rogan-Gladen estimator of $\pi$ is $\hat \pi_{SRGM}=(\hat \rho_{SRGM} + \hat \sigma_p - 1)/(\hat \sigma_e + \hat \sigma_p - 1)$, where $\hat \rho_{SRGM}=\sum_{j=1}^k \hat \rho_j (\hat \beta, z_j)\gamma_j$ and $\hat \beta$ is the maximum likelihood estimator of $\beta$. Estimating equation theory can again be used to derive large-sample properties by replacing the $k$ equations for $\rho_1,\dots,\rho_k$ from Section~\ref{sec:standardization} with $p$ equations for $\beta_1,\dots,\beta_p$ corresponding to the score equations from the binary regression.
Let $\theta_m=(\sigma_e,\sigma_p,\beta_1,\dots,\beta_p,\rho,\pi)^T$. The estimator $\hat \theta_m = (\hat \sigma_e, \hat \sigma_p, \hat \beta_1, \dots, \hat \beta_p, \hat \rho_{SRGM}, \hat \pi_{SRGM})^T$ is the solution to the vector $\sum \psi(X_i,Z_i; \delta_i,\theta_m)=\left(\sum \psi_e, \sum \psi_p, \sum \psi_\beta, \psi_\rho,\psi_\pi\right)^T=0$ of estimating equations where $\sum \psi_e, \sum \psi_p$, and $\psi_\pi$ are as in Section~\ref{sec:methods_rg}; $\sum \psi_\beta$ is a $p$-vector with $j$th element $\sum \psi_{\beta_j}=\sum I(\delta_i=3)
\left[
X_i-g^{-1}\left\{\beta h(Z_i)\right\}
\right]
h_j(Z_i)$; and $\psi_\rho=\sum_{j=1}^k g^{-1}\{\beta h(Z_j)\}\gamma_j-\rho$. It follows that $\hat \theta_m$ is consistent and asymptotically normal and $\sqrt{n}(\hat{\pi}_{SRGM} - \pi) \to_{d} \mathcal{N}\left(0, V_{\pi,SRGM}\right)$. The asymptotic variance $V_{\pi, SRGM}$ can be consistently estimated by $\hat V_{\pi, SRGM}$, the lower right element of the empirical sandwich variance estimator of the asymptotic variance of $\hat \theta_m$. A proof of asymptotic normality and the empirical sandwich variance estimator are given in Appendix~\ref{sec:proofs_srgm}.
\section{Simulation study}
\label{sec:sims}
A simulation study was conducted to compare $\hat \pi_{RG}$, $\hat \pi_{SRG}$, and $\hat \pi_{SRGM}$. Four data generating processes (DGPs) were considered, within which different scenarios were defined through full factorial designs that varied simulation parameters $\pi, \sigma_e, \sigma_p, n_1, n_2$, and $n_3$. These DGPs featured no selection bias (DGP 1), selection bias with two strata (DGP 2), and more realistic selection bias settings with 40 strata and 80 strata (DGPs 3 and 4).
For each DGP and set of simulation parameters, sensitivity and specificity validation samples of size $n_1$ and $n_2$ were generated with $X$ distributed Bernoulli with a mean of $\sigma_e$ or $1-\sigma_p$, respectively. In DGPs 1 and 2 a main study of size $n_3$ was then generated where $Y$ was Bernoulli with mean $\pi$ and $X \mid Y$ was Bernoulli with mean $\sigma_eY + (1 - \sigma_p)(1 - Y)$; in DGPs 3 and 4 $X$ was generated directly from the distribution of $X \mid Z$, as will be explained. Simulation parameter values were selected based on the seroprevalence studies described in Section~\ref{sec:data_analysis}. Sensitivity was varied in $\sigma_e \in \{.8, .99\}$, specificity in $\sigma_p \in \{.8, .95, .99\}$, and prevalence in $\pi \in \{.01, .02, \dots, .20\}$. Sample sizes were $n_1 = 40$, $n_2 = 250$, and $n_3 = 2500$. The full factorial design led to 120 scenarios per DGP, and within each scenario 1,000 simulations were conducted. Performance was measured across the 1,000 simulations by (a) mean bias, computed as $\hat \pi - \pi$ for each estimator $\hat \pi$; and (b) empirical coverage, i.e., whether the 95\% Wald-type CIs based on each variance estimator $\hat V_{\pi}$ contained the true prevalence.
\subsection{No selection bias}
\label{sec:sims_rg}
Simulations for DGP 1 were conducted to assess the performance of $\hat \pi_{RG}$ when no selection bias was present. The estimator $\hat \pi_{RG}$ was generally unbiased, as seen in Appendix Figure~\ref{fig:dgp1_bias}. Performance improved as $\sigma_e$ and $\sigma_p$ tended toward 1, with specificity $\sigma_p$ being a stronger determinant of bias. An exception to these results was for low prevalence $\pi$ (0.05 or lower), when $\hat \pi_{RG}$ overestimated the true prevalence. Wald CIs based on $\hat V_{\pi, RG}$ generally attained nominal coverage, as seen in Appendix Figure~\ref{fig:dgp1_coverage}. However, when prevalence $\pi$ was near zero, coverage of the 95\% CIs did not equal the nominal level. Coverage also decreased as $\sigma_p$ increased toward 1. These variable CI coverage results concord with previous simulation studies evaluating $\hat V_{\pi, RG}$ \citep{lang_confidence_2014}. Alternative methods for constructing CIs are discussed in Section~\ref{sec:discuss}.
\subsection{Low-dimensional selection bias}
\label{sec:dgp2}
In DGP 2, the target population was comprised of two strata defined by a covariate $Z \in \{z_1, z_2\}$ with proportions $\gamma_1=\gamma_2=.5$. Within the main study, $Z$ was generated from a multinomial distribution of sample size $n_3$ and sampling probability vector $(.2, .8)$. Individuals' serostatuses were generated from the conditional distribution $Y \mid Z$, which was such that $P(Y=1 \mid Z=z_1)=1.5 \pi$ and $P(Y=1 \mid Z=z_2)= 0.5 \pi$ for each value of $\pi$. In each simulation $\hat \pi_{RG}$ and $\hat \pi_{SRG}$ and their corresponding 95\% CIs were computed.
The nonparametric standardized estimator $\hat \pi_{SRG}$ was empirically unbiased for true prevalences $\pi \geq 0.05$, as seen in Appendix Figure~\ref{fig:dgp2_bias}, and 95\% CIs based on $\hat V_{\pi, SRG}$ attained approximately nominal coverage, as seen in Appendix Figure 4~\ref{fig:dgp2_coverage}. As with $\hat \pi_{RG}$ in DGP 1, CI coverage fell slightly below the nominal level for very low prevalences, and more noticeably for $\sigma_p = .99$. Appendix Figures~\ref{fig:dgp2_bias} and \ref{fig:dgp2_coverage} show that $\hat \pi_{RG}$ performed poorly under selection bias, with large negative bias and CI coverage far below the nominal level in most cases.
\subsection{More realistic selection bias } \label{sec:sims_realistic}
DGPs 3 and 4 compared $\hat \pi_{SRG}$ and $\hat \pi_{SRGM}$ in scenarios with larger numbers of strata.
\subsubsection{DGP 3} \label{sec:dgp3}
Three covariates were defined as $Z_{1}\in\{z_{10},z_{11}\}$, $Z_{2}\in \{z_{20},z_{21},z_{22},z_{23}\}$, and $Z_{3}\in\{z_{30},z_{31},z_{32},z_{33},z_{34}\}$, leading to $k=40$ strata with proportions $(\gamma_1, \dots, \gamma_{40})$. Within the main study, $Z$ was again generated as multinomial with size $n_3$ and known sampling probability vector. Figure~\ref{fig:dgp3_selectionbias}(a) shows the structure of selection bias in DGP 3 by comparing the stratum proportions and sampling probabilities. Some low-prevalence strata that frequently occur in the population were oversampled, while most remaining strata were undersampled. Individuals' test results were generated from the conditional distribution $X \mid Z$, where
\begin{equation*}
\begin{split}
\operatorname{logit}\{P(X=1\mid Z)\} &= \beta_0+\beta_{1}I(Z_1=z_{11})+\beta_{2}I(Z_2=z_{20})+\beta_{3}I(Z_2=z_{21}) \\
&+ \beta_4I(Z_3=z_{30})+\beta_5I(Z_3=z_{31}).
\end{split}
\end{equation*}
The parameters $\beta_1=-1, \ \beta_2 = -.6, \ \beta_3 = .8, \ \beta_4 = .6,$ and $\beta_5 = .4$ were set to reflect differential prevalences by stratum, while a ``balancing intercept'' $\beta_0$ \citep{rudolph_simulation_2021} was set to different values so that prevalence $\pi$ equaled (approximately) $\{.01, .02, \dots, .20\}$. The nonparametric estimator $\hat \pi_{SRG}$ and corresponding CI were computed using a restricted target population when random nonpositivity arose; the values of prevalence used to compute bias and coverage were based on the total (unrestricted) population, which is the parameter of interest. The parametric estimator $\hat \pi_{SRGM}$ was computed with a correctly-specified logistic regression model.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{figs/selbias.pdf}
\caption{Panels A and B represent selection bias in the simulation studies of DGPs 3 and 4, described in Sections~\ref{sec:dgp3} and \ref{sec:dgp4}, respectively. Circle size is proportional to prevalence. Points are jittered slightly for legibility, and the diagonal lines denote equality between $\gamma_j$ (stratum proportion) and $s_j$ (sampling probability).}
\label{fig:dgp3_selectionbias}
\end{figure}
Both $\hat \pi_{SRG}$ and $\hat \pi_{SRGM}$ performed well in this scenario. Figure~\ref{fig:dgp3_bias} shows that the estimators were generally empirically unbiased, though modest bias occurred when $\sigma_p=0.8$ and prevalence was low. Appendix Figure~\ref{fig:dgp3_coverage} shows 95\% CIs based on either $\hat V_{\pi, SRG}$ or $\hat V_{\pi, SRGM}$ attained nominal coverage, with slight under-coverage for $\pi < 0.05$. On average across all 120 scenarios, 89\% (range of 86\%-92\%) of simulated datasets had positivity. As in DGP 2, $\hat \pi_{RG}$ was seriously biased and the corresponding CIs did not attain nominal coverage.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{figs/dgp3.pdf}
\caption{Empirical bias of the Rogan-Gladen ($\hat \pi_{RG}$), nonparametric standardized ($\hat \pi_{SRG}$), and logistic regression standardized ($\hat \pi_{SRGM}$) estimators from simulation study for DGP 3, described in Section~\ref{sec:dgp3}. The six facets correspond to a given combination of sensitivity $\sigma_e$ (`Sens') and specificity $\sigma_p$ (`Spec').}
\label{fig:dgp3_bias}
\end{figure}
\subsubsection{DGP 4} \label{sec:dgp4}
Data were generated as in DGP 3, but the inclusion of a fourth covariate $Z_4 \in \{z_{40}, z_{41}\}$ now led to 80 strata. The conditional distribution $X \mid Z$ was such that $\operatorname{logit}\{P(X = 1 \mid Z)\} = \nu h(Z)$, where $h(Z)$ here contains the same terms as in DGP 3 plus a main effect for $I(Z_4 = z_{41})$ with corresponding coefficient $\nu_6$. Regression parameters were a balancing intercept $\nu_0$, $\nu_1 = -1$, $\nu_2 = 3.25$, $\nu_3 = .8$, $\nu_4 = .6$, $\nu_5 = .4$, and $\nu_6 = .1$. The larger value for $\nu_2$, as compared to $\beta_2$, led to a stronger relationship between $X$ and $Z$ than was present in DGP 3. Figure~\ref{fig:dgp3_selectionbias}(b) displays selection bias in DGP 4. Some of the highest-prevalence and most commonly-occurring strata were undersampled to a greater degree than occurred in DGP 3, so in this sense there was more selection bias in DGP 4.
Figure~\ref{fig:dgp4_bias} shows that only $\hat \pi_{SRGM}$ was empirically unbiased in this scenario, while $\hat \pi_{SRG}$ typically had a moderately negative bias. Nonpositivity almost always occurred (in either all or all but one of the simulations, for each of the 120 scenarios). The worse performance of $\hat \pi_{SRG}$ may be explained by restriction leading to bias under nonpositivity. CIs based on $\hat V_{\pi, SRGM}$ typically attained nominal coverage, unlike those based on $\hat V_{\pi, SRG}$ or $\hat V_{\pi, RG}$, as seen in Appendix Figure~\ref{fig:dgp4_coverage}. However, the trend of empirical coverage being below the nominal level for low prevalence and $\sigma_p = .99$ was more noticeable compared to other DGPs, perhaps due to greater selection bias. Note that $\hat V_{\pi, SRGM}$ was negative for two simulations in a single scenario, and these ``Heywood cases'' \citep{kolenikov_testing_2012} were ignored in the coverage calculation for that scenario.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{figs/dgp4.pdf}
\caption{Bias results from simulation study on DGP 4, described in Section~\ref{sec:dgp4}. Figure layout is as in Figure~\ref{fig:dgp3_bias}.}
\label{fig:dgp4_bias}
\end{figure}
In summary, both the nonparametric and parametric standardized estimators $\hat \pi_{SRG}$ and $\hat \pi_{SRGM}$ had low empirical bias and close to nominal 95\% CI coverage when there was little nonpositivity. As the number of covariates, amount of selection bias, and potential for nonpositivity increased, the (correctly-specified) parametric $\hat \pi_{SRGM}$ generally maintained its performance while $\hat \pi_{SRG}$ had greater empirical bias and the corresponding CIs did not attain nominal coverage levels.
\subsection{Model misspecification}
\label{sec:sims_misspec}
The performance of $\hat \pi_{SRGM}$ was also assessed in scenarios similar to DGPs 3 and 4, but under mild model misspecification. Here, the true conditional distributions of $Y \mid Z$ were $\operatorname{logit}\{P(Y = 1 \mid Z)\} = \beta h(Z)$ and $\operatorname{logit}\{P(Y = 1 \mid Z)\} = \nu h(Z)$, where $\beta h(Z)$ and $\nu h(Z)$ are the specifications used in the models for $P(X = 1 \mid Z)$ in DGPs 3 and 4, respectively. The test results $X$ were generated from $X \mid Y$ as in DGPs 1 and 2. The degree of misspecification in each scenario thus depended on the values of $\sigma_e$, $\sigma_p$, and $\pi$. Appendix Figures~\ref{fig:dgp5} through \ref{fig:dgp6_coverage} show that, in terms of bias and 95\% CI coverage, $\hat \pi_{SRGM}$ was robust to the misspecification considered.
\section{Applications}
\label{sec:data_analysis}
\subsection{Belgium seroprevalence study}
\label{sec:belgium}
The standardized Rogan-Gladen methods were applied to a nationwide SARS-CoV-2 seroprevalence study in Belgium conducted across seven week-long collection rounds between March and October 2020 \citep{herzog_seroprevalence_2021}. The final collection round took place before the first vaccine authorization in the European Union in December 2020. Residual sera were collected in a stratified random sample from private laboratories encompassing a wide geographical network, with stratification by age group (10 year groups from 0-9, 10-19, \dots, 90-plus), sex (male or female), and region (Wallonia, Flanders, or Brussels). The presence of SARS-CoV-2 IgG antibodies was determined using a semi-quantitative EuroImmun ELISA. Based on validation studies of $n_1 = 181$ RT-PCR confirmed COVID-19 cases and $n_2 = 326$ pre-pandemic negative controls, sensitivity and specificity were estimated to be $\hat \sigma_e = .851$ and $\hat \sigma_p = .988$ \citep[][Table S1.1]{herzog_seroprevalence_2021}. The number of samples for assessing seroprevalence varied between $n_3 = $ 2,960 and $n_3 = $ 3,910 across the seven collection rounds.
In our analysis, we estimated nationwide seroprevalence in Belgium during each collection round standardized by age group, sex, and province (11 total), using 2020 stratum proportion data from the Belgian \citet{federal_planning_bureau_federal_2021}. Province was used rather than region to match the covariates selected for weighting in \citet{herzog_seroprevalence_2021}. In six of the seven collection rounds nonpositivity arose, with between 2 to 15 of the 220 strata not sampled, so restricted target populations were used for computation of $\hat \pi_{SRG}$. For $\hat \pi_{SRGM}$, each logistic regression model had main effects for age group, sex, and province, as well as an interaction term between age group and sex.
Figure~\ref{fig:forest} displays point estimates and CIs for $\hat \pi_{RG}$, $\hat \pi_{SRG}$, and $\hat \pi_{SRGM}$ alongside those for the unadjusted, or naive, sample prevalence $\hat \rho$ for each collection round (with exact Clopper-Pearson 95\% CIs). The naive estimates $\hat \rho$ were typically greater than the other three estimates and had narrower CIs. The greatest differences were between $\hat \rho$ and $\hat \pi_{RG}$, which can be attributed to (estimated) measurement error in the assay. Both of the standardized estimates, $\hat \pi_{SRG}$ and $\hat \pi_{SRGM},$ were similar in value to $\hat \pi_{RG}$ in most collection periods. These estimates, in combination with the stratified random sampling design, suggest that the magnitude of measurement error in this study may have been larger than that of selection bias.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{figs/forest.pdf}
\caption{Estimates and corresponding 95\% confidence intervals for each of seven collection rounds for the 2020 Belgian seroprevalence study \citep{herzog_seroprevalence_2021}, described in Section~\ref{sec:belgium}.}
\label{fig:forest}
\end{figure}
Time-specific seropositivity was at its highest level in the third collection period in May 2020, and thereafter followed a general decreasing trend through September 2020. This trend could reflect a decrease in seroprevalence in Belgium during this calendar time period, perhaps due to a decrease in seroincidence and waning antibodies \citep{herzog_seroprevalence_2021}. Alternatively, the apparent trend could reflect changes in care seeking behaviors during the study timeframe which could have affected the composition of the population contributing residual sera in each collection period \citep{herzog_seroprevalence_2021}, although the standardized estimates should at least partially account for changes in the composition of individuals contributing sera over time.
\subsection{North Carolina seroprevalence study}
\label{sec:screennc}
The standardization methods of Section~\ref{sec:selection_bias} were also applied to ScreenNC, which tested a convenience sample of $n_3 = $ 2,973 asymptomatic patients age 20 and older in North Carolina (NC) for antibodies to SARS-CoV-2 between April to June 2020 \citep{barzin_sars-cov-2_2020}, before the authorization of vaccines in the United States. These patients were seeking unrelated medical care at eleven sites in NC associated with the University of North Carolina (UNC) Health Network. The presence of antibodies was determined with the Abbott Architect SARS-CoV-2 IgG assay. Based on validation studies of $n_1 = 40$ RT-PCR confirmed positive patients and $n_2 = 277$ pre-pandemic serum samples assumed to be negative, sensitivity was estimated as $\hat \sigma_e = 1$ and specificity as $\hat \sigma_p = 0.989$.
In our analysis, seroprevalence was estimated in two relevant target populations. First, we standardized to patients accessing the UNC Health Network during a similar timeframe (21,901 patients from February to June of 2020). The main study sample differed from this UNC target population in terms of age group, race, and sex characteristics, as seen in Table~\ref{tab:screennc}, and meta-analyses suggested that prevalence of COVID-19 infections differed between levels of these covariates in some populations \citep{pijls_demographic_2021, mackey_racial_2021}, supporting the covariates' use in standardization. Note that several racial classifications, including patient refused and unknown, were reclassified as `Other'. Second, we standardized to the 2019 NC population over the age of 20 (7,873,971 persons) using covariate data from the American Community Survey \citep{us_census_bureau_american_2019}. The assumption of equal sampling probabilities may be less reasonable for this target population because not all NC residents are in the UNC Health Network and because there were some geographic areas where no patients in the study sample were from. There was no sample data in the main study for two covariate strata that existed in the UNC Health Network, so restriction was used for $\hat \pi_{SRG}$. Logistic regression models with main effects for sex, race, and age group were used to compute $\hat \pi_{SRGM}$; interaction effects were not included as the small number of positive test results could have led to model overfit.
\begin{table}
\centering
\caption{Demographic comparisons of the ScreenNC study sample, UNC Hospitals patient population, and North Carolina population aged 20+. Data on the NC population are from the 2019 American Community Survey (ACS). Several racial classifications including Patient Refused and Unknown were reclassified as Other. Sample size is denoted by $n$. Some column totals do not sum to 100\% because of rounding.
\label{tab:screennc}}
{\tabcolsep=4.25pt
\begin{tabular}{llcccccc}
& & \multicolumn{2}{c}{ScreenNC}& \multicolumn{2}{c}{UNC Hospitals}& \multicolumn{2}{c}{NC ACS} \\
& & $n$ & \% & $n$ & \% & $n$ & \% \\
& & 2,973 & 100 & 31,095 & 100 & 7,873,971 & 100 \\ \hline
\multirow{2}{*}{Sex} & Female & 1,955 & 66 & 13,962 & 64 & 4,108,603 & 52\\
& Male & 1,018 & 34 & 7,975 & 36 & 3,765,368 & 48 \\ \hline
\multirow{4}{*}{Race} & Asian & 67 & 2 & 460 & 2 & 230,759 & 3 \\
& Black or African-American & 395 & 13 & 5,109 & 23 & 1,640,311 & 21 \\
& Other & 311 & 10 & 1,843 & 6 & 455,600 & 6 \\
& White or Caucasian & 2,200 & 74 & 21,074 & 68 & 5,547,301 & 70 \\ \hline
\multirow{7}{*}{Age} & 20-29 & 342 & 12 & 2060 & 9 & 1,400,918 & 18 \\
& 30-39 & 599 & 20 & 2,763 & 13 & 1,344,647 & 17 \\
& 40-49 & 518 & 18 & 3,382 & 15 & 1,351,156 & 17 \\
& 50-59 & 602 & 20 & 4,200 & 19 & 1,360,357 & 17 \\
& 60-69 & 489 & 17 & 4,548 & 21 & 1,228,123 & 16 \\
& 70-79 & 310 & 11 & 3,325 & 15 & 806,002 & 10 \\
& 80+ & 77 & 3 & 1,623 & 7 & 382,768 & 5
\end{tabular}}
\end{table}
The sample proportion of positive tests was $\hat \rho =24/2973=0.81\%$. The sample false positive rate was $1 - \hat \sigma_p = 1.08\%$, so the data are, at first appearance, consistent with a population prevalence of 0\%. Indeed, the Rogan-Gladen seroprevalence estimate was $\hat \pi_{RG}=0$\% (95\% CI 0\%, 1.00\%). Likewise, the UNC target population had nonparametric and parametric standardized estimates of $\hat \pi_{SRG}=0$\% (0\%, 1.11\%) and $\hat \pi_{SRGM}=0$\% (0\%, 1.13\%), and the NC target population had corresponding estimates of 0\% (0\%, 1.10\%) and 0\% (0\%, 1.11\%). All estimates were truncated into $[0,1]$. The closeness of the standardized and unstandardized results may be due to the small number of positive test results and similarities between the sample and the target populations.
\section{Discussion}
\label{sec:discuss}
We examined nonparametric and model-based standardized Rogan-Gladen estimators, deriving their large-sample properties and consistent variance estimators. While motivated by SARS-CoV-2 seroprevalence studies, the methods considered here are also applicable to prevalence estimation of any binary variable for settings where validation data can be used to estimate the measurement instrument's sensitivity and specificity and covariate data can be used for standardization. Simulation studies demonstrated that both standardized Rogan-Gladen methods had low empirical bias and nominal CI coverage in the majority of practical settings. The empirical results in Section~\ref{sec:sims} highlight the tradeoffs inherent in choosing which method to use for a seroprevalence study. The parametric standardized estimator $\hat \pi_{SRGM}$ was empirically unbiased even when the number of strata and covariates, and with them the potential for random nonpositivity, increased. A drawback to $\hat \pi_{SRGM}$ is the need to correctly specify the form of a regression model. On the other hand, the nonparametric standardized estimator $\hat \pi_{SRG}$ does not require model specification and performed well in scenarios with lower amounts of selection bias and nonpositivity. As the number of strata and covariates grew, however, $\hat \pi_{SRG}$ was empirically biased and its corresponding 95\% CIs did not attain nominal coverage.
For practical use of either method, careful choice of covariates is necessary. Including additional covariates may make the assumption of equal probability of sampling within strata more reasonable. However, such inclusion also makes covariate-defined strata smaller and random nonpositivity more likely. An alternative strategy is to collapse smaller strata with few or no persons to create larger strata, which may make random nonpositivity less likely. However, if strata with sufficiently different sampling probabilities were collapsed together, then the assumption of equal probability of sampling within strata would be violated.
Topics for future research and broader issues in SARS-CoV-2 seroprevalence studies merit mention. In this paper Wald-type confidence intervals are considered, which have known limitations \citep{brown_interval_2001, dean_evaluating_2015}; alternative types of confidence intervals could be considered based on the bootstrap \citep{cai_exact_2020}, Bayesian posterior intervals \citep{gelman_bayesian_2020}, or test inversion \citep{diciccio_confidence_2021}. While the approaches here estimate seroprevalence at a fixed point in time, seroprevalence is a dynamic parameter. For analysis of studies with lengthier data collection periods, extensions of the estimators in this paper could be considered which make additional assumptions (e.g., smoothness, monotonicity) about the longitudinal nature of seroprevalence. Another possible extension could consider variations in assay sensitivity, which may depend on a variety of factors such as: the type of assay used; the recency of exposure, infection, or vaccination of an individual; disease severity in infected individuals; the type and dose of vaccine for vaccinated individuals; and so forth. Where additional data are available related to these factors, then extensions of the standardized Rogan-Gladen estimators which incorporate these additional data could be developed. As an alternative to standardization, inverse probability of sampling weights \citep{lesko_generalizing_2017} or inverse odds of sampling weights \citep{westreich_invited_2010} could be considered. Standardization and weighting methods may possibly be combined to create a doubly robust Rogan-Gladen estimator.
\bibliographystyle{apalike}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,732
|
Validity of Prenuptial Agreements in New York
Here is an overview of some of the recent and important New York cases concerning prenuptial agreements:
Colello v. Colello, 9 A.D.3d 855, 780 N.Y.S.2d 450 (N.Y. App. Div. 4th Dep't, 2004)
Threat to cancel wedding. A man's alleged threat to cancel the wedding if his fiancée refused to sign a prenuptial agreement does not constitute duress that would invalidate the agreement. "As a matter of law, [the] exercise or threatened exercise of a legal right [does] not amount to duress."
Attorney chosen by other party. The allegation of plaintiff that her attorney was chosen and paid for by defendant does not by itself raise a triable issue of fact sufficient to sustain her claim of duress.
Breach of fiduciary duty. In order to establish a cause of action for breach of a fiduciary duty with respect to the execution of the agreement, plaintiff must establish the existence of a fiduciary relationship, misconduct by defendant, and that such misconduct "induced plaintiff to engage in the transaction in question," directly causing the loss about which plaintiff complains.
Unconscionability An unconscionable bargain is "one such as no [person] in his [or her] senses and not under delusion would make on the one hand, and as no honest and fair [person] would accept on the other." Defendant established that the agreement was not achieved by overreaching or fraud and was not "facially unfair." Plaintiff alleges that subsequent events have rendered the agreement unconscionable in its application to the parties' financial situation. According to plaintiff, defendant implemented the agreement by taking his compensation from his business in part in the form of durable assets that were titled in the name of the business, compensation that otherwise would have been marital income and would have been used to purchase marital property. Plaintiff alleges that defendant thereby purportedly insulated such assets, including the marital residence, from plaintiff's claims for equitable distribution, at least according to a strict and literal reading of the parties' agreement. Although courts are authorized to review whether maintenance agreements are unconscionable at the time of entry of a final judgment of divorce, they have no such authority concerning distribution of property. The agreement here concerns only property and is silent with respect to maintenance. Defendant therefore is entitled to summary judgment dismissing that cause of action.
Cron v. Cron, 8 A.D.3d 186;780 N.Y.S.2d 121 (1st Dept. 2004).
Fraud, etc. Defendant, in seeking rescission of the parties' prenuptial agreement, failed to carry her burden to demonstrate that the agreement was the product of fraud, duress or other inequitable conduct. Indeed, the record demonstrates that defendant was aware of plaintiff's earnings and substantial financial assets but nonetheless chose to sign the agreement, notwithstanding the contrary advice of her attorney, who represented her interests in a highly competent manner.
Fairness of agreement Court has the power to vary a prenuptial agreement in the interests of fairness (equity). Since the wife's waiver of maintenance did not place her in danger of becoming a public charge it was not unconscionable. But the agreement's housing provisions, when considered in light of defendant's current responsibilities as a custodial parent of two grade-school age children who have been raised in luxurious accommodations and attend school in an affluent community on Long Island's North Shore, were "plainly inequitable." In view of the overwhelming need to maintain a sense of continuity in the children's lives, including defendant's need to live in close proximity to the children's school, and the sharp rise in real estate values in that area since the 13-year-old prenuptial agreement was executed, there was virtually no prospect that defendant could find suitable housing within the $200,000 cap imposed by the prenuptial agreement. Also defendant has been out of the work force, at plaintiff's insistence, since the birth of the parties' first child, over 10 years ago, while plaintiff earned approximately $4 million a year and had assets of over $30 million. Under the circumstances, equity's intervention was warranted. Accordingly, the First Department modified the agreement to increase the amount of defendant's reasonable housing needs to $2 million. In light of defendant's meager resources and in order to maintain the children in the community in which they have lived all their lives, the husband was directed to maintain the home (with title in his name) until the children are emancipated or have moved elsewhere.
Kessler v. Kessler, 818 N.Y.S.2d 571 (App. Div. 2d Dept. 2006).
Attorneys fees and prenuptial agreements
"The determination as to whether or not a provision waiving the right to seek an award of an attorney's fee is enforceable must be made on a case-by-case basis after weighing the competing public policy interests in light of all relevant facts and circumstances both at the time the agreement was entered and at the time it is to be enforced. If, upon such an inquiry, the court determines that enforcement of the provision would preclude the non-monied spouse from carrying on or defending a matrimonial action or proceeding as justice requires, the provision may be held unenforceable. Also relevant to such a determination is the conduct of the parties over the course of the matrimonial action. Such a determination is frequently best made at the conclusion of the action. However, because an attorney's fee is authorized when needed to carry on or defend an action, it may be necessary to make such a determination at an earlier point in the litigation."
Werther v. Werther, 2005 NY Slip Op 51543(U) September 2, 2005 Supreme Court, Nassau County
Lack of Independent Counsel
Lack of independent counsel is not, without some extrinsic evidence of unconscionability, duress or fraud, sufficient in and of itself to overturn the agreement. This is especially true where one of the parties to the agreement makes a conscious decision not to retain an attorney despite being advised to do so.
Maintenance provisions
The maintenance provisions of the agreement were deemed unfair and unreasonable. General Obligations Law §5-311 provides that the parties to an agreement may relieve one another from the ability of support provided neither party is likely to become a public charge. While the Court acknowledges that defendant's waiver of maintenance would likely not result in her becoming a public charge, the waiver is not fair and reasonable in view of the current and prospective financial circumstances of the parties, same which are disparate. Accordingly, that portion of the agreement waiving maintenance is set-aside.
"Without question, a provision in an agreement eliminating a party's child support obligation is void as against public policy."
The Thirteen Factors in Property Divisions in New York
Relocation of Children Under New York Law
Enjoining Potential International Child Abduction
No-Fault Divorce: New York
Concealment of Value of Marital Assets - New York
When There is No Home State, Who Has Child Custody Jurisdiction?
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,163
|
use handlebars::{Handlebars, Helper, HelperDef, RenderContext, RenderError};
use toml;
use super::super::RenderResult;
#[derive(Clone, Copy)]
pub struct ToTomlHelper;
impl HelperDef for ToTomlHelper {
fn call(&self, h: &Helper, _: &Handlebars, rc: &mut RenderContext) -> RenderResult<()> {
let param = h.param(0)
.ok_or_else(|| RenderError::new("Expected 1 parameter for \"toToml\""))?
.value();
let bytes = toml::ser::to_vec(¶m)
.map_err(|e| RenderError::new(format!("Can't serialize parameter to TOML: {}", e)))?;
rc.writer.write_all(bytes.as_ref())?;
Ok(())
}
}
pub static TO_TOML: ToTomlHelper = ToTomlHelper;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,455
|
\section{Introduction}
\label{sec:intro}
One of the greatest challenges in computer science is the development
of rigorous methods for the specification and verification of reactive
systems, {\it{i.e.}}, systems that compute by interacting with their
environment. Typical examples include embedded systems, control
programs and distributed communication protocols. Over the last three
decades, process algebras, such as ACP~\cite{BaetenBR2010},
CCS~\cite{Mi89} and CSP~\cite{Ho85},
have been successfully used as common languages for the description of
both actual systems and their specifications. In this context,
verifying whether the implementation of a reactive system complies to
its specification reduces to proving that the corresponding
process terms are related by some notion of behavioural equivalence or
preorder~\cite{Glabbeek01}.
One approach to proving equivalence between two terms is to exploit
the equational style of reasoning supported by process algebras. In
this approach, one obtains a (ground-)complete axiomatization of the
behavioural relation of interest and uses it to prove the equivalence
between the terms describing the specification and the implementation
by means of equational reasoning, possibly in conjunction with proof
rules to handle recursively-defined process specifications.
Finding a ``finitely specified", (ground-)complete axiomatization of a
behavioural equivalence over a process algebra is often a highly non-trivial
task. However, as shown in~\cite{Aceto:1994:TSR:184662.184663} in the
setting of bisimilarity~\cite{Mi89,Pa81}, this process can be
automated for process languages with an operational semantics given in
terms of rules in the GSOS format of Bloom, Istrail and
Meyer~\cite{Bloom:1995:BCT:200836.200876}. In that reference, Aceto,
Bloom and Vaandrager provided an algorithm that, given a GSOS language
as input, produces as output a ``conservative extension" of the
original language with auxiliary operators together with a finite
axiom system that is sound and ground-complete with respect to
bisimilarity (see, {\it e.g.}, \cite{DBLP:conf/concur/Aceto94,GazdaFokkink2010,DBLP:conf/concur/1994,DBLP:journals/tcs/Ulidowski00} for further results in this line of research). As the operational specification of several operators
often requires a clear distinction between successful termination and
deadlock, an extension of the above-mentioned approach to the setting of GSOS with a predicate for termination was proposed in~\cite{DBLP:journals/jlp/BaetenV04}.
In this paper we contribute to the line of the work
in~\cite{Aceto:1994:TSR:184662.184663} and
\cite{DBLP:journals/jlp/BaetenV04}. Inspired by~\cite{DBLP:journals/jlp/BaetenV04}, we introduce the {\emph{preg}} rule
format, a natural extension of the GSOS format with an arbitrary collection of predicates such
as termination, convergence and divergence. We further adapt the
theory in~\cite{Aceto:1994:TSR:184662.184663} to this setting and give a procedure for
obtaining ground-complete axiomatizations for bisimilarity over
{\emph{preg}} systems.
More specifically, we develop a general procedure that, given a
{\emph{preg}} language as input, automatically synthesizes a conservative
extension of that language and a finite axiom system that, in
conjunction with an infinitary proof rule, yields a sound and
ground-complete axiomatization of bisimilarity over the extended
language. The work we present in this paper is based on the one
reported
in~\cite{Aceto:1994:TSR:184662.184663,DBLP:journals/jlp/BaetenV04}. However, handling more general predicates than immediate termination requires
the introduction of some novel technical ideas. In particular, the
problem of axiomatizing bisimilarity over a {\emph{preg}} language is
reduced to that of axiomatizing that relation over finite trees whose
nodes may be labelled with predicates. In order to do so, one needs to
take special care in axiomatizing negative premises in rules that may
have positive and negative premises involving predicates and
transitions.
The results of the current paper have been used for the implementation
of a Maude~\cite{DBLP:conf/maude/2007} tool \cite{pregax-calco-tools2011}
that enables the user to specify {\emph{preg}} systems in a uniform
fashion, and that automatically derives the associated
axiomatizations. The tool is available at {\footnotesize{\url{http://goriac.info/tools/preg-axiomatizer/}}}. This paves the way to checking bisimilarity
over process terms by means of theorem-proving techniques for a large
class of systems that can be expressed using {\emph{preg}} language specifications.
\paragraph{Paper structure.}
{ In Section~\ref{sec:prelim} we introduce the {\emph{preg}} rule format.
In Section~\ref{sec:fintree} we
introduce an appropriate ``core" language for expressing finite trees
with predicates. We also provide a ground-complete axiomatization for
bisimilarity over this type of trees, as our aim is to prove
the completeness of our final axiomatization by head normalizing
general {\emph{preg}} terms, and therefore by reducing the completeness problem for arbitrary languages to that for trees.
Head normalizing general {\emph{preg}} terms is not a straightforward
process. Therefore, following~\cite{Aceto:1994:TSR:184662.184663}, in
Section~\ref{sec:smooth} we introduce the notion of smooth and distinctive
operation, adapted to the current setting. These operations are
designed to ``capture the behaviour of general {\emph{preg}} operations",
and are defined by rules satisfying a series of syntactic constraints
with the purpose of enabling the construction of head normalizing axiomatizations. Such axiomatizations are based on a collection of equations that describe the interplay between smooth and distinctive operations, and the operations in the signature for finite trees.
The existence of a sound and ground-complete axiomatization characterizing
the bisimilarity of {\emph{preg}} processes is finally proven in
Section~\ref{sec:completeness}. A technical discussion on why it is important to handle predicates as first class notions, instead of encoding them by means of transition relations, is presented in
Section~\ref{sec:rationale}. In Section~\ref{sec:conclusions} we
draw some conclusions and provide pointers to future work.
\section{GSOS with predicates}
\label{sec:prelim}
In this section we present the {\emph{preg}} systems which are a generalization of GSOS \cite{Bloom:1995:BCT:200836.200876} systems.
Consider a countably infinite set $V$ of \emph{process variables} (usually denoted by $x$, $y$, $z$) and a signature $\Sigma$ consisting of a set of \emph{operations} (denoted by $f$, $g$). The set of \emph{process terms} ${\mathbb{T}}(\Sigma)$ is inductively defined as follows: each variable $x \in {\it V}$ is a term; if $f \in \Sigma$ is an operation of arity $l$, and if $S_1, \ldots, S_l$ are terms, then $f(S_1, \ldots, S_l)$ is a term. We write $T(\Sigma)$ in order to represent the set of \emph{closed process terms} (\textit{i.e.}, terms that do not contain variables), ranged over by $t, s$.
A \emph{substitution} $\sigma$ is a function of type $V \rightarrow {\mathbb T}(\Sigma)$. If the range of a substitution is included in $T(\Sigma)$, we say that it is a \emph{closed substitution}. Moreover, we write $[x \mapsto t]$ to represent a substitution that maps the variable $x$ to the term $t$.
Let $\vec{x} = x_1, \ldots, x_n$ be a sequence of pairwise distinct variables. A $\Sigma$-\emph{context} $C[\vec{x}]$ is a term in which at most the variables $\vec{x}$ appear. For instance, $f(x,f(x,c))$ is a $\Sigma$-context, if the binary operation $f$ and the constant $c$ are in $\Sigma$.
Let $\textnormal{$\mathcal A$}$ be a finite, nonempty set of \emph{actions} (denoted by $a$, $b$, $c$). A \emph{positive transition formula} is a triple $(S, a, S')$ written $S \xrightarrow{a} S'$, with the intended meaning: process $S$ performs action $a$ and becomes process $S'$. A \emph{negative transition formula} $(S, a)$ written $S \notranz{a}$, states that process $S$ cannot perform action $a$. Note that $S, S'$ may contain variables. The ``intended meaning" applies to closed process terms.
We now define {\emph{preg}} -- \emph{pr}edicate \emph{e}xtension of the \emph{G}SOS rule format.
Let $\textnormal{$\mathcal P$}$ be a finite set of \emph{predicates} (denoted by $P, Q$). A \emph{positive predicate formula} is a pair $(P, S)$, written $PS$, saying that process $S$ satisfies predicate $P$. Dually, a \emph{negative predicate formula} $\neg P\,S$ states that process $S$ does not satisfy predicate $P$.
\begin{definition}[{\emph{preg}} rule format]
\label{def:apreg}
Consider $\cal A$, a set of actions, and $\textnormal{$\mathcal P$}$, a set of predicates.
\begin{enumerate}\itemsep1pt
\setlength\itemsep{1ex}
\item A \emph{{\emph{preg}} transition rule} for an $l$-ary operation $f$ is a deduction rule of the form:
\[
\dfrac{
\begin{array}{c@{~~~}c}
\{ x_i \xrightarrow{a_{ij}} y_{ij} \mid i \in {\textnormal{$I^+$}}, j \in \actpar{i} \} &
\{ P_{ij} x_i \mid i \in {\textnormal{$J^+$}}, j \in \satpar{i} \}
\\
\{ x_i \notranz{b} \hspace{7pt} \mid i \in {\textnormal{$I^-$}}, b \in \textnormal{$\mathcal B$}_i \} &
\{ \neg Q x_i \mid i \in {\textnormal{$J^-$}}, Q \in \textnormal{$\mathcal Q$}_i \}
\end{array}
}
{
f(x_1, \ldots, x_l) \xrightarrow{c} C[\vec{x}, \vec{y}]
}
\]
where
\begin{enumerate}\itemsep1pt
\item $x_1, \ldots, x_l$ and $y_{ij}$ $(i \in {\textnormal{$I^+$}}, j \in {\textnormal{$J^+$}})$ are pairwise distinct variables;
\item ${\textnormal{$I^+$}},{\textnormal{$J^+$}},{\textnormal{$I^-$}},{\textnormal{$J^-$}} \subseteq {\textnormal{$L$}} = \{1,\ldots,l\}$ and each $\actpar{i}$ and $\satpar{i}$ is finite;
\item $a_{ij}, b$ and $c$ are actions in $\cal A$ (${\textnormal{$\mathcal B$}_i} \subseteq {\cal A}$); and
\item $P_{ij}$ and $Q$ are predicates in $\textnormal{$\mathcal P$}$ (${\textnormal{$\mathcal Q$}_i} \subseteq {\textnormal{$\mathcal P$}}$).
\end{enumerate}
\item A \emph{{\emph{preg}} predicate rule} for an $l$-ary operation $f$ is a deduction rule similar to the one above, with the only difference that its conclusion has the form $P(f(x_1, \ldots, x_l))$ for some $P \in \textnormal{$\mathcal P$}$.
\end{enumerate}
\end{definition}
Let $\rho$ be a {\emph{preg}} (transition or predicate) rule for $f$.
The symbol $f$ is the \emph{principal operation} of $\rho$. All the formulas above the line are \emph{antecedents} and the formula below is the \emph{consequent}. We say that a position $i$ for $\rho$ is \emph{tested positively} if $i \in {\textnormal{$I^+$}} \cup {\textnormal{$J^+$}}$ and $\actpar{i} \cup \satpar{i} \not= \emptyset$. Similarly, $i$ is \emph{tested negatively} if $i \in {\textnormal{$I^-$}} \cup {\textnormal{$J^-$}}$ and $\textnormal{$\mathcal B$}_i \cup \textnormal{$\mathcal Q$}_i \not= \emptyset$.
Whenever $\rho$ is a transition rule for $f$, we say that
$f(\vec{x})$ is the \emph{source}, $C[\vec{x}, \vec{y}]$ is the \emph{target}, and $c$ is the \emph{action} of $\rho$.
Whenever $\rho$ is a predicate rule for $f$, we call $f(\vec{x})$ the \emph{test} of $\rho$.
In order to avoid confusion, if in a certain context we use more than one rule, e.g. $\rho, \rho'$, we parameterize the corresponding sets of indices with the name of the rule, {\it e.g.},
$\actpar{\rho}$, $\nsatpar{\rho'}$.
\begin{definition}[{\emph{preg}} system]
\label{def:apregSys}
A {\emph{preg}} system is a pair $G = (\Sigma_G, \textnormal{${\cal R}_G$})$, where $\Sigma_G$ is a finite signature and $\textnormal{${\cal R}_G$} = \textnormal{$\setrules^{\calA}_G$} \cup \textnormal{$\setrules^{\calP}_G$}$ is a finite set of {\emph{preg}} rules over $\Sigma_G$
({\textnormal{$\setrules^{\calA}_G$}} and {\textnormal{$\setrules^{\calP}_G$}} represent the transition and, respectively, the predicate rules of $G$).
\end{definition}
Consider a {\emph{preg}} system $G$.
Formally, the operational semantics of the closed process terms in $G$ is fully characterized by the relations
$\sstrans{G} \subseteq T(\Sigma_G) \times {\textnormal{$\mathcal A$}} \times T(\Sigma_G)$ and $\sssat{G} \subseteq \textnormal{$\mathcal P$} \times T(\Sigma_G)$, called the (unique) \emph{sound and supported} transition and, respectively, predicate relations.
Intuitively, soundness guarantees that $\sstrans{G}$ and $\sssat{G}$ are closed with respect to the application of the rules in $\textnormal{$\cal R$}_G$ on $T(\Sigma_G)$, {\it i.e.}, $\sstrans{G}$ (resp. $\sssat{G}$) contains the set of all possible transitions (resp. predicates) process terms in $T(\Sigma_G)$ can perform (resp. satisfy) according to $\textnormal{$\cal R$}_G$.
The requirement that $\sstrans{G}$ and $\sssat{G}$ be supported means that all the transitions performed (resp. all the predicates satisfied) by a certain process term can be ``derived" from the deductive system described by $\textnormal{$\cal R$}_G$.
As a notational convention, we write $S \xrightarrow{a}_{G} S'$ and $P_{G} S$ whenever $(S,a,S') \in\, \rightarrow_{G}$ and $(P,S) \in \sssat{G}$. We omit the subscript $G$ when it is clear from the context.
\begin{lemma}
\label{lm:fin-bran}
Let $G$ be a {\emph{preg}} system. Then, for each $t \in T(\Sigma_G)$ the set $\{(a, t') \mid t \xrightarrow{a} t',\, a\in \textnormal{$\mathcal A$}\}$ is finite.
\end{lemma}
Next we introduce the notion of \emph{bisimilarity} -- the equivalence over processes we consider in this paper.
\begin{definition}[Bisimulation]
\label{def:bisimulation}
Consider a {\emph{preg}} system $G=(\Sigma_G, \textnormal{${\cal R}_G$})$.
A symmetric relation $R\, \subseteq T(\Sigma_G) \times T(\Sigma_G)$ is a \emph{bisimulation} iff:
\begin{enumerate}\itemsep1pt
\item for all $s, t, s' \in T(\Sigma_G)$, whenever $(s, t) \in\, R$ and $s \xrightarrow{a} s'$ for some $a \in {\cal A}$, then there is some $t' \in T(\Sigma_G)$ such that $t \xrightarrow{a} t'$ and $(s',t') \in\, R$;\label{def:bisim1}
\item whenever $(s,t) \in\, R$ and $Ps$ $(P \in \textnormal{$\mathcal P$})$ then $Pt$. \label{def:bisim2}
\end{enumerate}
Two closed terms $s$ and $t$ are \emph{bisimilar} (written $s \sim t$) iff there is a bisimulation relation $R$ such that $(s,t) \in\, R$.
\end{definition}
\begin{proposition}
\label{bis:equiv-congr}
Let $G$ be a {\emph{preg}} system. Then $\sim$ is an equivalence relation and a congruence for all operations $f$ of $G$.
\end{proposition}
\begin{definition}[Disjoint extension]
\label{def:disjExt}
A {\emph{preg}} system $G'$ is a disjoint extension of a {\emph{preg}} system $G$, written $G \sqsubseteq G'$, if the signature and the rules of $G'$ include those of $G$, and $G'$ does not introduce new rules for operations in $G$.
\end{definition}
It is well known that if $G \sqsubseteq G'$ then two terms in $T(\Sigma_G)$ are bisimilar in $G$ if and only if they are bisimilar in $G'$.
{
From this point onward, our focus is to find a \emph{sound and ground-complete axiomatization of bisimilarity on closed terms} for an arbitrary {\emph{preg}} system $G$, {\it i.e.}, to identify a (finite) axiom system $\textnormal{$\textit{E}$}_G$ so that
$
\textnormal{$\textit{E}$}_G \vdash s = t\,\, \it{iff}\,\, s \sim t \text{ for all } s, t \in T$$(\Sigma_G).
$
}
The method we apply is an adaptation of the technique in~\cite{Aceto:1994:TSR:184662.184663} to the {\emph{preg}} setting.
The strategy is to incrementally build a finite, head-normalizing axiomatization for general {\emph{preg}} terms, {\it{i.e.}}, an axiomatization that, when applied recursively, reduces the completeness problem for arbitrary terms to that for synchronization trees. This way, the proof of ground-completeness for $G$ reduces to showing the equality of closed tree terms.
\section{Preliminary steps towards the axiomatization}
\label{sec:fintree}
In this section we start by identifying an appropriate language for expressing finite trees with predicates. We continue in the style of \cite{Aceto:1994:TSR:184662.184663}, by extending the language with a kind of restriction operator used for expressing the inability of a process to perform a certain action or to satisfy a given predicate. (This operator is used in the axiomatization of negative premises.) We provide the structural operational semantics of the resulting language, together with a sound and ground-complete axiomatization of bisimilarity on finite trees with predicates.
\subsection{Finite trees with predicates}
The language for trees we use in this paper is an extension with predicates of the language BCCSP~\cite{Glabbeek01}. The syntax of BCCSP consists of closed terms built from
a constant $\delta$ (\emph{deadlock}), the binary operator $\_\hspace{-1.5pt}+\hspace{-2.2pt}\_$\, (\emph{nondeterministic choice}), and the unary operators $a.\_$ (\emph{action prefix}), where $a$ ranges over the actions in a set {\textnormal{$\mathcal A$}}. Let {\textnormal{$\mathcal P$}} be a set of predicates. For each $P \in \textnormal{$\mathcal P$}$ we consider a process constant $\textnormal{$\kappa$}_{P}$, which ``witnesses'' the associated predicate in the definition of a process. Intuitively, $\textnormal{$\kappa$}_P$ stands for a process that only satisfies predicate $P$ and has no transition.
A finite tree term $t$ is built according to the following grammar:
\begin{equation}
\label{treesgrammar}
t {\,\,::=\,\,} \delta \mid \textnormal{$\kappa$}_P ~ (\forall P \in \textnormal{$\mathcal P$}) \mid
a.t ~ (\forall a \in \textnormal{$\mathcal A$}) \mid t + t.
\end{equation}
Intuitively, $\delta$ represents a process that does not exhibit any behaviour, $s + t$ is the nondeterministic choice between the behaviours of $s$ and $t$, while $a.t$ is a process that first performs action $a$ and behaves like $t$ afterwards. The operational semantics that captures this intuition is given by the rules of BCCSP:
\begin{figure}[H]
\begin{center}
\begin{tabular}{c@{\hspace{6ex}}c@{\hspace{6ex}}c}
$\dfrac{}{a.x \xrightarrow{a} x}$ $(rl_1)$ &
$\dfrac{x \xrightarrow{a} x'}{x + y \xrightarrow{a} x'}$ $(rl_2)$ &
$\dfrac{y \xrightarrow{a} y'}{x + y \xrightarrow{a} y'}$ $(rl_3)$
\end{tabular}
\caption{The semantics of BCCSP}
\label{fig:BCCSP}
\end{center}
\end{figure}
As our goal is to extend BCCSP, the next step is to find an appropriate semantics for predicates. As can be seen in Fig.~\ref{fig:BCCSP}, action performance is determined by the shape of the terms.
Consequently, we choose to define predicates in a similar fashion.
Consider a predicate $P$ and the term $t=\textnormal{$\kappa$}_P$. As previously mentioned, the purpose of $\textnormal{$\kappa$}_P$ is to witness the satisfiability of $P$. Therefore, it is natural to consider that $\textnormal{$\kappa$}_P$ satisfies $P$.
Take for example the \emph{immediate termination} predicate $\downarrow$. As a term $s+s'$ exhibits the behaviour of both $s$ and $s'$, it is reasonable to state that $(s + s')\downarrow$ if $s\downarrow$ or $s'\downarrow$. Note that for a term $t=a.t'$ the statement $t\downarrow$ is in contradiction with the meaning of immediate termination, since $t$ can initially only execute action $a$. Predicates of this kind are called \emph{explicit predicates} in what follows.
Consider now the \emph{eventual termination} predicate \lightning. In this situation, it is proper to consider that $(s + t)\textnormal{\lightning}$ if $s\textnormal{\lightning}$ or $t\textnormal{\lightning}$ and, moreover, that
$a.s\textnormal{\lightning}$ if $s\textnormal{\lightning}$. We refer to predicates such as $\textnormal{\lightning}$ as \emph{implicit predicates} (that range over a set {\textnormal{$\mathcal{P^I}$}} included in \textnormal{$\mathcal P$}), since their satisfiability propagates through the structure of tree terms in an implicit fashion. We denote by $\textnormal{$\mathcal A$}_{P}$ (included in $\textnormal{$\mathcal A$}$) the set consisting of the actions $a$ for which this behaviour is permitted when reasoning on the satisfiability of predicate $P$.
The rules expressing the semantics of predicates are:
\begin{figure}[H]
\begin{center}
\begin{tabular}{c@{\hspace{3ex}}c@{\hspace{3ex}}c@{\hspace{3ex}}c}
$\dfrac{}{P\textnormal{$\kappa$}_P}$ $(rl_4)$ & $\dfrac{Px}{P(x + y)}$ $(rl_5)$ & $\dfrac{Py}{P(x+y)}$ $(rl_6)$ & $\dfrac{Px}{P(a.x)}, \forall P \in \textnormal{$\mathcal{P^I}$} \,\, \forall a \in \textnormal{$\mathcal A$}_{P} ~(rl_7)$ \\[4ex]
\end{tabular}
\caption{The semantics of predicates}
\label{fig:TTSz}
\end{center}
\end{figure}
The operational semantics of trees with predicates is given by the set of rules ($rl_1$)--($rl_7$) illustrated in Fig.~\ref{fig:BCCSP} and Fig.~\ref{fig:TTSz}.
For notational consistency, we make the following conventions.
Let $\textnormal{$\mathcal A$}$ be an action set and $\textnormal{$\mathcal P$}$ a set of predicates.
$\Sigma_{\textit{FTP}}$ represents the signature of finite trees with predicates.
$T(\Sigma_{\textit{FTP}})$ is the set of (closed) tree terms built over $\Sigma_{\textit{FTP}}$, and
${\textnormal{$\cal R$}}_{\textit{FTP}}$ is the set of rules ($rl_1$)--($rl_7$). Moreover, by $\textit{FTP}$ we denote the system $(\Sigma_{\textit{FTP}}, {\textnormal{$\cal R$}}_{\textit{FTP}})$.
\paragraph{Discussion on the design decisions.}
{
At first sight, it seems reasonable for our framework to allow for language specifications containing rules of the shape $\frac{}{P(x + y)}$, or just one of ($rl_5$) and ($rl_6$). We decided, however, to disallow them, as their presence would invalidate standard algebraic properties such as the idempotence and the commutativity of $\_\hspace{-1.5pt}+\hspace{-2pt}\_\,$.
Without loss of generality we avoid rules of the form $ \frac{}{P(a.x)} $. As far as the user is concerned, in order to express that $a.x$ satisfies a predicate $P$, one can always add the witness $\textnormal{$\kappa$}_P$ as a summand: $a.x + \textnormal{$\kappa$}_P$. This decision helped us avoid some technical problems for the soundness and completeness proofs for the case of the restriction operator $\onedagpar{\calB,\calQ}$, which is presented in Section~\ref{subsec:dagger}.
Due to the aforementioned restriction, we also had to leave out universal predicates with rules of the form $\frac{P x ~ P y}{P(x + y)}$. However, the elimination of universal predicates is not a theoretical limitation to what one can express, since a universal predicate can always be defined as the negation of an existential one.
As a last approach, we thought of allowing the user to specify existential predicates using rules of the form $\frac{P_1 x \ldots P_n x}{P(x+y)} (*)$ and $\frac{P_1 y \ldots P_n y}{P(x+y)} (**)$ (instead of $(rl_5)$ and $(rl_6)$). However, in order to maintain the validity of the axiom $x + x = x$ in the presence of rules of these forms, it would have to be the case that one of the predicates $P_i$ in the premises is $P$ itself. (If that were not the case, then let $t$ be the sum of the constants witnessing the $P_i$'s for a rule of the form $(*)$ above with a minimal set of set premises. We have that $t + t$ satisfies $P$ by rule $(*)$. On the other hand, $P t$ does not hold since none of the $P_i$ is equal to $P$ and no rule for $P$ with a smaller set of premises exists.) Now, if a rule of the form $(*)$ has a premise of the form $P x$, then it is subsumed by $(rl_{5})$ which we must have to ensure the validity of laws such as $\textnormal{$\kappa$}_P = \textnormal{$\kappa$}_P + \textnormal{$\kappa$}_P$.
}
\subsection{Axiomatizing finite trees}
\label{sec:axFinTrees}
In what follows we provide a finite sound and ground-complete axiomatization ({\textnormal{$\textit{E}_{\FINTREEPRED}$}}) for bisimilarity over finite trees with predicates.
The axiom system {\textnormal{$\textit{E}_{\FINTREEPRED}$}} consists of the
following axioms:
\begin{figure}[H]
\begin{center}
\begin{tabular}{r@{\hspace{3pt}}c@{\hspace{3pt}}lr@{\hspace{20pt}}r@{\hspace{3pt}}c@{\hspace{3pt}}lr}
$x + y$ & = & $y + x$ & $(A_1)$ &
$x + x$ & = & $x$ & $(A_3)$
\\[1ex]
$(x + y) + z$ & = & $x + (y + z)$ & $(A_2)$ &
$x + \delta$ & = & $x$ & $(A_4)$
\\[1ex]
\multicolumn{8}{c}{{$a.(x+\textnormal{$\kappa$}_P)$ = $a.(x+\textnormal{$\kappa$}_P) + \textnormal{$\kappa$}_P, \forall P \in \textnormal{$\mathcal{P^I}$} \,\, \forall a \in \textnormal{$\mathcal A$}_{P}$ $(A_5)$}}
\end{tabular}
\caption{The axiom system \textnormal{$\textit{E}_{\FINTREEPRED}$}}
\label{fig:ETz}
\end{center}
\end{figure}
Axioms $(A_1)$--$(A_4)$ are well-known %
\cite{Mi89}.
Axiom $(A_5)$ describes the propagation of witness constants for the case of implicit predicates.
We now introduce the notion of terms in \emph{head normal form}. This concept plays a key role in the proofs of completeness for the axiom systems generated by our framework.
\begin{definition}[{Head} Normal Form]
Let $\Sigma$ be a signature such that $\Sigma_{\textit{FTP}} \subseteq \Sigma$.
A term $t$ in $T(\Sigma)$ is in \emph{head normal form} (for short, h.n.f.) if
\label{def:hnf}
~\\
\[
t = {\sum_{i \in I}a_i.t_i + \sum_{j \in J}\textnormal{$\kappa$}_{P_j}}, \text{and the $P_j$ are all the predicates satisfied by $t$}.
\]
The empty sum $(I = \emptyset, J = \emptyset)$ is denoted by the deadlock constant $\delta$.
\end{definition}
\begin{lemma}
\label{lem:hnf_exists}
{\textnormal{$\textit{E}_{\FINTREEPRED}$}} is head normalizing for terms in $T(\Sigma_{\textit{FTP}})$.
That is, for all $t$ in $T(\Sigma_{\textit{FTP}})$, there exists $t'$ in $T(\Sigma_{\textit{FTP}})$ in h.n.f. such that $\textnormal{$\textit{E}_{\FINTREEPRED}$} \vdash t = t'$ holds.
\end{lemma}
\begin{proof}
{The reasoning is by induction on the structure of $t$.}
\end{proof}
\begin{theorem}
\label{thm:soundness_completeness}
$\textnormal{$\textit{E}_{\FINTREEPRED}$}$ is sound and ground-complete for bisimilarity on $T(\Sigma_{\textit{FTP}})$. That is, it holds that
$(\forall t, t' \in T(\Sigma_{\textit{FTP}}))\,.\, \textnormal{$\textit{E}_{\FINTREEPRED}$} \vdash t = t' \text{ iff } t \sim t.
$
\end{theorem}
\iffalse
\begin{proof}
\end{proof}
\fi
\subsection{Axiomatizing negative premises}
\label{subsec:dagger}
A crucial step in finding a complete axiomatization for {\emph{preg}} systems is the ``axiomatization'' of negative premises (of the shape $x \notranz{a},\, \neg P x$). In the style of \cite{Aceto:1994:TSR:184662.184663}, we introduce the restriction operator $\onedagpar{\calB,\calQ}$, where $\textnormal{$\mathcal B$} \subseteq \textnormal{$\mathcal A$}$ and $\textnormal{$\mathcal Q$} \subseteq \textnormal{$\mathcal P$}$ are the sets of initially forbidden actions and predicates, respectively. The semantics of $\onedagpar{\calB,\calQ}$ is given by the two types of transition rules in Fig.~\ref{fig:TSSdag}.
\begin{figure}[H]
\begin{center}
\begin{tabular}{c@{~~~}c}
$\dfrac{x \xrightarrow{a} x'}{\onedagpar{\calB,\calQ}(x) \xrightarrow{a} {\onedagpar{\emptyset, \textnormal{$\mathcal Q$} \cap \textnormal{$\mathcal{P^I}$} }(x')}} ~ \text{ if } a \not \in \textnormal{$\mathcal B$} ~ (rl_{8})$
&
$\dfrac{Px}{P(\onedagpar{\calB,\calQ}(x))} ~ \text{ if } P \not \in \textnormal{$\mathcal Q$} ~ (rl_{9})$
\end{tabular}
\caption{The semantics of $\onedagpar{\calB,\calQ}$}
\label{fig:TSSdag}
\end{center}
\end{figure}
Note that $\onedagpar{\calB,\calQ}$ behaves like the one step restriction operator in \cite{Aceto:1994:TSR:184662.184663} for the actions in $\cal B$, as the restriction on the action set disappears after one transition. On the other hand, for the case of predicates in $\cal Q$, the operator $\onedagpar{\calB,\calQ}$ resembles the CCS restriction operator~\cite{Mi89} since, due to the presence of implicit predicates, not all the restrictions related to predicate satisfaction necessarily disappear after one step,
as will become clear in what follows.
We write $\textnormal{$\textit{E}$}_{\textit{FTP}}^{\partial}$ for the extension of $\textnormal{$\textit{E}$}_{\textit{FTP}}$ with the axioms involving $\onedagpar{\calB,\calQ}$ presented in Fig.~\ref{fig:ETdag}. ${\textnormal{$\cal R$}}_{\textit{FTP}}^{\partial}$ stands for the set of rules $(rl_1)\!\!-\!\!(rl_{9})$, while $\textit{FTP}^{\partial}$ represents the system $(\Sigma_{\textit{FTP}}^{\partial}, {\textnormal{$\cal R$}}_{\textit{FTP}}^{\partial})$.
\begin{figure}[H]
\begin{center}
\begin{tabular}{r@{\hspace{3pt}}c@{\hspace{3pt}}lll@{\hspace{20pt}}r@{\hspace{3pt}}c@{\hspace{3pt}}lll}
$\onedagpar{\calB,\calQ}(\delta)$ & = & $\delta$ & & $(A_{6})$ & $\onedagpar{\calB,\calQ}(a.x)$ & = & $\sum_{{P \not \in \textnormal{$\mathcal Q$}, P(a.x)}} \textnormal{$\kappa$}_P$ & if $a \in \textnormal{$\mathcal B$}$ & $(A_{9})$\\[1ex]
$\onedagpar{\calB,\calQ}(\textnormal{$\kappa$}_P)$ & = & $\delta$ & if $P \in \textnormal{$\mathcal Q$}$ & $(A_{7})$ & $\onedagpar{\calB,\calQ}(a.x)$ & = & $\onedagpar{\emptyset,\calQ}(a.x)$ & if $a \not \in \textnormal{$\mathcal B$}$ & $(A_{10})$\\[1ex]
$\onedagpar{\calB,\calQ}(\textnormal{$\kappa$}_P)$ & = & $\textnormal{$\kappa$}_P$ & if $P \not \in \textnormal{$\mathcal Q$}$ & $(A_{8})$ & $\onedagpar{\emptyset,\calQ}(a.x)$ & = &
{$a.\onedagpar{\emptyset, \textnormal{$\mathcal Q$} \cap \textnormal{$\mathcal{P^I}$} }(x)$} &
& $(A_{11})$\\[1ex]
\multicolumn{10}{c}{{$\onedagpar{\calB,\calQ}(x+y)$ = $\onedagpar{\calB,\calQ}(x) + \onedagpar{\calB,\calQ}(y)$ $(A_{12})$}}
\end{tabular}
\caption{The axiom system $\textnormal{$\textit{E}$}_{\textit{FTP}}^{\partial} \setminus \textnormal{$\textit{E}_{\FINTREEPRED}$}$}
\label{fig:ETdag}
\end{center}
\end{figure}
Axiom $(A_{6})$ states that it is useless to impose restrictions on $\delta$, as $\delta$ does not exhibit any behaviour.
The intuition behind $(A_{7})$ is that since a predicate witness $\textnormal{$\kappa$}_P$ does not perform any action, inhibiting the satisfiability of $P$ leads to a process with no behaviour, namely $\delta$. Consequently, if the restricted predicates do not include $P$, the resulting process is $\textnormal{$\kappa$}_P$ itself (see $(A_{8})$).
Inhibiting the only action a process $a.t$ can perform leads to a new process that, in the best case, satisfies some of the predicates in $\textnormal{$\mathcal{P^I}$}$ satisfied by $t$ (by $(rl_7)$) if $\textnormal{$\mathcal Q$} \not= \textnormal{$\mathcal{P^I}$}$ (see $(A_{9})$).
Whenever the restricted action set $\textnormal{$\mathcal B$}$ does not contain the only action a process $a.t$ can perform, then it is safe to give up $\textnormal{$\mathcal B$}$ (see $(A_{10})$).
As a process $a.t$ only satisfies the predicates also satisfied by $t$, it is straightforward to see that $\onedagpar{\emptyset, \textnormal{$\mathcal Q$}}(a.t)$ is equivalent to the process obtained by propagating the restrictions on implicit predicates deeper into the behaviour of $t$ (see $(A_{11})$).
Axiom $(A_{12})$ is given in conformity with the semantics of
$\_\hspace{-1.5pt}+\hspace{-2pt}\_\,$ ($s+t$ encapsulates both the behaviours of $s$ and $t$).
\begin{remark}
For the sake of brevity and readability, in Fig.~\ref{fig:ETdag} we presented $(A_{9})$, which is a schema with infinitely many instances. However, it can be replaced by a finite family of axioms. See Appendix~D in the full version of the paper available at \textnormal{\footnotesize{\url{http://www.ru.is/faculty/luca/PAPERS/axgsos.pdf}}} for details.
\end{remark}
\begin{theorem}
\label{thm:soundness_hnf_dagger}
The following statements hold for $\textnormal{$\textit{E}_{\FINTREEPRED}^\partial$}$:
\begin{enumerate}\itemsep1pt
\item $\textnormal{$\textit{E}_{\FINTREEPRED}^\partial$}$ is sound for bisimilarity on $T(\Sigma_{\textit{FTP}}^{\partial})$. \label{stmt:sound}
\item $\forall t \in T(\Sigma_{\textit{FTP}}^{\partial}), \exists t' \in T(\Sigma_{\textit{FTP}}) ~s.t.~ \textnormal{$\textit{E}_{\FINTREEPRED}^\partial$} \vdash t = t'$. \label{stmt:hnf}
\end{enumerate}
\end{theorem}
\iffalse
\begin{proof}
\end{proof}
\fi
As proving completeness for $\textit{FTP}^{\partial}$ can be reduced to showing completeness for $\textit{FTP}$ (already proved in Theorem~\ref{thm:soundness_completeness}), the following result is an immediate consequence of Theorem~\ref{thm:soundness_hnf_dagger}:
\begin{corollary}
\label{cor:completeness_dagger}
$\textnormal{$\textit{E}_{\FINTREEPRED}^\partial$}$ is sound and complete for bisimilarity on $T(\Sigma_{\textit{FTP}}^{\partial})$.
\end{corollary}
\section{Smooth and distinctive operations}
\label{sec:smooth}
Recall that our goal is to provide a sound and ground-complete axiomatization for bisimilarity on systems specified in the {\emph{preg}} format. As the {\emph{preg}} format is too permissive for achieving this result directly, our next task is to find a class of operations for which we can build such an axiomatization by ``easily" reducing it to the completeness result for $\textit{FTP}$, presented in Theorem~\ref{thm:soundness_completeness}. In the literature, these operations are known as \emph{smooth and distinctive}~\cite{Aceto:1994:TSR:184662.184663}. As we will see, these operations are incrementally identified by imposing suitable restrictions on {\emph{preg}} rules. The standard procedure is to first find the \emph{smooth} operations, based on which one determines the \emph{distinctive} ones.
\begin{definition}[Smooth operation]
\label{def:smooth}
~
\begin{enumerate}\itemsep1pt
\setlength\itemsep{1ex}
\item \label{eq:smoothtranz} A {\emph{preg}} transition rule is \emph{smooth} if it is of the following format:
\begin{equation}
\notag
\dfrac{
\begin{array}{c@{~~~}c}
\{ x_i \xrightarrow{a_{i}} y_{i} \mid i \in {\textnormal{$I^+$}} \} &
\{ P_{i} x_i \mid i \in {\textnormal{$J^+$}} \}
\\
\{ x_i \notranz{b} \hspace{7pt} \mid i \in {\textnormal{$I^-$}}, b \in \textnormal{$\mathcal B$}_i \} &
\{ \neg Q x_i \mid i \in {\textnormal{$J^-$}}, Q \in \textnormal{$\mathcal Q$}_i \}
\end{array}
}
{
f(x_1, \ldots, x_l) \xrightarrow{c}
C[\vec{x}, \vec{y}]
}
\end{equation}
where
\begin{enumerate}\itemsep1pt
\item \label{def:smooth:premise}${\textnormal{$I^+$}},{\textnormal{$J^+$}},{\textnormal{$I^-$}},{\textnormal{$J^-$}}$ disjointly cover the set $L = \{1,\ldots,l\}$,
\item in the target $C[\vec{x}, \vec{y}]$ we allow only:
$y_i$ $(i \in {\textnormal{$I^+$}})$, $x_i$ $(i \in {\textnormal{$I^-$}} \cup {\textnormal{$J^-$}})$. \label{lb:no-x-target}
\end{enumerate}
\item \label{eq:smoothsat} A $\emph{preg}$ predicate rule is {\em smooth} if it has the form above, its premises satisfy condition (\ref{def:smooth:premise}) and its conclusion is $P(f(x_1,\ldots, x_l))$ for some $P \in\mathcal{P}$.
\item An operation $f$ of a {\emph{preg}} system is \emph{smooth} if all its (transition and predicate) rules are smooth.
\end{enumerate}
\end{definition}
By Definition~\ref{def:smooth}, a rule $\rho$ is smooth if it satisfies the following properties:
\begin{itemize}\itemsep1pt
\item a position $i$ cannot be tested both positively and negatively at the same time,
\item positions tested positively are either from {{\textnormal{$I^+$}}} or {{\textnormal{$J^+$}}} and they are not tested for the performance of multiple transitions (respectively, for the satisfiability of multiple predicates) within the same rule, and
\item if $\rho$ is a transition rule, then the occurrence of variables at positions $i \in {\textnormal{$I^+$}} \cup {\textnormal{$J^+$}}$ is not allowed in the target of the consequent of $\rho$.
\end{itemize}
\begin{remark}
Note that we can always consider a position $i$ that does not occur as a premise in a rule for $f$ as being negative, with the empty set of constraints (i.e. either $i \in {\textnormal{$I^-$}}$ and $\textnormal{$\mathcal B$}_i = \emptyset$, or $i \in {\textnormal{$J^-$}}$ and $\textnormal{$\mathcal Q$}_i = \emptyset$).
\end{remark}
\begin{definition}[Distinctive operation]
\label{def:smooth-distinctive}
An operation $f$ of a $\emph{preg}$ system is \emph{distinctive} if it is smooth and:
\begin{itemize}\itemsep1pt
\item for each argument $i$, either all rules for $f$ test $i$ positively, or none of them does, and
\item for any two distinct rules for $f$ there exists a position $i$ tested positively, such that one of the following holds:
\begin{itemize}\itemsep1pt
\item[-] both rules have actions that are different in the premise at position $i$,
\item[-] both rules have predicates that are different in the premise at position $i$,
\item[-] one rule has an action premise at position $i$, and the other rule has a predicate test at the same position $i$.
\end{itemize}
\end{itemize}
\end{definition}
According to the first requirement in Definition~\ref{def:smooth-distinctive}, we state that for a smooth and distinctive operation $f$, a position $i$ is \emph{positive} (respectively, \emph{negative}) for $f$ if there is a rule for $f$ such that $i$ is tested positively (respectively, negatively) for that rule.
The existence of a family of smooth and distinctive operations ``describing the behaviour" of a general {\emph{preg}} operation is formalized by the following lemma:
\begin{lemma}
\label{lm:distinctive-law-all}
Consider a {\emph{preg}} system $G$. Then there exist a {\emph{preg}} system $G'$, which is a disjoint extension of $G$ and $\textit{FTP}$, and a finite axiom system $\textnormal{$\textit{E}$}$ such that
\begin{enumerate}
\item $\textnormal{$\textit{E}$}$ is sound for bisimilarity over any disjoint extension $G''$ of $G'$, and
\item for each term $t$ in $T(\Sigma_{G})$ there is some term $t'$ in $T(\Sigma_{G'})$ such that $t'$ is built solely using smooth and distinctive operations and $\textnormal{$\textit{E}$}$ proves $t = t'$.
\end{enumerate}
\end{lemma}
\subsection{Axiomatizing smooth and distinctive {\emph{preg}} operations}
\label{sec:axiom-amoothAndSist}
To start with, consider, for the good flow of the presentation, that we only handle explicit predicates (\emph{i.e.}, we take $\textnormal{$\mathcal{P^I}$} = \emptyset$). Towards the end of the section we discuss how to extend the presented theory to implicit predicates.
We proceed in a similar fashion to~\cite{Aceto:1994:TSR:184662.184663} by defining a set of laws used in the construction of a complete axiomatization for bisimilarity on terms built over smooth and distinctive operations. The strength of these laws lies in their capability of reducing terms to their head normal form, thus reducing completeness for general {\emph{preg}} systems to completeness of $\textnormal{$\textit{E}_{\FINTREEPRED}$}$ (which has already been proved in Section~\ref{sec:axFinTrees}).
\begin{definition}
\label{def:axioms}
Let $f$ be a smooth and distinctive $l$-ary operation of a {\emph{preg}} system $G$, such that ${\textit{FTP}}^{\partial} \sqsubseteq G$.
\begin{enumerate}\itemsep1pt
\setlength\itemsep{1ex}
\item \label{enm:distr} For a positive position $i \in {\textnormal{$L$}} = \{1, \ldots, l\}$, the \emph{distributivity law} for $i$ w.r.t. $f$ is given as follows:
\[
f(X_1, \ldots, X'_i + X''_i, \ldots, X_l) =
f(X_1, \ldots, X'_i, \ldots, X_l) +
f(X_1, \ldots, X''_i, \ldots, X_l).
\]
\item \label{enm:actpred}
{For a rule $\rho \in \textnormal{$\cal R$}$ for $f$ the \emph{{trigger law}} is, depending on whether $\rho$ is a transition or a predicate rule:}
\[
f(\vec{X}) =
\left\{
\begin{array}{rll}
c.C[\vec{X}, \vec{y}] &,~\rho \in \textnormal{$\setrules^{\calA}$} &\emph{{(action law)}}\\
\textnormal{$\kappa$}_P &,~\rho \in \textnormal{$\setrules^{\calP}$} &\emph{{(predicate law)}}
\end{array}
\right.
\]
\noindent
where
\[
X_i \equiv
\left\{
\begin{array}{rl}
a_i.y_i &,~i \in {\textnormal{$I^+$}}\\
\textnormal{$\kappa$}_{P_i} &,~i \in {\textnormal{$J^+$}}\\
\onedagpar{{\textnormal{$\mathcal B$}_i}, {\textnormal{$\mathcal Q$}_i}}(x_i) &,~i \in {\textnormal{$I^-$}} \cup {\textnormal{$J^-$}}\\
\end{array}
\right..
\]
\item \label{enm:dead}
Suppose that for $i \in {\textnormal{$L$}}$, term $X_i$ is in one of the forms $\delta, z_{i}, \textnormal{$\kappa$}_{P_i}, a.z_i, a.z_i + z'_i$ or $\textnormal{$\kappa$}_{P_i} + z_i$.
Suppose further that for each rule for $f$ there exists $X_j \in \vec{X}$ ($j \in \{1, \ldots, l\}$) s.t. one of the following holds:
\begin{itemize}\itemsep1pt
\item $j \in {\textnormal{$I^+$}}$ and ($X_j \equiv \delta$ or $X_j \equiv b.z_j ~ (b \not= a_j)$ or $X_j \equiv \textnormal{$\kappa$}_Q$, for some $Q$),
\item $j \in {\textnormal{$J^+$}}$ and ($X_j \equiv \delta$ or $X_j \equiv \textnormal{$\kappa$}_{Q}~ \text{(}Q \not= P_j\text{)}$ or $X_j \equiv b.z_j$, for some $b$),
\item $j \in {\textnormal{$I^-$}}$ and $X_j \equiv b.z_j + z'_j$, where $b \in \textnormal{$\mathcal B$}_j$,
\item $j \in {\textnormal{$J^-$}}$ and $X_j \equiv \textnormal{$\kappa$}_Q + z_j$, where $Q \in \textnormal{$\mathcal Q$}_j$.
\end{itemize}
Then the \emph{deadlock law} is as follows:
\[
f(\vec{X}) = \delta.
\]
\end{enumerate}
\end{definition}
\begin{example}
\label{ex:toy}
Consider the \emph{right-biased sequential composition} operation $\_\hspace{1.5pt};^{r}\hspace{-3pt}\_$\hspace{2pt}, whose semantics is given by the rules $
\frac{x\downarrow ~ y ~\xrightarrow{a}~ y'}{x ~;^{r}~ y ~\xrightarrow{a}~ y'}$,\hspace{0.5ex}$
\frac{x\downarrow ~ y\downarrow}{(x ~;^{r}~ y)\downarrow}$, and\hspace{0.5ex}$
\frac{x\downarrow ~ y\uparrow}{(x ~;^{r}~ y)\uparrow}
$, where $\downarrow$ \,and $\uparrow$ are, respectively, the \emph{immediate termination} and \emph{immediate divergence} predicates.
$\_\hspace{1.5pt};^{r}\hspace{-3pt}\_$\hspace{2pt} is one of the auxiliary operations generated by the algorithm for deriving smooth and distinctive operations when axiomatizing the \emph{sequential composition} in the presence of the two mentioned predicates.
The laws derived according to Definition~\ref{def:axioms}\, for this system are:
\begin{center}
\begin{tabular}{rl@{\hspace{4ex}}rl}
$(x + y) ~;^{r} z$ &$=~~ x ~;^{r} z ~~+~~ y ~;^{r} z$&
$\delta ~;^{r} y$ &$=~~ \delta$\\
$x ~;^{r} (y + z)$ &$=~~ x ~;^{r} y ~~+~~ z ~;^{r} z$&
$k_{\uparrow} ~;^{r} y$ &$=~~ \delta$\\
$k_{\downarrow} ~;^{r} a.y$ &$=~~ a.y$&
$a.x ~;^{r} y$ &$=~~ \delta$\\
$k_{\downarrow} ~;^{r} k_{\downarrow}$ &$=~~ k_{\downarrow}$&
$x ~;^{r} \delta$ &$=~~ \delta$\\
$k_{\downarrow} ~;^{r} k_{\uparrow}$ &$=~~ k_{\uparrow}$&
\ldots
\end{tabular}
\end{center}
\end{example}
\medskip
\begin{theorem}
\label{thm:s&c-smooth-dist}
Consider $G$ a {\emph{preg}} system such that ${\textit{FTP}}^{\partial} \sqsubseteq G$. Let $\Sigma \subseteq \Sigma_G \setminus \Sigma_{\textit{FTP}}^{\partial}$ be a collection of smooth and distinctive operations of $G$. Let $\textnormal{$\textit{E}$}_G$ be the finite axiom system that extends $\textnormal{$\textit{E}_{\FINTREEPRED}^\partial$}$ with the following axioms for each $f \in \Sigma$:
\begin{itemize}\itemsep1pt
\item for each positive argument $i$ of $f$, a distributivity law (Definition~\ref{def:axioms}.\ref{enm:distr}),
\item for each transition rule for $f$, an action law (Definition~\ref{def:axioms}.\ref{enm:actpred}),
\item for each predicate rule for $f$, a predicate law (Definition~\ref{def:axioms}.\ref{enm:actpred}), and
\item all deadlock laws for $f$ (Definition~\ref{def:axioms}.\ref{enm:dead}).
\end{itemize}
The following statements hold for $\textnormal{$\textit{E}$}_G$, for any $G'$ such that $G \sqsubseteq G'$:
\begin{enumerate}\itemsep1pt
\item $\textnormal{$\textit{E}$}_G$ is sound for bisimilarity on $T(\Sigma_{G'})$. \label{stmt:sd-sound}
\item $\textnormal{$\textit{E}$}_G$ is head normalizing for $T(\Sigma \cup \Sigma_{\textit{FTP}}^{\partial})$. \label{stmt:sd-hnf}
\end{enumerate}
\end{theorem}
\begin{comment}
\begin{proof}
Soundness (statement \ref{stmt:sd-sound}.) follows in a standard fashion, by analyzing the action performance and the predicate satisfiability for the terms checked for bisimilarity.
For the head normalization part (statement \ref{stmt:sd-hnf}.), we proceed as follows.
By Theorem~\ref{thm:soundness_hnf_dagger} and Lemma~\ref{lem:hnf_exists}, $\textnormal{$\textit{E}$}_G \supseteq \textnormal{$\textit{E}_{\FINTREEPRED}^\partial$} \supseteq \textnormal{$\textit{E}_{\FINTREEPRED}$}$
is head normalizing for $T(\Sigma_{\textit{FTP}}^{\partial}) \supseteq T(\Sigma_{\textit{FTP}})$.
At this point, proving the head normalization of $T(\Sigma \cup \Sigma_{\textit{FTP}}^{\partial})$ reduces to proving the following claim:\\
\noindent
{\it Claim.} Let $f \in \Sigma$ be a smooth and distinctive operation with the arguments $\vec{t} = t_1, \ldots, t_l$ as closed terms over $\Sigma_G$ in head normal form. Then there exists a closed term $t'$ over $\Sigma_G$ in head normal form such that $\textnormal{$\textit{E}$}_G \vdash f(\vec{t}) = t'$.\\
The result follows by induction on the the combined size of $t_1, \ldots, t_l$.\\
\end{proof}
\end{comment}
Obtaining the soundness of the action law (Definition~\ref{def:axioms}.\ref{enm:actpred}) requires some care when allowing for specifications with implicit predicates ($\textnormal{$\mathcal{P^I}$} \not= \emptyset$).
Consider a scenario in which a transition rule for a smooth and distinctive operation $f$ is of the form $\frac{H}{f(\vec{X}) \xrightarrow{c} C[\vec{X}, \vec{y}]}$.
Assume the closed instantiation $\vec{X} = \vec{s}$, $\vec{y} = \vec{t}$ and assume that $P(c. C[\vec{s},\vec{t}])$ holds for some predicate $P$ in $\textnormal{$\mathcal{P^I}$}$. This means that $P(C[\vec{s},\vec{t}])$ holds. In order to preserve the soundness of the action law, $P(f(\vec{s}))$ should also hold, but this is impossible since $f$ is distinctive. One possible way of ensuring the soundness of the action law in the presence of implicit predicates is to stipulate some syntactic consistency requirements on the language specification. One sufficient requirement would be that if predicate rule $\frac{H'}{P(C[\vec{z},\vec{y}])}$ is derivable, then the system should contain a predicate rule $\frac{H''}{P(f[\vec{z}])}$ with $H'' \subseteq H'$. This is enough to guarantee that if the right-hand side of the action law satisfies $P$ then so does the left-hand side.
\section{Soundness and completeness}
\label{sec:completeness}
Let us summarize our results so far. By Theorem~\ref{thm:s&c-smooth-dist}, it follows that, for any {\emph{preg}} system $G \sqsupseteq {\textit{FTP}}^{\partial}$, there is an axiomatization that is head normalizing for $T(\Sigma \cup \Sigma_{\textit{FTP}}^{\partial})$, where
$\Sigma \subseteq \Sigma_G \setminus \Sigma_{\textit{FTP}}^{\partial}$ is a collection of smooth and distinctive operations of $G$.
Also, as hinted in Section~\ref{sec:smooth} (Lemma~\ref{lm:distinctive-law-all}), there exists a sound algorithm for transforming general {\emph{preg}} operations to smooth and distinctive ones.
So, for any {\emph{preg}} system $G$, we can build a {\emph{preg}} system $G' \sqsupseteq G$ and an axiomatization $\textnormal{$\textit{E}$}_{G'}$ that is head normalizing for $T(\Sigma_{G'})$. This statement is formalized as follows:
\begin{theorem}
\label{thm:s&c-fin-proc}
Let $G$ be a {\emph{preg}} system. Then there exist $G'\sqsupseteq G$ and a finite axiom system $\textnormal{$\textit{E}$}_{G'}$ such that
\begin{enumerate}\itemsep1pt
\item $\textnormal{$\textit{E}$}_{G'}$ is sound for bisimilarity on $T(\Sigma_{G'})$, \label{snd}
\item $\textnormal{$\textit{E}$}_{G'}$ is head normalizing for $T(\Sigma_{G'})$,
\end{enumerate}
and moreover, $G'$ and $\textnormal{$\textit{E}$}_{G'}$ can be effectively constructed from $G$.
\end{theorem}
\begin{proof}
{The result follows immediately by Theorem~\ref{thm:s&c-smooth-dist} and
by the existence of an algorithm used for transforming general {\emph{preg}} to smooth and distinctive operations}.
\end{proof}
\begin{remark}
Theorem~\ref{thm:s&c-fin-proc} guarantees ground-completeness of the generated axiomatization for well-founded {\emph{preg}} specifications, that is, {\emph{preg}} specifications in which each process can only exhibit finite behaviour.
\end{remark}
Let us further recall an example given in~\cite{Aceto:1994:TSR:184662.184663}. Consider the constant $\omega$, specified by the rule $\omega \xrightarrow{a} \omega$. Obviously, the corresponding action law $\omega = a.\omega$ will apply for an infinite number of times in the normalization process.
So the last step in obtaining a complete axiomatization is to handle infinite behaviour.
Let $t$ and $t'$ be two processes with infinite behaviour (remark that the infinite behaviour is a consequence of performing actions for an infinite number of times, so the extension to predicates is not a cause for this issue). Since we are dealing with finitely branching processes, it is well known that if two process terms are bisimilar at each finite depth, then they are bisimilar. One way of formalizing this requirement is to use the well-known \emph{Approximation Induction Principle} (AIP) \cite{Baeten:1991:PA:103272,Bergstra:1986:VAB:16663.16664}.
Let us first consider the operations $\pi_n(\cdot)$, $n \in \mathbb{N}$, known as \emph{projection operations}. The purpose of these operations is to stop the evolution of processes after a certain number of steps. The AIP is given by the following conditional equation:
\[
{x = y} \textnormal{ if } \pi_n(x) = \pi_n(y) ~ (\forall n \in \mathbb{N}).
\]
We further adapt the idea in \cite{Aceto:1994:TSR:184662.184663} to our context, and model the infinite family of projection operations $\pi_n(\cdot)$, $n \in \mathbb{N}$, by a binary operation $\cdot / \cdot$ defined as follows:
\begin{center}
\begin{tabular}{c@{~~~}c}
$\dfrac{x \xrightarrow{a} x' ~~ h \xrightarrow{c} h'}{x/h \xrightarrow{a} x'/h'} ~ (rl_{10})$
&
$\dfrac{Px}{P(x/h)} ~ (rl_{11})$
\end{tabular}
\end{center}
where $c$ is an arbitrary action. Note that $\cdot / \cdot$ is a smooth and distinctive operation.
The role of variable $h$ is to ``control" the evolution of a process, \textit{i.e.}, to stop the process in performing actions, after a given number of steps. Variable $h$ (the ``hourglass" in \cite{Aceto:1994:TSR:184662.184663}) will always be instantiated with terms of the shape $c^n$, inductively defined as: $c^0 = \delta$, $c^{n+1} = c.c^{n}$.
Let $G = (\Sigma_G, \textnormal{${\cal R}_G$})$ be a {\emph{preg}} system. We use the notation $G_{/}$ to refer to the {\emph{preg}} system $(\Sigma_G \cup \{\cdot/\cdot\}, \textnormal{${\cal R}_G$} \cup \{(rl_{10}), (rl_{11})\})$ -- the extension of $G$ with $\cdot / \cdot$~. Moreover, we use the notation $\textnormal{$\textit{E}$}_{\it{AIP}}$ to refer to the axioms for the smooth and distinctive operation $\cdot / \cdot$, derived as in Section~\ref{sec:axiom-amoothAndSist} -- Definition~\ref{def:axioms}.
\begin{comment}
Based on the fact that $\cdot / \cdot$ is a smooth and distinctive operation, we derive the following axioms, as in Section~\ref{sec:axiom-amoothAndSist} -- Definition~\ref{def:axioms}:
\begin{figure}[H]
\begin{center}
\begin{tabular}{r@{\hspace{3pt}}c@{\hspace{3pt}}ll}
$(x+y)/z$ & = & $x/z + y/z$ & $(A_{13})$\\[1ex]
$x/(y+z)$ & = & $x/y + x/z$ & $(A_{14})$\\[1ex]
$a.x/c.y$ & = & $a.(x/y)$ & $(A_{15})$\\[1ex]
$\delta/y$ & = & $\delta$ & $(A_{16})$\\[1ex]
$a.x/\delta$ & = & $\delta$ & $(A_{17})$\\[1ex]
$\textnormal{$\kappa$}_{P}/y$ & = & $\textnormal{$\kappa$}_{P}$ & $(A_{18})$
\end{tabular}
\caption{The axiom system $\textnormal{$\textit{E}$}_{\it{AIP}}$}
\label{fig:AIP}
\end{center}
\end{figure}
\end{comment}
We reformulate AIP according to the new operation $\cdot/\cdot$~:
\[
x = y \textnormal{ if } x/c^n = y/c^n ~ (\forall n \in \mathbb{N})
\]
\begin{lemma}
\label{lm:aip}
AIP is sound for bisimilarity on $T(\Sigma_{\textit{FTP}_{/}})$.
\end{lemma}
\iffalse
\begin{proof}
\end{proof}
\fi
\begin{comment}
\begin{remark}
Note that axiom $x/\delta = \delta$ (51) in \cite{Aceto:1994:TSR:184662.184663} is not sound for our approach, as by $(rl_{11})$ $x/\delta$ satisfies all the predicates satisfied by $x$, while by definition, $\delta$ satisfies no predicates. For the case of {\emph{preg}} systems, axiom (51) is ``encoded" by $(A_{17})$, and $(A_{18})$ for $y = \delta$.
\end{remark}
\end{comment}
In what follows we provide the final ingredients for proving the existence of a ground-complete axiomatization for bisimilarity on {\emph{preg}} systems. As previously stated, this is achieved by reducing completeness to proving equality in {\textit{FTP}}. So, based on AIP, it would suffice to show that for any closed process term $t$ and natural number $n$, there exists an {\textit{FTP}} term equivalent to $t$ at moment $n$ in time:
\begin{lemma}
\label{lm:aip-ftp}
Consider $G$ a {\emph{preg}} system. Then there exist $G' \sqsupseteq G_{/}$ and $\textnormal{$\textit{E}$}_{G'}$ with the property: $\forall t \in T(\Sigma_{G'}), \forall n \in \mathbb{N}, \exists t' \in T(\Sigma_{\textit{FTP}})$ s.t. $\textnormal{$\textit{E}$}_{G'} \vdash t/c^n = t'$.
\end{lemma}
At this point we can prove the existence of a sound and ground-complete axiomatization for bisimilarity on general $\emph{preg}$ systems:
\begin{theorem}[Soundness and Completeness]
\label{thm-s&c}
Consider $G$ a {\emph{preg}} system. Then there exist $G' \sqsupseteq G_{/}$ and $\textnormal{$\textit{E}$}_{G'}$ a finite axiom system, such that $\textnormal{$\textit{E}$}_{G'} \cup \textnormal{$\textit{E}$}_{\it{AIP}}$ is sound and complete for bisimilarity on $T(\Sigma_{G'})$.
\end{theorem}
\section{Motivation for handling predicates as first-class notions}
\label{sec:rationale}
In the literature on the theory of rule formats for Structural
Operational Semantics (especially, the work devoted to congruence
formats for various notions of bisimilarity), predicates are often
neglected at first and are only added to the considerations at a later
stage. The reason is that one can encode predicates quite easily by
means of transition relations. One can find a number of such encodings
in the literature---see, for instance,~\cite{assoc,Verhoef95}. In each
of these encodings, a predicate $P$ is represented as a transition
relation $\xrightarrow{P}$ (assuming that $P$ is a fresh action label)
with a fixed constant symbol as target. Using this translation, one can
axiomatize bisimilarity over {\emph{preg}} language specifications by first
converting them into ``equivalent'' standard GSOS systems, and then
applying the algorithm from~\cite{Aceto:1994:TSR:184662.184663} to
obtain a finite axiomatization of bisimilarity over the resulting GSOS
system.
In light of this approach, it is natural to wonder whether it is
worthwhile to develop an algorithm to axiomatize {\emph{preg}} language
specifications directly. One possible answer, which has been
presented several times in the literature~\cite{Verhoef95}, is that
often one does not want to encode a language specification with
predicates using one with transitions only. Sometimes, specifications
using predicates are the most natural ones to write, and one should
not force a language designer to code predicates using
transitions. (However, one can write a tool to perform the translation
of predicates into transitions, which can therefore be carried out
transparently to the user/language designer.) Also, developing an
algorithm to axiomatize GSOS language specifications with predicates
directly yields insight into the difficulties that result from the
first-class use of, and the interplay among, various types of
predicates, as far as axiomatizability problems are concerned. These
issues would be hidden by encoding predicates as
transitions. Moreover, the algorithm resulting from the encoding would
generate axioms involving predicate-prefixing operators, which are
somewhat unintuitive.
Naturalness is, however, often in the eye of the beholder. Therefore,
we now provide a more technical reason why it may be worthwhile to
develop techniques that apply to GSOS language specifications with
predicates as first-class notions, such as the {\emph{preg}} ones. Indeed,
we now show how, using predicates, one can convert any standard GSOS
language specification $G$ into an equivalent {\em positive} one with
predicates $G^+$.
Given a GSOS language $G$, the system $G^+$ will have the same
signature and the same set of actions as $G$, but uses predicates
$\text{cannot}(a)$ for each action $a$. The idea is simply that ``$x
\:\text{cannot}(a)$'' is the predicate formula that expresses that ``$x$
does not afford an $a$-labelled transition''. The translation works as
follows.
\begin{enumerate}
\item Each rule in $G$ is also a rule in $G^+$, but one replaces each
negative premise in each rule with its corresponding positive
predicate premise. This means that $x \notranz{a}$ becomes $x
\:\text{cannot}(a)$.
\item One adds to $G^+$ rules defining the predicates
$\text{cannot}(a)$, for each action $a$. This is done in such a way
that $p\:\text{cannot}(a)$ holds in $G^+$ exactly when $p\notranz{a}$
in $G$, for each closed term $p$ and action $a$. More precisely, we
proceed as follows.
\begin{enumerate}
\item For each constant symbol $f$ and action $a$, add the
rule
\[
\frac{}{f \:\text{cannot}(a)}
\]
whenever there is no transition rule in $G$ with $f$ as principal
operation and with an $a$-labelled transition as its consequent.
\item For each operation $f$ with arity
at least one and action $a$, let $R(f,a)$ be the set of rules in $G$ that
have $f$ as principal operation and an $a$-labelled transition as consequent. We want to
add rules for the predicate $\text{cannot}(a)$ to $G^+$ that allow us to prove
the predicate formula $f(p_1,\ldots,p_l) \:\text{cannot}(a)$ exactly when
$f(p_1,\ldots,p_l)$ does not afford an $a$-labelled transition in $G$. This
occurs if, for each rule in $R(f,a)$, there is some premise that
is not satisfied when the arguments of $f$ are $p_1,\ldots,p_l$. To
formalize this idea, let $H(R(f,a))$ be the collection of premises of
rules in $R(f,a)$. We say that a choice function is a function $\phi:
R(f,a) \rightarrow H(R(f,a))$ that maps each rule in $R(f,a)$ to one of its
premises. Let
\begin{eqnarray*}
\text{neg}(x \xrightarrow{a} x') & = & x \:\text{cannot}(a) \quad \text{and} \\
\text{neg}(x \notranz{a}) & = & x \xrightarrow{a} x' , \quad \text{for some } x'. \end{eqnarray*}
Then, for each choice function $\phi$, we add
to $G^+$ a predicate rule of the form
\[
\frac{\{\text{neg}(\phi(\xi)) \mid \xi\in
R(f,a)\}}{f(x_1,\ldots,x_l) \:\text{cannot}(a)} ,
\]
where the targets of the positive
transition formulae in the premises are chosen to be all different.
\end{enumerate}
\end{enumerate}
The above construction ensures the validity of the following lemma.
\begin{lemma}
\label{lm:GtoGplus}
For each closed term $p$ and action $a$,
\begin{enumerate}
\item $p \xrightarrow{a} p'$ in $G$ if, and only if, $p \xrightarrow{a} p'$ in $G^+$;
\item $p \:\text{cannot}(a)$ in $G^+$ if, and only if, $p \notranz{a}$
in $G^+$ (and therefore in $G$).
\end{enumerate}
\end{lemma}
This means that two closed terms are bisimilar in $G$ if, and only if,
they are bisimilar in $G^+$. Moreover, two closed terms are bisimilar
in $G^+$ iff they are bisimilar when we only consider the transitions
(and not the predicates $\text{cannot}(a)$).
The language $G^+$ modulo bisimilarity can be axiomatized using our
algorithm without the need for the exponentially many
restriction operators. The conversion to positive GSOS with predicates
discussed above does incur in an exponential blow-up in the number of
rules, but it gives an alternative way of generating ground-complete
axiomatizations for standard GSOS languages to the one proposed
in~\cite{Aceto:1994:TSR:184662.184663}. In general, it is useful to
have several approaches in one's toolbox, since one may choose the one
that is ``less expensive'' for the specific task at hand. Moreover,
using positive GSOS operations, one can also try to extend the methods
from the full version of the paper~\cite{DBLP:conf/concur/Aceto94}
(see Section 7.1 in the technical report available at
{\footnotesize{\url{http://www.ru.is/~luca/PAPERS/cs011994.ps}}}) to optimize these
axiomatizations. We are currently working on applying such methods to
positive {\emph{preg}} systems with universal as well as existential
predicates, and on extending our tool~ \cite{pregax-calco-tools2011}
accordingly.
It is worth noting that the predicates $\text{cannot}(a)$ are not implicit, therefore the restrictions presented at the end of Section~\ref{sec:axiom-amoothAndSist} need not to be imposed.
\section{Conclusions and future work}
\label{sec:conclusions}
In this paper we have introduced the {\emph{preg}} rule format, a natural extension of GSOS with arbitrary predicates. Moreover, we have provided a procedure (similar to the one in~\cite{Aceto:1994:TSR:184662.184663}) for deriving sound and ground-complete axiomatizations for bisimilarity of systems that match this format. In the current approach, explicit predicates are handled by considering constants witnessing their satisfiability as summands in tree expressions. Consequently, there is no explicit predicate $P$ satisfied by a term of shape $\Sigma_{i\in I}a_i.t_i$.
The procedure introduced in this paper has also enabled the implementation of a tool \cite{pregax-calco-tools2011} that can be used to automatically reason on bisimilarity of systems specified as terms built over operations defined by {\emph{preg}} rules.
Several possible extensions are left as future work. It would be worth investigating the properties of positive {\emph{preg}} languages. By allowing only positive premises we eliminate the need of the restriction operators (\onedagpar{\calB,\calQ}) during the axiomatization process. This would enable us to deal with more general predicates over trees, such as those that may be satisfied by terms of the form $a.t$ where $a$ ranges over some subset of the collection of actions.
Another direction for future research is that of understanding the presented work from a coalgebraic perspective. The extensions from \cite{Aceto:1994:TSR:184662.184663} to the present paper, might be thought as an extension from coalgebras for a functor $\mathscr{P}(\textnormal{$\mathcal A$} \times \textit{Id})$ to a functor $\mathscr{P}(\textnormal{$\mathcal P$}) \times \mathscr{P}(\textnormal{$\mathcal A$} \times \textit{Id})$ where $\mathscr{P}$ is the powerset functor, $\textnormal{$\mathcal A$}$ is the set of actions and $\textnormal{$\mathcal P$}$ is the set of predicates. Also the language {\textit{FTP}} coincides, apart from the recursion operator, with the one that would be obtained for the functor $\mathscr{P}(\textnormal{$\mathcal P$}) \times \mathscr{P}(\textnormal{$\mathcal A$} \times \textit{Id})$ in the context of Kripke polynomial coalgebras \cite{DBLP:conf/lics/BonsangueRS09}.
Finally, we plan to extend our axiomatization theory in order to reason on the bisimilarity of guarded recursively defined terms, following the line presented in \cite{DBLP:conf/concur/Aceto94}.
\paragraph{Acknowledgments.} The authors are grateful for the useful comments and suggestions from Alexandra Silva and three anonymous reviewers.
\bibliographystyle{eptcs}
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{
"redpajama_set_name": "RedPajamaArXiv"
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Borken (Westf) station () is the main station of the town of Borken and important transport hub of west Münsterland in the German state of North Rhine-Westphalia.
Borken station is a former railway junction on the Gelsenkirchen-Bismarck–Winterswijk railway, the Empel-Rees–Münster railway and the Borken–Steinfurt railway. Since 1996, it has been the terminus of the only section of the Gelsenkirchen-Bismarck–Winterswijk line that is still operating.
History
The Dutch Westphalian Railway (NWE) began to build its Gelsenkirchen-Bismarck–Winterswijk line in 1878. It opened it together with the line from Borken-Gemen station as a through station on 21 June 1880. It was named Borken (Westphalia) station on 6 March 1883.
The Prussian state railways (PSE) took over the line of the NWE in 1889. On 1 August 1902, it opened the Empel-Rees–Münster railway from Bocholt station, making Borken station into a railway junction. Just two months later, on 1 October 1902, the Westfälische Landes-Eisenbahn (Westphalian Lands Railway, WLE) opened its terminus of the Borken–Steinfurt line nearby. On 1 October 1904, the PSE opened the Borken-Coesfeld section of the Empel-Rees–Münster line.
Operations
Borken station is now served only by a single Regional-Express service. This is like a Stadt-Express service, as it stops on the section south of Gladbeck West only at the most important stations, while each station is served on the northern section:
Bus services
Notes
External links
Railway stations in North Rhine-Westphalia
Railway stations in Germany opened in 1880
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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Archive for July 7th, 2015
A player for all genres: Nick Cole's heroic video gamer assumes the mantle of a knight-errant in 'Soda Pop Soldier'
Nick Cole's 2014 novel, Soda Pop Soldier, is a fun science-fiction romp with literary value roughly equal to the nutritional value of — well, of soda.
Cole's vaguely realized protagonist doesn't even get a proper name; most of the time, he's known as PerfectQuestion, his in-game handle for the WarWorld video game competitions. At other times, others address the character as Wu, the moniker of the samurai he plays in an illicit fantasy video game.
Still, the plot is fairly compelling. Several decades in the future, Question has a job playing WarWorld games on his computer. The results have real-life consequences: Each victory on a given virtual front rewards the winning team's sponsor corporation with valuable real-world advertising space. Unfortunately, Question's sponsor, ColaCorp, has been losing battle after battle to the enemy WonderSoft corporation in a modern-warfare game set in a fictitious Southeast Asian country. (For ColaCorp, read Coca-Cola; for WonderSoft, Microsoft.) If things continue on this course, the entire team will be fired.
Tags: Aliens, Coca-Cola, computer games, Ernest Cline, fantasy adventure games, Game of Thrones, George R.R. Martin, George RR Martin, Harry Bosch, James Cameron, Michael Connelly, Microsoft, Nick Cole, Ready Player One, science fiction novel, Soda Pop Soldier, video games
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
\IEEEPARstart{T}{he} wireless communications channel may undergo the effect of the multipath and shadowing fading simultaneously [1]. Accordingly, many works have been recently dedicated to analyse the performance of the generalized composite models, such as, $\kappa-\mu$, $\eta-\mu$, and $\alpha-\mu$ distributions that are used to model the line-of-sight (LoS), the non-LoS (NLoS), and the non-linear wireless communication mediums, respectively [2], [3]. These generalized conditions can provide close results to the practical measurements and approximately comprise all the classical fading distributions, i.e., Rayleigh, Nakagami-$m$, Nakagami-$n$, Nakagami-$q$, one-sided Gaussian, Weibull, and Gamma. Hence, the probability density function (PDF), the cumulative distribution function (CDF), and the moment generating function (MGF) of the composite $\eta-\mu$/Gamma fading models were derived in [4]. The authors in [5] assumed that both $\kappa-\mu$ and $\eta-\mu$ fading conditions are shadowed by an inverse Gamma distribution. The fundamental statistics of the $\kappa-\mu$ shadowed fading in which the shadowing effect is represented by a Nakagami-$m$ distribution were given in [6] with applications to the outage probability (OP) and the average bit error probability (ABEP) of wireless communications systems. The composite of the $\alpha-\kappa-\mu$/Gamma distribution was analysed in [7]. The non-linear scenario of the Fisher-Snedecor $\mathcal{F}$ distribution [8], namely, $\alpha-\mathcal{F}$, was investigated in [9]. The PDF, the CDF, and the MGF of the $\alpha-\eta-\mathcal{F}$ and the $\alpha-\kappa-\mathcal{F}$ composite fading conditions were derived in [10]. These models were then unified in a single distribution which is named $\alpha-\eta-\kappa-\mathcal{F}$ composite fading [11].
\par Although different statistical properties have been reported in [4]-[11], they are derived in mathematically intractable expressions. This is because they are expressed in terms of either the hypergeometric or the modified Bessel functions. Therefore, the mathematical intricacy of the performance metrics of the wireless communications systems is high and this would to unclear insights into the system behaviour with the fading parameters. Moreover, this complexity is increased when the generalised multipath fading channels undergo to double shadowing impacts [12].
\par To overcome the aforementioned challenges, a mixture Gamma (MG) distribution has been widely used to approximate with high accuracy most of the composite generalised/Gamma fading conditions [13]-[16]. For instance, the average area under the receiver operating characteristics (AUC) curve of the energy detection (ED) based spectrum sensing was derived in [14]. The average channel capacity (ACC) of $\alpha-\eta-\lambda-\mu$/Gamma fading and the effective capacity (EC) of $\alpha-\eta-\mu$/Gamma fading were analyzed in [15] and [16], respectively, using a MG distribution.
\par Based on the above observations and motivated by the merits of a MG distribution, we propose a mixture Gamma shadowed (MGS) as a unified composite model where the shadowing is represented by an inverse Nakagami-$m$. To this end, the basic statistics of this distribution are mathematically simple and tractable. Consequently, a clear insight into the effects of the fading parameters on the performance of the wireless communications systems can be deduced when the channel is subjected to double shadowing impacts.
\par Our main contributions are summarized as follows:
\begin{itemize}
\item The exact and the asymptotic expressions at high average signal-to-noise ratio (SNR) values for novel unified mathematically tractable statistical characterizations of composite model that is based on MG and inverse Nakagami-$m$ distributions, namely, MGS distribution, are derived.
\item The derived statistics are employed to analyse the performance of the double shadowed $\alpha-\kappa-\mu$ fading channel in which the first and the second shadowing impacts are respectively followed the Nakagami-$m$ and the inverse Nakagami-$m$ distributions. Accordingly, the equivalent parameters of a MG distribution of composite $\alpha-\kappa-\mu$/Nakagami-$m$ fading channels are provided.
\item Capitalizing on the above, unified closed-form expressions for the ABEP, the ACC, the EC, and the average AUC are obtained.
\end{itemize}
\section{MG and Inverse Nakagami-$m$ Distributions}
The PDF of a MG distribution is given by [13, eq. (1)]
\label{eqn_1}
\setcounter{equation}{0}
\begin{align}
f(x)= \sum_{j=1}^{K} \sigma_j x^{\beta_j-1} e^{-\zeta_j x}
\end{align}
where $\sigma_j$, $\beta_j$, and $\zeta_j$ are the parameters of $j$th Gamma component and $K$ that stands for the number of terms is evaluated via using the mean square error (MSE) method between the exact PDF and its MG representation [13].
\par If $\xi$ is an inverse Nakagami-$m$ random variable (RV), its PDF is expressed as [12, eq. (50)]
\label{eqn_2}
\setcounter{equation}{1}
\begin{align}
f_{\xi}(r) = \frac{(m_s-1)^{m_s}}{\Gamma(m_s)} r^{m_s-1} e^{-\frac{(m_s-1)}{r}}. \quad m_s > 1
\end{align}
where $m_s$ refers to the shadowing severity index and $\Gamma(.)$ is the incomplete Gamma function [17, eq. (8.310.1)].
\section{Statistical Properties of a MGS Distribution}
\par Let $\gamma\sim \text{MGS}(\sigma_j, \beta_j, \zeta_j, m_s)$ for $j=1,\cdots, K$ is an MGS-distributed RV. Then, the PDF of $\gamma$ can be obtained by the product of MG and inverse Nakagami-$m$ RVs as [4, eq. (8)]
\label{eqn_3}
\setcounter{equation}{2}
\begin{align}
f_{\gamma}(\gamma) = \int_0^\infty \frac{1}{z}f\bigg(\frac{\gamma}{z}\bigg) f_{\xi} (z) dz.
\end{align}
\par Substituting (1) and (2) into (3) and making use of [17, eq. (3.381.4)] with some mathematical manipulations, this yields
\label{eqn_4}
\setcounter{equation}{3}
\begin{align}
f_{\gamma}(\gamma) = (m_s-1)^{m_s} \sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j} \gamma^{\beta_j-1}}{(m_s-1+\zeta_j \gamma)^{\beta_j+m_s}}.
\end{align}
where $(.)_n$ is the Pochhammer symbol.
\par When $\zeta_j \rightarrow 0$ for all $j = 1,\cdots,K$, the asymptotic of the PDF, $f^{\infty}_{\gamma}(\gamma)$, can be expressed as
\label{eqn_5}
\setcounter{equation}{4}
\begin{align}
f^{\infty}_{\gamma}(\gamma) \simeq \sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j} \gamma^{\beta_j-1}}{(m_s-1)^{\beta_j}}.
\end{align}
\par Inserting (4) in $F_{\gamma}(\gamma)=\int_0^\gamma f_{\gamma}(\gamma) d\gamma$ and recalling [17, eq. (3.194.1)], the CDF of a MGS distribution can be derived as
\label{eqn_6}
\setcounter{equation}{5}
\begin{align}
F_{\gamma}(\gamma) = &\sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j} {_2F_1}\big(\beta_j+m_s,\beta_j;\beta_j+1;-\frac{\zeta_j}{m_s-1}\gamma\big)}{ \beta_j ((m_s-1)\gamma^{-1})^{\beta_j}}.
\end{align}
where ${_2F_1}(.)$ is the Gauss hypergeometric function [17, eq. (9.14.1)].
\par Using the fact that ${_2F_1}(.,.;.;0) \simeq 1$ when $\zeta_j \rightarrow 0$ or plugging (5) in $F_{\gamma}(\gamma)=\int_0^\gamma f_{\gamma}(\gamma) d\gamma$, the asymptotic of the CDF, $F^{\infty}_{\gamma}(\gamma)$, can be evaluated as
\label{eqn_7}
\setcounter{equation}{6}
\begin{align}
F^{\infty}_{\gamma}(\gamma) \simeq \sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j} \gamma^{\beta_j}}{ \beta_j(m_s-1)^{\beta_j}}.
\end{align}
\par Using the Laplace transform and invoking [18, eq. (13.2.5)], the MGF of the MGS distribution can be obtained as
\label{eqn_8}
\setcounter{equation}{7}
\begin{align}
\mathcal{M}_{\gamma}(s) &= \sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j} \Gamma(\beta_j) U\big(\beta_j;1-m_s;\frac{m_s-1}{\zeta_j}s\big)}{ \zeta^{\beta_j}_j} .
\end{align}
where $U(.)$ is the Tricomi confluent hypergeometric
function of the second kind defined in [17, eq. (9.211.4)].
\par The asymptotic of the MGF, $\mathcal{M}^{\infty}_{\gamma}(s)$, can be deduced after applying Laplace transform for (5) and invoking [17, eq. (8.310.1)]. Thus, this yields
\label{eqn_9}
\setcounter{equation}{8}
\begin{align}
\mathcal{M}^{\infty}_{\gamma}(s) \simeq \sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j} \Gamma(\beta_j)}{ [(m_s-1)s]^{\beta_j}}.
\end{align}
\par The $n$-th moment, $\boldsymbol{\mu}_n$, of the MGS distribution can be found by using (4) and [17, eq. (3.194.3)] as
\label{eqn_10}
\setcounter{equation}{9}
\begin{align}
\boldsymbol{\mu}_n=\mathbb{E}\{\gamma^n\} = (m_s-1)^n \sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j}}{\zeta^{\beta_j+n}_j} B(\beta_j+n, m_s-n).
\end{align}
where $B(.)$ is the Beta function [17, eq. (8.380.1)].
\par It is worth interesting that the MGS distribution can used to model the $\kappa-\mu$/inverse Gamma [5, eq. (6)] with $\theta_j=\frac{e^{-\mu \kappa}\mu^{\mu+2(j-1)}\kappa^{j-1}}{\Gamma(\mu+j-1)\Gamma(j)\bar{\gamma}^{\mu+j-1}}$ which is used to compute $\sigma_j$, $\beta_j=\mu+j-1$, and $\zeta_j = \frac{\mu(1+\kappa)}{\bar{\gamma}}$. Additionally, the Fisher Senedcor $\mathcal{F}$ composite fading [8, eq. (6)] can be represented by (4)-(10) with $K = 1$, $\sigma_1 = \frac{m^m}{\Gamma(m) \bar{\gamma}^m}$, $\beta_1 = m$, and $\zeta_1 = \frac{m}{\bar{\gamma}}$.
\label{eqn_14}
\setcounter{equation}{13}
\begin{table*}[t]
\begin{align}
&f(x)= \frac{\alpha \mu m^m \kappa^\frac{1-\mu}{2} (1+\kappa)^\frac{1+\mu}{2}}{2 \Gamma(m) \exp(\mu \kappa) \bar{\gamma}^\frac{\alpha(1+\mu)}{4}} x^{\frac{\alpha(1+\mu)}{4}-1} \int_0^\infty z^{-\frac{\alpha(1+\mu)}{4}+m-1} e^{-\frac{(1+\kappa)\mu x^{\alpha/2}}{(\bar{\gamma} z)^{\alpha/2}}-m z} I_{\mu-1} \bigg(2\mu \sqrt{\frac{\kappa(1+\kappa) x^{\alpha/2} }{(\bar{\gamma} z)^{\alpha/2}}}\bigg) dz.
\end{align}
\hrulefill
\end{table*}
\section{Double Shadowed $\alpha-\kappa-\mu$ Fading Channels}
The received signal envelope, $R$, over double shadowed $\alpha-\kappa-\mu$ fading channel can be expressed as
\label{eqn_11}
\setcounter{equation}{10}
\begin{align}
R^\alpha= \xi^2 \sum_{l=1}^{\mu}(X_l+ \vartheta p_l)^2+(Y_l+ \vartheta q_l)^2
\end{align}
where the parameters of (11) are defined as follows:
\begin{enumerate}[label=\roman*)]
\item $\alpha>0$ denotes the non-linearity of the propagation
medium.
\item $\mu$ is a real-valued extension related to the number of multipath clusters.
\item $\xi$ and $\vartheta$ represent the RVs which are responsible for introducing the shadowing impacts that are modelled by inverse Nakagami-$m$ and Nakagami-$m$ distributions, respectively, with $\mathbb{E}[\xi^2] = \mathbb{E}[\vartheta^2] =1$, where $\mathbb{E}[.]$ stands for the expectation operator. It is worth mentioning that (11) becomes equivalent to example 1 of the double shadowed $\kappa-\mu$ type I model [12, eq. (22)], when $\alpha = 2$.
\item $X_l$ and $Y_l$ are mutually independent Gaussian random processes with mean $\mathbb{E}[X_l]$ and $\mathbb{E}[Y_l] = 0$ and variance $\mathbb{E}[X^2_l]=\mathbb{E}[Y^2_l] = \delta^2$.
\item $p_l$ and $q_l$ are the mean values of the in-phase and quadrature phase components of the multipath cluster $l$.
\end{enumerate}
\par From (11), one can note that the PDF of $R$ can be derived by averaging the PDF of the $\alpha-\kappa-\mu$ fading over the PDF of $\vartheta$ and $\xi$ RVs. However, the PDF of the $\alpha-\kappa-\mu$ fading is included the modified Bessel function of the first kind, $I_\upsilon(.)$ [17, eq. (8.445)] that would lead to mathematically intractable statistical properties (please refer to [12]). Therefore, to obtain simple closed-form statistics, the PDF of inducing the shadowing of the dominant component is approximated by using a MG distribution whereas the multiplicative shadowing is added by utilizing a MGS model.
\par The PDF of the instantaneous SNR, $\gamma$, over double shadowed $\alpha-\kappa-\mu$ fading is expressed as [7, eq. (4)]
\label{eqn_12}
\setcounter{equation}{11}
\begin{align}
f_\gamma(r)= \frac{\alpha \mu \kappa^\frac{1-\mu}{2} (1+\kappa)^\frac{1+\mu}{2}}{2 \exp(\mu \kappa) \bar{\gamma}^\frac{\alpha(1+\mu)}{4}} r^{\frac{\alpha(1+\mu)}{4}-1} e^{-\frac{(1+\kappa)\mu}{\bar{\gamma}^{\alpha/2}} r^{\alpha/2}} \nonumber\\
\times I_{\mu-1} \bigg(2\mu \sqrt{\frac{\kappa(1+\kappa)}{\bar{\gamma}^{\alpha/2}} r^{\alpha/2}}\bigg).
\end{align}
where $\kappa$ is the ratio between the total powers of the dominant components and scattered waves and $\bar{\gamma}$ is the average SNR.
\par The PDF of $\vartheta$ is given by [12, eq. (54)]
\label{eqn_13}
\setcounter{equation}{12}
\begin{align}
f_{\vartheta}(r) = \frac{m^m}{\Gamma(m)} r^{m-1} e^{-m r}.
\end{align}
where $m$ is the shadowing severity index of the Nakagami-$m$.
\par Substituting (12) and (13) into (3), we have (14) shown at the top of this page.
\par Using the substitution $y=\frac{(1+\kappa)\mu x^{\alpha/2}}{(\bar{\gamma} z)^{\alpha/2}}$ in (14), we obtain
\label{eqn_15}
\setcounter{equation}{14}
\begin{align}
f(x)= \Xi x^{m-1}\int_0^\infty e^{-y} h(y)dy.
\end{align}
where $\Xi=\frac{m^m \mu^{\frac{2}{\alpha}m-\frac{\mu-1}{2}} \kappa^\frac{1-\mu}{2} (1+\kappa)^{\frac{2}{\alpha}m}}{\Gamma(m) \exp(\mu \kappa) \bar{\gamma}^m}$ and $h(y) = y^{\frac{\alpha(1+\mu)}{4}-\frac{2}{\alpha}m-1} e^{-\frac{m ((1+\kappa)\mu)^{\alpha/2}}{\bar{\gamma} y^{2/\alpha}} x} I_{\mu-1} \big(2 \sqrt{\kappa \mu y}\big)$.
\par The integration in (15), $\Phi= \int_0^\infty e^{-y} h(y) dy$, can be approximated by using a Gaussian-Laguerre quadrature method, as $\Phi \approx \sum_{j=1}^{K} w_{j} h(y_j) $, where $w_{j}$ and $y_{j}$ are the weight factors and abscissas, respectively, given in [18]. Hence, (15) can be equivalently expressed by (1) with the following coefficients
\label{eqn_16}
\setcounter{equation}{15}
\begin{align}
\beta_j = m,& \quad \zeta_j = \frac{m ((1+\kappa)\mu)^{2/\alpha}}{\bar{\gamma} y^{2/\alpha}_j}, \quad \sigma_j=\frac{\theta_j}{\sum_{l=1}^{K} \theta_l \Gamma(\beta_l) \zeta^{-\beta_l}_l} \nonumber\\
&\theta_j= \Xi w_j y^{\frac{\alpha(1+\mu)}{4}-\frac{2}{\alpha}m-1}_j I_{\mu-1} \big(2 \sqrt{\kappa \mu y_j}\big).
\end{align}
\par From (16), one can see that $\zeta_j \rightarrow 0$ when $\bar{\gamma} \rightarrow \infty$.
\section{Performance Analysis using a MGS Model}
\subsection{Outage Probability}
The OP is defined as the probability of falling the values of the output SNR below a predefined threshold value $\varphi$.
\par The OP, $P_o$, can be computed by [1, eq. (1.4)]
\label{eqn_17}
\setcounter{equation}{16}
\begin{align}
P_o = F_\gamma(\varphi).
\end{align}
where $F_\gamma(.)$ is provided in (6).
\par The asymptotic of the OP, $P^{\infty}_o$, can be analysed by (7), i.e., $P^{\infty}_o = F^{\infty}_\gamma(\varphi)$. Furthermore, the $P^{\infty}_o$ may be closely represented as $P^{\infty}_o \simeq \bar{\gamma}^{-G_d}$ whereby $G_d$ denotes the diversity gain that demonstrates the increasing in the slope of the OP versus $\bar{\gamma}$. Hence, plugging $\sigma_j$ of (16) in (7), one can notice that the values of the $G_d$ of the MGS and double shadowed $\alpha-\kappa-\mu$ are proportional to $\beta_j$ and $m$, respectively.
\label{eqn_34}
\begin{table*}[t]
\setcounter{equation}{33}
\begin{align}
\bar{\mathcal{A}} = 1-\sum_{l=0}^{u-1} \sum_{i=0}^l {{l+u-1} \choose {l-i}} \frac{2^{-(l+i+u)}}{ i!} (m_s-1)^i \sum_{j=1}^{K} \sigma_j (m_s)_{\beta_j} \Gamma(\beta_j+i) U\bigg(\beta_j+i;i-m_s+1;\frac{m_s-1}{2\zeta_j}\bigg).
\end{align}
\hrulefill
\end{table*}
\subsection{Average Bit Error Probability}
The ABEP can be evaluated by [1, eq. (9.11)]
\label{eqn_18}
\setcounter{equation}{17}
\begin{align}
\bar{P}_e = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \mathcal{M}_\gamma \bigg(\frac{\rho}{\sin^2 \phi}\bigg) d\phi.
\end{align}
where $\rho=0.5$, $\rho=1$, and $\rho=0.715$ for coherent BFSK, BPSK, and BFSK with minimum correlation, respectively.
\par Substituting (8) into (18) and employing the identity [19, eq. (07.33.26.0004.01)], we have
\label{eqn_19}
\setcounter{equation}{18}
\begin{align}
\bar{P}_e = \frac{1}{\pi \Gamma(m_s)}\sum_{j=1}^{K} \frac{\sigma_j}{ \zeta^{\beta_j}_j} \int_0^{\frac{\pi}{2}} G^{2,1}_{1,2}\bigg[ \frac{(m_s-1)\rho}{\zeta_j \sin^2 \phi} \bigg\vert
\begin{matrix}
1-\beta_j\\
0, m_s\\
\end{matrix}
\bigg] d \phi.
\end{align}
\par By employing the definition of the Meijer's $G$-function [19, eq. (07.34.02.0001.01)] and $t=\sin^2 \phi$, (19) becomes
\label{eqn_20}
\setcounter{equation}{19}
\begin{align}
\bar{P}_e &= \frac{1}{2 \pi \Gamma(m_s)} \sum_{j=1}^{K} \frac{\sigma_j}{ \zeta^{\beta_j}_j} \int_0^1 \frac{1}{\sqrt{(1-t)t}}\frac{1}{2 \pi i}\int_{\mathcal{L}} \Gamma(r) \nonumber\\
& \Gamma(m_s+r) \Gamma(\beta_j-r) \bigg(\frac{(m_s-1)\rho}{t \zeta_j }\bigg)^{-r} dr dt.
\end{align}
where $i=\sqrt{-1}$ and $\mathcal{L}$ is the suitable contours in the $r$-plane from $\varrho-i\infty$ to $\varrho+i\infty$ with $\varrho$ is a constant value.
\par Changing the order of the integrations of (20) and then using [17, eq. (3.191.3)] for the linear integral, we obtain
\label{eqn_21}
\setcounter{equation}{20}
\begin{align}
\bar{P}_e = \frac{1}{2 \pi \Gamma(m_s)}&\sum_{j=1}^{K} \frac{\sigma_j}{ \zeta^{\beta_j}_j} \frac{1}{2 \pi i}\int_{\mathcal{L}} \Gamma(r) \Gamma(m_s+r) \Gamma(\beta_j-r) \nonumber\\
& B(r+0.5,0.5) \bigg(\frac{(m_s-1)\rho}{\zeta_j }\bigg)^{-r}dr.
\end{align}
\par Recalling the properties [17, eq. (8.384.1)/ eq. (8.338.2)] and using [19, eq. (07.34.02.0001.01)], (21) is expressed as
\label{eqn_22}
\setcounter{equation}{21}
\begin{align}
\bar{P}_e = \frac{1}{2 \sqrt{\pi} \Gamma(m_s)}\sum_{j=1}^{K} \frac{\sigma_j G^{2,1}_{2,2}\bigg[ \frac{(m_s-1)\rho}{\zeta_j} \bigg\vert
\begin{matrix}
1-\beta_j, 1\\
0, m_s, 0.5\\
\end{matrix}
\bigg]}{ \zeta^{\beta_j}_j} .
\end{align}
\par The asymptotic of the ABEP at high $\bar{\gamma}$ regime, $\bar{P}^{\infty}_e$, can be deduced after inserting (9) in (18) and using $t = \sin^2 \phi$ as
\label{eqn_23}
\setcounter{equation}{22}
\begin{align}
\bar{P}^{\infty}_e \simeq \frac{1}{\pi} \sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j} \Gamma(\beta_j)}{ [(m_s-1)\rho]^{\beta_j}} \int_0^{1} \frac{t^{\beta_j}}{\sqrt{(1-t)t}} dt.
\end{align}
\par The integration of (23) is recorded in [17, eq. (3.191.3)]. Thus, after some mathematical manipulations, this yields
\label{eqn_24}
\setcounter{equation}{23}
\begin{align}
\bar{P}^{\infty}_e \simeq \frac{1}{2\sqrt{\pi}}\sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j} \Gamma(\beta_j+0.5)}{ \beta_j [(m_s-1)\rho ]^{\beta_j}} .
\end{align}
\par It is evident from (24) that the $G_d$ of the MGS model is proportional to $\beta_j$, i.e., $\bar{P}^{\infty}_e \simeq \bar{\gamma}^{-\beta_j}$. Hence, the $G_d$ for the double shadowed $\alpha-\kappa-\mu$ fading is proportional to $m$.
\subsection{Average Channel Capacity}
According to Shannon's theorem, the normalized ACC, $\bar{C}$, can be determined by [8, eq. (26)]
\label{eqn_25}
\setcounter{equation}{24}
\begin{equation}
\bar{C} =\frac{1}{\ln(2)} \int_0^\infty \text{ln}(1+\gamma) f_\gamma(\gamma) d\gamma.
\end{equation}
\par Plugging (4) in (25) and making use the identity [19, eq. (01.04.26.0002.01)] to write the natural logarithm in terms of the Meijer's $G$-function, we obtain
\label{eqn_26}
\setcounter{equation}{25}
\begin{align}
\bar{C}=&\frac{(m_s-1)^{m_s}}{\ln(2)} \sum_{j=1}^{K} \sigma_j (m_s)_{\beta_j} \nonumber\\
&\times \int_0^\infty \frac{\gamma^{\beta_j-1}}{(m_s-1+\zeta_j \gamma)^{\beta_j+m_s}} G^{1,2}_{2,2}\bigg[ \gamma \bigg\vert
\begin{matrix}
1,1\\
1,0\\
\end{matrix}
\bigg] d\gamma.
\end{align}
\par Employing [17, eq. (7.811.5)], (26) can be expressed in exact closed-form as
\label{eqn_27}
\setcounter{equation}{26}
\begin{align}
\bar{C}=\frac{1}{\ln(2) \Gamma(m_s)} \sum_{j=1}^{K} \frac{ \sigma_j}{\zeta_j^{\beta_j} } G^{2,3}_{3,3}\bigg[ \frac{m_s-1}{\zeta_j} \bigg\vert
\begin{matrix}
1-\beta_j, 1,1\\
m_s,1,0\\
\end{matrix}
\bigg].
\end{align}
\par The asymptotic of the ACC for $\bar{\gamma}\rightarrow \infty$, $\bar{C}^{\infty}$, can be evaluated via [8, eq. (28)]
\label{eqn_28}
\setcounter{equation}{27}
\begin{align}
\bar{C}^{\infty} \simeq \frac{1}{\ln(2)}\frac{\partial}{\partial n} \mathbb{E}[\gamma^n] \bigg\vert_{n=0}.
\end{align}
\par Substituting (10) into (28), computing the partial derivative, and setting $n=0$, $\bar{C}^{\infty}$ over MGS model is deduced as
\label{eqn_29}
\setcounter{equation}{28}
\begin{align}
\bar{C}^{\infty} \simeq \sum_{j=1}^{K} \frac{ \sigma_j \Gamma(\beta_j) \bigg[\ln\big(\frac{m_s-1}{\zeta_j}\big)+\psi(\beta_j)-\psi(m_s)\bigg]}{\ln(2) \zeta_j^{\beta_j} } .
\end{align}
where $\psi(.)$ is the digamma function [17, eq. (8.360.1)].
\subsection{Effective Capacity}
In Shannon's theorem, the ACC has been measured under perfect quality of service (QoS). However, in the EC, the constraints of the QoS, such as, system delay, are taken into consideration [5].
\par The EC can be calculated by [15, eq. (4)]
\label{eqn_30}
\setcounter{equation}{29}
\begin{align}
\mathcal{R}=-\frac{1}{A}\text{log}_2 \bigg\{\int_0^\infty (1+\gamma)^{-A}f_\gamma(\gamma)d\gamma \bigg\}.
\end{align}
where $ A \triangleq \theta T B/\mathrm{ln}(2)$ with $\theta$, $T$, and $B$ denote the delay exponent, the time and the bandwidth of the channel.
\par Inserting (4) in (30) and employing [17, eq. (3.197.1)], the expression of the EC over MGS distribution is yielded as
\label{eqn_31}
\setcounter{equation}{30}
\begin{align}
\mathcal{R}&=-\frac{1}{A}\text{log}_2 \bigg\{\sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j} B(\beta_j,m_s+A)}{ (m_s-1)^{\beta_j}} \nonumber\\
&\times {_2F_1}\bigg(\beta_j+m_s,\beta_j;\beta_j+m_s+A;1-\frac{\zeta_j}{m_s-1}\bigg) \bigg\}.
\end{align}
\par Plugging (5) in (30) and recalling [17, eq. (3.194.3)], the asymptotic of the EC at $\bar{\gamma} \rightarrow \infty$, $R^{\infty}$, is deduced as
\label{eqn_32}
\setcounter{equation}{31}
\begin{align}
R^{\infty} \simeq -\frac{1}{A}\text{log}_2 \bigg\{\sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j}}{ (m_s-1)^{\beta_j}} B(\beta_j,A-\beta_j) \bigg\}.
\end{align}
\subsection{Average AUC of Energy Detection}
The AUC is a single figure of merit that is suggested as an alternative performance measure to the ROC curve [14].
\par The average AUC, $\bar{\mathcal{A}}$, can be computed by [20, eq. (20)/ eq. (21)]
\label{eqn_33}
\setcounter{equation}{32}
\begin{align}
\bar{\mathcal{A}} & = 1- \sum_{l=0}^{u-1} \sum_{i=0}^{l} \frac{{{l+u-1} \choose {l-i}}}{2^{l+i+u}i!} \int_0^\infty \gamma^i e^{-\frac{\gamma}{2}} f_\gamma(\gamma) d\gamma.
\end{align}
where ${a \choose b}=\frac{a!}{b! (a-b)!}$ is the binomial coefficient.
\par Plugging (6) in (33) and utilizing [18, eq. (13.2.5)], $\bar{\mathcal{A}}$ is yielded as in (34) given at the top of this page.
\par The average AUC at high $\bar{\gamma}$ value, $\bar{\mathcal{A}}^{\infty}$, can be evaluated after inserting (5) in (33) and invoking [17, eq. (3.381.4)] as
\label{eqn_35}
\setcounter{equation}{34}
\begin{align}
&\bar{\mathcal{A}}^{\infty} \simeq 1-\sum_{l=0}^{u-1} \sum_{i=0}^l \sum_{j=1}^{K} \frac{\sigma_j (m_s)_{\beta_j} {{l+u-1} \choose {l-i}} \Gamma(\beta_j+i)}{2^{l+u-\beta_j} (m_s-1)^{\beta_j} i!}.
\end{align}
\section{Analytical and Simulation Results}
In this section, the numerical and the asymptotic results of the derived performance measures are presented for different scenarios. To achieve MSE $\leq 10^{-5}$, we have used $K = 15$.
\par Figs. 1-4 demonstrate the OP for $\varphi = 5$ dB, the ABEP for BPSK modulation, the comparison between the normalized ACC and the EC for $A = 3.5$, and the average complementary of AUC (CAUC), $1-\bar{\mathcal{A}}$, for $u=3$ versus $\bar{\gamma}$, respectively. Three cases for the shadowing index $m_s$, namely, heavy ($m_s = 1.5$), moderate ($m_s = 5.5$), and light ($m_s = 50$) are analyzed.
\par From the provided results, it can be observed that the performance becomes better when $\alpha$, $\kappa$ and/or $\mu$ increase. This is because the high values of $\alpha$, $\kappa$, and $\mu$ indicate that the system tends to be linear, the total power of the dominant components is less than that of the scattered waves and large number of multipath clusters arrive at the receiver, respectively. Additionally, the increasing in $m$ and/or $m_s$ means the impacts of the first and/or the second shadowing on the received signal are low. However, $m$ has the high effect on the performance when $\bar{\gamma} \rightarrow \infty$ due to improving in the $G_d$.
\par In Fig. 3, one can note that the ACC is higher than the EC for the same scenario. This refers to the impact of the system delay on the EC whereas its ignored in the ACC. Moreover, the EC is related to the average AUC via $u$ [20]. This relationship explains how the low quality of the received signal by the unlicensed user would reduce the detectability of the ED.
\par In all figures, a perfect agreement between the exact and the simulation results as well as the asymptotic at high $\bar{\gamma}$ can be noticed which proves the validation of our expressions.
\label{Fig_1}
\begin{figure}[t]
\centering
\includegraphics[width=3.2 in, height=2.03 in]{Fig_1.eps}
\centering
\caption{OP versus normalized outage threshold for $\varphi = 5$ dB.}
\end{figure}
\label{Fig_2}
\begin{figure}[t]
\centering
\includegraphics[width=3.2 in, height=2.03 in]{Fig_2.eps}
\centering
\caption{ABEP for BPSK versus average SNR.}
\end{figure}
\label{Fig_3}
\begin{figure}[t]
\centering
\includegraphics[width=3.2 in, height=2.03 in]{Fig_3.eps}
\caption{Normalized ACC and EC with $A = 3.5$ versus average SNR.}
\end{figure}
\label{Fig_4}
\begin{figure}[t]
\centering
\includegraphics[width=3.2 in, height=2.03 in]{Fig_4.eps}
\centering
\caption{Complementary AUC for $u=3$ versus average SNR.}
\end{figure}
\section{Conclusions}
In this paper, a MGS distribution was proposed as highly accurate approximate unified representation where the shadowing impact was assumed to be an inverse Nakagami-$m$ RV. This distribution was then applied to the double shadowed $\alpha-\kappa-\mu$ fading channel. The exact and the asymptotic expressions of the OP, the ABEP, the ACC, and the ER of the wireless communications systems and the average AUC of ED over a MGS model were derived. The results for different values of the fading parameters were presented. The expressions of this work can be employed for many fading conditions, such as, $\alpha-\kappa-\mathcal{F}$ fading with $m \rightarrow \infty$.
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{
"redpajama_set_name": "RedPajamaArXiv"
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According to the Facebook bio, The Motel Pines' name "represents an older motel in an area where nice new businesses are springing up all around it. Not being phased by all the revitalization, it has stayed true to its old-school vibe." Drummer Mark Starling describes this as a fitting metaphor for his group because like the motel, the members are happy doing their own thing and aren't concerned with trying to make more contemporary music.
The Motel Pines is an indie rock band from Dallas, TX formed in 2012. The debut album A Sad History is available for preorder on Bandcamp and is due out on March 14th.
The LP's second single, #nolivesmatter discusses the common issues that have always plagued humans. This one is soothing, melodic, and yet it sounds so unorthodox due to its rhythms. My Abandoned Ship has the feel of classic Folk, but with a modern twist. It starts off with some marching tom-tom beats and serene flanger effects. The pace eventually picks up into something more energetic, but this doesn't prevent Michael Carrasco from maintaining the sincerity in his voice.
As a whole, the album is made up of sunny-disposition rock songs doused with light sprinklings of cynicism, depression, and anger. The discontent is most noticeable on certain songs such as The Heart and the Head and Of Mediocrity. Despite this, these tracks are optimistic to a fault. And while we're talking about positivity, a good closing note about these guys is that they seem to know exactly who they are and from what I can tell, they're perfectly happy about it.
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{
"redpajama_set_name": "RedPajamaC4"
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| 4,271
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\section*{Abstract}
{\bf
It is well-known that the symmetry group of a Feynman diagram can give important information on possible strategies for its evaluation, and the mathematical objects that will be involved.
Motivated by ongoing work on multi-loop multi-photon amplitudes in quantum electrodynamics,
here I will discuss the usefulness of introducing a polynomial basis of invariants of the symmetry group of a diagram in Feynman-Schwinger parameter space.
}
\section{Introduction: Schwinger parameter representation of Feynman diagrams}
\label{sec:intro}
The most universal approach to the calculation of Feynman diagrams uses Feynman-Schwinger parameters $ x_i$,
introduced through the exponentiation of the (Euclidean) scalar propagator,
$$ \frac{1}{p^2+m^2} = \int_0^\infty dx \,{\rm e}^{-x (p^2+m^2)} $$
For scalar diagrams, one finds the following universal structure for an arbitrary graph $ G$ with $ n$ internal lines and $ l$ loops
in $ D$ dimensions:
$$ I_G = \Gamma(n-l D/2) \int_{x_i \geq 0} d^n x \, \delta\Bigl(1-\sum_{i=1}^n x_i\Bigr) \frac{{\cal U}^{n - (l+1)\frac{D}{2}}}{{\cal F}^{n-l D/2}}$$
$ {\cal U}$ and $ {\cal F}$ are polynomials in the $ x_i$ called the first and second Symanzyk (graph) polynomials. There exist graphical methods for their construction.
For more general theories (involving not only scalar particles) the same graph will involve, in addition to these two polynomials, also a
{\it numerator polynomial} $ {\cal N}(x_1,\ldots,x_n)$.
\section{Symmetries of Feynman diagrams}
Many Feynman diagrams possess a non-trivial symmetry group, generated by interchanges of the internal lines that leave the
topology of the graph unchanged. Then all its graph polynomials must be invariant under the natural action of the
group on the set of polynomials ${\rm I\!R}[x_1,\ldots, x_n]$, $ g.P(X) \equiv P(g.X)$.
A nice example is the $ l-loop $ {\it banana graph} shown in Fig. \ref{fig-banana}.
\vspace{-80pt}
\begin{center}
\begin{figure}[htp]
\centerline{\includegraphics[width=5cm]{fig-banana.pdf}}
\vspace{-60pt}
\caption{$l-loop$ banana graph.}
\label{fig-banana}
\end{figure}
\end{center}
\vspace{-30pt}
This graph has full permutation symmetry in all the internal lines, so the symmetry group is $S_{l+1}$, and its graph polynomials must be symmetric functions of $x_1,\ldots, x_{l+1}$. As is well-known, this implies that they can be rewritten as
polynomials in the {\it elementary symmetric polynomials} $ S_1,\ldots,S_n$.
Perhaps less known is that this generalizes to the case of a general symmetry group as follows \cite{sturm-book}:
{\bf Theorem 1}:
Let $G$ be a finite group and let $\Gamma$ be an $n-$dimensional (real) representation.
\begin{enumerate}
\item There exist
$n= \dim (\Gamma)$ algebraically invariant polynomials
$P_1,\cdots, P_n$, called the {\it primitive invariants}, such that the Jacobian
$\frac {\partial (P_1,\ldots,P_n)}{\partial(x_1,\ldots,x_n)} \ne 0$.
\item
Denote $d_k=\deg(P_k)$ and ${\cal R} = \mathbb R[P_1,\cdots,P_n]$
the
subalgebra of polynomial invariants generated by the primitive invariants.
\item There exists $m= d_1 \cdots d_n/|G|$
secondary invariants polynomials $S_1,\cdots, S_m$.
\item The subalgebra of invariants $\mathbb R[x_1,\cdots,x_n]^\Gamma$ is a free ${\cal R}-$module with basis $(S_1,\cdots,S_m)$.
In particular this means that any invariant $I \in \mathbb R[x_1,\cdots,x_n]^\Gamma$ can be uniquely written as
$I=\sum\limits_{i=1}^m f_i(P_1,\cdots,P_n) S_i$,
where $f_i(P_1,\cdots,P_n), i=1,\cdots, m$ belong to ${\cal R}$, {\it i.e.}, are polynomials in $(P_1,\cdots,P_n)$.
\end{enumerate}
There exist computer algebra systems for the computation of $P_1,\ldots,P_n$ such as SINGULAR \cite{decker}.
\section{The Euler-Heisenberg Lagrangian at one loop}
In 1936 Heisenberg and Euler obtained their following famous representation of the
one-loop QED effective Lagrangian in a constant field
(`` Euler-Heisenberg Lagrangian'')
\begin{eqnarray}
{\cal L}^{(1)}(a,b)&=& - {1\over 8\pi^2}
\int_0^{\infty}{dT\over T^3}
\,\,{\rm e}^{-m^2T}
\biggl\lbrack
{(eaT)(ebT)\over {\rm tanh}(eaT)\tan(ebT)}
- {e^2\over 3}(a^2-b^2)T^2 -1
\biggr\rbrack
\label{ehspin}
\end{eqnarray}\noindent
Here $m$ is the electron mass, and $a,b$ are the two invariants of the Maxwell field,
related to $\bf E$, $\bf B$ by $a^2-b^2 = B^2-E^2,\quad ab = {\bf E}\cdot {\bf B}$.
This effective Lagrangian hold the information on the QED $N$ - photon amplitudes
in the low-energy limit where all photon energies are small
compared to the electron mass, $\omega_i\ll m$.
It corresponds to the Feynman diagrams shown in Fig. \ref{fig-EHL1loop}.
\begin{figure}[ht]
\centerline{\includegraphics[scale=.8]{fig-EHL1loop.pdf}}
\vspace{10pt}
\caption{Feynman diagrams corresponding to the Euler-Heisenberg Lagrangian. The photon legs are in the low-energy limit.}
\label{fig-EHL1loop}
\end{figure}
For the extraction of the amplitudes from the effective Lagrangian,
one expands it in powers of the Maxwell invariants,
\begin{eqnarray}
{\cal L} (a,b) = \sum_{k,l} c_{kl}\, a^{2k}b^{2l}
\end{eqnarray}\noindent
then fixes a helicity assignment and uses spinors helicity techniques \cite{56}.
\section{Imaginary part of the effective action}
Except for the purely magnetic case where $b=0$, the proper-time integral in \eqref{ehspin} has poles on the
integration contour at $ ebT = k\pi$ which create an imaginary part.
For the purely electric case one gets \cite{schwinger51}
\begin{eqnarray}
{\rm Im} {\cal L}^{(1)}(E) &=& \frac{m^4}{8\pi^3}
\beta^2\, \sum_{k=1}^\infty \frac{1}{k^2}
\,\exp\left[-\frac{\pi k}{\beta}\right]
\nonumber\\
\label{schwinger}
\nonumber
\end{eqnarray}
($\beta = eE/m^2$). We note:
\begin{itemize}
\item
The $ k$th term relates to coherent creation of $ k$ pairs in
one Compton volume.
\item
In the weak-field limit $\beta \ll 1$ the terms with $k\geq 2$ can be neglected.
\item
$ {\rm Im}{\cal L}(E)$ depends on $ E$ non-perturbatively (non-analytically), which is consistent with
Sauter's \cite{sauter} interpretation of pair creation as vacuum tunnelling (Fig. \ref{fig-pairtunnel}).
\begin{figure}[h]
\hspace{150pt}
{
\includegraphics[scale=0.6]{fig-pairtunnel.pdf}
}
\caption{Pair creation by an external field as vacuum tunnelling.}
\label{fig-pairtunnel}
\end{figure}
\end{itemize}
\section{Beyond one loop}
The two-loop correction to the Euler-Heisenberg Lagrangian due to one internal photon exchange (Fig. \ref{fig-2loopEHL}) has been analyzed \cite{ritusspin,ditreu-book,18},
and turned out to contain important information on the Sauter tunnelling picture \cite{lebrit},
on-shell versus off-shell renormalization\cite{ritusspin,ritusscal}, and the asymptotic properties of the QED photon
S-matrix \cite{111}.
\begin{figure}[h]
\hspace{120pt}
{
\includegraphics[scale=.7]{fig-EHL2loop.pdf}
}
\vspace{10pt}
\caption{Feynman diagrams contributing to the 2-loop EHL.}
\label{fig-2loopEHL}
\end{figure}
It leads to rather intractable two-parameter integrals.
However, in the electric case its imaginary part $ {\rm Im}{\cal L}^{(2)}(E)$ permits a decomposition
analogous to Schwinger's \eqref{schwinger} \cite{lebrit}. For single-pair production, this is now interpreted as a a tunnelling process where,
in the process of turning real, the electron-positron pair is already interacting at the one-photon exchange level.
Even for the imaginary part no completely explicit formulas are available. However, it
simplifies dramatically in the weak-field limit, where it just becomes an $\alpha\pi$ correction to the
one-loop contribution:
\begin{eqnarray}
{\rm Im} {\cal L}^{(1)} (E) +
{\rm Im}{\cal L}^{(2)} (E)
\,\,\,\, {\stackrel{\beta\to 0}{\sim}} \,\,\,\,
\frac{m^4\beta^2}{8\pi^3}
\bigl(1+\alpha\pi\bigr)
\,{\rm e}^{-{\pi\over\beta}}
\nonumber
\label{Im1plus2}
\end{eqnarray}\noindent
This suggests that higher loop orders might lead to an exponentiation, and indeed Lebedev and Ritus \cite{lebrit}
provided strong support for this hypothesis by showing that, assuming that
\begin{eqnarray}
{\rm Im} {\cal L}^{(1)} (E) +
{\rm Im}{\cal L}^{(2)} (E)
+
{\rm Im}{\cal L}^{(3)} (E) + \ldots
\,\,\,\, {\stackrel{\beta\to 0}{\sim}} \,\,\,\,
\frac{m^4\beta^2}{8\pi^3}
\,{\rm exp}\Bigl[ -{\pi\over\beta}+\alpha\pi \Bigr]
= {\rm Im}{\cal L}^{(1)}(E)\,\,{\rm e}^{\alpha\pi}
\nonumber
\end{eqnarray}\noindent
then the result can be interpreted in the tunnelling picture as the corrections to the
Schwinger pair creation rate due to the pair being created with a negative Coulomb
interaction energy
\begin{eqnarray}
m(E) \approx m + \delta m(E), \quad \delta m(E) = - \frac{\alpha}{2}\frac{eE}{m}
\nonumber
\end{eqnarray}\noindent
Moreover, the resulting field-dependent mass-shift $\delta m (E)$ is identical with
the {\it Ritus mass shift}, originally derived by Ritus in \cite{ritusmass} from the crossed process of
one-loop electron propagation in the field (Fig. \ref{fig-crossed}).
\vspace{5pt}
\begin{figure}[h]
\centering
\hspace{10pt}\includegraphics[scale=.5]{fig-uncrossed.pdf}
\raisebox{1.1 em} {$\hspace{20pt}\Longleftrightarrow\hspace{20pt}$}
\includegraphics[scale=.4]{fig-crossed.pdf}
\caption{Photon-corrected pair-creation vs. electron propagation in the field.}
\label{fig-crossed}
\end{figure}
Unbeknownst to the authors of \cite{lebrit}, for scalar QED the corresponding conjecture had already been established
two years earlier by Affleck, Alvarez and Manton \cite{afalma} using Feynman's worldline path integral formalism
and a semi-classical {\it worldline instanton} approximation.
\vspace{2pt}
\begin{figure}[h]
\hspace{70pt}
{
\includegraphics[scale=.6]{fig-aamdiag.pdf}
}
\caption{Feynman diagrams contributing to the exponentiation hypothesis.}
\label{AAMfeyn}
\end{figure}
Diagrammatically, we note the following features of the exponentiation formula (see Fig. \ref{AAMfeyn}):
\begin{itemize}
\item
It Involves diagrams with any numbers of loops and legs.
\item
Although not shown, also all the counter-diagrams from mass renormalization must contribute.
\item
It does {\it not} include diagrams with more than one fermion loop
(those get suppressed in the weak-field limit \cite{afalma}).
\item
Horizontal summation produces the Schwinger exponential $ {\rm e}^{-\frac{\pi}{\beta}}$.
\item
Vertical summation produces the Ritus-Lebedev/Affleck-Alvarez-Manton exponential $ {\rm e}^{\alpha\pi}$.
\end{itemize}
\section{QED in 1+1 dimensions}
The exponentiation conjecture has so far been verified only at two loops.
A three-loop check is in order, but calculating the three-loop EHL in $ D=4$ is presently hardly feasible.
Motivated by work by Krasnansky \cite{Kras} on Euler-Heisenberg in various dimensions,
in 2010 two of the authors with D.G.C. McKeon started investigating the analogous problem in 2D QED.
In \cite{81} we used the worldline instanton method to generalize the exponentiation conjecture to the 2D case,
resulting in
\begin{eqnarray}
{\rm Im}{\cal L}^{(all-loop)}_{2D}
\sim
\,{\rm e}^{-\frac{m^2\pi}{eE} + \tilde\alpha \pi^2 \kappa^2}
\label{exp2D}
\end{eqnarray}\noindent
where $\kappa = m^2/(2ef)$, $ f^2={1\over4} F_{\mu\nu}F^{\mu\nu}$,
and $\tilde\alpha = \frac{2e^2}{\pi m^2}$ is the two-dimensional analogue of the fine-structure constant.
Defining the weak-field expansion coefficients in $2D$ by
\begin{eqnarray}
{\cal L}^{(l)(2D)}(\kappa) &=& \frac{m^2}{2\pi} \sum_{n=1}^{\infty}(-1)^{l-1}c_{2D}^{(l)}(n) (i\kappa)^{-2n}
\label{defcn}
\end{eqnarray}\noindent
we then used Borel analysis to derive from (\ref{exp2D}) a formula for the limits of ratios of $l$ - loop to one - loop
coefficients:
\begin{eqnarray}
{{\rm lim}_{n\to\infty}} {c^{(l)}_{2D}(n)\over c^{(1)}_{2D}(n+l-1)}
&=& {(\tilde\alpha\pi^2)^{l-1}\over (l-1)!}
\label{AAM2Dcoeff}
\end{eqnarray}\noindent
Moreover, we calculated the 2D Euler-Heisenberg Lagrangian at one and two loops,
\begin{eqnarray}
{\cal L}^{(1)}(f) &=& -{m^2\over 4\pi} {1\over\kappa}
\Bigl[{\rm ln}\Gamma(\kappa) - \kappa(\ln \kappa -1) +
{1\over 2} \ln \bigl({\kappa\over 2\pi}\bigr)\Bigr]
\label{1loopEHL2D}\\
{\cal L}^{(2)}(f) &=& {m^2\over 4\pi}\frac{\tilde\alpha}{4}
\Bigl[ \tilde\psi(\kappa) + \kappa \tilde\psi'(\kappa)
+\ln(\lambda_0 m^2) + \gamma + 2 \Bigr]
\label{2loopEHL2D}
\end{eqnarray}\noindent \noindent
where $\tilde\psi (x) \equiv \psi(x) - \ln x + {1\over 2x}$, $\psi(x)=\Gamma^\prime(x)/\Gamma(x)$, and the
constant $\lambda_0$ comes from an IR cutoff. One finds from \eqref{1loopEHL2D} and \eqref{2loopEHL2D} that
\begin{eqnarray}
c_{2D}^{(1)}(n) &=& (-1)^{n+1} \frac{B_{2n}}{4n(2n-1)} \\
c_{2D}^{(2)}(n) &=& (-1)^{n+1} \frac{\tilde\alpha}{8}\frac{2n-1}{2n}B_{2n}
\end{eqnarray}\noindent
Using properties of the Bernoulli numbers $ B_n$ it is then easy to verify that
\begin{eqnarray}
\lim_{n\to\infty} {c_{2D}^{(2)}(n)\over c_{2D}^{(1)}(n+1)}
&=&
\tilde\alpha \pi^2
\nonumber
\end{eqnarray}\noindent
in accordance with \eqref{AAM2Dcoeff}.
\section{Three-loop EHL in 2D: diagrams}
At three loops, we face the task of computing the two diagrams shown in Fig. \ref{fig-AB}
(there are also diagrams involving more than one fermion-loop, including several that
involve Gies-Karbstein tadpoles \cite{GK}, but those can be shown to be subdominant in the asymptotic limit).
\begin{figure}[h]
\hspace{140pt}
{
\includegraphics[scale=.4]{fig-AB.pdf}
}
\caption{Three-loop diagrams contributing to the exponentiation conjecture.}
\label{fig-AB}
\end{figure}
Due to the super-renormalizability of $2D$ QED these diagrams are already UV finite.
They suffer from spurious IR - divergences, but those can be removed by going to the {\it traceless gauge} $\xi = -2$ \cite{121}.
The calculation of diagram A is relatively straightforward, thus we focus on the much more substantial task of
computing diagram B and its weak-field expansion coefficients.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=.7]{fig-DiagB.pdf}
\end{center}
\vspace{-10pt}
\caption{Parametrization of diagram B.}
\label{fig-DiagB}
\end{figure}
Introducing Schwinger parameters for this diagram as shown in Fig. \ref{fig-DiagB} leads to the integral representation \cite{121}
\begin{eqnarray}
{\cal L}^{3B} (f)
&=& \frac{\tilde\alpha^2m^2}{128\pi} \int_{0}^{\infty}dw dw' d\hat w d\bar w ~ I_B ~ \,{\rm e}^{-a} \nonumber\\
I_B &=&
\frac{\rho^3}{\cosh^2 \rho w \cosh^2 \rho w' \cosh^2 \rho \hat w\cosh^2 \rho \bar w}
\frac{B}{A^3C}
\nonumber\\&&
- \rho \frac{\cosh(\rho \tilde{w}) }{\cosh \rho w \cosh \rho w' \cosh \rho \hat w\cosh \rho \bar w}
\Bigl\lbrack
\frac{1}{A} - \frac{C}{G^2}\ln \Bigl(1+ \frac{G^2}{AC}\Bigr)\Bigr\rbrack
\nonumber
\label{Bw}
\end{eqnarray}
where
\begin{eqnarray}
B &=& (\rm tanh^2 z + \rm tanh^2 \hat z)(\rm tanh z' + \rm tanh \bar z) + (\rm tanh^2 z' + \rm tanh^2 \bar z)(\rm tanh z + \rm tanh \hat z) \nonumber\\
C &=& \rm tanh z\,\rm tanh z' \,\rm tanh\, \hat z + \rm tanh z\, \rm tanh z'\, \rm tanh \bar z + \rm tanh z\, \rm tanh \hat z\, \rm tanh \bar z + \rm tanh z'\, \rm tanh \hat z \,\rm tanh \bar z \nonumber\\
G &=& \rm tanh z\, \rm tanh \hat z - \rm tanh z'\, \rm tanh \bar z \nonumber
\end{eqnarray}\noindent
($ z = \rho w$ etc.).
Although for a three-loop diagram this is a fairly compact representation, an exact calculation is out of the question, and a straightforward expansion in powers of the external field
to get the weak-field expansion coefficients turns out to create huge numerator polynomials. To deal with those, we will now take advantage of the high symmetry
of the diagram.
\section{Integration-by-parts algorithm}
Introduce the operator
$\tilde d \equiv \frac{\partial}{\partial w}- \frac{\partial}{\partial w'}+ \frac{\partial}{\partial \hat w} -\frac{\partial}{\partial \bar w} $
which acts simply on the trigonometric building blocks of the integrand.
Integrating by parts with this operator, it is possible to write the integrand of the $n$-th coefficient $ \beta_n$ as a total derivative
$ \beta_n = \tilde d \theta_n$. Then, using once more the symmetry of the graph,
\begin{eqnarray}
\int_{0}^{\infty}dw dw' d\hat w d\bar w \,\,{\rm e}^{-a} \beta_n &=& \int_{0}^{\infty}dw d\bar w d\hat w dw'
\tilde d
\,\,{\rm e}^{-(w+w'+\hat w +\bar w)}
\theta_n
\nonumber\\
&=&
4\int_{0}^{\infty}dw dw' d\hat w \,\,{\rm e}^{-(w+w'+\hat w)}\, \theta_n\vert_{\bar w =0}
\nonumber
\end{eqnarray}\noindent
The remaining threefold integrals are already of a fairly standard type.
\section{Using the polynomial invariants of $D_4$}
Diagram $ B$ has the symmetries
\begin{eqnarray}
\qquad w &\leftrightarrow& \hat w \nonumber\\
\quad w' &\leftrightarrow& \bar w\nonumber\\
\quad (w,\hat w) &\leftrightarrow& (w',\bar w) \nonumber
\end{eqnarray}\noindent
Those generate the dihedral group $ D_4$. After a slight generalization to the inclusion of semi-invariants (invariants up to a sign) \cite{120},
Theorem 1 can be used to deduce that the numerator polynomials can be rewritten
as polynomials in the variable $ \tilde w = w - w' +\hat w -\bar w$ with coefficients that are polynomials
in the four $ D_4$ - invariants $ a,v,j,h$,
\begin{eqnarray}
a &=& w + w' + \hat w + \bar w \nonumber\\
v &=& 2 (w \hat w + w' \bar w) + (w + \hat w)(w' + \bar w) \nonumber\\
j &=& a \tilde w - 4 ( w \hat w - w' \bar w) \nonumber\\
h &=& a (ww'\hat w + ww'\bar w + w\hat w \bar w + w'\hat w\bar w) + ( w \hat w - w' \bar w)^2 \nonumber
\end{eqnarray}\noindent
These invariants are moreover chosen such that they are annihilated by $ \tilde d$.
Thus they are well-adapted to the integration-by-parts algorithm.
This rewriting leads to a very significant reduction in the size of the expressions generated by the expansion in the field.
\section{Results}
In this way we obtained the first two coefficients of the weak-field expansion analytically,
\begin{eqnarray}
\Gamma^B_0 &=& -\frac{3}{2}+\frac{7}{4}\zeta(3) \nonumber\\
\Gamma^B_1 &=& -\frac{251}{120} + \frac{35}{16}\zeta(3)\nonumber
\end{eqnarray}\noindent
and five more coefficients numerically. For a definite conclusion concerning the
exponentiation conjecture this is still insufficient, and the computation of further coefficients is in progress.
\section{Outlook}
\begin{itemize}
\item
Writing Feynman graph polynomials in terms of invariant polynomials is a universal option that, to the best of our knowledge,
has not previously been used, but we expect that it will be found very useful for multiloop calculations involving a large number of external legs.
\item
In particular, this is the case for the weak-field expansion of the QED effective Lagrangian starting from three loops (in any dimension).
\end{itemize}
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\section*{Introduction}
In Riemannian geometry, E. Cartan defined an axiom of $r$-planes as follows. \emph{A Riemannian manifold $M$ of dimension $n\geq3$ satisfies the axiom of $r$-planes, where $r$ is a fixed integer $2\leq r< n$, if for each point $p$ of $M$ and any $r$-dimensional subspace $S$ of the tangent space $T_{p}M$ there exists an $r$-dimensional totally geodesic submanifold $V$ containing $p$ such that $T_{p}V=S$.} He proved that if $M$ satisfies the axiom of $r$-planes for some $r$, then $M$ has constant sectional curvature, cf., \cite{C}. The axiom of $r$-spheres in Riemannian geometry was proposed by D. S. Leung and K. Nomizu as follows. \emph{For each point $p$ of $M$ and any $r$-dimensional subspace $S$ of $T_{p}M$, there is an $r$-dimensional umbilical submanifold $V$ with parallel mean curvature vector field such that $p\in V$ and $T_{p}V=S$.} They proved that if a Riemannian manifold $M$ of dimension $n\geq3$ satisfies the axiom of $r$-spheres for some $r$, $2\leq r< n$, then $M$ has constant sectional curvature, cf., \cite{LN}.
In \cite{AZ}, Akbar-Zadeh extends the Cartan's axiom of 2-planes to Finsler geometry as follows. \emph{A Finslerian manifold $M$ of dimension $n\geq3$ satisfies the axiom of 2-planes if for each point $p\in M$ and every subspace $E_{2}$ of dimension two of $T_{p}M$ there exists a totally geodesic surface $S$ passing through $p$ such that $T_{p}S=E_{2}$.} He proved that every Finsler manifold satisfying the axiom of 2-planes is of constant flag curvature, cf., \cite{AZ}, page 182.
Recently, a definition of circle in Finsler spaces is introduced by one of the present authors in a joint work with Z. Shen, cf., \cite{BS}. Based on the definition of a circle we will show later that a connected submanifold of a Finsler manifold is an \emph{extrinsic sphere} if and only if its circles coincide with circles of the ambient manifold. The proof will appear elsewhere.\\
In the present work, we propose in a natural way, the following axiom of $r$-spheres in Finsler geometry.\\
\\
\emph{{\bf{Axiom of $r$-spheres.}} Let $(M,F)$ be a Finsler manifold of dimension $n\geq3$. For each point $x$ in $M$ and any $r$-dimensional subspace $E_{r}$ of $T_{x}M$, there exists an $r$-dimensional umbilical submanifold $S$ with parallel mean curvature vector field such that $x\in S$ and $T_{x}S=E_{r}$.}\\
\\
We shall prove the following theorem.
\begin{thm}\label{main1}
If a Finsler manifold of dimension $n\geq3$ satisfies the axiom of $r$-spheres for some $r$, $2\leq r< n$, then $M$ has constant flag curvature.
\end{thm}
\section{Notations and preliminaries on Finsler submanifolds}
Let $M$ be a real n-dimensional manifold of class $C^{\infty}$. We denote by $TM$ the tangent bundle of tangent vectors, by $p :TM_{0}\longrightarrow M$ the fiber bundle of non-zero tangent vectors and by $p^*TM\longrightarrow TM_0$ the pulled-back tangent bundle.
Let $(x,U)$ be a local chart on $ M$ and $(x^i,y^i)$ the induced local coordinates on $p^{-1}(U)$. A \emph{Finsler structure} on M is a function $F: TM \longrightarrow [0,\infty )$, with the following properties:(i) $F$ is differentiable $C^{\infty}$ on $TM_{0}$; (ii) $F$ is positively homogeneous of degree one in $y$, that is, $F(x,\lambda y)=\lambda F(x,y)$, for all $\lambda >0$; (iii) The Finsler \emph{metric tensor} $g$ defined by the Hessian matrix of $F^{2}$, $(g_{ij})=(\frac{1}{2}[\frac{\partial^{2}}{\partial y^{i}\partial y^{j}}F^{2}])$, is positive definite on $TM_{0}$. A \emph{Finsler manifold} is a pair $(M,F)$ consisting of a differentiable manifold $M$ and a Finsler structure $F$ on $M$. We denote by $TTM_0$, the tangent bundle of $TM_0$ and by $\rho$, the canonical linear mapping $\rho:TTM_0\longrightarrow p^*TM,$ where, $ \rho=p_*$. There is the horizontal distribution $HTM$ such that we have the Whitney sum $TTM_0=HTM_{0}\oplus VTM_{0}.$ This decomposition permits to write a vector field $\hat{X}\in \chi(TM_0)$ into the horizontal and vertical parts in a unique manner, namely $\hat{X}=H\hat{X}+V\hat{X}$. In the sequel, we decorate the vector fields on $TM_0$ by hat, i.e. $\hat{X}$ and $\hat{Y}$ and the corresponding sections of $p^*TM$ by $X=\rho(\hat X)$ and $Y=\rho(\hat Y)$, respectively, unless otherwise specified, cf., \cite{AZ}. For all $X\in p^{*}TM$ we denote by $^h\!\hat{X}$ the horizontal lift of $X$ defined by the bundle morphism $\beta:p^{*}TM\longrightarrow HTM$ where, $\beta(\frac{\partial}{\partial x^i})=\frac{\delta}{\delta x^i}$, cf., \cite{A}. For another approach on geometry of Finslerian manifolds, one can refer to \cite{BM}.
\subsection{Finsler geometry of submanifolds}
Let $(M,F)$ be a Finsler manifold and $S\subset M$ a $k$-dimensional submanifold defined by the immersion $i:S\longrightarrow M$. We identify any point $x\in S$ by its image $i(x)$ and any tangent vector $X\in T_{x}S$ by its image $i_{*}(X)$, where $i_{*}$ is the linear tangent mapping. Thus $T_{x}S$ becomes a sub-space of $T_{x}M$. Let $TS_{0}$ be the fiber bundle of non-zero tangent vectors on $S$. $TS_{0}$ is a sub-vector bundle of $TM_{0}$ and the restriction of $p$ to $TS_{0}$ is denoted by $q:TS_{0}\longrightarrow S$. We denote by $\bar{T}(S)=i^{*}TM$, the pull back induced vector bundle of $TM$ by $i$. The Finslerian metric $g$ on $TM_{0}$ induces a Finslerian metric on $TS_{0}$, where, we denote it again by $g$.
At a point $x=qz\in S$, where $z\in TS_{0}$, the orthogonal complement of $T_{qz}S$ in $\bar{T}_{qz}S$ is denoted by $N_{qz}S$, namely, $\bar{T}_{x}(S)=T_{x}(S)\oplus N_{qz}S,$ where $T_{x}(S)\cap N_{qz}S=0$. We have the following decomposition:
\begin{equation}\label{1}
q^{*}\bar{T}S=q^{*}TS\oplus N,
\end{equation}
where, $N$ is called the normal fiber bundle. If $TTS_{0}$ is the tangent vector bundle to $TS_{0}$, we denote by $\varrho$, the canonical linear mapping $\varrho:TTS_{0}\longrightarrow q^{*}TS$. Let $\hat{X}$ and $\hat{Y}$ be the two vector fields on $TS_{0}$. For $z\in TS_{0}$, $(\nabla_{\hat{X}}Y)_{z}$ belongs to $\bar{T}_{qz}S$. Attending to (\ref{1}) we have
\begin{equation}\label{2}
\nabla_{\hat{X}}Y=\bar{\nabla}_{\hat{X}}Y+\alpha(\hat{X},Y),\quad Y=\varrho(\hat{Y}),\quad X=\varrho(\hat{X}),
\end{equation}
where, $\nabla$ is the covariant derivative of Cartan connection and $\alpha(\hat{X},Y)$ the second fundamental form of the submanifold $S$. It belongs to $N$ and is bilinear in $\hat{X}$ and $Y$. It results from (\ref{2}) that the induced connection $\bar{\nabla}$ is a metric compatible covariant derivative with respect to the induced metric $g$ in the vector bundle $q^{*}TS\longrightarrow TS_{0}$.
\subsection{Shape operator or Weingarten formula}
Let $S$ be an immersed submanifold of $(M,F)$. For any $\hat{X}\in \chi(TS_{0})$ and $W\in \Gamma(N)$ we set
\begin{equation}\label{4}
\nabla_{\hat{X}}W=-A_{W}\hat{X}+\bar{\nabla}^{^\perp}_{\hat{X}}W,
\end{equation}
where, $A_{W}\hat{X}\in \Gamma(q^{*}TS)$ and $\bar{\nabla}^{^\perp}_{\hat{X}}W\in \Gamma(N)$ and we have partially used notations of \cite{BF}. It follows that $\bar{\nabla}^{^\perp}$ is a linear connection on the normal bundle $N$. We also consider the bilinear map
\begin{align*}
A:&\Gamma(N)\otimes\Gamma(TTS_{0})\longrightarrow \Gamma(q^{*}TS),\\
&A(W,\hat{X})=A_{W}\hat{X}.
\end{align*}
For any $W\in \Gamma(N)$, the operator $A_{W}:\Gamma(TTS_{0})\longrightarrow \Gamma(q^{*}TS)$ is called the \emph{shape operator} or the \emph{Weingarten map} with respect to $W$. Finally, (\ref{4}) is said to be the \emph{Weingarten formula} for the immersion of $S$ in $M$. We have
\begin{align*}
g(\alpha(^h\!\hat{X},Y),W)=g(A_{W}{^h\!\hat{X}},Y),
\end{align*}
where, $g$ is the Finslerian metric of $M$, $X,Y\in \Gamma(q^{*}TS)$ and $^h\!\hat{X}$ is the horizontal lift of $X$, cf., \cite{A}.
\subsection{Totally umbilical submanifolds in Finsler spaces}
The \emph{mean curvature} vector field $\eta$ of the isometric immersion $i:S\longrightarrow M$ is defined by
\begin{equation}\label{MC}
\eta=\frac{1}{n}tr_{g}\alpha(^h\!\hat{X},Y),
\end{equation}
where, $X,Y\in \Gamma(q^{*}TS)$ and $^h\!\hat{X}$ is the horizontal lift of $X$, cf., \cite{A}. We say that the mean curvature vector field $\eta$ is parallel in all directions if $\bar{\nabla}^{^\perp}_{^h\!\hat{X}}\eta=0$ for all $X\in \Gamma(q^{*}TS)$.
\begin{defn}
\cite{A} A submanifold of a Finsler manifold is said to be totally umbilical, or simply umbilical, if it is equally curved in all tangent directions.
\end{defn}
More precisely, let $i:S\longrightarrow M$ be an isometric immersion. Then $i$ is called totally umbilical if there exists a normal vector field $\xi \in N$ along $i$ such that its second fundamental form $\alpha$ with values in the normal bundle satisfies
\begin{equation}\label{TU}
\alpha(^h\!\hat{X},Y)=g(X,Y)\xi,
\end{equation}
for all $X,Y\in \Gamma(q^{*}TS)$, where $^h\!\hat{X}$ is the horizontal lift of $X$. Equivalently, $S$ is umbilical in $M$ if $A_{W}=g(W,\xi)I$ for all $W\in\Gamma(N)$ where, $I$ is the identity transformation, cf., \cite{A}.
To give an example of a totally umbilical submanifold in Finsler space, we refer to a theorem on totally umbilical submanifolds given in \cite{HYZ}. There is shown that if $(\tilde{M}^{n+1},\tilde{\alpha}+\tilde{\beta})$ is a Randers space, where $\tilde{\alpha}$ is an Euclidean metric and $\tilde{\beta}$ is a closed 1-form, then any complete and connected n-dimensional totally umbilical submanifold of $(\tilde{M}^{n+1},\tilde{\alpha}+\tilde{\beta})$ must be either a plane or an Euclidean sphere. The latter case happens only when there exist a point $\tilde{x}_0$ and a function $\lambda(\tilde{x})$ on $M^{n+1}$ such that $\tilde{\beta}=\lambda(\tilde{x})d(\parallel \tilde{x}-\tilde{x}_0\parallel_{\tilde{\alpha}}^{2})$ and the sphere is centered at $\tilde{x}_0$.\\
\hspace{-0.6cm}{\bf{Example 1.1.}} \cite{HYZ} Let $(\tilde{M}^{n+1},\tilde{F})$ be a Randers space with $\tilde{F}=\tilde{\alpha}+\tilde{\beta}$, where,
\begin{equation}
\tilde{\alpha}=\sqrt{\sum_{k=1}^{n+1}(\tilde{y}^k)^{2}},\quad \tilde{\beta}
=\sum_{k=1}^{n+1}\frac{b\tilde{x}^{k}d\tilde{x}^{k}}{\sqrt{\sum_k(\tilde{x}^k)^{2}}},\quad \forall (\tilde{x},\tilde{y})\in T\tilde{M}_0,\nonumber
\end{equation}
$b$ is a constant and $0<\mid b\mid<1$. One can see that $d\tilde{\beta}=0$. Let
\begin{equation}
M=\{\tilde{x}\in \tilde{M}^{n+1}:\sum_{k=1}^{n+1}(\tilde{x}^{k}-\tilde{x}_{0}^{k})^2=r^2\},\nonumber
\end{equation}
and $f:(M,F)\longrightarrow (\tilde{M}^{n+1},\tilde{F})$ be an isometric immersion where $F=\alpha+\beta$ such that
\begin{equation}
\alpha=\sqrt{\sum_{k=1}^{n+1}\frac{\partial f^k}{\partial x^i}\frac{\partial f^k}{\partial x^j}y^{i}y^{j}},\quad \beta=\sum_{k=1}^{n+1}\frac{\partial f^k}{\partial x^i}\frac{b\tilde{x}^{k}y^i}{\sqrt{\sum_k(\tilde{x}^k)^{2}}},\nonumber
\end{equation}
where, $(x,y)\in TM_0$. It is obvious that $\sum_{k=1}^{n+1}(f^k(x)-\tilde{x}^{k}_{0})\frac{\partial f^k}{\partial x^i}=0$ hence if $\tilde{x}_0=0$ then $\beta=0$. On the other hand from theorem mentioned above we see that $(M,F)$ is a totally umbilical submanifold of $(\tilde{M}^{n+1},\tilde{F})$ if $\tilde{x}_0=0$. Therefore if $\tilde{x}_0=0$, the Euclidean sphere $(M,\alpha)$ is a totally submanifold of Randers space $(\tilde{M}^{n+1},\tilde{\alpha}+\tilde{\beta})$. For more details on totally umbilical Finsler submanifolds one can refer to \cite{L}.
\begin{rem}\label{RIM}
Let $i:S\longrightarrow M$ be an isometric immersion. If $S$ is totally umbilical then the normal vector field $\xi$ is equal to the mean curvature vector field $\eta$.
\end{rem}
\subsection{Codazzi equation for Finsler submanifolds }
Consider a vector field $\hat{X}\in \Gamma(TTM_{0})$. We have locally $\hat{X}=X^{i}\frac{\delta}{\delta x^{i}}+\dot{X}^{i}\frac{\partial}{\partial y^{i}}$ where, $\{\frac{\delta}{\delta x^i},\frac{\partial}{\partial y^i}\}$ are horizontal and vertical bases of $TM$. Then one can define
\begin{align*}
Q:&\Gamma(TTM_{0})\longrightarrow \Gamma(TTM_{0}).\\
&Q\hat{X}:=\dot{X}^{i}\frac{\delta}{\delta x^{i}}+X^{i}\frac{\partial}{\partial y^{i}}.
\end{align*}
By means of the Cartan connection $\nabla$ on $(M,F)$ and the operator $Q$, one can define a linear connection on the manifold $TM_{0}$ by
\begin{equation*}
D_{\hat{X}}\hat{Y}:=\nabla_{\hat{X}}Y+Q\nabla_{\hat{X}}Q(H\hat{Y}),
\end{equation*}
for all $\hat{X},\hat{Y}\in \Gamma(TTM_{0})$. $D$ is called the \emph{associated linear connection} to $\nabla$ on $TM_{0}$. The torsion tensor field $T^{D}$ of $D$ is given by
\begin{equation*}
T^{D}(\hat{X},\hat{Y}):=\tau(\hat{X},\hat{Y})+Q(\nabla_{\hat{X}}Q(H\hat{Y})-\nabla_{\hat{Y}}Q(H\hat{X})-H[\hat{X},\hat{Y}]),
\end{equation*}
where, $\tau$ is the torsion tensor field of $\nabla$, cf., \cite{BF}. Let $R$ be the $hh$-curvature tensor of the Cartan connection $\nabla$, $\bar{\nabla}$ the induced connection on the submanifold $S$, $D$ the associated linear connection to the induced connection $\bar{\nabla}$ and $\bar{\nabla}^{^\perp}$ the linear connection on the normal bundle $N$. Let $A$ be the shape operator. One can define a covariant derivative $\nabla'$ of $A$ as follows, cf., \cite{BF}.
\begin{equation}\label{111}
(\nabla'_{\hat{X}}A)(W,\hat{Y}):=\bar{\nabla}_{\hat{X}}(A_{W}\hat{Y})-A_{\bar{\nabla}^{^\perp}_{\hat{X}}W}\hat{Y}-A_{W}(D_{\hat{X}}\hat{Y}),
\end{equation}
for any $\hat{X},\hat{Y}\in \Gamma(TTS_{0})$ and $W\in\Gamma(N)$. The \emph{$A$-Codazzi equation} for the Finsler submanifold $S$ with respect to the connection $\nabla$ on Finsler manifold $(M,F)$ is written
\begin{align}\label{codazzi}
g(R(X,Y)W,Z)&=g((\nabla'_{H\hat{Y}}A)(W,H\hat{X})-(\nabla'_{H\hat{X}}A)(W,H\hat{Y}),Z)\nonumber\\&-g(A_{W}(T^{D}(H\hat{X},H\hat{Y})),Z),
\end{align}
where, $W\in\Gamma(N)$, $X=\varrho(\hat{X})$, $Y=\varrho(\hat{Y})$ and $X,Y,Z\in \Gamma(q^{*}TS)$, cf., \cite{BF}, page 84.
\subsection{Sectional and flag curvatures}
Let $G_{2}(M)$ be the fiber bundle of 2-planes on $M$. Denote by $\pi^{-1}G_{2}(M)\longrightarrow SM$ the fiber induced on $SM$ by $\pi:SM\longrightarrow M$, where $SM$ is the unit sphere bundle. Let $P\in \pi^{-1}G_{2}(M)$ be a 2-plane generated by vectors $X,Y\in T_{x}M$ linearly independent at $x=\pi z\in M$ where, $z\in SM$. By means of $hh$-curvature tensors of Berwald and Cartan connection Akbar-Zadeh defined two \emph{sectional curvatures} denoted by $K_1$ and $K_2$ respectively. Here in this work we are dealing with Cartan connection and related sectional curvature $K_{2}:\pi^{-1}G_{2}(M)\longrightarrow \mathbb{R}$ defined by
\begin{equation}\label{OOO}
K_{2}(z,X,Y)=\frac{g(R(X,Y)Y,X)}{\parallel X\parallel^{2}\parallel Y\parallel^{2}-g(X,Y)^{2}},
\end{equation}
where, $R$ is the $hh$-curvature tensor of Cartan connection. The scalar $K_{2}$ is called the \emph{sectional curvature} at $z\in SM$. If the vector field $Y$ is replaced by the canonical section $v$ then sectional curvature is called flag curvature and does not depend on the choice of connection. If we denote the flag curvature by $K$ then we have
\begin{equation*}
K_{2}(z,v,X)=K(z,v,X),
\end{equation*}
where, $v$ is the canonical section, cf., \cite{AZ}, page 156.\\
Akbar-Zadeh as a \emph{generalization of Schur's theorem} has proved the following theorem.
\begin{theorem}\label{SS}
\cite{AZ} $K_{2}(z,P)$ is independent of 2-plane $P(X,Y)$ (dim $M>2$) if and only if the curvature tensor $R$ of the Cartan connection satisfies
\begin{equation*}
R(X,Y)Z=K[g(Y,Z)X-g(X,Z)Y],
\end{equation*}
where $K$ is a constant and $X,Y,Z\in T_{x}M$.
\end{theorem}
\section{Main results}
\begin{lem}\label{NNN}
Let $(M,F)$ be a Finsler manifold of dimension $n\geq 3$ satisfying the axiom of $r$-spheres for some $r$, $2\leq r< n$, then
\begin{equation*}
g(R(X,Y)Z,X)=0,
\end{equation*}
where, $X,Y,Z\in T_{x}M$ are three orthonormal vectors.
\end{lem}
\begin{proof} Let $(M,F)$ be a Finsler manifold which satisfies the axiom of $r$-spheres. Consider the Cartan connection $\nabla$ on the pulled-back bundle $p^{*}TM$, the induced connection $\bar{\nabla}$ on $S$ and the normal connection $\bar{\nabla}^{^\perp}$ on normal bundle. Let $X,Y$ and $Z$ be the three orthonormal vectors at $x=pz,z\in TM_0$. Consider the $r$-dimensional subspace $E_{r}$ of $T_{x}M$ which is normal to $Z$ and contains $X$ and $Y$. By assumption there exists an $r$-dimensional umbilical submanifold $S$ with parallel mean curvature vector field $\eta$ such that $x\in S$ and $T_{x}S=E_{r}$. It is well known for every point $x$ in a Finsler manifold there is a sufficiently small neighborhood $U$ on $M$ such that every pair of points in $U$ can be joined by a unique minimizing geodesic, see for instance \cite{BCS}, page 160. Hence there is a specific neighborhood $U$ of $x$ such that for each point $u\in U$ there exists a unique minimizing geodesic from $x$ to $u$. Let $W_{u}\in N_{u}S$ be the normal vector at $u$ which is parallel to $Z$ with respect to the normal connection $\bar{\nabla}^{^\perp}$ along the geodesic from $x$ to $u$ in $U$. The Finslerian metric $g$ on $TM_0$ defined by $F$, induces a Finslerian metric on $TS_0$, where we denote it again by $g$. By means of metric compatibility of Cartan connection, along each geodesic $\gamma$ from $x$ to any point in $U$ we have
\begin{equation}\label{ad}
\frac{d}{dt}g(W,\eta)=g(\nabla_{^h\!\hat{\dot{\gamma}}}W,\eta)+g(W,\nabla_{^h\!\hat{\dot{\gamma}}}\eta),
\end{equation}
where, $^h\!\hat{\dot{\gamma}}$ is the horizontal lift of the tangent vector field $\dot{\gamma}$. By means of the Weingarten formula (\ref{4}), rewrite (\ref{ad}) as follows
\begin{align}\label{add}
\frac{d}{dt}g(W,\eta)&=g(-A_{W}(^h\!\hat{\dot{\gamma}})+\bar{\nabla}^{^\perp}_{^h\!\hat{\dot{\gamma}}}W,\eta)+g(W,-A_{\eta}(^h\!\hat{\dot{\gamma}})+\bar{\nabla}^{^\perp}_{^h\!\hat{\dot{\gamma}}}\eta)\\
&=g(-A_{W}(^h\!\hat{\dot{\gamma}}),\eta)+g(\bar{\nabla}^{^\perp}_{^h\!\hat{\dot{\gamma}}}W,\eta)+g(W,-A_{\eta}(^h\!\hat{\dot{\gamma}}))+g(W,\bar{\nabla}^{^\perp}_{^h\!\hat{\dot{\gamma}}}\eta).\nonumber
\end{align}
Since $-A_{W}(^h\!\hat{\dot{\gamma}})$ and $-A_{\eta}(^h\!\hat{\dot{\gamma}})$ belong to $T_{x}S$ and on the other hand $\eta$ and $W$ are normal to $T_{x}S$ we have
\begin{equation*}
g(-A_{W}(^h\!\hat{\dot{\gamma}}),\eta)=g(W,-A_{\eta}(^h\!\hat{\dot{\gamma}}))=0.
\end{equation*}
By assumption the submanifold $S$ has parallel mean curvature vector field, that is, $\bar{\nabla}^{^\perp}_{^h\!\hat{\dot{\gamma}}}\eta=0$, hence $g(W,\bar{\nabla}^{^\perp}_{^h\!\hat{\dot{\gamma}}}\eta)=0$. By definition the vector $W$ is parallel along the geodesic $\gamma$ with respect to the normal connection $\bar{\nabla}^{^\perp}$, i.e. $\bar{\nabla}^{^\perp}_{^h\!\hat{\dot{\gamma}}}W=0$, hence $g(\bar{\nabla}^{^\perp}_{^h\!\hat{\dot{\gamma}}}W,\eta)=0$. Therefore by means of (\ref{add}) we have $\frac{d}{dt}g(W,\eta)=0$ and $g(W,\eta)=\lambda$ is constant along each geodesic. Keeping in mind $S$ is a totally umbilical submanifold of $M$, we have $A_{W}=g(W,\eta)I=\lambda I$ at every point of $U$. Rewriting (\ref{111}) for the horizontal lift $^h\!\hat{X}$ of $X$ leads
\begin{equation}\label{222}
(\nabla'_{^h\!\hat{X}}A)(W,\hat{Y})=(\nabla^{*}_{^h\!\hat{X}}A_{W})(\hat{Y})-A_{\bar{\nabla}^{^\perp}_{^h\!\hat{X}}W}\hat{Y},
\end{equation}
where, we have put, $(\nabla^{*}_{^h\!\hat{X}}A_{W})(\hat{Y}):=\bar{\nabla}_{^h\!\hat{X}}(A_{W}\hat{Y})-A_{W}(D_{^h\!\hat{X}}\hat{Y})$ which can be considered as a covariant derivative of $A_{W}$. Plugging $A_{W}=\lambda I$ in the last equation leads
\begin{equation}\label{333}
\nabla^{*}_{^h\!\hat{X}}A_{W}=0.
\end{equation}
Similarly for the horizontal lift $^h\!\hat{Y}$ of $Y$ we have
\begin{equation}\label{444}
\nabla^{*}_{^h\!\hat{Y}}A_{W}=0.
\end{equation}
On the other hand, by means of metric compatibility of Cartan connection and the fact that $g(W,\eta)$ is constant we have $g(\nabla_{^h\!\hat{X}}W,\eta)+g(W,\nabla_{^h\!\hat{X}}\eta)=0$. By means of the Weingarten formula (\ref{4}) the last equation leads
\begin{equation}\label{LL}
g(-A_{W}(^h\!\hat{X}),\eta)+g(\bar{\nabla}^{^\perp}_{^h\!\hat{X}}W,\eta)+g(W,-A_{\eta}(^h\!\hat{X}))+g(W,\bar{\nabla}^{^\perp}_{^h\!\hat{X}}\eta)=0.
\end{equation}
Since $A_{W}(^h\!\hat{X})$ and $A_{\eta}(^h\!\hat{X})$ belong to $T_{x}S$ and on the other hand $\eta$ and $W$ are normal to $T_{x}S$ we have
\begin{equation*}
g(-A_{W}(^h\!\hat{X}),\eta)=g(W,-A_{\eta}(^h\!\hat{X}))=0.
\end{equation*}
By assumption the submanifold $S$ has parallel mean curvature vector field, that is, $\bar{\nabla}^{^\perp}_{^h\!\hat{X}}\eta=0$, hence $g(W,\bar{\nabla}^{^\perp}_{^h\!\hat{X}}\eta)=0$. Therefore (\ref{LL}) reduces to $g(\bar{\nabla}^{^\perp}_{^h\!\hat{X}}W,\eta)=0$. By non-degeneracy of the metric tensor $g$ at $x\in S$ we have
\begin{equation}\label{5555}
\bar{\nabla}^{^\perp}_{^h\!\hat{X}}W=0.
\end{equation}
Similarly at $x\in S$ for vector $Y$ we obtain
\begin{equation}\label{555}
\bar{\nabla}^{^\perp}_{^h\!\hat{Y}}W=0.
\end{equation}
Therefore plugging (\ref{333}), (\ref{444}), (\ref{5555}) and (\ref{555}) in (\ref{222}) at $x\in S$ we obtain
\begin{equation*}
\nabla'_{^h\!\hat{X}}A=\nabla'_{^h\!\hat{Y}}A=0.
\end{equation*}
Now the Codazzi equation (\ref{codazzi}) implies
\begin{align}\label{codazzi2}
g(R(X,Y)W,X)=-g(A_{W}(T^{D}(^h\!\hat{X},^h\!\hat{Y})),X).
\end{align}
By assumption $A_{W}=g(W,\eta)I$. Thus we have
\begin{equation}\label{ne}
g(A_{W}(T^{D}(^h\!\hat{X},^h\!\hat{Y})),X)=g(T^{D}(^h\!\hat{X},^h\!\hat{Y}),X)g(W,\eta).
\end{equation}
Plugging (\ref{ne}) in (\ref{codazzi2}) we obtain
\begin{equation}\label{codazzi3}
g(R(X,Y)W,X)+g(T^{D}(^h\!\hat{X},^h\!\hat{Y}),X)g(W,\eta)=0.
\end{equation}
The first term $g(R(X,Y)W,X)$ is symmetric with respect to $Y$ and $W$, cf., \cite{AZ}, page 187. By means of the fact that $\eta$ is normal to $S$ we have $g(Y,\eta)=0$. Therefore we conclude
\begin{equation*}
g(T^{D}(^h\!\hat{X},^h\!\hat{Y}),X)g(W,\eta)=g(T^{D}(^h\!\hat{X},^h\!\hat{W}),X)g(Y,\eta)=0.
\end{equation*}
Thus (\ref{codazzi3}) becomes
\begin{equation*}
g(R(X,Y)W,X)=0.
\end{equation*}
Hence for orthonormal vectors $X,Y\in T_{x}S$ and $Z\in N_{x}S$ we have
\begin{equation*}
g(R(X,Y)Z,X)=0.
\end{equation*}
This completes the proof.
\end{proof}
\begin{lem}\label{MMMM}
Let $(M,F)$ be a Finsler manifold of dimension $n\geq 3$. If $g(R(X,Y)Z,X)=0$ whenever $X,Y$ and $Z$ are three orthonormal tangent vectors of $M$, then $M$ has constant flag curvature.
\end{lem}
\begin{proof}
If we put
\begin{equation*}
Y'=\frac{(Y+Z)}{\sqrt{2}}\quad,\quad Z'=\frac{(Y-Z)}{\sqrt{2}},
\end{equation*}
then since $X,Y$ and $Z$ are orthonormal, the vectors $X,Y'$ and $Z'$ are again orthonormal. By means of assumption
\begin{equation*}
g(R(X,Y')Z',X)=0.
\end{equation*}
By replacing $Y'$ and $Z'$ we obtain
\begin{equation}\label{1000}
g(R(X,Y)Y,X)=g(R(X,Z)Z,X).
\end{equation}
From which we can conclude from (\ref{OOO}), $K_{2}(z,X,Y)=K_{2}(z,X,Z)$. Thus the sectional curvature $K_{2}$ does not depend on the 2-plane $P(X,Y)$. By generalization of Schur's Theorem \ref{SS}, $M$ has constant sectional curvature and hence constant flag curvature. This completes the proof.
\end{proof}
{\bf Proof of Theorem \ref{main1}. } Let $(M,F)$ be a Finsler manifold which satisfies the axiom of $r$-spheres. By means of Lemmas \ref{NNN} and \ref{MMMM} we conclude that $M$ has constant flag curvature.
\hspace{\stretch{1}}$\Box$
|
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| 3,207
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{"url":"https:\/\/neurips.cc\/Conferences\/2020\/ScheduleMultitrack?event=18085","text":"Timezone: \u00bb\n\nSpotlight\nConstant-Expansion Suffices for Compressed Sensing with Generative Priors\nConstantinos Daskalakis \u00b7 Dhruv Rohatgi \u00b7 Emmanouil Zampetakis\n\nTue Dec 08 08:10 AM -- 08:20 AM (PST) @ Orals & Spotlights: Dynamical Sys\/Density\/Sparsity\n\nGenerative neural networks have been empirically found very promising in providing effective structural priors for compressed sensing, since they can be trained to span low-dimensional data manifolds in high-dimensional signal spaces. Despite the non-convexity of the resulting optimization problem, it has also been shown theoretically that, for neural networks with random Gaussian weights, a signal in the range of the network can be efficiently, approximately recovered from a few noisy measurements. However, a major bottleneck of these theoretical guarantees is a network \\emph{expansivity} condition: that each layer of the neural network must be larger than the previous by a logarithmic factor. Our main contribution is to break this strong expansivity assumption, showing that \\emph{constant} expansivity suffices to get efficient recovery algorithms, besides it also being information-theoretically necessary. To overcome the theoretical bottleneck in existing approaches we prove a novel uniform concentration theorem for random functions that might not be Lipschitz but satisfy a relaxed notion which we call `pseudo-Lipschitzness.'' Using this theorem we can show that a matrix concentration inequality known as the \\emph{Weight Distribution Condition (WDC)}, which was previously only known to hold for Gaussian matrices with logarithmic aspect ratio, in fact holds for constant aspect ratios too. Since WDC is a fundamental matrix concentration inequality in the heart of all existing theoretical guarantees on this problem, our tighter bound immediately yields improvements in all known results in the literature on compressed sensing with deep generative priors, including one-bit recovery, phase retrieval, and more.","date":"2022-05-24 12:22:50","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\": 0.8478127717971802, \"perplexity\": 732.709959900781}, \"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-2022-21\/segments\/1652662572800.59\/warc\/CC-MAIN-20220524110236-20220524140236-00063.warc.gz\"}"}
| null | null |
Course of Studies and Examination
Course of studies includes Religious knowledge, moral science, English language, English literature, Hindi, Sanskrit, History of India, Geography, mathematics, Physics, Chemistry, Biology, Geometry & Drawing, Economics, Structure of commerce, Principles of Accounts, Computer studies, Art education, Physical Training, Handicraft, Music & many curricular activities.
No pupil can opt for subjects which are normally not taught in the school or provided for in the approved subjects of studies.
The progress reports shall strictly be shown to the parents after the FAs & Sas and duly got signed in proof thereof.
Promotion will be based on the performance in FA1, FA2, FA3, FA4, SA1 & SA2.
Students who absents themselves from an examination for reasons, whatsoever, will not be re-examined.
a. Willful breach of any of the regulations or the conduct of the examination is punishable by expulsion from the examination hall and if need be, by cancelling the paper subsequently.
b. Candidates who are caught indulging in unfair means or found giving or obtaining or attempting to give or obtain through unfair methods or who otherwise are detected to have indulged in dishonest means, whatsoever, will be expelled from the examination hall forthwith and may be refused admission in remaining papers or could be expelled from the school as well.
a. Report cards could be withheld for nonpayment of school dues, non-submission of necessary documents etc.
b. Unless and until the report cards are collected by the students their names will not be appearing in the promotion list.
Absence from a subject of Exam precludes a student from being recognized in the order of merit in the examination.
A student will be considered for promotion only if he/ she obtains a minimum attendance of 75% of the total working days of the Year.
To be considered for promotion, a student has to obtain minimum pass marks prescribed in all major subjects of English, Hindi, Science, Math's & social Studies.
The pupils of classes 1st to 5th must secure 45% marks for a pass and those of 6th to 9th must secure 40% marks for a pass.
Promotion once refused will not be reconsidered.
A student who fails shall forfeit his/ her right for any concessions in fees for the next year and may even be advised to leave the school; particularly it's so, if his/ her conduct has not been satisfactory in the school. If a student fails twice he/she will have to leave the institution invariably.
Games are compulsory for all the classes.
All kinds of fees are deposited by cheque in the school office between 09:00 a.m. to 1:00 p.m. on all working days as per fee schedule.
The fees must be paid for all months during which the pupil's name has been carried on the register, even if the pupil has been absent during such months.
The fees for the term must be paid in full, even if the pupil leaves before the term is over. The School usually collects fees for 10 months in a Year.
The fees could also be paid in advance for the full term or the whole Year. In case of those who pays monthly, the fees of the current month must be paid strictly on or before the dates mentioned in the Fee Payment Schedule.
No reduction of fees is allowed merely of midsummer and winter holidays or other vacations of temporary absence.
The children whose school fee falls into arrears are liable to be debarred from taking the terminal tests and/ or attending school, unless the fees are paid in full. Neither the report cards nor transfer certificates will be issued, till all the arrears are settled to the satisfaction of the School.
While paying fees, please write the details of the ward behind the cheque leaf in the following format.
Name: Class: Section:-
|
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| 7,662
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Freedom™ Network Services
Global Antenna Network
LINKS™ Electronically Steered Array
Deep Space (ISCN)
ATLAS Space Operations Announces Partnership with Aevum and Contribution to ASLON-45 Space Lift
TRAVERSE CITY, Mich. – Nov. 11, 2019 – ATLAS Space Operations, Inc., a leading innovator in ground communications for the space industry, and Aevum, Inc., a principal provider of launch and space logistics services, today publicly announced their partnership and collaboration on The Agile Small Launch Operational Normalizer (ASLON)-45 space lift mission.
The partnership expands on the existing collaboration between ATLAS and Aevum with the $4.9 million ASLON-45 mission, which provides orbital launch services to the Department of Defense (DOD) Space Test Program and other government agencies. By facilitating experimental satellites in low-Earth orbit, ATLAS and Aevum will help the DOD improve their real-time threat warnings.
"We're ecstatic to announce our long-term relationship with Aevum," said Sean McDaniel, CEO and co-founder of ATLAS. "This partnership will push the envelope of capabilities that are available to the space community. Beyond the ASLON-45 mission, ATLAS and Aevum are looking forward to conducting many successful launches and continuing to empower global access to space."
ATLAS' Freedom™ Ground Network currently has 31 operational and planned antennas placed strategically around the world. This extensive ground communications network perfectly complements Aevum's scalable launch service, which allows their unmanned, autonomous lift vehicles to launch from virtually any runway in the world and not be limited to range. This combination of services gives the partners and their customers global accessibility to space operations.
A unique advantage of the partnership will be the seamless transition in implementing ATLAS' global communications capabilities. Because Aevum already uses ATLAS for telemetry, tracking, and command, launch support to post-launch support will be a simple transition for customers. Furthermore, the partnership will allow Aevum to lower the cost of launches for small satellite missions.
"By choosing to horizontally integrate with ATLAS, we're shifting the risk away from technology and capital to execution- which involves a lot of trust," said Jay Skylus, founder and CEO of Aevum. "Vertical integration may reduce execution risk but often heightens cost and technology development risks. Our choice to integrate with ATLAS for ground communications provides our customers with more benefits and robust services."
Aevum will now have access to ATLAS' extensive ground station network, enabling the service to conduct launches all over the globe. ATLAS and Aevum's scalable partnership will allow each company to continue to produce cutting-edge capabilities for the space community and better serve their customers.
About ATLAS Space Operations
ATLAS Space Operations, Inc., based in Traverse City, Michigan, empowers global access to space through Freedom™, a simple solution for processing and analyzing data from space, through a global antenna network, powered by a revolutionary cloud-based software. ATLAS's forward-thinking communications solutions are transforming the space industry by making ground communications simple, affordable and scalable than ever before. For more information on ATLAS Space Operations, please visit https://www.atlasground.com/.
About Aevum Space Logistics
Aevum, Inc. provides comprehensive space logistics services, including launch, to enable commercial and government customers to deploy small payloads in low Earth orbit. Aevum's reusable, fully-autonomous launch system drastically reduces launch costs, provides high mission flexibility, and enables rapid launch capabilities in as low as 180 minutes – from anywhere in the world. https://www.aevumspace.com/
Media Contact Information:
Dan Carey
Marketing Director, ATLAS Space Operations
+1 (231) 598-6184 ext 105
dcarey@atlasground.com
Aevum Media
media@aevum.us
http://www.aevumspace.com
Spread the awesome news!
Explore More News & Updates
ATLAS Ushers Smart Space Comms Into Future With SBIR
ATLAS Space Operations has advanced its machine-to-machine learning capability through a Small Business Innovation Research (SBIR) grant. The ATLAS SBIR involves the use of over one billion data points from all aspects of the ATLAS communications platform to improve reliability and performance.
atlasspaceoperations November 17, 2021
ATLAS Named to Military Times "Best for Vets"
ATLAS Space Operations has been named to the Military Times Best for Vets list, an annual publication of the country's top veteran-friendly employers. ATLAS ranked #5 among small and medium sized businesses, and #7 in the state of Michigan.
atlasspaceoperations November 5, 2021
ATLAS Team Takes Two Spots on Coveted 20 Under 35
ATLAS is proud to share that team members Dan Carey, Director of Marketing, and Sam Lewis, Senior Software Development Engineer, are among this year's awardees. This represents the third such win for ATLAS in as many years, after Software Development Engineer Ryan Clulo received the same honor in 2019.
atlasspaceoperations September 24, 2021
Connecting Humanity
Through Space™
Get the latest updates from ATLAS
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ATLAS Space Operations, Inc.
10850 E Traverse Highway, Ste. 3375 Traverse City, Michigan 49684
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You probably owned a classic pocket knife as a kid and maybe your father or grandfather kept one handy. Gerber's decades of experience crafting tools and Bear Grylls's adventure knowledge come together in our Grandfather Knife. Based on a knife that was passed down from Bear's grandfather to his dad and then to Bear himself, we've honored the classic style and utility of Bear's knife by designing ours with 7 utilitarian components while maintaining the vintage appearance of his original knife. The Grandfather Knife is the quintessential do-it-all pocket knife with the benefit of modern construction.
Like a good hand tool, the first thing you'll notice about the Grandfather Knife is its comfortable heft and easy handling. It weighs half of what most multi-tools weigh and embodies the essence of pocket knife: equal parts utility and zest for adventure. The tool selection is simple by design, offering what you need to save the day at the picnic when there's no wine opener in sight or to adjust the throttle on your mower without making a trip back to the garage. The fine edge blade, corkscrew, flat and Phillips head drivers, small file and all-important bottle opener are easy to deploy and give you flexibility in your moments of need. At 3.75 inches long, the closed knife is only slightly bigger than a disposable lighter. Still, you get a 2.75-inch fine edge blade, big enough for substantial tasks yet compact enough to carry daily.
The Grandfather knife disappears in your pocket until you need to put it to work. Like a perfectly weighted skipping stone, it feels great in your hand, too. The rubber-molded grip adds to its easy-to-hold shape and offers a little added security in cold, wet conditions. Don't let the knife's vintage look fool you; it's meant to get used today. The fine edge blade is easy to sharpen and there's a lanyard ring for adding a small strap or hanging it from your belt.
- The knife's 7 components include a fine edge blade, flat driver, Phillips driver, corkscrew, bottle opener, file and lanyard ring.
- The stainless steel Bear Grylls badge plays on the knife's vintage origins and lends it a classic look.
- The rubber-over-mold grip gives the knife a comfortable feel and adds security to your grip.
- Includes a copy of Bear Grylls "Priorities of Survival" Pocket Guide for easy reference.
|
{
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The Continuing Long Hard Slog for Streetlights in City Heights
May 10, 2013 by Anna Daniels
By Anna Daniels
"Emerald said there are $26 million in unfunded streetlights [in City Heights]. She called for a survey of existing lights to identify dark spots. She wants to then cross reference that information with police reports to see where new lights could deter crime." SpeakCityHeights 1/13
There isn't any mystery as to why residents expect to have streetlights in their respective communities. It's important to be able to see where you are walking at night; streetlights are an essential element of crime deterrence; and they contribute to our perceptions of personal safety.
City Heights is a transit dependent community and residents don't tend to work bankers hours. Many of my neighbors go to work while it is still dark or return home when it is dark. Many of these commuting workers are women working in the hospitality and food service industries or providing in home personal care.
This is also a community that sustains elevated incidents of assault, robbery and break-ins. City Heights should be one of the best lit neighborhoods in the City of San Diego simply on the basis of need and yet it is unfunded $26 million for streetlights.
The City of San Diego does not get a free pass on this issue because of the economy. City Heights was starved of streetlights twenty five years ago when I moved here and it is still starved of that critical infrastructure investment. The real story here has little to do with the economy.
Good economic times have come and gone and too many parts of City Heights are still in the dark. Residents of City Heights deserve an accounting for why that is. It isn't as though residents here don't care. We have never ceased advocating for streetlights. City Height's Capital Improvement List for FY-14 includes streetlights.
The Battle of the Bug Lights
In the early 1990's a small group of residents in my Teralta neighborhood banded together to contest the use of low sodium lights in the area and also to advocate for mid-block streetlights. Low sodium lights are no longer used and it is worth noting that these lights reduced any color normally discernible at night to the same muddy grey.
That meant that when you called the police to report a break-in of a car in front of your house, you described a car that was grey and a grey perp who was wearing grey clothes. We were told by the City that these bug lights were the only option.
Except they weren't the only option. The Gas Lamp district downtown was being developed and that area had high pressure sodium lighting that did not reduce everything to a monotone grey. Sweet. We wanted that.
Neighbor Mary Laiuppa, music teacher by day and Teralta troubadour by night, composed a series of songs for us. We sang under the streetlights in the neighborhood. We weren't permitted to sing at the City Council hearing on the issue, but we passed out the lyrics before we provided our testimony.
"There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy…"
There was unanticipated resistance. San Diego State's astronomy department was against the use of high pressure sodium because of the impact of light pollution on the Mt Palomar Observatory. We sure didn't see that one coming.
The astronomy department's position would have had credibility with us if they were just as strongly pushing back against the light pollution created by the business community and governmental entities. Take a look at the billboards you pass at night. Are the lights mounted on the top of the billboard facing down, or at the bottom of the billboard shining up?
I called the two billboard companies that advertise in City Heights–Gannet and Outdoor Advertising. They defended the use of the upward shining lights because they provided the clear illumination that their customers expected.
Pearson Ford had a car lot that took up a city block on El Cajon Blvd. The lot was lit up like a circus after the business closed and the lights shot straight up into the night sky all night long as a crime deterrent.
San Diego State itself did not follow their own requirement for the use of low sodium, opting for high sodium instead on the campus parking lots, as did the City of San Diego in its own maintenance yards.
I contacted the City of Tucson, which had the most comprehensive municipal ordinances regarding light pollution in the country at the time– they too have an observatory. Businesses and governmental entities did not receive an automatic variance. They didn't receive any variance. All lights were shielded and pointed down instead of up. This seemed like a reasonable approach to light pollution and it got no traction here.
It was too easy to pick the dark skies fight in poor communities with crime issues and cast the debate as the Astronomers v the Mud People. The City was not about to take on the business community and public entities were not about to re-think their own hypocrisy and some areas like the Gas Lamp ended up being more equal in the lighting debate. This is how poor communities get screwed– the essential discussion about social justice and equity is never considered.
A compromise was eventually hammered out. Communities south of 8 would have high sodium streetlights. It was the end of low sodium bug lights in City Heights, but there remained the problem of the lack of mid block streetlights. Our group asked the City what was necessary to get these lights. We were provided petitions and covered a four block area to get the sign off for mid block lights and authorization by property owners to permit a pass through on their property of electrical lines to the streetlight.
The majority of City Heights property owners don't live in City Heights. This is a rental community. Property owners, if you can find them, live all over the country. We passed around our petitions and tried to identify an actual home owner about mid-block who would authorize the pass through.
The City lost our petitions, and no, we didn't make copies of the originals. Naivete comes at a high cost in this community. So we passed around the petitions all over again. And waited. That was around 1992.
I can't remember exactly when a mid-block streetlight appeared on 45th street. I would say around 2002.
That was a ten year wait. In the interim, we approached the City Heights CDC for assistance and to expand the advocacy to other areas that also needed streetlights. Board member John Stump presented the idea that we should pursue an additional approach. "Light the Heights" was born.
The CHCDC applied for CDBG funds during that same time period (1993) and bought dusk to dawn lights that could be installed directly on resident's homes and roof lines of apartments. The installation was free and residents were asked to maintain the lights in return and pay for the electricity. Over four hundred of these lights were installed, and Frank Gormlie who worked for the CHCDC at the time oversaw the project.
It was a remarkable effort which did more than provide partial illumination into the public right of way in high crime areas. Property owners were also asked to paint out graffiti, repair broken windows and remove trash and debris as part of the agreement. And the project also used local electricians for the installation.
Twenty years later, the City Heights Town Council is currently initiating an "adopt-a-light" program to put solar powered flood lights wherever there are dark spaces in the community.
At a recent City Heights Area Planning Committee, Matthew Hervey of Price Charities discussed and described a proposal to fund and install eight street lights in areas that are dark and that are noted for nuisances and crimes. The total cost is expected to be $80,000, and the City has agreed to pay for electricity and maintenance.
Philanthropic and citizen efforts are laudatory, but it begs the question why City Heights is still underfunded $26 million in streetlights. Since that time in the early 90's when residents and businesses organized around the streetlight issue, our representation on the City Council changed from John Hartley to Christine Kehoe to Toni Atkins to Todd Gloria to Marti Emerald today.
In that same period of time, Susan Golding was mayor, then Jerry Sanders and now it's Bob Filner. The lack of streetlights in City Heights is rooted in the inability or unwillingness of elected representatives throughout the city and over a period of many years to articulate an argument for fairness and equity in how and where public infrastructure investments are made.
It is time to have that discussion now. The economy can not be used as an argument for shared sacrifice when there are such disproportionate impacts on low income communities. It should be cast into the heap with all the other past arguments which have been so successful in keeping us in the dark.
Anna Daniels was a past president and board member of the City Heights Community Development Corporation
Anna Daniels
I left a moribund Western Pennsylvania mill town the year that Richard M. Nixon was not impeached for crimes against the American people, and set off in search of truth, beauty, justice and a beat I could dance to. Here I am.
Latest posts by Anna Daniels (see all)
Bob Dorn: Rest in Power! - December 3, 2018
The Urgency of Transgender Safety, Legal Protections - November 27, 2018
Chaos and Cruelty at the Border - November 26, 2018
Filed Under: Activism, City Heights: Up Close & Personal, Culture, Government, Politics Tagged With: City Heights
About Anna Daniels
« The Starting Line – All is Not Clear on the Low Wage Front: San Diego Fundraiser for a Walmart Strike Fund Planned
Should We Have Saved AIG and Other Wall Street Banks? »
Mark Tran says
Another great article! I love reading your work.
Buffy Budz says
Right on, Anna.
judi says
Interesting, Anna. My foreign students, over 380 of them, have often asked me why the street lights are so dim around my Pt. Loma neighborhood. They all take the bus from school at night, and walking through dark alleys where they cannot tell the difference in colors of cars, trash cans, etc. has been a frightening thing for them – male or female. I was told it was because the observatory on Palomar Mountain wanted the "darkness" to see the skies but after your article I am wondering it that is truly the case. After all, if retail establishments can have bright lights focused upwards, then that negates the excuse that it is the observatory that wants the dim lights in neighborhoods. Looks like this problem may be city-wide. Hmmm
I think astronomers are rightfully concerned about light pollution, but have chosen not to take the lead on the issue to come up with more rigorous policies that apply to businesses too. I spoke to someone in the City a few days ago and asked for an update. More stringent lighting laws are applied closer to the the observatory.
Do you have City lights in the alleys around you? I can not imagine walking home at night through the alley behind my house. The lights I am talking about are all on the streets (or not.)
There are no lights in the alleys here, but they are "backyards". The alley the students walk down starts at the bus stop, and goes past a church and preschool. One the other "side" of the alley are basketball courts where there are "impromptu" games played nightly. From the end of the alley to the house are two street lights, with very dim bulbs. (Kind of like the City fathers that made the decision to put in low-lights in the first place.) It still is not safe – and we have had a rash of burglaries in the area recently because there is a crack house up the street.
Bright street lights are the number one factor in deterring crime. The city should start installing them along with cameras in crime ridden areas.
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Марк Вільямс (; 22 серпня 1959 Бромсгроув, Вустершир, Велика Британія) — англійський актор, комік і сценарист. Найбільш відомий ролями Артура Візлі (в серії фільмів про Гаррі Поттера) і отця Брауна (в однойменному телесеріалі).
Життєпис
Марк Вільямс народився 22 серпня 1959 року в Бромсгроувi, Англія.
У 1996 році в парі з Гленн Клоуз (Лютелла де Віль) і Г'ю Лорі (Джаспер Бякін) знявся у фільмі «101 далматинець», виконавши роль Горація/Хоріса Бякіна.
У 2007 році взяв участь в зйомках фільму «Зоряний пил», за сценарієм Ніла Ґеймана. Також в екранізації були задіяні такі актори, як Мішель Пфайффер, Роберт Де Ніро і Клер Дейнс.
Вільямс також представив кілька документальних фільмів: «Великі вибухи Марка Вільямса» (про історію вибухових речовин), «Марк Вільямс на рейках», «Промислові відкриття» та «Більше промислових одкровень».
Є вболівальником футбольних клубів «Астон Вілла» і «Брайтон енд Гоув Альбіон».
З'явився в 7 сезоні науково-фантастичного телесеріалу «Доктор Хто» в ролі Брайна Вільямса. Знімається в головній ролі в детективному телесеріалі «Отець Браун» телеканалу BBC One про католицького священика отця Брауна, який веде розслідування в невеликому англійському містечку Кемблфорд в 50-х роках XX століття.
Фільмографія
Примітки
Посилання
Британські телеактори
Актори озвучування Великої Британії
Актори XX століття
Актори XXI століття
Лауреати премії Гільдії кіноакторів США
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\section{Introduction}
\label{s:intro}
\paragraph{Zeta functions derived from endomorphisms.}
Throughout, rings are assumed to be commutative and unital.
We say that a ring $R$ has \bfemph{polynomial submodule
growth} if the following holds for every finitely generated $R$-module $M$:
for each $m \ge 1$, the number of submodules of additive index $m$ of $M$ is finite and
polynomially bounded as a function of $m$.
Recall that $R$ is \bfemph{semi-local} if it contains only
finitely many maximal ideals.
\begin{thm}[{\cite[Thm~1]{Seg97}}]
\label{thm:segal}
Let $R$ be a ring which is finitely generated over $\mathbf{Z}$ or semi-local with
finite residue fields.
Then $R$ has polynomial submodule growth if and only if it has Krull dimension at most $2$.
\end{thm}
Let $R$ be a ring with polynomial submodule growth,
let $M$ be a finitely generated
$R$-module, and let $A \in \End_R(M)$.
For $m \ge 1$, let $a_m(A,R)$ denote the number of $A$-invariant
$R$-submodules $U \le M$ with $\idx{M:U} = m$.
We define a zeta function
\[
\zeta_{A,R}(s) := \sum_{m=1}^\infty a_m(A,R) m^{-s}
\]
and we let $\alpha_{A,R} < \infty$ denote its abscissa of convergence;
it is well-known that $\alpha_{A,R}$ is precisely the degree of polynomial
growth of the partial sums $a_1(A,R) + \dotsb + a_m(A,R)$ as a function of $m$.
The zeta functions $\zeta_{A,R}(s)$ belong to the larger
theory of subobject zeta functions; for a recent survey of the area, see
\cite{VollSurvey}.
Indeed, using the terminology from \cite{topzeta}, $\zeta_{A,R}(s)$ is the
submodule zeta function $\zeta_{R[A] \ensuremath{\curvearrowright} M}(s)$ of the enveloping algebra
$R[A] := \sum\limits_{i=0}^\infty R \ensuremath{\,\cdotp} A^i \subset \End_R(M)$ of $A$ acting on $M$.
The main results of this article, Theorems~\ref{Thm:global}--\ref{Thm:poles},
constitute a rather exhaustive analysis of the zeta functions $\zeta_{A,R}(s)$
in the cases that $R$ is the ring of ($S$-)integers of a number field
or a (generic) completion of such a ring.
In particular, our findings provide further evidence in support of the
author's general conjectures on submodule zeta functions stated in \cite[\S 8]{topzeta}.
\paragraph{Related work: invariant subspaces.}
The study of subspaces invariant under an endomorphism has a long history.
For a finite-dimensional vector space $V$ over the real or complex numbers and
$A \in \End(V)$, Shayman~\cite{Sha82}
investigated topological properties of the compact analytic space $S_A$ of $A$-invariant subspaces of
$V$.
In particular, if $A$ is nilpotent, then he found the subspace $S_A(d) \subset
S_A$ of $d$-dimensional $A$-invariant subspaces of $V$ to be connected but
usually singular.
For an arbitrary ground field $F$ and a fixed number $n$, Ringel and
Schmidmeier~\cite{RS08} studied the category of triples $(V,U,T)$, where $V$ is
a finite-dimensional vector space over $F$, $T\in \End_F(V)$ satisfies $T^n =0$,
and $U\le V$ is $F$-invariant.
While their point of view is rather different from ours, we would like to
point out that they found the case of exponent $n \ge 7$ to involve instances of
so-called ``wild'' representation type.
\paragraph{Ideal zeta functions.}
In our study of the zeta functions $\zeta_{A,R}(s)$, we will frequently encounter
another special case of submodule zeta functions, namely ideal zeta functions.
Let $R$ be a ring with polynomial submodule growth and let $\mathsf A$ be a
possibly non-associative $R$-algebra whose underlying $R$-module is finitely
generated.
We write $\mathsf I \triangleleft_R \mathsf A$ to indicate that $\mathsf I$ is a
two-sided ideal of $\mathsf A$ which is also an $R$-submodule.
The \bfemph{ideal zeta function} (cf.\ \cite{GSS88}) of $\mathsf A$ is
$$\zeta_{\mathsf A}(s) := \sum_{\substack{\mathsf I \triangleleft_R \mathsf A
\\\idx{\mathsf A:\mathsf I}<\infty}} \idx{\mathsf A:\mathsf I}^{-s}.$$
For example, the ideal zeta function of the ring of integers of a number field $k$
is precisely the Dedekind zeta function of $k$.
In particular, the ideal zeta function of $\mathbf{Z}$ is the Riemann zeta function
$\zeta(s)$.
As explained in \cite[Rem.\ 2.2(ii)]{topzeta}, ideal zeta functions are in fact
a special case of the submodule zeta functions discussed below.
\paragraph{Global setup, Euler products, and growth rates.}
For the remainder of this article, let~$k$ be a number field with ring of
integers $\mathfrak{o}$.
Let $\ensuremath{\mathcal V}_k$ denote the set of non-Archimedean places of $k$.
For $v \in \ensuremath{\mathcal V}_k$, let $k_v$ be the $v$-adic completion of $k$ and
let $\mathfrak{o}_v$ be its valuation ring.
For $S\subset \ensuremath{\mathcal V}_k$,
let $$\mathfrak{o}_S = \bigcap\limits_{v\in\ensuremath{\mathcal V}_k\setminus S} \mathfrak{o}_v \cap k$$
be the usual ring of $S$-integers of $k$.
In the following, we investigate $\zeta_{A,R}(s)$, where $A\in \End_R(M)$ and
$R = \mathfrak{o}_v$ or $R = \mathfrak{o}_S$ for $v \in \ensuremath{\mathcal V}_k$ or a finite set $S\subset
\ensuremath{\mathcal V}_k$, respectively.
The techniques that we use are predominantly local and valid for almost all
places of~$k$ (i.e.\ for all but finitely many places);
the exclusion of a finite number of exceptional places is common and
frequently unavoidable in the theory of subobject zeta functions.
If $M$ is a finitely generate $\mathfrak{o}_S$-module,
then $M \otimes_{\mathfrak{o}_S} \mathfrak{o}_v$ is a free $\mathfrak{o}_v$-module for almost all $v\in
\ensuremath{\mathcal V}_k\setminus S$.
We thus lose little by henceforth assuming that $M = \mathfrak{o}_S^n$ and
$A \in \Mat_n(\mathfrak{o}_S)$, where $\Mat_n(R)$ denotes the algebra of $n\times n$
matrices over a ring $R$.
Note that if $A \in \Mat_n(k)$, then $A \in \Mat_n(\mathfrak{o}_v)$ for almost all $v \in
\ensuremath{\mathcal V}_k$.
In order to exclude trivialities, unless otherwise stated, we always assume that $n > 0$.
Being instances of submodule zeta functions,
the zeta functions $\zeta_{A,\mathfrak{o}_S}(s)$ admit natural Euler product factorisations.
\begin{prop*}[Cf.\ {\cite[Lemma~2.3]{topzeta}}]
Let $A \in \Mat_n(\mathfrak{o}_S)$ for finite $S \subset \ensuremath{\mathcal V}_k$.
Then $$\zeta_{A,\mathfrak{o}_S}(s) = \prod\limits_{v \in \ensuremath{\mathcal V}_k\setminus S} \zeta_{A,\mathfrak{o}_v}(s).$$
\end{prop*}
The following is a consequence of deep results of du~Sautoy and Grunewald on
subobject zeta functions expressible in terms of what they call ``cone integrals''.
\begin{thm}[{Cf.\ \cite[\S 4]{dSG00}}]
\label{thm:dSG}
Let $A \in \Mat_n(\mathfrak{o}_S)$ for finite $S \subset \ensuremath{\mathcal V}_k$. Then:
\begin{enumerate}
\item
\label{thm:dSG1}
The abscissa of convergence $\alpha_{A,\mathfrak{o}_S}$ of $\zeta_{A,\mathfrak{o}_S}(s)$ is a rational number.
\item
\label{thm:dSG2}
$\zeta_{A,\mathfrak{o}_S}(s)$ admits meromorphic continuation to $\{ s \in \mathbf{C} : \mathrm{Re}(s) >
\alpha_{A,\mathfrak{o}_S} - \delta\}$ for some $\delta > 0$.
This continued function is regular on the line $\mathrm{Re}(s) = \alpha_{A,\mathfrak{o}_S}$
except for a pole at $s = \alpha_{A,\mathfrak{o}_S}$.
\item
\label{thm:dSG3}
Let $\beta_{A,\mathfrak{o}_S}$ denote the multiplicity of the pole of
(the meromorphic continuation of) $\zeta_{A,\mathfrak{o}_S}(s)$ at $\alpha_{A,\mathfrak{o}_S}$.
Then there exists a real constant $c_{A,\mathfrak{o}_S} > 0$ such that
\[
a_1(A,\mathfrak{o}_S) + \dotsb + a_m(A,\mathfrak{o}_S) \sim c_{A,\mathfrak{o}_S} \ensuremath{\,\cdotp} m^{\alpha_{A,\mathfrak{o}_S}} (\log m)^{\beta_{A,\mathfrak{o}_S}-1}.
\]
where $f(m) \sim g(m)$ signifies that $f(m)/g(m) \to 1$ as $m \to \infty$.
\end{enumerate}
\end{thm}
\paragraph{Matrices, polynomials, and partitions.}
Prior to stating our main results, we need to establish some notation and recall
some terminology.
By a \bfemph{partition} of an integer $n \ge 0$, we mean a non-increasing sequence
$\bm\lambda = (\lambda_1,\dotsc,\lambda_r)$ of positive integers with $n =
\lambda_1 + \dotsb + \lambda_r$;
for background, we refer to \cite{Mac79}.
We write $\abs{\bm\lambda} := n$,
$\len{\bm\lambda} := r$, and $\lambda_{-1} := \lambda_r$.
We write $\bm\lambda \vdash n$ to signify that $\bm\lambda$ is a partition of $n$.
For $i \ge 0$, define $\ensuremath{\mathsf{\sigma}}_i(\bm\lambda) := \lambda_1 + \dotsb + \lambda_i$.
For $1\le j \le \abs{\bm\lambda}$, let $\ind{\bm\lambda} j$ be the unique
number $i \in \{ 1,\dotsc, \len{\bm\lambda} \}$ with $\ensuremath{\mathsf{\sigma}}_{i-1}(\bm\lambda) < j \le \ensuremath{\mathsf{\sigma}}_i(\bm\lambda)$;
equivalently,
$\ind{\bm\lambda} j = \min\Bigl( i \in \{ 1,\dotsc, \len{\bm\lambda}\} : j \le \ensuremath{\mathsf{\sigma}}_i(\bm\lambda)\Bigr)$.
The \bfemph{dual partition} of $\bm\lambda$ is denoted by $\bm\lambda^*$.
Thus, if $\abs{\bm\lambda} >0$, then $\bm\lambda^* = (\mu_1,\dotsc,\mu_t)$, where $t = \lambda_1$ and
$\mu_i = \#\bigl\{ i \in \{ 1,\dotsc,\len{\bm\lambda} \} : \lambda_i \ge i\bigr\}$.
For a monic polynomial $f = X^m + a_{m-1}X^{m-1} + \dotsb + a_0$,
let
\[
\ensuremath{\mathsf{C}}(f) =
\begin{bmatrix}
0 & 1\\
& \ddots & \ddots \\
& & 0 & 1\\
-a_0 & \hdots & -{a_{m-2}} & -a_{m-1}
\end{bmatrix}
\]
be its companion matrix.
Let $A \in \Mat_n(k)$.
It is well-known that there are monic irreducible polynomials $f_1,\dotsc,f_e
\in k[X]$
and partitions $\bm\lambda_1, \dotsc,\bm\lambda_e$ of
positive integers $n_1$,$\dotsc$, $n_e$ such that $n = \deg(f_1) n_1 + \dotsb + \deg(f_e) n_e$
and $A$ is similar to its (primary) rational canonical form
\[
\diag\Bigl(
\ensuremath{\mathsf{C}}\Bigl(f_1^{\lambda_{1,1}}\Bigr),
\dotsc,
\ensuremath{\mathsf{C}}\Bigl(f_1^{\lambda_{1,\len{\bm\lambda_1}}}\Bigr),
\,\,\dotsc\dotsc, \,\,
\ensuremath{\mathsf{C}}\Bigl(f_e^{\lambda_{e,1}}\Bigr),
\dotsc\,
\ensuremath{\mathsf{C}}\Bigl(f_e^{\lambda_{e_,\len{\bm\lambda_e}}}\Bigr)
\Bigr)
\]
over $k$.
We call $( (f_1,\bm\lambda_1), \dotsc, (f_e,\bm\lambda_e))$ an
\bfemph{elementary divisor vector} of $A$ over $k$;
any two elementary divisor vectors of $A$ coincide up to reordering.
\paragraph{Main results.}
Recall that $k$ is a number field with ring of integers $\mathfrak{o}$.
Throughout, $\mathfrak{p}_v \in \Spec(\mathfrak{o})$ denotes the prime ideal corresponding
to a place $v \in \ensuremath{\mathcal V}_k$ and $q_v = \card{\mathfrak{o}/\mathfrak{p}_v}$ denotes the residue field size
of $k_v$.
Our global main result is the following.
\begin{thmabc}
\label{Thm:global}
Let $S\subset \ensuremath{\mathcal V}_k$ be finite and $A \in \Mat_n(\mathfrak{o}_S)$.
Let $((f_1,\bm\lambda_1), \dotsc, (f_e,\bm\lambda_e))$ be an elementary divisor
vector of $A$ over $k$.
Write $k_i = k[X]/(f_i)$.
Let $\mathfrak{o}_i$ denote the ring of integers of $k_i$.
Let $S_i = \{ w \in \ensuremath{\mathcal V}_{k_i} : \exists v \in S. \divides w v\}$
and write $\mathfrak{o}_{i,S_i} := (\mathfrak{o}_i)_{S_i}$.
Then the following hold:
\begin{enumerate}
\item
\label{Thm:global1}
There are finitely many places $w_1,\dotsc,w_\ell \in \ensuremath{\mathcal V}_k \setminus S$ and
associated rational functions $W_1,\dotsc,W_\ell \in \mathbf{Q}(X)$ such that
\begin{equation}
\label{eq:global}
\zeta_{A,\mathfrak{o}_S}(s) = \prod_{u=1}^\ell W_u(q_{w_u}^{-s}) \times \prod_{i=1}^e
\prod_{j=1}^{\abs{\bm\lambda_i}} \zeta_{\mathfrak{o}_{i,S_i}}\bigl( (\bm\lambda_i^*)^{-1}(j) \ensuremath{\,\cdotp} s - j + 1\bigr).
\end{equation}
In particular, $\zeta_{A,\mathfrak{o}_S}(s)$ admits meromorphic continuation to the
complex plane.
\item
\label{Thm:global2}
The abscissa of convergence $\alpha_{A,\mathfrak{o}_S}$ of $\zeta_{A,\mathfrak{o}_S}(s)$ satisfies
$\alpha_{A,\mathfrak{o}_S} = \max\limits_{1\le i \le e} \len{\bm\lambda_i} \in \mathbf{N}$.
\item
\label{Thm:global3}
Let $I := \bigl\{ i \in \{ 1,\dotsc,e\} : \len{\bm\lambda_i} =
\alpha_{A,\mathfrak{o}_S}\bigr\}$.
Then the multiplicity $\beta_{A,\mathfrak{o}_S}$ of the pole of $\zeta_{A,\mathfrak{o}_S}(s)$
at $\alpha_{A,\mathfrak{o}_S}$ satisfies $\beta_{A,\mathfrak{o}_S} = \sum\limits_{i\in I} \lambda_{i,-1}$.
\end{enumerate}
\end{thmabc}
As we will see, part~(\ref{Thm:global1}) is in fact a consequence of a similar
formula~\eqref{eq:local} which is valid for almost all local zeta functions $\zeta_{A,\mathfrak{o}_v}(s)$.
The exceptional factors $W_u(q_{w_u}^{-s})$ in~\eqref{eq:global}
cannot, in general, be omitted, see Example~\ref{ex:exceptional} below.
We note that the special case $A = 0_n$ in Theorem~\ref{Thm:global} is
consistent with the well-known formula $\zeta_{\mathfrak{o}_S}(s) \zeta_{\mathfrak{o}_S}(s-1)
\dotsb \zeta_{\mathfrak{o}_S}(s - (n-1))$ for the zeta function enumerating all
finite-index submodules of $\mathfrak{o}_S^n$.
We further note that the shape of the right-hand side of \eqref{eq:global} is
rather similar to that of Solomon's formula \cite[Thm~1]{Sol77} for the zeta
function enumerating submodules of finite index of a $\mathbf{Z} G$-lattice for a
finite group~$G$.
\medskip
Local functional equations under ``inversion of the residue field size'' are a
common, but not universal, phenomenon in the theory of subobject zeta functions;
see \cite{Vol10,Vol16}.
For an extension of number fields $k'/k$ and $v \in \ensuremath{\mathcal V}_k$, let
$\ensuremath{\mathrm g}_v(k')$\label{lab:noplaces} denote the number of places of $k'$ which divide $v$.
\begin{thmabc}
\label{Thm:FEqn}
Let $A \in \Mat_n(k)$ and let $((f_1,\bm\lambda_1),\dotsc,(f_e,\bm\lambda_e))$ be an
elementary divisor vector of $A$ over $k$.
Write $\bm\mu_i := \bm\lambda_i^*$.
Then, for almost all $v \in \ensuremath{\mathcal V}_k$,
\begin{equation}
\label{eq:feqn}
\zeta_{A,\mathfrak{o}_v}(s) \Bigg\vert_{q_v^{\phantom 1}\to q_v^{-1}}
= (-1)^{\sum\limits_{i=1}^e \abs{\bm\lambda_i} \ensuremath{\,\cdotp} \ensuremath{\mathrm g}_v(k[X]/(f_i))}
\ensuremath{\,\cdotp}
q_v^{\sum\limits_{i=1}^e \deg(f_i) \binom{\abs{\bm\lambda_i}}{2} -
\Bigl(\sum\limits_{i=1}^ e \deg(f_i) \sum\limits_{j=1}^{\lambda_{i1}}j
\mu_{ij}\Bigr)s}
\ensuremath{\,\cdotp} \zeta_{A,\mathfrak{o}_v}(s).
\end{equation}
\end{thmabc}
The operation of inverting $q_v$ can be interpreted using \eqref{eq:local} or,
in far greater generality, in terms of suitable explicit formulae as
in \cite{Vol10}.
We note that in the special case that $(A - a 1_n)^n = 0$ for some $a \in k$,
the functional equation \eqref{eq:feqn} follows
from \cite[Thm\ 1.2]{Vol16} (see \cite[Rem.\ 1.5]{Vol16}).
\medskip
It is natural to ask what properties of $A$ can be inferred from its associated
zeta functions.
We will make frequent use of the following elementary observation.
\begin{lemma*}
Let $A,B\in \Mat_n(k)$.
Suppose that
$k[A]$ and $k[B]$ are similar (i.e.\ $\GL_n(k)$-conjugate).
Then for almost all $v \in \ensuremath{\mathcal V}_k$,
$\zeta_{A,\mathfrak{o}_v}(s) = \zeta_{B,\mathfrak{o}_v}(s)$. \qed
\end{lemma*}
The following is another consequence of our explicit formulae.
\begin{thmabc}
\label{Thm:sim}
Let $A \in \Mat_n(k)$ and $B \in \Mat_m(k)$ be nilpotent.
The following are equivalent:
\begin{enumerate}
\item
\label{Thm:sim1}
$n = m$ and $A$ and $B$ are similar.
\item
\label{Thm:sim2}
For almost all $v \in \ensuremath{\mathcal V}_k$, $\zeta_{A,\mathfrak{o}_v}(s) =
\zeta_{B,\mathfrak{o}_v}(s)$.
\item
\label{Thm:sim3}
There exists a finite $S \subset \ensuremath{\mathcal V}_k$ such that $A$ and $B$ both have
entries in $\mathfrak{o}_S$ and such that $\zeta_{A,\mathfrak{o}_S}(s) = \zeta_{B,\mathfrak{o}_S}(s)$.
\end{enumerate}
\end{thmabc}
The nilpotency condition in Theorem~\ref{Thm:sim} cannot, in general, be
omitted, see Remark~\ref{rem:nilpotency_required}.
\medskip
The author previously conjectured \cite[\S 8.3]{topzeta} that generic local submodule zeta functions
associated with nilpotent matrix algebras have a simple pole at zero.
In the present case, our explicit formulae allow us to deduce the following.
\begin{thmabc}
\label{Thm:poles}
Let $A \in \Mat_n(k)$.
Then for almost all $v \in \ensuremath{\mathcal V}_k$, $\zeta_{A,\mathfrak{o}_v}(s)$ has a pole at zero.
Moreover, the following are equivalent:
\begin{enumerate}
\item For almost all $v \in \ensuremath{\mathcal V}_k$, $\zeta_{A,\mathfrak{o}_v}(s)$ has a \underline{simple} pole at
zero.
\item
There exists $a \in k$ with $(A - a 1_n)^n = 0$.
\end{enumerate}
\end{thmabc}
\paragraph{Behaviour at zero in general---a conjecture.}
We use this opportunity to state a generalisation of our conjecture on
the behaviour at zero of local submodule zeta functions (see \cite[Conj.\ IV and
\S 8.3]{topzeta});
this generalisation disposes of the mysterious nilpotency assumption found in its precursor.
For a ring $R$ with polynomial submodule growth, a finitely generated
$R$-module $M$, and $\Omega \subset \End_{R}(M)$,
the submodule zeta function $\zeta_{\Omega \ensuremath{\curvearrowright} M}(s)$ is the Dirichlet
series enumerating $\Omega$-invariant $R$-submodules of finite index of $M$
(cf.\ \cite[Def.\ 2.1(ii)]{topzeta}).
Let $V$ be a finite-dimensional vector space over $k$ and let $\mathcal A\subset
\End_k(V)$ be an associative, unital subalgebra.
Let $\rad(\mathcal A)$ denote the (nil)radical of $\mathcal A$.
By the Wedderburn-Malcev Theorem \cite[Thm~72.19]{CR62},
there exists a subalgebra $\mathcal S
\subset \mathcal A$ such that $\mathcal A = \rad(\mathcal A) \oplus
\mathcal S$ as vector spaces (whence $\mathcal S \approx_k \mathcal
A/\rad(\mathcal A)$ is semisimple);
moreover, $\mathcal S$ is unique up to conjugacy under $(1 + \rad(\mathcal A))
\le \mathcal A^\times$.
Choose $\mathfrak{o}$-forms $\mathsf V \subset V$, $\mathsf A \subset \End_{\mathfrak{o}}(\mathsf
V)$ and $\mathsf S \subset \End_{\mathfrak{o}}(\mathsf V)$ of $V$, $\mathcal A$, and
$\mathcal S$, respectively.
We write $\mathsf X_v := \mathsf X \otimes_{\mathfrak{o}} \mathfrak{o}_v$ in the following.
\begin{conjabc}
\label{conj:zero}
For almost all $v \in \ensuremath{\mathcal V}_k$,
\[
\frac{\zeta_{\mathsf A_v \ensuremath{\curvearrowright} \mathsf V_v}(s)}
{\zeta_{\mathsf S_v \ensuremath{\curvearrowright} \mathsf V_v}(s)}
\Biggm\vert_{s=0} = 1.
\]
\end{conjabc}
This conjecture reduces to the behaviour predicted in \cite[\S 8.3]{topzeta} in
the ``nilpotent case'' $\mathcal A = \rad(\mathcal A) \oplus k 1_V$.
In order to make Conjecture~\ref{conj:zero} more explicit, we recall Solomon's
formula for $\zeta_{\mathsf S_v \ensuremath{\curvearrowright} \mathsf V_v}(s)$.
Let $\mathcal S = \mathcal S_1 \oplus \dotsb \oplus \mathcal S_r$ be the Wedderburn
decomposition of the semisimple algebra $\mathcal S$ (so that each $\mathcal
S_i$ is simple).
Let $W_i$ be a simple $\mathcal S_i$-module and
decompose $V = V_1 \oplus \dotsb \oplus V_r$, where $V_i$ is isomorphic to
$W_i^{m_i}$ and $\mathcal S$ acts diagonally on $V$.
Let $k_i$ be the centre of $\mathcal S_i$ and let $\mathfrak{o}_i$ be the ring of
integers of $k_i$.
Finally, let $e_i$ be the Schur index of the central simple $k_i$-algebra
$\mathcal S_i$ and define $n_i$ by $\dim_{k_i}(\mathcal A_i)= n_i^2$.
\begin{thm}[{\cite[\S 4]{Sol77}}]
\label{thm:solomon}
For almost all $v \in \ensuremath{\mathcal V}_k$,
\begin{equation}
\label{eq:solomon}
\zeta_{\mathsf S_v \ensuremath{\curvearrowright} \mathsf V_v}(s)
= \prod_{i=1}^r \prod_{j=1}^{m_i e_i}
\prod_{\substack{w \in \ensuremath{\mathcal V}_{k_i}\\\divides w v}}
\zeta_{\mathfrak{o}_{i,w}}(n_is - j + 1).
\end{equation}
\end{thm}
The special case $\mathcal A = k[\alpha]$ ($\alpha \in \End_k(V)$) of
Conjecture~\ref{conj:zero} follows from Theorem~\ref{thm:solomon} and
Theorem~\ref{thm:local} below.
For a more abstract interpretation of Conjecture~\ref{conj:zero}, note that we
may identify $\mathcal S$ acting on $V$ with $\mathcal A/\rad(\mathcal A)$
acting (faithfully) on the semi-simplification of $V$ as an $\mathcal A$-module (i.e.\ the
direct sum of the composition factors of $V$ as an $\mathcal A$-module).
\paragraph{Overview.}
In order to derive Theorems~\ref{Thm:global}--\ref{Thm:poles}, we proceed as
follows.
In \S\ref{s:redprimary}, we reduce the computation of $\zeta_{A,\mathfrak{o}_S}(s)$ to
the case that the minimal polynomial of $A$ over $k$ is a power of an
irreducible polynomial.
In \S\ref{s:rednilpotent}, we then further reduce to the case that $A$ is
nilpotent.
The heart of this article, \S\ref{s:nilpotent}, is then devoted to the explicit
determination of $\zeta_{A,\mathfrak{o}_v}(s)$ for nilpotent $A$ and almost all $v \in \ensuremath{\mathcal V}_k$;
as a by-product, in Theorem~\ref{thm:ZZX}, we compute the ideal zeta
function of the $2$-dimensional ring $\mathbf{Z}\ensuremath{[\![ } X\ensuremath{]\!] }$.
We then combine our findings and derive
Theorems~\ref{Thm:global}--\ref{Thm:poles} in \S\ref{s:proofs}.
Finally, as an application, in \S\ref{s:app}, we use Theorem~\ref{Thm:global} to
compute the abscissae of convergence of some (largely unknown) submodule and
ideal zeta functions.
\subsection*{Acknowledgment}
I would like to thank Christopher Voll for interesting discussions.
\subsection*{\textit{Notation}}
Throughout, $\mathbf{N} = \{ 1,2,\dotsc\}$ and $\delta_{ij}$ denotes the Kronecker symbol.
The symbol ``$\subset$'' indicates not necessarily proper inclusion.
We use $\approx_R$ to denote both the similarity of matrices over $R$ and the
existence of an $R$-isomorphism.
Matrices act by right-multiplication on row vectors.
Matrix sizes are indicated by single subscripts for square matrices and double
subscripts in general; in particular, $1_n$ and $0_{m,n}$ denote the $n\times n$
identity and $m\times n$ zero matrix, respectively.
We say that a property depending on $S$ holds for sufficiently large finite
$S \subset \ensuremath{\mathcal V}_k$, if there exists a finite $S_0 \subset
\ensuremath{\mathcal V}_k$ such that the property holds for all finite $S \subset \ensuremath{\mathcal V}_k$
with $S \supset S_0$.
Given $v \in \ensuremath{\mathcal V}_k$,
we write $\abs{\ensuremath{\,\cdotp}}_v$ for the $v$-adic absolute value on $k_v$ with
$\abs{\pi}_v = q_v^{-1}$ for $\pi \in \mathfrak{p}_v\setminus \mathfrak{p}_v^2$.
By a $p$-adic field, we mean a finite extension $K$ of the $p$-adic numbers
$\mathbf{Q}_p$ for some prime $p$.
We let $\mathfrak{O}_K$ denote the valuation ring of $K$ and write $q_K$ for the
residue field size of $K$.
Furthermore, $\nu_K$ and $\abs{\ensuremath{\,\cdotp}}_K$ denote the additive valuation and
absolute value on $K$, respectively, normalised such that any uniformiser $\pi$
satisfies $\nu_K(\pi) = 1$ and $\abs{\pi}_K = q^{-1}_K$.
When the reference to $K$ is clear, we occasionally omit the
subscript ``$K$''.
\section{Reduction to the case of a primary minimal polynomial}
\label{s:redprimary}
By the following, up to enlarging $S$,
we may reduce the computation of $\zeta_{A,\mathfrak{o}_S}(s)$ to the case where the
minimal polynomial of $A$ over $k$ is primary (i.e.\ a power of an irreducible
polynomial).
\begin{prop}
\label{prop:primary}
Let $A \in \Mat_n(k)$.
Let $f = f_1 \dotsb f_e$ be a factorisation of the minimal
polynomial $f$ of $A$ over $k$ into a product of pairwise coprime monic polynomials
$f_i \in k[X]$.
Let $A_i \in \Mat_{n_i}(k)$ denote the matrix of $A$ acting on $\Ker(f_i(A))$
with respect to an arbitrary $k$-basis.
Then for almost all $v \in \ensuremath{\mathcal V}_k$, $$\zeta_{A,\mathfrak{o}_v}(s) = \prod_{i=1}^e \zeta_{A_i,\mathfrak{o}_v}(s).$$
\end{prop}
\begin{proof}
It is well-known that $k^n = \Ker(f_1(A)) \oplus \dotsb \oplus \Ker(f_e(A))$
is an $A$-invariant decomposition
into subspaces of dimensions $n_1,\dotsc,n_e$, say,
and $f_i$ is the minimal polynomial of $A_i$.
We may thus assume that $A = \diag(A_1,\dotsc,A_e)$.
By the Chinese remainder theorem, for each $i = 1,\dotsc,e$, there exists $g_i \in
k[X]$ with $g_i \equiv \delta_{ij} \bmod {f_j}$ for $j = 1,\dotsc,e$.
Hence, $g_i(A) = \diag( \delta_{i1} 1_{n_1},\dotsc,\delta_{ie}
1_{n_e}) \in k[A]$.
Choose a finite set $S\subset \ensuremath{\mathcal V}_k$ with $A_i \in \Mat_{n_i}(\mathfrak{o}_S)$ and $g_i
\in \mathfrak{o}_S[X]$ for $i = 1,\dotsc,e$.
Let $v \in \ensuremath{\mathcal V}_k \setminus S$.
Write $V := \mathfrak{o}_v^n$.
The block diagonal shape of $A$ yields an $A$-invariant decomposition
$V = V_1 \oplus \dotsb \oplus V_e$ into free $\mathfrak{o}_v$-modules of ranks $n_1,\dotsc,n_e$.
Note that $A$ acts as~$A_i$ on each $V_i$
and that each $g_i(A)$ acts as the natural map $V \twoheadrightarrow V_i \hookrightarrow V$.
Let $U \le V$ be an $\mathfrak{o}_v$-submodule.
If $U$ is $A$-invariant, then it decomposes as $U = U_1 \oplus \dotsb \oplus
U_e$ for $A_i$-invariant submodules $U_i \le V_i$.
We conclude that $(U_1,\dotsc,U_e) \mapsto U_1 \oplus \dotsb \oplus U_e$
defines a bijection from
\[
\Bigl\{ (U_1,\dotsc,U_e) : U_i \le_{\mathfrak{o}_v} V_i \text{ and } U_i A_i \le U_i \text{ for } i = 1,\dotsc, e
\Bigr\}
\]
onto the set of $A$-invariant submodules of $V$ whence
$\zeta_{A,\mathfrak{o}_v}(s) = \zeta_{A_1,\mathfrak{o}_v}(s) \dotsb \zeta_{A_e,\mathfrak{o}_v}(s)$.
\end{proof}
\section{Reduction to the case of a nilpotent matrix}
\label{s:rednilpotent}
Recall that $\ensuremath{\mathsf{C}}(f)$ denotes the companion matrix of a polynomial $f$.
Given a partition $\bm\lambda = (\lambda_1,\dotsc,\lambda_r)$, let
$$\ensuremath{\mathsf{N}}(\bm\lambda) := \diag(\ensuremath{\mathsf{C}}(X^{\lambda_1}),\dotsc,\ensuremath{\mathsf{C}}(X^{\lambda_r})).$$
Suppose that the minimal polynomial of $A \in \Mat_n(k)$ is a power of an
irreducible polynomial $f$;
we then say that $A$ is \bfemph{($f$-)primary}.
The elementary divisors of $A$ are $f^{\lambda_1},\dotsc,f^{\lambda_r}$
for a unique partition $\bm\lambda = (\lambda_1,\dotsc,\lambda_r)$ of $n / \deg(f)$.
We call $\bm\lambda$ the \bfemph{type} of~$A$.
For an extension $k'/k$ of number fields and $S\subset \ensuremath{\mathcal V}_k$, define
\[
\ensuremath{\mathcal D}_{k'/k}(S) = \{
w \in \ensuremath{\mathcal V}_{k'} : \exists v \in S. \divides w v
\}.
\]
Hence, using the notation from Theorem~\ref{Thm:FEqn}, $\#\ensuremath{\mathcal D}_{k'/k}(S) = \sum\limits_{v \in S} \ensuremath{\mathrm g}_v(k')$.
In this section, we prove the following.
\begin{thm}
\label{thm:rednil}
Let $f \in k[X]$ be monic and irreducible.
Let $A \in \Mat_n(k)$ be an $f$-primary matrix of type $\bm\lambda$.
Let $k' = k[X]/(f)$, and let $\mathfrak{o}'$ be the ring of integers of $k'$.
Then for almost all $v \in \ensuremath{\mathcal V}_k$,
\[
\zeta_{A,\mathfrak{o}_v}(s) = \prod_{\substack{w \in \ensuremath{\mathcal V}_{k'} \\\divides w v}} \zeta_{\ensuremath{\mathsf{N}}(\bm\lambda),\mathfrak{o}'_w}(s).
\]
Hence, for all sufficiently large finite $S \subset \ensuremath{\mathcal V}_k$,
setting $S' = \ensuremath{\mathcal D}_{k'/k}(S)$.
\[
\zeta_{A,\mathfrak{o}_S}(s) = \zeta_{\ensuremath{\mathsf{N}}(\bm\lambda),\mathfrak{o}'_{S'}}(s).
\]
\end{thm}
\begin{rem}
In \cite[\S 3]{Sha82}, the study of the variety of subspaces invariant under an
endomorphism of a finite-dimensional real or complex vector space is
reduced to the case of a nilpotent endomorphism.
Shayman proceeds by first reducing to the case of a primary
endomorphism (\cite[Thm~2]{Sha82}) and our Proposition~\ref{prop:primary}
proceeded along the same lines.
In his setting, the minimal polynomial of a primary endomorphism is a power of
a linear or quadratic irreducible and he considers these cases separately.
His reasoning is similar to arguments employed in our proof of
Theorem~\ref{thm:rednil} below.
We may regard the factorisation of $\zeta_{A,\mathfrak{o}_v}(s)$ obtained by
combining Proposition~\ref{prop:primary} and Theorem~\ref{thm:rednil}
as an arithmetic analogue of the factorisation of the space of $A$-invariant
subspaces in \cite[Thm~3]{Sha82}.
In \cite[\S 4]{Sha82}, Shayman then proceeds to study invariant subspaces of
nilpotent matrices in Jordan normal form.
For our purposes, a slightly different normal form, introduced in
\S\ref{ss:nf}, will prove advantageous.
\end{rem}
Our proof of Theorem~\ref{thm:rednil} requires some preparation.
\subsection{A generalised Jordan normal form for primary matrices}
Let $\otimes$ denote the usual Kronecker product $[a_{ij}]\otimes B = [a_{ij}B]$
of matrices.
The following result is a special case of the ``separable Jordan
normal form'' in \cite[\S 6.2]{Nor12};
it can also be obtained by restriction of scalars from the usual Jordan normal
form of an $f$-primary matrix over a minimal splitting field of $f$ over $k$.
\begin{prop}
\label{prop:jnf}
Let $f \in k[X]$ be monic and irreducible of degree $d$.
Let $A \in \Mat_n(k)$ be $f$-primary of type $\bm\lambda$.
Write $m := n/d$.
Then $A\approx_k 1_m \otimes \ensuremath{\mathsf{C}}(f) + \ensuremath{\mathsf{N}}(\bm\lambda) \otimes 1_d$.
\end{prop}
\begin{lemma}
\label{lem:diagin}
Let $f \in k[X]$ be monic and irreducible of degree $d$, $\bm\lambda \vdash m > 0$,
and $A = 1_m \otimes \ensuremath{\mathsf{C}}(f) + \ensuremath{\mathsf{N}}(\bm\lambda) \otimes 1_d$.
Then $1_m \otimes \ensuremath{\mathsf{C}}(f) = \diag(\ensuremath{\mathsf{C}}(f),\dotsc,\ensuremath{\mathsf{C}}(f)) \in k[A]$.
\end{lemma}
\begin{proof}
Write $\gamma := \ensuremath{\mathsf{C}}(f)$ and $e := \lambda_1$;
note that $X^e$ is the minimal polynomial of $\ensuremath{\mathsf{N}}(\bm\lambda)$ over every field.
We may naturally regard $A$ as an $m\times m$ matrix over the field $k':=
k[\gamma]$. Moreover, we may identify $k' = k[1_m \otimes \ensuremath{\mathsf{C}}(f)]$ as $k$-algebras.
Thus, $k[A, 1_m \otimes \ensuremath{\mathsf{C}}(f)] = k'[\gamma 1_m + \ensuremath{\mathsf{N}}(\bm\lambda)] =
k'[\ensuremath{\mathsf{N}}(\bm\lambda)]$
whence the $k$-dimension of $k[A, 1_m \otimes \ensuremath{\mathsf{C}}(f)]$ is $\idx{k':k} e = de$.
As $f^e$ is the minimal polynomial of $A$ over $k$,
the number $de$ is also the $k$-dimension of $k[A]$ whence the claim follows.
\end{proof}
Regarding the transition from the number field $k$ to the local ring $\mathfrak{o}_v$,
we note that the enveloping algebras of companion matrices take the expected
forms over \textup{\textsf{UFD}}{}s.
\begin{lemma}
\label{lem:evalCf}
Let $R$ be a \textup{\textsf{UFD}}{} and let $f \in R[X]$ be monic.
Then evaluation at $\ensuremath{\mathsf{C}}(f)$ induces an isomorphism $R[X]/(f) \approx_R R[\ensuremath{\mathsf{C}}(f)]$.
\end{lemma}
\begin{proof}
Let $K$ denote the field of fractions of $R$.
The kernel of the natural map $R[X] \to R[\ensuremath{\mathsf{C}}(f)]$ is $I := R[X] \cap f
K[X]$ and, clearly, $f R[X] \subset I$.
Let $h \in I$ so that $h = fg$ for some $g \in K[X]$.
By \cite[Thm~7.7.2]{Coh03}, there exists $a \in K^\times$ with
$af,a^{-1}g \in R[X]$.
As $f$ is monic (hence primitive), $a \in A$ whence
$g = a(a^{-1}g) \in R[X]$
and $h \in fR[X]$.
\end{proof}
\subsection{Properties of $S$-integers and their completions}
\label{ss:S}
\begin{lemma}
\label{lem:o'}
Let $k'/k$ be an extension of number fields.
Let $\mathfrak{o}'$ be the ring of integers of~$k'$.
Let $S \subset \ensuremath{\mathcal V}_k$ be finite and $S' = \ensuremath{\mathcal D}_{k'/k}(S)$.
Then $\mathfrak{o}' \otimes_{\mathfrak{o}} \mathfrak{o}_S \approx_{\mathfrak{o}} \mathfrak{o}'_{S'}$.
\end{lemma}
\begin{proof}
The following argument is taken from \cite{Con}:
if $h$ is the class number of $k$ and $a \in \mathfrak{o}$ generates
the principal ideal $\prod_{v\in S} \mathfrak{p}_v^h$,
then $\mathfrak{o}_S = \mathfrak{o}[1/a]$.
We conclude that $\mathfrak{o}' \otimes_{\mathfrak{o}} \mathfrak{o}_S = \mathfrak{o}'[1/a] = \mathfrak{o}'_{S'}$.
\end{proof}
\begin{lemma}
\label{lem:eqorder}
Let $f \in k[X]$ be monic and irreducible.
Let $k' = k[X]/(f)$ with ring of integers $\mathfrak{o}'$.
Then the following holds for all sufficiently large finite $S \subset \ensuremath{\mathcal V}_k$:
\begin{enumerate}
\item
\label{lem:eqorder1}
$\mathfrak{o}_S[X]/(f) \approx_{\mathfrak{o}_S} \mathfrak{o}'_{S'}$, where $S'= \ensuremath{\mathcal D}_{k'/k}(S)$.
\item
\label{lem:eqorder2}
$\mathfrak{o}_v[X]/(f)
\approx_{\mathfrak{o}_v}
\prod\limits_{\substack{w\in\ensuremath{\mathcal V}_{k'}\\\divides w v}} \mathfrak{o}'_w$
for $v \in \ensuremath{\mathcal V}_k\setminus S$.
\end{enumerate}
\end{lemma}
\begin{proof}
We freely use the exactness of localisation and completion;
see \cite[Prop.\ 2.5, Thm~7.2]{Eis95}.
Let $S_0 \subset \ensuremath{\mathcal V}_k$ be finite with $f \in \mathfrak{o}_{S_0}[X]$.
If $S \supset S_0$, then $\mathfrak{o}_{S_0}[X]/(f)
\otimes_{\mathfrak{o}_{S_0}} \mathfrak{o}_S \approx_{\mathfrak{o}_S} \mathfrak{o}_S[X]/(f)$.
As $\mathfrak{o}_{S_0}[X]/(f)$ and $\mathfrak{o}'$ both become isomorphic to $k'$ after base change to~$k$, for
sufficiently large finite $S \supset S_0$,
$\mathfrak{o}_S[X]/(f) \approx_{\mathfrak{o}_S} \mathfrak{o}'_{S'}$ by Lemma~\ref{lem:o'}.
This proves the first part.
For the second part, first note that, using (\ref{lem:eqorder1}) and Lemma~\ref{lem:o'},
\begin{equation}
\label{eq:fov}
\mathfrak{o}_v[X]/(f)
\approx_{\mathfrak{o}_v}
\mathfrak{o}_S[X]/(f) \otimes_{\mathfrak{o}_S} \mathfrak{o}_v
\approx_{\mathfrak{o}_v}
\mathfrak{o}'_{S'} \otimes_{\mathfrak{o}_S} \mathfrak{o}_v
\approx_{\mathfrak{o}_v}
\mathfrak{o}' \otimes_{\mathfrak{o}} \mathfrak{o}_v.
\end{equation}
Write $\mathfrak{o}_{(v)} := \mathfrak{o}_v \cap k$ for the $v$-adic valuation ring of $k$.
It is easy to see that we may naturally identify $\mathfrak{o}' \otimes_{\mathfrak{o}} \mathfrak{o}_{(v)}$ with the
integral closure of $\mathfrak{o}_{(v)}$ in $k'$.
The key observation here is that if $a \in k'$ is a root of a monic
polynomial $f(X) \in \mathfrak{o}_{(v)}[X]$, then there exists $m \in \mathfrak{o}$ with $v(m) =
0$ and $ma \in \mathfrak{o}'$.
Indeed, as in the proof of Lemma~\ref{lem:o'}, we find $m \in \mathfrak{o}$ such that
for all $w \in \ensuremath{\mathcal V}_k$,
$w(m) > 0$ if and only if some coefficient $c$ of $f(X)$ satisfies $w(c) < 0$.
By replacing $m$ by a suitable power, we can ensure that all coefficients of $m f(X)$
belong to $\mathfrak{o}$ whence $ma$ is integral over $\mathfrak{o}$ and thus belongs to $\mathfrak{o}'$.
We conclude (see \cite[Ch.~II, \S 8, Exerc.\ 4]{Neu99})
that the canonical isomorphism $k' \otimes_k k_v \approx_{k_v}
\prod\limits_{\divides w v} k'_w$ (\cite[Ch.~II, Prop.\ 8.3]{Neu99}) induces an isomorphism
$\mathfrak{o}' \otimes_{\mathfrak{o}} \mathfrak{o}_v
\approx_{\mathfrak{o}_v}
\prod\limits_{\divides w v} \mathfrak{o}'_w$.
Part (\ref{lem:eqorder2}) thus follows from the latter isomorphism and \eqref{eq:fov}.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:rednil}}
Recall that $a_m(A,R)$ denotes the number of $A$-invariant $R$-submodules of
$R^n$ of index~$m$, where $A \in \Mat_n(R)$.
\begin{prop}
\label{prop:ringprod}
Let $R_1,\dotsc,R_r$ be rings with polynomial submodule growth.
\begin{enumerate}
\item
\label{prop:ringprod1}
$R := R_1 \times \dotsb \times R_r$ has polynomial submodule growth.
\item
\label{prop:ringprod2}
(Cf.\ \cite[Lem.\ 1]{Sol77}.)
Let $A \in \Mat_n(R)$ and
let $A_i$ denote the image of $A$ under the map $\Mat_n(R) \to \Mat_n(R_i)$
induced by the projection $R \to R_i$.
Then $a_m(A,R) = a_m(A_1,R_1) \dotsb a_m(A_r,R_r)$ for each $m \in \mathbf{N}$.
Thus, $\zeta_{A,R}(s) = \zeta_{A_1,R_1}(s) \dotsb \zeta_{A_r,R_r}(s)$.
\end{enumerate}
\end{prop}
\begin{proof}
Decompose $R^n = R_1^n \times \dotsb \times R_r^n$ with $R$ acting diagonally
on $R^n$.
Multiplication by $e_i = (\delta_{1i},\dotsc,\delta_{ni}) \in R$ acts as the natural
map $R^n \to R_i^n \to R^n$.
Given an $R_i$-submodule $U_i \le R_i^n$ for $i = 1,\dotsc,r$, we
obtain an $R$-submodule $U = U_1 \times \dotsb \times U_r$ of $R^n$ and
it is easy to see that every $R$-submodule of $R^n$ is of this form in a
unique way.
Evidently, $U$ has finite index in $R^n$ if and only if each $U_i$ has finite
index in $R_i^n$.
Part (\ref{prop:ringprod1}) is immediate and (\ref{prop:ringprod2}) follows since $A$ acts as $A_i$ on $R_i^n$.
\end{proof}
\begin{proof}[{Proof of Theorem~\ref{thm:rednil}}]
Assuming that the finite set $S \subset \ensuremath{\mathcal V}_k$ is sufficiently large,
we can make the following assumptions for all $v \in \ensuremath{\mathcal V}_k \setminus S$:
\begin{itemize}
\item[\texttt{(NOR)}] $A = 1_m \otimes \ensuremath{\mathsf{C}}(f) + \ensuremath{\mathsf{N}}(\bm\lambda) \otimes 1_d \in \Mat_n(\mathfrak{o}_v)$ for
$d = \deg(f)$ and
$\bm\lambda \vdash m$ (Proposition~\ref{prop:jnf}).
\item[\texttt{(DIA)}] $1_m \otimes \ensuremath{\mathsf{C}}(f) \in \mathfrak{o}_v[A]$ (Lemma~\ref{lem:diagin}).
\item[\texttt{(INT)}]
$\mathfrak{o}_v[X]/(f) \approx_{\mathfrak{o}_v} \prod\limits_{\substack{w \in \ensuremath{\mathcal V}_{k'}\\ \divides w v}} \mathfrak{o}'_w$
(Lemma~\ref{lem:eqorder}).
\end{itemize}
Let $v \in \ensuremath{\mathcal V}_k\setminus S$.
First note that as an $\mathfrak{o}_v$-module, $\mathfrak{o}_v[\ensuremath{\mathsf{C}}(f)]$ is freely generated
by $(1_d,\ensuremath{\mathsf{C}}(f),\dotsc,\ensuremath{\mathsf{C}}(f)^{d-1})$.
It follows easily that $\mathfrak{o}_v^n$ is free of rank $m$ as an $\mathfrak{o}_v[\ensuremath{\mathsf{C}}(f)]$-module.
Using Lemma~\ref{lem:evalCf},\texttt{(INT)} allows us to identify
$\mathfrak{o}_v[\ensuremath{\mathsf{C}}(f)] = \mathfrak{o}_v[X]/(f) = \prod_{\divides w v} \mathfrak{o}'_w =: R_v$.
Thanks to \texttt{(NOR)}, we may then regard $A$ as an $m \times m$ matrix over $R_v$.
It follows from \texttt{(DIA)} that $A$-invariant $\mathfrak{o}_v$-submodules of
$\mathfrak{o}_v^n$ coincide with $A$-invariant $R_v$-submodules of $R_v^m$.
Using \texttt{(DIA)} once more, the latter $R_v$-submodules are precisely those
invariant under $A - \ensuremath{\mathsf{C}}(f) \ensuremath{\,\cdotp} 1_m = \ensuremath{\mathsf{N}}(\bm\lambda)$.
Therefore, $\zeta_{A,\mathfrak{o}_v}(s) = \zeta_{\ensuremath{\mathsf{N}}(\bm\lambda),R_v}(s)$.
Noticing that the $(0,1)$-matrix $\ensuremath{\mathsf{N}}(\bm\lambda)$ is preserved
by each projection $R_v \to \mathfrak{o}'_w$, Proposition~\ref{prop:ringprod} shows that
$\zeta_{\ensuremath{\mathsf{N}}(\bm\lambda),R_v}(s) = \prod\limits_{\divides w v} \zeta_{\ensuremath{\mathsf{N}}(\bm\lambda),\mathfrak{o}'_w}(s)$
which concludes the proof.
\end{proof}
\section{The case of a nilpotent matrix}
\label{s:nilpotent}
Let $\bm\lambda \vdash n$.
Recall the definitions
of $\ind{\bm\lambda} j$ from the introduction
and of $\ensuremath{\mathsf{N}}(\bm\lambda)$ from \S\ref{s:rednilpotent}.
\begin{defn*}
$W_{\bm\lambda}(X,Y) = 1 / \prod\limits_{j=1}^n \bigl( 1 - X^{j-1}
Y^{\ind{\bm\lambda}j}\bigr)
\in \mathbf{Q}(X,Y)$.
\end{defn*}
Equivalently, $W_{\bm\lambda}(X,Y) = 1
/\prod\limits_{i=1}^{\len{\bm\lambda}}\prod\limits_{j=1}^{\lambda_i}
\bigl(1-X^{\ensuremath{\mathsf{\sigma}}_{i-1}(\bm\lambda)+j-1} Y^i\bigr)$.
This section is devoted to proving the following.
\begin{thm}
\label{thm:nilpotent}
Let $\bm\lambda \vdash n$ and
let $K$ be a $p$-adic field.
Then $$\zeta_{\ensuremath{\mathsf{N}}(\bm\lambda^*),\mathfrak{O}_K}(s) = W_{\bm\lambda}(q_K,q_K^{-s}).$$
\end{thm}
Prior to giving a proof of Theorem~\ref{thm:nilpotent},
we record a few consequences.
\begin{cor}
\label{cor:nilpotent_global}
Let $A \in \Mat_n(k)$ be nilpotent of type $\bm\lambda$ (see \S\ref{s:rednilpotent}).
Then for all sufficiently large finite sets $S \subset \ensuremath{\mathcal V}_k$,
\[
\zeta_{A,\mathfrak{o}_S}(s) = \prod\limits_{j=1}^n \zeta_{\mathfrak{o}_S}\Bigl(\ind{(\bm\lambda^*)}
j \ensuremath{\,\cdotp} s - j + 1\Bigr).
\]
If $A \in \Mat_n(\mathfrak{o})$ and $A \approx_{\mathfrak{o}} \ensuremath{\mathsf{N}}(\bm\lambda)$,
then we may take $S = \varnothing$.
\qed
\end{cor}
As an application, we can determine the ideal zeta function of
$\mathbf{Z}[X]/(X^n)$. Recall that $\zeta(s)$ denotes the Riemann zeta function.
\begin{cor}
\label{cor:ideals_Xn}
For every prime $p$,
$$\zeta_{\mathbf{Z}_p[X]/(X^n)}(s) = 1 / \prod\limits_{j=1}^n(1-p^{j-1 - js}).$$
In particular,
$$\zeta_{\mathbf{Z}[X]/(X^n)}(s) = \prod\limits_{j=1}^n
\zeta(js - j + 1).$$
\end{cor}
\begin{proof}
The matrix of multiplication by $X$ acting on $\mathbf{Z}[X]/(X^n)$ with respect to the
basis $(1,X,\dotsc,X^{n-1})$, i.e.\ the companion matrix of $X^n$, is
precisely $\ensuremath{\mathsf{N}}((n))$.
\end{proof}
\begin{rem*}
The subalgebra zeta functions of $\mathbf{Z}_p[X]/(X^n)$ are known only for $n \le 4$
and sufficiently large primes $p$. Moreover, the author's computation of these
zeta functions for $n = 4$ relied on fairly involved machine calculations;
see \cite[\S 9.2]{padzeta}.
(The formula for $\zeta_{\mathbf{Z}_p[X]/(X^4)}(s)$ in \cite{padzeta} takes up
about a page in total.)
\end{rem*}
Subobject zeta functions over rings other than $\mathfrak{o}_S$ or $\mathfrak{o}_v$ have received
little attention so far.
We obtain the following.
\begin{thm}
\label{thm:ZZX}
\quad
\begin{enumerate}
\item
\label{thm:ZZX1}
$\mathbf{Z}\ensuremath{[\![ } X\ensuremath{]\!] }$ has polynomial submodule growth.
\item
\label{thm:ZZX2}
$\zeta_{\mathbf{Z}\ensuremath{[\![ } X \ensuremath{]\!] }}(s) = \prod\limits_{j=1}^\infty \zeta(js - j + 1)$
for $\Real(s) > 1$.
\end{enumerate}
\end{thm}
\begin{proof}
It is well-known that the maximal ideals of $\mathbf{Z}\ensuremath{[\![ } X \ensuremath{]\!] }$ are precisely of
the form $(X,p)$ for a rational prime $p$.
It follows that $X$ acts nilpotently on every $\mathbf{Z}\ensuremath{[\![ } X\ensuremath{]\!] }$-module of finite
length.
Hence, if $U \le_{\mathbf{Z}\ensuremath{[\![ } X\ensuremath{]\!] }} \mathbf{Z}\ensuremath{[\![ } X\ensuremath{]\!] }^d$ has finite index,
then $U$ contains $X^n \mathbf{Z}\ensuremath{[\![ } X\ensuremath{]\!] }^d$ for some $n \ge 1$.
As $\mathbf{Z}\ensuremath{[\![ } X\ensuremath{]\!] }$ is Noetherian, $U$ thus corresponds to a $\mathbf{Z}[X]$-submodule
of $\mathbf{Z}[X]^d/X^n\mathbf{Z}[X]^d$.
In particular, (\ref{thm:ZZX1}) follows since $\mathbf{Z}[X]$ has polynomial
submodule growth by Theorem~\ref{thm:segal}.
Moreover, Corollary~\ref{cor:ideals_Xn} implies the identity in
(\ref{thm:ZZX2}) on the level of formal Dirichlet series.
In order to establish (absolute) convergence, let $s > 1$ be real.
By well-known facts on infinite products,
$\prod_{j=1}^\infty \zeta(js-j+1)$ converges (absolutely) if and only if the
same is true of $F(s) := \sum_{j=1}^\infty (\zeta(js-j+1)-1)$.
Using the non-negativity of the coefficients of each Dirichlet series
$\zeta(js-j+1)$, we obtain
\begin{align*}
F(s) & = \sum_{j=1}^\infty \sum_{n=2}^\infty n^{j-1}(n^j)^{-s}
= \sum_{n=2}^\infty g_n n^{-s},
\end{align*}
where
$$g_n := n \ensuremath{\,\cdotp} \sum_{\substack{m \ge 2,j \ge 1\\n = m^j}} \frac 1 m.$$
We see that for $N \ge 2$,
\begin{align*}
\sum_{n=2}^N g_n & \le N \sum_{\substack{m \ge 2, j \ge 1\\m^j \le
N}} \frac 1 m \le N \sum_{m=2}^N \frac{2 \log N} m = \mathcal O(N (\log
N)^2)
= \mathcal O(N^{1+\varepsilon})
\end{align*}
for every $\varepsilon > 0$.
In particular, $F(s)$ and $\zeta_{\mathbf{Z}\ensuremath{[\![ } X\ensuremath{]\!] }}(s)$ both converge for $\Real(s) > 1$.
\end{proof}
\begin{rem}
Note, in particular, that $\zeta_{\mathbf{Z}\ensuremath{[\![ } X\ensuremath{]\!] }}(s)$ has an
essential singularity at $s = 1$ and therefore does not admit meromorphic
continuation beyond its abscissa of convergence.
This illustrates that Theorem~\ref{thm:dSG}(\ref{thm:dSG2}) does not carry over
to general ground rings with polynomial submodule growth.
\end{rem}
In order to prove Theorem~\ref{thm:nilpotent}, we employ the $p$-adic
integration machinery from \cite{GSS88}.
For a ring $R$, let $\Tr_n(R)$ denote the $R$-algebra of upper triangular
$n\times n$-matrices over $R$.
Recall that an element of a ring is \bfemph{regular} if it is not a zero divisor.
Write $\Tr_n^\ensuremath{\mathrm{reg}}(R) = \{ \bm x \in \Tr_n(R) : \det(\bm x) \in R \text{ is regular}\}$.
For a $p$-adic field $K$, let $\mu_K$ denote the Haar measure on $K^n$ with
$\mu_K(\mathfrak{O}_K^n) = 1$.
\begin{prop}[{\cite[\S 3]{GSS88}}]
\label{prop:coneint}
Let $K$ be a $p$-adic field
and $A \in \Mat_n(\mathfrak{O}_K)$.
Define
$V_K(A) := \bigl\{
\bm x \in \Tr^\ensuremath{\mathrm{reg}}_n(\mathfrak{O}_K) : \mathfrak{O}_K^n \bm x A \subset \mathfrak{O}_K^n \bm x
\bigr\}$
to be the set of upper-triangular $n\times n$ matrices over~$\mathfrak{O}_K$ whose rows
span an $A$-invariant $\mathfrak{O}_K$-submodule of finite index of $\mathfrak{O}_K^n$.
Then
\begin{equation}
\label{eq:GSS}
\zeta_{A,\mathfrak{O}_K}(s) = (1-q_K^{-1})^{-n} \int_{V_K(A)} \abs{x_{11}}_K^{s-1}
\abs{x_{22}}_K^{s-2} \dotsb \abs{x_{nn}}_K^{s-n} \dd\mu_K(\bm x).
\end{equation}
\end{prop}
\paragraph{Strategy.}
In order to prove Theorem~\ref{thm:nilpotent}, we proceed as follows.
First, in \S\ref{ss:nf}, we define a matrix $\ensuremath{\mathsf{A}}(\bm\lambda)$ which is similar
(over $\mathbf{Z}$) to $\ensuremath{\mathsf{N}}(\bm\lambda^*)$ so that $\zeta_{\ensuremath{\mathsf{N}}(\bm\lambda^*),\mathfrak{O}_K}(s) =
\zeta_{\ensuremath{\mathsf{A}}(\bm\lambda),\mathfrak{O}_K}(s)$.
As we will see in \S\ref{ss:recursion},
the advantage of $\ensuremath{\mathsf{A}}(\bm\lambda)$ over $\ensuremath{\mathsf{N}}(\bm\lambda^*)$ is that the sets
$V_K(\ensuremath{\mathsf{A}}(\bm\lambda))$ in Proposition~\ref{prop:coneint} exhibit a natural,
recursive structure.
Specifically, we will define $\ensuremath{\mathsf{d}}{\bm\lambda} := (\lambda_2,\dotsc,\lambda_{\len{\bm\lambda}})$
and find that $V_K(\ensuremath{\mathsf{A}}(\bm\lambda))$ can be described in terms of
$V_K(\ensuremath{\mathsf{A}}(\ensuremath{\mathsf{d}}{\bm\lambda}))$ and membership conditions for generic vectors
in generic sublattices.
In \S\ref{ss:membership},
the geometry of such membership conditions is elucidated by means of suitable
(birational) changes of coordinates.
Finally, in \S\ref{ss:proof_nilpotent}, we combine all these ingredients and
prove Theorem~\ref{thm:nilpotent}.
\subsection{A dual normal form for nilpotent matrices}
\label{ss:nf}
\begin{defn*}
Let $\bm\lambda = (\lambda_1,\dotsc,\lambda_r) \vdash n \ge 0$.
Define $\ensuremath{\mathsf{d}}{\bm\lambda} := (\lambda_2,\dotsc,\lambda_r)$.
We recursively define $\ensuremath{\mathsf{A}}(\bm\lambda) \in \Mat_n(\mathbf{Z})$ as follows:
\begin{enumerate}
\item If $r \le 1$, define $\ensuremath{\mathsf{A}}(\bm\lambda) = 0_n$.
\item
If $r > 1$,
define
\sbox0{$\begin{matrix}1&2\\4&5\end{matrix}$}
\sbox1{$\begin{matrix}1\\2\end{matrix}$}
\sbox2{$\begin{matrix} 1_{\lambda_2} \\ 0_{\lambda_1-\lambda_2,\lambda_2} \end{matrix}$}
\sbox3{$\begin{matrix} 1 & 2 & 3\\
1 & 2 & 3 \\
1 & 2 & 3 \end{matrix}$}
\begin{equation}
\label{eq:A}
\ensuremath{\mathsf{A}}(\bm\lambda) =
\left[
\begin{array}{c|cc}
\makebox[\wd0]{\large $0_{\lambda_1}$}
& \usebox{2} & 0_{\lambda_1,\lambda_3+\dotsb+\lambda_r} \\\hline
&
\vphantom{\usebox{3}}
\makebox{\large $\ensuremath{\mathsf{A}}(\ensuremath{\mathsf{d}}{\bm\lambda})$}
\end{array}
\right].
\end{equation}
\end{enumerate}
\end{defn*}
In other words,
\begin{equation}
\label{eq:A_alt}
\ensuremath{\mathsf{A}}(\bm\lambda) =
\left[
\begin{array}{ccccc}
\sbox0{$\begin{matrix}1&2\\4&5\\7&8\end{matrix}$}
\vphantom{\usebox{0}} \makebox{$0_{\lambda_1}$} &
\begin{array}{|c|} \hline \makebox{$1_{\lambda_2}$}\\ \hline \makebox{$0_{\lambda_1-\lambda_2,\lambda_2}$}\\\hline\end{array}
\\
& \vphantom{\usebox{0}} \makebox{$0_{\lambda_2}$} &
\begin{array}{|c|} \hline \makebox{$1_{\lambda_3}$}\\ \hline \makebox{$0_{\lambda_2-\lambda_3,\lambda_3}$}\\\hline\end{array}
\\
& & \ddots & \ddots \\
& &&\ddots &
\begin{array}{|c|} \hline \makebox{$1_{\lambda_r}$}\\ \hline \makebox{$0_{\lambda_{r-1}-\lambda_r,\lambda_r}$}\\\hline\end{array}
\\
& & & & \vphantom{\usebox{0}}\makebox{$0_{\lambda_r}$}
\end{array}
\right]
\end{equation}
By the following, the $\ensuremath{\mathsf{A}}(\bm\lambda)$ parameterise similarity
classes of nilpotent matrices.
\begin{prop}
\label{prop:permconj}
$\ensuremath{\mathsf{A}}(\bm\lambda^*)$ and $\ensuremath{\mathsf{N}}(\bm\lambda)$ are conjugate by permutation matrices.
\end{prop}
\begin{proof}
Let $T(\bm\lambda)$ be the Young diagram of $\bm\lambda$ and
let $V(\bm\lambda)$ be the $\mathbf{Z}$-module freely generated by the cells
of $T$;
we use ``English notation'' for $T(\bm\lambda)$ and draw each row
underneath its predecessor (if any).
Define $\Theta(\bm\lambda)$ to be the endomorphism of $V(\bm\lambda)$ (acting on the
right) which sends each cell to its right neighbour if it exists and to zero otherwise.
We consider two orderings on the cells of $T(\bm\lambda)$ and describe the
associated matrices representing $\Theta(\bm\lambda)$.
The \itemph{horizontal order} is defined by traversing the cells of $T(\bm\lambda)$
from left to right within each row, proceeding from top to bottom.
Clearly, $\ensuremath{\mathsf{N}}(\bm\lambda)$ is the matrix of $\Theta(\bm\lambda)$ with respect to
this order.
The \itemph{vertical order} is obtained by traversing the cells of $T(\bm\lambda)$
from top to bottom within each column, proceeding from left to right.
Write $\bm\mu := \bm\lambda^*$, say $\bm\mu = (\mu_1,\dotsc,\mu_\ell)$.
We now show by induction on $\ell$ that the matrix of $\Theta(\bm\lambda)$ with respect
to the vertical order is $\ensuremath{\mathsf{A}}(\bm\mu)$---it then follows, in particular,
that $\ensuremath{\mathsf{A}}(\bm\mu)$ and $\ensuremath{\mathsf{N}}(\bm\lambda)$ are conjugate as claimed.
If $\ell \le 1$, then $\Theta(\bm\lambda) = 0$ and $\ensuremath{\mathsf{A}}(\bm\mu) = 0$ so let $\ell > 1$.
Let $t_1,\dotsc,t_n$ be the cells of $T(\bm\lambda)$ according to the vertical order.
Then $t_i \Theta(\bm\lambda) = t_{\mu_1+i}$ for $1\le i \le \mu_2$
and $t_i \Theta(\bm\lambda) = 0$ for $\mu_2 < i \le \mu_1$.
Let $\tilde{\bm\lambda} := (\ensuremath{\mathsf{d}}{\bm\mu})^*$
and $\tilde V :=\mathbf{Z} t_{\mu_1+1} \oplus \dotsb \oplus \mathbf{Z} t_n$.
We may naturally identify the endomorphism of $\tilde V$ induced by
$\Theta(\bm\lambda)$ with $\Theta(\tilde{\bm\lambda})$ acting on $V(\tilde{\bm\lambda})$;
the defining basis of $\tilde V$ is then ordered vertically.
By induction, the matrix of $\Theta(\bm\lambda)$ acting on $\tilde V$
with respect to the basis $(t_{\mu_1+1},\dotsc,t_n)$ is therefore
$\ensuremath{\mathsf{A}}(\ensuremath{\mathsf{d}}{\bm\mu})$ whence the claim follows from the recursive description of
$\ensuremath{\mathsf{A}}(\bm\mu)$ in \eqref{eq:A}.
\end{proof}
For $\abs{\bm\lambda} > 0$, let $\ensuremath{\mathsf{B}}(\bm\lambda) \in
\Mat_{\abs{\bm\lambda},\abs{\ensuremath{\mathsf{d}}{\bm\lambda}}}(\mathbf{Z})$ denote the matrix obtained
by deleting the first $\lambda_1$ columns of $\ensuremath{\mathsf{A}}(\bm\lambda)$.
The following consequence of \eqref{eq:A_alt} will be useful below.
\begin{lemma}
\label{lem:B}
$\ensuremath{\mathsf{B}}(\bm\lambda)$ contains precisely $\lambda_1$ zero rows and
by deleting these, the $\abs{\ensuremath{\mathsf{d}}{\bm\lambda}}\times\abs{\ensuremath{\mathsf{d}}{\bm\lambda}}$ identity matrix is
obtained. \qed
\end{lemma}
\subsection{Recursion}
\label{ss:recursion}
In this subsection, we give a recursive description of $V_K(\ensuremath{\mathsf{A}}(\bm\lambda))$
(see Proposition~\ref{prop:coneint}).
\begin{lemma}
\label{lem:rec}
Let $\bm\lambda = (\lambda_1,\dotsc,\lambda_r) \vdash n$
and let $X$ be the generic upper triangular $n\times n$ matrix.
Partition $X$ in the form
\[
X = \left[\begin{array}{c|c}
\makebox{$\begin{matrix} X^{\mathrm{I}}_{\lambda_2} &
* \\ 0_{\lambda_1-\lambda_2,\lambda_2} &
X^{\mathrm{II}}_{\lambda_1-\lambda_2}\end{matrix}$}
& \makebox{$\bar X_{\lambda_1,\abs{\ensuremath{\mathsf{d}}{\bm\lambda}}}$}\\\hline
0 & \makebox{$X^{\prime}_{\abs{\ensuremath{\mathsf{d}}{\bm\lambda}}}$}
\end{array}\right],
\]
where subscripts are added to denote block sizes.
Then
\[
X \ensuremath{\mathsf{A}}(\bm\lambda) =
\left[
\begin{array}{c|c}
\makebox{$0_{\lambda_1}$} &
\begin{array}{c|c}
\begin{array}{c}
\makebox{$X^{\mathrm I}$} \\
\makebox{$0$}
\end{array} &
\makebox{$\bar X \ensuremath{\mathsf{B}}(\ensuremath{\mathsf{d}}{\bm\lambda})$}
\end{array}
\\\hline
0 & X' \ensuremath{\mathsf{A}}(\ensuremath{\mathsf{d}}{\bm\lambda})
\end{array}
\right].
\]
\end{lemma}
\begin{proof}
This follows easily from \eqref{eq:A}.
\end{proof}
By Lemmas \ref{lem:B}--\ref{lem:rec},
the $\lambda_1\times \abs{\ensuremath{\mathsf{d}}{\bm\lambda}}$ submatrix obtained by considering the first $\lambda_1$ rows of
$X\ensuremath{\mathsf{A}}(\bm\lambda)$ and then deleting the first $\lambda_1$ columns is of the
form
\begin{equation}
\label{eq:Xlambda}
X^{\bm\lambda} :=
\begin{bmatrix}
x_{1,1} & \hdots & x_{1,\lambda_2} & * & \hdots & *\\
& \ddots & \vdots & \vdots & \ddots & \vdots\\
& & x_{\lambda_2,\lambda_2} & * & \hdots & * \\
& & & \vdots & \ddots & \vdots \\
& & & * & \hdots & *
\end{bmatrix},
\end{equation}
where the entries marked ``$*$'' indicate unspecified but \itemph{distinct} variables taken from $\bar X$.
\begin{cor}
\label{cor:rec}
Let $\bm\lambda \vdash n$ and let $K$ be a $p$-adic field.
For $\bm x \in \Tr_n(K)$, define $\bm x'$ and $\bm x^{\bm\lambda}$ by specialising
$X'$ and $X^{\bm\lambda}$ from Lemma~\ref{lem:rec} and \eqref{eq:Xlambda},
respectively, at $\bm x$.
Then
\begin{align}
\label{eq:Vrec}
V_K(\ensuremath{\mathsf{A}}(\bm\lambda)) = \Bigl\{ \bm x \in \Tr^\ensuremath{\mathrm{reg}}_n(\mathfrak{O}_K) :
\text{(i) } & \text{each row of $\bm x^{\bm\lambda}$ belongs to
$\mathfrak{O}_K^{\abs{\ensuremath{\mathsf{d}}{\bm\lambda}}} \bm x '$
and } \nonumber \\ \text{ (ii) }&
\bm x' \in V_K(\ensuremath{\mathsf{A}}(\ensuremath{\mathsf{d}}{\bm\lambda}))
\Bigr\}.
\end{align}
\end{cor}
\begin{proof}
Let $\bm x \in \Tr_n^\ensuremath{\mathrm{reg}}(\mathfrak{O}_K)$.
Clearly,
$\bm x \in V_K(\ensuremath{\mathsf{A}}(\bm\lambda))$ if and only if every row of $\bm x
\ensuremath{\mathsf{A}}(\bm\lambda)$ is contained in the $\mathfrak{O}_K$-span of the rows of $\bm x$.
By Lemma~\ref{lem:rec} and since $\det(\bm x) \not= 0$, the first $\lambda_1$
rows of $\bm x \ensuremath{\mathsf{A}}(\bm\lambda)$ satisfy this condition if and only if every row
of $\bm x^{\bm\lambda}$ is contained in the $\mathfrak{O}_K$-span of the rows of $\bm x'$.
Similarly, the rows numbered $\lambda_1 + 1,\dotsc,n$ of
$\bm x \ensuremath{\mathsf{A}}(\bm\lambda)$ are contained in the $\mathfrak{O}_K$-span of $\bm x$ if and
only if each row of $\bm x' \ensuremath{\mathsf{A}}(\ensuremath{\mathsf{d}}{\bm\lambda})$ is contained in the
$\mathfrak{O}_K$-span of $\bm x'$ or, equivalently, if $\bm x' \in V_K(\ensuremath{\mathsf{A}}(\ensuremath{\mathsf{d}}{\bm\lambda}))$.
\end{proof}
\subsection{Characterising submodule membership}
\label{ss:membership}
Condition (i) in \eqref{eq:Vrec} leads us to investigate pairs $(\ensuremath{\bm x},\ensuremath{\bm y}) \in R^n
\times \Tr_n(R)$ (where $R$ is a ring) such that $\bm x$ is contained in the row
span of $\ensuremath{\bm y}$ over $R$.
In this subsection, we study the set of all such pairs $(\ensuremath{\bm x},\ensuremath{\bm y})$
in the case that $R = \mathfrak{O}_K$ for a $p$-adic field $K$.
We write $\mathbf{A}^n = \Spec(\mathbf{Z}[X_1,\dotsc,X_n])$ and $\Tr_n = \Spec(\mathbf{Z}[Y_{ij} :
1\le i \le j \le n])$.
Let
\begin{equation}
\label{eq:EnR}
E_n(R) := \bigl\{
(\ensuremath{\bm x}, \ensuremath{\bm y}) \in R^n \times \Tr_n(R) : \ensuremath{\bm x} \in R^n \ensuremath{\bm y} \bigr\}.
\end{equation}
We identify $\mathbf{A}^n\times\Tr_n = \Spec(\mathbf{Z}[X_1,\dotsc,X_n,
Y_{11},\dotsc,$ $Y_{1n},Y_{22}, \dotsc, Y_{nn}])$.
Define
\begin{equation}
\label{eq:Cn}
\mathcal{C}_n :=
\bigl\{ (\alpha,\omega) \in \RR_{\ge 0}^n \times \Tr_n(\RR_{\ge 0}) :
\omega_{ii} \le \alpha_i \text{ for } 1 \le i \le n
\bigr\}.
\end{equation}
For a $p$-adic field $K$,
we extend $\nu_K$ to families of elements of $K$ via $\nu_K(a_1,\dotsc,a_m) = (\nu_K(a_1),\dotsc,\nu_K(a_m))$
and write
\[
\mathcal{C}_n(K) := \Bigl\{ (\ensuremath{\bm x},\ensuremath{\bm y}) \in K^n \times \Tr_n(K) : (\nu_K(\ensuremath{\bm x}),\nu_K(\ensuremath{\bm y}))
\in \mathcal{C}_n\Bigr\}
\subset \mathfrak{O}_K^n \times \Tr_n^\ensuremath{\mathrm{reg}}(\mathfrak{O}_K).
\]
The following lemma
will play a key role in our proof of Theorem~\ref{thm:nilpotent}.
It shows that away from sets of measure zero, a suitable $\mathbf{Z}$-defined change of
coordinates (defined independently of $K$) transforms $E_n(\mathfrak{O}_K)$ into $\mathcal{C}_n(K)$.
\begin{lemma}
\label{lem:coc}
There exist
\begin{itemize}
\item closed subschemes $V_n,V_n' \subset \mathbf{A}^n \times \Tr_n$ of the
form $f_n = 0$ and $f_n'= 0 $, respectively, where $f_n,f_n' \in \mathbf{Z}[\bm X,
\bm Y]$ are non-zero non-units, and
\item an isomorphism
$\varphi_n \colon (\mathbf{A}^n\times\Tr_n)\setminus V_n \to
(\mathbf{A}^n\times\Tr_n)\setminus V_n'$
\end{itemize}
such that the following conditions are satisfied:
\begin{enumerate}
\item
\label{lem:coc1}
For each $p$-adic field $K$,
$\varphi_n^K( E_n(\mathfrak{O}_K) \setminus V_n(\mathfrak{O}_K)) = \mathcal{C}_n(K) \setminus
V_n'(\mathfrak{O}_K)$,
where $\varphi_n^K$ denotes the map induced by $\varphi_n$ on $K$-points.
\item
\label{lem:coc2}
The Jacobian determinant of $\varphi_n$ is identically $1$.
\item
\label{lem:coc3}
$\varphi_n$ commutes with (the restriction to its domain of) the projection of $\mathbf{A}^n \times \Tr_n$ onto
$\Tr_n$ and (the restriction of) the projection onto the first coordinate of $\mathbf{A}^n$.
\end{enumerate}
\end{lemma}
\begin{ex*}[$n=2$]
Let $K$ be a $p$-adic field; we drop the subscripts ``$K$'' in the following.
Let $x,y,a,b,c \in \mathfrak{O}$ and suppose that
$x (ay - bx) a b c\not= 0$.
Define $y' := y - \frac x a b \in K$ and note that $y' \not= 0$.
Then $(x,y) \in \mathfrak{O}^2 \ensuremath{\,\cdotp} \bigl[\begin{smallmatrix} a & b \\ 0 &
c\end{smallmatrix}\bigr]$ if and only if $\nu(a) \le \nu(x)$ and
$(x,y) - \frac x a (a,b) = (0,y') \in \mathfrak{O} (0,c)$; the latter
condition is equivalent to $\nu(c) \le \nu(y')$ and implies that $y' \in \mathfrak{O}$.
We see that the map $((x,y),\bigl[\begin{smallmatrix} a & b\\0 & c\end{smallmatrix}\bigr])
\mapsto ((x,y'),\bigl[\begin{smallmatrix} a & b\\0 &
c\end{smallmatrix}\bigr])$
has the properties of $\varphi_2$ stated in Lemma~\ref{lem:coc}.
\end{ex*}
\begin{proof}[Proof of Lemma~\ref{lem:coc}]
We proceed by induction.
For $n = 1$, we let $f_1 = f_1' = X_1 Y_{11}$ and define $\varphi_1$ to be the identity.
Clearly, (\ref{lem:coc1})--(\ref{lem:coc3}) are satisfied.
Let $n > 1$ and suppose that $\varphi_{n-1}$ with the stated properties has
been defined.
Let~$K$ be a $p$-adic field
and let $(\ensuremath{\bm x},\ensuremath{\bm y}) \in K^n\times \Tr_n(K)$ with $x_1 y_{11} \not= 0$.
We again drop the subscripts ``$K$''.
Gaussian elimination shows that $(\ensuremath{\bm x},\ensuremath{\bm y}) \in E_n(\mathfrak{O})$ if and only if the
following conditions are satisfied:
\begin{enumerate}
\item[(a)]
$x_i, y_{ij} \in \mathfrak{O}$ for $1\le i \le j \le n$,
\item[(b)]
$\frac{x_1}{y_{11}} \in \mathfrak{O}$, and
\item[(c)]
$\bigl(x_2 - \frac{x_1}{y_{11}} y_{12}, \dotsc, x_n - \frac{x_1}{y_{11}}
y_{1n}\bigr) \in \mathfrak{O}^{n-1} \ensuremath{\,\cdotp} \Bigl[ y_{ij} \Bigr]_{2\le i \le j \le
n}$.
\end{enumerate}
We will now simplify (c) using a change of coordinates.
For $2 \le j \le n$, let $x_j' := x_j - \frac{x_1}{y_{11}} y_{1j}$.
Write $x_1' := x_1$ and $\ensuremath{\bm x}' := (x_1',\dotsc,x_n')$.
Note that $(\ensuremath{\bm x},\ensuremath{\bm y}) \mapsto (\ensuremath{\bm x}',\ensuremath{\bm y})$ is an
automorphism of the complement of $Y_{11} = 0$ in $\mathbf{A}^n \times \Tr_n$
and that the Jacobian determinant of this map is identically $1$.
Assuming that $y_{ij} \in \mathfrak{O}$ for $1\le i \le j \le n$
and $\frac{x_1}{y_{11}} \in \mathfrak{O}$, we see that $x_j \in \mathfrak{O}$ if and only if
$x_j' \in \mathfrak{O}$.
Hence, $(\ensuremath{\bm x},\ensuremath{\bm y}) \in E_n(\mathfrak{O})$ if and only if (b) and the following two
conditions are satisfied:
\begin{enumerate}
\item[(a')] $x_i', y_{ij} \in \mathfrak{O}$ for $1\le i \le j \le n$,
\item[(c')] $(x_2', \dotsc, x_n') \in \mathfrak{O}^{n-1} \ensuremath{\,\cdotp} \Bigl[ y_{ij}
\Bigr]_{2\le i \le j \le n}$.
\end{enumerate}
After excluding suitable hypersurfaces,
our inductive hypothesis allows us to perform another change of coordinates,
replacing $x_2', \dotsc,x_n'$ by $x_2'', \dotsc,x_n''$, say, such that
$(\ensuremath{\bm x},\ensuremath{\bm y}) \in E_n(K)$ if and only if the following conditions are
satisfied:
\begin{enumerate}
\item[(a'')] $x_i'', y_{ij} \in \mathfrak{O}$ for $1\le i \le j \le n$ (where
$x_1'' := x_1' = x_1$) and
\item[(c'')] $\nu(y_{ii}) \le \nu(x_i'')$ for $1 \le i \le n$;
\end{enumerate}
note that (b) is implied by the case $i = 1$ of (c'').
For (\ref{lem:coc1}), assuming that the product of all $x_i''$ and $y_{ij}$ is
non-zero, conditions (a'') and (c'') are both satisfied if and only
if $(\ensuremath{\bm x}'',\ensuremath{\bm y}) \in \mathcal{C}_n(K)$, where $\ensuremath{\bm x}'' := (x_1'',\dotsc,
x_n'')$.
The change of coordinates $\bm x \mapsto \bm x''$ is defined over $\mathbf{Z}$,
does not depend on $K$, and, does not modify the $x_1$- or $\bm
y$-coordinate, as required for (\ref{lem:coc3});
part (\ref{lem:coc2}) follows since $\varphi_n$ is defined as a composite of maps,
the Jacobian determinant of each of which is identically $1$.
\end{proof}
\begin{rem}
\label{rem:jacobian}
It follows from Lemma~\ref{lem:coc}(\ref{lem:coc2})
that the change of variables afforded by $\varphi_n$ does not affect $p$-adic measures.
Moreover, it is well-known that if $0\not= f \in \mathfrak{O}_K[X_1,\dotsc,X_n]$, then
the zero locus of $f$ in $\mathfrak{O}_K^n$ has measure zero.
We conclude that $V_n$ and $V_n'$ in Lemma~\ref{lem:coc} are without relevance
for the computation of the integral in Proposition~\ref{prop:coneint}.
\end{rem}
\subsection{Final steps towards Theorem~\ref{thm:nilpotent}}
\label{ss:proof_nilpotent}
By combining Corollary~\ref{cor:rec} and Lemma~\ref{lem:coc}, we may reduce
the computation of the integral in Proposition~\ref{prop:coneint} for $A =
\ensuremath{\mathsf{A}}(\bm\lambda)$ to a purely combinatorial problem.
\begin{prop}
\label{prop:make_monomial}
Let $\bm\lambda = (\lambda_1,\dotsc,\lambda_r) \vdash n$ and let $K$ be a
$p$-adic field.
Then
\begin{equation}
\label{eq:int_linear}
\zeta_{\ensuremath{\mathsf{A}}(\bm\lambda),\mathfrak{O}_K}(s) = (1-q_K^{-1})^{-n} \int_{V_{\bm\lambda}(\mathfrak{O}_K)} \prod_{i=1}^n
\abs{x_i}_K^{s-i} \dd\mu(\ensuremath{\bm x}),
\end{equation}
where
$V_{\bm\lambda}(\mathfrak{O}_K)$ consists of those $\ensuremath{\bm x} \in \mathfrak{O}_K^{n(n+1)/2}$ satisfying the
following divisibility conditions, where the $y_{i,j,\ell}$ below denote \underline{distinct}
variables among the $x_{n+1},\dotsc,x_{n(n+1)/2}$:
\begin{itemize}
\item
For $2 \le i \le r$ and $1 \le j \le \lambda_{i}$,
\[
{x_{\ensuremath{\mathsf{\sigma}}_{i-1}(\bm\lambda) + j}}\mathrel{\Big|}{x_{\ensuremath{\mathsf{\sigma}}_{i-2}(\bm\lambda)+j}, y_{i,j,1},\dotsc,y_{i,j,j-1}}.
\]
\item
For $3 \le i \le r$ and $\ensuremath{\mathsf{\sigma}}_{i-1}(\bm\lambda) < j \le n$,
\[
{x_j}\mathrel{\Big|}{y_{i,j,n+1},\dotsc,y_{i,j,n+\lambda_{i-2}}}.
\]
\end{itemize}
\end{prop}
\begin{rem*}
Since the $y_{i,j,\ell}$ do not appear in the integrand in the right-hand side
of \eqref{eq:int_linear}, it is of no consequence precisely which of the
$x_{n+1},\dotsc,x_{n(n+1)/2}$ each $y_{i,j,\ell}$ refers to provided that
distinct triples $(i,j,\ell)$ yield different $y_{i,j,\ell}$.
\end{rem*}
\begin{proof}[Proof of Proposition~\ref{prop:make_monomial}]
If $r \le 1$, the claim is trivially true so let $r \ge 2$.
As our first step, we combine Corollary~\ref{cor:rec} and Lemma~\ref{lem:coc} in order to
transform the membership condition (i) in \eqref{eq:Vrec} into the given
divisibility conditions for $i = 2$ and $i =3$, respectively;
here, $x_1,\dotsc,x_n$ correspond to the diagonal entries
$x_{11},\dotsc,x_{nn}$ in Proposition~\ref{prop:coneint}.
This transformation does not affect the integrand in \eqref{eq:GSS} thanks
to condition (\ref{lem:coc3}) in Lemma~\ref{lem:coc}.
Subsequent steps then recursively apply the same procedure
in order to express the condition $\bm x' \in V_K(\ensuremath{\mathsf{A}}(\ensuremath{\mathsf{d}}{\bm\lambda}))$ in
Corollary~\ref{cor:rec} in terms of the stated divisibility conditions,
taking into account the evident shifts of variable indices.
Crucially, in doing so, none of the diagonal coordinates $x_1,\dotsc,x_n$ will ever be
modified, again thanks to condition (\ref{lem:coc3}) in Lemma~\ref{lem:coc}.
Therefore, the divisibility conditions obtained during earlier steps will
never be altered by subsequent ones.
The claim thus follows by induction.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:nilpotent}]
We once again omit subscripts ``$K$'' in the following.
Moreover, we will make repeated use of the identity
\begin{equation}
\label{eq:x_mid_y}
\int\limits_{\{ (x,y) \in \mathfrak{O}^2 : \divides x y\}} \abs{x}^r \abs{y}^s
\dd\mu(x,y) = \int\limits_{\mathfrak{O}^2}\abs{x}^{r+s+1} \abs{y}^s \dd\mu(x,y)
\end{equation}
which follows by performing a change of variables $y = xy'$ on the
left-hand side.
We will furthermore use the well-known identity $\int_{\mathfrak{O}}\abs{x}^s\dd\mu(x)
= (1-q^{-1})/(1-q^{-s-1})$.
By repeatedly applying \eqref{eq:x_mid_y}, we can eliminate all the
$y_{i,j,\ell}$ variables and rewrite~\eqref{eq:int_linear} as an integral over
$\mathfrak{O}^n$.
In order to record the effect of this procedure on the integrand, we use
$\bm\lambda$ to index $x_1,\dotsc,x_n$ as follows.
Let $f(i,j) := \ensuremath{\mathsf{\sigma}}_{i-1}(\bm\lambda) + j$ and, for $\ensuremath{\bm x} = (x_1,\dotsc,x_n)$, write
$x_{ij} := x_{f(i,j)}$.
Define
\[
U_{\bm\lambda}(\mathfrak{O}) := \Bigl\{ \ensuremath{\bm x} \in \mathfrak{O}^n : \divides{x_{i,j}}{x_{i-1,j}}
\text{ for } 2 \le i \le r \text{ and } 1 \le j \le \lambda_i
\Bigr\}
\]
Proposition~\ref{prop:permconj} and repeated applications of \eqref{eq:x_mid_y} to \eqref{eq:int_linear}
show that
$$
\zeta_{\ensuremath{\mathsf{N}}(\bm\lambda^*),\mathfrak{O}}(s) =
\zeta_{\ensuremath{\mathsf{A}}(\bm\lambda),\mathfrak{O}}(s) = (1-q^{-1})^{-n}\int\limits_{U_{\bm\lambda}(\mathfrak{O})}
F_{\bm\lambda}(\ensuremath{\bm x})\dd\mu(\ensuremath{\bm x}),$$
where
\begin{align*}
F_{\bm\lambda}(\ensuremath{\bm x}) & = \prod_{i=1}^r \prod_{j=1}^{\lambda_i} \abs[\big]{x_{ij}}^{s -
f(i,j)}
\times
\prod_{i=2}^r \prod_{j=1}^{\lambda_i} \abs[\big]{x_{ij}}^{j-1}
\times
\prod_{a=3}^r \prod_{i=a}^r \prod_{j=1}^{\lambda_i}
\abs[\big]{x_{ij}}^{\lambda_{a-2}}
\\
& =
\prod_{j=1}^{\lambda_1} \abs[\big]{x_{1j}}^{s-j} \times
\prod_{i=2}^r \prod_{j=1}^{\lambda_i} \abs[\big]{x_{ij}}^{s - (\lambda_{i-1}+1)};
\end{align*}
the second equality follows since
$s - f(i,j) + j - 1 + \sum_{a=3}^i \lambda_{a-2} = s - (\lambda_{i-1} + 1)$
for $2 \le i \le r$ and $1 \le j \le \lambda_i$.
Another sequence of applications of \eqref{eq:x_mid_y} can be used to remove
the divisibility conditions in $U_{\bm\lambda}(\mathfrak{O})$, yielding
\pushQED{\qed}
\begin{align*}
(1-q^{-1})^n
\zeta_{\ensuremath{\mathsf{A}}(\bm\lambda),\mathfrak{O}}(s) & =
\int_{\mathfrak{O}^n} \prod_{j=1}^{\lambda_1} \abs[\big]{x_{1j}}^{s-j} \times
\prod_{i=2}^r \prod_{j=1}^{\lambda_i} \abs[\big]{x_{ij}}^{s-j + i - 1 +
\sum\limits_{a=1}^{i-1}(s - (\lambda_a+1))} \dd\mu(\ensuremath{\bm x})\\
& =
\int_{\mathfrak{O}^n} \prod_{i=1}^r \prod_{j=1}^{\lambda_i} \abs[\big]{x_{ij}}^{is
- (\ensuremath{\mathsf{\sigma}}_{i-1}(\bm\lambda) + j)} \dd\mu(\ensuremath{\bm x})\\
& = (1-q^{-1})^n \ensuremath{\,\cdotp} \prod_{i=1}^r \prod_{j=1}^{\lambda_i} \Bigl(1-q^{-is +
\ensuremath{\mathsf{\sigma}}_{i-1}(\bm\lambda) + j - 1}\Bigr)^{-1} \\
&= (1-q^{-1})^n \ensuremath{\,\cdotp} W_{\bm\lambda}(q,q^{-s}). \qedhere \popQED
\end{align*}
\end{proof}
\section{Proofs of Theorems~\ref{Thm:global}--\ref{Thm:poles}}
\label{s:proofs}
At the heart of our proofs of Theorems~\ref{Thm:global}--\ref{Thm:poles} lies
the following local version of Theorem~\ref{Thm:global}.
\begin{thm}
\label{thm:local}
Let $S\subset \ensuremath{\mathcal V}_k$ be finite and $A \in \Mat_n(\mathfrak{o}_S)$.
Let $((f_1,\bm\lambda_1), \dotsc, (f_e,\bm\lambda_e))$ be an elementary divisor
vector of $A$ over $k$.
Write $k_i = k[X]/(f_i)$.
Let $\mathfrak{o}_i$ denote the ring of integers of $k_i$.
Then for almost all $v \in \ensuremath{\mathcal V}_k$,
\begin{equation}
\label{eq:local}
\zeta_{A,\mathfrak{o}_v}(s) = \prod_{i=1}^e\prod_{j=1}^{\abs{\bm\lambda_i}}
\prod_{\substack{w \in \ensuremath{\mathcal V}_{k_i}\\\divides{w} v}}
\zeta_{\mathfrak{o}_{i,w}}\bigl( (\bm\lambda_i^*)^{-1}(j) \ensuremath{\,\cdotp} s - j + 1\bigr).
\end{equation}
\end{thm}
\begin{proof}
Combine Proposition~\ref{prop:primary},
Theorem~\ref{thm:rednil}, and Theorem~\ref{thm:nilpotent}.
\end{proof}
The following is a consequence of Proposition~\ref{prop:coneint} and well-known
rationality results from $p$-adic integration.
\begin{prop}[{Cf.\ \cite[\S 3]{GSS88}}]
\label{prop:rationality}
Let $K$ be a $p$-adic field and let $A \in \Mat_n(\mathfrak{O}_K)$.
Then $\zeta_{A,\mathfrak{O}_K}(s) \in \mathbf{Q}(q_K^{-s})$.
Hence, $\zeta_{A,\mathfrak{O}_K}(s)$ admits meromorphic continuation to all of
$\mathbf{C}$.
\end{prop}
In order to deduce parts (\ref{Thm:global2})--(\ref{Thm:global3}) of
Theorem~\ref{Thm:global}, we will use the following corollary to the detailed
analysis of analytic properties of subobject zeta functions in \cite{dSG00}.
\begin{lemma}
\label{lem:alpha}
Let $S' \subset \ensuremath{\mathcal V}_k$ be finite, $S\subset S'$, and let
$A \in \Mat_n(\mathfrak{o}_S)$.
Then $\alpha_{A,\mathfrak{o}_S} = \alpha_{A,\mathfrak{o}_{S'}}$
and $\beta_{A,\mathfrak{o}_S} = \beta_{A,\mathfrak{o}_{S'}}$.
\end{lemma}
\begin{proof}
We first argue that $\alpha_{A,\mathfrak{o}_v} < \alpha_{A,\mathfrak{o}_S}$ for each $v \in
\ensuremath{\mathcal V}_k\setminus S$.
The zeta function $\zeta_{A,\mathfrak{o}_S}(s+n)$ is an Euler product of cone integrals
(cf.\ Proposition~\ref{prop:coneint}) in the sense of \cite[Def.\ 4.2]{dSG00};
cf.\ \cite[Cor.\ 5.6]{dSG00}.
Using the notation from \cite{dSG00},
by \cite[Cor.\ 3.4]{dSG00} (which is correct despite a minor, fixable mistake
in \cite[Prop.\ 3.3]{dSG00}, see \cite[Rem.\ 4.6]{AKOV13}),
it follows that each $\alpha_{A,\mathfrak{o}_v}$ for $v \in
\ensuremath{\mathcal V}_k\setminus S$ is a number of the form $n-B_j/A_j$ for $j = 1,\dotsc,q$.
Hence, by combining \cite[Cor.\ 4.14, Lem.\ 4.15]{dSG00},
for each $v \in \ensuremath{\mathcal V}_k\setminus S$,
$$\alpha_{A,\mathfrak{o}_v} < n + \max_{k=1,\dotsc,q} \frac{1-B_k}{A_k} =
\alpha_{A,\mathfrak{o}_S}.$$
Clearly, $0 < \alpha_{A,\mathfrak{o}_{S'}} \le \alpha_{A,\mathfrak{o}_S}$.
Define $F(s) = \prod_{v\in S'\setminus S}\zeta_{A,\mathfrak{o}_v}(s)$
so that
$\zeta_{A,\mathfrak{o}_S}(s) = F(s)\zeta_{A,\mathfrak{o}_{S'}}(s)$
for all $s \in \mathbf{C}$ with $\mathrm{Re}(s) > \alpha_{A,\mathfrak{o}_S} - \delta$ and some
constant $\delta > 0$ (see Theorem~\ref{thm:dSG}).
By the above, every real pole of $F(s)$ is less than $\alpha_{A,\mathfrak{o}_S}$.
Since $F(s)$ is a non-zero Dirichlet series with non-negative coefficients,
we conclude that $F(\alpha_{A,\mathfrak{o}_S}) > 0$.
In particular, since $\zeta_{A,\mathfrak{o}_S}(s)$ has a pole at $\alpha_{A,\mathfrak{o}_S}$,
the same is true of $\zeta_{A,\mathfrak{o}_{S'}}(s)$ whence $\alpha_{A,\mathfrak{o}_{S'}} \ge
\alpha_{A,\mathfrak{o}_S}$.
Moreover, $F(\alpha_{A,\mathfrak{o}_S}) > 0$ clearly also implies that $\beta_{A,\mathfrak{o}_S} = \beta_{A,\mathfrak{o}_{S'}}$.
\end{proof}
\begin{rem}
\label{rem:alphainv}
\quad
\begin{enumerate}
\item
\label{rem:alphainv1}
The corresponding statement for subalgebra and submodule zeta functions
(proved in the same way) is certainly well-known to experts in the area.
Unfortunately, it does not seem to have been spelled out in the literature.
For a similar statement in the context of representation zeta functions,
see~\cite[Thm~1.4]{AKOV16}.
\item
While in \cite{dSG00} only the case $k = \mathbf{Q}$, $S = \varnothing$ is discussed,
their arguments carry over to the present setting in the expected way
(cf.\ \cite{AKOV13} and \cite[\S 4]{DV15}).
\end{enumerate}
\end{rem}
\begin{proof}[Proof of Theorem~\ref{Thm:global}]
Part (\ref{Thm:global1}) follows from Theorem~\ref{thm:local} and Proposition~\ref{prop:rationality}.
Let $\bm\mu \vdash n$.
We now determine the largest real pole,
$\alpha$ say, and its multiplicity, $\beta$ say, of
\[
\ensuremath{\mathsf{Z}}(s) := \prod_{j=1}^n \zeta_{\mathfrak{o}_S}(\bm\mu^{-1}(j) \ensuremath{\,\cdotp} s - j + 1).
\]
Write $r = \len{\bm\mu}$.
Since $\zeta_{\mathfrak{o}_S}(s)$ has a unique pole at $1$ (with multiplicity $1$) and
$\zeta_{\mathfrak{o}_S}(s_0) \not= 0$ for real $s_0 > 1$,
\begin{align*}
\alpha &
= \max_{1\le j \le n} \frac j {\bm\mu^{-1}(j)}
= \max_{1\le i \le r} \max_{1\le j \le \lambda_i} \frac {\ensuremath{\mathsf{\sigma}}_{i-1}(\bm\mu) +j} i
= \max_{1\le i \le r} \frac {\ensuremath{\mathsf{\sigma}}_{i}(\bm\mu)} i = \mu_1 = \len{\bm\mu^*},
\end{align*}
where the penultimate equality follows since
$i \mu_{i+1} \le \ensuremath{\mathsf{\sigma}}_i(\bm\mu)$
and thus $\frac {\ensuremath{\mathsf{\sigma}}_i(\bm\mu)} i \ge
\frac{\ensuremath{\mathsf{\sigma}}_{i+1}(\bm\mu)}{i+1}$ for $1\le i \le r - 1$.
Next, $\beta$ is precisely the
number of $i \in \{1,\dotsc,r\}$ with $\mu_1 = \frac{\ensuremath{\mathsf{\sigma}}_i(\bm\mu)} i$ or,
equivalently, the largest $\ell \ge 1$ with $\mu_1 = \dotso = \mu_\ell$.
In other words, $\beta = \mu^*_{-1}$.
Parts (\ref{Thm:global2})--(\ref{Thm:global3}) of Theorem~\ref{Thm:global}
now follow from Lemma~\ref{lem:alpha} and the observation that
$\ensuremath{\mathsf{Z}}(s) > 0$ for $s > \alpha$.
\end{proof}
\begin{ex}
\label{ex:exceptional}
The presence of the exceptional factors $W_u(q_{w_u}^{-s})$ in Theorem~\ref{Thm:global}
is in general unavoidable.
For a simple example, let $a \in \mathfrak{o}$ be non-zero and define $A
= \bigl[\begin{smallmatrix} 0 & a \\ 0 & 0 \end{smallmatrix}\bigr]$.
Using Proposition~\ref{prop:coneint}, a simple computation reveals that for $v \in \ensuremath{\mathcal V}_k$,
\begin{equation}
\label{eq:exceptional}
\zeta_{A,\mathfrak{o}_v}(s) = \frac{1 - q_v^{1-2s} + q_v^{(1-s)(v(a)+1)} \ensuremath{\,\cdotp} (q_v^{-s}-1)}{1-q_v^{1-s}} \ensuremath{\,\cdotp} \zeta_{\mathfrak{o}_v}(s) \zeta_{\mathfrak{o}_v}(2s - 1);
\end{equation}
note that $\zeta_{A,\mathfrak{o}_v}(s) = \zeta_{\mathfrak{o}_v}(s) \zeta_{\mathfrak{o}_v}(2s - 1)$
whenever $v(a) = 0$.
We further note that the exceptional factor in \eqref{eq:exceptional}
in fact belongs to $\mathbf{Z}[q_v^{-s}]$ and is thus regular at $s = 1$;
this is consistent with the general fact that for subobject zeta functions,
each local abscissa of convergence is strictly less than the associated global
one (see the proof of Lemma~\ref{lem:alpha}).
Finally note the failure of \eqref{eq:feqn} for the finitely many $v \in \ensuremath{\mathcal V}_k$ with $v(a) > 0$.
\end{ex}
\begin{rem}
In view of a conjecture of Solomon proved by Bushnell and Reiner~\cite{BR80},
it is natural to ask if the $W_u \in \mathbf{Q}(X)$ in Theorem~\ref{Thm:global} are
in fact always elements of $\mathbf{Z}[X]$.
\end{rem}
\begin{proof}[Proof of Theorem~\ref{Thm:FEqn}]
The claim follows by combining Theorem~\ref{thm:local} and the following simple
observation.
Let $k'/k$ be an extension of number fields, let $\mathfrak{o}'$ be the ring of
integers of~$k'$, and let $v \in \ensuremath{\mathcal V}_k$ be unramified in $k'$.
If $w \in \ensuremath{\mathcal V}_{k'}$ divides $v$,
define $\mathfrak f(w/v)$ by $q_w = q_v^{\mathfrak f(w/v)}$.
Define
\[
\ensuremath{\mathsf{Z}}_v(s) = \prod_{\substack{w \in \ensuremath{\mathcal V}_{k'}\\\divides w v}}
\zeta_{\mathfrak{o}'_w}(s)
= \prod_{\substack{w \in \ensuremath{\mathcal V}_{k'}\\\divides w v}} \bigl (1-q_v^{-\mathfrak
f(w/v)s} \bigr)^{-1}.
\]
Then, recalling the definition of $\ensuremath{\mathrm g}_v(k')$ from p.\
\pageref{lab:noplaces}
and using $\sum\limits_{\divides w v} \mathfrak f(w/v) = \idx{k':k}$,
\[
\ensuremath{\mathsf{Z}}_v(s) \Bigm\vert_{q_v\to q_v^{-1}} = (-1)^{\ensuremath{\mathrm g}_v(k')}
q_v^{-\idx{k':k} s} \ensuremath{\,\cdotp} \ensuremath{\mathsf{Z}}_v(s). \popQED
\]
\end{proof}
\begin{lemma}
Let $S\subset \ensuremath{\mathcal V}_k$ be finite.
Let $\ensuremath{\mathsf{Z}}(s)$ and $\ensuremath{\mathsf{Z}}'(s)$ be two Dirichlet series with finite abscissae of
convergence.
Suppose that $\ensuremath{\mathsf{Z}}(s) = \prod_{v \in \ensuremath{\mathcal V}_k\setminus S}\ensuremath{\mathsf{Z}}_v(s)$ and
$\ensuremath{\mathsf{Z}}'(s) = \prod_{v \in \ensuremath{\mathcal V}_k\setminus S}\ensuremath{\mathsf{Z}}'_v(s)$, where
each $\ensuremath{\mathsf{Z}}_v(s)$ and $\ensuremath{\mathsf{Z}}'_v(s)$ is a series in $q_v^{-s}$ with non-negative
real coefficients.
Suppose that $\ensuremath{\mathsf{Z}}(s) = \ensuremath{\mathsf{Z}}'(s)$ and that $W(X,Y),W'(X,Y) \in \mathbf{Q}(X,Y)$ satisfy
$\ensuremath{\mathsf{Z}}_v(s) = W(q_v,q_v^{-s})$ and $\ensuremath{\mathsf{Z}}'_v(s) = W'(q_v,q_v^{-s})$ for almost
all $v\in \ensuremath{\mathcal V}_k \setminus S$.
Then $W(X,Y) = W'(X,Y)$.
\end{lemma}
\begin{proof}
Let $S_0$ be the set of rational primes which are divisible by at least one element of~$S$.
For a rational prime~$p\not\in S_0$,
define $\ensuremath{\mathsf{Z}}_p(s) = \prod_{{v \in
\ensuremath{\mathcal V}_k, \divides v p}} \ensuremath{\mathsf{Z}}_v(s)$ and define $\ensuremath{\mathsf{Z}}'_p(s)$ in the same
way.
Assuming that $\ensuremath{\mathsf{Z}}(s) = \ensuremath{\mathsf{Z}}'(s)$,
it is well-known that the coefficients of the Dirichlet series $\ensuremath{\mathsf{Z}}(s)$ and
$\ensuremath{\mathsf{Z}}'(s)$ coincide.
We conclude that $\ensuremath{\mathsf{Z}}_p(s) = \ensuremath{\mathsf{Z}}'_{p}(s)$ for $p \not\in S_0$.
By Chebotarev's density theorem, there exists an infinite set of rational
primes $P$ such that each $p\in P$ splits completely
in $k$.
Writing $d = \idx{k:\mathbf{Q}}$, for almost all $p \in P$,
we thus have $W(p,p^{-s})^d = \ensuremath{\mathsf{Z}}_p(s) = \ensuremath{\mathsf{Z}}'_p(s) =
W'(p,p^{-s})^d$ which easily implies $W(X,Y)^d = W'(X,Y)^d$.
Thus, $W(X,Y)/W'(X,Y)$ is a $d$th root of unity in $\mathbf{R}(X,Y)$ and hence in
$\mathbf{R}$, for the latter is algebraically closed in the former (see \cite[Prop.\
11.3.1]{Coh03}).
The non-negativity assumptions on the coefficients of $\ensuremath{\mathsf{Z}}_v(s)$ and
$\ensuremath{\mathsf{Z}}'_v(s)$ as series in $q_v^{-s}$ now imply $W(X,Y) = W'(X,Y)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Thm:sim}]
The implications
``(\ref{Thm:sim1})$\Rightarrow$(\ref{Thm:sim2})$\Rightarrow$(\ref{Thm:sim3})''
in Theorem~\ref{Thm:sim} are obvious.
Suppose that (\ref{Thm:sim3}) holds.
Let $\bm\lambda$ and $\bm\mu$ be the types of the matrices $A$ and $B$,
respectively.
By Theorem~\ref{thm:local} and the preceding lemma,
$W_{\bm\lambda}(X,Y) = W_{\bm\mu}(X,Y)$.
It is easy to see that the binomials $1-X^aY^b$ for $a \ge 0$ and $b \ge 1$
freely generate a free abelian subgroup of $\mathbf{Q}(X,Y)^\times$.
Hence, $\bm\lambda = \bm\mu$ and $A$ and $B$ are similar.
\end{proof}
\begin{rem}
\label{rem:nilpotency_required}
If $A$ is nilpotent and $\alpha \in k^\times$, then $A$ and $A + \alpha 1_n$
give rise to the same local and global zeta functions without $A$ and $A +
\alpha 1_n$ being similar.
In general, equality of local and global zeta functions associated with
non-nilpotent matrices $A$ and $B$ does not suffice to even conclude that
the algebras $k[A]$ and $k[B]$ are similar.
We give two examples to illustrate this behaviour, the first being arithmetic
and the second of combinatorial origin.
\begin{enumerate}
\item By \cite{Per77}, there are monic irreducible polynomials $f,g\in
\mathbf{Z}[X]$ of the same degree such that the number fields $\mathbf{Q}[X]/(f)$ and
$\mathbf{Q}[X]/(g)$ are non-isomorphic but have the same Dedekind zeta
functions; moreover, as explained in \cite[\S 1]{Per77}, every rational prime
has the same ``splitting type'' in each of these two number fields.
Consequently, $\zeta_{\ensuremath{\mathsf{C}}(f),\mathbf{Z}_p}(s) = \zeta_{\ensuremath{\mathsf{C}}(g),\mathbf{Z}_p}(s)$ for almost
all primes~$p$
\item
Recall the definition of $W_{\bm\lambda}$ from \S\ref{s:nilpotent}.
A simple calculation shows that $$W_{(2,2,1)} \ensuremath{\,\cdotp} W_{(3,1)} =
W_{(2,2)} \ensuremath{\,\cdotp} W_{(3,1,1)}.$$
Let $a,b \in k^\times$ be distinct and choose $A,B \in \Mat_9(k)$
to have elementary divisor vectors $((X-a,(3,2), (X-b,(2,1,1)))$ and
$((X-a,(2,2)),(X-b,(3,1,1)))$, respectively.
Then $k[A]$ and $k[B]$ are not similar but $\zeta_{A,\mathfrak{o}_v}(s) =
\zeta_{B,\mathfrak{o}_v}(s)$ for almost all $v \in \ensuremath{\mathcal V}_k$.
\end{enumerate}
\end{rem}
\begin{rem}
We further note that even for nilpotent $A$, the family of associated functional
equations \eqref{eq:feqn} in Theorem~\ref{Thm:FEqn} does not determine $A$ up to
similarity; an example is given by two nilpotent $7\times 7$-matrices with types
$(3,1,1,1,1)$ and $(2,2,2,1)$, respectively.
\end{rem}
\begin{proof}[Proof of Theorem~\ref{Thm:poles}]
By Theorem~\ref{thm:local},
$\zeta_{A,\mathfrak{o}_v}(s)$ has a pole at zero for almost all $v \in \ensuremath{\mathcal V}_k$.
Moreover, again for almost all $v \in \ensuremath{\mathcal V}_k$, this pole is simple if and
only if $e = 1$ and almost all places of $k$ remain inert in $k[X]/(f_1)$;
the latter condition is equivalent to $f_1$ being linear.
\end{proof}
\section{Applications}
\label{s:app}
\subsection{Submodules for unipotent groups}
Let $S \subset \ensuremath{\mathcal V}_k$ be finite,
let $\mathsf M$ be a finitely generated $\mathfrak{o}_S$-module, and let $\Omega
\subset \End_{\mathfrak{o}_S}(\mathsf M)$.
We let $\alpha_{\Omega \ensuremath{\curvearrowright} \mathsf M}$ denote the abscissa of convergence of
$\zeta_{\Omega\ensuremath{\curvearrowright} \mathsf M}(s)$.
As a special case (cf.\ \cite[Rem.~2.2(ii)]{topzeta})), given a possibly
non-associative $\mathfrak{o}_S$-algebra $\mathsf A$ whose underlying $\mathfrak{o}_S$-module is
finitely generated, we let $\alpha_{\mathsf A}$ denote the abscissa of
convergence of its ideal zeta function $\zeta_{\mathsf A}(s)$, as defined in the introduction.
We now illustrate how Theorem~\ref{Thm:global} can sometimes be used to
determine $\alpha_{\Omega \ensuremath{\curvearrowright} \mathsf M}$ or $\alpha_{\mathsf A}$ without computing
the corresponding zeta function.
The key observation is that if $\omega \in \Omega$, then $\alpha_{\Omega \ensuremath{\curvearrowright}
\mathsf M} \le \alpha_{\omega,\mathfrak{o}}$;
by Theorem~\ref{Thm:global}(\ref{Thm:global2}),
the latter number can be easily read off from an elementary divisor vector of
$\omega \otimes_{\mathfrak{o}_S} k$.
We let $\Uni_n$ denote the group scheme of upper unitriangular $n\times n$
matrices.
For $\bm\lambda = (\lambda_1,\dotsc,\lambda_r) \vdash n$, we regard $\Uni_{\bm\lambda} :=
\Uni_{\lambda_1}\times \dotsb \times \Uni_{\lambda_r}$ as a subgroup scheme of
$\Uni_n$ via the natural diagonal embedding.
The case $\len{\bm\lambda} = 1$ of the following provides an affirmative answer to \cite[Question~9.7]{padzeta}.
\begin{prop}
Let $\bm\lambda \vdash n$.
Then $\alpha_{\Uni_{\bm\lambda}(\mathfrak{o}) \ensuremath{\curvearrowright} \mathfrak{o}^n} = \len{\bm\lambda}$.
\end{prop}
\begin{proof}
Using the characterisation of $\Uni_m(k)$ as the centraliser of a maximal
flag of subspaces of $k^m$, we see that $\mathfrak{o}^n$ contains an
$\Uni_{\bm\lambda}(\mathfrak{o})$-invariant submodule $N$ such that $\Uni_{\bm\lambda}(\mathfrak{o})$ acts
trivially on $\mathfrak{o}^n/N$ and $\mathfrak{o}^n/N \approx_{\mathfrak{o}} \mathfrak{o}^{\len{\bm\lambda}}$.
We conclude that $\alpha_{\Uni_{\bm\lambda}(\mathfrak{o}) \ensuremath{\curvearrowright} \mathfrak{o}^n} \ge \len{\bm\lambda}$.
For an upper bound, note that $(1 + \ensuremath{\mathsf{N}}(\bm\lambda)) \in \Uni_{\bm\lambda}(\mathfrak{o})$
whence $\alpha_{\Uni_{\bm\lambda}(\mathfrak{o}) \ensuremath{\curvearrowright} \mathfrak{o}^n} \le \alpha_{\ensuremath{\mathsf{N}}(\bm\lambda),\mathfrak{o}}
= \len{\bm\lambda}$.
\end{proof}
For $\abs{\bm\lambda} \le 5$ and almost all $v \in \ensuremath{\mathcal V}_k$,
explicit formulae for $\zeta_{\Uni_{\bm\lambda}(\mathfrak{o}_v)\ensuremath{\curvearrowright} \mathfrak{o}_v^n}(s)$ have been
obtained by the author (see \cite[\S 9.4]{padzeta} and the database
included with \cite{Zeta});
the only unknown case for $\len{\bm\lambda} = 6$, namely
$\bm\lambda = (6)$, seems out of reach at present.
In addition to their global abscissae of convergence, the
$\zeta_{\Uni_{\bm\lambda}(\mathfrak{o}_v)\ensuremath{\curvearrowright} \mathfrak{o}_v^n}(s)$ are known to
generically satisfy local functional equations under inversion of $q_v$ by
\cite[\S 5.2]{Vol16}.
\subsection{Lie algebras of maximal class}
Let $\bm\ensuremath{\mathfrak g}$ be a finite-dimensional Lie $k$-algebra.
For finite $S\subset \ensuremath{\mathcal V}_k$, by an \bfemph{$\mathfrak{o}_S$-form} of $\bm\ensuremath{\mathfrak g}$,
we mean a Lie $\mathfrak{o}_S$-algebra $\ensuremath{\mathfrak g}$ whose underlying module is free and such
that $\ensuremath{\mathfrak g} \otimes_{\mathfrak{o}_S} k \approx_k \bm\ensuremath{\mathfrak g}$.
Let $\bm\ensuremath{\mathfrak g} = \bm\ensuremath{\mathfrak g}^1 \supset \bm\ensuremath{\mathfrak g}^2\supset \dotsb$ be the lower central
series of $\bm\ensuremath{\mathfrak g}$.
Recall that $\bm\ensuremath{\mathfrak g}$ has \bfemph{maximal class} if $\bm\ensuremath{\mathfrak g}$ is nilpotent of class
$\dim_k(\bm\ensuremath{\mathfrak g})-1$.
Equivalently, $\bm\ensuremath{\mathfrak g}$ has maximal class if and only if $\dim_k(\bm\ensuremath{\mathfrak g}^1/\bm\ensuremath{\mathfrak g}^{2}) = 2$ and $\dim_k(\bm\ensuremath{\mathfrak g}^i/\bm\ensuremath{\mathfrak g}^{i+1}) =
1$ for $1 \le i \le \dim_k(\bm\ensuremath{\mathfrak g}) - 1$.
\begin{prop}
\label{prop:maxclass}
Let $\ensuremath{\mathfrak g}$ be an $\mathfrak{o}_S$-form of a non-abelian finite-dimensional Lie
$k$-algebra of maximal class.
Then $\alpha_{\ensuremath{\mathfrak g}} = 2$.
\end{prop}
A proof of Proposition~\ref{prop:maxclass} using Theorem~\ref{Thm:global} will be given below.
We note that Proposition~\ref{prop:maxclass} is consistent with explicit calculations
carried out for specific $\mathbf{Z}$-forms of the
Lie algebras $M_3$,$M_4$,$M_5$, and $\mathrm{Fil}_4$ of maximal class and
dimension at most $5$ over the rationals; see \cite[Ch.\ 2]{dSW08}.
\begin{lemma}
Let $S\subset \ensuremath{\mathcal V}_k$ be finite.
Let $\ensuremath{\mathfrak g}$ be an $\mathfrak{o}_S$-form of a nilpotent Lie $k$-algebra of finite
dimension $n$.
Let $\mathsf A$ be the enveloping unital associative algebra of
$\mathrm{ad}(\ensuremath{\mathfrak g})$ within $\End_{\mathfrak{o}_S}(\ensuremath{\mathfrak g})$.
\begin{enumerate}
\item For each $\varphi \in \mathsf A$, there exists $c \in \mathfrak{o}_S$ with $(\varphi - c
1_{\ensuremath{\mathfrak g}})^n = 0$; thus, $\varphi \otimes_{\mathfrak{o}_S} k$ is primary.
\item
Let $\varphi \in \mathsf A$ have type $\bm\lambda$ over $k$.
Then $\alpha_{\ensuremath{\mathfrak g}} \le \len{\bm\lambda}$.
\end{enumerate}
\end{lemma}
\begin{proof}
The first part follows from Engel's theorem and the second part is then an
immediate consequence of Theorem~\ref{Thm:global}(\ref{Thm:global2}).
\end{proof}
\begin{lemma}
\label{lem:goodbasis}
Let $\bm\ensuremath{\mathfrak g}$ be an $(n+2)$-dimensional non-abelian Lie $k$-algebra of maximal class.
Then there exists a $k$-basis $(x_1,x_2,y_1,\dotsc,y_n)$ of $\bm\ensuremath{\mathfrak g}$ such that
$[x_1,x_2] = y_1$,
$[x_1,y_i] = y_{i+1}$ for $1 \le i \le n-1$, and $[x_1,y_n] = 0$.
\end{lemma}
\begin{proof}
Consider the graded Lie algebra $\bigoplus_{i\ge 1} \bm\ensuremath{\mathfrak g}^i/\bm\ensuremath{\mathfrak g}^{i+1}$
associated with $\bm\ensuremath{\mathfrak g}$.
We claim that there exists an element $a \in \bm\ensuremath{\mathfrak g}/\bm\ensuremath{\mathfrak g}^2$ such
that $[a,\ensuremath{\,\cdotp}]$ maps $\bm\ensuremath{\mathfrak g}^i/\bm\ensuremath{\mathfrak g}^{i+1}$ onto
$\bm\ensuremath{\mathfrak g}^{i+1}/\bm\ensuremath{\mathfrak g}^{i+2}$ for each $i \ge 1$.
To see that, first note that $[\bm\ensuremath{\mathfrak g}/\bm\ensuremath{\mathfrak g}^2,\bm\ensuremath{\mathfrak g}^i/\bm\ensuremath{\mathfrak g}^{i+1}] =
\bm\ensuremath{\mathfrak g}^{i+1}/\bm\ensuremath{\mathfrak g}^{i+2}$ for each $i \ge 1$.
Let $(u,v)$ be a $k$-basis of $\bm\ensuremath{\mathfrak g}/\bm\ensuremath{\mathfrak g}^2$.
Then $[u,v]$ spans $\bm\ensuremath{\mathfrak g}^2/\bm\ensuremath{\mathfrak g}^3$.
Moreover, if $w_i$ spans $\bm\ensuremath{\mathfrak g}^i/\bm\ensuremath{\mathfrak g}^{i+1}$,
then the image of at least one of $[u,w_i]$ and $[v,w_i]$ spans
$\bm\ensuremath{\mathfrak g}^{i+1}/\bm\ensuremath{\mathfrak g}^{i+2}$.
Consequently, we may take $a = u + cv$ for almost all $c \in k$.
Given $a$ as above, choose $b \in \bm\ensuremath{\mathfrak g}/\bm\ensuremath{\mathfrak g}^2$ such that $(a,b)$ is a basis of
$\bm\ensuremath{\mathfrak g}/\bm\ensuremath{\mathfrak g}^2$.
Let $x_1, x_2 \in \bm\ensuremath{\mathfrak g}$ be preimages of $a$ and $b$, respectively.
Then, if we define $y_1 = [x_1,x_2]$ and $y_{i+1} = [x_1,y_i]$, we obtain a
basis $(x_1,x_2,y_1,\dotsc,y_n)$ of the desired form.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:maxclass}]
By Lemma~\ref{lem:alpha} and Remark~\ref{rem:alphainv}(\ref{rem:alphainv1}),
we are free to enlarge $S$ as needed.
In particular, we may assume that $\ensuremath{\mathfrak g}/\ensuremath{\mathfrak g}^2 \approx_{\mathfrak{o}_S} \mathfrak{o}_S^2$ whence
$\alpha_{\ensuremath{\mathfrak g}} \ge 2$ follows.
Moreover, we may assume
that $\ensuremath{\mathfrak g}$ possesses an $\mathfrak{o}_S$-basis $(x_1,x_2,y_1,\dotsc,y_n)$ as in by
Lemma~\ref{lem:goodbasis}.
The matrix of $[x_1,\ensuremath{\,\cdotp}]$ with respect to the basis
$(x_2,y_1,\dotsc,y_n,x_1)$ is precisely $\ensuremath{\mathsf{N}}((n+1,1))$ whence
$\alpha_{\ensuremath{\mathfrak g}} \le 2$ follows from Theorem~\ref{Thm:global}.
\end{proof}
{
\bibliographystyle{abbrv}
\tiny
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,334
|
\section{Introduction}
The distribution of quadratic functionals for Gaussian random functions is an interesting and intensively developing topic in connection with the demands of asymptotic problems of empirical processes and the theory of small deviations in $L_2$. We will be interested in different Gaussian random functions with equally distributed quadratic norms.
By virtue of the Karhunen--Lo\`eve expansion, such norms can be represented as infinite quadratic Gaussian forms, whose coefficients are the eigenvalues of the corresponding covariance operators. Therefore, to prove the identity in law for $L_2$-norms of two Gaussian random functions, it suffices to verify the coincidence of spectra for their covariance operators (excluding zero). In this case we prefer to call two random Gaussian random functions ${\bf X}$ and ${\bf Y}$ {\it spectrally equivalent} and write ${\bf X \sim Y}$.
It should be noted that the equality of $L_2$-norms does not entail the equality of Gaussian processes or fields in law. Consider the typical example of this kind. Let $W$ be the standard Brownian motion on $[0,1]$, and denote by $B$ the standard Brownian bridge on the same interval.
The following spectral equivalence is well known:
\begin{equation}
\label{Wa}
\qquad W(t)- \int\limits_0^1 W (s)\,ds \sim \, B(t).
\end{equation}
For instance, Donati-Martin and Yor \cite{Dona} proved (\ref{Wa}) using the Fubini--Wiener technique, while in \cite{BNO} this equivalence was proved by the direct calculation of spectra. On the other hand, the Gaussian processes in (\ref{Wa}) have different covariances:
\begin{align*}
&\mathbb{E}\Big (W(s)- \int\limits_0^1 W (u)\,du\Big)\Big( W(t)- \int\limits_0^1 W (u)\,du\Big)= \min(s,t) - \frac {2s-s^2}2 - \frac {2t-t^2}2 + \frac 13;\\
&\mathbb{E}B(s) B(t) = \min(s,t) - st.
\end{align*}
Peccati and Yor \cite{PecY}, Deheuvels, Peccati and Yor \cite{DPY} and Deheuvels \cite{Deh} extended spectral equivalence (\ref{Wa}) to the Gaussian fields on the unit square. Let ${\bf W}$ and ${\bf B}$ denote, respectively, the classical Brownian sheet and the bivariate Brownian bridge or pinned Brownian sheet, see \cite{DPY}. The authors of \cite{PecY} and \cite{DPY} obtained several spectral equivalences generalizing (\ref{Wa}). The simplest of them has the form:
\begin{equation*}
\label{bivar}
\qquad {\bf W}(t_1,t_2)- \int\limits_0^1 \int\limits_0^1 {\bf W} (s_1,s_2) ds_1 ds_2 \sim {\bf B}(t_1,t_2).
\end{equation*}
Deheuvels, Peccati and Yor stated these results in dimension $d= 2$, indicating that similar equivalences can be written out in the case $d > 2$ \ ``at the price of minor additional technicalities.'' However, these formulations never appeared.
Our aim is to prove far reaching and sometimes unexpected generalizations of these and similar spectral equivalences in a very short and compact way which differs from that of \cite{DPY} and \cite{PecY}. The originality of our approach is due to the use of the methods of operator theory in Hilbert space. The spectral equivalence of various $d$-parametric Brownian functions takes on a simple and uniform perspective. We argue that the operator language is the most correct and convenient for writing identities in law for quadratic norms of Gaussian random functions.
Moreover, we demonstrate that similar relations are also valid for {\it integrated} and {\it multiply integrated} Gaussian fields and processes, something that has been done for the first time. We expect future applications to nonparametric statistics (in particular, to goodness-of-fit testing and testing of independence), and to the theory of Brownian functionals.
\medskip
The structure of the paper is as follows. First, we introduce the necessary facts from functional analysis in Hilbert space. Next, we represent the operations of centering, integration and bridge construction for Gaussian random functions in operator terms and prove several theorems about spectral equivalence of various processes and fields. The last section deals with the spectral properties of the kernel in the case of multivariate $\omega^2$-statistic.
The Gaussian processes and operators acting in the space of functions of one variable are denoted by capital letters, the multivariate Gaussian fields and the corresponding operators -- by bold letters. If we need to mention that, say, an operator $T$ acts on functions of variable $x_k$, we write $T_k$.
\section{Some functional analytic preliminaries}
The following statement is well known, see, e.g. \cite[Section 3.10]{BS10}.
\medskip
{\bf Proposition 1.} {\it Let $A$ and $B$ be compact operators in the Hilbert space $H$. Then the non-zero eigenvalues of the operators $AB$ and $BA$ coincide (with the multiplicities).
}
\medskip
Recall that if $A$ and $\widetilde A$ are operators in the Hilbert spaces $H$ and $\widetilde H$ respectively, we can define their tensor product $A\otimes\widetilde A$ in the Hilbert space $H\otimes\widetilde H$. If $A$ and $\widetilde A$ are integral operators with kernels ${\cal A}(x,y)$ and $\widetilde {\cal A}(s,t)$ then $A\otimes\widetilde A$ is also an integral operator with kernel $\mathbb{A}((x,s),(y,t))={\cal A}(x,y)\widetilde {\cal A}(s,t)$.
As the eigenvalues of the tensor product $A\otimes\widetilde A$ are the products of the eigenvalues $\lambda_i(A)$ and $\lambda_j(\widetilde A)$, the following statement is obvious.
\medskip
{\bf Proposition 2.} {\it Let $A_k$ and $B_k$ be compact operators in the Hilbert spaces $H_k$, $k=1,\dots,d$. Assume that non-zero eigenvalues of the operators $A_k$ and $B_k$ coincide (with the multiplicities) for every $k$. Then the non-zero eigenvalues of the tensor products
$$
A=\underset{k=1}{\overset{d}\otimes} A_k\qquad \text{and} \qquad B=\underset{k=1}{\overset{d}\otimes} B_k
$$
coincide (with the multiplicities).
}
\medskip
We define some operators in $L_2([0,1])$: operators of integration from the left and from the right
$$
(Tu)(x)=\int\limits_0^{x}u(t)\,dt, \qquad (T^*u)(x)=\int\limits_{x}^1 u(t)\,dt,
$$
the orthogonal projector onto the subspace of constants and the multiplication operators
$$
(Pu)(x)=\int\limits_0^1 u(t)\,dt, \qquad (S_fu)(x)=f(x)u(x).
$$
Also, we define multidimensional operators which are tensor products of the corresponding one-dimensional operators:
$$
{\bf T}=\underset{k=1}{\overset{d}\otimes} T_k, \qquad
{\bf T}^*=\underset{k=1}{\overset{d}\otimes} T_k^*, \qquad
{\bf P}=\underset{k=1}{\overset{d}\otimes} P_k
$$
Direct calculation shows that the covariance operator $K_W$ of the Wiener process $W(t)$ and the covariance operator $K_{W_1}$ of the inverted Wiener process $W_1(t)\equiv W(1-t)$ are given by
$$
K_W=TT^* \qquad \mbox{and} \qquad K_{W_1}=T^*T,
$$
respectively.
Furthermore, the covariance operator $K_B$ of the Brownian bridge allows for the following representation:
$$
K_B=T(I-P)T^*=T^*(I-P)T
$$
(here $I$ stands for the identity operator).
As a corollary, we obtain representations of the covariance operators of $d$-variate Brownian sheet ${\bf W}({\bf x})$, of $d$-variate inverted Brownian sheet ${\bf W}_1({\bf x})={\bf W}({\bf 1}-{\bf x})$, and of $d$-variate Brownian pillow ${\bf B}_*({\bf x})$ (each of them is the tensor product of the corresponding Gaussian processes, see \cite{KNN}):
$$
{\bf K}_{\bf W}={\bf T}{\bf T}^*, \qquad {\bf K}_{{\bf W}_1}={\bf T}^*{\bf T},\qquad {\bf K}_{{\bf B}_*}=
{\bf T}\cdot\underset{k=1}{\overset{d}\otimes}(I-P_k)\cdot{\bf T}^*={\bf T}^*\cdot\underset{k=1}{\overset{d}\otimes}(I-P_k)\cdot{\bf T}.
$$
The following statements can be also easily verified.
\medskip
{\bf Lemma 1}. {\it The covariance operator of $d$-dimensional pinned Brownian sheet
$$
{\bf B}({\bf x})={\bf W}({\bf x})-{\bf W}({\bf 1})\prod_{k=1}^d x_k
$$
has the following the representation}
$$
{\bf K}_{\bf B}={\bf T}({\bf I}-{\bf P}){\bf T}^*.
$$
{\bf Lemma 2}. {\it Let the Gaussian field $\bf X$ on $[0,1]^d$ have the covariance operator ${\bf K}_{\bf X}$. Then
1. The covariance operator of the centered field
$$
\overline{\bf X\vphantom{^1}}({\bf x})={\bf X}({\bf x})-\int\limits_{[0,1]^d}{\bf X}({\bf y})\,d{\bf y}
$$
has the following representation
$$
{\bf K}_{\overline{\bf X\vphantom{^1}}}=({\bf I}-{\bf P}){\bf K}_{\bf X}({\bf I}-{\bf P}).
$$
2. The covariance operators of the (left- and right-) integrated fields
$$
{\bf X}^{[0]}({\bf x})=\int\limits_0^{x_1}\dots\int\limits_0^{x_d}{\bf X}({\bf y})\,dy_1\dots dy_d \quad \text{and}\quad
{\bf X}^{[1]}({\bf x})=\int\limits_{x_1}^1\dots\int\limits_{x_d}^1{\bf X}({\bf y})\,dy_1\dots dy_d
$$
has the following representation}
$$
{\bf K}_{{\bf X}^{[0]}}={\bf T}{\bf K}_{\bf X}{\bf T}^*;\qquad {\bf K}_{{\bf X}^{[1]}}={\bf T}^*{\bf K}_{\bf X}{\bf T}.
$$
For the brevity we also introduce the notation for the integrated centered field:
$$
{\bf X}^{\{0\}}({\bf x})=\big({\overline{\bf X\vphantom{^1}}}\big)^{[0]}({\bf x}); \qquad
{\bf X}^{\{1\}}({\bf x})= \big({\overline{\bf X\vphantom{^1}}}\big)^{[1]}({\bf x}).
$$
\section{Spectral equivalence of certain Gaussian fields}
We begin with two generalizations of relation (\ref{Wa}).
Consider the stochastic integral of a non-random function $f$, see \cite[Ch.4, \S 2]{Rev},
$$
{\mathfrak F}_W(x) = \int\limits_0^x f(t) \,dW(t), \quad 0\leq x \leq 1.
$$
{\bf Theorem 1}. {\it Let $f\in L_2(0,1)$. Then the following relation is true:}
$$
\overline{{\mathfrak F}_W\vphantom{^1}}(x)\equiv {\mathfrak F}_W(x)-\int\limits_0^1{\mathfrak F}_W(y)\,dy \sim f(x)B(x).
$$
{\it Proof}. Lemmata 1 and 2 imply
$$
\aligned
K_{\overline{{\mathfrak F}_W\vphantom{^1}}}= &\, \big[(I-P)TS_f\big]\cdot\big[S_fT^*(I-P)\big];\\
K_{fB}=S_fT^*(I-P)TS_f= &\, \big[S_fT^*(I-P)\big]\cdot\big[(I-P)TS_f\big].
\endaligned
$$
The application of Proposition 1 completes the proof.
\hfill$\square$
\medskip
Further, we introduce a fractional counterpart of relation (\ref{Wa}). Recall that the Riemann--Liouville process is defined as the stochastic integral
$$
R_\alpha(x)=\alpha\int\limits_0^x (x-t)^{\alpha-1}\,dW(t),\qquad \alpha>\frac 12,
$$
see \cite[Sec. 3.2]{Lif} (the normalizing factor is chosen for convenience).
We also define the Riemann--Liouville bridge $R_\alpha^\circ(x)$ in a standard way
$$
R_\alpha^\circ(x)=R_\alpha(x)-x^{\alpha} R_\alpha(1).
$$
{\bf Theorem 2}. {\it The Riemann--Liouville bridge is spectrally equivalent to the centered Riemann--Liouville process:}
$$
R_\alpha^\circ(x)\sim \overline{R_\alpha\vphantom{^1}}(x).
$$
{\it Proof}.
It is easy to check that the covariance operators of the Riemann--Liouville process and the inverted Riemann--Liouville process have the following representations, respectively:
$$
K_{R_\alpha}=T_\alpha {T_\alpha}\!\!^*,\qquad K_{R_\alpha(1-x)}={T_\alpha}\!\!^*T_\alpha,
$$
where
$$
(T_\alpha u)(x)=\alpha\int\limits_0^x (x-t)^{\alpha-1}u(t)\,dt.
$$
Therefore, we have, as in Lemmata 1 and 2,
$$
\aligned
K_{R_\alpha^\circ}= T_\alpha(I-P){T_\alpha}\!\!^*= &\, \big[T_\alpha(I-P)\big]\cdot\big[(I-P)T_\alpha^*\big];\\
K_{\overline{R_\alpha\vphantom{^1}}(1-x)}= &\, \big[(I-P){T_\alpha}\!\!^*\big]\cdot\big[T_\alpha(I-P)\big],
\endaligned
$$
and Proposition 1 yields $R_\alpha^\circ(x)\sim \overline{R_\alpha\vphantom{^1}}(1-x)$. The relation $\overline{R_\alpha\vphantom{^1}}(x)\sim \overline{R_\alpha\vphantom{^1}}(1-x)$ is trivial by symmetry, and the statement follows.\hfill$\square$
\medskip
Turning to the Gaussian fields, we give a direct multivariate analog of the relation (\ref{Wa}).\medskip
{\bf Theorem 3}. {\it The pinned Brownian sheet is spectrally equivalent to the centered Brownian sheet:}
$$
{\bf B}({\bf x})\sim \overline{\bf W\vphantom{^1}}({\bf x}).
$$
{\it Proof}. For $d=1$, this relation reads $B\sim \overline{W\vphantom{^1}}$, see (\ref{Wa}). For $d=2$ it was proved in \cite{PecY} by the stochastic Fubini theorem, see (\ref{bivar}). In fact, Lemmata 1 and 2 imply for any $d$
$$
{\bf K}_{{\bf B}}={\bf T}({\bf I}-{\bf P}){\bf T}^*=\big[{\bf T}({\bf I}-{\bf P})\big]\cdot\big[({\bf I}-{\bf P}){\bf T}^*\big];
\quad {\bf K}_{\overline{{\bf W\vphantom{^1}}_1}}=\big[({\bf I}-{\bf P}){\bf T}^*\big]\cdot\big[{\bf T}({\bf I}-{\bf P})\big],
$$
and Proposition 1 yields ${\bf B}({\bf x})\sim \overline{{\bf W\vphantom{^1}}_1}({\bf x})=\overline{\bf W\vphantom{^1}}({\bf 1}-{\bf x})$. The relation $\overline{\bf W\vphantom{^1}}({\bf x})\sim \overline{\bf W\vphantom{^1}}({\bf 1}-{\bf x})$ is trivial by symmetry, and the statement follows.\hfill$\square$
\medskip
In the same way we derive a more general relation
\begin{equation}\label{a}
{\bf W}({\bf x})-a{\bf W}({\bf 1})\prod_{k=1}^d x_k
\sim {\bf W}({\bf x})-a\int\limits_{[0,1]^d}{\bf W}({\bf y})\,d{\bf y}, \qquad a\in\mathbb R.
\end{equation}
For $d=1$ it was proved in \cite{Dona}. If we denote the left-hand side of (\ref{a}) by ${\bf B}_a({\bf x})$ and the right-hand side by ${\bf W}_a({\bf x})$ then it is not difficult to see that
$$
{\bf K}_{{\bf B}_a}=\big[{\bf T}({\bf I}-a{\bf P})\big]\cdot\big[({\bf I}-a{\bf P}){\bf T}^*\big];
\quad {\bf K}_{{\bf W}_a({\bf 1}-{\bf x})}=\big[({\bf I}-a{\bf P}){\bf T}^*\big]\cdot\big[{\bf T}({\bf I}-a{\bf P})\big],
$$
and the statement follows.
\medskip
{\bf Remark 1}. Also it is not difficult to obtain a multivariate counterpart of Theorem 1 using the stochastic integral with respect to the Brownian sheet, see \cite{Cai, Doz},
$$
{\mathfrak F}_{\bf W}({\bf x}) = \int\limits_0^{x_1}\dots\int\limits_0^{x_d}f({\bf y})\,d{\bf W}({\bf y}), \quad {\bf x}\in [0,1]^d.
$$
It holds that
$$
\overline{{\mathfrak F}_{\bf W}\vphantom{^1}}({\bf x})\equiv {\mathfrak F}_{\bf W}({\bf x})-\int\limits_{[0,1]^d}{\mathfrak F}_{\bf W}({\bf y})\,d{\bf y} \sim f({\bf x}){\bf B}({\bf x}).
$$
\medskip
It is appropriate here to return to the bivariate norm identities written out in \cite{DPY}. Formula (3.26) reads:
\begin{equation}
\label{Paul}
{\bf B}_*({\bf x}) \sim {\bf W}({\bf x}) - \int\limits_0^1{\bf W}({\bf x})\,dx_1 - \int\limits_0^1{\bf W}({\bf x})\,dx_2 + \int\limits_0^1 \int\limits_0^1 {\bf W}({\bf x})\,dx_1dx_2,\quad {\bf x}\in [0,1]^2.
\end{equation}
To give a new proof of (\ref{Paul}), observe that the covariance operator
on the left-hand side is the tensor product of two covariance operators $K_B$. At the same time the covariance operator of the Gaussian field in the right side of (\ref{Paul}) is the tensor product of two covariance operators $K_{\overline{W\vphantom{^1}}}$. As $B$ and $\overline{W\vphantom{^1}}$ are spectrally equivalent, see (\ref{Wa}), it remains to apply our Proposition 2.
Quite similarly, in the $d$-variate case we have
$$
{\bf K}_{{\bf B}_*}=
{\bf T}\cdot\underset{k=1}{\overset{d}\otimes}(I-P_k)\cdot{\bf T}^*= \big[{\bf T}\cdot\underset{k=1}{\overset{d}\otimes}(I-P_k)\big]\cdot\big[\underset{k=1}{\overset{d}\otimes}(I-P_k)\cdot{\bf T}^*\big].
$$
Using Proposition 1, we obtain the field ${\bf Z}$ with the covariance operator
$$
{\bf K}_{\bf Z}=
\big[\underset{k=1}{\overset{d}\otimes}(I-P_k)\cdot{\bf T}^*\big]\cdot\big[{\bf T}\cdot\underset{k=1}{\overset{d}\otimes}(I-P_k)\big],
$$
spectrally equivalent to the $d$-variate Brownian pillow.
The direct expression of ${\bf Z}$ is more complicated and contains partial integrals with respect to all variables. For instance, the trivariate analog of (\ref{Paul}) reads
\begin{equation*}
\aligned
{\bf B}_*({\bf x}) \sim &\, {\bf W}({\bf x}) - \int\limits_0^1{\bf W}({\bf x})\,dx_1 - \int\limits_0^1{\bf W}({\bf x})\,dx_2 - \int\limits_0^1{\bf W}({\bf x})\,dx_3\\
+ &\, \int\limits_0^1 \int\limits_0^1 {\bf W}({\bf x})\,dx_1dx_2 + \int\limits_0^1 \int\limits_0^1 {\bf W}({\bf x})\,dx_1dx_3 + \int\limits_0^1 \int\limits_0^1 {\bf W}({\bf x})\,dx_2dx_3\\
- &\, \int\limits_0^1 \int\limits_0^1 \int\limits_0^1 {\bf W}({\bf x})\,dx_1dx_2dx_3,
\qquad {\bf x}\in [0,1]^3.
\endaligned
\end{equation*}
Another identity from \cite{DPY} concerns the Kiefer field ${\mathfrak K}$ whose covariance operator is the tensor product of the Wiener process and of the Brownian bridge covariances. Identity (3.27) reads
\begin{equation}
\label{Kie}
\mathfrak K({\bf x}) \sim {\bf W}({\bf x}) - \int\limits_0^1 {\bf W}({\bf x})\, dx_2, \qquad {\bf x} \in [0,1]^2.
\end{equation}
To establish the operator proof of (\ref{Kie}) we observe that the covariance operator in the right-hand side is the tensor product of the covariance operators of $W$ and $\overline{W}$. It remains to apply the spectral equivalence (\ref{Wa}) and Proposition 2.\medskip
{\bf Remark 2}. It is not difficult to derive some weighted analogs of obtained results, similar to \cite{DPY}.
\section{Integrated fields}
Now we pass to the integrated fields. \medskip
{\bf Theorem 4}. {\it The} ({\it left/right}){\it-integrated pinned Brownian sheet is spectrally equivalent to the centered} ({\it left/right}){\it-integrated Brownian sheet:}
\begin{equation}
\label{Int1}
{\bf B}^{[0]}({\bf x})\sim \overline{{\bf W}^{[0]}\vphantom{^{1^1}}}({\bf x});
\qquad {\bf B}^{[1]}({\bf x})\sim \overline{{\bf W}^{[1]}\vphantom{^{1^1}}}({\bf x}).
\end{equation}
{\it Proof}. We prove the first equivalence in (\ref{Int1}), the second one can be proved in the same way. For $d=1$, this relation reads $B^{[0]}\sim \overline{W^{[0]}\vphantom{^1}}$
and was discovered in \cite{BNO}. Once again, Lemmata 1 and 2 imply that for any $d$
$$
{\bf K}_{{\bf B}^{[0]}}={\bf T}^2({\bf I}-{\bf P}){\bf T}^{*2}=\big[{\bf T}^2({\bf I}-{\bf P})\big]\cdot\big[({\bf I}-{\bf P}){\bf T}^{*2}\big];
\quad {\bf K}_{\overline{{\bf W}^{[0]}\vphantom{^{1^1}}}({\bf 1}-{\bf x})}=\big[({\bf I}-{\bf P}){\bf T}^{*2}\big]\cdot\big[{\bf T}^2({\bf I}-{\bf P})],
$$
and Proposition 1 yields ${\bf B}^{[0]}({\bf x})\sim \overline{{\bf W}^{[0]}\vphantom{^{1^1}}}({\bf 1}-{\bf x})$. The relation $\overline{{\bf W}^{[0]}\vphantom{^{1^1}}}({\bf x})\sim \overline{{\bf W}^{[0]}\vphantom{^{1^1}}}({\bf 1}-{\bf x})$ is trivial by symmetry, and the statement follows.\hfill$\square$
\medskip
{\bf Remark 3}. Similar relations hold for $n$-times integrated fields, for instance,
$$
{\bf B}^{[0^n]}({\bf x})\sim \overline{{\bf W}^{[0^n]}\vphantom{^{1^1}}}({\bf x}).
$$
{\bf Theorem 5}. {\it The} ({\it left/right}){\it-integrated pinned centered Brownian sheet is spectrally equivalent to the centered} ({\it left/right}){\it-integrated centered Brownian sheet:}
\begin{equation}
\label{Int2}
{\bf B}^{\{0\}}({\bf x})\sim \overline{{\bf W}^{\{0\}}\vphantom{^{1^1}}}({\bf x});
\qquad {\bf B}^{\{1\}}({\bf x})\sim \overline{{\bf W}^{\{1\}}\vphantom{^{1^1}}}({\bf x}).
\end{equation}
{\it Proof}. We again restrict ourselves to the first equivalence in (\ref{Int2}). Lemmata 1 and 2 imply that for any $d$
$$
\aligned
{\bf K}_{{\bf B}^{\{0\}}}={\bf T}({\bf I}-{\bf P}){\bf T}({\bf I}-{\bf P}){\bf T}^*({\bf I}-{\bf P}){\bf T}^*= &\, \big[{\bf T}({\bf I}-{\bf P})\big]^2\cdot\big[({\bf I}-{\bf P}){\bf T}^*\big]^2;\\
{\bf K}_{\overline{{\bf W}^{\{0\}}\vphantom{^{1^1}}}({\bf 1}-{\bf x})}=({\bf I}-{\bf P}){\bf T}^*({\bf I}-{\bf P}){\bf T}^*{\bf T}({\bf I}-{\bf P}){\bf T}({\bf I}-{\bf P})= &\, \big[({\bf I}-{\bf P}){\bf T}^*\big]^2\cdot\big[{\bf T}({\bf I}-{\bf P})]^2,
\endaligned
$$
and Proposition 1 yields ${\bf B}^{\{0\}}({\bf x})\sim \overline{{\bf W}^{\{0\}}\vphantom{^{1^1}}}({\bf 1}-{\bf x})$. The relation $\overline{{\bf W}^{\{0\}}\vphantom{^{1^1}}}({\bf x})\sim \overline{{\bf W}^{\{0\}}\vphantom{^{1^1}}}({\bf 1}-{\bf x})$ is trivial by symmetry, and the statement follows.\hfill$\square$
\medskip
{\bf Remark 4}. Similar relations hold for $n$-times integrated fields, for instance,
$$
{\bf B}^{\{0^n\}}({\bf x})\sim \overline{{\bf W}^{\{0^n\}}\vphantom{^{1^1}}}({\bf x}).
$$
For $d=1$, this relation reads $B^{\{0^n\}}\sim \overline{W^{\{0^n\}}\vphantom{^1}}$ and was discovered in \cite[Sec. 4]{N09}.
\medskip
{\bf Theorem 6}. {\it The centered right-integrated pinned Brownian sheet is spectrally equi\-valent to the right-integrated centered Brownian sheet:}
$$
\overline{{\bf B}^{[1]}\vphantom{^{1^1}}}({\bf x})
\sim {\bf W}^{\{1\}}({\bf x}).
$$
{\it Proof}. For $d=1$ this relation reads $\overline{B^{[1]}\vphantom{^{1^1}}}\sim W^{\{1\}}$. Notice that, by symmetry of both Brownian bridge and centered Wiener process, we can change the right-integration to the left-integration that is not the case for $d>1$. The spectral equivalence $\overline{B^{[0]}\vphantom{^{1^1}}}\sim W^{\{0\}}$ was also first observed in \cite{BNO}.
Lemmata 1 and 2 imply that for any $d$
$$
\aligned
{\bf K}_{\overline{{\bf B}^{[1]}\vphantom{^{1^1}}}}= &\, \big[({\bf I}-{\bf P}){\bf T}^*\big]\cdot\big[{\bf T}({\bf I}-{\bf P}){\bf T}^*{\bf T}({\bf I}-{\bf P})\big];\\
{\bf K}_{{\bf W}^{\{1\}}({\bf 1-{\bf x}})}={\bf T}({\bf I}-{\bf P}){\bf T}^*{\bf T}({\bf I}-{\bf P}){\bf T}^*= &\, \big[{\bf T}({\bf I}-{\bf P}){\bf T}^*{\bf T}({\bf I}-{\bf P})\big]\cdot\big[({\bf I}-{\bf P}){\bf T}^*\big],
\endaligned
$$
and Proposition 1 yields $\overline{({\bf B})^{[1]}\vphantom{^{1^1}}}({\bf x})
\sim {\bf W}^{\{1\}}({\bf 1-{\bf x}})$. Once again, the statement follows by symmetry.\hfill$\square$ \medskip
{\bf Remark 5}. Similar (but more intricate) relations hold for multiply integrated fields, for instance,
$$
\overline{({\bf B})^{[011]}\vphantom{^{1^1}}}({\bf x})\sim \big(\overline{{\bf W}^{[0]}\vphantom{^{1^1}}}\big)^{[11]}({\bf x}).
$$
\section{Detrended processes of high order}
For the last example we restrict ourselves to the univariate case. For the Gaussian process $X$ on $[0,1]$, consider the $n$-th order detrended process, see \cite{Ai, Pe}:
$$
X_{\langle n\rangle}(t)=X(t)-\sum\limits_{j=0}^na_jt^j,
$$
where $a_j$ are defined by the relations
$$
\int\limits_0^1 X_{\langle n\rangle}(t)t^j\,dt=0,\qquad j=0,\dots,n.
$$
{\bf Theorem 7}. {\it The $n$-th order detrended $n$-times integrated Wiener process is spectrally equivalent to the conditional (``bridged'') $n$-times integrated Wiener process} \cite{Lach}:
$$
(W^{[0^n]})_{\langle n\rangle}(t)\sim \mathbb{B}_n(t)\equiv \Big(W^{[0^n]}(t)\ \Big|\ W^{[0^m]}(1)=0,\ \ m=0,\dots,n\Big).
$$
{\it Proof}. For $n=0$ this relation coincides with $\overline{W\vphantom{^1}}\sim B$. Note that the $n$-th order detrending operation can be considered as the projection onto the subspace of $L_2([0,1])$ orthogonal to the polynomials with degree not greater than $n$. Therefore, the covariance operator $K_{(W^{[0^n]})_{\langle n\rangle}}$ has the following representation
$$
K_{(W^{[0^n]})_{\langle n\rangle}}=\big[(I-P_{\langle n\rangle})T^n\big]\cdot\big[T^{*n}(I-P_{\langle n\rangle})\big],
$$
where $P_{\langle n\rangle}$ is the orthogonal projector in $L_2([0,1])$ onto the subspace ${\cal P}_n$ of polynomials with degree not greater than $n$.
On the other hand, the direct calculation shows that the covariance operator $K_{\mathbb{B}_n}$ can be written as
$$
K_{\mathbb{B}_n}=T^n(I-P_{\langle n\rangle})T^{*n}=\big[T^n(I-P_{\langle n\rangle})\big]\cdot\big[(I-P_{\langle n\rangle})T^{*n}\big]=\big[T^{*n}(I-P_{\langle n\rangle})\big]\cdot\big[(I-P_{\langle n\rangle})T^n\big]
$$
(the last equality holds by symmetry of $\mathbb{B}_n$), and the statement follows from Proposition 1.\hfill$\square$
\section{An application to goodness-of-fit tests}
Here we give an application of the obtained results to the classical goodness-of-fit problem. Consider the sample $X_1,...,X_n$ with continuous distribution function $F$ in $R^d$, $d\ge2$. We are testing the simple null hypothesis $H_0$: $F=F_0$. Via the well-known Rosenblatt transform \cite{Ro} we reduce testing $H_0$ to testing uniformity on the unit cube $[0,1]^d$ using the transformed sample $\mathcal{S} = ({\bf x}^1,\dots,{\bf x}^n)$.
Let $F_n$ be the empirical distribution function based on this sample. The famous test statistic $\omega_n^2$, see \cite{SW} for its history and properties, in our case has the form
$$
\omega_n^2 = \int\limits_{[0,1]^d} \Big( F_n ({\bf z}) - \prod_{i=1}^d z_i \Big)^2 d{\bf z},
$$
which can be also written as
\begin{equation}
\label{omega}
\omega_n^2 = \Big(\,\frac 13\,\Big)^d -\frac2n \sum_{{\bf x}\in \mathcal{S}}\prod_{k=1}^d \Big(\frac{1-x_k^2}{2}\Big) + \frac{1}{n^2}\sum_{{\bf x}, {\bf x}'\in \mathcal{S}}\prod_{k=1}^d \big(1 - \max\{x_k, x'_k\}\big).
\end{equation}
We can interpret (\ref{omega}) as a degenerate $V$-statistic which has the limiting distribution depending on the eigenvalues of its kernel
\begin{equation*}
{\cal Q}({\bf x,y})= \prod_{k=1}^d\big(1-\max\{x_k,y_k\}\big)-2^{-d}\prod_{k=1}^d(1-x_k^2)-2^{-d}\prod_{k=1}^d(1-y_k^2)+3^{-d},
\end{equation*}
see, e.g., \cite[Ch.4]{Koro}. An old and well-known problem consists in finding the spectrum of this kernel ${\cal Q}$, i.e. the eigenvalues of the problem
\begin{equation*}\label{eigen}
({\bf Q}u)({\bf x}):=\int\limits_{[0,1]^d}{\cal Q}({\bf x,y}) u({\bf y})\,d{\bf y}=\lambda u({\bf x}),\qquad {\bf x}\in[0,1]^d,
\end{equation*}
In particular, the first eigenvalue is of special interest because it is important for the Bahadur approximate efficiency calculation of the omega-square test \cite{Bah}. It is also indispensable when evaluating the logarithmic large deviation asymptotics of $\omega_n^2$ statistic.
It is not difficult to see that
\begin{equation*}
{\cal Q}({\bf x,y})={\cal K}_1({\bf x,y})-\int\limits_{[0,1]^d}{\cal K}_1({\bf x,y})\,d{\bf y}
-\int\limits_{[0,1]^d}{\cal K}_1({\bf x,y})\,d{\bf x}+\int\limits_{[0,1]^d}\int\limits_{[0,1]^d}{\cal K}_1({\bf x,y})\,d{\bf x}d{\bf y},
\end{equation*}
where
\begin{equation*}
{\cal K}_1({\bf x,y})=\prod_{k=1}^d\big(1-\max\{x_k,y_k\}\big)
\end{equation*}
is the covariance function of the inverted Brownian sheet ${\bf W}_1$. Therefore ${\bf Q}$ is the covariance operator of the centered inverted Brownian sheet $\overline{{\bf W}\vphantom{^1}_1}$.
It follows from the proof of Theorem 3 that $\overline{{\bf W}\vphantom{^1}_1}({\bf x}) \sim {\bf B}({\bf x})$. Thus, the spectrum of $\cal Q$ coincides with the spectrum of the pinned Brownian sheet ${\bf B}({\bf x})$ which was studied in many sources.
Durbin \cite{Dur} was the first to investigate the spectrum of ${\bf B}({\bf x})$ for $d=2$ and gave the list of the first $30$ reciprocal eigenvalues beginning with $\lambda_1^{-1}\approx 15.814...$. For $d=3$, the spectrum was described in \cite{Kri}, and Martynov reported the first $10$ reciprocal eigenvalues beginning by $\lambda_1^{-1}\approx 30.196...$ in \cite[\S 5]{Mart}. To the best of our knowledge, the numerical values of eigenvalues for $d>3$ are unknown.
\subsection*{Acknowledgement}
The authors are indebted to Professor I.A. Ibragimov and Professor M.A. Lifshits for valuable comments and suggestions. This work was supported by the Russian Foundation of Basic Research Grant 20-51-12004.
\small
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,652
|
Crash Bandicoot N Sane Trilogy Coming To PC
Everyone's favourite half naked marsupial moron, Crash is making his way to the PC at long last in the form of the Crash Bandicoot N. Sane Trilogy remasters. Originally released on PlayStation 4, the classic three games got the usual lick of paint and upgrades to bring them more in line with modern games. Check out the press release below for the usual chunk of text about an upcoming game release. Here's hoping he doesn't live up to his name once he's on our beloved platform.
Crash Bandicoot N. Sane Trilogy launches on PC on 10 July 2018.
WUMPA THERE IT IS! CRASH BANDICOOT TO MAKE HISTORY THIS SUMMER!
Crash Bandicoot™ N. Sane Trilogy — #1 Selling Remastered Collection in PS4™ History¹ – Arrives on Nintendo Switch™, Xbox One and Steam for the First Time on 10th July
London, UK – 9th March 2018 – Fans asked, and Activision, a wholly owned subsidiary of Activision Blizzard, Inc. (NASDAQ: ATVI), answered by bringing the Crash Bandicoot™ N. Sane Trilogy to all major gaming platforms. Already available on PlayStation® 4, Crash fans worldwide can experience the beloved '90s videogame icon like never before when everyone's favorite marsupial arrives on Nintendo Switch, Xbox One and Steam® on 10th July 2018. The arrival of the Crash Bandicoot N. Sane Trilogy on new platforms this summer marks the first time that the original three games – Crash Bandicoot™, Crash Bandicoot™ 2: Cortex Strikes Back, and Crash Bandicoot™ 3: Warped – will be playable on Nintendo, Microsoft and PC. The Crash Bandicoot N. Sane Trilogy is available now for pre-orders on all new platforms.
As recently announced as part of Nintendo Direct, Nintendo Switch players will make history as the first three original Crash Bandicoot games arrive on a Nintendo platform. With Nintendo Switch, fans can spin, jump, wump and repeat as they take on the epic challenges and adventures from the Crash Bandicoot N. Sane Trilogy either on the go in handheld mode or from the comfort of their living rooms in TV mode.
Giving fans more UMPH in their WUMP, 10th July also marks the first time that the original three Crash Bandicoot games will appear on the family of Xbox One devices from Microsoft, including the Xbox One X. The last Crash Bandicoot game that was available on an Xbox platform was 10 years ago. Now the Xbox community — that has been craving to get their hands on the trilogy — will soon be able to experience Crash on their preferred console.
What's more, Crash Bandicoot is making his debut on PC when the Crash Bandicoot N. Sane Trilogy launches on Steam. The release of the Crash Bandicoot N. Sane Trilogy on Steam ushers in a new way for gamers to play with a variety of control options including optimised Steam Controller support, as well as the ability to play with mouse and keyboard.
The Crash Bandicoot N. Sane Trilogy is available now on PlayStation 4. Fans can pre-order the Crash Bandicoot N. Sane Trilogy for all new platforms now. Check out Crashbandicoot.com for more details.
Activision and Vicarious Visions are honouring the heritage of Crash throughout the trilogy in a variety of ways. With more than 100 levels to explore, the Crash Bandicoot N. Sane Trilogy fully remastered game offers brand-new lighting, animations, textures, models and recreated cinematics—all in dazzling "N. Hanced Fur-K." The game's soundtrack is packed with all the didgeridoos, xylophones and thumpin' bass lines you can handle, as well as newly recorded dialogue from some of the familiar voice actors who appear in the original Crash Bandicoot games, including Jess Harnell, Lex Lang and Debi Derryberry, among others.
New features of the trilogy include playing as Coco, Crash's smart and spirited little sister who comes complete with her own set of special attacks. Full analog stick support, a unified save system and checkpoint system makes it easier for new fans to enjoy the classic adventures, while improved bonus levels and time trials in this epic trilogy will challenge the hardest of the 'Coot core!
For more information about the Crash Bandicoot N. Sane Trilogy follow @CrashBandicoot on Twitter, Facebook and Instagram.
Day of the Devs 2020
No related articles.
K-putt
Posted 12 Mar 2018, 13:49
Who would've thunk 50 years ago that we'd get a Bandicoot game on PC. Now Spyro please.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,467
|
#ifndef __ARCH_ARM_SRC_SAM34_SAM_UDP_H
#define __ARCH_ARM_SRC_SAM34_SAM_UDP_H
/************************************************************************************
* Included Files
************************************************************************************/
#include <nuttx/config.h>
#include <nuttx/usb/usbdev.h>
#include <stdint.h>
#include "chip.h"
#include "chip/sam_udp.h"
/************************************************************************************
* Public Functions
************************************************************************************/
#ifndef __ASSEMBLY__
#undef EXTERN
#if defined(__cplusplus)
#define EXTERN extern "C"
extern "C" {
#else
#define EXTERN extern
#endif
/************************************************************************************
* Name: sam_udp_suspend
*
* Description:
* Board logic must provide the sam_udp_suspend logic if the UDP driver is
* used. This function is called whenever the USB enters or leaves suspend mode.
*
* When 'resume' is false, this function call provides an opportunity to perform
* board-specific power-saving actions so that less power is consumed while the
* USB is suspended.
*
* Certain power-saving operations are performed by the UDP driver when it enters
* suspend mode: The USB device peripheral clocks are be switched off. MCK and
* UDPCK are switched off and the USB transceiver is disabled.
*
* When 'resume' is true, normal clocking and operations must all be restored.
*
************************************************************************************/
void sam_udp_suspend(FAR struct usbdev_s *dev, bool resume);
#undef EXTERN
#if defined(__cplusplus)
}
#endif
#endif /* __ASSEMBLY__ */
#endif /* __ARCH_ARM_SRC_SAM34_SAM_UDP_H */
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,422
|
THE CITY is an independent, nonprofit news outlet dedicated to hard-hitting reporting that serves the people of New York.
Reporting For
Open Newsroom
De Blasio Files
nycha July 26, 2019
Public Housing's Federal Monitor Presented a $20 Million Bill
By Greg B. Smith@Gregbsmithnyc
NYCHA federal monitor Bart Schwartz. Photo: Britney Anne Majure/NYCHAMonitor.com
The Housing Authority's new federal monitor initially proposed a first-year budget of nearly $20 million with little explanation of how all that taxpayer money would be spent, THE CITY has learned.
Bart Schwartz — who released his first report Monday — has brought in an extensive team of 19 employees from his firm, Guidepost Solutions. He's also recruited seven third-party experts, two consultant companies and the head of the Manhattan Democratic Committee for his "city expertise."
Five months after his appointment, however, Schwartz has yet to officially inform taxpayers how big of a check they'll be writing.
Housing Secretary Ben Carson announces a deal with Mayor Bill de Blasio in January to have a federal monitor. Photo: Ben Fractenberg/THE CITY
Sources familiar with the internal discussions about Schwartz's plans say he submitted a request last month for almost $20 million for his first year to the U.S. Department of Housing & Urban Development (HUD) and Manhattan U.S. Attorney's office.
He provided a brief document with a barebones explanation and few specifics on how the money would be spent, the sources said. The federal agencies returned it to him with instructions to provide a more extensive breakdown of his planned expenses.
Since then, he's come back to HUD and federal prosecutors with a far more detailed proposal, the sources said. But to date no budget has been publicly revealed and no contract has been submitted to the city comptroller to be registered. The final estimated cost remains a mystery.
Schwartz also has yet to provide City Hall with any details on his spending plan. Mayor Bill de Blasio agreed as part of a deal worked out in January that the city — not NYCHA — would foot the bill for the federal monitor and his team.
None of the governmental agencies that must sign off on Schwartz's budget — the U.S. Attorney, HUD and City Hall — would comment. Montieth Illingworth, a spokesperson for Schwartz, also declined comment.
Could Pay for Elevators and More
A budget of $20 million would cover the cost of installing seven new boilers at Brooklyn's Marcy Houses or replacing all 49 elevators in the sprawling Queensbridge North Houses in Long Island City.
Both of these planned projects are on hold because Gov. Andrew Cuomo has yet to release more than $450 million in promised state funds for NYCHA boiler and elevator fixes.
The Queensbridge Houses, May 14, 2019. Photo: Ben Fractenberg/THE CITY
Schwartz's behind-the-scenes budget dance takes place as Gregory Russ, the new $403,000-a-year NYCHA chairperson is set to arrive next month and as the authority struggles to clean up lead paint from apartments where children live. NYCHA, home to 400,000 New Yorkers, is in need of an estimated $32 billion in repairs.
The monitor's report released Monday noted that despite NYCHA's promises, more than 1,500 apartments targeted for cleanup still have not been declared "clear" of lead. At least 18 more children tested for high levels of lead in their blood, the report says.
Schwartz was appointed to settle a lawsuit filed last year by Manhattan U.S. Attorney Geoffrey Berman that documented how NYCHA managers for years covered-up unhealthy, unsafe apartment conditions such as lead paint, toxic mold and faulty elevators.
The monitor began work Feb. 28 and promised to provide City Hall with a detailed budget. But he kept delaying, and ultimately said he first needed to submit his plan to the Manhattan U.S. attorney, according to internal emails obtained by THE CITY.
On March 11, city Law Department staffers began requesting a budget from Anthony Colura, the chief financial officer of Schwartz's firm, Guidepost Solutions. Collura initially promised a budget "within the next two weeks," the emails show.
Minneapolis Public Housing Authority leader Gregory Russ is set to lead NYCHA. Photo: Minneapolis Public Housing Authority
That date came and went. On April 4, Georgia Pestana, chief of staff to Corporation Counsel Zachary Carter, asked again, this time stating city budget deadlines were fast approaching that required the Law Department to spell out its annual expenses.
Collura again promised he would come through by the week's end. Soon after, that changed to the "end of the month."
That came and went, and by May 23, Department of Law staffer Ashley Iodice wrote: "I was told today that our window to add your figures to our budget is rapidly closing. We need to get this information from you this week if possible."
Iodice got no response, so she tried again the next day, noting what she described as "the time crunch."
That's when another Guidepost executive revealed the Manhattan U.S. Attorney's Office was reviewing the budget, and promised to come through after they finished their analysis in a few days. That deadline passed, and on May 29, City Hall made one last plea:
"We need to submit to [the Office of Management and Budget] today," Iodice wrote. "Any update?"
A Growing Lineup
Sources familiar with the matter say, since then, City Hall has been speaking with federal prosecutors who expressed some concerns about the amount of money Schwartz says he needs and the lack of specificity about how he intends to spend it.
The final bill is likely to be significant, given the long list of lawyers, investigators and various experts Schwartz has already brought in. Besides himself and 19 of his Guidepost colleagues, he's hired a former Bloomberg-era deputy mayor, Anthony Coles, as an outside counsel.
He's also retained the services of ex-Harlem Assemblymember Keith Wright, the current head of the Manhattan Democratic Committee and a registered lobbyist with Davidoff Hutcher Citron LLP. Wright's job description is listed in Schwartz's report as "governmental communications and city expertise."
"Mr. Wright will be an important asset to the Monitor in his overall and on-going communications with the community, city and state leaders on matters relating to NYCHA's compliance with the Agreement," the report states.
Schwartz also has hired David Jacobs, a nationally known expert on lead paint; Carl Bornstein, a former inspector general of the city School Construction Authority: and two consultant firms, the U.K.-based Turner & Townsend and the New York-based Newmark-Knight, to provide construction manager advice.
Sign up for "THE CITY Scoop," our daily newsletter where we send you stories like this first thing in the morning.
Want to republish this story? See our republication guidelines.
nycha,
public housing,
gregory russ,
bart schwartz,
bill de blasio,
ben carson,
keep up with the city
Sign up for THE CITY Scoop, our daily newsletter, and stay on top of essential New York news.
New Dyslexia Risk Screening at Two Brooklyn Schools — For $2,000
Subway Door Problems Went Beyond Two Incidents, Records Show
LIC Library Reveals Looming Shut-Down Date, With No New Home
Or an idea to share? We want to hear from you in The Bronx, Brooklyn, Manhattan, Queens and Staten Island. Email us at tips@thecity.nyc.
traffic 6:08 PM
NYC Congestion Pricing Plan Needs Open Doors, Watchdogs Say
New York State's Committee on Open Government is urging the MTA to make congestion pricing process more transparent.
education 4:32 PM
Find Your NYC School's 2019 Graduation Rate
Chalkbeat
City students graduated at a record high last school year. Chalkbeat made a searchable database to find any public or charter school graduation rate.
ankle support Jan. 17, 2020
City Scrambling to Get Electronic Monitors for New Bail Rules
Even with new rules in effect, the city is looking at ankle bracelets and apps. Meanwhile, some defendants are being set free, sans monitoring.
bill's bills Jan. 16, 2020
Mayor's $95.3 Billion Proposed Budget Adds Costs, Not Programs
Despite no new big-ticket items, Bill de Blasio's no-frills fiscal plan piles on $3.1 billion personnel and other costs as state money woes loom.
wrong track Jan. 16, 2020
'Shoddy' Outreach to Homeless Led to Missed Subway Goals, Comptroller Says
A new audit from State Comptroller Thomas DiNapoli blames surge in subway homelessness on poor oversight of nonprofit tasked with outreach.
succession Jan. 16, 2020
Likely New Brooklyn Dems Boss Gets Big Bucks From Groups Her Bills Aid
Assemblymember Rodneyse Bichotte proves a fundraising powerhouse with a boost from doctors, lawyers and entrepreneurs who seek her help.
ny-9 Jan. 16, 2020
Two Congressional Hopefuls Sharing the Same Brooklyn Coworking Space
Politics makes strange office-fellows. Isiah James and Alex Hubbard, long-shots in the NY-9 race, have headquarters just steps away from each other.
nycha Jan. 15, 2020
HUD Secretary Carson Claims Fewer 'Horrible Things' in NYC Public Housing
The housing boss argued there's been "substantial improvement" at NYCHA — even as tab to fix the ailing home to 400k New Yorkers rises to $42 billion.
ill timed Jan. 15, 2020
Bloomberg's Firm Violated Sick Leave Law He Vetoed as Mayor, but Now Supports
The billionaire presidential hopeful's media company was cited for a policy requiring workers to provide a doctor's note after one or two sick days.
helping hands Jan. 15, 2020
Puerto Ricans in New York Help Homeland, Using Lessons from Maria
Grassroots aid efforts sprout following the Jan. 7 earthquake as local politicians pledge support and critics say Washington is MIA.
jails Jan. 14, 2020
Correction Board Seeks Probe Into 'Misuse' of Rikers Body Scanners
The city jails oversight body recommends "an immediate investigation" — citing "risk of radiation exposure" due to a lack of staff training.
the count Jan. 14, 2020
City Reveals $40 Million Plan of Attack on 2020 Census
The effort will rely on outreach to traditionally undercounted communities with Congressional representation and federal funding at stake.
immigration Jan. 14, 2020
Some Undocumented Students Find Letdown at End of NY DREAM Act Rainbow
The NY DREAM Act opened college help to undocumented students. But the confusing process ended with unexpected — and unexplained — bad news for some.
stand clear Jan. 14, 2020
Rising Reports of Subway Door Surprise Openings Still Rare But Jarring
After MTA pulled new cars because of door issues, THE CITY found 139 reports system-wide over the last two years — including 64 in 2019.
schools Jan. 13, 2020
State Could Ease Requirements for Officers Hearing Special Education Cases
To help clear over 10,000 backlogged cases, New York state officials are considering eliminating requirement of law license — and upping pay.
© 2020, THE CITY. ALL RIGHTS RESERVED
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,261
|
Universitět () je stanice moskevského metra Sokolnické linky. Svůj název má podle známé univerzity.
Charakter stanice
Stanice je ražená, trojlodní, založená hluboko pod zemí, 26,5 m. Její střední loď má průměr 9,5 m, boční lodě pak 8,5 m. Všechny tři lodě jsou spojené prostupy a podpírané pak masivními pilíři.
Výstupy jsou dva, jeden vede z prostředku střední lodě kolmo přes jednu z kolejí a druhý vychází přímo z jednoho konce této lodě. Vedou po eskalátorových tunelech, každý do vlastního vestibulu. Oba vestibuly jsou povrchové a mají kruhový půdorys; svým vzhledem připomínají válce.
Obklad stanice tvoří bílé a černé keramické dlaždice (u stěn za nástupištěm), pilíře samotné jsou obložené bílým mramorem. Podlahu pak tvoří standardně žula.
Universitet byl otevřen 12. ledna 1959 jako poslední stanice úseku Park Kultury – Universitet. Do roku 1963 plnil funkci konečné stanice, jižním směrem za nástupištěm se nacházejí dvě odstavné koleje, které mohou být případně využity i dnes.
Externí odkazy
Profil stanice na stránkách Mymetro.ru (rusky)
Fotogalerie na stránkách Metrowalks.ru (rusky)
Informace na stránkách metro.ru, včetně fotografie vestibulu (rusky)
Stanice metra v Moskvě
Postaveno v Rusku 1959
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,344
|
% osx_joystick mask initialization callback helper function
% This function should not be called directly.
function [JoyLocKey,pA,pB,pP,pO,label,mss,vals] = osx_joystick_MaskInitFcn( blk, cbA, cbB, cbP, cbO )
% ASSUMPTION: UserData has been validated by LoadFcn
ud = get_param( blk, 'UserData' );
vals = get_param( blk, 'MaskValues' );
mss = ud.MaskStyleString;
% If we have selected the empty (NULL) joystick
if ud.SelectedJoystick==0 || isempty(ud.list) || ud.SelectedJoystick > size(ud.list,1)
JoyLocKey = int32( 0 );
vals{1} = '0: None';
if cbA; pA=1; else pA=0; end
if cbB; pB=1; else pB=0; end
if cbP; pP=1; else pP=0; end
if cbO; pO=1; else pO=0; end
else
JoyLocKey = ud.list{ ud.SelectedJoystick, 2 };
vals{1} = ud.list{ ud.SelectedJoystick, 1 };
if ud.sizes(1); pA=1; else pA=0; end
if ud.sizes(2); pB=1; else pB=0; end
if ud.sizes(3); pP=1; else pP=0; end
if cbO; pO=1; else pO=0; end
end
if ud.saving
ud.saving = 0;
vals{1} = '0: None';
set_param( blk, 'UserData', ud );
end
% Update the port labels
label = sprintf('image( imread( ''osx-sl-joystick.png'') );\n');
portnum = 1;
if pA
if ud.SelectedJoystick~=0; dims=sprintf(' [%i]',ud.sizes(1)); else dims=''; end
label = [label, sprintf('port_label(''output'',%i,''Axes%s'');\n',portnum,dims)];
portnum = portnum+1;
end
if pB
if ud.SelectedJoystick~=0; dims=sprintf(' [%i]',ud.sizes(2)); else dims=''; end
label = [label, sprintf('port_label(''output'',%i,''Buttons%s'');\n',portnum,dims)];
portnum = portnum+1;
end
if pP
if ud.SelectedJoystick~=0; dims=sprintf(' [%i]',ud.sizes(3)); else dims=''; end
label = [label, sprintf('port_label(''output'',%i,''POVs%s'');\n',portnum,dims)];
end
if pO
if ud.SelectedJoystick~=0; dims=sprintf(' [%i]',ud.sizes(4)); else dims=''; end
label = [label, sprintf('port_label(''input'',1,''Outputs%s'');\n',dims)];
end
% Copyright (c) 2012, Zebb Prime and The University of Adelaide
% All rights reserved.
%
% Redistribution and use in source and binary forms, with or without
% modification, are permitted provided that the following conditions are met:
% * Redistributions of source code must retain the above copyright
% notice, this list of conditions and the following disclaimer.
% * Redistributions in binary form must reproduce the above copyright
% notice, this list of conditions and the following disclaimer in the
% documentation and/or other materials provided with the distribution.
% * Neither the name of the organization nor the
% names of its contributors may be used to endorse or promote products
% derived from this software without specific prior written permission.
%
% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
% ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
% WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
% DISCLAIMED. IN NO EVENT SHALL ZEBB PRIME OR THE UNIVERSITY OF ADELAIDE BE LIABLE FOR ANY
% DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
% (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
% LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
% ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
% (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
% SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,703
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Sulphur Springs – miasto w Stanach Zjednoczonych, w stanie Teksas, w hrabstwie Hopkins. W 2000 roku liczyło 14 551 mieszkańców.
Z Sulphur Springs pochodzi Colleen Hoover, amerykańska pisarka.
Przypisy
Miasta w stanie Teksas
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,455
|
Q: What can I do about 99% disk usage (only 0.9 MB/s)? My new laptop is two months old, the first day I got this laptop I noticed this but I assumed it was a bug in Windows 8. However I still have this issue today after reformatting and installing all updates.
The problem is that the disk shows 99% usage from an application...
In this case Dropbox, but the usage is never over 2 MB/s; in this case it is 0.8 MB/s.
What can I do about this?
A: I've seen a similar behavior on some computers at work. You have a pretty good chance to be infected with a trojan malware. I'll suggest to run malware and virus scans on your PC.
A: I had the same issue. It got resolved after updating Intel® Chipset Device Software. Download this from Intel website for your chipset and install to see if it resolves the issue.
This link will help get the right drivers http://www.intel.com/content/www/us/en/support/detect.html
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,264
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Lleida (Spaans: Lérida) is een stad en gemeente in de Spaanse provincie Lleida. Lleida ligt in het westen van de autonome regio Catalonië, op 155 meter boven zeeniveau, en beslaat een oppervlakte van 212 km². In 2013 had de stad een inwonersaantal van 139.809. Het is ook de hoofdstad van de comarca Segrià. Lleida heeft een eigen universiteit en een groot aantal professionele sportverenigingen.
Geschiedenis
Ooit was Lleida een nederzetting van een Iberisch volk, genaamd de Ausetanen. De toenmalige koning heette Mandoni, en hij verdedigde de stad gedurende lange tijd tegen de inval van de Romeinen. In het jaar 205 v. Chr viel de stad desondanks in Romeinse handen en ze kreeg de naam 'Llerda'. Aan het einde van de 3e eeuw kreeg de stad een stenen brug als verdediging, deze werd echter vernield door Germaanse barbaren, jaren later. Rond het jaar 716 werd de stad veroverd door de Moren, en kreeg de naam Lareda. In 1149 kwam de stad in handen van de Catalaanse graaf Ramón Berenguer IV. In 1300 werd de universiteit van Lleida opgericht, de eerste stad van het toenmalige Rijk van Aragón. In 1717 werd de universiteit echter gesloten en verwoest door koning Felipe V, om zo de Catalaanse onafhankelijkheid en macht te onderdrukken. Ook andere Catalaanse instituties in Lleida werden compleet vernietigd. Uit deze periode vindt men nu nog alleen de kathedraal "La Seu Vella".
Tijdens de Spaanse Burgeroorlog werd Lleida gebruikt als een verdedigingswal tegen de Spaanse troepen van Francisco Franco voor de rest van Catalonië. De stad werd dan ook zwaar gebombardeerd en kwam uiteindelijk onder het dictatoriale regime van Franco terecht.
Demografische ontwikkeling
Bron: INE; 1857-2011: volkstellingen
Opm: Bevolkingscijfers in duizendtallen
Taal
Lleida is een van de steden binnen Catalonië waar het meest Catalaans wordt gesproken. Het Catalaans wordt in deze streek met een onmiskenbaar accent gesproken, wat bekendstaat als het dialect Noordwest Catalaans, of in de volksmond 'lleidatà'. Het gaat dan vooral om verschillen met het 'Standaardcatalaans' qua uitspraak, maar er zijn ook een aantal grammaticale verschillen te onderscheiden. De meerderheid van de bevolking is niettemin tweetalig en spreekt ook Spaans.
Klimaat
Lleida heeft een landklimaat en géén mediterraan klimaat. De stad heeft daarmee een behoorlijk ander weerbeeld dan de Middellandse Zeekust en steden zoals Barcelona en Tarragona. Dat betekent dat de zomers erg heet zijn met temperaturen die soms weken achter elkaar boven de 35 graden stijgen. De winters zijn echter koud, met minimumtemperaturen rond en soms onder het vriespunt, iets wat aan de kust zeer zeldzaam is. De lucht is droog en er valt zeer weinig neerslag in Lleida en omgeving.
Ligging
Afstanden naar andere steden
Andorra: 152 kilometer
Barcelona: 160 kilometer
Bilbao: 448 kilometer
Gerona: 215 kilometer
Madrid: 470 kilometer
Tarragona: 101 kilometer
Toulouse: 295 kilometer
Zaragoza: 149 kilometer
Cultuur
In Lleida worden een groot aantal traditionele Catalaanse feesten gevierd. De belangrijkste daarvan zijn het "Festa Major" en de "Fira de Sant Miquel", waar mensen uit de verre omgeving jaarlijks op af komen. Verder organiseert de stad elk jaar het Festival van Latijns-Amerikaanse films en in mei vindt Animac, een festival van animatiefilms, plaats.
Bezienswaardigheden
De kathedraal "La Seu Vella" is een kerk die zowel in gotische als romaanse stijl is gebouwd. In de 18e eeuw werd de kathedraal als militaire uitvalsbasis gebruikt.
Het historisch museum "Institut d'Estudis Ilerdencs" is een van de hoogtepunten van de stad. Het was van oudsher een ziekenhuis, gebouwd in gotische stijl.
Het stadhuis "La Paeria" is een historisch monument met intacte overblijfselen uit de middeleeuwen en uit Romeinse en Moorse tijden.
Het kasteel "Gardeny" dat in de middeleeuwen werd gebruikt door de Tempeliers
De tuinen "Camps Elisis" (Catalaans voor "Champs Élysées", de beroemde lanen in Parijs). Deze tuinen werden al gebruikt in de tijd van de Romeinen. In de tuin bevindt zich de fontein van de sirene.
Het bisschopspaleis, waarin men een museum vindt dat romaanse en barokarchitectuur tentoonstelt.
Transport
Lleida heeft een eigen station: Lleida Pirineus, en is aangesloten op de zogenaamde AVE, een hogesnelheidslijn naar Madrid, die onderweg stopt in Zaragoza, Calatayud en Guadalajara. Sinds 20 februari 2008 gaat deze hogesnelheidslijn via Tarragona door naar Barcelona. Daarnaast bestaan er reguliere treinverbindingen met de Catalaanse hoofdstad.
De stad beschikt sinds 17 januari 2010 over een eigen vliegveld "Lleida-Alguaire", gelegen op 15 kilometer van de stad in de gemeente Alguaire.
Partnersteden
Ferrara (Italië)
Foix (Frankrijk)
Lérida (Colombia)
Geboren in Lleida
Enrique Granados (1867-1916), pianist/componist
Ricardo Viñes (1875-1943), pianist
Enric Gensana (1936-2005), voetballer
Núria Añó (1973), schrijver
Emilio Alzamora (1973), motorcoureur
Sergi Escobar Roure (1974), wielrenner
Albert Costa (1975), tennisser
Xavier Estrada (1976), voetbalscheidsrechter
Sergej Milinković-Savić (1995), Servisch-Spaans voetballer
Zie ook
Provincie Lleida
|
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Asiaen
18&UP
The Number of Māori Babies Being Taken by the State is on the Rise in New Zealand
For every 10,000 Māoris born in New Zealand in 2018, 102 were taken away from their parents—more than quadruple the rate for the rest of the population.
by Gavin Butler
Image via Wikimedia
Last week, the New Zealand state tried to remove a newborn Māori baby from his family. The boy—whose parents were allegedly prone to domestic violence and transient housing arrangements—was deemed "high risk" by authorities at Hawkes Bay Hospital, in the north island city of Hastings, based on limited evidence and an order granted by the Family Court. Police and officials from the Ministry for Children attempted to "uplift" the child, citing concern for his wellbeing, but were ultimately rebuffed by the young mother's family, midwives, and iwi (tribe).
"I felt like I was in jail and I just got released… [but] I felt good walking out of the hospital with baby," the 19-year-old mother told Newsroom. "It's a shame that I couldn't find this help when [the Ministry for Children] took my last baby."
The young Māori woman had another newborn "uplifted" by authorities last year—and every week, around the country, three Māori babies are being taken into state custody.
It's a number that's on the rise, according to The Conversation. Between 2015 and 2018, the number of Māori babies removed by the New Zealand state increased by 33 percent. For every 10,000 Māoris born in 2018, 102 were removed—more than quadruple the rate for the rest of the population. And despite being subject to a royal commission of inquiry into the abuse of children in its care, the state continues to insist it can do a better job of looking after these young children than the hundreds of Māori parents it intervenes upon.
Chief district court judge Jan-Marie Doogue suggests otherwise. Speaking at the Law Foundation's Ethel Benjamin Commemorative Address last year, Jan-Marie asserted that placing children in state care can greatly increase their risk of becoming criminals and chronic offenders, Radio New Zealand reported.
"What is clear to Family Court Judges is that state care can have disastrous implications on a child or young person's development and can greatly increase the risk of future offending," she said. "A staggering two-thirds of young people in youth justice residences meet the criteria for substance abuse disorder. Even more are reported to be heavy drinkers. These figures are worse for Māori who, on average, start using alcohol and drugs from an earlier age."
Jan-Marie went on to declare that "our judges will need to be able to both understand the disadvantage that those children and young persons who come into the Court have faced, as well as recognise how their whānau [extended family], hapū [collection of extended family groups], and iwi can be part of the solution."
While the problem of young children being taken from their families appears to be on the rise in New Zealand, the government does seem to acknowledge that this recognition of whānau, hapū, and iwi might go some way toward addressing Māori rights and interests in child welfare. Legislation set to take effect on July 1 intends to emphasise the obligation of the Ministry for Children to involve these groups in decision-making, and to recognise a child's wider family—rather than just the parents—in care arrangements.
While last week's incident mercifully ended with the family's own arrangements prevailing—the mother and baby will now go to a safe environment for young mothers, where they'll have the constant support of whānau—the altercation could likely have been avoided if the Ministry for Children had taken a broader, more generous view of the situation. As Dominic O'Sullivan, Associate Professor of Political Science at Charles Sturt University, notes: "the important moral and political principle is that the family, except when it is demonstrably and irreparably dysfunctional, is prior to the state."
Midwife Jean Te Huia, for one, believes the whole ordeal was completely unnecessary. "To me this situation was clearly the result of Oranga Tamariki [the Ministry for Children] believing they are a law unto themselves," she said. "[The Ministry for Children are] not willing to work with the whānau or professional midwives, and clearly and evidently not committed to working with parents and families and a range of professionals to fulfil their duty to keep children safe."
Follow Gavin on Twitter or Instagram
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\section{Introduction}
\par
An ultracold Fermi gas is well known as a system with high
tunability of various physical parameters\cite{Gurarie,Bloch}.
For example, one can experimentally tune the strength of a pairing interaction associated with a
Feshbach resonance.
This has enabled us to systematically study how superfluid properties
continuously change from the weak-coupling BCS (Bardeen-Cooper-Schrieffer
)-type to the BEC (Bose-Einstein condensation) of tightly bound
molecular boson with increasing the interaction strength, which
is also referred to as the BCS-BEC crossover in the literature.
In the crossover region, pairing fluctuations are expected to be strong
near the superfluid phase transition temperature $T_{\rm c}$, so that the
so-called pseudogap phenomenon has been discussed there\cite{Stewart,Gaebler}.
\par
Another example of the high tunability is the realization of a low
dimensional Fermi gas by using an optical lattice technique.
Since pairing fluctuations are enhanced by the low dimensionality of the
system, together with the tunable pairing interaction, this is also use
for the study of strong-coupling physics in a systematic manner.
In particular, two-dimensional Fermi gases have recently attracted much
attention in this
field\cite{Martiyanov,Feld,Frohlich,Sommer,Makhalov,Ries,Murthy2,Fenech},
because the quasi-long-range superfluid order, called the
Berezinskii-Kosterlitz-Thouless (BKT) phase is expected there\cite{Berezinskii,Kosterlitz}.
Indeed, various physical quantities, such as photoemission
spectra\cite{Feld} , as well as thermodynamic
quantities\cite{Makhalov,Fenech}, have been measured in this system, and
the observation of BKT transition has recently been reported in a
two-dimensional $^6$Li Fermi gas\cite{Ries,Murthy2}.
\par
In the three-dimensional case, a (non-self-consistent) $T$-matrix
approximation (the detail of which is explained in Sec. 2) has
been
extensively
used to successfully explain various interesting phenomena
observed in the BCS-BEC crossover region \cite{Tsuchiya,Chen,HuiHu}.
In this regard, however, when TMA is applied to the two-dimensional
case, while it is valid for the strong-coupling regime, it overestimates
strong-coupling effects in the weak-coupling
regime\cite{Marsiglio,Matsumoto1}.
For example, TMA does not give free-particle density of states even in
the weak-coupling regime, when the pairing interaction is very weak.
Thus, in the current stage of research, it is a crucial theoretical
issue to improve TMA so that one can correctly deal with the
weak-coupling regime.
\par
In this paper, we show that the self-consistent $T$-matrix approximation
(SCTMA), which involves higher-order pairing fluctuations than TMA,
meets our demand.
Within this framework, we clarify how the so-called pseudogap phenomenon
disappears in a two-dimensional Fermi gas as one approaches the
weak-coupling regime from the strong-coupling side.
Comparing SCTMA results with TMA ones, we also discuss the reason why
the above mentioned problem in TMA can be eliminated in SCTMA.
\par
Throughout this paper, we take $\hbar =k_{\rm B}=1$ and the two-dimensional system area is taken to be unity, for simplicity.
\par
\section{Formulation}
\par
We consider a two-dimensional uniform Fermi atomic gas consisting of two atomic hyperfine states, described by the BCS Hamiltonian,
\begin{align}
H=\sum_{\bm{p},\sigma}\xi_{\bm{p}}c^{\dagger}_{\bm{p},\sigma}c_{\bm{p},\sigma}
-U\sum_{\bm{p},\bm{p}'\bm{q}}c^{\dagger}_{\bm{p}+\bm{q}/2,\uparrow}c^{\dagger}_{-\bm{p}+\bm{q}/2,\downarrow}
c_{-\bm{p}'+\bm{q}/2,\downarrow}c_{\bm{p}'+\bm{q}/2,\uparrow}. \label{Hamiltonian}
\end{align}
Here, $c^{\dagger}_{\bm{p},\sigma}$ is a creation operator of a Fermi
atom with pseudospin $\sigma =\uparrow,\downarrow$ and the two-dimensional
momentum $\bm{p}=(p_{x},p_{y})$. $\xi_{\bm p}=p^2/(2m)-\mu$ is the
kinetic energy, measured from the Fermi chemical potential $\mu$, where
$m$ is an atomic mass. The pairing interaction $-U$ ($<0$) is assumed to
be tunable by adjusting the threshold energy of a Feshbach resonance. As
usual, we measure the interaction strength in terms of the
two-dimensional $s$-wave scattering length $a_{2{\rm D}}$, which is
related to $U$ as\cite{Morgan},
\begin{equation}
\frac{1}{U}=\frac{m}{2\pi}\ln{(k_{\rm F}a_{2{\rm D}})}+\sum_{p\geq
k_{\rm F}}\frac{m}{p^2}, \label{U}
\end{equation}
where $k_{\rm F}=\sqrt{2\pi N}$ is
the Fermi momentum, with $N$ being the total number of Fermi atoms.
Using this scale, $\ln{(k_{\rm F}a_{2{\rm D}})} \ll
-1$ ($ \gg 1$)
corresponds to the strong-coupling (weak-coupling) regime.
$-1\lesssim \ln{(k_{\rm F}a_{2{\rm D}})} \lesssim 1$ is the crossover region.
\par
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\textwidth]{fig1.eps}
\caption{
(a) Self-energy $\Sigma({\bm p},i\omega_n)$ in the self-consistent
$T$-matrix approximation (SCTMA).
The double-solid line shows the dressed Green's function $G$ in SCTMA.
The dotted line represents the pairing interaction $-U$.
We also show the self-energy $\Sigma^{\rm TMA}({\bm p},i\omega_n)$ in
the non-self-consistent $T$-matrix approximation (TMA) in (b),
where all $G$
appearing in (a) are replaced
by the free propagator $G_{0}=1/(i\omega_{n}-\xi_{\bm{p}})$.}
\label{diagram}
\end{center}
\end{figure}
\par
Many-body corrections to Fermi single-particle excitations can be
conveniently incorporated into the theory by considering the self-energy $\Sigma(\bm{p},i\omega_{n})$ in the single-particle thermal Green's function,
\begin{equation}
G(\bm{p},i\omega_{n})=
{1 \over i\omega_n-\xi_{\bm p}-\Sigma(\bm{p},i\omega_{n})}.
\label{Green}
\end{equation}
Here, $\omega_n$ is the fermion Matsubara frequency. The self-energy
$\Sigma(\bm{p},i\omega_{n})$ in the
self-consistent $T$-matrix approximation (SCTMA) is diagrammatically described as
Fig. \ref{diagram}(a), which gives \cite{Haussmann,Bauer,Mulkerin},
\begin{equation}
\Sigma({\bm p},i\omega_n)
=T\sum_{\bm{q},i\nu_{n}}
\Gamma(\bm{q},i\nu_{n})G(\bm{q}-\bm{p},i\nu_{n}-i\omega_{n}).
\label{Sigma}
\end{equation}
Here, $\nu_{n}$ is the boson Matsubara frequency.
The particle-particle scattering matrix $\Gamma(\bm{q},i\nu_{n})$ in
SCTMA has the form (see the second line in Fig.\ref{diagram}(a))
\begin{equation}
\Gamma(\bm{q},i\nu_{n})=-\frac{U}{1-U\Pi(\bm{q},i\nu_{n})},
\label{Gamma}
\end{equation}
where
\begin{equation}
\Pi(\bm{q},i\nu_{n})=T\sum_{\bm{p},i\omega_{n}}G\left(\bm{p}+\frac{\bm{q}}{2},i\nu_{n}+i\omega_{n}\right)G\left(-\bm{p}+\frac{\bm{q}}{2},-i\omega_{n}\right)
\label{PII}
\end{equation}
is a pair-correlation function, describing fluctuations in the Cooper
channel.
\par
The self-energy $\Sigma^{\rm TMA}(\bm{p},i\omega_{n})$ in the non-self-consistent $T$-matrix
approximation (TMA) is
given by replacing all the
the dressed Green's functions in the SCTMA $\Sigma(\bm{p},i\omega_{n})$
by the
free Fermi Green's functions
$G_{0}(\bm{p},i\omega_{n})=(i\omega_{n}-\xi_{\bm{p}})^{-1}$, as shown in
Fig. \ref{diagram} (b).
That is,
\begin{equation}
\Sigma^{\rm TMA}({\bm p},i\omega_n)
=T\sum_{\bm{q},i\nu_{n}}
\Gamma^{\rm
TMA}(\bm{q},i\nu_{n})G_0(\bm{q}-\bm{p},i\nu_{n}-i\omega_{n}),
\label{Sigma_TMA}
\end{equation}
where
$
\Gamma^{\rm TMA}(\bm{q},i\nu_{n})=-U/(1-U\Pi^{\rm
TMA}(\bm{q},i\nu_{n}))
$
and the TMA pair correlation function is given by
$
\Pi^{\rm
TMA}(\bm{q},i\nu_{n})=T\sum_{\bm{p},i\omega_{n}}G_0\left(\bm{p}+\frac{\bm{q}}{2},i\nu_{n}+i\omega_{n}\right)G_0\left(-\bm{p}+\frac{\bm{q}}{2},-i\omega_{n}\right)
$.
Because of this simplification, in contrast to SCTMA,
strong coupling corrections to Fermi single-particle excitations, as
well as the resulting pseudogap phenomenon, are completely ignored in
evaluating the TMA particle-particle scattering matrix $\Gamma^{\rm
TMA}(\bm{q},i\nu_{n})$.
We will show how this ignorance leads to the overestimation of the
pseudogap phenomenon in the weak-coupling case when $\ln{(k_{\rm
F}a_{2{\rm D}})}\gtrsim 0$.
\par
In both SCTMA and TMA, the Fermi chemical potential $\mu$ is determined from
the equation of the total number $N$ of Fermi atoms,
\begin{equation}
N=2T\sum_{\bm{p},i\omega_{n}}G(\bm{p},i\omega_{n}).
\label{Number}
\end{equation}
We then examine the pseudogap appearing in the single-particle
density of states $\rho(\omega)$, given
by
\begin{equation}
\rho(\omega)=-\frac{1}{\pi}\sum_{\bm{p}}{\rm
Im}G(\bm{p},i\omega_{n}\rightarrow \omega +i\delta). \label{Dos}
\end{equation}
\par
We briefly note that neither SCTMA nor TMA can describe the BKT phase
transition temperature $T_{\rm BKT}$.
Thus, this paper only deals with the normal phase above $T_{\rm BKT}$.
\par
\section{Pseudogap Phenomena in the crossover regime}
\par
\par
\par
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.54\textwidth]{fig2.eps}
\caption{ Calculated density of states $\rho(\omega)$ in a
two-dimensional Fermi gas.
The solid line and the dashed line show the results in SCTMA and TMA,
respectively.
$\rho_{0}=m/2\pi$ is the density of state in a two-dimensional free
Fermi gas.
We set $\ln{(k_{\rm F}a_{2{\rm D}})}=0.57$, and
$T=T^{\rm exp}_{\rm BKT}=0.146T_{\rm F}$, where
$T^{\rm exp}_{\rm BKT}$ is the observed BKT phase transition
temperature at this interaction strength in a $^6$Li Fermi
gas\cite{Ries,Murthy2}. (Color figure online.)}
\label{Dos_Tbkt}
\end{center}
\end{figure}
\par
\par
Figure \ref{Dos_Tbkt} shows the density of states (DOS) $\rho(\omega)$
in a two-dimensional Fermi gas,
when
$\ln{(k_{\rm F}a_{2{\rm D}})}=0.57$ (in the crossover region) at the
observed BKT phase transition temperature $T^{\rm exp}_{\rm
BKT}=0.146T_{\rm F}$ (where $T_{\rm F}$ is
the Fermi temperature) in a $^6$Li Fermi gas\cite{Ries,Murthy2}.
We see that SCTMA gives a pseudogap, that is, a dip structure
around $\omega=0$.
As discussed in the three-dimensional case \cite{Tsuchiya}, this dip
structure originates from pairing fluctuations around the Fermi surface,
and the resulting formation of preformed Cooper pairs.
Such an anomalous structure is also seen in the case of TMA, as shown in
Fig. \ref{Dos_Tbkt}.
However, the pseudogap structure in this case is much more remarkable
than that in the case of SCTMA, and the overall structure is rather
close to the BCS-type superfluid density of states
with the coherence peaks of the gaps
edges (although the system in the present case is still in the normal
state).
At this interaction strength, the binding energy $E_{\rm b}=1/(ma_{\rm
2D}^2)$
of a two-body bound state equals $E_{\rm b}=0.64\varepsilon_{\rm F}$
(where $\varepsilon_{\rm F}$ is the Fermi energy).
While this value is comparable to the pseudogap size seen in
$\rho(\omega)$ in SCTMA in Fig. \ref{Dos_Tbkt}, the peak-to-peak energy
in $\rho(\omega)$ in TMA ($\gtrsim 4\varepsilon_{\rm F}$) is much larger
than $E_{\rm b}$.
This implies that the pseudogap size in TMA does not reflect the binding
energy of a preformed pair in this regime.
\par
\par
Figure \ref{Dos_nearTbkt} shows the interaction dependence of the
density of states $\rho(\omega)$
when $T=0.15T_{\rm F}$.
In the case of SCTMA shown in panel (a), the dip structure becomes less
remarkable with decreasing the interaction strength, as
expected.
According to the preformed pair scenario for the pseudogap phenomenon
\cite{Tsuchiya}, the pseudogap gradually disappears when $T\gtrsim
E_{\rm b}$.
Noting that
$E_{\rm b}(\ln{(k_{\rm F}a_{2{\rm D}})}=1.23)=0.17\varepsilon_{\rm F}$,
and $E_{\rm b}(\ln{(k_{\rm F}a_{2{\rm D}})}=1.72)=0.064\varepsilon_{\rm
F}$, one finds that the interaction dependence of the pseudogap
structure seen in Fig. \ref{Dos_nearTbkt} (a) is consistent with this
scenario.
However, a large gap still remains
in the case of TMA
even in the weak coupling case when
$\ln{(k_{\rm F}a_{2{\rm D}})}=1.72$, as shown in
Fig. \ref{Dos_nearTbkt} (b).
This is clearly contradict with the ordinary pseudogap case because
$T=0.15T_{\rm F}$ is much larger than the binding energy
$E_{\rm b}=0.064\varepsilon_{\rm F}$ at this interaction strength.
\par
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.95\textwidth]{fig3.eps}
\caption{Calculated density of states $\rho(\omega)$ in a
two-dimensional Fermi gas at $T=0.15T_{\rm F}$ (a) SCTMA. (b) TMA.
(Color figure online.)
}
\label{Dos_nearTbkt}
\end{center}
\end{figure}
\par
\par
To explain the reason why TMA gives very different results form those in
SCTMA in the weak-coupling regime ($\ln{(k_{\rm F}a_{2{\rm D}})}\gtrsim
0$), it is instructive to consider
the weak-coupling limit [$\ln{(k_{\rm F}a_{2{\rm D}})}\rightarrow \infty$] at $T=0$,
where the system should become a free Fermi gas with no pseudogap.
In a two-dimensional uniform system,
although the Hohenberg-Mermin-Wagner theorem\cite{Mermin,Hohenberg}
prohibits the long-range superfluid order at $T> 0$,
it may be realized at $T=0$, when the
Thouless criterion \cite{Thouless},
\begin{equation}
\Gamma^{-1}(\bm{q}=\bm{0},i\nu_{n}=0)=0 \label{Thouless},
\end{equation}
is satisfied.
When one use $\Gamma^{\rm TMA}(\bm{q},i\nu_{n})$ given below Eq. (\ref{Sigma_TMA}), the
TMA Thouless criterion gives the chemical potential
$\mu_{\rm TMA}(T=0)=-E_{\rm b}/2$, indicating that all the
Fermi atoms form two-body bound molecules with the binding energy
$E_{\rm b}=1/ma_{\rm 2D}^2$.
Even not in the weak-coupling limit but at
$\ln{(k_{\rm F}a_{2{\rm D}})}=0.57$,
$\mu_{\rm TMA}(T)$ is found to approach $-E_{\rm b}/2\simeq
-0.32\varepsilon_{\rm F}$ at low temperatures, as seen in Fig. \ref{mu}.
When the Thouless criterion in Eq. (\ref{Thouless}) is satisfied,
one may approximate the self-energy in Eq. (\ref{Sigma_TMA})
to $\Sigma^{\rm TMA}(\bm{p},i\omega_{n})\simeq -\Delta^2_{\rm
PG}G_{0}(-\bm{p},-i\omega_{n})$,
where
$\Delta_{\rm PG}=\sqrt{-T\sum_{\bm{q},i\nu_{n}}\Gamma(\bm{q},i\nu_{n})}$
is sometimes referred to as the pseudogap parameter in the literature\cite{Tsuchiya,Chen,Matsumoto1}.
In this so-called static approximation,
the TMA Green's function is approximated to
\begin{equation}
G_{\rm
TMA}(\bm{p},i\omega_{n})=-\frac{i\omega_{n}+\xi_{\bm{p}}}{\omega^2_{n}+\xi^2_{\bm{p}}+\Delta^2_{\rm
PG}}. \label{Green_TMA}
\end{equation}
Equation (\ref{Green_TMA}) just has the same form as the diagonal
component of the mean-filed BCS Green's function \cite{Schrieffer}, so that one
has a clear energy gap with $E_{\rm PG}=2\sqrt{|\mu_{\rm TMA}|^2+\Delta_{\rm
PG}^2}$.
In addition, substituting Eq. (\ref{Green_TMA}) into the number equation
(\ref{Number}) at $T=0$, one obtains $\Delta_{\rm PG}=2\sqrt{\varepsilon_{\rm F}(\varepsilon_{\rm
F}-\mu_{\rm TMA})}$, unphysically giving the large (pseudo) gap size as
$E_{\rm PG}=4\varepsilon_{\rm F}\gg 2E_{\rm b}$, even in the
weak-coupling limit.
In the case of SCTMA, the static approximation for the SCTMA Green's function
gives,
\begin{equation}
G_{\rm
SCTMA}(\bm{p},i\omega_{n})=-\frac{i\omega_{n}+\xi_{\bm{p}}}{\omega^2_{n}+\xi^2_{\bm{p}}+2\Delta^2_{\rm
PG}\left[1+\sqrt{1+\frac{4\Delta^2_{\rm
PG}}{\omega^2_{n}+\xi^2_{\bm{p}}}}\right]^{-1} }. \label{Green_SCTMA}
\end{equation}
Apart from the factor $2\left[1+\sqrt{1+\frac{4\Delta^2_{\rm
PG}}{\omega^2_{n}+\xi^2_{\bm{p}}}}\right]^{-1}$,
Eq. (\ref{Green_SCTMA}) is still close to the diagonal component of the
mean-filed BCS Green's function.
Indeed, when one uses Eq. (\ref{Green_SCTMA}) to evaluate the number
equation (\ref{Number}), together with the Thouless criterion in
Eq. (\ref{Thouless}), the resulting coupled equations are found to be
formally close to the number equation at the gap equation in the
mean-field BCS theory at $T=0$\cite{Miyake}, giving $\mu_{\rm
SCTMA}=\varepsilon_{\rm F}$, and $\Delta_{\rm PG}=0$, as expected.
\par
\par
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.54\textwidth]{fig4.eps}
\caption{Calculated chemical potential as a function of
temperature, when $\ln{(k_{\rm F}a_{2{\rm D}})}=0.57$.
$\mu_{\rm SCTMA}$ and $\mu_{\rm TMA}$ show the solution for the number
equation (\ref{Number}) in SCTMA and TMA, respectively.
$\mu^{\rm Th}_{\rm SCTMA} (\mu_{\rm TMA}^{\rm Th})$ satisfies the
Thouless criterion in Eq. (\ref{Thouless}) in SCTMA (TMA). (Color
figure online.)
}
\label{mu}
\end{center}
\end{figure}
\par
\par
The above discussions at $T=0$ may be also applicable to the
weak-coupling regime at the finite temperatures.
In this case, although the Thouless criterion is, exactly speaking, not
satisfied, the TMA chemical potential $\mu_{\rm TMA}$ becomes very
close to the value $\mu^{\rm Th}_{\rm TMA}$, (which satisfies
Eq. (\ref{Thouless}) (where $\Gamma^{\rm TMA}$ is used).) at low
temperatures, as shown in Fig. \ref{mu}.
In the case of Fig. \ref{mu}, the static approximation is considered
to be valid for $T\lesssim 0.4T_{\rm F}$, where an unphysically large
pseudogap is expected in TMA density of states.
In the case of SCTMA, $\mu_{\rm SCTMA}$ becomes close to the value
$\mu^{\rm Th}_{\rm SCTMA}$, which satisfies the Thouless criterion in
Eq (\ref{Thouless}) (where $\Gamma$ in SCTMA is used),
only when $T/T_{\rm F}\lesssim 0.1$, so that the pseudogap structure in
this case is not so remarkable as that in the TMA case.
Physically, when the Fermi chemical potential approximately satisfies
the Thouless criterion $(\mu \simeq \mu^{\rm Th})$,
strong pairing fluctuations cause a dip structure in the density of
states $\rho(\omega)$ around $\omega=0$.
However, in the weak-coupling regime, since preformed pairs are
dominantly formed around the Fermi surface, the appearance of the
pseudogap would also suppress pairing fluctuations, as well as the
pseudogap phenomenon.
Such a feedback effect is, however, completely ignored in TMA, because
the free propagator $G_{0}$ with no TMA self-energy is used in
evaluating the particle-particle scattering matrix
$\Gamma(\bm{q},i\nu_{n})$.
In this case, SCTMA treats pairing fluctuations in a consistent
manner, so that the expected pseudogap behavior of the density of
states is correctly obtained in the weak-coupling regime.
\par
\section{Summary}
\par
To summarize, we have discussed the pseudogap phenomenon in a
two-dimensional ultracold Fermi gas in the crossover region, as well as
in the weak-coupling regime.
We showed that the pseudogap phenomenon associated with pairing
fluctuations in this regime can correctly be treated by the
self-consistent $T$-matrix approximation (SCTMA).
In contrast to the ordinary (non-self-consistent) $T$-matrix
approximation (TMA), which unphysically gives a large pseudogap in
the density of states even in the weak-coupling regime,
SCTMA gives a expected small pseudogap, which gradually
disappears as one approaches the weak-coupling regime.
We also pointed out the importance of a feedback effect in theoretically
dealing with pairing fluctuations in this regime, which is completely
ignored in TMA.
\par
\par
\begin{acknowledgements}
We thank H. Tajima, T. Yamaguchi, P. van Wyk , and D. Kagamihara for discussions.
M. M. was supported by Graduate School Doctoral Student Aid Program
from Keio University.
R. H. was supported by a Grant-in-Aid for JSPS fellows.
D. I. was supported by Grant-in-Aid for Young Scientists (B)
(No.16K17773) from JSPS in Japan.
This work was supported by the KiPAS project in Keio university. Y.O
was supported by Grant-in-Aid for Scientific Research from MEXT and
JSPS in Japan (No.15K00178, No.15H00840, No.16K05503).
\end{acknowledgements}
\par
\par
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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START_ATF_NAMESPACE
namespace ATL
{
#pragma pack(push, 8)
struct CCritSecLock
{
_RTL_CRITICAL_SECTION *m_cs;
bool m_bLocked;
};
#pragma pack(pop)
}; // end namespace ATL
END_ATF_NAMESPACE
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,727
|
Q: (iPhone)How to change font size of a row in tableview Hi I am now working on changing font size of a row. There seem to be no such settings in inspector. And I dont know which delegate method to override. Thx for your help :)
A: Try something like this in your cellForRowAtIndexPath method:
//Configure the cell...
cell.textLabel.font = [UIFont systemFontOfSize:14]; //Change this value to adjust size
cell.textLabel.numberOfLines = 2; //Change this value to show more or less lines.
cell.textLabel.text = @"This is my text";
A: //font size
cell.textLabel.font = [UIFont systemFontOfSize:14];
A: I am not sure if this answers your question, but inside your cellForRowAtIndexPath delegate you can set cell.textLabelfont to a UIFont.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,587
|
The VM202 MP3 Jukebox Module allows you to play your MP3 music files stored on a USB memory stick and have full control of your music. With pushbutton control for Next, Previous, and Play/Pause you control what you want to listen to. The stereo line level output allows you to connect this module to either your home entertainment system or to your car.
For automobile use(12V), put a 68 ohm/1W resistor in a series with the (+) of the power supply to avoid excess dissipation.
Dimensions 1.77" x 1.57" x 0.31"
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,902
|
British Columbia.. V3K 521.. 604-468-8160.. 604 468 3292.. Richards Pkg. 7100 River Road..Ryco Packaging .. 11422 Miracle Hills Drive, Suite 201.. Omaha.. NE.. 68154..
Paul. Ryco Conveyors, Inc. ( Laundry Only) North Vancouver, British Columbia. Liftsafe Inspections, Inc.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,707
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HomeBhopalGreat show of resilience: Madhya Pradesh Chief Minister on Indian women hockey team's maiden entry into Olympic semifinals
Great show of resilience: Madhya Pradesh Chief Minister on Indian women hockey team's maiden entry into Olympic semifinals
A brave and determined Indian women's hockey team etched its name in the history books by entering the Olympic Games semifinals for the first time, stunning three-time champions and world no.2 Australia 1-0 in an intense last-eight tie in Tokyo on Monday.
PTIUpdated: Monday, August 02, 2021, 12:42 PM IST
Players of India celebrate after defeating Australia 1-0 in their women's quarter-final match of the Tokyo 2020 Olympic Games field hockey competition, at the Oi Hockey Stadium in Tokyo, on August 2, 2021. | Photo: AFP
Bhopal (Madhya Pradesh): Madhya Pradesh Chief Minister Shivraj Singh Chouhan congratulated the Indian women's hockey team for advancing to its first ever semifinal in the Olympic Games on Monday, and described the team's win in the quarter-finals as a "great show of tenacity, resilience and confidence".
It triggered an outpouring of emotions as people from all walks of life congratulated the team.
"Great show of tenacity, resilience and confidence! I congratulate the Indian Women's Hockey team for cruising into the semifinal of of #Hockey in #Tokyo2020. I am sure you will succeed in the next game too. My best wishes!" Chouhan tweeted.
A day after the Indian men's team entered the Olympic semifinals following a 49-year gap, the world no. 9 women's side also produced a phenomenally gritty performance against the world no. 2 Australia in an intense quarter-final match.
Let us know! 👂
What type of content would you like to see from us this year?
— HubSpot (@HubSpot)
Bhopal: CM congratulates Sindhu on winning bronze in Olympics
Madhya Pradesh: Four dead while trying to steal scrap form closed mine in Shahdol
Madhya Pradesh: Ailing female cheetah at Kuno park much better now, says official
Madhya Pradesh: MP police have curbed network of SIMI terrorists, finished terror of dacoits and...
Madhya Pradesh: Students reach CM House to watch PM's 'Pariksha Pe Charcha'
Khelo India Youth Games 2022: A Sen with a different Lakshya
|
{
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}
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|
Q: The integral $\int_{-\infty}^\infty \frac{e^{-x^2}}{(1+e^x)^2} dx$. Let
$$T(n) = \int_{-\infty}^\infty \frac{e^{-x^2}}{(1+e^{x})^n} dx.$$
We have that
$$ T(0) = \sqrt{\pi} \text{ and } T(1) = \frac{\sqrt{\pi}}{2}$$
and also that
$$ T(3) = \tfrac{3}{2} T(2) - \frac{\sqrt{\pi}}{4}.$$
Can we find a closed form for $T(2)$? It would also give us $T(3)$. Perhaps something in terms of special functions?
A: An expression for $T(2)$ as a series of Bernoulli numbers:
\begin{align}
T(2) &= \int_{-\infty}^\infty \frac{e^{-x^2}}{(1+e^{x})^2} d\,dx\\
&=\frac{1}{4}\int_{-\infty}^\infty \frac{e^{-x^2-x}}{\cosh^2 \frac{x}{2}} \,dx\\
&=\frac{1}{2}\int_{-\infty}^\infty \frac{e^{-4x^2-2x}}{\cosh^2 x} \,dx\\
&=\frac{1}{2}\int_{-\infty}^\infty \frac{e^{-4x^2+2x}}{\cosh^2 x} \,dx
\end{align}
Summing the last two expressions and using the formula $\cosh 2x=2\cosh^2x-1$, it comes
\begin{align}
T(2)&=\frac{1}{2}\int_{-\infty}^\infty \frac{\cosh 2x}{\cosh^2 x}e^{-4x^2}\,dx\\
&=\int_{-\infty}^\infty e^{-4x^2}\,dx-\frac{1}{2}\int_{-\infty}^\infty \frac{e^{-4x^2}}{\cosh^2 x}\,dx\\
&=\frac{\sqrt{\pi}}{2}-\frac{1}{2}\int_{-\infty}^\infty \frac{e^{-4x^2}}{\cosh^2 x}\,dx
\end{align}
We integrate by parts,
\begin{equation}
T(2)=\frac{\sqrt{\pi}}{2}-4\int_{-\infty}^\infty xe^{-4x^2}\tanh x\,dx
\end{equation}
Now, using the series expansion for $\tanh$ DLMF:
\begin{equation}
\tanh z=\sum_{n=1}^\infty\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}
\end{equation}
one can express,
\begin{align}
T(2)&=\frac{\sqrt{\pi}}{2}-4\sum_{n=1}^\infty\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}\int_{-\infty}^\infty x^{2n}e^{-4x^2}\,dx\\
&=\frac{\sqrt{\pi}}{2}-2\sqrt{\pi}\sum_{n=1}^\infty\frac{2^{2n}-1}{2^{2n}}\frac{B_{2n}}{n!}
\end{align}
where we have used the tabulated formula
\begin{equation}
\int_0^\infty x^{2n}e^{-px^2}\,dx=\frac{(2n-1)!!}{2(2p)^n}\sqrt{\frac{\pi}{p}}
\end{equation}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,107
|
Fairview Park City Schools Hall of Fame Newsletter
The Hall of Fame Committee welcomed three new members in 2016. Rebecca Tweedle, Donna Martins, and John Jefferson bring their talents in areas of
technology, marketing, communication, and fundraising to help us reach more people and expand our scholarship fund. We are grateful for their time and commitment to the Hall of Fame.
The founders of the Fairview Park City Schools Hall of Fame will be recognized for their work at our 2017 Induction banquet on May 6, 2017 at the Airport Marriott on W. 150th St. in Cleveland. Fundraising to provide dinners and plaques for this special group of people has raised over $4000 from generous friends of the Hall of Fame.
In March of 2016, the Hall of Fame Committee reviewed all nominations and voted to select twelve inductees to the Alumni, Athletic, and Faculty/Staff Halls of Fame.
We are proud to announce the Fairview Park City Schools 2017 Hall of Fame Inductees:
Alumni: Tudor Dimofte, Rene Hernandez, Mary Nock, Stephen Senderoff.
Athletic: Pat Hoy, Jim Leffingwell, Bryan Weir, Jeremy Zarins
Faculty/Staff: Margaret S. Goebelt, Tony DiBiasio, Tom Kairis, Judi Norton
Hall of Fame Scholarship - The Hall of Fame Committee awards a $500 scholarship to a graduating senior each May.
Fundraising has been enhanced with a GoFundMe page. A link to it is provided on our webpage. A goal of $2000 has been set.
Donations can also be mailed to: Fairview Park Hall of Fame Committee, P.O. Box 26288, Fairview Park, OH 44126.
Please help us meet our goal with your tax-deductible contribution!
Our webpage is currently being updated to include biographies of inductees, an online candidate nomination form, and coming soon, a link to PayPal. We'll continue to add more features in the future.
Do you know of someone who should be considered for induction in the Hall of Fame?
We encourage you to submit a nomination! Use the online form on our webpage to provide thorough information about a candidate. Gather facts describing the
person's achievements and tell us what puts them above and beyond others to be considered for induction. Be sure to include contact information for
you as well as your nominee. Nominations are accepted for each induction cycle and considered 'active' for a total of three cycles, or nine years. If your candidate is not
inducted in that time, you may re-submit your nomination. We have already received several submissions for the 2020 Induction.
Thanks for your support of the Fairview Park City Schools Hall of Fame.
Committee members: Joel Chermonte, Chairman, Tom Kairis, Vice-Chairman, Jean Scothon, Treasurer, Mary Kay Wysong, Secretary, Karen Welter, Craig Fawcett, Sandy Bennhoff, Kathy Rosol, Tony DiBiasio, Becky Tweedle and John Jefferson. We meet monthly and welcome new members. If you'd like to join the committee, please contact Joel at 440-915-4509.
Hello, Everyone! We wanted to keep you informed about the activities of the Hall of Fame Committee since the most recent Induction Banquet in 2014.
Thanks to your generosity, our fundraising campaign has netted over $6,000! As a result, we have quite a few new names to include in our program for the next banquet. The FPCS Hall of Fame Committee is now a tax-exempt organization, and gladly accepts
donations any time.
-The Hall of Fame Committee will sponsor a student scholarship, starting this year, in the amount of $250. It will be awarded to a student planning to major in education.
-The Fairview Park City Schools have named Bill Wagner as their new superintendent. He has expressed interest in working with the Hall of Fame Committee and we welcome him and his family to Fairview. If you haven't been to Fairview in a while, it might surprise you to know that Garnett and Coffinberry schools are now just a memory. The "new" Gilles-Sweet opened in 2007, while Parkview is now the administrative/ kindergarten building.
-Our website is up and running. Visit: http://www.fairviewparkschoolshalloffame.com/ for a list of inductees, nomination forms, and upcoming events. We are accepting nominations for induction.
-Our next Induction Banquet will be Saturday, May 6, 2017, at the Marriott on W.150th St. Be sure to save the date!
-The Hall of Fame Committee will support the Fairview Education Foundation by sponsoring a hole at their golf outing Saturday, August 1, 2015, at Grey Hawk Golf Course. Lunch is 11:00, with a shotgun start at noon. Dinner and awards follow golf. Cost is $100/golfer, or $30 for dinner only. For reservations contact: Sally Lisowski 440-779-1417 sally.lisowski @yahoo.com Denise Devine 440-343-5389 yayadevine@gmail.com Bill Wagner 440-331-5500 bwagner@fairview.k12.oh.us
-Sadly, we note the passing of our friends, Ned Livengood, John Cawley, and David Misener. We send condolences to their families. Their service to Fairview Park will long be remembered.
-Do you remember Garnett Gatherings, Field Day at Parkview, Red Days at the High School, Homecoming bonfires, or spaghetti dinners for the marching band? Do you have memories of traditions in Fairview? Do you have news to share about inductees? We'd love to hear from you! You can email Mary Kay at ddwysong@yahoo.com with your news or favorite memories.
|
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}
| 9,677
|
Professor Ataur Belal
The Department of Accounting
Professor in Accounting and Sustainability
a.belal@bham.ac.uk
Ataur is a Professor of Accounting and Sustainability at the Department of Accounting within the Birmingham Business School. Before coming to Birmingham he held professorships at Aston and Sheffield Universities where he also held various senior leadership positions including a stint as as member of Senior Management Team, Head of Department and Research Development Director. He has a longstanding research interests in the field of social and environmental accounting in emerging economies. His research continues in this field.
Other significant latest research interests include accounting for Sustainable Development Goals (SDGs) and NGO accountability with a focus on beneficiary accountability. His teaching interests is closely aligned with his current research interests. So far, he has supervised nine PhD students to completion.
His work in area of sustainability has led to a number of international large project collaborations funded by the British Council, British Academy and CIMA. He has published extensively in international leading peer reviewed journals, contributed a book and various book chapters. He has presented his work at various conferences around the world. He has also held visiting positions in various universities around the world including Kobe University, Japan and University of Ottawa, Canada.
PhD The University of Sheffield
FCCA UK
FHEA Advances HE UK
MBA University of Wales, Cardiff
B.Com. (Hons), M. Com. University of Chittagong, Bangladesh
Ataur is a Professor in Accounting and Sustainability. He joined Birmingham Business School in August 2022 having previously been a professor in the Department of Accounting at Aston University. During his 16 years of service at Aston, he also served as the Head of the Department and member of the Senior Management Team. Before that he was an Associate Professor at the University of Chittagong, Bangladesh. In Birmingham, he is a member of the Department of Accounting located within the Birmingham Business School.
He has held visiting positions and taught at the universities of Toulouse (France), Regensburg (Germany), University of Ottawa (Canada), Kobe University, Japan and De los Andes (Colombia). His editorial positions include former Editorship of Advances in Environmental Accounting and Management, Associate Editorship of Accounting Forum and editorial Board Memberships in Accounting, Auditing and Accountability Journal, Critical Perspectives on Accounting and Business Ethics: A European Review. His Google Scholar H Index is 26 and has nearly 4500 citations.
His principal research interest lies in the area of social and environmental accounting from the context of developing countries such as Bangladesh, India and Vietnam. His work in this area has led to a number of international large project collaborations funded by the British Council, British Academy and CIMA. He has presented his research in leading international conferences and workshops in various countries. His research papers have appeared in leading international journals in the field such as Work, Employment and Society, Accounting, Auditing and Accountability Journal, Journal of Business Ethics, Critical Perspectives on Accounting, Accounting Forum, Financial Accountability and Management, and Accounting and Business Research.
Ataur's teaching is research informed and adopts a critical perspective on accounting. He brings in the latest insights from his research field in social and environmental accounting. He also brings guest lecturers in from the industry to provide practical insights to the students.
His role is mainly to facilitate students' teaching and learning process by invoking and sometimes provoking their latent critical minds. Ataur's teaching ethos includes looking at accounting from an alternative perspective which promotes the role of accounting in capturing the social and environmental impacts and thereby promotes the transparency and accountability of organisations. This is informed by the view that accounting has a much broader role to serve wider social interests.
This is a point of departure from the narrow and traditional perspectives on accounting which applies it mainly as a tool to aid corporate decision making with a view to maximise shareholders' wealth. He has taught various accounting modules to undergraduate, postgraduate and post experience audience. Ataur has the experience of running CPD sessions for professional accountants internationally.
Of late, he has been specialising in teaching sustainability accounting in the UK and abroad including Kobe University in Japan.
Accounting, Auditing and Accountability Journal, Critical Perspectives on Accounting, Accounting Forum and Business Ethics, Environment and Responsibility
Expert reviewer of research proposals and reports for ESRC and British Council. Recently invited to join the peer review college of UKRI Future Leaders Funding Scheme.
International advisor
"SESAMI Program (Strategic Entrepreneurship and Sustainability Alliance Management Initiatives)",Kobe University, Japan.
Panel member of 13 university professors, ESG Investing, UK for ESG related accreditation for investment funds.
Member, UKRI peer review colleague for Future Leaders Scheme.
Academic Fellow, Humanitarian Academy for Development, Birmingham UK.
Dewi, MK, Manochin, M & Belal, A 2021, 'Towards a conceptual framework of beneficiary accountability by NGOs: an Indonesian case study', Critical Perspectives on Accounting, vol. 80, 102130. https://doi.org/10.1016/j.cpa.2019.102130
Dewi, M, Manochin, M & Belal, AR 2019, 'Marching with the Volunteers: Their Role and Impact on Beneficiary Accountability in an Indonesian NGO', Accounting, Auditing and Accountability Journal. https://doi.org/10.1108/AAAJ-10-2016-2727
Ataur has experience of working with a range of Bangladeshi policy makers in the area of sustainability accounting. Some of them include: the Central Bank of Bangladesh; Dhaka Stock Exchange; the Institute of Chartered Accountants of Bangladesh and the Institute of Cost and Management Accountants of Bangladesh.
His research informed the green banking guidelines issued by the Central Bank of Bangladesh. He engaged with the senior management of the Central Bank of Bangladesh to inform the Green Reporting Guidelines developed by the Central Bank. The Governor of the Central Bank confirmed the impact via an endorsement letter dated 3rd June, 2012. Since then, the Central Bank of Bangladesh requires all banks in Bangladesh to undertake green reporting. Ataur's research has also influenced the national corporate reporting award scheme organised by the Institute of Cost and Management Accountants of Bangladesh as evidenced by a letter from the President of the Institute.
|
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| 7,452
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Q: Connect to internet through DC with 2 network interfaces I have a network like this :
Client 1 <== Wireless ==> Access Point <== Wire ==> DC <== Wireless ==> ADSL Modem
Client 1 :
IP : 192.168.1.181
DG : 192.168.1.100
DNS : 192.168.1.100
Access Point :
IP : 192.168.1.10
DG : 192.168.1.100
DC :
IP : 192.168.1.100
DG : 192.168.1.1
DNS : 127.0.0.1
ADSL Modem :
IP : 192.168.1.1
I can ping yahoo.com from client1, but cannot browse the internet.
UPDATE1 : my DC has two network interfaces that I bridge together.
UPDATE2 : I powered down the DC firewall
UPDATE3 : I set a forwarder for my DC to 8.8.8.8 (Google dns)
A: By DC, i assume you mean Domain Controller? If so, you are doing it wrong. The DC should handle your DNS, but not be a gateway. Put a proper router into your network topology.
Internet > ADSL modem > Router/firewall/gateway device > Switch > Servers/clients/access points/etc
A: Since all your machines are numbered in the same subnet, you are bridging. But you cannot bridge to a WiFi client connection (that's why WDS has to be enabled on both sides). If the DC is connected to the ADSL modem by wireless as a client, it cannot bridge to additional machines on the same network.
WiFi is not wireless Ethernet. It's its own protocol with its own rules. You can't treat a client link to an access point like a wired connection. Unfortunately, WiFi is enough like Ethernet that it's easy to think it's a drop in replacement for a wired link. It is not.
A: Your DNS is wrongly configured on the DC it should be 127.0.0.1 pointing to itself as the DNS server
|
{
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\section*{Introduction}
Transient heating induced by laser pulse absorption has been intensely studied to induce phase transitions of interest such as metal-insulator in VO$_2$ \cite{XBV11}, amorphous-crystalline in Ge$_2$Sb$_2$Te$_5$ \cite{FWN00}, and charge density waves in 1{\it T}-TaS$_2$ \cite{HEE16}, all of which hold strong potential for applications in next-generation electronic and optical data storage \cite{WY07}. An interesting possibility arises when considering phase transitions in components of artificial structures known as metamaterials, which grant us access into a broader range of optical properties extending beyond those encountered in naturally occurring materials. Actually, fascinating applications have been demonstrated by using metamaterial properties, including negative refraction \cite{V1968,P00}, superlensing \cite{FLS05}, optical cloaking \cite{SMJ06}, and superfocusing \cite{LBP02}. More recently, hyperbolic metamaterials have been found to exhibit an effective uniaxial anisotropic permittivity tensor that produces peculiar hyperbolic topology in the isofrequency dispersion contours and enables engineering of the decay rates of optical emitters \cite{KJN12,LKF14}, as well as hyperbolic waveguiding \cite{PN05} and far-field subwavelength imaging \cite{JAN06,LLX07}, among other feats. Hyperbolic metamaterials can be realized, for example, by simply stacking layers composed by alternating dielectric and plasmonic materials \cite{LKF14,KJN12,PIB13}. Among the latter, graphene offers additional appealing properties compared with noble metals, such as remarkably low optical losses \cite{WLG15,NMS18}, exceptionally small electronic heat capacity \cite{ESG18}, extraordinary photothermal response \cite{paper313}, and the ability to tune its optical conductivity through electrical gating \cite{NGM04,FV07,FP07_2,CGP09}. In fact, hyperbolic metamaterials based on layered graphene/dielectric stacks \cite{IMS13,OGC13} are being intensely explored as an excellent platform for applications involving infrared light \cite{BPA19}.
When subject to ultrafast optical pulse irradiation, the optical energy absorbed by graphene is first deposited in its conduction electrons, which can be heated to an elevated temperature that remains during $\sim0.5-1\,$ps before transferring a substantial fraction of heat to the lattice \cite{JUC13,GPM13,RWW10}. Notably, due to the small heat capacity of electrons in graphene \cite{paper286,ESG18} compared with the lattice, the latter remains close to ambient temperature for optical pump-pulse fluences as high as $\sim2\,$mJ/cm$^2$ and peak electron temperatures of 1000's\,K \cite{LMS10,paper330}, partially assisted by the weak electron-phonon coupling that characterizes this material \cite{ESG18,paper313}. A strong thermo-optical response is then triggered thanks to the strong temperature dependence of the graphene optical conductivity. Consequently, we expect that topological transitions of isofrequency contours could be triggered in metamaterials formed by layered graphene/dielectric stacks under ultrafast optical pumping, similar to light-driven phase-transitions in natural materials.
\begin{figure*}
\noindent \begin{centering}
\includegraphics[width=0.85\textwidth]{Fig1}
\par\end{centering}
\caption{(a) Schematic of a metamaterial made of graphene/dielectric stacks. The right inset shows a vertical unit cell with graphene located in the center. In this work, we take a unit cell thickness $d=20\,$nm, a permittivity of the dielectric $\epsilon_{{\rm d}}=4$, and a graphene Fermi energy $E_{{\rm F}}=0.4\,$eV. (b) Real and imaginary parts of the effective in-plane (i.e., for polarization in the $x$-$y$ plane) permittivity $\epsilon_{\parallel}$ as a function of optical frequency at different graphene electron temperatures $T_{{\rm e}}$, as indicated by labels. (c) Isofrequency dispersion contours at a fixed frequency of 42\,THz [indicated as a dash-dotted vertical line in panel (b)] for different electron temperatures. We compare results obtained from electromagnetic numerical simulations (solid curves) and an effective medium model (dashed curves). The topology of the contour transforms from elliptic to hyperbolic as $T_{{\rm e}}$ increases.}
\label{Fig1}
\end{figure*}
In this Letter, we theoretically investigate ultrafast photothermal manipulation of a topological transition in layered metamaterials composed by graphene/dielectric stacks, driven by light absorption in graphene. Based on a realistic description of the temperature-dependent optical properties of graphene, in combination with the spatiotemporal heat flow within its electron/lattice subsystems under ultrafast laser pumping, we predict a topological transition of the isofrequency dispersion contours of the metamaterial in the infrared domain, whereby the topology transforms from elliptic to hyperbolic within an ultrafast timescale. We show that the spatiotemporal dynamics of this topological transition can be probed by a delayed free electron beam, and further find the transient hyperbolic phase to last $\sim1\,$ps. Our results enable several exotic phenomena in the ultrafast regime, such as dynamical engineering of the decay rate of optical emitters in the vicinity of the metamaterial, as well as directional beam steering by carving the metamaterial into a cylindrical lens, which we show to be useful for subwavelength far-field image encoding/decoding.
\section*{Results}
In Fig.\ \ref{Fig1}(a), we sketch a metamaterial composed of graphene/dielectric stacks with graphene located in the center of the dielectric-graphene-dielectric unit cell (see inset). We take a unit cell thickness $d=20\,$nm, a permittivity of the dielectric material $\epsilon_{{\rm d}}=4$, and a graphene Fermi energy $E_{{\rm F}}=0.4\,$eV throughout this work. Given the small value of $d$ compared with the infrared light wavelengths considered in this work, an effective permittivity should accurately describe the dielectric response of the metamaterial. Because of the system symmetry, we then have an effective uniaxial anisotropic material with two different permittivities $\epsilon_{\parallel}$ and $\epsilon_{\perp}$ along in-plane ($x$-$y$ plane) and out-of-plane ($z$ axis) directions, respectively. Additionally, due to the two-dimensional (2D) nature of graphene, we have $\epsilon_{\perp}=\epsilon_{{\rm d}}$. It is well known that optical pulse pumping can elevate the temperature of graphene electrons up to $\sim5000\,$K within a ultrafast timescale \cite{JUC13,GPM13,TSJ13} without causing any damage to the material because of its exceptionally small electronic heat capacity \cite{ESG18}. We present the spectral dependence of the in-plane effective permittivity $\epsilon_{\parallel}=\epsilon_{{\rm d}}+\ii 4\pi\sigma/\omega d$ for different electron temperatures $\Te$ in Fig.\ \ref{Fig1}(b), where $\sigma$ is the temperature-dependent surface conductivity of graphene and $\omega$ is the angular frequency. It should be noted that the epsilon-near-zero frequency (${\rm Re}\{ \epsilon_{\parallel} \}=0$, see solid curves) is first redshifted and then blueshifted as $\Te$ is increased, which is a consequence of the nontrivial $\Te$-dependence of the graphene chemical potential \cite{paper313}. More specifically, the ${\rm Re}\{ \epsilon_{\parallel} \}=0$ condition leads to a frequency $\approx\sqrt{4e^2\mu^{\rm D}/\hbar^2\epsilon_{{\rm d}}d}$, where $\mu^{\rm D}$ is the temperature-dependent effective Drude weight in the graphene conductivity \cite{paper313}. Additionally, ${\rm Im}\{ \epsilon_{\parallel} \}$ (dashed curves) increases with $\Te$ as a consequence of the enhancement in the inelastic scattering rate of graphene electrons \cite{YYM19}.
We now explore the isofrequency dispersion contours at different electron temperatures for p-polarized electromagnetic fields, which is shown in Fig.\ \ref{Fig1}(c) at a frequency of 42\,THz [indicated by a vertical black dash-dotted line in Fig.\ \ref{Fig1}(b)]. The figure reveals two distinct regimes represented by the topology of the isofrequency contour, which evolves from elliptic to hyperbolic as $\Te$ increases. Specifically, within the $\Te=300-3000\,$K range, the isofrequency contour remains elliptic, despite some variations in shape. When $\Te$ is further increased, a dramatic variation of the contour topology occurs, which results in an emerging hyperbolic metamaterial because ${\rm Re}\{ \epsilon_{\parallel} \}<0$ when $\Te>3000\,$K. These results demonstrate that controlling $\Te$ in graphene can be an efficient and ultrafast route toward inducing topological transitions through optical pulse pumping. Incidentally, we compare the isofrequency dispersion contour calculated with the transfer matrix method in Fig.\ \ref{Fig1}(c) (solid curves) with that obtained from the effective medium theory (dashed curves), as determined by
\begin{align}
\frac{k_{\parallel}^2}{\epsilon_{\perp}}+\frac{k_{\perp}^2}{\epsilon_{\parallel}}=k_0^2,
\label{disp}
\end{align}
where $k_0$ is the free-spade light wave vector, while $k_{\parallel}$ and $k_{\perp}$ are the wave vectors along in- and out-of plane directions, respectively. We attribute the discrepancies between these two methods observed at large values of $k_{\parallel}$ to the lack of validity of the effective medium model when $k_{\parallel}d \ll 1$ no longer holds.
\begin{figure*}
\noindent \begin{centering}
\includegraphics[width=0.85\textwidth]{Fig2}
\par\end{centering}
\caption{(a-c) Temperature-dependent enhancement of the local density of optical states (LDOS) as a function of frequency for metamaterials composed of different numbers $N$ of vertical periods. The probing point dipole has in-plane polarization and is placed 50\,nm away from the surface. (d-f) Same as (a-c) for out-of-plane polarization. We compare the results obtained from a fully numerical calculation (solid curves) with the effective medium model (dashed curves).}
\label{Fig2}
\end{figure*}
A promising application of hyperbolic metamaterials relates to their ability to manipulate the decay rate of proximal emitters by enhancing the local density of optical states (LDOS) \cite{LKF14}, which is given by \cite{LK1977,NH06}
\begin{align}
\frac{{\rm LDOS}_{\parallel}}{{\rm LDOS}_0}=1+\frac{3}{4}\int_{0}^{\infty}\frac{k_xdk_x}{k_0^3}{\rm Re} \left \{\left (\frac{k_0^2 r_s}{k_z}-k_z r_p \right ) {\rm e}^{2\ii k_z z_0}\right \}
\label{LDOS1}
\end{align}
and
\begin{align}
\frac{{\rm LDOS}_{\perp}}{{\rm LDOS}_0}=1+\frac{3}{2}\int_{0}^{\infty}\frac{k_x^3dk_x}{k_0^3}{\rm Re} \left \{ \frac{r_p}{k_z} {\rm e}^{2\ii k_z z_0} \right \}
\label{LDOS2}
\end{align}
for in- and out-of-plane polarization, respectively, where $k_z=\sqrt{k_0^2-k_x^2}$, $z_0$ is the separation distance between the emitter and the metamaterial surface, and $r_s$ and $r_p$ are the reflection coefficients at the metamaterial-air interface for s- and p-polarization. These expressions are normalized to the projected LDOS in free space ${\rm LDOS}_0=\omega^2/3\pi^2 c^3$, where $c$ is the speed of light. The temperature-dependence of the LDOS enhancement is presented in Fig.\ \ref{Fig2} at a distance $z_0=50\,$nm above the upper surface of the metamaterial [see inset in Fig.\ \ref{Fig2}(a)]. A large LDOS enhancement ($>10^3$) is found as a result of strong confinement of the photonic modes supported by the metamaterial. In general, as the electron temperature $\Te$ increases, the spectral range with high LDOS extends to lower frequencies. More specifically, an enhancement of two orders of magnitude can be obtained at $\sim 48\,$THz when increasing $\Te$ (see color-coded labels) due to the topological transformation of the isofrequency contour described in Fig.\ \ref{Fig1}. This means that one can control the emitter decay rate in an ultrafast manner through rising the electron temperature by means of optical pulse pumping. Our findings are robust against the number $N$ of vertical unit cells composing the metamaterial film, as shown in Fig.\ \ref{Fig2}(a-c) for a dipolar emitter polarized parallel to the surface, as calculated from Eq.\ \ref{LDOS1}. Similar results and conclusions are found for out-of-plane polarization, as shown in Fig.\ \ref{Fig2}(d-f), calculated from Eq.\ \ref{LDOS2}. The deviation between the results calculated by using the transfer matrix method (solid curves) or the effective medium model (dashed curves) are again related to the mismatch at large values of $k_{\parallel}$.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=0.5\textwidth]{Fig3}
\par\end{centering}
\caption{(a) Schematic of a pump-probe configuration involving a normally-incident optical pump pulse and a probing free electron beam passing parallel to the surface at a distance of 50\,nm. We consider a metamaterial film consisting of $N=10$ vertical periods. (b) Calculated spatiotemporal dynamics of the graphene electron temperature assumed to be uniform across the thin metamaterial film. The spatial coordinate indicates the distance to the center of the axisymmetric optical Gaussian pulse (600\,nm beam width). (c,d) Spatiotemporal dynamics of the EELS signal for an electron frequency loss of 42\,THz, as obtained through full electromagnetic calculations (c) or using an effective medium model for the metamaterial (d). The occurrence of the topological transition of the isofrequency contour, taking place at $\Te \approx 3560\,$K for the optical frequency under consideration, is indicated by blue-dashed curves in (b-d), which enclose the spatiotemporal domain characterized by a hyperbolic response.}
\label{Fig3}
\end{figure}
In order to resolve the ultrafast spatial and temporal dynamics of the topological transition, an electron probe moving in free space, can be employed to spatially image the topological transition in the so-called aloof configuration \cite{paper149} after a short optical pulse pumping, as illustrated in Fig.\ \ref{Fig3}(a). This type of experiment can be performed with state-of-the-art ultrafast electron microscopes, relying on pulsed optical pumping and electron probing \cite{BFZ09,FES15,PLQ15,paper311,DNS19}. The probe electron can provide direct information about the topological transition through the electron energy-loss spectroscopy (EELS) signal, the probability of which is given by \cite{paper149}
\begin{align}
\frac{\Gamma(\omega)}{L}=\frac{2e^2}{\pi \hbar v^2}\int_{0}^{\infty}\frac{dk_y}{k_{\parallel}^2}{\rm Re} \left \{ \left( \frac{k_y^2 v^2}{k_z^2 c^2}r_s-r_p \right) k_z {\rm e}^{2\ii k_z z_0} \right \},
\label{EELS}
\end{align}
where $L$ is the length of the electron trajectory, $v$ is the electron velocity, $z_0$ is the separation distance between the electron beam and the metamaterial surface, $k_{\parallel}=\sqrt{\omega^2/v^2+k_y^2}$, and $k_z=\sqrt{k_0^2-k_{\parallel}^2}$. We note that the loss probability given by Eq.\ (\ref{EELS}) bears a close relation to the momentum decomposition of LDOS along the electron trajectory \cite{paper102}. In what follows, we set the electron velocity to $v=0.5c$ ($\approx100\,$keV energy) and the separation to $z_0=50\,$nm. We further consider a Gaussian pump pulse of 100\,fs duration, 2\,mJ/cm$^2$ fluence, and 600\,nm beam width, which is realistic for lasers in the visible range. Following optical pumping of a metamaterial film composed of $N=10$ vertical unit cells, we calculate the resulting spatiotemporal dynamics of the electron temperature using a two-temperature model \cite{paper313,paper330}. The results are shown in Fig.\ \ref{Fig3}(b), where the in-plane radial distance is referred from the Gaussian beam center. The electron temperature is a maximum value $\sim 4000\,$K at the beam center immediately after pumping and then decreases as time evolves. This range of high electron temperatures is currently achievable in state-of-the-art experiments \cite{JUC13,GPM13,TSJ13}. In a way that is consistent with its intimate relation to the LDOS, the spatiotemporal dynamics of the EELS signal follows closely that of $\Te$ at a fixed frequency loss of 42\,THz, as shown in Fig.\ \ref{Fig3}(c,d), where the results obtained from the effective medium model (Fig.\ \ref{Fig3}(d)) match quite well those obtained from the full calculation (Fig.\ \ref{Fig3}(c)). The blue-dashed curves in Fig.\ \ref{Fig3}(b-d) enclose a spatiotemporal regime in which the isofrequency contour is transformed to be hyperbolic, giving rise to an enhanced EELS signal. Within $\sim1\,$ps timescale, the topology of the isofrequency contour transforms from elliptic to hyperbolic, and then back to elliptic, thus encompassing an ultrafast topological transition.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=0.5\textwidth]{Fig4}
\par\end{centering}
\caption{(a) Spatial distribution of the optical magnetic field in a plane perpendicular to a cylindrical metamaterial lens (bounded by the two black semicircles of radii $\lambda/5$ and $6\lambda/5$, respectively) at room temperature $\Te=300\,$K when it is excited by two magnetic current sources (represented by two small black circles close to the inner lens surface) separated by a subwavelength distance $\lambda/4$. We consider a light wavelength $\lambda=7.8\,\mu$m (i.e., 38.2\,THz frequency). The permittivity of the medium inside the inner circle and outside the outer one is taken to be 1 and 4, respectively. (b) Same as (a) when the graphene electron temperature in the metamaterial is raised to $\Te=1000\,$K. (c) Normalized far-field emitted energy as a function of azimuthal angle, spanning a range from $-90^\circ$ to $90^\circ$, at the two electron temperatures considered in (a,b). We compare results obtained from full electromagnetic simulations (solid curves) and the effective medium model (dashed curves).}
\label{Fig4}
\end{figure}
As a final example of application of the ultrafast topological phase transition discussed above, we investigate light steering and super-resolution imaging. It has been demonstrated that a perfect lens can be realized through negative refraction and amplification of evanescent waves in negative-index metamaterials \cite{P00}. Hyperbolic media with nearly flat isofrequency dispersion contours are also capable of producing subwavelength imaging of a point source \cite{JAN06,LLX07}. Here, we study the radiated power distribution emanating from two magnetic line current sources (black circles in Fig.\ \ref{Fig4}(a,b), separated by $\lambda/4$, where $\lambda=7.8\,\mu$m is the optical frequency corresponding to a 38.2\,THz frequency). These sources are placed in front of a cylindrical lens made of a curved version of the layered graphene/dielectric under consideration. At room temperature $\Te=300\,$K, the isofrequency contour is indeed hyperbolic at 38.2\,THz [see Fig.\ \ref{Fig1}(b)]. However, because of the curved nature of the isofrequency contour, several guided modes are excited with different wave vectors $k_{r}$ along the radial direction \cite{BS06}. As a result, multiple lobes show up in the spatial distribution of the magnetic fields [Fig.\ \ref{Fig4}(a)], which also transmit into the far-field [red curves in Fig.\ \ref{Fig4}(c)]. When $\Te$ is increased to 1000\,K, the so-called canalization condition \cite{BSI05,SE06_2} ${\rm Re} \{ \epsilon_\theta \}\approx0$ is satisfied leading to a nearly flat isofrequency contour. Here, $\epsilon_\theta$ is the effective permittivity of the cylindrical lens along the azimuthal direction. This results in two highly directive lobes in the spatial distribution of the magnetic field [Fig.\ \ref{Fig4}(b)], thus demonstrating that the cylindrical lens is capable of ultrafast beam steering driven by the transient elevation of $\Te$-rising upon optical pulse pumping. Additionally, two clear angular peaks associated with those two individual point sources can be identified in the far-field regime [see blue curves in Fig.\ \ref{Fig4}(c)], further supporting the potential for far-field subwavelength imaging in a dynamical and ultrafast manner. Once more, the azimuthal distribution of the far-field signal obtained from full numerical simulations (solid curves) matches well the result obtained from the effective medium model (dashed curves), as shown in Fig.\ \ref{Fig4}(c). Our results further suggest a novel subwavelength image encoding/decoding mechanism, whereby an elevated electron temperature induced by optical pumping is the key to resolve encoded subwavelength images in the far-field regime.
\section*{Conclusion}
In summary, we have shown that transient heating can significantly modify the topology of the isofrequency dispersion contours in metamaterials formed by layered graphene/dielectric stacks by exploiting the remarkably small electronic heat capacity of graphene, which allows us to efficiently elevate the electron temperature through ultrafast optical pumping. By examining the spatiotemporal dynamics of the EELS signal obtained by using an electron-beam probe, we have found that the contour topology can transform between elliptic and hyperbolic shapes within a sub-picosecond timescale, that is, the characteristic time over which the elevated electron temperature evolves in real space. We can thus manipulate the LDOS enhancement in the proximity of the metamaterial, reaching variations in the calculated LDOS of a few orders of magnitude at the frequency in which this topological transition appears. Additionally, this type of transition enables ultrafast beam steering, which we have illustrated by illuminating a cylindrical lens made of metamaterials with two line sources, in which dynamical far-field subwavelength imaging has been demonstrated. Our findings open a promising route toward ultrafast control of light emission, beam steering, and optical image processing.
\section*{Acknowledgements}
This work has been supported in part by the ERC (Advanced Grant 789104-eNANO), the Spanish MINECO (MAT2017-88492-R and SEV2015-0522), the Catalan CERCA Program, and Fundaci\'o Privada Cellex. R.A. acknowledges the support of the Alexander von Humboldt Foundation through the Feodor Lynen Fellowship. R.W.B. acknowledges support through the Natural Sciences and Engineering Research Council of Canada, the Canada Research Chairs program, and the Canada First Research Excellence Fund.
\bibliographystyle{apsrev}
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Lake Gregory is a reservoir in the San Bernardino National Forest of the San Bernardino Mountains in San Bernardino County, California. The lake and the surrounding area make up the Lake Gregory Regional Park adjacent to Crestline, California.
The area, originally known as Houston Flat, was developed by and named for its developer, Redlands citrus grower Arthur Gregory, Sr. Gregory bought and developed land in an area known today as Valley of the Moon. He erected a sawmill at Valley of the Moon to cut wood for crating his "Orange Blossom" brand of citrus fruit. Gregory was also instrumental in creating the Crest Forest County Water District (CFCWD), which, in turn, was necessary to acquire federal aid in order to develop the area. Although the lake is in Crestline proper, Crestline is not a part of the Crest Forest District, but rather the Crestline Water District, which purchases water from CFCWD.
Work began in 1937 under a Works Progress Administration (WPA) grant to dam the east and west forks of Houston Creek, whose waters drained into tributaries of the Mojave River, thereby "going to waste". The project was nearly completed by March 1938, but federal funds had run out. Gregory financed the completion of the project, lending money to the district for the completion. The eventual cost was US$225,000, of which US$160,000 came from the federal government, with the balance funded by the water district.
Heavy rains that March put the dam to its first test. It had been estimated that it would take three years to fill the lake. So heavy were the rains that the lake filled in only three days. An unconfirmed, but plausible urban legend claims that the construction equipment left on the lake bed in 1938 during the rains remains at the bottom of the lake today.
A road built over the dam (present-day Lake Drive) completed the project in January 1939.
Today, the Lake Gregory Recreational Park with swimming and water slides is at the west end of the lake and a walking trail encircles the lake. The south shore of the lake is a popular fishing destination. The regional park is the site of Crestline's Independence Day celebration.
The southeast shore is the site of the private beach and Tyrolean-styled clubhouse of the San Moritz Lodge, once known as Club San Moritz. The original club was built in the Valley of the Moon in 1926 along the shore of now-drained Moon Lake, today the site of Lake Gregory Community Church. A fire of suspicious origin destroyed the building in 1950 and the lake was drained in the early 1960s over increasing problems with mosquitoes. The new club, built along Lake Gregory in 1950, was intended to be a members-only resort and club for property owners in the area. Today, the only remaining buildings are the restaurant, used as the site of special events such as weddings, and a bathhouse, now used as a senior center.
Private vessels and power boats are not allowed on Lake Gregory, although rowboats, paddle boards, pedal-powered "water trikes" and paddle boats are available for rental. The rowboats may be affixed with the renter's own electric trolling motor.
From 2011 to 2013, the park experienced operating losses of $1.4 million. In 2014, the San Bernardino County partnered with The California Parks Company, now named Basecamp Hospitality, after six years of decline due to the flagging economy. In 2018, San Bernardino County put the concessionaire contract up for bid to find a new managing company for Lake Gregory. Basecamp Hospitality won the bid as the only concessionaire applying, but struggled to come to financial terms with the longterm contract and renewed a two-year agreement instead. In 2019, San Bernardino County ended its relationship with Basecamp Hospitality.
The California Office of Environmental Health Hazard Assessment (OEHHA) has developed a safe eating advisory for Lake Gregory based on levels of mercury found in fish caught from this water body.
See also
List of dams and reservoirs in California
List of lakes in California
References
San Bernardino Mountains Early History from californiamountains.net, Jan. 06, 2011
http://www.sbsun.com/lifestyle/20140323/lake-gregory-getting-new-water-play-structure-seasonal-pass-varieties
External links
Official San Bernardino County Parks page
Official Lake Gregory Park page
The California Parks Company
San Bernardino Mountains
Gregory
1937 establishments in California
Works Progress Administration in California
San Bernardino National Forest
Gregory
Gregory
Gregory
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San Pietro in Valle ima više značenja:
San Pietro in Valle, Bolzano
San Pietro in Valle, Isernia
San Pietro in Valle, Verona
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\section{Introduction}
Let $(M,g)$ be a smooth compact manifold without boundary and let $\Delta_g$ be the associated Laplace-Beltrami operator. Let $W \in L^{\infty} (M\times \Rb)$ be a nonnegative function. Consider the damped wave equation with time dependent damping
\begin{equation}\label{DWE}
\begin{cases}
(\p_t^2 -\Delta_g + W(x,t)\p_t)u=0 \\
(u,u_t)|_{t=0} =(u_0, u_1).
\end{cases}
\end{equation}
With initial data $(u_0,u_1) \in \Hc := H^1(M) \oplus L^2(M)$ where $\Hc$ has the natural norm
$$
\hcnorm{(u_0,u_1)}^2 = \hp{u_0}{1}^2 + \ltwo{u}^2.
$$
The standard object of study is the energy of the solution.
$$
E(u,t) = \frac{1}{2} \int_M |\nabla_g u(x,t)|^2 + |\p_t u(x,t)|^2 dx_g.
$$
It is straightforward to compute
$$
\frac{d}{dt} E(u,t) = -2\Re \int W(x,t) |\p_t u(x,t)|^2 dx_g \leq 0.
$$
Where the sign is guaranteed by $W(x,t) \geq 0$. Because of this the energy is non-increasing but there is no indication on the rate of decay as $t \ra \infty$.
The most straightforward type of decay is uniform stabilization. That is, there exists a function $r(t)\ra 0$ as $t \ra \infty$ such that
$$
E(u,t) \leq r(t) E(u,0).
$$
When $W$ depends only on $x \in M$ uniform stabilization is equivalent to $W$ satisfying the Geometric Control Condition (GCC) \cite{Ralston1969, RauchTaylor1975}. The GCC is satisfied if there exists some $T>0$ such that every geodesic with length at least $T$ intersects the set $\{W>0\}$.
There is an equivalent condition to the geometric control condition, c.f. \cite{Lebeau1996}. Let $\phi_t(x_0,\xi_0)$ be the geodesic flow starting from $(x_0, \xi_0)$. The GCC is equivalent to the existence of $T_0,\Cm>0$ such that for all $(x_0, \xi_0) \in S^* M$ and $T \geq T_0$
$$
\frac{1}{T}\int_0^T W(\phi_t(x_0,\xi_0)) dt \geq \Cm.
$$
That is, there is a uniform lower bound on the long time averages of the damping along any geodesic.
In this paper, I show that the appropriate generalization of this condition to the time dependent setting implies exponential decay for smooth damping.
\begin{assumption}\label{timegcc} (Time-dependent geometric control condition)
Let $\phi_t(x,\xi)$ be the geodesic flow starting from $(x,\xi) \in S^*M$. Assume there exists $T_0, C >0$ such that for all $(x_0,\xi_0) \in S^* M, \alpha \in \Rb$ and $T \geq T_0$
$$
\frac{1}{T} \int_0^T W(\phi_t(x_0,\xi_0), \alpha + t) dt \geq C.
$$
\end{assumption}
\begin{theorem}\label{mainresult}
If $W(x,t) \in C^{\infty}_b(M \times \Rb),$ that is $W$ is smooth and there are $L^{\infty}$ bounds for all its derivatives, and $W$ satisfies Assumption \ref{timegcc}, then there exists $C,c>0$ such that all solutions to \eqref{DWE} satisfy
$$
E(u,t) \leq C e^{-ct} E(u,0).
$$
\end{theorem}
\begin{remark}
The assumption that $W$ is smooth can be relaxed to $W \in C^k_b$ for $k$ large enough depending on the dimension of $M$. The proof relies on compositions of pseudodifferential operators defined in terms of $W$, which produces error terms. These error terms are controlled if a large, but finite number of derivatives of $W$ are bounded. For ease of exposition the valeu of $k$ is not explicitly computed.
\end{remark}
When $M$ is compact without boundary and $W \in C^0(M)$, \cite{RauchTaylor1975} show that the GCC implies exponential decay. The techniques of \cite{Ralston1969} can be used to see that without the GCC exponential decay cannot occur. The best possible exponential decay rate was computed in \cite{Lebeau1996} in terms of long time averages of the damping and the spectral abscissa of the stationary equation. When $M$ has boundary \cite{BardosLebeauRauch1992} proved that the GCC implies exponential decay, while \cite{BurqGerard1997} show that the GCC is necessary for exponential decay. When $M$ is compact without boundary and $W$ is a 0th order pseudodifferential operator \cite{KeelerKleinhenz2020} show that the GCC is equivalent to exponential decay and compute the sharp exponential decay rate.
\begin{remark}\label{whynotsmoothremark}
In the stationary case it is only necessary to assume $W \in C^0$, because $W$ can be replaced with a $\underline{W} \in \Ci$ such that $\underline{W} \leq W$ and $\underline{W}$ still satisfies the geometric control condition. This is in part due to the fact that energy decay is equivalent to resolvent estimates on the associated stationary operator $-h^2 \Delta + i h W - 1$. Replacing $W$ by $\underline{W}$ in the stationary equation creates an error term $h(\underline{W}-W)$ which can be controlled as a perturbation. For a more detailed exposition of this classical argument see Section 2.3 of \cite{thesis}.
Although it is possible to construct such a $\underline{W} \in \Ci_b$ when $W \in C^0_b$, it is not straightforward to perform the replacement. Because $W$ is time dependent the problem cannot be reduced to a stationary equation and replacing $W$ by $\underline{W}$ in $(\p_t^2 -\Delta + W \p_t)u =0$ produces an error term $(W-\underline{W}) \p_t u$, which is incompatible with the very exact form the intermediate estimates take.
\end{remark}
There are a variety of results when the damping is allowed to depend on time, although many of them apply only in Euclidean space, require the damping to never vanish and establish polynomial decay rates for damping tending to 0 as time goes to infinite \cite{Matsumura1977, Uesaka1980, MochizukiNakazawa2001, Ikehata2003, HirosawaNakazawa2003, Yamazaki2006, Wirth2004, Wirth2006, Wirth2007, Kenigson2011, Hatvani2018}.
There are fewer energy decay results on manifolds when the damping term is allowed to vanish. If the damping is time-periodic then $W$ satisfying a version of the GCC implies exponential decay \cite{RousseaLebeauPeppinoTrelat}. That paper also applies in the case where $M$ has a boundary. Note that any damping which satisfies the GCC hypotheses of \cite{RousseaLebeauPeppinoTrelat} also satisfies Assumption \ref{timegcc}.
\textbf{Acknowledgements} I would like to thank Andras Vasy for proposing this question to me and for his helpful comments throughout the course of this project.
\section{Outline of Proof}\label{intermediatesection}
Let $\Omega_{\alpha}=M \times (\alpha,\alpha+T)$ where $T \geq T_0$. For ease of notation, norms with a subscript $\alpha$ are taken over $\Omega_{\alpha}$ for example $\ltwoo{u} = \left(\int_{\Omega_{\alpha}} |u(x,t)|^2 dx dt \right)^{1/2}$.
The following three results together can be used to prove the main result.
\begin{proposition}\label{poincwirt}
Let $\Pi_c$ be orthogonal projection onto $\ker(\Delta) \times \{0\}$ in $\Hc$ and suppose $W, \p_t W \in C^0_b(M \times \Rb)$.
If $u(t)$ solves \eqref{DWE} with initial data $(u_0,u_1) \in H^1 \times L^2$ and $\tilde{u}(t)$ solves \eqref{DWE} with initial data $(Id-\Pi_c)(u_0, u_1)$ then $u(t)= \tilde{u}(t)+ \Pi_c u_0$ and $E(u,t)=E(\tilde{u},t)$.
Furthermore $\tilde{u}(t)$ satisfies a Poincar\'e inequality, that is there exists a constant $C_{p}$ such that for all $t$
\begin{equation}\label{eq:poincwirt}
C_{p} \ltwom{\tilde{u}(\cdot, t)} \leq \ltwom{\nabla \tilde{u}(\cdot, t)}.
\end{equation}
\end{proposition}
\begin{proposition}\label{weirdenergy}
Suppose $W, \p_t W \in C^0_b(M \times \Rb)$. Let $t_1, t_2$ be such that $t_2-t_1>2$ and let $B_1=\max(3,\lp{W}{\infty}^2)$. For any $\e>0$ and $u$ solving \eqref{DWE}
with initial data of the form $(Id-\Pi_c)(u_0,u_1)$
$$
\frac{\e}{2B_1} (t_2-t_1-2) E(u)(t_2) - \frac{C_{p}^2 \e^2}{2B_1} (t_2-t_1) E(u)(t_1) \leq \int_{t_1}^{t_2} \ltwo{u_t}^2.
$$
\end{proposition}
Propositions \ref{poincwirt} and \ref{weirdenergy} are proved in Section \ref{splitsection}.
\begin{proposition}\label{propofsing} Assume $W \in C_b^{\infty}(M \times \Rb)$ satisfies Assumption \ref{timegcc}, then there exists $B_2>0$ such that for all $\alpha \in \Rb$ and all solutions $u$ of \eqref{DWE}
$$
\ltwoo{u_t}^2 \leq B_2 \ltwoo{W^{\frac{1}{2}} u_t}^2.
$$
\end{proposition}
Proposition \ref{propofsing} is proved in Section \ref{propsection}
\begin{proof}[Proof of Theorem \ref{mainresult}]
If $u$ solves \eqref{DWE} with initial data $(u_0, u_1)$, let $\ti{u}(t)$ be the solution of \eqref{DWE} with initial data $(Id-\Pi_c)(u_0, u_1)$. Then $u(t)=\ti{u}(t) + \Pi_c u_0$ and $E(u,t)=E(\ti{u},t)$. So without loss of generality assume $u$ has initial data of the form $(Id-\Pi_c)(u_0,u_1)$.
Recall
$$
\frac{d}{dt} E(u,t) = - \int_M W(x,t) |u_t|^2 dx_g.
$$
Integrating in $t$ from $\alpha$ to $\alpha+T$
$$
E(u,\alpha+T) - E(u)(\alpha) = - \ltwoo{W^{\frac{1}{2}} u_t}^2.
$$
Applying Proposition \ref{propofsing}
$$
E(u,\alpha+T) \leq E(u)(\alpha) - \frac{1}{B_2} \ltwoo{u_t}^2.
$$
Now applying Proposition \ref{weirdenergy} with $t_1=\alpha, t_2=\alpha+T$
$$
E(u,\alpha+T) \leq E(u)(\alpha) + \frac{C_{p}^2\e^2 T}{2 B_1 B_2} E(u)(\alpha) - \frac{\e (T-2)}{2 B_1 B_2 } E(u)(\alpha+T).
$$
Grouping like terms
$$
\left(1+\frac{\e (T-2)}{2B_1 B_2}\right) E(u)(\alpha+T) \leq \left( 1 + \frac{ C_{p}^2 \e^2T }{2 B_1 B_2} \right) E(u)(\alpha)
$$
So
$$
E(u)(\alpha+T) \leq \frac{2B_1 B_2 + C_{p}^2 \e^2 T}{2B_1B_2 +\e (T-2)} E(u)(\alpha).
$$
Now note that when $\e<\frac{(T-2)}{C_{p}^2 T}$
$$
\frac{2B_1 B_2 + C_{p}^2 \e^2 T}{2B_1B_2 +\e (T-2)}< 1,
$$
and so there exists a $c<1$ such that
$$
E(u)(\alpha+T) \leq c E(u)(\alpha) ,
$$
for any $\alpha$. Therefore
$$
E(u)(\alpha+kT) \leq c^k E(u)(\alpha).
$$
This along with the fact that the energy is non-increasing means that there exists $C,c>0$ such that
$$
E(u)(t) \leq C e^{-ct} E(u)(0),
$$
which completes the proof of Theorem \ref{mainresult}.
\end{proof}
\section{Proof of Proposition \ref{weirdenergy}}\label{splitsection}
This section contains the proofs of Proposition \ref{poincwirt} and Proposition \ref{weirdenergy}. The inequality \eqref{eq:poincwirt} is a key ingredient in the proof of Proposition \ref{weirdenergy}. Note that in this section I only require $W, \p_t W \in C^0_b$.
\subsection{Proof of Proposition \ref{poincwirt}}
Because $u(t)$ and $\ti{u}(t) + \Pi_c u_0$ each solve \eqref{DWE} with the same initial data $u(t)=\ti{u}(t) + \Pi_c u_0$. Furthermore by definition of the Energy
$$
E(u,t)= E(\ti{u}+\Pi_c u_0, t) = E(\ti{u},t).
$$
It remains to be seen that $\ti{u}(t)$ satisfies \eqref{eq:poincwirt} for all $t$. The inequality holds at $t=0$ by the Poincar\'e-Wirtinger inequality, because $\ti{u}(0)$ is orthogonal to constants and thus has average value 0. To see that \eqref{eq:poincwirt} holds with a constant uniform in $t$ it is enough to show that $\ti{u}(t)$ is orthogonal to constants for all times $t$.
To do so define the initial data space with position data orthogonal to constants, $\Hcd=(Id-\Pi_c)\Hc$, and equip it with the norm
$$
\hcdnorm{(u_0,u_1)}= \ltwo{\nabla u_0}^2 + \ltwo{u_1}^2.
$$
Note that this is indeed a norm as $\hcdnorm{(u_0,u_1)}^2=0$ is equivalent to $(u_0, u_1) \in ker(\Delta) \times \{0\} = \Pi_c \Hc$.
Now define $\Ac(t) = \begin{pmatrix} 0 & Id \\ \Delta & -W(t) \end{pmatrix}$ with $D(\Ac(t))=H^2(M) \times H^1(M)$ and define $\Acd(t)= \Ac(t)|_{\Hcd}$. Solutions of
\begin{equation}\label{katoeq}
\begin{cases} \p_t U = \Acd(t) U \\
U(0) = \begin{pmatrix} u_0 \\ u_1 \end{pmatrix} \in \Hcd,
\end{cases}
\end{equation}
are equivalent to solutions of \eqref{DWE} with initial data in $\Hcd$. A formal solution of \eqref{katoeq} is given by $V(t,0) \begin{pmatrix} u_0 \\ u_1 \end{pmatrix}$. The following proposition (\cite{Kato1953} Theorem 4) can be used to show that $V(t,0): \Hcd \ra \Hcd$. That is solutions with initial position data orthogonal to constants are orthogonal to constants for all time.
\begin{proposition}\label{katothm}
Assume that
\begin{enumerate}
\item The domain of $\Acd(t)$ is independent of $t$.
\item $\Acd(t)$ is defined for $a \leq t \leq b$ and for each $t,$ $\Acd(t)$ is the generator of a semigroup.
\item Define $B(t,s)= (I-\Acd(t))(I-\Acd(s))^{-1}$. There exists $M>0$ such that
$$
\|B(t,s)\| \leq M
$$
for every $s,t \in [a,b]$.
\item There exists $N>0$ such that for every partition $a<t_0 <t_1 < \cdots < t_n=b$ of the interval $(a,b)$ and for some $s$
$$
\sum_{j=1}^n \| B(t_j, s) - B(t_{j-1}, s) \| \leq N.
$$
\item $B(t,s)$ is weakly continuous in $t$ for some $s \in [a,b]$
\item $B(t,s)$ is weakly differentiable in $t$ and $\frac{\p B}{\p t}(t,s)$ is strongly continuous in $t$ for some $s \in [a,b]$.
\end{enumerate}
Then there exists $V(t,s)$ a bounded operator on $\Hcd$ with norm $\leq 1$ such that $U(t)=V(t,a) U(a)$, solves \eqref{katoeq}.
\end{proposition}
I will now show that $\Acd(t)$ satisfies the hypotheses of Proposition \ref{katothm}. By construction, the domain of $\Acd(t)$ does not depend on $t$, so Assumption \ref{timegcc} is satisfied. Now fix $[a,b]$ and $t\in [a,b]$.
To check Assumption 2 it is sufficient to show that $\Acd(t)$ is a maximal dissipative operator by the Lumer-Phillips Theorem \cite{LumerPhillips1961}. To see $\Acd(t)$ is dissipative consider $U=(u_0,u_1) \in \Dc(\Acd(t))$ and compute
\begin{align*}
\< \Acd(t) U, U \>_{\Hcd} &= \< \begin{pmatrix} 0 & Id \\ \Delta & -W(t) \end{pmatrix} \begin{pmatrix} u_0 \\ u_1 \end{pmatrix}, \begin{pmatrix} u_0 \\ u_1 \end{pmatrix}\>_{\Hcd} = \<\nabla u_1, \nabla u_0\>_{L^2} + \<\Delta u_0, u_1\>_{L^2} - \<W(t) u_1, u_1\>_{L^2} \\
&=-\<W(t) u_1, u_1\> \leq 0.
\end{align*}
So $\Acd(t)$ is dissipative. To see $\Acd(t)$ is maximally dissipative it is enough to see $(Id-\Acd(t))$ is onto. By \cite{AnantharamanLeautaud2014} Lemma 4.2 $(Id-\Ac(t))$ is onto. So for all $F \in \Hcd \subset \Hc$ there exists $U \in \Hc$ such that $(Id-\Ac(t))U=F$. Apply $(Id-\Pi_c)$ to both sides to see
\begin{align*}
(Id-\Pi_c) (Id-\Ac(t))U=F \\
(Id-\Acd(t))(Id-\Pi_c) U - \Ac(t)\Pi_c U + \Pi_c \Ac(t) U = F.
\end{align*}
So to see that $Id-\Acd(t)$ is surjective it is enough to see that $(\Pi_c \Ac(t) -\Ac(t) \Pi_c) U =0$. To do so it is convenient to introduce a new operator, related to $\Acd(t)$ that $\Pi_c$ commutes with.
\begin{lemma}\label{a0lemma}
Let $\Ac_0=\begin{pmatrix} 0 & Id \\ \Delta & 0 \end{pmatrix}$, then $\Pi_c \Ac_0 = \Ac_0 \Pi_c$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{a0lemma}]
By \cite{AnantharamanLeautaud2014} Lemma 4.2, $Sp(\Ac_0)$ consists of only isolated eigenvalues and
$$
Sp(\Ac_0) \subset \{0 + 0 i\} \cup \Rb.
$$
Let $\gamma$ be a positively oriented circle centered on $0$ with radius so small that $0$ is the only eigenvalue of $\Ac_0$ in the interior of $\gamma$. Define the spectral projector onto the 0 eigenspace of $\Ac_0$
$$
\Pi_0 = \frac{1}{2\pi i} \int_{\gamma} (\Ac_0-zId)^{-1} dz.
$$
By \cite{HislopSigal2012} Proposition 6.9, $\Pi_0 \Ac_0 = \Ac_0 \Pi_0$. Now I claim that $\Pi_0$ is orthogonal projection onto $\ker(\Delta) \times \{0\}$, that is $\Pi_0=\Pi_c$. Showing this will complete the proof of the lemma.
To see $\Pi_0=\Pi_c$, I will first show $\Pi_0=\Pi_0^*$, that is $\Pi_0$ is orthogonal projection onto its range. First note that $\Ac_0$ is skew adjoint, that is $\Ac_0^*=-\Ac_0$. Writing $R_{\Ac_0}(\lambda)=(\Ac_0-\lambda)^{-1}$ to represent the resolvent of $\Ac_0,$ then by the skew adjointness of $\Ac_0,$ $\Rc_{\Ac_0}(\lambda)^*=-\Rc_{\Ac_0}(-\bar{\lambda})$.
So now recognizing $\gamma$ as the contour $re^{i \theta}$ for $\theta \in [-\pi, \pi)$ and some small $r>0$
\begin{align*}
\Pi_0^* &= \frac{1}{2\pi i} \int_{\gamma} \Rc_{\Ac_0}(\lambda)^* dz = \frac{1}{2\pi i} \int_{\gamma} -(\Ac_0+\bar{\lambda} Id)^{-1} d\lambda \\
&=\frac{1}{2\pi} \int_{-\pi}^{\pi} -(\Ac_0 + re^{-i \theta})^{-1} r d \theta \\
&= \frac{1}{2\pi } \int_{0}^{2\pi} (\Ac_0-re^{i\vphi})^{-1} r d \vphi \text{ via change of variables } \vphi=-\theta+\pi \\
&= \Pi_0.
\end{align*}
Thus $\Pi_0$ is orthogonal.
To see that $ran(\Pi_0) \subset \ker(\Ac_0)=\ker(\Delta)\times\{0\}$ compute
\begin{align*}
\Ac_0 \Pi_0& = \frac{1}{2\pi i} \int_{\gamma} \Ac_0 (\Ac_0-\lambda)^{-1} d\lambda \\
&=\frac{1}{2\pi i} \int_{\gamma} \lambda(\Ac_0-\lambda)^{-1} d \lambda.
\end{align*}
Because $\Ac_0$ is skew adjoint it is also normal and thus $\| (\Ac-\lambda)^{-1}\| \leq d(\lambda, \sigma(\Ac_0))^{-1}$.
Therefore $\lambda(\Ac_0-\lambda)^{-1}$ is an analytic operator valued function and satisfies
$$
\| \lambda(\Ac_0-\lambda)^{-1} \|\leq \frac{|\lambda|}{d(\lambda, \sigma(\Ac_0))}.
$$
Taking $\gamma$ small enough so that $0$ is the closest point of $\sigma(\Ac_0)$ to $\gamma$
$$
\| \lambda(\Ac_0-\lambda)^{-1} \|\leq 1.
$$
Thus $\lambda(\Ac_0-\lambda)^{-1}$ is uniformly bounded on the interior of $\gamma$ minus $\{0\}$ so it extends to an analytic function on all of the interior of $\gamma$. Thus by Cauchy's theorem the integral form of $\Ac_0 \Pi_0$ vanishes so $\Ac_0 \Pi_0=0$ and $Ran(\Pi_0) \subset \ker(\Ac_0)$. Therefore $\Pi_0$ is orthogonal projection onto $\ker(\Delta)\times\{0\}$ and $\Pi_0=\Pi_c$.
\end{proof}
Now to see that $\Pi_c \Ac(t) - \Ac(t) \Pi_c=0$ note that $\Ac(t)= \begin{pmatrix} 0 & Id \\ \Delta & -W(t) \end{pmatrix} = \Ac_0 + \begin{pmatrix} 0 & 0 \\ 0 & -W(t) \end{pmatrix}$.
By Lemma \ref{a0lemma}, $\Pi_c$ commutes with $\Ac_0$ and so
\begin{align*}
(\Pi_c \Ac(t) -\Ac(t) \Pi_c) U &= \Pi_c\begin{pmatrix} 0 & 0 \\ 0 & -W(t) \end{pmatrix} U - \begin{pmatrix} 0 & 0 \\ 0 & -W(t) \end{pmatrix} \Pi_c U \\
&=\Pi_c \begin{pmatrix} 0 \\ -W(t)u_2 \end{pmatrix} - \begin{pmatrix} 0 & 0 \\ 0 & -W(t) \end{pmatrix} \begin{pmatrix} C \\ 0\end{pmatrix} \\
&=0.
\end{align*}
Therefore $(\Acd-Id)(Id-\Pi_c) U =F$ so $\Acd-Id: \Dc(\Acd) \ra \Hcd$ is onto as desired and by the Lumer-Phillips Theorem $\Acd(t)$ generates a contraction semigroup for each fixed $t$. Thus Assumption 2 is satisfied.
So now to check Assumption 3 is satisfied recall
$$
B(t,s) = (I - \Acd(t))(I - \Acd(s))^{-1}.
$$
Fix $s \in [a,b]$ and let $P_s=-\Delta +W(x,s) + 1$. For ease of notation I will write $W_s$ and $W_t$ to mean $W(x,s)$ and $W(x,t)$, respectively, for the remainder of this proof. Then by \cite{AnantharamanLeautaud2014} Lemma 4.2
$$
(I-\Acd(s))^{-1} = \begin{pmatrix}
P_s^{-1}(W_s+Id) & P_s^{-1} \\
P_s^{-1}(W_s+Id)-Id & P_s^{-1}
\end{pmatrix}.
$$
Therefore taking $(f_0, f_1) \in \Hcd$
\begin{align*}
(Id-\Acd(t)) (Id-\Acd(s))^{-1} \begin{pmatrix} f_0 \\ f_1 \end{pmatrix} &= \begin{pmatrix} Id & -Id \\ -\Delta & Id + W_t \end{pmatrix}\begin{pmatrix}
P_s^{-1}(W_s+Id)f_0+ P_s^{-1}f_1 \\
P_s^{-1}(W_s+Id)f_0-f_0 + P_s^{-1}f_1
\end{pmatrix}\\
&= \begin{pmatrix}
f_0 \\
(-\Delta + Id + W_t) P_s^{-1} ((W_s+Id)f_0 + f_1) - (Id + W_t)f_0
\end{pmatrix}
\end{align*}
Taking the norm of this in $\Hcd$
$$
\hcdnorm{B(t,s) \begin{pmatrix} f_0 \\ f_1 \end{pmatrix}}\leq \ltwo{\nabla f_0} + \ltwo{(-\Delta+Id+W_t)P_s^{-1}((W_s+Id)f_0+f_1)} + \ltwo{(Id+W_t)f_0}.
$$
The first term is controlled by $\hcdnorm{(f_0,f_1)}$. The third term is controlled by
$$\left(1+\lp{W}{\infty} \right) \ltwo{\nabla f_0} \leq C \hcdnorm{(f_0,f_1)}.$$ Therefore it remains to control the middle term.
To do so note that $-\Delta P_s^{-1} = Id - (W_s+Id)P_s^{-1}$ so the middle term becomes
\begin{align*}
\ltwo{(\Delta+Id+W_t)P_s^{-1}((W_s+Id)f_0+f_1)}& = \ltwo{(Id + (W_t-W_s)P_s^{-1}) ((W_s+Id)f_0+f_1)} \\
&\leq C(1+ \lp{W}{\infty}) \ltwo{(W_s+Id)f_0 +f_1} \\
&\leq C(1+\lp{W}{\infty})^2 \ltwo{f_0} + C(1+\lp{W}{\infty}) \ltwo{f_1}.
\end{align*}
Where I used that $P_s^{-1}$ is bounded on $L^2$ (c.f. \cite{AnantharamanLeautaud2014} Lemma 4.2). Now since $f_0$ is orthogonal to constants it satisfies a Poincar\'e inequality, i.e. $\ltwo{f_0} \leq C \ltwo{\nabla f_0}$ and so
$$
\hcdnorm{B(t,s)\begin{pmatrix}f_0\\f_1\end{pmatrix}} \leq C(1+\lp{W}{\infty})^2 \ltwo{\nabla f_0} + C(1+\lp{W}{\infty}) \ltwo{f_1} \leq C (1+\lp{W}{\infty})^2 \hcdnorm{\begin{pmatrix} f_0 \\ f_1 \end{pmatrix}}.
$$
Since $W \in C_b^0$, $\lp{W}{\infty}$ is finite and Assumption 3 is satisfied.
Now to see that Assumption 4 is satisfied recall that
$$
B(t,s) \begin{pmatrix} f_0 \\ f_1 \end{pmatrix} = \begin{pmatrix}
f_0 \\
(-\Delta + Id + W_t) P_s^{-1} ((W_s+Id)f_0 + f_1) - (Id + W_t)f_0
\end{pmatrix}
$$
Therefore
\begin{equation}\label{bdifference}
\left(B(t_j,s)-B(t_{j-1},s) \right) \begin{pmatrix} f_0 \\ f_1 \end{pmatrix} = \begin{pmatrix} 0 \\ (W_{t_j}-W_{t_{j-1}}) (P_s^{-1} ((W_s+Id)f_0 + f_1) -f_0)
\end{pmatrix}
\end{equation}
Thus
\begin{align*}
\hcdnorm{\left(B(t_j,s)-B(t_{j-1},s) \right) \begin{pmatrix} f_0 \\ f_1 \end{pmatrix} } &\leq \ltwo{(W_{t_j}-W_{t_{j-1}}) (P_s^{-1} ((W_s+Id)f_0 + f_1) -f_0)} \\
&\leq \lp{W_{t_j}-W_{t_{j-1}}}{\infty} \left( C(1+\lp{W}{\infty}) \ltwo{\nabla f_0} + \ltwo{f_1} \right).
\end{align*}
So
$$
\| B(t_j,s) - B(t_{j-1}, s)\| \leq C( 1 + \lp{W}{\infty}) \lp{W_{t_j} - W_{t_{j-1}}}{\infty}.
$$
And so
$$
\sum_{j=1}^n \| B(t_j, s) -B(t_{j-1}, s)\| \leq C(1+ \lp{W}{\infty})) \sum_{j=1}^n \lp{W_{t_j} - W_{t_{j-1}}}{\infty}
$$
Since $W, \p_t W\in C_b^0$ then $W \in BV(t, L^{\infty}(M))$ and so this right hand side is bounded by some large $N$. Thus Assumption 4 is satisfied.
By an analogous argument using \eqref{bdifference}, $B(t,s)$ is continuous and differentiable in $t$ so long as $W(t)$ is, so Assumptions 5 and 6 are satisfied. Note that checking Assumptions 4,5 and 6 are the parts of this section where $W \in C^0_b(M \times \Rb)$ is not sufficient and necessitate the assumption $\p_t W \in C^0_b$.
Since the assumptions of Proposition \ref{katothm} are satisfied $V(t,a): \Hcd \ra \Hcd$. That is solutions with initial position data orthogonal to constants are orthogonal to constants for all time and so \eqref{eq:poincwirt} holds with a constant uniform in $t$.
\subsection{Proof of Proposition \ref{weirdenergy}}
By Proposition \ref{poincwirt} it is enough to prove this result for $u$ with initial data in $(1-\Pi_c)\Hc$.
Let $g \in \Cs(t_1,t_2)$ be nonnegative and identically 1 on $(t_1+1, t_2-1)$. Insert $g$ into the following intergal and integrate by parts inside the $L^2$ norm
\begin{align*}
\int_{t_1+1}^{t_2-1} \ltwo{\nabla u}^2 dt &\leq \int_{t_1}^{t_2} g(t) \ltwo{\nabla u}^2 dt \\
&= \int_{t_1}^{t_2} g(t) \<-\Delta u, u\>_{L^2} dt.
\end{align*}
Now apply \eqref{DWE} to replace $-\Delta u=(-\p_t^2 -W \p_t)u$
$$
\int_{t_1+1}^{t_2-1} \ltwo{\nabla u}^2 dt \leq \int_{t_1}^{t_2} g(t) \<(-\p_t^2 - W \p_t) u, u\> dt.
$$
Integrate by parts the $\p_t^2$ term. Note there are no boundary terms because $g$ is compactly supported in $(t_1,t_2)$.
$$
\int_{t_1+1}^{t_2-1} \ltwo{\nabla u}^2 dt \leq \int_{t_1}^{t_2} g'(t) \<u_t, u\> + g \ltwo{u_t}^2 - g\<Wu_t, u\> dt.
$$
Now take absolute values of both sides and choose $g$ such that $|g| \leq 1$ and $|g'|\leq 2$
$$
\int_{t_1+1}^{t_2-1} \ltwo{\nabla u}^2 dt \leq \int_{t_1}^{t_2} 2|\<u_t,u\>| + \ltwo{u_t}^2 + |\<Wu_t, u\>| dt.
$$
Let $\e>0$ and use Young's inequality for products on the first and third terms,
\begin{align*}
\int_{t_1+1}^{t_2-1} \ltwo{\nabla u}^2 dt &\leq \int_{t_1}^{t_2} \frac{2}{\e} \ltwo{u_t}^2 + \frac{\e}{2} \ltwo{u}^2 + \ltwo{u_t}^2 + \frac{1}{2\e} \ltwo{Wu_t}^2 + \frac{\e}{2} \ltwo{u}^2. \\
&\leq \left(1+ \frac{4+\lp{W}{\infty}^2}{2\e}\right) \int_{t_1}^{t_2} \ltwo{u_t}^2 dt + \e \int_{t_1}^{t_2} \ltwo{u}^2.
\end{align*}
Now add $\int_{t_1+1}^{t_2-1} \ltwo{u_t}^2 dt$ to both sides
$$
\int_{t_1+1}^{t_2-1} E(u)(t) dt = \int_{t_1+1}^{t_2-1} \ltwo{\nabla u}^2 + \ltwo{u_t}^2 \leq \left(2+ \frac{4+\lp{W}{\infty}^2}{2\e} \right) \int_{t_1}^{t_2} \ltwo{u_t}^2 + \e \int_{t_1}^{t_2} \ltwo{u}^2.
$$
Multiply through by $\e$
$$
\e \int_{t_1+1}^{t_2-1} E(u)(t) dt \leq \left(2\e+\frac{\lp{W}{\infty}^2}{2}+2\right) \int_{t_1}^{t_2} \ltwo{u_t}^2 + \e^2 \int_{t_1}^{t_2} \ltwo{u}^2.
$$
Now let $C_{p}$ be the Poincar\'e constant from Proposition \ref{poincwirt} and subtract $C_{p}^2 \e^2 \int_{t_1}^{t_2} E(u)(t) dt$ from both sides.
$$
\e\int_{t_1+1}^{t_2-1} E(u)(t) - C_{p}^2 \e^2 \int_{t_1}^{t_2} E(u)(t) \leq \left(2\e+\frac{\lp{W}{\infty}^2}{2}+2 - C_{p}^2 \e^2\right) \int_{t_1}^{t_2} \ltwo{u_t}^2 + \e^2 \int_{t_1}^{t_2} \ltwo{u}^2 - C_{p}^2 \ltwo{\nabla u}^2 dt.
$$
By Proposition \ref{poincwirt} $\ltwo{u}^2 \leq C_{p}^2 \ltwo{\nabla u}^2$. Therefore the second integral on the right hand side is negative. That integral and the $-C_{p}^2 \e^2 \int \|u_t\|$ term on the right hand side can both be dropped
$$
\e\int_{t_1+1}^{t_2-1} E(u)(t) - C_{p}^2 \e^2 \int_{t_1}^{t_2} E(u)(t) \leq \left(2\e+\frac{\lp{W}{\infty}^2}{2}+2\right) \int_{t_1}^{t_2} \ltwo{u_t}^2.
$$
Now, assuming $\e<\frac{1}{2}$ let $B_1=\max(3,\lp{W}{\infty}^2)$ and then
\begin{equation}\label{prenonincrease}
\e\int_{t_1+1}^{t_2-1} E(u)(t) - C_{p}^2 \e^2 \int_{t_1}^{t_2} E(u)(t) \leq 2B_1 \int_{t_1}^{t_2} \ltwo{u_t}^2.
\end{equation}
Since the Energy is non-increasing
$$
\int_{t_1+1}^{t_2-1} E(u)(t) \geq (t_2-t_1-2) E(t_2-1) \geq (t_2-t_1-2)E(t_2),
$$
and
$$
\int_{t_1}^{t_2} E(u)(t) \leq (t_2-t_1) E(u)(t_1).
$$
Plugging these back into \eqref{prenonincrease} gives
$$
\e(t_2-t_1-2) E(t_2) - C_{p}^2 \e^2 (t_2-t_1) E(t_1) \leq 2B_1 \int_{t_1}^{t_2} \ltwo{u_t}^2.
$$
Finally dividing both sides by $2B_1$ gives the desired inequality.
\section{Proof of Propagation of Singularities: Proposition \ref{propofsing}}\label{propsection}
To begin, recall $\Omega_{\alpha} = M \times [\alpha, \alpha+T]$. I will prove estimates on three pieces of $T^* \Omega_{\alpha}$ and then combine them back together to get the desired estimate.
The three pieces are a damped region, an elliptic region, and a propagating region. Recall $\Cm$ from Assumption \ref{timegcc},
\begin{align*}
U_W &= \{W > \frac{\Cm}{200}\} \cap T^* \Omega_{\alpha} \\
U_E &= \{ |\tau^2 -|\xi|^2| \geq \d \<\zeta\>^2 \} \cap T^* \Omega_{\alpha} \\
U_p&=\{ |\tau^2 -|\xi|^2| \leq 2\d \<\zeta\>^2 \} \cap \{W <\frac{\Cm}{100}\} \cap T^* \Omega_{\alpha}.
\end{align*}
It is straightforward to prove the desired estimate on the damped region, $U_W$ and the elliptic region $U_E$ but the value of $\d$ is determined in the proof of the estimate on $U_p$. Because of this I will prove the estimate on $U_p$ first.
The proof of the estimate on $U_p$ follow the classical proof of Propagation of Singularities c.f. \cite{Taylor1981} Theorem 2.1. The unique feature is that the escape function is explicitly constructed in product coordinates around individual null bicharacteristics. This makes it possible to ensure constants in the eventual estimate are uniform in the starting time $\alpha$, this uniformity is the key to proving the decay result.
Consider $u$ solving \eqref{DWE}. Then let $v= \p_t u$ and differentiate both sides of \eqref{DWE} to obtain
\begin{equation}\label{DDWE}
Pv:=(\p_t^2 -\Delta + W \p_t + W') v =0.
\end{equation}
Note that the principal symbol of $P$ is $p=\tau^2 -|\xi|^2$ so the projection of null bicharacteristics of $P$ to $M\times \Rb$ are exactly the geodesics of $M \times \Rb$. Throughout $\Op$ will denote the Weyl Quantization.
\begin{lemma}\label{lemmabigrprop}
There exists $C,c>0$ and $r \in S^0(T^* (M \times \Rb))$ with $r \geq 0, r\geq c$ on $U_p$ such that if $R=\Op(r)$ for $v$ solving $Pv =0$
\begin{equation}\label{propregion}
\ltwoo{Rv}^2 \leq C \hhalfo{v}^2 + C (T_0+1) \lp{W}{\infty} \ltwoo{W^{\frac{1}{2}} v}^2.
\end{equation}
\end{lemma}
To begin consider $\rho \in U_{p,0}:=p^{-1}(0) \cap W^{-1}([0,\frac{\Cm}{100}]) \cap S^*\Omega_{\alpha}$ and let $\gamma_{\rho}$ be the null bicharacteristic of $P$ through $\rho$.
By abuse of notation I will write $W(\gamma_{\rho}(s))$ to be $W$ evaluated at the $(x,t)$ coordinate of $\gamma_{\rho}(s)$.
I will refer to the base variables, $(x,t)$, together as $z$ and the fiber variables, $(\xi,\tau)$ together as $\zeta$. I will also use $\Pi_z, \Pi_{\zeta}$ to denote projection onto these variables respectively and write $\rho_z=\Pi_z \rho, \rho_{\zeta}=\Pi_{\zeta} \rho$.
I will now construct coordinates around these null bicharacteristics. Rephrasing Assumption \ref{timegcc} in terms of $\gr$, there exists $\Cm, T_0>0$ such that for all $T\geq T_0$
$$
\frac{1}{T}\int_0^T W(\gr(s)) ds \geq \Cm.
$$
Now for a fixed $\rho$, let $\Tr$ be the smallest $T$ such that
\begin{equation}\label{Trdef}
\frac{1}{\Tr} \int_0^{\Tr} W(\gr(s)) ds = \Cm.
\end{equation}
Since $W(\gr(0)) \leq \frac{\Cm}{100}$ and $W$ is uniformly continuous there exists a constant $T_1$ such that for all $\rho$, $T_1 \leq \Tr \leq T_0$.
Note $\gr(\Tr) \geq \Cm$, otherwise a smaller $T$ would satisfy \eqref{Trdef}. Now by the uniform continuity of $W$ there exists $\e>0$ such that $|z_1-z_2| \leq 2\et $ implies $|W(z_1)-W(z_2)| \leq \frac{\Cm}{4}$. Then
$$
|\Pi_z(\gr(\Tr)-\gr(\Tr+\et))| \leq 2\et,
$$
because null bicharacteristics with $\rho \in S^*\Omega_{\alpha}$ travel at unit speed in $z$. Therefore
\begin{equation}\label{wontr}
W(\gr(s)) \geq \frac{3\Cm}{4} \quad \text{ for } s \in [\Tr, \Tr + \e].
\end{equation}
Let $\Sigma_{\rho}$ be a hypersurface in $M\times \Rb$ transverse to $\gr$ and let $O_{\rho} \subset \Sigma_{\rho}$ be a neighborhood of $\rho_z$, $V_{\rho}$ a neighborhood of $\rho_{\zeta}$ in $\Pi_{\zeta}( S^*(M \times \Rb))$. Choose these neighborhoods small enough so that
$$
\Phi_{\rho} : \left[-\frac{1}{2}, \Tr+\et\right] \times O_{\rho} \times V_{\rho} \ra S^*(M \times \Rb), \quad \Phi_{\rho}(s,z,\zeta) = \exp(s H_p)(z,\zeta),
$$
is a diffeomorphism onto its image. Also choose $O_{\rho}, V_{\rho}$ small enough so that
\begin{equation}\label{orvrconstruct}
|W(\Phir(s,z_1,\zeta_1))-W(\Phir(s,z_2,\zeta_2))| \leq \frac{\Cm}{4},
\end{equation}
for any $s \in [-\frac{1}{2}, \Tr+\et], z_1, z_2 \in O_{\rho}, \zeta_1,\zeta_2 \in V_{\rho}$. Because $\Tr$ is taken from a compact interval, $S^* \Omega_{\alpha}$ is compact, and $W$ is uniformly continuous there are uniform lower bounds on how small $O_{\rho}$ and $V_{\rho}$ must be taken.
Now extend $\Phi_{\rho}$ to $T^*(M \times \Rb)$ so that its image is conic. Let $\sigma_{\lambda}(z,\zeta) = (z,\lambda \zeta)$ and define
$$
\Phi_{\rho} : \left[-\frac{1}{2}, \Tr+\et\right] \times O_{\rho} \times V_{\rho} \times [1, \infty) \ra T^* (M \times \Rb) \backslash \{0\}, \quad \Phi_{\rho}(s,z,\zeta, \lambda) = \sigma_{\lambda} \left( \exp(s H_p)(z,\zeta) \right).
$$
Then $\Phi_{\rho}$ defines product coordinates $(s,z,\zeta,\lambda)$ on its image.
Now using this coordinate system I will construct an escape function. In this lemma inner products are all taken with respect to the $L^2_{\alpha}$ norm. Let $\Hpo$ be the Hamilton vector field of $p=\tau^2-\xi^2$.
\begin{lemma}\label{lemmaescape}
There exist constants $C_1,C_2, C_3, c>0$ such that for any $\gr$, a null bicharacteristic as constructed above, there exist symbols $\are, \rr, \qr \in S^0(T^*(M \times \Rb))$ such that if $\Ar=\Op(\are)$ and $v$ solves $P v=0$ then
\begin{enumerate}
\item $\are \geq 0$ and
$$\<\Op(\Hpo \are^2) v,v\> - C_1 \ltwoo{\Ar v}^2 \leq C_2 \hhalfo{v}^2.$$
\item $\qr\geq0, \qr^2 \leq C_3 (T_0+1) \lp{W}{\infty} W$.
\item $\rr \geq 0$, $\rr$ is homogeneous of degree $0$ and $\rr\geq c$ on a neighborhood in $S^*(M \times \Rb)$ of uniform width (i.e. not depending on $\rho$) around $\gr(s=0)$
\item $\rr^2 \leq \Hpo \are^2 - C_1 \are^2 + \qr^2$.
\item
$$
\ltwoo{\Op(\rr) v}^2 \leq C_2 \hhalfo{v}^2 + C_3 (T_0+1) \lp{W}{\infty} \ltwoo{W^{\frac{1}{2}} v}^2.
$$
\end{enumerate}
\end{lemma}
\begin{proof}
To begin multiply $P$ on the left by an elliptic operator $E \in \Psi^{-1}$. Note that $char(EP)=char(P)$ and if $p_1=\sigma(EP)$ then $\Hpo \sim H_{p_1}$. Because of this for the rest of the proof I can replace $P$ by $EP$, so notably $P \in \Psi^1$.
Define $\phir \in \Cs(O_{\rho}), \chir \in \Cs(V_{\rho})$ to be nonnegative and 1 near $\rho_z$ and $\rho_{\zeta}$ respectively. Note that by the uniform lower bound on the size of $V_{\rho}, \chir$ can be constructed so that there exists a $\ti{\d}>0$ which does not depend on $\rho$ such that
\begin{equation}\label{chirdef}
\chi_{\rho}(\zeta) > \frac{1}{2} \text{ on } |\zeta-\rho_{\zeta}| < \ti{\d}, \zeta \in V_{\rho}.
\end{equation}
Now define
$$
\grk(s,z,\zeta,\lambda)= \left( \Cm - {W}(\gr(s))\right) \phir(z) \chir(\zeta).
$$
Let $\ar$ be the solution of
$$
\begin{cases}
\Hpo \ar =\grk \quad s \in [0,\Tr] \\
\ar = \Cm \et^2 \text{ on } \Phi_{\rho}(s=0,z,\zeta,\lambda).
\end{cases}
$$
In the product coordinates from $\Phi_{\rho}$
$$
\Hpo \ar(s,z,\zeta, \lambda) = \frac{\p}{\p s} \ar(s,z,\zeta,\lambda).
$$
Therefore if
\begin{equation*}\label{arkdef}
\alphar(s) = \Cm s - \int_0^s {W}(\gr(\ti{s})) d\ti{s} + \Cm \et^2,
\end{equation*}
then $\ar(s,z,\zeta,\lambda)=\alphar(s) \phir(z) \chir(\zeta)$ for $s \in [0,\Tr]$. Note that $\alpha(s) \geq 0$ if $\Cm \geq \frac{1}{s} \int_0^s W(\gr(\ti{s}) d\ti{s}$, and by definition of $\Tr$ this holds for all $s \in (0,\Tr]$.
Now extend $\alpha_{\rho}$ to be a smooth function compactly supported in $s \in [-\frac{1}{2},\Tr+\et]$. $\alphar$ can be constructed so that it has non-negative derivative on $[-\frac{1}{2},0]$, and so that there exists $\e_1>0$ such that for all $\rho$, on $[-\e_1,0],$
\begin{equation}\label{eq:alpharneg}
\alphar\geq \frac{\Cm \et^2}{2} \text{ and } \alphar'\geq \frac{\Cm}{100}.
\end{equation}
Note that $\alphar$ does not actually depend on $\rho$ in $[-\e_1,0]$, because $\e_1$ and $\Cm \et^2$, which are the key constants used to define it there, do not depend on $\rho$. Furthermore, because $\e$ and $\Cm \e^2$ are uniform in $\rho$ and $\p_s \alphar = \Cm - W(\gr(s))$ is uniformly bounded by $2 \lp{W}{\infty}$, $\alphar$ can be constructed so that there exists $C^* \geq 1$ such that for all $\rho$
\begin{equation}\label{eq:atret}
|\p_s \alpha_{\rho}| \leq C^* \lp{W}{\infty} \text{ for } s \in [\Tr, \Tr+\et].
\end{equation}
Now let $\ti{\phir}(z) \in \Cs(O_{\rho}), \ti{\chir}(\zeta)\in \Cs(V_{\rho}) $ be nonnegative and identically 1 on $\supp \phir, \sup \chir$ respectively. Use these to define $\aro(s,z,\zeta,\lambda) = s \ti{\phir}(z) \ti{\chir}(\zeta)$ and note $\aro \in S^0$ and $\Hpo \aro \equiv 1$ on $\supp \ar$.
Using these define
$$
\are(s,z,\zeta,\lambda) = \ar e^{\eta \aro} = \alphar(s) \phir(z) \chir(\zeta) e^{\eta \aro(s,z,\zeta,\lambda)} \in S^0(T^* (M \times \Rb)),
$$
where $\eta$ is to be chosen later, and note that $\are \geq 0$.
To see the inequality in 1) recall that $Pv=0$ and compute
\begin{equation}\label{eq:commmain}
0=\Im\<\Ar P v, \Ar v\> = \Im \<[\Ar, \Re P]v, \Ar v\> + \Re \<[\Ar,\Im P]v, \Ar v\> + \Re \<\Im P \Ar v, \Ar v\>
\end{equation}
Note $\Im P= E W \p_t \in \Psi^0$ so there exists $C_1>0$ such that
\begin{equation}\label{eq:notcomm}
|\<\Im P \Ar v, \Ar v\>| \leq C_1 \ltwoo{\Ar v}^2.
\end{equation}
Now note $\Ar^* [\Ar, \Re P]$ has principal symbol $\frac{i}{2} \Hpo \are^2 \in S^0$ therefore there exists $C_2 >0$ such that
\begin{equation}\label{eq:comm}
| \<[\Ar, \Re P] u, \Ar u\> | \geq \frac{1}{2} \< \Op(\Hpo \are^2)u,u\> - C_2 \hhalfo{u}^2.
\end{equation}
Furthermore $\Ar^* [\Ar, \Im P] \in \Psi^{-1}$ so there exists $C_3$ such that
\begin{equation}\label{eq:imcomm}
|\<[\Ar, \Im P]u, \Ar u\>| \leq C_3 \hhalfo{u}^2.
\end{equation}
Note that the $C_j$ depend on finitely many derivatives of $p$ and $\are$. By the construction of $\are$ there is control, uniform in $\rho$, on these derivatives and so the $C_j$ are uniform in $\rho$ as well.
Combining \eqref{eq:commmain}, \eqref{eq:notcomm}, \eqref{eq:comm} and \eqref{eq:imcomm}, then relabeling constants gives exactly part 1).
Note that $\ar$ was defined on $S^* (M \times \Rb)$ and then extended via 0-homogeneity to the image of $\Phir$. This leaves a compact subset, in $\zeta$, of $T^* (M \times \Rb)$ on which the symbols are undefined. However this region only contributes $S^{-\infty}$ terms to the symbols, which become $\Psi^{-\infty}$ terms after being quantized. These contribute $H^{-\infty}$ terms in estimates which are then absorbed into the $H^{-\frac{1}{2}}$ terms. This same argument will hold for the construction of $\qr$ and $\rr$.
For part 2), define $\ti{W} \in \Cs(\{{W}>\frac{\Cm}{4}\}),$ a nonnegative function such that $\ti{W}={W}$ on $\{{W} \geq \frac{\Cm}{2}\}$ and define
$$
\qr(s,z,\zeta, \lambda) = \left(2 C^* (T_0+1) \lp{W}{\infty} \ti{W}(\gr(s)) \right)^{\frac{1}{2}} \phir(z) \chir(\zeta) e^{\eta \aro}.
$$
Note that $\gr(s)=\Phir(s,\rho_z,\rho_{\zeta},1)$ and so by \eqref{orvrconstruct}
$$
|W(\gr(s))-W(\Phir(s,z,\zeta,\lambda))| \leq \frac{\Cm}{2}.
$$
Therefore $W(\gr(s)) \leq 2 W(\Phir(s,z,\zeta,\lambda))$ on $W \geq \frac{\Cm}{4}$. Thus there exists $C_3>0$ such that for all $\rho$
$$
\qr^2 \leq C_3(T_0+1) \lp{W}{\infty} W,
$$
which is 2).
Now to show 3) first recall $\alphar(0)=\Cm \et^2$. Also recall by \eqref{eq:alpharneg} there exists $\e_1>0$ such that for all $\rho, \alphar \geq \frac{\Cm \et^2}{2}$ on $[-\e_1,0]$. On the other hand, by the uniform continuity of $W$ there exists $\e_2>0$ such that for all $\rho, \alphar \geq \frac{\Cm \et^2}{2}$ on $[0,\e_2]$. So letting $\e_3=\min(\e_1,\e_2)$ define $\mu(s) \in \Cs(-\e_3,\e_3)$ such that $\mu(0)=\frac{\Cm \et^2}{4}$ is a maximum and $\mu(s) \leq \alphar(s)$ everywhere. Then define
$$
\rr(s,z,\zeta,\lambda) = \phir(z) \chir(\zeta) \left(\frac{\Cm}{100} \mu(s)\right)^{\frac{1}{2}} e^{\eta \aro},
$$
so $\rr \geq 0$. Note that because $\rr$ does not depend on $\lambda$ it is homogeneous of degree $0$. By the uniform lower bound on the size of $O_{\rho}, V_{\rho}$ and since $\mu$ does not depend on $\rho$ there exists a uniformly large set around $\gr(s=0)$ in $S^*(M\times \Rb)$ such that $\rr \geq \frac{\Cm \et^2}{8}$ there. This completes 3).
Now to show 4) consider
\begin{align}\label{eq:arqraboverr}
\Hpo \are^2 - C_1 \are^2 + \qr^2 &= 2 e^{2 \eta \aro} \ar \Hpo \ar + 2\eta \are^2 - C_1 \are^2 + \qr^2 \nonumber \\
&\geq 2 e^{2 \eta \aro} \ar \Hpo \ar + \qr^2 & \text{ choosing } \eta \geq C_1 \nonumber\\
&= 2 e^{2 \eta \aro} \phir^2 \chir^2 \alphar \alphar' + \qr^2 \nonumber \\
&= 2 e^{2 \eta \aro} \phir^2 \chir^2\left( \alphar \alphar'+C^* (T_0+1) \lp{W}{\infty} \ti{W}\right)
\end{align}
Note that $\rr^2= e^{2 \eta \aro}\phir^2 \chir^2 \frac{\Cm}{100} \mu(s)$ so to show $\rr$ lies underneath \eqref{eq:arqraboverr} it is enough to show
\begin{equation}\label{eq:undersimple}
\frac{\Cm}{100} \mu(s) \leq \alphar \alphar'+C^* (T_0+1) \lp{W}{\infty} \ti{W}.
\end{equation}
It is convenient to consider the following 3 cases
\begin{enumerate}[i]
\item $s \in [-\frac{1}{2},0],$
\item $s \in [0,\Tr],$
\item $s \in [\Tr, \Tr+\et]$.
\end{enumerate}
In case i) combining $\mu(s) \leq \alphar(s)$ everywhere and \eqref{eq:alpharneg} gives
$\frac{\Cm}{100} \mu(s) \leq 2\alphar \alphar'$ on $[-\frac{1}{2},0]$, as desired.
In case ii) there are two subcases $\{W \leq \frac{3 \Cm}{4}\}$ and $\{W \geq \frac{\Cm}{2}\}$.
When $W \leq \frac{3 \Cm}{4}$ then $\alphar' = \Cm - W(\gr) \geq \frac{\Cm}{4}$ and so combined with \eqref{eq:arqraboverr}
\begin{equation}\label{eq:subcasea}
\alphar \alphar'+C^* (T_0+1) \lp{W}{\infty} \ti{W} \geq \frac{\Cm}{4} \alphar.
\end{equation}
When $W \geq \frac{\Cm}{2}$ then $\ti{W}=W$ and so
$$
\alphar \alphar'+C^* (T_0+1) \lp{W}{\infty} \ti{W} = \alphar \alphar'+C^* (T_0+1) \lp{W}{\infty}W.
$$
By definition $\alphar \leq (T_0+1) \lp{W}{\infty}$ and so
$$
2\alphar(\Cm - W(\gr)) + 2C^* (T_0+1) \lp{W}{\infty} W(\gr) \geq \alphar \Cm.
$$
Combining this with \eqref{eq:subcasea} on $[0,\Tr]$
$$
2\alphar \alphar'+2C^* (T_0+1) \lp{W}{\infty} \ti{W} \geq \frac{\Cm}{2} \alphar \geq \frac{\Cm}{100} \mu,
$$
as desired.
In case iii) by \eqref{wontr}, $W(\gr(s)) \geq \frac{3 \Cm}{4}$. By construction $0 \leq \alphar \leq \Cm \et^2$ and by \eqref{eq:atret} $|\alphar'| \leq C^* \lp{W}{\infty}$ there. Also $\ti{W}=W$ there and so
\begin{align*}
2 \alphar \alphar' +2C^* (T_0+1) \lp{W}{\infty} W(\gr) &\geq 2 C^* \lp{W}{\infty}(-\Cm \et^2 + W(\gr)) \\
&\geq 2 C^* \lp{W}{\infty} \frac{\Cm}{2} \geq 0 = \frac{\Cm}{100} \mu(s),
\end{align*}
for $s \in [\Tr, \Tr + \et]$. Altogether this shows 4).
Finally 5) follows from applying 4) to a solution of $Pv=0$, then pairing with $v$ and applying 1), 2) and the G\r{a}rding inequality (\cite{Taylor1981} Theorem 8.1) and relabeling constants.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lemmabigrprop}]
Consider $\{\rr \geq {c} \}$, which by Lemma \ref{lemmaescape} part 3 contains a neighborhood of uniform width around $\rho$. Therefore
$$
\bigcup_{\rho \in U_{p_0}} \left\{\rr \geq {c} \right\} \supset W^{-1}\left(\left[0,\frac{\Cm}{100}\right]\right) \cap p^{-1}(0) \cap S^* \Omega = U_{p_0}
$$
Now by \eqref{chirdef} there exists $\ti{\d}>0$ such that $\chi_{\rho}(\zeta) > \frac{1}{2}$ on $\{\zeta \in S^*_{\rho_z} (M \times \Rb); |\zeta-\rho_{\zeta}|<\ti{\d}\}$. By the product construction of $\rr$, $\rr \geq \frac{c}{2}$ on $\{\zeta \in S^*_{\rho_z}(M \times \Rb); |\zeta - \rho_{\zeta}| < \ti{\d}\}$.
Now because $U_{p_0} \subset p^{-1}(0)$, $\d$ can be choose small enough so that
$$
\{\zeta \in S^*(M \times \Rb); |\tau^2-|\xi|^2| \leq 2 \delta \} \subset \bigcup_{\rho \in U_{p_0}} \{\zeta \in S^*_{\rho_{z}} (M\times \Rb); |\zeta - \rho_{\zeta}| < \ti{\d} \}.
$$
Therefore
$$
\bigcup_{\rho} \{\rr \geq \frac{c}{2}\} \supset \{\zeta; |\tau^2 - |\xi|^2| \leq 2 \delta \} \cap S^* \Omega \cap W^{-1}([0,\frac{\Cm}{100}]).
$$
Note this $\d>0$ is the one used to define $U_p$ and $U_e$.
Now since $S^* \Omega_{\alpha}$ is compact there exists a finite subcover
$$
\bigcup_{j=1}^n \left\{\rrj \geq {c} \right\} \supset \{\zeta; |\tau^2 - |\xi|^2| \leq 2 \delta \} \cap S^* \Omega \cap W^{-1}([0,\frac{\Cm}{100}]).
$$
Note that since $\{\rr \geq c\}$ is uniformly bounded from below in size there is uniform control (in $\alpha$) of the number of sets, $n$, it takes to form this cover.
Now since the $\rr$ are homogeneous of degree $0$ this holds on the conic extension of these sets to $T^* \Omega$ (away from a compact set in $\zeta$, call it $K$)
$$
\bigcup_{j=1}^n \left\{\rrj \geq {c} \right\} \supset \{\zeta; |\tau^2 - |\xi|^2| \leq 2 \delta \<\zeta\>^2 \} \cap (T^* \Omega \backslash K) \cap W^{-1}([0,\frac{\Cm}{100}]).
$$
Therefore there exists $r_{\infty} \in S^{-\infty}$ such that $\sum_{j=1}^n \rrj^2 + r_{\infty} \geq \frac{c}{4}$ on $U_p$. Letting $r = \left( \sum_{j=1}^n \rrj + r_{\infty} \right)^{\frac{1}{2}}$ then $r \geq \frac{c}{2}$ on $U_p$. Now applying part 5 of Lemma \ref{lemmaescape}.
\begin{align*}
\ltwo{\Op(r) v}^2 &= \sum_{j=1}^n \< \Op(\rrj^2) v, v\> + C \hp{v}{-\frac{1}{2}}^2 \\
&\leq n \left( C \hp{v}{-\frac{1}{2}}^2 + C_3 (T_0+1) \lp{W}{\infty} \ltwo{W^{\frac{1}{2}} v}^2 \right),
\end{align*}
since $n$ is uniform with respect to $\alpha$ this is exactly the desired conclusion.
\end{proof}
It is now possible to obtain estimates on $U_W$ and $U_E$.
Define $\chi_W \in \Cs(\{W>\frac{\Cm}{400}\})$ so that $\chi_W \equiv 1$ on $\{W > \frac{\Cm}{200}\}$ then $\chi_W \leq \frac{400}{\Cm} W$ and so for all $\alpha$
\begin{equation}\label{dampregion}
\ltwoo{\chi_W^{\frac{1}{2}} v}^2 \leq \frac{400}{\Cm} \ltwoo{W^{\frac{1}{2}} v}.
\end{equation}
For $U_E$, by classical elliptic theory of pseudodifferential operators there exists $S \in \Psi^0$ such that $\sigma(s) \equiv 1$ on $\{|\tau^2-|\xi|^2| \geq 2 \d \<\zeta\>^2\}$, $\sigma(S) \geq 0$ and $WF'(S) \subset \{|\tau^2-|\xi|^2| \geq \d \<\zeta\>^2\}$ and there exists $C>0$ such that for all $\alpha$
\begin{equation}\label{elliptic}
\ltwoo{Sv} \leq C \hhalfo{v}.
\end{equation}
Now, note that $S^2 + R^2 + \chi_W \geq C$ on $T^* \Omega_{\alpha}$. Therefore by the G\r{a}rding inequality (\cite{Taylor1981} Theorem 8.1) and \eqref{propregion}, \eqref{dampregion} and \eqref{elliptic} there exists $C>0$ such that for all $\alpha$ and all $v$ solving $Pv=0$
\begin{equation}\label{propsingwitherror}
\ltwoo{v}^2 \leq C \hhalfo{v}^2 +C \ltwoo{W^{\frac{1}{2}}v}^2.
\end{equation}
\qedhere
Note also, that by an argument analogous to the proof of Lemmas \ref{lemmabigrprop} and \ref{lemmaescape} there exists $C>0$ such that for all $\alpha$ and all $u$ solving \eqref{DWE}
\begin{equation}\label{eq:propofsingu}
\|u\|_{H^1_{{\alpha}}}^2 \leq C \|u\|_{H^{\frac{1}{2}}_{{\alpha}}}^2 +C \| W^{\frac{1}{2}} u\|_{H^1_{{\alpha}}}^2.
\end{equation}
To prove Proposition \ref{propofsing} it remains to eliminate the $\hp{v}{-\frac{1}{2}}$ on the right hand side of \eqref{propsingwitherror}.
To do so I will adapt the analogous proof on page 15 from \cite{RousseaLebeauPeppinoTrelat}. This relies on the triviality of solutions which do not interact with the damping.
\begin{definition}For $\chi \in C^0(M \times (0,T))$ define invisible solutions
$$
N_T = \{v \in H^1(M \times (0,T)); (\p_t^2 -\Delta) v =0, (v_0,v_1) \in H^1(M) \times L^2(M) \text{ and } \chi(x,t) \p_t v =0\}.
$$
\end{definition}
\begin{lemma}\label{invisiblelemma}
Fix $T>0$ and $\chi \in C^0(M \times (0,T))$. Suppose that for all unit speed geodesics $x(s)$ there exists $s \in (0,T)$ such that $\chi(x(s),s)>0$, then $N_T=\{0\}$.
\end{lemma}
This is a restatement of Lemma 2.3 of \cite{RousseaLebeauPeppinoTrelat}. The hypothesis on geodesics meeting the positive set of $\chi$ is a simplified version of their time-dependent geometric control condition. I can now eliminate the $H^{-\frac{1}{2}}$ error term from \eqref{propsingwitherror}.
\begin{proof}[Proof of Proposition \ref{propofsing}]
For technical reasons it is easier to prove a slightly stronger result.
The stronger result I will show is that there exists $C>0$ such that for all $\alpha$ and all $u$ solving \eqref{DWE}
\begin{equation*}
\|u\|_{H^1_{{\alpha}}}^2 + \ltwoo{\p_t u}^2 \leq C \left( \ltwoo{W^{\frac{1}{2}}\p_t u}^2 + \| W^{\frac{1}{2}} u\|_{H^1_{{\alpha}}}^2\right).
\end{equation*}
Assume the desired conclusion does not hold, so there exists $\alpha_j \in \Rb, u_j \in H^1_{\alpha_j}$ solutions of \eqref{DWE} such that
\begin{equation}\label{eq:tobecontradict}
\|u_j\|_{H^1_{\Omega_{\alpha_j}}}^2 + \ltwooj{\p_t u_j}^2=1 , \quad \ltwooj{W^{\frac{1}{2}}\p_t u_j}^2 + \| W^{\frac{1}{2}} u_j\|_{H^1_{{\alpha_j}}}^2 \ra 0.
\end{equation}
Now let $(u_{0,j},u_{1,j}) \in H^1(M) \times L^2(M)$ be $(u, \p_t u)|_{t=\alpha_j+T}=(u_{0,j}, u_{1,j})$ and note that because energy of solutions to the damped wave equation is non-increasing $(u_{0,j}, u_{1,j})$ are bounded in $H^1(M) \times L^2(M)$.
Since $(u_{0,j}, u_{1,j})$ is bounded in $H^1(M) \times L^2(M)$ there exists a weakly convergent subsequence with limit $(u_0, u_1) \in H^1(M) \times L^2(M)$.
Now let $W_j(x,t) =W(x,t+\alpha_j)$ so if $\ti{u}_j(x,t) = u_j(x,t+\aj)$ then
$$
\ltwaj{W^{\frac{1}{2}} \p_t u_j} = \ltwooo{W_j^{\frac{1}{2}} \p_t \tiuj}
$$
Note also that for any $k, W_j$ is bounded in $H^k(M \times (0,T))$, so there exists a weakly convergent subsequence with limit $\Winf$ in $H^k (M \times (0,T))$. By Rellich-Kondrachov this convergence is strong in $H^{k-1} (M \times (0,T))$.
Now let $\ti{u}$ solve
$$
\begin{cases}
(\p_t^2 - \Delta + \Winf \p_t) \ti{u} = 0 \\
(\ti{u}, \p_t \ti{u})|_{t=T} = (u_0, u_1).
\end{cases}
$$
Then $\p_t \tiuj \ra \p_t \ti{u}$ weakly in $L^2(M \times (0,T))$. Therefore
$$
\ltwooo{\Winf^{\frac{1}{2}} \p_t \ti{u}} \leq \lim_{n \ra \infty} \inf \ltwooo{\Winf^{\frac{1}{2}} \p_t \tiuj}.
$$
And note
$$
\ltwooo{\Winf^{\frac{1}{2}}\p_t \tiuj} \leq \ltwooo{W_j^{\frac{1}{2}} \p_t \tiuj} + \ltwooo{(\Winf^{\frac{1}{2}}-W_j^{\frac{1}{2}}) \p_t \tiuj} \leq \ltwooo{W_j^{\frac{1}{2}} \p_t \tiuj} + \|\Winf^{\frac{1}{2}}-W_j^{\frac{1}{2}}\|_{L^{\infty}_0} \ltwooo{\p_t \tiuj}.
$$
Both terms on the right hand side go to $0$ and so $\ltwoo{\Winf^{\frac{1}{2}} \p_t \ti{u}} =0$.
I now claim that $\Winf$ satisfies the t-GCC hypothesis of Lemma \ref{invisiblelemma}. To see this choose $J$ big enough so that $\|\Winf-W_j\|_{L^{\infty}_0} < \frac{\Cm}{2}$ for $j \geq J$. Consider a unit speed geodesic $x(s)$, then since $T \geq T_0$ by Assumption \ref{timegcc}
$$
\frac{1}{T} \int_0^T \Winf(x(s),s)ds = \frac{1}{T} \int_0^T W_j(x(s),s) ds + \frac{1}{T} \int_0^T \Winf(x(s),s)-W_j(x(s),s)ds \geq \Cm - \frac{\Cm}{2}=\frac{\Cm}{2}.
$$
Because the average of $\Winf$ on $(0,T)$ is positive $\Winf$ must be positive at some point $(x(s),s)$. Then since $\ti{u}$ is invisible with respect to $\Winf$, by Lemma \ref{invisiblelemma}, $\ti{u}\equiv 0$. Therefore $(u_j,\p_t u_j) \ra 0$ weakly in $H^1(M \times (0,T)) \times L^2(M \times (0,T))$ and thus strongly in $H^{1/2}(M \times (0,T) ) \times H^{-1/2} (M \times (0,T))$.
Combining this with \eqref{propsingwitherror}, \eqref{eq:propofsingu} and \eqref{eq:tobecontradict}.
$$
1=\ltwoo{\p_t u_j}^2 + \|u_j\|_{H^1_{\Omega_{\alpha}}}^2 \leq C \hhalfo{\p_t u_j}^2 + C \|u_j\|_{H^{\frac{1}{2}}_{\Omega_{\alpha}}}^2 +C \ltwoo{W^{\frac{1}{2}} \p_t u_j}^2 +C \| W^{\frac{1}{2}} u_j\|_{H^1_{\Omega_{\alpha}}}^2 \ra 0,
$$
which is a contradiction.
\end{proof}
\bibliographystyle{alpha}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,101
|
\section{Introduction}
This paper is concerned with the following scenario. A stationary emitter is located at an unknown location on the surface of the earth. The frequency of its transmissions is known. An unmanned aerial vehicle (UAV) or a low earth orbit satellite (LEOS) receives the transmission. Using the Doppler shift, it is required to determine a curve, depicted using a 3D map tool, on which the transmitter must lie.
An idealized version of this scenario might assume that the earth is a sphere, or possibly an ellipsoid, that all measurements (of position, velocity and frequency) were noise free and that the rotation of the earth can be neglected. In this case, the UAV or LEO receiver data serves to determine a right circular semi-infinite cone on which the emitter must lie, the axis coinciding with the velocity vector direction, the semi-angle derived from the Doppler shift, and the apex obtained from the receiver position. The intersection of this cone with the sphere/ellipsoid corresponding to the earth then provides the solution to the problem.
There are however a number of complicating factors. These include
\begin{enumerate}
\item
the multiple related but differing representations of the earth surface, including the WGS84 ellipsoid, defined by a constant gravitational equipotential surface; the EGM96 geoid, as an approximate ellipsoid with fine structure topographical variations capturing the height above sea level of points on the earth's surface \cite{wgs84}.
\item
the different types of intersection/non-intersection that can occur between a cone and a ellipsoid. There may be no intersection; there may be a tangency; there may be a single curve; there may be two separate, i.e. non-intersecting, curves (one of which may be in the 'shadow` of the earth as seen from the receiver), etc.
\item
the data sources available to represent the earth, e.g. the parameters of the WGS84 model \cite{wgs84}, the DTED data base \cite{DTED}, which indicates for samples spaced at a known interval the height of a point on the surface of the earth above the geoid. DTED is a matrix of terrain elevation values which provides basic quantitative data for systems and applications that require terrain elevation, slope, and/or surface roughness information. DTED data is uniformly spaced in angle (not distance).
\item
the fact that the earth is rotating about a polar axis, which means that given two points that are stationary with respect to the earth's surface except for the two poles, there is actually a relative velocity; roughly speaking this is a Coriolis effect. The value depends on the position of the two points. In consequence, it would seem that the determination of a cone using a Doppler shift, which amounts to indirect use of velocity data of the receiver (and assuming a stationary emitter), ought to take into account this relative velocity.
\item
the fact that a nominal rather than exact value for the emitter frequency may be known. (By way of example, a mobile phone tower may will an precisely known value, but a mobile phone may only have a nominal value). When a nominal value is used, there will be an error in the computation of Doppler shift and therefore of the cone and it is intersection with the Earth's surface is moved.
\item
the fact that transmissions between a ground-located transmitter and a LEO receiver pass through a sufficiently long distance that an assumption of uniform value of the refractive index, or equivalently straight line propagation, is dangerous. (In contrast, if a UAV rather than LEO receives the transmission, the assumption of straight line propagation is almost certainly reasonable.) The refractive index varies with altitude. This variation will be significant between a ground receiver and a satellite, but minor between a ground receiver and a low-altitude UAV.
\end{enumerate}
A significant part of the contents of this paper seeks to deal, at least to some degree, with these issues.
Our work is of course not the first work on this subject. Prior contributions, some with more idealized assumptions than those we seek to have, certainly exist. In \cite{RN1152, RN1153, lee2007doppler}, geolocation of radio frequency (RF) emitters using Doppler shift measurements from mobile vehicles was considered. These works investigated scenarios assuming the vehicle path and the emitter are located in a plane, and so without considering the shape of the earth's surface. In \cite{ho1997geolocation}, a set of solutions from the combination of time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements for localizing an emitter with known altitude above the earth surface was proposed. However, the method assumed an ellipsoid earth model without considering any height information. This restriction also appeared in \cite{pattison2000sensitivity, musicki2010mobile}, which had the similar problem settings to those in \cite{ho1997geolocation}.
The structure of the paper is as follows. In the next section we provide a general review of various coordinate systems used for representing points on or above earth's surfaces, in this paper, relating them to the World Geodetic System (WGS) 1984 , and its related Geoid system for representing the earth's terrain. We introduce Digital Terrain Elevation Data (DTED), which is used as a data reference to index specific points on the earth's terrain. We also review standard coordinate systems that represent points on or above the earth surface, including earth-centered-earth-fixed coordinate systems, geographic coordinate systems and body coordinate systems, and explain their connections. In section III, we present fundamental Doppler-shift equations and identify two possible sources of Doppler shift in our scenario, viz. the motion relative to the earth's surface of the vehicle and the effect of the earth's rotation, and reveal the fact that the earth's rotation actually does not affect the Doppler shift, due to the fulfillment of a certain orthogonality condition. Though in our scenario, we consider a stationary emitter, the conclusion that the earth's rotation does not affect the Doppler shift is not dependent on this staitonarity, as the later derivation shows. We present the equations to build a particular right circular cone with the knowledge of the sensing vehicle position, velocity and the measured Doppler shift. In Section IV-A, we present an algorithm to find the intersections of a right circular cone with the WGS84 ellipsoid and identify different types of intersections. In Section IV-B, we present our developed method to find the cone-earth-terrain intersection curve where the emitter lies, by using the formerly built cone-earth-ellipsoid intersections and the DTED. We demonstrate the effectiveness of this method with some examples \footnote{Before moving to the next section, we refer the readers to the appendix for the notation and abbreviations used throughout this paper.}.
\section{Preliminaries}
\subsection{WGS84 ellipsoid and EGM96 geoid}
In general, global geodetic applications require three different surfaces to be clearly defined:
\begin{enumerate}
\item
The earth's topographic surface, which includes the landmass topography and the ocean bottom topography
\item
A geometrical or mathematical reference surface, which is an ellipsoid with known semi-axes.
\item
An equipotential surface, called the geoid. (Potential refers to gravity). Because of height variation, and because of density variation within the solid earth, this is bumpy rather than a smooth ellipsoid, and it does not coincide with the topographic surface.
\end{enumerate}
The World Geodetic System (WGS) is a standard for use in cartography, geodesy, and satellite navigation including GPS. The latest revision is WGS84(G1762), established in 1984 and last revised in 2013. It comprises a standard coordinate system for the Earth, a standard ellipsoidal reference surface (the datum or reference ellipsoid) for raw altitude data, and a gravitational equipotential surface (the geoid) that defines the nominal sea level.
The coordinate system defined in WGS84 is depicted in the next subsection.
The WGS84 reference ellipsoid is a mathematically defined surface that approximates the truer shape of the Earth. It is used as a preferred surface on which geodetic computations are performed and point coordinates such as latitude, longitude, and elevation are defined. The ellipsoid is defined in terms of the semi-major axis value $a=6378137.0\text{m}$ and a flattening coefficient $f=1/298.257223563$ (from which the semi-minor axis value $b=6356752.314245\text{m}$ can be determined). The fact that an ellipsoid rather than a sphere is involved gives rise to an important observation. Whereas in normal life, the notion of a vertical direction is considered to be identical with the notion of the direction of the force exerted by gravity on a body on or above the earth's surface (disregarding sign), these two directions cannot strictly be identified.
The geoid is a smooth but highly irregular surface and has been built as a mathematical representation of the surface of the earth's gravity field. The geoid is widely used to describe mean sea level (MSL) since the surface of the gravity field coincides approximately with the mean sea level. Disregarding the terrain elevation on continents, the geoid is a much more accurate description of the true physical shape of the earth, than the WGS84 ellipsoid, though it is not of course identical. The WGS84 reference ellipsoid is the baseline of EGM96 geoid, i.e., the geoid undulations are with respect to the WGS84 ellipsoid. EGM96 geoid is a data grid overlay onto the WGS84 ellipsoid and can be used to find MSL for almost any point on the Earth at any given latitude and longitude.
\subsection {Coordinate systems}
In this subsection, we review standard material concerning the different coordinate systems used for representing points on or above the earth's surface
The first of these systems is an \textit{earth-centered, earth-fixed (ECEF) cartesian coordinate system}. A stationary point on the surface of the earth has constant ECEF coordinates. The origin is taken to be the earth's center of mass. The $z$-axis passes from the origin through a point on the surface termed the reference pole (effectively the north pole), and the $x$-axis passes through a zero meridian (effectively the longitudinal line through Greenwich, UK). The $y$-axis is chosen to ensure a right-handed orthogonal coordinate system.
The second coordinate system is the geographic coordinate system, which enables every location on Earth to be specified by a triple consisting of latitude $\phi$, longitude $\lambda$ and height $h$. The latitude of a point on or above the earth's surface (ellipsoid surface) is the angle between the equatorial plane and the normal at that point. Note that the normal at a point on an ellipsoid surface does not pass through the centre, except for points on the equator or at the poles. The height of a point is the vertical distance from the point to some surfaces (e.g. WGS84 ellipsoid or EGM96 geoid).
The third coordinate system is the body coordinate system, which is directly defined on the body of the vehicle. Its origin is located at the center of gravity of the vehicle. The $x$-axis of the body coordinate system points froward; the y-axis points to the right of the $x$-axis, perpendicular to the $x$-axis; the $z$-axis points down through the bottom the vehicle, perpendicular to the $x$-$y$ plane.
The last coordinate system is the vehicle-carried navigation coordinate system, which is usually viewed as recording east, north and up displacements from some agreed point of origin. We denote the vehicle-carried navigation coordinate system with the term ENU coordinate system in this paper. Its origin is defined as the center of the gravity of the vehicle as well. A tangent plane is fitted to this fixed origin point, with the east, north and up axes pointing in the obvious directions.
\subsection{Connection between coordinate systems}
Navigation instruments such as the global positioning system (GPS) and the inertial navigation system (INS) may well determine location (latitude, longitude and height) and attitude (roll, pitch, yaw) in relation to some agreed vehicle and some system is needed to marry the use of measurements derived from such systems and ECEF measurements.
We first consider the connection between ECEF and geographic coordinate systems. The output of a GPS receiver is latitude $\phi$, longitude $\lambda$ and height $h$ in the geographic coordinate system, denoted by $(\phi, \lambda, h)$ (the vertical baseline is the WGS84 ellipsoid) \cite{kaplan2005understanding}. The relevant equations for converting $(\phi, \lambda, \text{h})$ to $(x, y, z)$ in the ECEF coordinate system are \cite{gerdan1999transforming}:
\begin{equation}
\begin{split}
x&=\left(\frac{a}{\chi}+h\right)\cos \phi\cos\lambda \\
y&=\left(\frac{a}{\chi}+h\right)\cos \phi\sin\lambda \\
z&=\left(\frac{a(1-e^2)}{\chi}+h\right)\sin\phi
\end{split}
\label{eq:geotocartesian}
\end{equation}
where $e^2=6.69437999014\times 10^{-3}$ is the square of the first eccentricity, and
\begin{equation}
\chi=\sqrt{1-e^2\sin^2\phi}
\end{equation}\textbf{}
The inverse transformation from ECEF coordinates to geographic coordinates can be described by the following equations \cite{iliffe2000datums}:
\begin{equation}
\begin{split}
&\lambda = \arctan{\left(\frac{y}{x}\right)}\\
&\phi = \arctan\left(\frac{z(1-f)+(2f-f^2)a\sin^3\mu}{(1-f)(p-(2f-f^2)a\cos^3\mu)}\right)\\
&h = p\cos\phi + z\sin\phi - a\sqrt{1-(2f-f^2)sin^2\phi}
\label{eq:cartesiantogeo}
\end{split}
\end{equation}
where $p = \sqrt{x^2 + y^2}$, $r = \sqrt{p^2 + z^2}$, $f$ is the flattening of the ellipsoid and $\mu$ is a parameter calculated according to
\begin{align*}
\mu = \arctan\left(\frac{z}{p}(1-f) + \frac{(2f-f^2)az}{rp} \right)\\
\end{align*}
We now present the connections between ECEF and ENU coordinate systems. Suppose that the origin for ENU coordinates is defined by a point $(x,y,z)$ in ECEF coordinates. Suppose that a point near to the specified point is defined by the coordinate differences $(dx,dy,dz)$ between the coordinates of the nearby point, viz, $(x+dx,y+dy,z+dz)$ and the coordinates of the specified origin point $(x,y,z)$. The ENU coordinates of the nearby point are $(de,dn,du)$, the origin point of course having ENU coordinates $(0,0,0)$. The question arises: how is $(de,dn,du)$ related to $(dx,dy,dz)$. This can be answered by understanding that the orientation of ENU coordinates is determined by rotating the ECEF coordinates; the first rotation is about the $z$-axis, by $\lambda$ degrees (corresponding to the longitude), and then rotating about the new $x$-axis (obtained from the old $x$-aixs through rotation of $\lambda$ degrees in the $x$-$y$ or equatorial plane) by $\phi$ degrees. Consequently we have
\resizebox{0.85\linewidth}{!}
{
\begin{minipage}{\linewidth}
\begin{eqnarray}
\left[\begin{array}{c}
de\\dn\\du
\end{array}\right]
&=&\left[\begin{array}{ccc}
1&0&0\\
0&-\sin\phi&\cos\phi\\
0&\cos\phi&\sin\phi\end{array}\right]
\left[\begin{array}{ccc}
-\sin\lambda&\cos\lambda&0\\
\cos\lambda&\sin\lambda&0\\
0&0&1
\end{array}\right]\left[\begin{array}{c}
dx\\dy\\dz
\end{array}\right] \nonumber \\
&=&\left[\begin{array}{ccc}
-\sin\lambda&\cos\lambda&0\\
-\sin\phi\cos\lambda&-\sin\phi\sin\lambda&\cos\phi\\
\cos\phi\cos\lambda&\cos\phi\sin\lambda&\sin\phi\end{array}\right]
\left[\begin{array}{c}
dx\\dy\\dz
\end{array}\right]
\label{eq:rotation_enu_ecef}
\end{eqnarray}
\end{minipage}
}
We now state the connections between the vehicle-carried navigation system and the body coordinate system. The output of INS is the Euler angles, the set of roll $\alpha$, pitch $\beta$, raw $\gamma$, with respect to the vehicle-carried navigation coordinate system. The rotation matrix $\mathbf{R}^{\text{ENU}}_{\text{Body}}$ from the ENU coordinate system to the body coordinate system can be described by using the Euler angles, with the formula described by \eqref{eq:rotation_enu_body} at the top of the next page \cite{grewal2007global}:
\newcounter{mytempeqncnt}
\begin{figure*}[t]
\normalsize
\setcounter{mytempeqncnt}{\value{equation}}
\setcounter{equation}{4}
\begin{align}
\mathbf{R}^{\text{ENU}}_{\text{Body}} =
\begin{bmatrix}
\sin{\gamma}\cos{\beta} & \cos{\alpha}\cos{\gamma}+\sin{\alpha}\sin{\gamma}\sin{\beta} & -\sin{\alpha}\cos{\gamma} + \cos{\alpha}\sin{\gamma}\sin{\beta} \\
\cos{\gamma}\cos{\beta} & -\cos{\alpha}\sin{\gamma}+\sin{\alpha}\cos{\gamma}\sin{\beta} & \sin{\alpha}\sin{\gamma} + \cos{\alpha}\cos{\gamma}\sin{\beta} \\
\sin{\gamma} & -\sin{\alpha}\cos{\beta} & -\cos{\alpha}\cos{\beta} \\
\end{bmatrix}\label{eq:rotation_enu_body}
\end{align}
\setcounter{equation}{\value{mytempeqncnt}}
\hrulefill
\vspace*{4pt}
\end{figure*}
\subsection{Dealing with height data}
Height data is available publicly and has been specified precisely in what is known as DTED (Digital Terrain Elevation Data) \cite{durland2009defining}. As noted earlier, DTED is a digital elevation model, consisting of a matrix of terrain elevation values, corresponding to the height of the ground above the geoid. The DTED format has three levels, termed level 0, level 1 and level 2, corresponding respectively to spacing of approximately 900 meters, 90 meters and 30 meters. These line spacings correspond to specific arcsecond changes in latitude and longitude.
DTED data is evidently indexed to specific points on the earth's terrain. The horizontal datum is referenced to the WGS84 ellipsoid and the vertical datum is referenced to the EGM96 geoid. For a particular point above the ground, there are three types of height: orthometric height is defined as the height of the terrain above the geoid: ellipsoid height is defined as the height of the terrain above the WGS84 ellipsoid; geoid height is defined as the height of the geoid above the WGS84 ellipsoid. The relationship of orthometric height, ellipsoid height and geoid height is depicted in Fig. \ref{fig:geoid_height}, where an up arrow indicates a positive value and down arrow indicates a negative value. It is straightforward to see that the orthometric height at a given point with latitude $\phi$ and longitude $\lambda$ can be expressed by
\stepcounter{equation}
\begin{equation}\label{eq:transformation_h}
h(\phi,\lambda) = H(\phi,\lambda) + N(\phi,\lambda)
\end{equation}
\begin{figure}
\centering
\includegraphics[width=60mm]{Geoid.jpg}
\caption{\label{fig:geoid}A geoid. The figure is taken from \cite{durland2009defining}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm]{DTED.pdf}
\caption{\label{fig:geoid_height}The relationship between orthometric height, ellipsoid height and geoid height. Note that $N$ assumes a negative value when the geoid is below the ellipsoid. The figure is taken from \cite{durland2009defining}}
\end{figure}
\subsection{Problem statement}
Our task is one of passing from a set of measurements to a presentation in a useful (visual) form of information provided by those measurements. More specifically, our task is to define on a representation such as Google map a curve corresponding to those points consistent with a single FDOA measurement taken by an aircraft.
This task can be broken down as follows:
\begin{enumerate}
\item
Establish the equation of a right circular cone with fixed apex (corresponding to vehicle position), axis (corresponding to vehicle heading) and semi-angle (corresponding to FDOA measurement) in the ECEF coordinate system.
\item
Establish a procedure for determining an array from which the curve of intersection of the two surfaces (the FDOA cone established in the first step and the WGS84 ellipsoid) can be constructed.
\item
Relate the ECEF coordinates to geographic coordinates.
\item
Make an adjustment to cope with ground height above sea level according to DTED.
\end{enumerate}
\section{Doppler shift and earth rotation}
\subsection{Locus of points of constant Doppler}
We now consider the problem of defining the possible emitter locations corresponding to a particular measured Doppler shift. The scenario is that there is a transmitter on the surface of the earth, and a receiver above the earth that can receive signals.
Throughout this section, we make several explicit assumptions. However, with warning, we will relax specific assumptions at certain points. The assumptions are:
\begin{enumerate}
\item
The earth's surface is modelled as the WGS84 ellipsoid
\item
A transmitter is located on the surface of the earth and a receiver mounted in a sensing vehicle is located above the surface of the earth, with the transmitter being visible from the receiver.
\item
The transmitter is stationary, while the sensing vehicle is moving.
\item
The velocity of light is constant and known; as a consequence, straight line propagation occurs. (This assumption is in a sense one of the least justifiable, and its relaxation will be explored later.)
\item
The exact position and velocity of the sensing vehicle are known at the time it measures a Doppler-shifted signal.
\item
The sensing vehicle can infer the Doppler shift associated with a received signal, because it has precise knowledge of the unshifted frequency at the transmitter.
\end{enumerate}
It is straightforward to make a general statement about the nature of the locus of the points corresponding to a particular Doppler shift. Knowledge of the Doppler shift associated with a received signal implies knowledge of a circular cone on which the transmitter must lie. The cone's apex is at the sensing vehicle position, and axis aligned with the velocity vector of the sensing vehicle. The direction of the axis can be either the same or reverse of the velocity direction, depending on the sign of the measured Doppler shift. If the measured Doppler shift is positive, then the axis of the cone is defined with the same direction of the velocity vector, otherwise the direction of axis is the reverse of the velocity vector, see Fig. \ref{fig:fdoa_cone}. The semi-angle, call it $\psi$, can be computed from knowledge of the magnitude of the velocity $|\mathbf{v}|$ of the sensing vehicle, the speed of light $c$, the unshifted transmitter frequency $f_0$, and the Doppler shift $\delta$: \footnote{The transmitter is assumed to be stationary, and for the moment we neglect any contribution to the relative velocity of the receiver with respect to the transmitter due to the earth's rotation. This will be dealt with subsequently. }
\begin{equation} \label{eq:cone_semi_angle}
\cos \psi=(\delta/f_0)/(|\mathbf{v}|/c)
\end{equation}
The fractional Doppler shift equals the fraction relative to the speed of light of the component of the sensing vehicle's velocity along the direction to the transmitter.
The FDOA cone and WGS84 ellipsoid are both two-dimensional surfaces in a 3D space. They intersect in at least one point, namely the point where the transmitter is located. Generically, if there is a nonempty intersection, it will be a one-dimensional smooth set, a curve in space in fact. Exceptionally, as a kind of limiting case, the intersection may be a single point. This would arise if the tangent plane to the ellipsoid coincided with the tangent plane to the cone at the common point of intersection.
Our ultimate goal is to the intersection curve.
\begin{figure}[tb]
\centering
\includegraphics[width=80mm]{Doppler_cone}
\caption{\label{fig:fdoa_cone}The two cases of the FDOA Cone. The left-side figure indicates that the direction of axis of the FDOA cone has the same direction as the receiver velocity if the measured Doppler shift is positive. Otherwise, the direction of the axis of the FDOA cone is the reverse of the direction of velocity of the receiver, as illustrated in the right-side figure.}
\end{figure}
\subsection{Coriolis Effect}
In this subsection, we highlight the need to consider the fact that the earth is rotating, when considering the determination of the cone associated with a Doppler shift. We demonstrate that the operative kinematics constraints imply that the relative velocity component between an emitter and a receiver due to the earth's rotation makes zero contribution to the Doppler shift.
Viewed from a point in a coordinate frame that is not earth fixed, but which views the earth as rotating, any two points on the earth's surface (other than the poles) can be seen to have velocities associated with the rotation of the earth about its axis. Further, any two distinct points excluding the two poles have \textit{different} velocities on this account. (The yearly motion of the earth around the sun is discounted in making these observations.) It follows that any two points on (or above) the earth's surface which in an ECEF coordinate frame is concerned would be considered stationary actually have a relative velocity (whose magnitude can be substantial, e.g. some tens of m/sec), and thus potentially gives rise to a Doppler shift between the transmit/receive frequency of an RF signal propagating from one point to the other. Further, if one of the points is fixed on or above the earth's surface, and the other is moving (relative to the earth's surface), the relative velocity can be regarded as coming from two causes, the motion relative to the earth's surface of one point, and the effect of the earth's rotation (`Coriolis effect'). It is because a coordinate frame fixed to the earth is actually rotating, albeit with a fixed angular velocity, rather than just translating at a uniform velocity, that this Coriolis effect arises.
We now establish how the relative velocity can be determined, and seek to determine the associated Doppler shift. For this purpose, it is the case to attribute the Coriolis component to rotation of an earth-centred earth-fixed coordinate basis.
In ECEF coordinates we denote the emitter coordinates by using a vector $\mathbf{p}:=p^x \mathbf{e}_x+p^y\mathbf{e}_y+p^z\mathbf{e}_z$ and the receiver coordinates by $\mathbf{r}:=r^x\mathbf{e}_x+r^y\mathbf{e}_y+r^z\mathbf{e}_z$. Where $\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z$ denote unit vectors aligned with the ECEF coordinate axes. Due to the rotation of the earth, these vectors are not stationary, when seen from an inertial frame. The transmitter frequency $f_r$ measured at the receiver is
\begin{equation}\label{eq:generalDoppler}
f_r = f_0 \left(1- \frac{1}{c}\left[\frac{d(\mathbf{p}-\mathbf{r})}{dt}\right]\cdot\frac{\mathbf{p}-\mathbf{r}}{\|\mathbf{p}-\mathbf{r}\|}\right)
\end{equation}
In computing the derivative, we must allow for motion relative to the ECEF system (values of the coordinates of $\mathbf{p},\mathbf{r}$ change) and, separately, motion of the coordinate axes.
In the ECEF coordinate system the basis vectors rotate so that \cite{goldstein2002classical}
\[
\frac{d\mathbf{e}_i}{dt} = \mathbf{\Omega}\times \mathbf{e}_i\;\;i\in\{x,y,z\} \quad,
\]
where $\Omega$ is the magnitude of the angular velocity. Consequently the relative velocity between the emitter and the receiver is the sum of the coordinate velocity and the Coriolis effect:
\begin{eqnarray}
\frac{d\left(\mathbf{p} - \mathbf{r}\right)}{dt} &=&
\sum_{i\in\{x,y,z\}}\frac{d\left(p^i-r^i\right)}{dt}\mathbf{e}_i + \left(p^i-r^i\right)\mathbf{\Omega}\times \mathbf{e}_i \nonumber \\
&=& \left( \sum_{i} \frac{d \left(p^i-r^i\right)}{dt} \mathbf{e}_i\right) + \mathbf{\Omega}\times\left(\mathbf{p}-\mathbf{r}\right)
\label{eq:dpr}
\end{eqnarray}
The second term on the right hand side of~\eqref{eq:dpr} is due to the Coriolis effect. Note that this vector is perpendicular to $\mathbf{p}-\mathbf{r}$ and hence doesn't contribute to the Doppler effect:
\begin{align*}
f = f_0 \left( 1 - \frac{1}{c} \left( \sum_i \frac{d \left(p^i -r^i\right)}{dt} \mathbf{e}_i \cdot \frac{\mathbf{p}-\mathbf{r}}{\|\mathbf{p}-\mathbf{r} \|}\right)\right)
\end{align*}
This is exactly the formula we would write down if we failed to take into account the rotation of the earth, or put another, we can simply neglect the effect associated with rotation of the earth.
\subsection{Equation of an arbitrary right circular cone}
In this section, our aim is to find an equation representing a right circular cone with apex coordinates $(r_x,r_y,r_z)$ and unit vector in the direction of velocity given by $(\alpha,\beta,\gamma)$, i.e. subject to $\alpha^2+\beta^2+\gamma^2=1$, in ECEF coordinate system. Note that a mobile vehicle can determine its attitude $(\alpha,\beta,\gamma)$ with the help of the INS devices. The velocity direction vector $(\alpha,\beta,\gamma)$ cannot be directly obtained but can be calculated by transforming a unit vector $(1,0,0)$ from the vehicle's body coordinate system to the ECEF coordinate system by using the inverse of the rotation matrices provided in \eqref{eq:rotation_enu_ecef} and \eqref{eq:rotation_enu_body}.
Our starting point is that, as is well-known and indeed easily checked, (the surface of) a cone with axis corresponding to the $z$-axis and apex at the origin is given by an equation of the form
\begin{equation}\label{eq:original_cone}
\frac{x^2}{d^2}+\frac{y^2}{d^2}-z^2=0
\end{equation}
The semi-angle is $\tan^{-1}d$.
Our immediate goal now is to understand how to handle translation of the apex and rotation of the axis of the cone. If the apex of the cone is at $(r_x,r_y,r_z)$, then the above equation is replaced by
\begin{equation}
\frac{(x-r_x)^2}{d^2}+\frac{(y-r_y)^2}{d^2}-(z-r_z)^2=0
\end{equation}
To understand how to handle rotation, we first observe a simple Lemma.
\begin{lem}
Let $\alpha,\beta,\gamma$ be a set of direction cosines of a real 3-vector, i.e. $\alpha^2+\beta^2+\gamma^2=1$. Then the following matrix is an orthogonal rotation matrix:
\begin{equation}\label{eq:rotation_matrix}
\mathbf{R}=\left[\begin{array}{ccc}
\frac{\alpha\gamma}{(\alpha^2+\beta^2)^{1/2}}&-\frac{\beta}{(\alpha^2+\beta^2)^{1/2}}&\alpha\\
\frac{\beta\gamma}{(\alpha^2+\beta^2)^{1/2}}&\frac{\alpha}{(\alpha^2+\beta^2)^{1/2}}&\beta\\
-(\alpha^2+\beta^2)^{1/2}&0&\gamma\end{array}
\right]
\end{equation}
\end{lem}
By way of an outline proof, we observe first that it is easily verified that each of the columns is a vector of length 1, and the columns are mutually orthogonal. Further, a straightforward calculation shows that the determinant is 1, assuring that the matrix is a rotation matrix.
When $\alpha=\beta=0$, four entries of the matrix are not well-defined, and so the continuity of the matrix comes into question. However, let $\theta$ be such that $\alpha=\sqrt{1-\gamma^2}\cos\theta$ and $\beta=\sqrt{1-\gamma^2}\sin\theta$. Note that given the direction cosines $\alpha, \beta$ and $ \gamma$, such a $\theta$ always exists and is unique. Let us regard the matrix $R$ as a function of $\theta$ and $\gamma$. Then we can write
\begin{equation}
\mathbf{R}(\theta,\gamma)=\left[\begin{array}{ccc}
\gamma\cos\theta&-\sin\theta&\sqrt{1-\gamma^2}\cos\theta\\
\gamma\sin\theta&\cos\theta&\sqrt{1-\gamma^2}\sin\theta\\
-\sqrt{1-\gamma^2}&0&\gamma\end{array}\right]
\end{equation}
Now if $\alpha,\beta$ tend continuously to zero while obeying $\alpha^2+\beta^2<0$ (except in the limit), i.e. $\gamma<1$ except in the limit, and if they tend to zero in such a way that $\beta/\alpha$ also approaches a limit, it is evident that $R(\theta,\gamma)$ will approach the limit
\begin{equation}
\mathbf{R}(\theta,1)=\left[\begin{array}{ccc}
\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\
0&0&1
\end{array}
\right]
\end{equation}
Now suppose a right-circular cone with apex at the origin is such that the axis of the cone has direction cosines $\alpha,\beta,\gamma$. Assume temporarily that a coordinate basis with coordinates $\bar{\mathbf{p}}=[\bar x ~ \bar y ~ \bar z]^\top$ is established with the same origin and with the $\bar z$-axis coinciding with the axis of the cone. The equation of the cone in the new coordinate basis is given by
\begin{equation}\label{eq:standard cone}
\frac{\bar x^2}{d^2}+\frac{\bar y^2}{d^2}-\bar z^2=0
\end{equation}
for suitably chosen $d$. Write this equation as
\begin{equation}\label{eq:niceaxiscone}
\bar{\mathbf p}^{\top}\bar \Lambda\bar{\mathbf p}=0
\end{equation}
where
\begin{equation}
\bar\Lambda=\mbox{diag}[d^{-2},d^{-2},-1]
\end{equation}
It is clear that we want correspondence between the line $\bar x=0, \bar y=0$, i.e. the axis of the cone in $\bar x, \bar y, \bar z$ space and the axis of the cone in $x,y,z$ space, which is defined by direction cosines $\alpha,\beta, \gamma$. Because of the structure of the last column of $\mathbf{R}$, this correspondence is assured if we take $\mathbf{p}=\mathbf{R}\bar{\mathbf{p}}$ or
\begin{equation}
\left[\begin{array}{c}
x\\y\\z\end{array}\right]=\mathbf{R}\left[\begin{array}{c}
\bar x\\ \bar y\\ \bar z
\end{array}\right]
\end{equation}
Equivalently, the equation in $x,y,z$ space of the cone follows from \eqref{eq:niceaxiscone} as
\begin{equation}
\mathbf{p}^{\top}\mathbf{R}\bar\Lambda \mathbf{R}^{\top}\mathbf{p}=0
\end{equation}
The matrix $\mathbf{R}(\theta,1)$ is relevant in a special case. If $\alpha=0,\beta=0, \gamma=1$ define the directions in which the axis of the cone should point, then it is evident that the original cone, with $z$-axis coinciding with the direction of the cone is already correctly positioned. Thus one might reasonably suppose that the transforming rotation matrix should be the identity matrix. The formula for $\mathbf{R}(\theta,1)$ defines a rotation matrix which leaves that axis invariant, as it causes rotation only of the $x,y$ coordinate plane about the $z$-axis, which is normal to it.
Combining the results on translation and rotation, we have the following Lemma.
\begin{lem}
Consider a right circular cone with apex at $\mathbf{p}=[r_x,r_y,r_z]^{\top}$, with axis given by the unit norm vector of direction cosines $[\alpha,\beta,\gamma]^{\top}$ and with semi-angle given by $\tan^{-1}d$. Then the equation of this cone is
\begin{equation}
\left[x-r_x\;y-r_y\;z-r_z\right]\mathbf{R} \diag[d^{-2},d^{-2},-1]\mathbf{R}^{\top}\left[\begin{array}{c}
x-r_x\\
y-r_y\\
z-r_z
\end{array}\right]=0
\label{eq:transformed_cone}
\end{equation}
\end{lem}
Our next major task is to study the intersection of this conical surface with the ellipsoid defined by the earth.
\section{Determining the intersection of the FDOA cone with the earth's terrain}
Here is a summary of what is required.
The equation of WGS84 ellipsoid (with the positive $z$-axis identified with a line starting at the center of the earth and passing through the north pole) in ECEF coordinates is
\begin{equation}\label{eq:earth_ellipsoid}
\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{b^2}=1
\end{equation}
where $a$ represents semi-major axis value and $b$ represents semi-minor axis value.
One seeks to determine the intersection between an arbitrary right circular cone (with parameters corresponding to typical values obtained from a UAV or LEOS) and the WGS84 ellipsoid.
This intersection will in general be a curve, or more accurately, may comprise one or more disjoint curves. To see that for example two (non-intersecting) curves could be obtained, suppose that the apex of the cone lay outside the ellipsoid, say above the north pole, with the axis of the cone passing through the centre of the ellipsoid. Then if the semiangle of the cone were sufficiently small, the cone would intersect the upper hemisphere of the ellipsoid in a closed curve, and separately it would intersect the lower hemisphere of the ellipsoid in a second closed curve. Of course, this is not the only possibility.
For the purposes of establishing and recording the intersection of the cone and the ellipsoid, one could imagine choosing a set of neighboring values of $x$ (or $y$ or $z$), and for each value determining the real intersection points (which will normally be 0, 2 or 4 in number) of the two equations. (We say normally, because the number may be different if there is a point at which the two surfaces touch and have some sort of common tangency.)
Given the possibility of the intersection comprising two separate curves, it would be important to determine, for neighboring values of $x$, which intersection points were on the same curve, and which were not. When the intersection points are remote from one another, this will be easy. It will be less straightforward if curves merge, touch, etc.
\subsection{Find the intersections of a cone with an ellipsoid}
To parametrize a right circular cone, we can divide the cone surface into two parts, an apex $\mathbf{r}=(r_x, r_y, r_z)$ and a set $\Phi$ of straight rays which start from the apex and ends in the infinity. Suppose $\alpha_c$, $\beta_c$ and $\gamma_c$ are direction cosines of one of the rays in $\Phi$. The ray equation can be given by
\begin{equation}\label{eq:cone_para}
\begin{split}
x = r_x + \alpha_c s\\
y = r_y + \beta_c s\\
z = r_z + \gamma_c s\\
\end{split}
\end{equation}
where $s\in[0, \infty)$ is the range of the ray. The constraint of non-negative value on $s$ ensures that \eqref{eq:cone_para} describes a right circular cone rather than a double cone.
\begin{figure}
\centering
\includegraphics[width=60mm]{cone_coordinates}
\caption{\label{fig:cone_coordinates}Illustration of $\zeta$ and $\eta$.}
\end{figure}
Now the problem of finding the intersections of two quadratic surfaces (a right circular cone and an ellipsoid) has been transformed to finding the intersections of a set of rays described by \eqref{eq:cone_para} with the ellipsoid \eqref{eq:earth_ellipsoid}. According to \eqref{eq:cone_para}, it is observed that the direction cosines $\alpha_c$, $\beta_c$ and $\gamma_c$ can be obtained by transforming the direction cosines of the rays on the surface of the original cone \eqref{eq:original_cone}. We first pick one ray on the surface of the cone \eqref{eq:original_cone} and identify its direction cosines $\alpha_r$, $\beta_r$ and $\gamma_r$ as follows:
\begin{equation}
\begin{split}
\alpha_r &= \cos(\zeta)\cos(\eta)\\
\beta_r &= \cos(\zeta)\sin(\eta)\\
\gamma_r &= \sin(\zeta)\\
\end{split}
\end{equation}
where $\zeta$ denotes the angle from the ray to its projection on $x-y$ plane, and $\eta$ denotes the angle between the $x-{\text{axis}}$ and the projection of the ray on $x-y$ plane. See Fig. \ref{fig:cone_coordinates} for the illustrations of $\zeta$ and $\eta$. Let $\mathbf{d}_r: = [\alpha_r ~ \beta_r ~ \gamma_r]^\top$ be an unit direction vector of an arbitrary ray on cone \eqref{eq:original_cone}. The unit direction vector $\mathbf{d}_r: = [\alpha_t ~ \beta_t ~ \gamma_t]^\top$ of the mapping of the ray on the transformed cone $\eqref{eq:transformed_cone}$ can be directly calculated by
\begin{equation}
\mathbf{d}_t = \mathbf{R}\mathbf{d}_r.
\end{equation}
Here $\mathbf{R}$ is the rotation matrix defined in \eqref{eq:rotation_matrix}.
Then for any particular ray, $r_x, r_y, r_z, \alpha_c, \beta_c$ and $\gamma_c$ are known. By substituting \eqref{eq:cone_para} into the ellipsoid equation \eqref{eq:earth_ellipsoid}, we have a second-order equation in $s$ with the form
\begin{equation}\label{eq:s}
a_s s^2 + b_s s + c_s = 0
\end{equation}
where
\begin{equation}\label{eq:abc}
\begin{split}
a_s &= q_1\alpha_c^2 + q_2\beta_c^2 + q_3\gamma_c^2\\
b_s &= 2q_1r_x\alpha_c + 2q_2r_y\beta_c + 2q_3r_z\gamma_c\\
c_s &= q_1r_x^2 + q_2r_y^2 + q_3r_z^2 + q_0;\\
\end{split}
\end{equation}
are all known parameters with $q_0 = -1$, $q_1=q_2=1/a^2$ and $q_3 = 1/b^2$. The real, non-negative solutions of \eqref{eq:s} define points on the ray corresponding to its intersections with ellipsoid. Note that the number of solutions of \eqref{eq:s} can be zero, one or two, depends on the value of the "quadratic discriminant":
\begin{equation}
D_s = b_s^2-4a_s c_s
\end{equation}
The number of solutions of \eqref{eq:s} also represents the number the intersection points of the ray with the ellipsoid.
\subsection{Find the intersections of the FDOA cone with the earth's terrain}
The aim of this subsection is to extend the calculation of the previous subsection involving the WGS84 ellipsoid to provide an algorithm to find the intersections of the FDOA cone with the earth terrain and plot these intersection points on a 3D map tool (e.g. google earth). We assume that the vehicle's measurements of its position, velocity and the Doppler shift are all accurate, i.e. without error.
Since the earth terrain is uneven, we cannot find equations to describe the terrain, which means we cannot directly calculate the intersections of a cone with the terrain by solving mathematical equations. Motivated by the observations that the earth terrain can be modelled by the discrete point set DTED, our idea is to find the intersection of the cone with the WGS84 ellipsoid and map these intersection points to the DTED dataset, according to the rays generated between the cone apex and the intersection points on the ellipsoid surface.
Our algorithm is consisted of the following steps:
\begin{enumerate}
\item Introduce $\mathrm{D}$ as a set of points comprising the DTED. Thus
\begin{align*}
\mathrm{D} := \{(\phi_i,\lambda_i,H_i): (\phi_i,\lambda_i,H_i) ~\text{is in the DTED},\notag\\
i=1,2,\ldots\}
\end{align*}
Note that $\phi_i$ and $\lambda_i$ are defined with respect to the WGS84 ellipsoid, while $H_i$ is defined with respect to the EGM96 geoid \cite{durland2009defining} . For further use, we need to transfer $H_i$ (orthometric height) to $h_i$ (ellipsoid height) by using \eqref{eq:transformation_h}. We denote this new set as $\hat{\mathrm{D}}$.
\item By applying $\eqref{eq:geotocartesian}$ on each point in $\hat{\mathrm{D}}$, we obtain a new set $\hat{D}_c$, in which the coordinates for each point are defined with respect to the ECEF Cartesian coordinate system.
\item By using the method introduced in the previous subsection, the vehicle determines a set of the intersection points of the FDOA cone (associated with one single measurement) with the WGS84 ellipsoid. Introduce
\begin{equation*}
\mathrm{A} := \{(x_i,y_i,z_i): (x_i,y_i,z_i)~\text{is on the intersection curve.}\} \end{equation*}
to denote the set of intersection points. Note that the coordinates associated with the points in $\mathrm{A}$ are defined with respect to the ECEF Cartesian coordinate system.
\item Select a point $p_{\mathrm{A}}^i$ in $\mathrm{A}$. By associating the coordinates of $p_{\mathrm{A}}^i$ with the receiver coordinates $\mathbf{r}$, we can define a line described by the following equation
\begin{equation}
\frac{x_i - r_x}{a_i} = \frac{y_i - r_y}{b_i} = \frac{z_i - r_z}{c_i}
\end{equation}
where $a_i, b_i, c_i$ can be obtained by substituting the coordinates of $p_A^i$ and $\mathrm{r}$ into the above equation. For later use, we denote this line as $l_i$. It is one of the rays that comprise the surface of the FDOA cone.
\item Now we have a transformed DTED point set $\hat{\mathrm{D}}_c$ and the line $l_i$. For each point $p_k, k=1,2,\ldots$ in $\hat{\mathrm{D}}_c$, we can calculate its Euclidean distance $d_k^i$ to $l_i$. We then select a fixed threshold $tr_i$ (the threshold is determined according to the spacing accuracy of the DTED sets) and figure out a new point set $\hat{D}_i$, defined as
\begin{equation}
\hat{D}_i := \{(x_k,y_k,z_k): (x_k,y_k,z_k)\in\hat{D}_c, d_k^i\leq tr_i\}
\end{equation}
The mapping of $p_A^i$ onto $\hat{D}_c$ is thus included in $\hat{D}_i$.
\item For each point $\hat{D}_i$, we calculate its Euclidean distance to the receiver, the point $\tilde{p}$ associated with the minimum distance to the receiver is the mapping point of $p_A^i$ onto $\hat{\mathrm{D}}_c$.
\item By repeating the steps 4)-6) for each $l_i$, we can finally find the cone-terrain intersection points.
\item For each cone-terrain intersection point, we use \eqref{eq:cartesiantogeo} to transform its ECEF coordinates into the geographical coordinates to depict it in a 3d map tool.
\end{enumerate}
In the following, we use an example associated with UAV cases to illustrate the performance of our algorithms. The UAV position is selected to be close to the city of Adelaide, Australia. The FDOA cone, the intersections of the FDOA cone with the earth terrain, and the UAV are depicted in Fig. \ref{fig:simulation_1}. We also provide Fig. \ref{fig:simulation_1_intersection} for the purpose of clear observations on how the intersection points are distributed on the earth's terrain.
\begin{figure*}[t]
\centering
\includegraphics[width=170mm]{Simulation_2.jpg}
\caption{\label{fig:simulation_1}The performance of the method for finding the intersections of the FDOA cone with the earth's terrain. In the simulation, the instantaneous UAV parameters are selected as: roll \ang{0}, pitch \ang{-30}, yaw \ang{190}, latitude \ang{-34.6462}, longitude \ang{138.833} and height 2000m. The semi-angle of the FDOA cone is assumed to be \ang{26.56}. The white rays represent the surface of the FDOA cone. The yellow place-marks indicate the intersections of the FDOA cone with the earth's terrain. The red place-marks represent the intersections of the FDOA cone with the WGS84 ellipsoid. Note that these red place-marks should be under the earth terrain since their ellipsoid heights are zero. The ellipsoid heights of the cone-ellipsoid intersection points are increased manually so that we can compare the difference between the cone-terrain intersections and the cone-ellipsoid intersections.}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=170mm]{Simulation_2_intersection.jpg}
\caption{\label{fig:simulation_1_intersection}\color{blue}The intersection points on the terrain.}
\end{figure*}
\section{Error analysis}
This section aims to discuss the curve shift associated with different errors raised in practical scenarios. We identify two potential error sources, 1) the fact that the nominal frequency of the emitter may be different from its true frequency; 2) the changing refractive index of the atmosphere, which results in a variation of the speed of the electromagnetic wave. We provide numerical examples to illustrate the curve shifts associated with different errors. Since UAV and LEOS are operated in different environments, we divide the discussions into two cases. For LEOS case, we further consider the relativistic Doppler effect. For all the examples in this section, we assume that the measured frequency at the receiver and the navigation (position, velocity and attitude) data of the vehicle are accurate, i.e. without errors. Note that for calculation simplicity, we assume that the nominal frequency of the emitter is 299,792,458$\text{Hz}$.
\subsection{Error sources}
We now discuss the first error source. The carrier frequency of the emitter is generated according to its built-in oscillators. The output frequency of an oscillator has a drift due to its internal changes with respect to time plus changes in the environment. So there is potentially an offset between the nominal frequency and the true frequency of the emitter. Before this section, we assumed that the true frequency rather than the nominal frequency of the emitter is precisely known to the receiver. This can only be assumed if, for example, there is a separate stationary receiver providing the information to the moving receiver, or the emitter is actually reflecting signals due to its being illuminated by a radar, with the position, velocity and precise frequency of the radar transmitter known to the moving receiver.
We then explain the second error source. It is well-known that the speed of electromagnetic wave in a medium different to vacuum, is different to its speed in the vacuum. Since the speed of electromagnetic wave changes with respect to the air refractive index when travelling in the air, a small error will be introduced when calculating the semi-angle of the FDOA cone using \eqref{eq:cone_semi_angle}. The speed of electromagnetic wave in the air is normally calculated using:
\begin{equation}
c_a = c/n
\label{eq:light_speed_in_air}
\end{equation}
where $n$ is the refractive index of the air. The refractive index of the air varies according to air temperature and pressure, which leads to a refractive index gradient in the atmosphere \cite{edlen1966refractive}. Note that air temperature and pressure can also be described with respect to the altitude. In this paper, we refer to an air refractive index model described by a function of altitude proposed in \cite{neda2003flatness}.
In the following, we will use numerical examples to illustrate the curve shift associated with different errors. In all the figures, the curves that represent the intersections of the true FDOA cones with the WGS84 ellipsooid are marked in yellow and the curves calculated with respect to the errors are marked in red.
\subsection{The UAV cases}
According to the U.S. Department of Defense \cite{dempsey2010eyes}, UAVs can be conveniently classified into five categories according to their sizes (see Appendix for the classifications) . In our application scenarios, a medium- or large-size UAV is capable of carrying on the tasks. In this subsection, we assume that the UAV speed and height are 50$\text{m/s}$ (180$\text{km/h}$) and 2000$\text{m}$, respectively. The yaw angle of the UAV is selected as \ang{0}, the pitch angle is selected as \ang{-30} and the roll angle of the UAV is selected as \ang{0}. We consider three situations associated with different magnitudes (close to \ang{30}, close to \ang{0} and close to \ang{90}) of the semi-angle of the true FDOA cone, to better describe the situations in practical scenarios.
\subsubsection{Curve shift associated with the error between the nominal frequency and the true frequency}
\begin{itemize}
\item \textbf{UAV example 1:}
Assume that the measured frequency at the receiver is 299,792,501.33$\text{Hz}$, and the true frequency of the emitter is 299,792,468$\text{Hz}$. There is a 10Hz error between the nominal frequency and the true frequency of the emitter. The semi-angle of the true FDOA cone is calculated according to the true frequency of the emitter, which can be obtained as \ang{48.243}. Since the receiver has only the knowledge of the nominal frequency of the emitter, the semi-angle of the FDOA cone calculated at the receiver is \ang{29.934}. We plot the two intersection curves in Google Earth, see Fig. \ref{fig:E1}, and observe that the magnitude of the curve shift ranges from approximately 730m to 4,000km \footnote{Note that we do not consider the fact that the signal from the emitter can be blocked due to the eccentricity of the earth surface. The curve shift does not represent the error of the position of the emitter.}.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{uav_ex_1.jpg}
\caption{\label{fig:E1}UAV example 1.}
\end{figure}
\item \textbf{UAV example 2:}
We still assume that the true frequency of the emitter is 299,792,468$\text{Hz}$. By assuming that the measured frequency at the receiver is 299,792,507.93Hz, the Doppler shift calculated according to the nominal frequency and the true frequency of the emitter are 49.93Hz and 39.93Hz, respectively. The semi-angle of the FDOA cone associated with the true Doppler shift is \ang{3}. The semi-angle calculated according to the difference between the measured frequency and the nominal frequency is \ang{37.0}. The two intersection curves for this example are depicted in Fig. \ref{fig:E2}. The minimum magnitude of the curve shift is approximately 2,250m and the maximum magnitude of the curve shift is too large to be determined in Google Earth. This is because the size of the yellow curve is very small (the radius is about 800m) and the red curve in Fig. \ref{fig:E2} has a similar shape as the red curve in Fig. \ref{fig:E1}, with a much bigger size.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{uav_ex_2.jpg}
\caption{\label{fig:E2}UAV example 2. }
\end{figure}
\item \textbf{UAV example 3:}
Assume that the measured frequency at the receiver is 299,792,470.615Hz, and the true frequency of the emitter is 299,792,468Hz. The true Doppler shift is 2.615Hz, thus the semi-angle of the true FDOA cone is \ang{87}. The Doppler shift calculated according to the nominal frequency of the emitter is 12.615 Hz and its associated FDOA cone has a semi-angle with the value of \ang{75.386}. The magnitude of the curve shift measured in Goolge Earth ranges from approximately 500m to 2,600km
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{uav_ex_3.jpg}
\caption{\label{fig:E3}UAV example 3.}
\end{figure}
\end{itemize}
By observing the above three UAV examples, we can conclude that there is a significant curve shift associated with the error between the nominal frequency and the true frequency of the emitter. When applying our algorithm on the UAV platforms, one must be very careful to ensure that the knowledge of the transmission frequency of the emitter is accurate enough.
\subsubsection{Curve shift associated with the changing refractive index of the atmosphere}
The UAV operation altitude in our application scenarios is less than 5km. According to the air refractive model proposed in \cite{neda2003flatness}, the air refractive index can be regarded as a constant with a value 1.0003. We will use this value in the following examples.
\begin{itemize}
\item \textbf{UAV example 4:}
Assume that the Doppler shift measured at the receiver is 43.3Hz. Then the semi-angle of the FDOA cone calculated using the speed of electromagnetic wave in the vacuum is \ang{30}. By considering the refractive index of the air, we can directly substitute \eqref{eq:light_speed_in_air} into equation \eqref{eq:cone_semi_angle} and obtain that the semi-angle of the FDOA cone considering the refractive index of the air is \ang{29.973}. We depeict the two intersections curves in Fig. \ref{fig:E4}. The range of the curve shift varies from 3m to 6,500m.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{uav_ex_4.jpg}
\caption{\label{fig:E4}UAV example 4}
\end{figure}
\item \textbf{UAV example 5:}
In this example, we assume that the Doppler shift measured at the receiver is 49.93Hz. Thus the semi-angle of the FDOA cone calculated using the speed of the electromagnetic wave in the vacuum is \ang{3}. Considering the speed of the electromagnetic wave in the air, the new magnitude of the semi-angle of the FDOA cone can be obtained as \ang{2.688}. Fig. \ref{fig:E5} shows the two intersection curves. The curve shift of the two curves varies from 20m to 40m.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{uav_ex_5.jpg}
\caption{\label{fig:E5}UAV example 5}
\end{figure}
\item \textbf{UAV example 6:}
In this example, we assume that the Doppler shift measured at the receiver is 2.615Hz. The semi-angle of the FDOA cone calculated using the speed of an electromagnetic wave in a vacuum is \ang{87}. Considering the speed of an electromagnetic wave in the air, the magnitude of the semi-angle of the true FDOA cone is \ang{86.993}. The curve shift of the two curves varies from 3m to 1,500m.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{uav_ex_6.jpg}
\caption{\label{fig:E6}UAV example 6}
\end{figure}
\end{itemize}
{According to the three examples for UAV cases considering the changing refractive index of the atmosphere, we can find that the magnitude of the curve shift is actually minor. Though the maximum curve shifts in example 4 and 6 are 1500m and 3500m, they both appear at the positions far away from the receiver (the distance can be more than 4,000km). And it is obvious that signal emitted at those positions cannot be received by the receiver. In conclusion, we may neglect the effect of the changing refractive index of the atmosphere when applying our algorithm in practice.}
\subsection{The LEOS cases}
\subsubsection{Curve shift associated with the error between the nominal frequency and the true frequency of the emitter}
In all the examples for LEOS, we assume that the LEOS is above the equator and the roll, pitch and yaw are all selected as \ang{0}. We also assume that the satellite speed is 7800m/s (28,080km/h). The height of the satellite is 200,000m (200km). These two parameters are typical for LEOS. In all the examples provided below, we assume that the error between the nominal frequency and the true frequency is 60Hz.
\begin{itemize}
\item\textbf{LEOS example 1:}
The true transmission frequency of the emitter is 299,792,518Hz. Suppose that the measured frequency at LEOS is 299,798,033.4329Hz. Thus the true Doppler shift is 5515.4329Hz. The semi-angle of the FDOA cone calculated according to the true frequency of the emitter is \ang{45}. The semi-angle of the FDOA cone calculated according to the nominal frequency of the emitter $45.62^o$. The two curves are shown in Fig. \ref{fig:LE1}. The curve shift of the two curves varies from approximately 5,000m to 15,000m.
\begin{figure}[tb]
\centering
\includegraphics[width=0.9\linewidth]{leos_ex_1.jpg}
\caption{\label{fig:LE1}LEOS example 1}
\end{figure}
\item\textbf{LEOS example 2:}
Since the LEOS orbit is very close to circular and the moving direction of LEOS is close to be parallel to the plane of the horizon. If we consider the situation that the semi-angle of the FDOA cone is close to \ang{0}, then it is impossible to find an emitter on the earth's surface to generate such the FDOA cone. We do not need to provide illustrations to this example.
\item\textbf{LEOS example 3:}
We still assume that the true transmission frequency of the emitter is 299,792,518Hz. Assume that the Doppler shift measured at the LEOS is 347.94Hz. Because of the error between the nominal frequency and the true frequency, the output of the Doppler shift at the LEOS is 407.94Hz. Then the semi-angle of the FDOA cone calculated according to the nominal frequency of the emitter is $\ang{87.433}$. The semi-angle of the FDOA cone calculated according to the true transmission frequency of the emitter is \ang{87}. The two curves are depicted in Fig. \ref{fig:E6}. The curve shift of the two curves varies from 2,000m to 100,000m.
\begin{figure}[tb]
\centering
\includegraphics[width=0.9\linewidth]{leos_ex_3.jpg}
\caption{\label{fig:LE3}LEOS exmaple 3}
\end{figure}
\end{itemize}
By observing the above LEOS examples, we can find out that the magnitude of the curve shift is not negligible, though the change of the semi-angle of the FDOA cone is minor. This is due to the fact that LEOS are operated at a high altitude. When applying our algorithms on the LEOS, it is still required to have a accurate knowledge of the true transmission frequency of the emitter.
\subsubsection{Curve shift associated with the changing refractive index of the atmosphere}
The refractive index of air varies according to a function of temperature and pressure, which leads to a refractive index gradient in the atmosphere. Although this gradient is very small, atmospheric refraction of electromagnetic waves is observable due to the large distances traveled in the atmosphere between the transmitter and the emitter.
The earth atmosphere can be divided into several layers, including Troposphere, Stratosphere, Mesosphere and Ionosphere. From the viewpoint of atmospheric refraction, only the first two layers, the troposphere and the stratosphere, are important. In the upper layers of the atmosphere the air is so rarefied that the refractive index can be considered to be unity.
In this subsection, we roughly estimate the curve shift by assuming a simple two layer atmosphere model instead of the complicated model described in \cite{neda2003flatness}. Our model is described as follows: assume that the height of the Troposphere is $20km$, and the Stratosphere starts at $20km$ and ends at $50km$. The refractive index is assumed to be a constant for each layer, i.e, the refractive index $n_s$ of the Stratosphere is equal to 1 and the refractive index $n_t$ of Troposphere is equal to 1.0003.
Now we describe our scenario to calculate an estimated curve shift. The graphical illustration of this scenario is in Fig. \ref{fig:f10}. Suppose the LEOS is moving horizontally and its calculated semi-angle of the FDOA cone associated with one single measurement is \ang{30}. According to the above 2-layer atmosphere model and the well-known Snell's law, it is straightforward to obtain that the curve shift is about 41m, which is small when compared to the curve shift associated with the error between the nominal and the true frequencies.
\begin{figure}[tb]
\centering
\includegraphics[width=0.9\linewidth]{Atmosphere_refraction.pdf}
\caption{\label{fig:f10}Graphical illustration of the assumed scenario of the atmosphere.}
\end{figure}
\subsubsection{Curve shift associated with the relativistic Doppler effect}
The relativistic Doppler effect is the change in frequency of electromagnetic waves, caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects described by the special theory of relativity. The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity and do not involve the medium of propagation as a reference point.
Due to relativistic effects, clocks on the receiver are time dilated relative to clocks at the source. The measured frequency in the receiver, denoted by $f_{r}$, can be described by
\begin{equation}
f_{r} = \rho f
\label{eq:relativistic_doppler}
\end{equation}
where $f$ is the classical Doppler shift described by \eqref{eq:generalDoppler}. $\rho$ is called the Lorentz factor, which is defined as
\begin{equation}
\rho = \frac{1}{\sqrt{1-\varsigma^2}}
\end{equation}
in which $\varsigma:=v_r/c$ is the velocity of the receiver in terms of the speed of light.
We now use an example to illustrate the curve shift associated with the relativistic Doppler shift. We use the parameters selected in LEOS example 1 in this example. We assume that there exists a FODA cone with an accurate semi-angle of \ang{45} associated with a single Doppler shift measurement at the LEO satellite. Now we consider the relativistic Doppler effect. According to \eqref{eq:relativistic_doppler}, it can be obtained that $\rho$ is equal to a value that is very close to 1 with the error less than $10^{-10}$. By using this value, we can obtain that the change of the semi-angle of FDOA cone is $1.94\times 10^{-7}$ degrees. By assuming that the earth surface is flat and there is no refraction through the air, it can be directly calculated that the curve shift is about $0.0001352m$, which is small enough to be neglected.
\section{Conclusions and future work}
This paper investigates the problem of determining a curve on the earth's surface on which a stationary emitter must lie based only on the measured frequency on a mobile vehicle (UAV or LEOS) and the vehicle's navigation data (position and velocity). Our investigated scenario considers the following two facts: 1) Coriolis effect; 2) the bumpy earth's surface. We prove that the relative velocity component due to the Coriolis effect makes zero contribution to the Doppler shift. We then provide methods for building equations to describe a FDOA cone and finding its intersection curve with the WGS84 ellipsoid. The curve comprises points, which are mapped into a DTED dataset to find the curve on the earth's terrain. We further provide numerical examples to illustrate how the errors resulting from the nonconstant refractive index of the atmosphere and from lack of precise knowledge of the transmitter frequency affect the positions of curves. Future work will focus on estimating the true frequency of the emitter using multiple frequency measurements and their associated intersection curves on the surface of the WGS84 ellipsoid.
\bibliographystyle{ieeetr}
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Q: Flink Kafka Source Asynchronous auto-commit of offsets failed Flink version: v1.15.2
I had a problem with Apache Flink: Flink failed to submit offset when the Kafka Source table of the Flink task used the same group ID as other Kafka consumers. The problem scenario is described as follows:
*
*I have a Java application that is a Kafka consumer, using the consumer group 'TopicA' to consume data from the topic 'topic_a'
*There is a Flink task, and the Kafka consumer group used by its Kafka Source table is also 'TopicA', but consumes the data of the topic 'topic_b'
At this point, the following error appears in the log information of the Flink task: Asynchronous auto-commit of offsets {topic_b-0=OffsetAndMetadata{offset=xxx, leaderEpoch=0, metadata=''}} failed: Commit cannot be completed since the group has already rebalanced and assigned the partitions to another member. This means that the time between subsequent calls to poll() was longer than the configured max.poll.interval.ms, which typically implies that the poll loop is spending too much time message processing. You can address this either by increasing max.poll.interval.ms or by reducing the maximum size of batches returned in poll() with max.poll.records.
A: The Java application uses the Spring-Kafka framework, which uses the subcribe() method by default.
The Flink Kafka Connector uses the assign() method.
They use the same consumer group ID, so it is inevitable to report an error when submitting an Offset in Flink.
The solution is to specify the partitions that need to be consumed on the KafkaListener annotation.
e.g:
@KafkaListener(topicPartitions = {@TopicPartition(topic = "topic_a", partitions = {"0"})})
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{"url":"http:\/\/gmat.kmf.com\/question\/91jbhj.html","text":"# GMAT \u8003\u6ee1\u5206\u9898\u5e93\n\nThe cost of 10 pounds of apples and 2 pounds of grapes was $12. What was the cost per pound of apples? (1)The cost per pound of grapes was$2\n\n(2)The cost of 2 pounds of apples was less than the cost of 1 pound of grapes.\n\u2022 AStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient.\n\u2022 BStatement (2) ALONE is sufficient, but statement (1) alone is not sufficient.\n\u2022 CBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.\n\u2022 DEach statement ALONE is sufficient.\n\u2022 EStatements (1) and (2) TOGETHER are NOT sufficient.\n\n|\n\n\u2022 \u6309\u70ed\u5ea6\n\u2022 \u6309\u987a\u5e8f","date":"2020-12-05 05:50:09","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\": 0.4545826315879822, \"perplexity\": 3400.643692221254}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141746320.91\/warc\/CC-MAIN-20201205044004-20201205074004-00170.warc.gz\"}"}
| null | null |
Harlow Chorus appeal after being silenced by government
Covid-19 / Wed 26th May 2021 pm31 12:12pm
HARLOW Chorus has appealed to their local MP after the government told them that they still can't meeting indoors.
The chorus, which meets at St John's Arc in Old Harlow, resumed on Monday May 17th in accordance with the loosening of lockdown measures.
But a day later, without warning, the government updated its guidance to say that, in England, non-professional singing could take place only in groups of up to six people indoors.
The new rules – published the day after a significant relaxation of Covid restrictions, and contravening musicians' expectations – were met with anger and despair.
Harlow Chorus member Peter Sandell has told YH that the chorus is upset but hope the powers that be change their minds.
Peter said: "We had been meeting on zoom for a long long time which has not been ideal and so we were looking forward to meeting again.
"We had the one session at the Arc in which we complied with the social distancing regulations.
"But then to be told the next day that we had to go back to a group of six was very frustrating.
"We have been in contact with MP Robert Halfon. He attended a zoom meeting and was very supportive. We must now sit and wait.
Harlow MP Robert Halfon said:
"We are lucky to have many wonderful choirs in Harlow including Harlow Chorus who I was pleased to meet with last night to discuss the challenges facing them further.
"The past year has been extremely difficult for many of these groups and I am therefore extremely disappointed by the Government's decision to continue to restrict non-professional choral activities to only six people indoors – despite restrictions on other indoor activities, such as gym classes, allowing up to 30 people to participate.
"I have written to both the Culture and Health Secretaries to ask them to immediately review this decision and to ensure that the restrictions on choirs are in line with the restrictions on other indoor activities.
"I will continue to do all I can to help Harlow Chorus and other choir groups in Harlow."
No Comments for Harlow Chorus appeal after being silenced by government:
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\section{Introduction}
The demand for wireless spectrum is projected to continue growing well into the future, and
will only worsen the currently felt spectrum crunch. For instance, the annual data traffic generated by mobile devices is expected to surpass $130$ exabits by 2020 \cite{khan2011mmwave}. To address the problem of spectrum scarcity for cellular communications, it is envisioned that in $5$G cellular systems certain portions of the mmWave band will be used, spanning the spectrum between $30$ GHz to $300$ GHz \cite{rappaport2013millimeter}. However, before mmWave communications can become a reality, it faces significant challenges such as high propagation losses. The propagation loss at the mmWave band is much higher than the sub-6 GHz frequencies due to a variety of factors including atmospheric absorption, basic Friis transmission-effect, and low penetration. For instance, materials
such as brick can attenuate mmWave signals by as much as $40$ to $80$ dB \cite{zhao201328} and the
human body itself can result in a $20$ to $35$ dB loss \cite{lu2012modeling}.
In order to compensate for high propagation losses, large antenna arrays with high directivity are needed. In fact, one of the main drivers behind the emergence of mmWave mobile communications
is the emergence of large antenna arrays that can be deployed in relatively small chip areas. In this case, the mean end-to-end channel
gain is amplified by the product of the gains of the transmitter and receiver antennas. These large antenna-arrays, however, cause several other issues such as high energy consumption, mainly because of the analog-to-digital converters (ADCs) and power
amplifiers. For instance,
consumption in ADCs is substantial and it can be written as $P^{\text{(ADC)}} = c_\text{ox} W 2^{r_{\text{ADC}}}$, where $W$ is the bandwidth of the mmWave signal, $r_{\text{ADC}}$ is the quantization rate in bits/sample, and constant $c_\text{ox}$ depends on the gate-oxide capacitance of the converter. At a sampling rate of $1.6$ Gsamples/sec, an $8$-bit quantizer consumes $\approx 250$mW of power. During active transmissions, this would constitute up to $50$\% of the overall power consumed for a typical smart phone.
Table \ref{tab:power-consumpiton} compares the representative values for power consumption in mmWave against that in sub-6 GHz \cite{orhan2015low,phan2010reconfigurable,liang20070}.
From this table and the fact that ADC power consumption increases proportionally with bandwidth, we conclude that \emph{the large bandwidths afforded by mmWave channels present an issue for the components due to the need for a proportionally high power, whereas the achievable data rate increases only logarithmically as a function of the bandwidth. }
\begin{table}[t]\centering
\renewcommand{\arraystretch}{1.3}
\ra{1.3}
\begin{tabular}{@{}p{4.5cm}cc@{}}\toprule
\textbf{Component} & \textbf{mmWave Technology} & \textbf{RF Technology} \\ \toprule
Analog to Digital Converter (ADC) & 250 mW & 0.14 mW \\ \midrule
Low Noise Amplifier (LNA) & 39 mW & 3-10 mW \\ \midrule
Frequency Mixer & 16.8 mW & 0.5-8 mW \\ \midrule
Phase Shifter & 19.8 mW & -- \\
\bottomrule
\end{tabular}
\caption{Representative values for different component power consumption in the RF and mmWave technologies}
\label{tab:power-consumpiton}
\end{table}
In addition to power consumption, in designing a communication system, one of the main objectives is to maximize the achievable rate (bits/sec). However, there is a law of diminishing returns, when it comes to the achievable rate, with increasing bandwidth. Indeed, for a wideband coherent communications system, the rate of increase in achievable rate varies as $\frac{\text{SNR}}{W}$ as a function of the bandwidth $W$. Therefore, it is often the case that the achievable rate per unit power is a non-increasing function of available bandwidth beyond a threshold. To illustrate this point, we calculate the achieved bits/sec/watt for mmWave and sub-6 GHz interfaces in Fig.~\ref{fig:shannon_with_components}, incorporating the consumption by the components. In order to plot this figure, we have used a SISO model and typical values for consumption provided in~\cite{rappaport2011state,phan2010reconfigurable,liang20070} for an 8-bit quantizer and power spectral density of white noise taken to be the product of the temperature and the Boltzman constant. The overall transmission power is taken to be $10$ dB higher for sub-6 GHz and the loss in the channel is also taken into account. Note that the span of the bandwidth values is taken to be between $10$ KHz to $10$ MHz for sub-6 GHz and between $0.7$ MHz to $7$ GHz for mmWave.
\begin{figure}[t]
\centerline{\includegraphics[height=2.5in]{fig/bits_per_joule_new}}
\vspace{-0.2in}
\caption{\small{Achievable rate per unit power with the component energy consumption taken into account.}}
\label{fig:shannon_with_components}
\end{figure}
We have some interesting trends here. Firstly, the achievable rate per unit bandwidth is not a monotonic function: for mmWave, it tends to decrease for large bandwidth values due to the increased consumption by the ADCs. The amount of increase in rate decreases inversely with the bandwidth, while the ADC power consumption increases linearly with bandwidth. This leads to the reduction in rate per unit power in the wideband regime. On the contrary, in a large band of values, sub-6 GHz interface becomes increasingly energy efficient as the bandwidth increases, due to the relatively low consumption in ADCs and other components.
Thus, the mmWave interface should be utilized for data communication up to a certain bandwidth (above $\sim 800$MHz in Fig.~\ref{fig:shannon_with_components}). From this example, we conclude that \emph{even though a large bandwidth is available in mmWave band, it may be more energy efficient to utilize only a part of it.} Beyond that point, sub-6~GHz starts to become more energy-efficient per bit transmitted due to the relatively low consumption in ADCs.
\subsection{Our Contributions}
In order to fully exploit the abundant yet intermittent mmWave bandwidth, we consider an integrated architecture in which the sub-6 GHz and mmWave interfaces coexist and act in cohesion \cite{hashemi2017hybrid}. This paper is aimed to optimally allocate the power and bandwidth jointly across the interfaces. To this end, first we formulate an optimization problem to maximize the achievable sum rate under power constraints at the transmitter and receiver. We investigate the solution space under various conditions depending on the availability of channel state information at the transmitter. Next, we consider the energy efficiency optimization problem, and demonstrate that this problem, in fact, can be reduced to the sum rate maximization. The important point is that our formulation explicitly takes into account the energy consumption in integrated-circuit components. Our results show that despite the availability of huge mmWave bandwidth, it may not be the most energy efficient to utilize it fully at all times. In particular, we prove that:
\begin{enumerate}
\item The ratio of optimal power allocated to the bandwidth utilized for each interface is proportional to the channel state of that interface. Therefore, whenever the channel condition of one of the interfaces degrades, the ratio of power to bandwidth for that interface should decrease as well.
\item As the ADC components become less energy efficient (i.e., a higher power consumption), it is optimal to decrease the utilized mmWave bandwidth.
\item As the total power budget increases, it is optimal to expand the span of the utilized mmWave bandwidth.
\item Since the sub-6 GHz bandwidth is much smaller than the mmWave bandwidth by several orders of magnitude, component consumption in sub-6 GHz becomes negligible compared with mmWave. Hence, it is almost optimal to fully utilize the sub-6 GHz bandwidth.
\item When the system operates at low SNR regime (i.e., low power budget), the input power is allocated to the interface that has a better channel condition.
\end{enumerate}
In summary, the main contributions of this work are as follows: (i) we provision an integrated sub-6 GHz/mmWave transceiver model, and formulate two optimization problems to jointly allocate the bandwidth and power across the interfaces in order to maximize the achievable sum rate and energy efficiency; (ii) we investigate the optimal solution space under various conditions such as when there is no channel state information or when there is only partial channel state information available at the transmitter; and (ii) we analytically confirm the experimental observations in that it may not be optimal (in terms of achievable sum rate) to allocate full bandwidth to both interfaces.
\subsection{Related Work}
We classify existing and related work across the following thrusts: (i) energy efficient mmWave architectures, and (ii) integrated mmWave systems.
\textbf{(I) Energy efficient mmWave architectures:} Energy efficient transceiver architectures such as the use of low resolution ADCs and hybrid analog/digital combining has attracted significant interest. The limits of communications over additive white Gaussian channel
with low resolution (1-3 bits) ADCs at the receiver is studied
in \cite{singh2009limits}. The bounds on the capacity of the MIMO channel with 1-bit ADC at high and low SNR regimes are derived in \cite{mo2014high} and \cite{mezghani2007ultra},
respectively. The joint optimization
of ADC resolution with the number of antennas in a MIMO channel is studied in \cite{bai2013optimization}. While \cite{el2014spatially} provides efficient hybrid precoding and combining algorithms for sparse mmWave channels that performs close to full digital solution, \cite{alkhateeb2014mimo} combines efficient channel estimation
with the hybrid precoding and combining algorithm in \cite{el2014spatially}.
Although there has been extensive amount of work to optimize the mmWave receivers architecture (e.g., in terms of ADCs), the effect of bandwidth on the mmWave performance has not been fully investigated. To the best of our knowledge, only the authors in \cite{orhan2015low} have studied the effect of bandwidth on the performance of \emph{standalone mmWave systems}. Compared with \cite{orhan2015low}, we consider an \emph{integrated sub-6 GHz/mmWave architecture} in which the transmitter and receiver are power constrained. In this case, an optimal power and bandwidth allocation is derived to maximize the achievable sum rate and energy efficiency. In addition, the authors in \cite{orhan2015low} consider the number of ADC quantization bits as an optimization parameter, while we assume that the ADC architecture is fixed and the transmit power and bandwidth are optimized for a MIMO architecture. We derive the closed form expressions for the optimal power and bandwidth allocation across the sub-6 GHz and mmWave interfaces. To the best of our knowledge, there is no previous work that investigates the joint effect of bandwidth and transmission power on the performance of integrated sub-6 GHz/mmWave systems.
\textbf{(II) Integrated sub-6 GHz/mmWave systems:}
Beyond the classical mmWave communications and beamforming methods \cite{rappaport2013millimeter,collonge2004influence,roh2014millimeter,adhikary2014joint}, recently, there have been proposals on leveraging out-of-band information in order to enhance the mmWave performance. The authors in \cite{aliestimating} propose a transform method to translate the spatial correlation matrix at the sub-6 GHz band into the correlation matrix of the mmWave channel. The authors in \cite{nitsche2015steering} consider the $60$ GHz indoor WiFi network, and investigate the correlation between the estimated angle-of-arrival (AoA) at the sub-6 GHz band with the mmWave AoA in order to reduce the beam-steering
overhead. The authors in \cite{ali2017millimeter} propose a compressed beam selection method which is based on out-of-band spatial information obtained at sub-6 GHz band. Our work is distinguished from the above cited works as we investigate the optimal physical layer resource allocation across the sub-6 GHz and mmWave interfaces. In our proposed hybrid sub-6 GHz/mmWave architecture, both interfaces can be simultaneously used for data transfer. In \cite{hashemi2017hybrid}, we investigated the problem of optimal load devision and scheduling across the sub-6 GHz and mmWave interfaces.
\subsection{Notations} We use the following notation throughout the paper. Bold uppercase and lowercase letters are used for matrices and vectors, respectively, while non-bold letters are used for scalers. In addition, $(.)^H$ denotes the conjugate transpose; $\text{tr}(.)$ denotes the matrix trace operator; and $\mathds{E}[.]$ denotes the expectation operator. The sub-6 GHz and mmWave variables are denoted by $(.)_\text{sub-6}$ and $(.)_\text{m}$, respectively.
\section{System Model and Problem Formulation}
\subsection{System Model}
Figure \ref{fig:system} illustrates the components of our proposed architecture that integrates the sub-6 GHz and mmWave interfaces. In \cite{hashemi2017hybrid}, we demonstrated that sub-6 GHz spatial information enables the mmWave beamforming fully in the analog domain without incurring large delay overhead, and thus it can remedy the high energy consumption issue by reducing the number of ADC components that are otherwise needed for digital beamforming.
\begin{figure*}[t!]
\begin{center}
\includegraphics[scale=.33, trim = 0cm 3.7cm 0cm 3.2cm, clip]{fig/system_architect.pdf}
\caption{Integrated sub-6 GHz and mmWave system with optimal power and bandwidth allocation across the interfaces. The goal is to maximize the achievable sum rate and energy efficiency. }
\label{fig:system}
\end{center}
\end{figure*}
In this paper, in addition to beamforming, we leverage the sub-6 GHz interface for communications and data transfer, and assume that the sub-6 GHz/mmWave transmitter and receiver are power constrained. The power constraint at the transmitter dictates the optimal power allocation across the interfaces, while the receiver power constraint determines the optimal bandwidth allocation. Without loss of generality, we assume that the transmitter and receiver constraints are jointly considered as a single constraint, and the problem is expressed in joint power and bandwidth allocation across the interfaces with the total power budget $P_{\text{max}}$. In this case, the results will qualitatively be parallel to the scenario where we impose constraints on the consumed power at the transmitter and at the receiver separately.
\begin{figure}[t]
\centering
\includegraphics[scale=.4, trim = 0cm 1.5cm 0cm 2.8cm, clip]{fig/RF_architect.pdf}
\caption{Sub-6 GHz system model based on digital beamforming}
\label{fig:RF-model}
\end{figure}
\subsection{Sub-6 GHz System and Channel Model}
The sub-6 GHz system model is shown in Fig. \ref{fig:RF-model} where we use digital beamforming. As a result, the received signal at the receiver can be written as:
\begin{equation}
\label{eq:rec_signals}
\mathbf{y}_{\text{sub-6}}=\mathbf{H}_{\text{sub-6}} \cdot
\mathbf{x}_{\text{sub-6}}+\mathbf{n}_{\text{sub-6}},
\end{equation}
where $\mathbf{H}_{\text{sub-6}}$ is the sub-6 GHz channel matrix and $\mathbf{x}_{\text{sub-6}}$ is the transmitted signal vector in sub-6 GHz. Entries of circularly symmetric white Gaussian noise $\mathbf{n}_{\text{sub-6}}$ are normalized to have unit variance.
\begin{figure}[t]
\centering
\includegraphics[scale=.4, trim = 0cm 1.2cm 0cm 0.5cm, clip]{fig/mmWave_architect.pdf}
\caption{mmWave system model with analog beamforming}
\label{fig:mmWave-model}
\end{figure}
In the proposed system, we assume that the sub-6 GHz interface can utilize the total bandwidth of $W_\text{sub-6}^{\max}$. Moreover, the transmission power of the sub-6 GHz interface is denoted by $P_\text{sub-6}=\text{tr}({\mathbf{K}_{\textbf{xx}}})$ in which $\mathbf{K}_{\textbf{xx}}$ is the covariance matrix of signal $\mathbf{x}_\text{sub-6}$. We assume that the sub-6 GHz system includes $n_t$ transmit and $n_r$ receive antennas.
\subsection{MmWave System and Channel Model}
The mmWave system model is shown in Fig. \ref{fig:mmWave-model}. Unlike sub-6 GHz, we use analog combining for mmWave via a single ADC. Consequently, the signal at the input of the decoder is a scalar, identical to a weighted combination of signal $x_{\text{m}}$ across all antennas. Thus, the received signal at the mmWave receiver can be written as:
\begin{equation}
\label{eq:rec_signals}
y_{\text{m}}=\vc{w}_r^H \mathbf{H}_{\text{m}} \vc{w}_t \cdot x_{\text{m}}+\tilde{n}_{\text{m}},
\end{equation}
where $\vc{w}_r$ and $\vc{w}_t$ are the receive and transmit beamforming vectors.
The $\tilde{n}_{\text{m}}$ term denotes the effective Gaussian noise and is normalized to have unit variance. The mmWave interface is assigned with the total bandwidth of $W_\text{m}^{\max}$, and mmWave transmit power is denoted by $P_\text{m}$.
\subsection{Problem Formulation}
In mmWave communications, due to the large consumption of the components and the losses in the channel, energy is
a scarce resource, and thus we assume that $P_{\max}$ is the maximum power available for \emph{data transmission} and \emph{component consumption} (i.e., ADC components) across the sub-6 GHz and mmWave interfaces. Moreover, the ADC power consumption scales proportionally with bandwidth, i.e., $P^{\text{(ADC)}} = a W$, where $W$ denotes the bandwidth and $a$ is a constant for a given ADC with fixed quantization rate (i.e., $a = c_{ox} 2^{r_{ADC}}$). As mentioned earlier and from Fig. \ref{fig:shannon_with_components}, it may not be the most energy efficient to utilize the available bandwidth fully at all times, and thus bandwidth allocated to each interface needs to be optimized in addition to the power allocated to each interface. Hence, we define Problem 1 as follows in order to maximize the achievable sum rate across the interfaces with a given constraint on the joint transmitter and receiver power consumption.
\begin{itemize}
\item \textbf{Problem 1} \textbf{(Power-Constrained Sum Rate Maximization):} \emph{We consider the problem of maximizing the sum rate of the sub-6 GHz and mmWave interfaces subject to a power constraint, i.e.,:}
\begin{subequations}
\small
\begin{align}
& \displaystyle{\max_{\substack{W_{\text{m}},W_{\text{sub-6}} \\ P_{\text{m}}, P_{\text{sub-6}}}}} \displaystyle{\Erf{ W_{\text{sub-6}} \log \det \left( \mathbf{I}+ \frac{\mathbf{H}_{\text{sub-6}}^H \mathbf{K}_{\mathbf{xx}} \mathbf{H}_{\text{sub-6}}}{W_{\text{sub-6}}} \right)}} \displaystyle{+\Emm{W_{\text{m}} \log \left( 1 + \frac{P_{\text{m}} \left|\vc{w}_r^H \mathbf{H}_{\text{m}} \vc{w}_t\right|^2}{W_{\text{m}}} \right)}} \label{problem-1} \\
& \ \text{subject to:} \ \ \ P_{\text{sub-6}} \large{\mathds{1}}_{W_\text{sub-6} > 0} + n_r aW_{\text{sub-6}}+P_{\text{m}} \large{\mathds{1}}_{W_\text{m} > 0}+ a W_{\text{m}} \leq P_{\max}, \label{power-constraint}\\
& \hspace{2.2cm} 0 \leq W_\text{sub-6} \leq W_\text{sub-6}^{\max}; \ 0 \leq W_{\text{m}} \leq W_\text{m}^{\max}; \label{bandwidth-constraint}\\ & \hspace{2.2cm} 0 \leq P_\text{m}, P_\text{sub-6}.
\end{align}
\end{subequations}
\normalsize
In Section \ref{sec:problem1-solution}, we investigate the optimal solution of Problem 1.
\end{itemize}
\begin{itemize}
\item \textbf{Problem 2 (Energy Efficiency Maximization):}
\emph{The second problem that we explore is maximization of energy efficiency (sum rate per unit power expenditure), i.e.,:}
\begin{subequations}
\small
\begin{align}
& \displaystyle{\max_{\substack{W_{\text{m}},W_{\text{sub-6}} \\ P_{\text{m}}, P_{\text{sub-6}}}}} \frac{\displaystyle{\Erf{W_{\text{sub-6}} \log \det \left( \mathbf{I}+ \frac{\mathbf{H}_{\text{sub-6}}^H \mathbf{K}_{\mathbf{xx}} \mathbf{H}_{\text{sub-6}}}{W_{\text{sub-6}}}\right)} + \Emm{W_{\text{m}} \log \left( 1+ \frac{P_{\text{m}}}{W_{\text{m}}}\left|\vc{w}_r^H \mathbf{H}_{\text{m}} \vc{w}_t\right|^2 \right)}}}{P_{\text{sub-6}} \large{\mathds{1}}_{W_\text{sub-6} > 0} + n_r aW_{\text{sub-6}} + P_{\text{m}} \large{\mathds{1}}_{W_\text{m} > 0}+ a W_{\text{m}}}, \label{eq:problem2} \\
& \ \text{subject to:} \ \ 0 \leq W_\text{sub-6} \leq W_\text{sub-6}^{\max}; \ 0 \leq W_{\text{m}} \leq W_\text{m}^{\max}; \label{eq:problem2-bandwidth-constraint}\\ & \hspace{2cm} 0 \leq P_\text{m}, P_\text{sub-6}. \label{eq:problem2-power-constraint}
\end{align}
\end{subequations}
\normalsize
In Section \ref{sec:problem2-solution}, we consider the solution space of Problem 2.
\end{itemize}
\section{Power-Constrained Sum Rate Maximization}
\label{sec:problem1-solution}
In this section, we investigate the optimal solution of Problem 1. To this end, we note that Problem 1 is that of the convex optimization since the objective function is concave and the constraint is linear. In addition, the objective function is concave and increasing in the variables $W_{\text{sub-6}}, P_\text{sub-6}, W_{\text{m}}$, and $P_\text{m}$. It is straightforward to show that the objective function is increasing in $P_\text{m}$ and $P_\text{sub-6}$. Moreover, by taking the derivative of the function $f(x) = x\log(1+\frac{1}{x})$, one can see that $f(x)$ is increasing in $x$, and thus \eqref{problem-1} is increasing in $W_\text{sub-6}$ and $W_\text{m}$ as well. Using the second order derivatives, we can show that the objective function is concave in $W_\text{sub-6}$ and $W_\text{m}$.
From Problem 1, we note that there is a tradeoff in bandwidth allocation: it is desirable to set the sub-6 GHz and mmWave bandwidth variables to $W_\text{sub-6}^{\max}$ and $W_\text{m}^{\max}$ in order to increase the objective value. However, due to the \eqref{power-constraint} constraint, high bandwidth reduces the \emph{transmission power} (due to the power-hungry ADC components), which in turn, reduces the objective value. Similarly, there is a tradeoff in allocating the transmission power $P_\text{sub-6}$ and $P_\text{m}$.
In order to optimally balance this tradeoff, we solve the sum rate optimization problem using the convex optimization tools under two scenarios: (i) when no channel state information at the transmitter (CSIT) is available, and (ii) when partial CSIT is available.
\subsection{No Channel State Information at the Transmitter}
When the channel matrix $\mathbf{H}_\text{sub-6}$ is random and there is no channel state information at the transmitter (CSIT) available, the optimal power allocation across the $n_t$ antenna elments of the sub-6 GHz interface (different than the optimal power allocation across the interfaces), is uniform \cite{tse2005fundamentals}. Therefore, the covariance matrix $\mathbf{K}_\mathbf{xx}$ can be written as:
\begin{equation}
\mathbf{K}_\mathbf{xx} = \frac{P_\text{sub-6}}{n_t} \mathbf{I}_{n_t},
\label{eq:optimal-covariance-no-csi}
\end{equation}
\noindent and thus, the first term in \eqref{problem-1} is simplified to:
\begin{equation}
\displaystyle{W_{\text{sub-6}} {\log \det \left( \mathbf{I}+ \frac{P_\text{sub-6}}{W_{\text{sub-6}} n_t} \mathbf{H}_{\text{sub-6}}^H \mathbf{H}_{\text{sub-6}} \right)}}.
\label{rf-capacity}
\end{equation}
We assume that $\lambda_1 \geq \lambda_2 \geq ... \geq \lambda_{n}$ denote the ordered singular values of the sub-6 GHz channel matrix $\mathbf{H}_{\text{sub-6}}$ where $n = \min(n_t, n_r)$. Therefore, from the determinant and singular value properties, we can rewrite \eqref{rf-capacity} as:
\begin{equation}
W_{\text{sub-6}}\sum_{i=1}^{n} \log \left(1+\frac{P_\text{sub-6}}{W_\text{sub-6} n_t} \lambda_i ^2 \right).
\label{eq:RF-simple-form}
\end{equation}
As a result, \eqref{problem-1} is simplified by replacing its first term with \eqref{eq:RF-simple-form}. Due to the fact that problem \eqref{problem-1} is convex, the
Karush–Kuhn–Tucker (KKT) conditions are necessary and sufficient for the optimality of the solution \cite{boyd2004convex}. In order to derive the KKT conditions, we form the following Lagrangian function:
\vspace{-.6cm}
\small{
\begin{align}
\mathcal{L} (W_\text{m}, W_\text{sub-6}, P_\text{m}, P_\text{sub-6}, \boldsymbol{\mu}) = W_{\text{sub-6}}\displaystyle{\sum_{i=1}^{n} \log \left(1+\frac{P_\text{sub-6}}{W_\text{sub-6} n_t} \lambda_i ^2 \right) }
\displaystyle{+{W_{\text{m}} \log \left( 1 + \frac{P_{\text{m}}}{W_{\text{m}}} \left|\vc{w}_r^H \mathbf{H}_{\text{m}} \vc{w}_t\right|^2 \right)}} \nonumber \\
+ \mu_0\bigg(P_\text{max} - \displaystyle{P_{\text{sub-6}} - n_r a W_{\text{sub-6}}-P_{\text{m}}- a W_{\text{m}}}\bigg) + \mu_1 \left(W_\text{sub-6}^{\max} - W_\text{sub-6}\right) + \mu_2 (W_\text{m}^{\max} - W_\text{m}) + \mu_3 P_\text{sub-6} + \mu_4 P_\text{m} & ,
\label{lagrange}
\end{align}}
\normalsize
\vspace{-.5cm}
\noindent for the Lagrange multiplier vector $\boldsymbol{\mu} = (\mu_0, ..., \mu_4)$. From the KKT stability conditions, we have:
\begin{equation}
\left\{
\begin{array}{l l}
n \log \frac{P_\text{sub-6}}{W_\text{sub-6} n_\text{t}} + \sum_{i=1}^{n}
\log \lambda_i^2 - \frac{P_\text{sub-6} \lambda_i ^2}{W_\text{sub-6} n_t + P_\text{sub-6} \lambda_i^2} & = n_r a \mu_0 + \mu_1,\\ \vspace{.1cm}
W_\text{sub-6} \sum_{i=1}^{n} \frac{\lambda_i^2}{W_\text{sub-6} n_t + P_\text{sub-6} \lambda_i^2} & = \mu_0 - \mu_3, \\ \vspace{.1cm}
\log \frac{P_\text{m}}{W_\text{m}} + \log A - \frac{P_\text{m}A}{W_\text{m} + P_\text{m} A} &= a \mu_0 + \mu_2, \vspace{.14cm} \\
\frac{W_\text{m} A }{W_\text{m} + P_\text{m} A} & = \mu_0 - \mu_4.
\end{array} \right.
\label{kkt}
\end{equation}
For the sake of notations, we define $A~:= ~\left|\vc{w}_r^H \mathbf{H}_{\text{m}} \vc{w}_t\right|^2$.
From the KKT complimentary slackness conditions, we have:
\begin{equation}
\left\{
\begin{array}{l l}
\mu_0\bigg(P_\text{max} - \displaystyle{P_{\text{sub-6}} - n_r aW_{\text{sub-6}}-P_{\text{m}}- a_{\text{m}}W_{\text{m}}}\bigg) & = 0, \vspace{.14cm} \\
\mu_1 \left(W_\text{sub-6}^{\max} - W_\text{sub-6}\right) & = 0, \vspace{.14cm} \\
\mu_2 (W_\text{m}^{\max} - W_\text{m}) & = 0, \vspace{.14cm} \\
\mu_3 P_\text{sub-6} & = 0, \vspace{.14cm} \\
\mu_4 P_\text{m} & = 0.
\end{array} \right.
\label{kkt-slackness}
\end{equation}
From the second and fourth equations in \eqref{kkt}, we conclude that $\mu_0 - \mu_3 > 0 \Rightarrow \mu_0 > \mu_3 \geq 0$, and thus $\mu_0 > 0$. However, from the first equation in \eqref{kkt-slackness}, we conclude that $$P_\text{max} - \displaystyle{P_{\text{sub-6}} - n_r aW_{\text{sub-6}}-P_{\text{m}}- a_{\text{m}}W_{\text{m}}} = 0,$$ and thus the power constraint is satisfied with equality.
Intuitively, we can also argue that the objective function in Problem 1 is an increasing function of the bandwidth parameters $W_\text{sub-6}$ and $W_\text{m}$, and thus the objective function can be improved by increasing the value of $W_\text{sub-6}$ and $W_\text{m}$. In the case that constraint \eqref{bandwidth-constraint} becomes passive, the objective function can be improved by increasing the transmit power $P_\text{sub-6}$ and $P_\text{m}$ since the objective function is increasing in terms of these variables as well. Therefore, the power constraint \eqref{power-constraint} is always satisfied with equality at the optimal solution. Next, in order to characterize the optimal solution we note that since the transmit power $P_\text{sub-6}$ and $P_\text{m}$ cannot be zero, $\mu_3 = \mu_4 =0$ should hold to satisfy the complimentary slackness conditions. For the Lagrange multipliers $\mu_1$ and $\mu_2$, we have the following cases.
\textbf{(I) Full Sub-6 GHz Bandwidth Allocation ($\boldsymbol{\mu_1 >0}$):} Depending on the channel conditions, if $\mu_1 > \mu_2$ holds, since the Lagrange multiplier $\mu_2$ is nonnegative, it results in $\mu_1>0$. Thus, from the complimentary slackness condition, we conclude that
$W_\text{sub-6}^* = W_\text{sub-6}^{\max}$. Moreover, the condition $\mu_1 > \mu_2$ implies that $n_r a \mu_0 + \mu_1 > a \mu_0 + \mu_2$ as well. As a result, from \eqref{kkt}, we can see that if the sub-6 GHz interface has a better channel condition than the mmWave channel, then the whole available bandwidth $W_\text{sub-6}^{\max}$ should be utilized. This results in a system of equations from which the optimal value for $W_\text{m}^*, P_\text{m}^*,$ and $P_\text{sub-6}^*$ are calculated as follows:
\begin{equation}
P_\text{sub-6}^* = \max\left(0, \frac{n n_r a W_\text{sub-6}^\text{max}}{B}\right),
\label{eq:power-optimal}
\end{equation}
\begin{equation}
P_\text{m}^* = \max\left(0, \frac{P_\text{max} - CW_\text{sub-6}^\text{max}}{B+1}\right),
\end{equation}
\begin{equation}
W_\text{m}^* = \max\left(0, \frac{P_\text{max} - C W_\text{sub-6}^\text{max}}{a+\frac{a}{B}}\right),
\label{eq:BW-optimal}
\end{equation}
where:
$$
B = \omega \left(\log(a A) -1 \right) \ \text{and} \ \ C = n_r a + \frac{n n_r a}{B},
$$
in which $\omega(.)$ is the Wright omega function.
Note that in order to make the calculation more tractable, we assume that the system operates at high SNR regime, and thus the following approximations hold:
$$
1+\frac{P_\text{sub-6}}{W_\text{sub-6} n_t} \lambda_i ^2 \approx \frac{P_\text{sub-6}}{W_\text{sub-6} n_t} \lambda_i ^2,
\quad \text{and} \quad
1 + \frac{P_{\text{m}}}{W_{\text{m}}} A \approx \frac{P_{\text{m}}}{W_{\text{m}}} A.
$$
Moreover, the KKT conditions in \eqref{kkt} result in the following equations:
\begin{equation}
\frac{A }{1 + \frac{P_\text{m}}{W_\text{m}} A} = \sum_{i=1}^{n} \frac{\frac{\lambda_i^2}{n_t}}{1 + \frac{P_\text{sub-6}}{W_\text{sub-6} n_t} \lambda_i^2};
\label{eq:kkt-result-1}
\end{equation}
\begin{equation}
\frac{\log \frac{P_\text{m}}{W_\text{m}} + \log A}{\frac{P_\text{m}}{W_\text{m}} + a} = \frac{\log \frac{P_\text{sub-6}}{W_\text{sub-6}} + \sum_{i=1}^{n}\log \lambda_i^2}{\frac{P_\text{sub-6}}{W_\text{sub-6}} + n_r a},
\label{eq:kkt-result-2}
\end{equation}
in which $A = ~\left|\vc{w}_r^H \mathbf{H}_{\text{m}} \vc{w}_t\right|^2$ captures the mmWave channel conditions.
\emph{From \eqref{eq:kkt-result-1} and \eqref{eq:kkt-result-2}, we observe that whenever the channel condition of one of the interfaces degrades, the ratio of power to bandwidth for that interface should decrease as well. }
\begin{figure}[t!]
\begin{center}
\subfigure[MmWave bandwidth vs. ADC consumption]
{
\label{fig:mmWave_bandwidh_ADC}
\includegraphics[scale=.21, trim = 1cm 0cm 3cm 1cm, clip]{fig/optimal_mmWave_bandwidth_vs_AD_consumption}
}\hspace{1cm}
\subfigure[Sum rate vs. ADC consumption]
{
\label{fig:sum_rate_ADC}
\includegraphics[scale=.21, trim = 0cm 0cm 3cm 1cm, clip]{fig/optimal_sum_rate_vs_AD_consumption}
}
\end{center}
\vspace{-0.15in}
\caption{Optimal solution as a function of ADC energy consumption for $P_{\max} = 1000$ mW. In these figures, ADC energy consumption refers to the constant $a \times 10^{8}$. }
\vspace{-0.2in}
\label{HB_effect}
\end{figure}
\begin{figure}[t!]
\begin{center}
\subfigure[MmWave bandwidth vs. input power] {
\label{fig:mmWave_bandwidth_power}
\includegraphics[scale=.28, trim = 3cm 2.8cm 3cm 2.5cm, clip]{fig/optimal_mmWave_bandwidth_vs_power}
} \hspace{.2cm}
\subfigure[Sum rate vs. input power]
{
\label{fig:mmWave_sum_rate}
\includegraphics[scale=.28, trim = 3cm 2.3cm 3cm 2.5cm, clip]{fig/optimal_sum_rate_vs_power}
}
\end{center}
\vspace{-0.15in}
\caption{Optimal solution as a function of input power budget for $a=10^{-9}$}
\vspace{-0.2in}
\label{HB_effect}
\end{figure}
In order to investigate behavior of the optimal power and bandwidth allocation as a function of the physical characteristic of ADC (i.e., $a=c_{ox}2^{r_{ADC}}$), we note that
$W_\text{m}^*$ decreases as the power consumption by ADC component increases. It is straightforward to show this by
taking the derivative of $W_\text{m}^*$ with respect to $a$ and see that
$
\frac{\partial W_\text{m}^*}{\partial a} < 0.
$
Therefore, the optimal bandwidth allocated to the mmWave interface decreases as the power consumption by ADC increases.
Figure \ref{fig:mmWave_bandwidh_ADC} shows the optimal mmWave bandwidth allocation as a function of the scaling factor $a$, noting that a larger $a$ indicates that the ADC consumes more power for a fixed bandwidth. Moreover, Fig. \ref{fig:sum_rate_ADC} demonstrates the achievable sum rate as a function of $a$. From the results, we observe that the optimal bandwidth allocation and the achievable sum rate decreases as $a$ increases. In Fig. \ref{fig:mmWave_bandwidth_power} and \ref{fig:mmWave_sum_rate}, we investigate behavior of the optimal bandwidth allocation and achievable sum rate as a function of the power budget $P_{\max}$. From the results, we observe that the optimal allocated bandwidth to the mmWave interface and the achievable sum rate increases as the input power increases, as expected.
\textbf{(II) Full mmWave Bandwidth Allocation ($\boldsymbol{\mu2 > 0}$):} Similar to Case 1, if due to the channel conditions, the inequality $\mu_2 > \mu_1$ holds,
then from the complimentary slackness conditions, we conclude that the mmWave bandwidth should be fully allocated, i.e., $W_\text{m}^* = W_\text{m}^\text{max}$. In this case, the optimal solution is obtained as follows:
\begin{equation}
P_\text{m}^* = \max\left(0, \frac{aW_\text{m}^\text{max}}{D}\right),
\end{equation}
\begin{equation}
P_\text{sub-6}^* = \max\left(0, \frac{P_{\max}-EW_\text{m}^{\max}}{D+1}\right),
\end{equation}
\begin{equation}
W_\text{sub-6}^* = \max\left(0, \frac{P_\text{max} - E W_\text{m}^{\max}}{n_r E}\right),
\end{equation}
where
$$
D = \omega \left(\frac{\sum_{i=1}^n \log(\lambda_i^2)}{n} - \log\frac{n_t}{a} - 1 \right)\ \text{and} \ \ E=a+\frac{a}{D}.
$$
Similar to Case 1, we note that only one of the interfaces (i.e., mmWave) utilizes its full bandwidth, and the bandwidth allocated to the other interface (i.e., sub-6 GHz) decreases as the energy consumption by the ADC component increases.
\textbf{(III) Special case of negligible ADC consumption in sub-6 GHz:} In the case that ADC power consumption in the sub-6 GHz interface is negligible compared with the mmWave interface, it is optimal to always allocate full bandwidth to the sub-6 GHz interface. In particular, the power constraint \eqref{power-constraint} is simplified to:
$$
P_{\text{sub-6}} \large{\mathds{1}}_{W_\text{sub-6} > 0} +P_{\text{m}} \large{\mathds{1}}_{W_\text{m} > 0}+ a W_{\text{m}} \leq P_{\max}.
$$
In this case, the optimal solution always falls back to Case 1 discussed earlier where we have $\mu_1 > 0$, resulting in allocating full bandwidth to the sub-6 GHz interface, as expected. Moreover, it should be noted if the power consumption by the mmWave ADC is not taken into account, then the optimal solution allocates full bandwidth to both interfaces, as it has been prevalent in the previous works.
\textbf{(IV) Low SNR regime:} In order to derive the optimal values in \eqref{eq:power-optimal}--\eqref{eq:BW-optimal}, we assumed that the system operates at high SNR regime. We can extend the previous results into the low SNR setting as well. We apply the approximation $\log (1+x) \approx x \log_2 e$ for $x$ small, and obtain the following approximations for the sub-6 GHz and mmWave achievable rates at low SNR regime:
$$
\mathcal{R}_\text{sub-6} \approx \sum_{i=1}^n \frac{P_\text{sub-6}}{n_t}\lambda_i^2 \log_2 e,
\quad \text{and} \quad
\mathcal{R}_\text{m} \approx P_\text{m} \left|\vc{w}_r^H \mathbf{H}_{\text{m}} \vc{w}_t\right|^2 \log_2 e.
$$
As a result, the KKT conditions are obtained as follows:
\begin{equation}
\left\{
\begin{array}{l l}
n_r a \mu_0 + \mu_1 & =0 ,\\ \vspace{.1cm}
\frac{\log_2 e}{n_t} \sum_{i=1}^{n}\lambda_i^2 & = \mu_0 - \mu_3, \\ \vspace{.1cm}
a \mu_0 + \mu_2 & =0, \vspace{.14cm} \\
\left|\vc{w}_r^H \mathbf{H}_{\text{m}} \vc{w}_t\right|^2 \log_2 e& = \mu_0 - \mu_4.
\end{array} \right.
\label{kkt-low-SNR}
\end{equation}
Similar to the high SNR setting, depending on the channel conditions, we consider different scenarios under which only one of the interfaces becomes active. In the first scenario, we assume that the mmWave channel has a better channel condition than the sub-6 GHz channel. Therefore, from \eqref{kkt-low-SNR} we have:
$
\mu_0 - \mu_3 < \mu_0 - \mu_4 \Rightarrow 0 \leq \mu_4 < \mu_3.
$
However, from the complementary slackness condition, we have $P_\text{sub-6} \mu_3 = 0$. Therefore, the optimal allocated power to the sub-6 GHz interface should be zero since $0<\mu_3$. This is in agreement with intuition that when the system operates at low SNR regime, the input power is allocated to the interface that has a better channel condition. A similar argument holds when the sub-6 GHz channel has a better channel condition compared with the mmWave interface, which results in zero power allocation to the mmWave interface and transmission across the sub-6 GHz interface only.
\subsection{Partial Channel State Information at the Transmitter}
\label{sec:partial-CSIT}
In MIMO systems, calculating the achievable rate boils down to finding the transmit covariance
matrices that includes finding the
optimum transmit directions and the optimum power
allocation across the transmit antenna array. When the channel is changing over time due to fading, and perfect and instantaneous CSI is known both at the receiver and at the transmitter side, the optimal power allocation across the antenna elements extend to water-filling over both time and space (i.e., the eigenmodes). However, in most of the MIMO wireless communications scenarios, it is unrealistic
to assume that the transmitter side has the perfect knowledge
of the instantaneous CSI. In such cases, it might be more
realistic to assume that only the receiver side can perfectly
estimate the instantaneous CSI, while the transmitter side has
only a partial knowledge of the channel.
Thus far, we studied the optimal power and bandwidth allocation assuming that the sub-6 GHz transmitter does not have channel state information. In this section, we extend the previous results to the case in which there is partial CSIT.
In order to optimally allocate power across the sub-6 GHz and mmwave interfaces, we require that the instantaneous power constraint to be satisfied. Therefore, the optimal covariance matrix $\mathbf{K}_{\mathbf{xx}}$ should satisfy the power constraint at all times, independent of the channel state. Under this assumption, using the water-filling method and deriving the optimal covariance matrix becomes challenging. In order to resolve this issue, we obtain the optimal power allocation for the ``worst and best'' sub-6 GHz channel conditions. Specifically, we derive an upper and lower bound on the sub-6 GHz achievable rate and then obtain the optimal covariance matrix.
\textbf{Partial CSIT model:} Similar to the model presented in \cite{abdelaziz2017compound}, we assume that the sub-6 GHz transmitter has access to \emph{partial CSIT} such that the transmitter still does not know the \emph{exact realization of the channel}, but rather, it knows that the channel matrix belongs to a known compact set defined as follows:
$$
\mathcal{S} = \left\{\mathbf{H}_\text{sub-6}: \mathbf{H}_\text{sub-6} = \bar{\mathbf{H}}_\text{sub-6} + \Delta\mathbf{H}_\text{sub-6}, \ \ |\Delta\mathbf{H}_\text{sub-6}|_2 \leq \epsilon, \ \ \bar{\mathbf{H}}_\text{sub-6} = \lambda^{1/2} \mathbf{v}\mathbf{u}^H \right\},
$$
where $\mathbf{v} \in \mathcal{C}^{n_r\times 1}$ and $\mathbf{u} \in \mathcal{C}^{n_t\times 1}$. In this model, $\bar{\mathbf{H}}_\text{sub-6}$ is the mean channel information that corresponds to the LOS component. In general, LOS component causes the channel variations to be centered around a mean value with unit rank whose gain depends mainly on the distance between the transmitter and receiver, array configuration and respective array orientation. Hence, we assume that $\bar{\mathbf{H}}_\text{sub-6}$ is of unit rank.
On the other hand, $\Delta\mathbf{H}_\text{sub-6}$ captures the NLOS components and represents the variations around the mean value such that $|\Delta\mathbf{H}_\text{sub-6}|_2 \leq \epsilon$. We further assume that the channel error matrix $
\Delta \mathbf{H}_\text{\text{sub-6}}$ has zero mean
and i.i.d. Gaussian entries, each with variance of $\sigma_e ^2$.
Therefore, we have the following mapping between the channel matrices:
$$
\bar{\mathbf{H}}_\text{sub-6} \Longleftrightarrow \mathbf{H}_\text{sub-6}^{\text{LOS}} \ \ \text{and} \ \ \Delta\mathbf{H}_\text{sub-6} \Longleftrightarrow \mathbf{H}_\text{sub-6}^{\text{NLOS}}.
$$
\textbf{(I) Lower Bound on the sub-6 GHz Achievable Rate:} Given that the transmitter has partial CSI according to the presented model, we obtain a lower bound on the achievable rate of the sub-6 GHz interface, and derive the corresponding optimal covariance matrix. In order to derive a lower bound on the sub-6 GHz achievable rate, we consider the \emph{worst sub-6 GHz channel condition} that renders a lower bound on the sub-6 GHz achievable rate. Then, we optimize the physical layer resource allocation for the worst sub-6 GHz channel. From \cite{abdelaziz2017compound}, we adopt the following result.
\begin{theorem}
Assuming the presented CSIT model, and for any nonnegative definite covariance matrix $\mathbf{K}_\mathbf{xx}$, if the achievable data rate of the sub-6 GHz interface is denoted by $\mathcal{R}_\text{sub-6}(\mathbf{H}_\text{sub-6}, \mathbf{K}_\mathbf{xx})$, then we have:
\begin{equation}
\mathcal{R}_\text{sub-6}(\mathbf{H}_\text{sub-6}, \mathbf{K}_\mathbf{xx}) \geq \mathcal{R}_\text{sub-6}({\mathbf{H}}^*_\text{sub-6}, \mathbf{K}_\mathbf{xx}),
\end{equation}
in which ${\mathbf{H}}^*_\text{sub-6}$ is the \textbf{worst sub-6 GHz channel} such that ${\mathbf{H}}_\text{sub-6}^{*H}{\mathbf{H}}^*_\text{sub-6} = [(\lambda^{1/2} - \epsilon)^+]^2\mathbf{uu}^H$ where $(x)^+ = \max(0,x)$.
\label{theorem-lower-bound}
\end{theorem}
\begin{proof}
\small
\begin{align}
\mathcal{R}_\text{sub-6}(\mathbf{H}_\text{sub-6}, \mathbf{K}_\mathbf{xx}) = W_{\text{sub-6}} \log \det \left( \mathbf{I}+ \frac{\mathbf{H}_{\text{sub-6}}^H \mathbf{K}_\mathbf{xx} \mathbf{H}_{\text{sub-6}}}{W_{\text{sub-6}}} \right) \stackrel{(a)}{=} \sum_{i=1}^{n_t} W_\text{sub-6} \log\left(1+\frac{1}{W_\text{sub-6}}\sigma_i\left(\mathbf{H}_\text{sub-6}^H \mathbf{K}_\mathbf{xx} \mathbf{H}_\text{sub-6}\right)\right) \nonumber &\\ \stackrel{(b)}{=} \sum_{i=1}^{n_t} W_\text{sub-6} \log\left(1+\frac{1}{W_\text{sub-6}}\lambda_i\left((\bar{\mathbf{H}}_\text{sub-6}+\Delta\mathbf{H}_\text{sub-6})\mathbf{K}_\mathbf{xx}^{1/2}\right)^2\right) \stackrel{(c)}{\geq} W_\text{sub-6} \log\left(1+\frac{1}{W_\text{sub-6}}[(\lambda^{1/2}-\epsilon)^+]^2\lambda_1(\mathbf{K}_\mathbf{xx}) \right) &\nonumber\\ = \mathcal{R}_\text{sub-6}({\mathbf{H}}^*_\text{sub-6}, \mathbf{K}_\mathbf{xx})&,
\label{eq:partial-csi}
\end{align}
\normalsize
where $\sigma_i (\mathbf{X})$ and $\lambda_i (\mathbf{X})$ denote the $i$-th eigenvalue and singular value of matrix $\mathbf{X}$, respectively. In this case, (a) follows from the determinant properties that the determinant of a matrix is equal to the product of its eigenvalues. Moreover, (b) follows from the fact that:
$$\sigma_i\left(\mathbf{H}_\text{sub-6}^H\mathbf{K}_\mathbf{xx} \mathbf{H}_\text{sub-6}\right)~=~\lambda_i\left(\left(\bar{\mathbf{H}}_\text{sub-6}+\Delta\mathbf{H}_\text{sub-6}\right)\mathbf{K}_\mathbf{xx}^{1/2}\right)^2.
$$
Finally, (c) is concluded from the following singular value inequality \cite{schaefer2015secrecy}, and the fact that $\lambda_i(\bar{\mathbf{H}}_\text{sub-6}) = 0 \ \forall i > 1$.
\begin{lemma}[Singular value inequality]
Let $\mathbf{A}, \mathbf{B}$ and $\mathbf{C}$ be $n \times m$ and $m \times m$ matrices, and $\lambda_i (\mathbf{X})$ denotes the $i$-th singular value of matrix $\mathbf{X}$ such that $\{\lambda_i\}$ is in decreasing order. Then,
\begin{equation}
\lambda_i \left( (\mathbf{A}+\mathbf{B})\mathbf{C} \right) \geq \left(\lambda_i(\mathbf{A}) - \lambda_1 (\mathbf{B}) \right)^+ \lambda_i(\mathbf{C})
\end{equation}
\end{lemma}
Proof is provided in \cite{schaefer2015secrecy}.
\end{proof}
\emph{Theorem \ref{theorem-lower-bound} implies that the optimal input covariance matrix $\mathbf{K}_{\mathbf{xx}}$ for the worst sub-6 GHz channel has to be of unit rank.} That is because the matrix ${\mathbf{H}}^{*H}{\mathbf{H}}^*$ is shown to be of unit rank, and thus it has only one eigenvalue equals to $P_\text{sub-6}$. Its corresponding eigenvector $\mathbf{q}$ with $||\mathbf{q} ||=1$ satisfies the sub-6 GHz interface transmit power constraint. Therefore, the optimal covariance matrix $\mathbf{K}_\mathbf{xx}$ is written as follows:
\begin{equation}
\mathbf{K}_\mathbf{xx} = P_\text{sub-6} \mathbf{q q}^H \ \ \text{s.t.} \ \ ||\mathbf{q} ||=1,
\end{equation}
where the optimal $\mathbf{q}$ can be obtained by the null steering solution \cite{friedlander1989performance}.
\emph{Compared with \eqref{eq:optimal-covariance-no-csi}, we note that there is a power gain of factor $n_t$ when the transmitter can track the mean channel information.} Without channel knowledge at the transmitter, the transmit energy is spread out equally across all directions in $\mathcal{C}^{n_t}$. With transmitter knowledge of the mean channel information (i.e., LOS direction), the energy can now be focused on only one direction. However, it should be noted that the power gain accounts for the lower bound on the achievable rate. From the result on the optimal sub-6 GHz covariance matrix for the worst sub-6 GHz channel, we can apply the same method as in the scenario with no CSIT, and obtain the optimal power and bandwidth allocation across the interfaces.
\textbf{ (II) Upper Bound on the sub-6 GHz Achievable Rate:} In this section, we derive an upper bound on the sub-6 GHz achievable rate and express the optimal covariance matrix that achieves the upper bound. Similar to the previous section, the physical layer resources can then be optimally allocated across the sub-6 GHz and mmWave interfaces. We assume that the model of partial CSIT is valid such that the transmitter CSI includes a known component $\bar{\mathbf{H}}_\text{sub-6}$ that corresponds to the LOS component, and an error term $\Delta\mathbf{H}_\text{sub-6}$ that corresponds to the NLOS component.
For the purpose of our optimal resource allocation across the sub-6 GHz and mmWave interfaces, we require to obtain the optimal covariance matrix that is expressed in terms of total sub-6 GHz transmit power $P_\text{sub-6}$. In this case, we can show that:
$$
\mathcal{R}_\text{sub-6}(\mathbf{H}_\text{sub-6}, \mathbf{K}_\mathbf{xx}) \leq W_{\text{sub-6}} {\log \det \left( \mathbf{I}+ \frac{1}{W_{\text{sub-6}}} \mathbf{K}_\mathbf{xx}(\bar{\mathbf{H}}_{\text{sub-6}}^H \bar{\mathbf{H}}_{\text{sub-6}} + \sigma_e^2 \mathbf{I})\right)},
$$
As a result, the transmitter observes the equivalent channel covariance matrix $\boldsymbol{\Sigma} := \bar{\mathbf{H}}_{\text{sub-6}}^H \bar{\mathbf{H}}_{\text{sub-6}} + \sigma_e^2 \mathbf{I}$, and thus eigenvectors of the optimal covariance matrix $\mathbf{K}_\mathbf{xx}$ are equal to the eigenvectors of matrix $\boldsymbol{\Sigma}$. Given the knowledge of the channel covariance matrix, the authors in \cite{soysal2007optimum} derived the optimal power allocation across the transmit antennas such that the power $p_i$ is allocated to the $i$-th transmit antenna:
\begin{equation}
p_i = \frac{p_i E_i (\mathbf{p})}{\sum_{j=1}^{n_t} p_j E_j (\mathbf{p})} P_\text{sub-6},
\label{eq:optimal-power-RF}
\end{equation}
where $\mathbf{p} = (p_1, ..., p_{n_t})$ is the vector of optimal power allocated across the $n_t$ sub-6 GHz transmit antennas, and $E_i(\mathbf{p})$ is expressed in terms of eigenvalues of the channel covariance matrix and the power allocation vector $\mathbf{p}$, i.e.,:
$$
E_i(\mathbf{p}) = E \left[ \frac{\sigma_i({\boldsymbol{\Sigma}}) \mathbf{z}_i ^ H \mathbf{A}_i ^ {-1} \mathbf{z}_i}{1+p_i \sigma_i({\boldsymbol{\Sigma}}) \mathbf{z}_i ^ H \mathbf{A}_i ^ {-1} \mathbf{z}_i}\right],
$$
where $\mathbf{A}_i = \mathbf{I} + \sum_{j=1}^{n_t} p_j \sigma_j({\boldsymbol{\Sigma}}) \mathbf{z}_j \mathbf{z}_j^H - p_i \sigma_i({\boldsymbol{\Sigma}}) \mathbf{z}_i \mathbf{z}_i^H$, and $\mathbf{z}_i$ is the $i$-th column of $\mathbf{Z}$ that is used to convert the channel covariance matrix $\boldsymbol{\Sigma}$ to the channel matrix $\bar{\mathbf{H}}_\text{sub-6}$ by $\bar{\mathbf{H}}_\text{sub-6} = \mathbf{Z} \boldsymbol{\Sigma}^{1/2}$. Entries of $\mathbf{Z}$ are i.i.d., zero-mean, unit-variance complex Gaussian random variables \cite{soysal2007optimum}. Finally, from \eqref{eq:optimal-power-RF}, the optimal power allocation to the sub-6 GHz transmit antennas can be obtained by a fixed point algorithm \cite{soysal2007optimum}. Therefore, the optimal covariance matrix $\mathbf{K}_\mathbf{xx}$ can be expressed in terms of the total sub-6 GHz transmit power $P_\text{sub-6}$, and similar to the previous section, we derive the optimal power and bandwidth allocation across the sub-6 GHz and mmWave interfaces.
\section{Energy Efficiency Maximization}
\label{sec:problem2-solution}
Problem 2 formulated in \eqref{eq:problem2} through \eqref{eq:problem2-power-constraint} is aimed to optimize the normalized sum rate in terms of bit/sec/watt or bit/joule. Our observations on the solution of this problem for the SISO case indicate that (i) if the transmission power is much lower than component consumption in any interface, the objective function is decreasing in bandwidth for that interface. This suggests that the optimal solution is not to allocate any bandwidth for that interface; (ii) on the flip side, if the transmission power is much higher than the component consumption, then the objective function is increasing in bandwidth for that platform. This suggests that the optimal solution is to allocate full bandwidth for that platform; (iii) otherwise, the objective function is non-monotonic, increasing up to a certain bandwidth at which point it starts to decrease. Indeed, in Fig.~\ref{fig:shannon_with_components}, we observe such a case for the mmWave interface.
The problem of normalized sum rate maximization can be viewed as an instance of energy efficiency (EE) problem defined as the ratio between the amount of data transmitted and the corresponding incurred cost in terms of power. In order to solve this problem, we use fractional programming that solves the problem in the following general form:
\begin{align}
\max_{\mathbf{x}} \ \frac{f(\mathbf{x})}{g(\mathbf{x})}; \\
\text{s.t.} \ \ \mathbf{x} \in \mathcal{X},
\end{align}
with $f: \mathcal{C} \subseteq \mathds{R}^n \rightarrow \mathds{R}$, $g: \mathcal{C} \subseteq \mathds{R}^n \rightarrow \mathds{R}_+$, and $\mathcal{X} \subseteq \mathcal{C} \subseteq \mathds{R}^n$. In general, obtaining optimal solution for the fractional problems is challenging since the objective function is not concave, and thus the
strong results of convex optimization theory (i.e., KKT conditions) do not apply. However, under some assumptions on $f$ and $g$, the objective falls into the class of generalized concave functions, which makes it feasible to obtain the global solution. Considering the formulation in \eqref{eq:problem2}, since logarithm of identity matrix plus an Hermitian positive-semidefinite matrix is a matrix-concave function, the numerator is concave. Denumerator is a linear function of the sub-6 GHz and mmWave power and bandwidth variables. Therefore, the objective function is pseudo-concave. As a result, each stationary point of the objective function is a global maximizer, and KKT conditions are necessary and sufficient for optimality \cite{zappone2015energy}.
In order to solve pseudo-concave problems, some solution approaches have been proposed. Dinkelbach's algorithm proposed in \cite{dinkelbach1967nonlinear,jagannathan1966some} is a parametric algorithm that instead of solving the original problem, solves a sequence of easier problems that converge to the global solution. Specifically, let us assume that the set of feasible solutions of Problem 2 is denoted by $\mathcal{S}$. Then, the main result of the Dinkelbach's algorithm is the relation between the original problem and the following function:
\begin{equation}
F(\beta) = \max_{\mathbf{x} \in \mathcal{X}} \left\{f(\mathbf{x}) - \beta g(\mathbf{x}) \right\}.
\end{equation}
In this case, $F$ exists and is continuous provided that $f$ and $g$ are continuous and $\mathcal{X}$ is compact. In addition, it is possible to show that $F$ is convex on $\mathds{R}$, and is strictly monotone decreasing on $\mathds{R}$ \cite{zappone2015energy}. The connection between $F(\beta)$ and the original problem is as follows. Consider $\mathbf{x}^* \in \mathcal{X}$ and $\beta^* = \frac{f(\mathbf{x}^*)}{g(\mathbf{x}^*)}$, then $\mathbf{x}^*$ is the optimal solution for the original problem if and only if:
\begin{equation}
\mathbf{x}^* = \argmax_{\mathbf{x} \in \mathcal{X}} \{f(\mathbf{x}) - \beta^* g(\mathbf{x}) \}.
\label{eq:dinkelbach}
\end{equation}
Therefore, it can be shown that solving the fractional programming is equivalent to finding the roots of $F(\beta)$. Thus, Dinkelbach's algorithm achieves this goal as provided in Algorithm 1.
Optimality and convergence results of Dinkelbach's algorithm are shown in \cite{zappone2015energy}.
\begin{algorithm}[t]
\caption{Dinkelbach's Algorithm}
\begin{varwidth}{\dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
\begin{algorithmic}[1]
\State $\delta > 0$, $n=0$, $\beta_n =0$
\While{$F(\beta_n) > \delta$}
\State $\mathbf{x}^*_n = \argmax_{\mathbf{x} \in \mathcal{X}} \{f(\mathbf{x}) - \beta_n g(\mathbf{x}) \}$
\State $F(\beta_n) = \max_{\mathbf{x} \in \mathcal{X}} \left\{f(\mathbf{x^*}) - \beta_n g(\mathbf{x^*}) \right\}.$
\State $\beta_{n+1} = \frac{f(\mathbf{x}^*_n)}{g(\mathbf{x}_n^*)}$
\State $n=n+1$
\EndWhile
\end{algorithmic}
\end{varwidth}%
\end{algorithm}
By considering the energy efficiency problem as a pseudo-concave fractional problem and applying Dinkelbach's method, we in fact reach to the same formulation as Problem 1. In particular, Eq. \eqref{eq:dinkelbach} resembles the same equation as the Lagrange equation in \eqref{lagrange}. Therefore, the solution space will be similar, while Dinkelbach's algorithm iteratively solves the Lagrange equation instead of finding the solution in one-shot. Similar to Problem 1, various CSIT scenarios can be considered in order to first simplify the objective value in EE problem.
\section{Numerical Results}
In this section, we numerically investigate the performance of our proposed resource allocation scheme. For the mmWave technology, the carrier frequency is $30$ GHz and the \emph{total available bandwidth} is $1$ GHz. The number of transmit and receive antennas are $64$ and $16$, respectively. For the sub-6 GHz interface, the carrier frequency is $3$ GHz, the \emph{total available bandwidth} is $1$ MHz, and the number of antennas is the same as in mmWave. MmWave and sub-6 GHz channel matrices are extracted from the experimental data in \cite{mezzavilla20155g}. In the simulation results, we obtain the performance as a function of the total available power and the parameter $a = c_\text{ox} 2^{r_{\text{ADC}}}$. The former dictates the power constraint, while the latter determines the power consumption of ADC components.
In the first example, we consider an extreme scenario in which the ADC power consumption is very high. For this purpose, we set the scaling factor of ADC power consumption to be $a=10^{-7}$. Table \ref{tab:sum-rate} demonstrates the performance of our proposed optimal solution compared with the full bandwidth allocation. From the results, we observe that in the case of utilizing the whole available spectrum, the power consumption by ADC components exceeds the total available power. Thus, the transmit power becomes zero, and the achievable rate drops to zero as well. On the other hand, if only about $4$ MHz of the mmWave bandwidth is utilized, then the total sum rate of $2.55$ Kbps is achievable.
\begin{table}[h]\centering
\renewcommand{\arraystretch}{1.3}
\ra{1.3}
\caption{Achievable sum rate by: (i) Full sub-6 GHz and mmWave bandwidth allocation, and (ii) Optimal allocation.}
\begin{threeparttable}
\begin{tabular}{@{}p{4cm}cc@{}}\toprule
\textbf{Resource} & \textbf{Full Bandwidth Allocation} & \textbf{Optimal Bandwidth Allocation} \\ \toprule
Sub-6 GHz Bandwidth & $1$ MHz & $1$ MHz \\ \midrule
mmWave Bandwidth & $1$ GHz & $4.0597$ MHz \\ \midrule
Sub-6 GHz \emph{Transmission} Power & $0$\tnote{1} &
$394$ mW \\ \midrule
mmWave \emph{Transmission} Power & $0$\tnote{1} & $100$ mW \\ \midrule
\textbf{Achievable Sum Rate} & \textbf{0} & \textbf{$\boldsymbol{2.55\times 10^3}$} \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item[1] Due to high power consumption of the mmWave component, \emph{transmit power} becomes zero.
\end{tablenotes}
\end{threeparttable}
\label{tab:sum-rate}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[scale=.3]{fig/optimal_sum_rate_vs_full_allocation}
\caption{Sum rate comparison between the optimal scheme and the full bandwidth allocation paired with the waterfilling power allocation across the sub-6 GHz and mmWave interfaces.}
\label{fig:optimal_sum_rate_vs_power}
\end{figure}
In the second scenario, we investigate the optimal solution of Problem 1 with no CSIT and as a function of available power (i.e., input power) and the power consumption of ADC components. The results are shown in Fig. \ref{fig:optimal_sum_rate_vs_power} and Fig. \ref{fig:optimal_sum_rate_vs_ADC}, respectively. \emph{From the results, we observe that under low power scenario, it is optimal to partially use the available bandwidth to achieve a higher sum rate compared with the full bandwidth utilization. Moreover, when the ADC energy consumption increases, it is more energy efficient to partially use the available bandwidth.}
\begin{figure}
\centering
\includegraphics[scale=.3]{fig/optimal_sum_rate_vs_full_allocation_AD_consumption}
\caption{Sum rate comparison between the optimal scheme and the full bandwidth allocation paired with the waterfilling power allocation across the sub-6 GHz and mmWave interfaces. In this figure, ADC energy consumption refers to the constant $a\times 10^8$. }
\label{fig:optimal_sum_rate_vs_ADC}
\end{figure}
Figure \ref{fig:problem-2} numerically demonstrates the optimal solution of energy efficient resource allocation. From the results, we observe that after some threshold on the mmWave bandwidth, the energy efficiency in terms of bits/sec/watt or bits/joule decreases for the mmWave interface. However, under our optimal resource allocation scheme, the increasing trend of the energy efficiency as a function of bandwidth is preserved due to partially utilizing the bandwidth.
\begin{figure}
\centering
\includegraphics[scale=.3]{fig/rate_per_unit_power_vs_optimal_linear.eps}
\caption{Optimal solution of Problem 2 vs. full bandwidth allocation}
\label{fig:problem-2}
\end{figure}
\section{Conclusion}
\label{conclusion}
In this paper, we considered an integrated sub-6 GHz/mmWave architecture and proposed a joint power and bandwidth allocation framework in order to maximize the achieving sum rate and energy efficiency in the sub-6 GHz/mmWave system. In order to maximize the achievable sum rate under the transmitter and receiver power constraints, our formulation explicitly takes into account the energy consumption in integrated-circuit components. In addition, we investigated the optimal solution under various conditions such as when there is no channel state information at the transmitter or when there is only partial information available. From our optimal results, \emph{we observe that despite the availability of huge bandwidths at the mmWave interface, under some circumstances (e.g., low input power or ADC components with high energy consumption), it is more efficient to partially utilize the mmWave bandwidth.} We emphasize that physical layer resource allocation is of great importance in mmWave systems since the energy consumption by components increases with bandwidth, and thus they can incur large overhead in terms of power consumption. Hence, the power and bandwidth should be optimally allocated in order to avoid the heavy burden of components power consumption.
\bibliographystyle{ieeetr}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
Low-rank matrix approximations (LRA) are key problems in numerical linear algebra, and have become a central tool in data analysis and machine learning; see, e.g.,~\cite{UHZB14}.
A possible formulation for LRA is the following: given a matrix $M \in \mathbb{R}^{m \times n}$ and a factorization rank $r$, solve
\begin{equation} \label{lowrankapp}
\min_{X \in \Omega} \; ||M - X|| \quad \text{ such that } \quad \rank(X) \leq r,
\end{equation}
for some given (pseudo) norm $||.||$. In this paper, we focus on unconstrained variants, that is, $\Omega = \mathbb{R}^{m \times n}$, although there exists many important variants of~\eqref{lowrankapp} that take constraints into account, e.g.,
nonnegative matrix factorization~\cite{LS99},
independent component analysis~\cite{C94} and
sparse principal component analysis~\cite{d2007direct}.
When the norm $||.||$ is the Frobenius norm, that is, $||M - X||_F^2 = \sum_{i,j} (M_{ij}-X_{ij})^2$,
the problem can be solved using the singular value decomposition and is closely related to principal component analysis~\cite{golub2012matrix}.
In practice, it is often required to use other norms, e.g.,
the $\ell_1$ norm which is more robust to outliers~\cite{SWZ17},
weighted norms that can be used when data is missing~\cite{GZ79, KBV09},
$\sum_j ||M(:,j)-X(:,j)||_2^p$ for $p \geq 1$ which can model different situations depending on the value of $p$~\cite{CW15}, and
Kullback-Leibler (KL) divergence when Poisson noise is present~\cite{chi2012tensors}.
However, as soon as the norm is not the Frobenius norm, \eqref{lowrankapp}
becomes difficult in general;
in particular it was proved to be NP-hard for all the previously listed cases except for the KL divergence~\cite{GV15c, GG10c, SWZ17}.
(For the KL divergence, proving NP-hardness is, to the best of our knowledge, an open problem.)
In this paper, we focus on the variant with the component-wise $\ell_{\infty}$ norm:
\begin{equation} \label{lowrankinf}
\min_{X \in \mathbb{R}^{m \times n}} \; ||M - X||_{\infty} \quad \text{ such that } \quad \rank(X) \leq r,
\end{equation}
where $||M - X||_{\infty} = \max_{i,j} |M_{ij} - X_{ij}|$. We will refer to this problem as $\ell_{\infty}$ LRA. It should be used when the noise added to the low-rank matrix follows an \emph{i.i.d.\@ uniform distribution}. We will also use the notation $\ell_p$ LRA for the LRA problem where the norm used is the component-wise $\ell_p$ norm.
\subsection{Previous results and applications}
When $m=n$ and $r=\min(m,n)-1$, \eqref{lowrankinf} was studied in~\cite{PR93} and corresponds to the problem of distance to robust nonsingularity which can be stated as follows: what is the rank-deficient matrix $X$ that is the closest to $M$ in the component-wise infinity norm? In this particular case, \eqref{lowrankinf} was shown to be NP-hard~\cite{PR93}.
In~\cite{GT01, GT11}, Goreinov and Tyrtyshnikov obtained an approximate solution to $\ell_{\infty}$ LRA using the so-called rank-$r$ skeleton approximation which uses $r$ columns and $r$ rows of the given matrix.
Juditsky et al.~\cite{juditsky2011low} linked a variant of~\eqref{lowrankinf} (where the row range of $X$ is constrained to be contained in the row range of $M$) with the synthesis problem of compressed sensing and provided randomized algorithms with optimality guarantees.
Very recently, Chierichetti et al.~\cite{CGKe17} proposed
provably good approximation algorithms for $\ell_p$ LRA for any $p \geq 1$ using a subset of the columns of the input matrix $M$ to span the columns of $X$.
An application of~\eqref{lowrankinf} is the recovery of a low-rank matrix from a quantization~\cite{LJ17} (roughly speaking, its rounding to some precision); see, e.g.,~\cite{gersho1992vector} for more details on quantization.
For example, assume we are given a real rank-$r$ matrix where each entry has been rounded to the nearest integer. Given such a matrix $M$ (which in general will have full rank), the problem is to find a rank-$r$ $X$ such that $||M-X||_{\infty} \leq 0.5$ (note that rounding can be assimilated to noise distributed uniformly in the interval [-0.5,0.5]); see Section~\ref{algoappl} for some examples.
As far as we know, there is not as much literature on $\ell_{\infty}$ LRA~\eqref{lowrankinf} compared to other variants.
A plausible explanation is that, in practice, and especially in data analysis applications,
using the $\ell_{\infty}$ norm is not very useful and is in particular extremely sensitive to outliers.
Moreover, even if $M \neq 0$, the zero matrix could be an optimal solution of~\eqref{lowrankinf} (which is not possible for any other $\ell_{p}$-norm).
\begin{example} \label{ex1}
Let
\[
M = \left( \begin{array}{cc}
1 & 1 \\ 1 & -1
\end{array} \right).
\]
We have
\[
\min_{\rank(X) \leq 1} ||M - X||_{\infty}
\; = \;
\min_{u \in \mathbb{R}^2,v \in \mathbb{R}^2} ||M-uv^T||_{\infty} \; = \; ||M||_{\infty} \; = \; 1.
\]
In fact, assume the minimum is strictly smaller than 1. This requires $u_1 v_i > 0$ $(i=1,2$), $u_2 v_1 > 0$ and $u_2 v_2 < 0$ which is not possible.
\end{example}
However, there are several applications as mentioned above; namely, distance to singularity, compressed sensing, and recovery of a quantized low-rank matrix.
Except for the case $m=n$ and $r = n-1$ which was proved to be NP-hard by Poljak and Rohn~\cite{PR93}, there is, to the best of our knowledge, not a very good understanding of the computational complexity of~\eqref{lowrankinf}.
The main goal of this paper is to shed some light on this question; in particular proving NP-completeness of~\eqref{lowrankinf} when $r=1$ (Theorem~\ref{thrnpcom}).
\subsection{Outline of the paper and contributions}
In this paper we mainly focus on rank-one $\ell_{\infty}$ LRA, that is, \eqref{lowrankinf} with $r=1$.
In Section~\ref{pblinNP}, we show that the decision version of rank-one $\ell_{\infty}$ LRA is in NP.
In fact, we show that if the sign pattern of $X$ is known, then the decision version of rank-one $\ell_{\infty}$ LRA can be solved in polynomial time by finding a solution to a system of linear inequalities.
This is an important result because it turns out that, in most cases, the number of possible sign patterns of $X$ can be reduced drastically; e.g., $X$ can be assumed to be nonnegative if $M$ is.
In Section~\ref{NPcompl}, we prove that rank-one $\ell_{\infty}$ LRA is NP-complete using a reduction from `not all equal 3SAT',
with an intermediate problem on directed graphs (namely, the problem of making, if possible, a directed graph acyclic by reversing the direction of a particular subset of the edges).
Finally, in Section~\ref{algoappl}, we propose a simple heuristic algorithm and apply it on the recovery of quantized low-rank matrices.
\section{Decision version of rank-one $\ell_{\infty}$ LRA} \label{pblinNP}
Let us define formally the decision version of rank-one $\ell_{\infty}$ LRA:
\begin{problem} \label{probDellinf} (D-$\ell_\infty$-R1A($M$,$k$))
\noindent Given: A real $m$-by-$n$ matrix $M$ and a real number $k \geq 0$.
\noindent Question: Does there exist $u \in \mathbb{R}^{m}$ and $v \in \mathbb{R}^{n}$ such that $\max_{i,j} |M_{ij} - u_i v_j| \leq k$?
If yes, output a solution $(u,v)$.
\end{problem}
\begin{lemma} \label{lemsignpat}
If the sign pattern of $u$ or $v$ is given as a part of the data,
then D-$\ell_\infty$-R1A($M$,$k$) can be solved in polynomial time in $m$ and $n$,
namely in $O\big( mn (m+n)^3 \log mn \big)$ arithmetic operations.
\end{lemma}
\begin{proof}
First, we can assume without loss of generality (w.l.o.g.) that for each row (resp.\@ column) of $M$, there is at least one entry whose absolute value is larger than $k$, that is, for all $i$ (resp.\@ $j$), there exists $l$ (resp.\@ $p$) such that $|M_{il}| > k$ (resp.\@ $|M_{pj}| > k$).
In fact, if it is not the case, then one can trivially choose $u_i = 0$ (resp.\@ $v_j = 0$) in a solution of D-$\ell_\infty$-R1A($M$,$k$) and reduce the problem to a submatrix of $M$.
This implies that we can assume w.l.o.g.\@ that $u \neq 0$ and $v \neq 0$ since $|M_{ij}| > k \Rightarrow u_i v_j \neq 0$.
Second, if we assume that the sign pattern of $u$ is known we can assume w.l.o.g.\@ that $u > 0$.
In fact, if some entries of $u$ are negative, we can flip their signs along with the signs of the entries of the corresponding rows of $M$ to obtain an equivalent problem.
Therefore, if the sign pattern of $u$ is known, D-$\ell_\infty$-R1A($M$,$k$) can be reduced to finding $u > 0$ and $v$ such that
\[
-k \leq |M_{ij} - u_i v_j| \leq k
\quad \iff \quad
M_{ij}-k \leq u_i v_j \leq M_{ij}+k.
\]
Moreover, by the scaling degree of freedom (that is, $uv^T = (\alpha u)(\alpha^{-1} v)^T$ for any $\alpha \neq 0$), we can assume w.l.o.g.\@ that $u \leq 1$.
Defining $s_i = (u_i)^{-1} \geq 1$,
this problem is equivalent to finding $s \geq 1$ and $v$ such that
\[
s_i \, (M_{ij}-k)
\; \leq \;
v_j \; \leq \;
s_i \, (M_{ij}+k) \quad \text{ for all } i,j.
\]
This is a system of $2mn+m$ linear inequalities with $m+n$ variables,
where each inequality contains at most two variables,
and can be solved in $O\big( I N^3 \log I \big) = O\big( mn (m+n)^3 \log mn \big)$ arithmetic operations where $I=2mn+m$ is the number of inequalities and $N=m+n$ the number of variables~\cite{megiddo1983towards}.
If the sign pattern of $v$ is known, by symmetry ($||M-uv^T||_{\infty} = ||M^T - v u^T||_{\infty}$), the same result holds. This completes the proof.
\end{proof}
\begin{corollary} \label{lemMnonneg}
If $M \geq 0$, D-$\ell_\infty$-R1A($M$,$k$) can be solved in polynomial time.
\end{corollary}
\begin{proof}
In fact, in that case, one can assume w.l.o.g.\@ that $u \geq 0$ and $v \geq 0$ so that the result follows from Lemma~\ref{lemsignpat}.
\end{proof}
\begin{remark}
It is interesting to note that some rank-one LRA problems are NP-hard even when $M \geq 0$; e.g., for the $\ell_1$ norm~\cite{GV15c} and weighted norms~\cite{GG10c}.
\end{remark}
Lemma~\ref{lemsignpat} implies that D-$\ell_\infty$-R1A($M$,$k$) can be solved in $O\big( 2^{\min(m,n)} mn (m+n)^3 \log mn \big)$ operations since one can try all the possible sign patterns for $u$ (or $v$ if $n \leq m$).
It is possible to achieve a better complexity result by identifying the connected component of a particular bipartite graph.
\begin{definition}
Given $M$ and $k$, $G_b(M,k) = (V_1 \cup V_2, E)$ is the bipartite graph with
$V_1 = \{v_1,v_2,\dots,v_m\}$,
$V_2 = \{v_1',v_2',\dots,v_n'\}$,
and
$(v_i,v_j') \in E \iff |M_{ij}| > k$.
\end{definition}
\begin{lemma} \label{lemconcomp}
For D-$\ell_\infty$-R1A($M$,$k$), the number of possible sign patterns in a solution $u$ can be reduced to $2^{d-1}$ where $d$ is the number of connected components of $G_b(M,k)$ discarding the isolated vertices.
\end{lemma}
\begin{proof}
Note that the isolated vertices of $G_b(M,k)$ correspond to rows and columns of $M$ whose entries have absolute value smaller than $k$ and for which one can set w.l.o.g.\@ the corresponding entry of $u$ or $v$ to zero; see the proof of Lemma~\ref{lemsignpat}.
The rest of the proof follows from the fact that any solution $(u,v)$ of D-$\ell_\infty$-R1A($M$,$k$) must satisfy the following:
\[
|M_{ij}| > k \quad \Rightarrow \quad \sign(u_i v_j) = \sign(M_{ij}),
\]
where
\[
\sign(x) = \left\{
\begin{array}{cc}
1 & \text{ if $x > 0$,} \\
0 & \text{ if $x = 0$,} \\
-1 & \text{ if $x < 0$.}
\end{array}
\right.
\]
Therefore, fixing the sign of an entry of $u$ imposes the sign for all the entries of $u$ and $v$ contained in the same connected component. This makes $2^{d}$ possible sign patterns for $u$ and $v$.
It can be reduced to $2^{d-1}$ since $uv^T = (-u)(-v)^T$ hence half of the possible sign patterns can be discarded
(this can be achieved for example by imposing arbitrarily the sign of one entry of $u$ or $v$).
\end{proof}
For any submatrix of $M$ corresponding to a connected component of $G_b(M,k)$, there must exist a completion with $\pm 1$ of the entries smaller than $k$ such that its sign pattern has rank one, otherwise the answer to D-$\ell_\infty$-R1A($M$,$k$) is NO;
see, e.g., Example~\ref{ex1} for any $k < 1$.
This observation can be used to quickly obtain a lower bound on $k$ in order for the answer to D-$\ell_\infty$-R1A($M$,$k$) to possibly be YES
(hence also a lower bound for rank-one $\ell_{\infty}$ LRA).
\begin{theorem} \label{theoremNP}
D-$\ell_\infty$-R1A($M$,$k$) is in NP,
and can be solved in $O\big( 2^{d} mn (m+n)^3 \log mn \big)$
operations where $d$ is the number of connected components of $G_b(M,k)$ discarding isolated vertices.
\end{theorem}
\begin{proof}
This follows directly from Lemmas~\ref{lemsignpat} and \ref{lemconcomp}.
\end{proof}
Theorem~\ref{theoremNP} implies that for D-$\ell_\infty$-LRA to be a difficult problem, the number of connected components has to be high.
This will motivate our construction in our NP-completeness proof where we will use a square matrix for which only the diagonal entries are larger than $k$ so that the number of connected components is maximal, namely $d=m=n$.
Theorem~\ref{theoremNP} also implies that D-$\ell_\infty$-R1A($M$,$k$) can be solved in polynomial time
if the number of connected components $d$ satisfies $d = O\big(\log(\min(m,n)) \big)$.
\section{NP-completeness of D-$\ell_\infty$-R1A($M$,$k$)} \label{NPcompl}
The goal of this section is to prove that D-$\ell_\infty$-R1A is NP-hard. In order to do this, we construct a polynomial time reduction from the problem known as NOT-ALL-EQUAL 3-SAT. Recall that a \textit{literal} associated with a set $X$ of Boolean variables is either an element of $X$ or a negation of it.
\begin{problem}\label{prob12} (NOT-ALL-EQUAL $3$-SAT)
\noindent Given: A set $X$ of variables and a set $L$ of $3$-tuples of literals.
\noindent Question: Does there exist an assignment of the variables in $X$ to $\{0,1\}$ for which every tuple in $L$ has at least one false literal and at least one true literal?
\end{problem}
Since NOT-ALL-EQUAL $3$-SAT is NP-complete (see~\cite{Karp}), constructing a polynomial time reduction from it to D-$ell_\infty$-R1A would mean the NP-hardness of the latter problem. In order to present such a reduction, we need to recall the definitions of some basic concepts in graph theory. An \textit{oriented graph} $G$ is defined as a pair of sets $V$ and $E\subset V^2$. The elements of $V$ are called \textit{vertices}, a pair $(a,b)\in E$ is an \textit{edge passing from} $a$ \textit{to} $b$, and vertices $a,b$ are \textit{adjacent} if there is an edge passing between them. We assume that $V=\{1,\ldots,n\}$ and that at most one of the pairs $(a,b)$, $(b,a)$ belongs to $E$ for all $a,b \in V$.
A sequence $(a_0,\ldots,a_k)$ of vertices is called a \textit{cycle} if $a_0=a_k$ and there is an edge passing from $a_{i-1}$ to $a_i$ for all $i$.
A \textit{two-coloring} of $G$ is a partition of $V$ into the union of two disjoint sets $W$ and $B$. A subset $U\subset V$ is called \textit{monochromatic} with respect to $(W,B)$ if either $U\subset W$ or $U\subset B$. Let us introduce the auxiliary problem which we use as a tool in our NP-hardness proof.
\begin{problem} \label{probgraph} \hspace{0.1cm} \,
\noindent Given: An oriented graph $G=(V,E)$ and a set $D$ of pairs of non-adjacent vertices.
\noindent Question: Is there a two-coloring $(W,B)$ of $V$ such that
\begin{itemize}
\item[(i)] no pair in $D$ is monochromatic, and
\item[(ii)] the graph obtained from $G$ by reversing the edges passing between $W$ and $B$ has no cycle.
\end{itemize}
\end{problem}
\begin{lemma}\label{lemgraph}
Problem~\ref{probgraph} is NP-complete.
\end{lemma}
\begin{proof}
Let us construct the graph $G$ depending on an instance $(X,L)$ of Problem~\ref{prob12}:
\begin{itemize} \setlength{\itemindent}{0.5cm}
\item[Step 1.] For every variable $x\in X$, we create two vertices corresponding to $x$ and the negation of $x$, and we add to $D$ the pair containing these two.
\item[Step 2.] For every tuple $(y_1,y_2,y_3)\in L$, we create three new vertices corresponding to $y_1$, $y_2$ and $y_3$, and we draw a cycle on these vertices.
\item[Step 3.] We add a pair in $D$ containing a vertex created in Step~1 with a vertex created in Step~2 if they correspond to literals that are negations of each other.
\end{itemize}
Clearly, this graph can be constructed in polynomial time, and a two-coloring of $V$ leaves no pair in $D$ monochromatic if and only if it assigns the same color for every occurrence of a literal $y$ and the other color for the negation of $y$.
If $y_1$, $y_2$ and $y_3$ have all the same color and $(y_1,y_2,y_3)\in L$, then the item (ii) in Problem~\ref{probgraph} does not require changes in the edge directions between $y_1$, $y_2$ and $y_3$, so these vertices remain a cycle. This implies that any acceptable two-coloring of $G$ for Problem~\ref{probgraph} will not have $y_1$, $y_2$ and $y_3$ of the same color hence will correspond to a valid assignment for $(X,L)$.
On the other hand, any valid assignment of $(X,L)$ corresponds to a coloring in which $y_1$, $y_2$ and $y_3$ have different colors
for all $(y_1,y_2,y_3)\in L$. In other words, two of the three edges passing between vertices in $y_1$, $y_2$ and $y_3$ will change their directions as in item~(ii) in Problem~\ref{probgraph}, which means that the resulting graph will possess no cycle hence any valid assignment of $(X,L)$ corresponds to an acceptable two-coloring of $G$.
\end{proof}
We are now ready to present a reduction from Problem~\ref{probgraph} to D-$ell_\infty$-R1A.
\begin{definition}
Let $G=(V,E)$ and $D$ be defined as in the formulation of Problem~\ref{probgraph}.
We define the matrix $\mathcal{M}=\mathcal{M}(G,D) \in \{-1,0,1\}^{|V| \times |V|}$ with rows and columns indexed by elements of $V=\{1,2,\dots,n\}$ as follows:
\begin{itemize} \setlength{\itemindent}{0cm}
\item[(1)] $\mathcal{M}_{ii}=2$ for all $i$,
\item[(2)] $\mathcal{M}_{ij}=\mathcal{M}_{ji}=-1$ if $\{i,j\}\in D$,
\item[(3)] $\mathcal{M}_{ij}=-1$, $\mathcal{M}_{ji}=1$ if $(i,j)\in E$,
\item[(4)] $\mathcal{M}_{ij}=0$ otherwise.
\end{itemize}
\end{definition}
Before we can prove that $\mathcal{M}$ leads to a reduction,
we need a result stating that any partial order relation on a set can be extended to a total order relation. In terms of graphs, this result can be stated as follows.
\begin{obs}\label{obserpermut}
Let $G=(V,E)$ be an oriented graph without cycles. Then there exists a total ordering $\succ$ of $V$ such that $(a,b)\in E$ implies $a\succ b$.
\end{obs}
\begin{theorem}\label{thrred}
The pair $(G,D)$ is a yes-instance of Problem~\ref{probgraph} if and only of there are real numbers
$\{u_i\}_{i=1}^{|V|}$ and $\{v_j\}_{i=1}^{|V|}$ such that $|\mathcal{M}_{ij}-u_iv_j| \leq 3/2-0.001 |V|^{-6}$ for all $i,j$.
\end{theorem}
\begin{proof}
Assume $(G,D)$ admits a valid coloring $(W,B)$ as in Problem~\ref{probgraph},
and let $G'$ be the directed graph obtained after the transformation in item~(ii) of Problem~\ref{probgraph}.
Consider the matrix $N$ defined by $N_{ij}=\mathcal{M}_{ij}$ if $\{i,j\}$ is monochromatic and $N_{ij}=-\mathcal{M}_{ij}$ otherwise.
The matrix $N$ has $2$'s on the diagonal, $-1$'s at every position $(i,j)$ which is an edge of $G'$, and zeros and ones everywhere else.
Since $G'$ has no cycle, by Observation~\ref{obserpermut}, there is a permutation matrix $C$ (corresponding to an ordering) such that all the $-1$'s are located below the main diagonal of the matrix $C^{-1} N C$. Since the permutations of rows and columns and multiplications of them by $-1$ do not change our ability or inability to approximate a matrix, it is sufficient to prove the existence of real numbers $\{u_i\}_{i=1}^{|V|}$ and $\{v_j\}_{i=1}^{|V|}$ such that $|M_{ij}-u_iv_j| \leq 3/2-0.001 |V|^{-6}$
for any $n\times n$ matrix $M$ with the $2$'s on the diagonal, $-1$'s at some positions below the diagonal, and zeros and ones everywhere else. Towards this end, one can check that it suffices to set
$$
u_i=\frac{1}{\sqrt{2}}-i\varepsilon,\,\,\,
v_j=\frac{1}{\sqrt{2}}+j\varepsilon+\varepsilon^{1.5},$$
where $\varepsilon=0.1 |V|^{-4}$.
Conversely, assume that the numbers $\{u_i\}_{i=1}^{|V|}$ and $\{v_j\}_{i=1}^{|V|}$ are such that
$|\mathcal{M}_{ij}-u_iv_j| \leq 3/2-0.001 |V|^{-6} < 3/2$ for all $i,j$.
Since $\mathcal{M}_{ii}=2$ for all $i$, we have $|u_iv_i-2|<3/2$
which implies that $u_i$ and $v_i$ are non-zero and have the same sign. A relabeling of indices does not change the properties we discuss, so we can assume $|v_1|\leq \ldots\leq |v_n|$. We define $W$ (resp.\@ $B$) as the set of all $i$ such that $u_i>0$ (resp.\@ $u_i<0$), and we are going to prove that $(W,B)$ is a valid coloring of $G$ as in Problem~\ref{probgraph}.
We define the matrix $N$ by multiplying the rows and columns of $\mathcal{M}$ with indices in $B$ by $-1$, and define $u'$ and $v'$ by multiplying the entries in $u$ and $v$, respectively, with indices in $B$ by $-1$.
We have $\left|N_{ij}-u'_iv'_j\right|<3/2$ and $u' \geq 0$ and $v' \geq 0$.
For $j>i$, $v'_j\geq v'_i$ which implies $u'_iv'_j \geq u'_i v'_i > 1/2$, so $N_{ij}\neq-1$ if $j>i$. In other words, all the $-1$'s of $N$ are located below the main diagonal, and we have $N_{ab}=N_{ba}=1$ for all the positions $(a,b)$ in $D$. This means that the item~(i) of Problem~\ref{probgraph} is satisfied, and the graph as in item~(ii) has indeed no cycle because the edges $(a,b)$ of this graph correspond to the positions of the $-1$'s in $N$ which are all located below the main diagonal.
\end{proof}
Now we can determine the complexity status of D-$ell_\infty$-R1A by proving that it is NP-complete.
\begin{theorem}\label{thrnpcom}
The D-$ell_\infty$-R1A problem is NP-complete.
\end{theorem}
\begin{proof}
The function $(G,D)\to(\mathcal{M},3/2-0.001|V|^{-6})$ can be computed in polynomial time, and Theorem~\ref{thrred} proves that it is a reduction from Problem~\ref{probgraph} to D-$ell_\infty$-R1A. Therefore, D-$ell_\infty$-R1A is NP-hard by Lemma~\ref{lemgraph}. The membership of D-$ell_\infty$-R1A in NP is stated in Theorem~\ref{theoremNP}.
\end{proof}
\begin{example}
For the matrix
\[
M = \left( \begin{array}{ccccc}
2 & 0 & 1 & 1 & -1 \\
-1 & 2 & -1 & -1 & 0 \\
-1 & 1 & 2 & -1 & -1 \\
-1 & 1 & 1 & 2 & -1 \\
1 & -1 & 0 & 1 & 2 \\
\end{array} \right),
\]
there exist a permutation and a sign flip of the rows and columns so that the negative entries are below the main diagonal:
$N_{ij} = M_{\pi_i \pi_j} s_{\pi_i} s_{\pi_j}$ with $s = (1,-1,1,1,1)$ and $\pi = (5,2,1,4,3)$, with
\[
N =
\left( \begin{array}{ccccc}
2 & 1 & 1 & 1 & 0 \\
0 & 2 & 1 & 1 & 1 \\
-1 & 0 & 2 & 1 & 1 \\
-1 & -1 & -1 & 2 & 1 \\
-1 & -1 & -1 & -1 & 2 \\
\end{array} \right).
\]
We have $\min_{u,v} ||M-uv^T||_{\infty} = 1.3456 < 3/2$.
For the matrix
\[
M = \left( \begin{array}{ccccc}
2 & 1 & 1& -1 & 1 \\
-1 & 2 & -1 & -1 & 0 \\
-1 & 0 & 2 & 1 & 1 \\
0 & 1 & -1 & 2 & -1 \\
-1 &-1 & -1 & 1 & 2 \\
\end{array} \right)
\]
there is no such sign flip and permutation, and $\min_{u,v} ||M-uv^T||_{\infty} = 3/2$.
\end{example}
\section{Heuristic algorithm and application to the recovery of quantized low-rank matrices} \label{algoappl}
This goal of this section is to describe a simple heuristic algorithm for $\ell_{\infty}$ LRA and apply it for the recovery of quantized low-rank matrices. It will allow us to get some more insight on this problem.
The algorithm is available from \url{https://sites.google.com/site/nicolasgillis/} and allows the interested reader to tackle $\ell_{\infty}$ LRA (in particular the examples presented in this paper can be run directly).
In this section, all tests are performed using Matlab on a laptop Intel dual CORE i5-3210M CPU @2.5GHz 6Go RAM.
\subsection{Block coordinate descent method}
A popular approach in optimization is block coordinate descent (BCD): fix a subset of the variables and optimize over the other variables;
see~\cite{wright2015coordinate} for a recent survey. An crucial aspect of BCD is to make the subproblem easy (and fast) to solve.
For $\ell_{\infty}$ LRA, a judicious choice is to optimize alternatively over the columns of $U$ and the rows of $V$;
see Algorithm~\ref{altopt}.
In fact, the corresponding subproblems are convex and separable, that is, each entry in a column of $U$ (resp.\@ in a row of $V$) can be optimized independently of the other entries in the same column (resp.\@ row); this is described in the next section.
\begin{algorithm}
\caption{$(U,V)$ = BCD $\ell_{\infty}$ LRA $(M,U_0,V_0)$}
\label{altopt}
\begin{algorithmic}[1]
\STATE INPUT: $M \in \mathbb{R}^{m \times n}$, $U_0 \in \mathbb{R}^{m \times r}$ and $V_0 \in \mathbb{R}^{r \times n}$.
\STATE OUTPUT: $U \in \mathbb{R}^{m \times r}$, $V \in \mathbb{R}^{r \times n}$ so that $||M-UV||_{\infty}$ is minimized.
\STATE $U = U_0$, $V = V_0$.
\FOR{iter $=1,2,\dots$}
\STATE $R = M - UV$.
\FOR{$p=1,2,\dots,r$}
\STATE $R = R + U(:,p) V(p,:)$.
\STATE For all $i$, update $U(i,p) = \argmin_{x} \max_{ \{ j | V(p,j) \neq 0 \} } |R(i,j) - x \, V(p,j)|$.
\STATE For all $j$, update $V(p,j) = \argmin_{y} \max_{ \{ i | U(i,p) \neq 0 \} } |R(i,j) - U(i,p) \, y|$.
\STATE $R = R - U(:,p) V(p,:)$.
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
To initialize Algorithm~\ref{altopt}, we use the optimal solution of $\ell_{2}$ LRA.
It would be an interesting direction of research to use more sophisticated initialization strategies such as the approximation algorithm proposed in~\cite{CGKe17} that can be refined by Algorithm~\ref{altopt}.
\subsection{Secant method for the subproblem}
Let us focus on the rank-one subproblem in $v$ (by symmetry, the subproblem in $u$ can be solved in the same way).
It can be decoupled into $n$ problems in one variable: for $1 \leq j \leq n$,
we need to solve
\begin{equation} \label{subpbl1}
\min_{v_j} \; \max_{i} |M_{ij} - u_i v_j|.
\end{equation}
The optimal solution is not necessarily unique. Non-uniqueness may happen when $u_i = 0$ and $|M_{ij}|$ is large for some $i$, while uniqueness is guaranteed if $u_i \neq 0$ for all $i$ because the objective function is piece-wise linear with nonzero slopes.
To make the problem well posed, it makes sense to consider
\begin{equation} \label{subpbl}
\min_{v_j} \max_{ \{ i | u_i \neq 0 \} } |M_{ij} - u_i v_j|,
\end{equation}
with a unique solution which is also optimal for~\eqref{subpbl1}. (Note that if $u = 0$, any $v_j$ is optimal.)
Let us focus w.l.o.g.\@ on the case $u \geq 0$ by flipping the signs of the rows of $M$ accordingly.
The global minima is the intersection of two linear functions of the form $f^i_{\pm}(v_j) = \pm(M_{ij} - u_i v_j)$
($1 \leq i \leq m$), one with negative slope and one with positive slope.
Therefore, the optimal solution $v_j^*$ of~\eqref{subpbl} satisfies
\[
v_j^* \in \left\{ \frac{M_{i_1j} + M_{i_2j}}{u_{i_1} + u_{i_2}} \ \big| \ 1 \leq i_1 \neq i_2 \leq m, u_{i_1} > 0, u_{i_2} > 0 \right\} .
\]
Hence solving~\eqref{subpbl} can be done by identifying the two indices $i_1$ and $i_2$ corresponding to the optimal solution. This could be performed by inspection since there are $\frac{m(m-1)}{2}$ possible pairs.
A more efficient approach is described in the following. Let us define
\[
i_l = \argmin_{\{ i | u_i \neq 0 \}} \frac{M_{ij}}{u_i}
\quad
\text{ and }
\quad
i_u = \argmax_{\{ i | u_i \neq 0 \}} \frac{M_{ij}}{u_i}.
\]
We have $v_l = \frac{M_{i_l j}}{u_{i_l}} \leq v_j^* \leq \frac{M_{i_u j}}{u_{i_u}} = v_u$. Since the objective function is convex,
it is rather straightforward to implement the following \textit{secant method}:
\begin{enumerate}
\item Initialize $(i_1,i_2) = (i_l, i_u)$.
\item Intersect the two lines corresponding to the indices $i_1$ (with negative slope) and $i_2$ (with positive slope) to obtain the point $v_c = \frac{M_{i_1j} + M_{i_2j}}{u_{i_1} + u_{i_2}}$.
\item Compute the objective function of~\eqref{subpbl} in $v_j=v_c$ and identify the index $i_a$ that is active, that is,
the index $i_a$ such that $u_{i_a}\neq 0$ and $|M_{i_a j} - u_{i_a} v_j| = \max_{ \{ i | u_i \neq 0 \} } |M_{ij} - u_i v_j|$.
If the slope in $i_a$ is positive, replace $i_2$ by $i_a$; otherwise replace $i_1$ by $i_a$.
If the two indices $(i_1, i_2)$ are active together, the algorithm has converged: return $v_j^* = \frac{M_{i_1j} + M_{i_2j}}{u_{i_1} + u_{i_2}}$; otherwise, return to 2.
\end{enumerate}
It turns out that this secant method performs surprisingly well in the sense that it needs a very small number of iterations to terminate. Let us illustrate this on randomly generated instances.
\begin{example}[Numerical experiment on the secant method for~\eqref{subpbl1}] \label{ex2}
We have run the above secant method to solve $10^4$ problems of the form~\eqref{subpbl1} for different values of $m$, generating each entry of $M$ and $u$ using the normal distribution $N(0,1)$.
Table~\ref{tableGauss} reports the distribution of the number of iterations needed to solve~\eqref{subpbl1} for the $10^4$ problems,
along with the total computational time to solve them.
\begin{center}
\begin{table}[ht!]
\begin{center}
\caption{
Repartition of the number of iterations performed by the secant method to solve~\eqref{subpbl1} among $10^4$ instances and for different values of $m$, generating each entry of $M$ and $u$ using the normal distribution $N(0,1)$.
\label{tableGauss}
}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c||c|}
\hline
m / \# it. & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & Time (s.) \\
\hline
10 & 2019 & 1594 & 3277 & 2742 & 363 & 5 & 0 & 0 & 0 & 5.1 \\
$10^2$ & 199 & 204 & 487 & 4237 & 4099 & 747 & 27 & 0 & 0 & 6.55 \\
$10^3$ & 20 & 19 & 50 & 2668 & 5200 & 1885 & 154 & 4 & 0 & 8.13 \\
$10^4$ & 2 & 1 & 11 & 1739 & 5118 & 2769 & 351 & 9 & 0 & 22.03 \\
$10^5$ & 0 & 0 & 0 & 1258 & 4785 & 3403 & 525 & 28 & 1 & 108.11 \\
$10^6$ & 0 & 0 & 0 & 916 & 4618 & 3714 & 712 & 38 & 2 & 1685.40 \\
$10^7$ & 0 & 0 & 0 & 687 & 4192 & 4115 & 945 & 60 & 1 & 16793.87 \\ \hline
\end{tabular}
\end{center}
\end{table}
\end{center}
We observe that the secant method requires in average 5 iterations to terminate while 9 are necessary in the worst case.
\end{example}
\paragraph{Computational cost} Each iteration of the secant method requires $O(m)$ operations hence solving~\eqref{subpbl} requires $O(mK)$ operations where $K$ is the number of iterations performed by the secant method. In practice, as illustrates by Table~\ref{tableGauss}, $K$ will be rather small
(in all cases we have generated randomly, $K$ is smaller than 9).
In the worst case, the secant could potentially have to go through all indices with $K = O(m)$ and require $O(m^2n)$ operations. Note that sorting the values $\frac{M_{ij}}{u_i}$ ($1 \leq i \leq m$) in $O(m\log(m))$ operations would allow to perform a bisection in $O(m \log(m))$ operations (getting rid of about $m/2$ indices per iteration). Finally, the total computational cost of Algorithm~\ref{altopt} is $O(mnrK)$ per iteration.
\begin{remark}[$\ell_{\infty}$ LRA with nonnegativity constraints]
Algorithm~\ref{altopt} can easily be adapted to incorporate nonnegativity constraints on $U$ and $V$. In fact, the optimal solution of~\eqref{subpbl1} with the constraint $v_j \geq 0$ is $\max(v_j^*,0)$ where $v_j^*$ is the optimal solution of the unconstrained problem~\eqref{subpbl1}.
\end{remark}
\subsection{Recovery of quantized low-rank matrices}
We apply in this section Algorithm~\ref{altopt} to recover a quantized low-rank matrix. First, let us start with a toy example.
\begin{example} \label{ex3}
Let us generate a simple example with $m = 8$, $n = 5$ and $r = 3$ where $M = UV$ with each entry of $U$ and $V$ generated using the normal distribution $N(0,1)$. We obtain (with two digits of accuracy)
\[
M =
\left( \begin{array}{cccccccc}
0.35 & 0.65 & 0.15 & 0.54 & 1.49 \\
1.17 & -0.90 & -1.50 & -0.52 & -0.44 \\
1.03 & -1.12 & -3.48 & -1.41 & -0.17 \\
4.46 & -1.81 & 3.58 & 2.24 & -2.23 \\
-1.53 & -0.60 & -2.73 & -1.94 & -1.10 \\
-2.53 & 2.79 & 1.02 & 0.75 & 3.59 \\
3.38 & -0.90 & -1.16 & 0.53 & 0.86 \\
-0.66 & 0.42 & 0.71 & 0.20 & 0.14 \\
\end{array} \right)
\]
and a quantization (here we simply use its rounding to the nearest integer)
\[
M_q =
\left( \begin{array}{cccccccc}
0 & 1 & 0 & 1 & 1 \\
1 & -1 & -1 & -1 & 0 \\
1 & -1 & -3 & -1 & 0 \\
4 & -2 & 4 & 2 & -2 \\
-2 & -1 & -3 & -2 & -1 \\
-3 & 3 & 1 & 1 & 4 \\
3 & -1 & -1 & 1 & 1 \\
-1 & 0 & 1 & 0 & 0 \\
\end{array} \right) ,
\]
which has rank 5, and with $||M - M_q||_{\infty} = 0.498$ (entry on second row, third column).
The optimal solution $X_{\text{svd}}$ of rank-3 $\ell_2$ LRA gives $||M_q-X_{\text{svd}}||_{\infty} = 0.57$
while Algorithm~\ref{altopt} provides a solution $X^*$ with $||M_q-X^*||_{\infty} = 0.39$. Note that, for this problem,
the set
\[
\{ X | \rank(X) = 3, ||M-X||_\infty \leq 1/2\}
\]
is rather large and does not only contain a small neighborhood around the matrix $M$.
It would be an interesting direction for further research to identify conditions for this problem to be well posed
(e.g., sufficiently many entries of $M-M_q$ close to $\pm 1/2$, or using additional constraints on $X$).
\end{example}
Let us construct instances exactly as in Example~\ref{ex3} except that we use $m=n=200$ and $r=1,2,5,10,20$.
Note that the advantage of using quantized low-rank matrices is that we know there there exists a solution with error smaller than 1/2.
We stop the execution of Algorithm~\ref{altopt} only when 1000 iterations are performed or when the relative error between two iterates is smaller than $10^{-6}$, that is,
when $e_{t}-e_{t+1} \leq 10^{-6} ||M_q||_{\infty}$, where $e_t$ is the error at iteration $t$.
Table~\ref{tableQantG} provides
the smallest, average and largest value of the error of Algorithm~\ref{altopt} (second column), the number of solutions found with error smaller than 0.5 (third column),
the smallest, average and largest number of iterations needed to converge (fourth column),
the average computational time (fifth column),
and the smallest, average and largest value of the $\ell_{\infty}$ error for $\ell_2$ LRA which we used as an initialization for Algorithm~\ref{altopt} (last column).
\begin{center}
\begin{table}[h!]
\begin{center}
\caption{
Results of Algorithm~\ref{altopt} on Gaussian random instances of the recovery of quantized low-rank matrices.
\label{tableQantG}
}
\begin{tabular}{|c||c|c|c|c||c|}
\hline
$r$ & Error of Alg.~\ref{altopt} & \# runs & \# it. of Alg.~\ref{altopt} & Average & Error of $\ell_2$ LRA (init.) \\
& min , mean , max & error $\leq 0.5$ & min , mean , max & time (s.) & min , mean , max \\ \hline
1 & 0.50 , 0.50 , 0.50 & 100/100 & 14 , \;49 , 110 & 0.67 & 0.92 , 0.96 , 0.97 \\
2 & 0.53 , 0.69 , 0.87 & 0/100 & \;3 , \;\;5 , \;36 & 0.16 & 0.82 , 0.94 , 0.97 \\
5 & 0.52 , 0.54 , 0.62 & 0/100 & \;3 , \;19 , 127 & 1.47 & 0.65 , 0.70 , 0.87 \\
10 & 0.50 , 0.52 , 0.56 & 0/100 & \;5 , \;90 , 265 & 15.19 & 0.70 , 0.75 , 0.85 \\
20 & 0.48 , 0.49 , 0.53 & 93/100 & 14 , 175 , 333 & 58.69 & 0.74 , 0.81 , 0.91 \\ \hline
\end{tabular}
\end{center}
\end{table}
\end{center}
We observe the following:
\begin{itemize}
\item In all cases, Algorithm~\ref{altopt} is able to significantly improve the initial solution computed with $\ell_2$ LRA (second column vs.\@ last column of Table~\ref{tableQantG}).
\item In many cases, Algorithm~\ref{altopt} converges in a relative few number of iterations (sometimes in 3 iterations); see the fourth column of Table~\ref{tableQantG}. We believe the reason is that the objective function landscape is rather peaky hence it is more likely for the algorithm to terminate rapidly.
\item For $r=1$, Algorithm~\ref{altopt} is always able to recover a solution with error smaller than 1/2.
This is not surprising since the problem is not difficult: in fact, the graph $G_b(M_q,0.5)$ contains a single connected component since most entries of $|M_q|$ are larger than 0.5 (see Theorem~\ref{theoremNP}).
We observed in practice that, in this case, Algorithm~\ref{altopt} is always able to converge to an optimal solution. Optimality can be verified by checking that the answer to D-$\ell_\infty$-R1A($M_q$,$f^*-\epsilon$) is NO, where $f^*$ is the objective function value of Algorithm~\ref{altopt} at convergence and $\epsilon$ is a small positive constant. (We have included this function in the code available online.)
\item When $r > 1$, the problem becomes more difficult and Algorithm~\ref{altopt} is not able to identify a solution with error smaller than 1/2 (it is never able to do it for $r=2,5,10$). This leads to an interesting question:
is $\ell_{\infty}$ LRA hard for $r > 1$ (say $r=2$) even when $G_b(M_q,0.5)$ contains a unique connected component (or when $M \geq 0$)?
Table~\ref{tableQantG} suggests that it is the case. At least, we observed that there are many local minima: for example, for $r=2$, using 100 random initializations does not allow in general to obtain a solution with error smaller than 1/2, while the solution $M$ cannot be improved; see Table~\ref{tableQantGv2}.
Note also that the solutions obtained with random initializations have error significantly larger than with $\ell_2$-LRA initialization.
\item Surprisingly, when $r = 20$, Algorithm~\ref{altopt} is able in most cases to recover a solution with error smaller than 1/2. We believe the reason is that the number of degrees of freedom is large hence the optimal solution has error smaller than 0.5.
This is confirmed by the results in Table~\ref{tableQantGv2} where we have initialized Algorithm~\ref{altopt} using the original rank-$r$ matrix $M$ (hence the initial error is close to 0.5). We see that, quite naturally, it is able to identify better solutions than when initialized with the solution of $\ell_2$ LRA.
\begin{center}
\begin{table}[h!]
\begin{center}
\caption{
Results for the BCD Algorithm~\ref{altopt} initialized using the solution to $\ell_2$ LRA of matrix $M_q$ (exactly as for the second and last row of Table~\ref{tableQantG}) and with the solution to $\ell_2$ LRA of $M = UV$, where each entry of $U$ and $V$ is generated using the Gaussian distribution $N(0,1)$.
\label{tableQantGv2}
}
\begin{tabular}{|c||c|c|c|c|}
\hline
& Error of BCD & \# runs & \# it. of BCD & Average \\
& min , mean , max & error $\leq 0.5$ & min , mean , max & time (s.) \\ \hline
$r=2$, init.\@ $\ell_2$ & 0.52 , 0.67 , 0.88 & 0/100 & 3 , 6 , 35 & 0.19 \\
$r=2$, init.\@ $M$ & 0.50 , 0.50 , 0.50 & 100/100 & 2 , 10 , 51 & 0.31 \\ \hline
$r=20$, init.\@ $\ell_2$ & 0.48 , 0.49 , 0.51 & 96/100 & 71 , 178 , 261 & 56.12 \\
$r=20$, init.\@ $M$ & 0.46 , 0.46 , 0.46 & 100/100 & 66 , 96 , 135 & 30.60 \\ \hline
\end{tabular}
\end{center}
\end{table}
\end{center}
\end{itemize}
\section{Conclusion}
In this paper, we have analyzed the component-wise $\ell_{\infty}$ low-rank matrix approximation problem. We proved that, even in the rank-one case, the decision version of this problem is NP-complete using a reduction from `NOT ALL EQUAL 3SAT' (Theorem~\ref{thrnpcom}).
However, in the rank-one case when $M \geq 0$ or when $G_b(M,k)$ contains a few number of connected components,
the problem can be solved in polynomial time (Theorem~\ref{theoremNP}).
We then described a simple block coordinate descent method and applied it for the recovery of quantized low-rank matrices.
We observed that, as expected, the algorithm is able to recover an optimal solution when $r=1$.
However, as soon as $r > 1$, the problem becomes more difficult even when $G_b(M,k)$ contains a single connected component.
\section*{Acknowledgment}
We are grateful to Laurent Jacques for pointing us out to the quantized low-rank matrix problem and for insightful discussions. Part of this paper was written at the University of Mons during a visit of the second author who is grateful for the invitation and for being introduced to this problem.
\small
\bibliographystyle{siam}
|
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Cause destruction and mayhem in this action packed sequel! Dummy Crusher 2 is here, and it's absolutely crazy! Everything about it is improved, from the amount of action on screen to the variety of weapons you have! You can even control a heat seeking bee. That's pretty rad in my book. Can you beat all the levels in this challenging game? Good luck!
Smileys War is an action packed platform shooting game. You control a smiley and you have to take down your enemies usign a variety of guns. Many game modes, many maps, lots of fun!
|
{
"redpajama_set_name": "RedPajamaC4"
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| 8,060
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{"url":"https:\/\/cs.stackexchange.com\/questions\/12718\/is-it-decidable-if-a-language-described-by-number-of-occurences-is-regular","text":"# Is it decidable if a language described by number of occurences is regular?\n\nIt is known that the language of words containing equal number of 0 and 1 is not regular, while the language of words containing equal number of 001 and 100 is regular (see here).\n\nGiven two words $w_1,w_2$, is it decidable if the language of words containing equal number of $w_1$ and $w_2$ is regular?\n\n\u2022 Can you give other examples of regular languages so defined, other than $1^i0$ and $01^i$, or $0^i1$ and $10^i$ ? What about an example on a 3 symbols alphabet? \u2013\u00a0babou Jun 19 '13 at 13:54\n\u2022 If $w_1$ is a strict subword of $w_2$, there is a big chance the language is empty, therefore regular. I don't know other examples. \u2013\u00a0sdcvvc Jun 19 '13 at 14:39\n\u2022 I baguely suspect that the above examples are the only ones, which would make the problem decidable.If you specify only two substrings, I would guess it is CF ... depending on what you can specify regarding occurences. You do not make precise enough what you mean by \"described by number of occurences\". \u2013\u00a0babou Jun 19 '13 at 15:32\n\u2022 The question body is precise enough IMO. \u2013\u00a0sdcvvc Jun 19 '13 at 15:37\n\u2022 the solutions so far for special cases seem to hinge on the idea that occurrences of substrings of $w_1$ guarantee only single occurrences of intervening $w_2$. so somehow assuming current answers are correct [it is not clear to me yet] it seems there is some relation between $w_1$, $w_2$ that guarantees in the middle of scanning the string that one can be in either states \"equal\" or \"unequal\", but only off by a max finite number for the \"unequal\" case. \u2013\u00a0vzn Jun 19 '13 at 22:16\n\nGiven two words $w_1$,$w_2$, is it decidable if the language $L$ of words containing equal number of $w_1$ and $w_2$ is regular?\n\nFirst some definitions:\nThey could be made more concise, and the notations could be improved if they are to be used in proofs. This is only a first draft.\n\nGiven two words $w_1$ and $w_2$, we say that:\n\n\u2022 $w_1$ always occurs with $w_2$, noted $w_1\\triangleleft w_2$, iff\n\n1. for any string $s$ such that $s=xw_2y$ with $\\mid x\\mid,\\, \\mid y\\mid\\ \\geq \\mid w_1\\mid +\\mid w_2\\mid$ and $|x|_0,|x|_1|,|y|_0,|y|_1| \\geq 1$ there is another decomposition $s=x'w_1y'$.\nNote: The condition that $x$ and $y$ each contain at least a 0 and a 1 is required by a pathological case (found by @sdcvvc): $w_1=1^i0$, $w_2=v1^{i+j}$ and $y\\in1^*$, and its symetrical variants.\n2. there is a string $s=xw_2y$ with $\\mid x\\mid,\\, \\mid y\\mid\\ \\geq \\mid w_1\\mid +\\mid w_2\\mid$ such that there is at most one decomposition $s=x'w_1y'$\n\u2022 $w_1$ always cooccurs with $w_2$, noted $w_1\\triangleleft \\triangleright\\,w_2$, iff each always occur with the other,\n\n\u2022 $w_1$ and $w_2$ occur independently, noted $w_1\\triangleright \\triangleleft\\,w_2$, iff neither one always occur with the other,\n\n\u2022 $w_1$ always occurs $m$ times or more than $w_2$, noted $w_1\\triangleleft_m w_2$, iff for any string $s$ such that $s=xw_2y$ with $\\mid x\\mid,\\ \\mid y\\mid|\\ \\geq \\mid w_1\\mid +\\mid w_2\\mid$ there are $m$ other decompositions $s=x_iw_1y_i$ for $i\\in[1,m]$ such that $i\\neq j$ implies $x_i\\neq x_j$.\n\nThese definitions are constructed so that we can ignore what happens at the ends of the string where $w_1$ and $w_2$ are supposed to occur. Boundary effects at the end of the string have to be analyzed separately, but they represent a finite number of cases (actually I think I forgot one or two such boundary sub-cases in my first analysis below, but it does not really matter). The definitions are compatible with overlap of occurrences.\n\nThere are 4 main cases to consider (ignoring the symetry between $w_1$ and $w_2$):\n\n1. $w_1\\triangleleft \\triangleright\\,w_2$\nBoth words come necessarily together, except possibly at the ends of the string. This concerns only pairs of the form $1^i0$ and $01^i$, or $0^i1$ and $10^i$. This is easily recognized by a finite automaton that only checks for lone occurences at both ends of the string to be recognized, to make sure there is a lone occurrence at both ends or at neither end. There is also the degenerate case when $w_1=w_2$: then the language L is obviously regular.\n\n2. $w_1\\triangleleft w_2$, but not $w_2\\triangleleft w_1$\nOne of the 2 words cannot occur without the other, but the converse is not true (except possibly at the ends of the string). This happens when:\n\n\u2022 $w_1$ is a substring of $w_2$:then a finite automaton can just check that $w_1$ does not occur outside an instance of $w_2$.\n\n\u2022 $w_1=1^i0$ and $w_2=v1^j$ for some word $v\\in\\{0,1\\}^*$, $v\\neq01^i$: then a finite automaton check as in the previous case that $w_1$ does not occur separated from $w_2$. However, the automaton allows counting one extra instance of $w_1$ that will allow acceptance if $w_2$ is a suffix of the string. There are three other symetrical cases (1-0 symmetry and left-right symetry).\n\n3. $w_1\\triangleleft_2 w_2$\nOne of the 2 words occurs twice in the other. That can be recognized by an a finite automation that checks that the smaller word never occurs in the string. The is also a slightly more complex variant that combines the two variations of case 2. In this case the automaton checks that the smaller string $1^i0$ never occurs, except possibly as part of $v$ in the larger one $v1^j$ coming as a suffix of the string (and 3 other cases by symetry).\n\n4. $w_1\\triangleright \\triangleleft\\,w_2$\nThe 2 words can occur independently of each other. We build a generalized-sequential-machine (gsm) $G$ that output $a$ when it recognizes an occurrence of $w_1$ and $b$ when recognizing an occurrence of $w_2$, and forgets everything else. The language $L$ is regular only if the language $G(L)$ is regular. But $G(L)=\\{w\\in\\{a,b\\}^*\\mid\\ \\mid w\\mid_a=\\mid w\\mid_b\\}$ which is clearly context-free and not regular. Hence $L$ is not regular.\nActually we have $L=G^{-1}(G(L))$. Since regular languages and context-free languages are closed under gsm mapping and inverse gsm mapping, we know also that $L$ is context free.\n\nOne way to organize a formal proof could be the following. First build a PDA that recognizes the language. Actually it can be done with a 1-counter machine, but it is easier to have two stack symbols to avoid duplicating the finite control. Then, for the cases where it should be a FA, show that the counter can be bounded by a constant that depends only on the two words. For the other cases show that the counter can reach any arbitrary value. Of course, the PDA should be organized so that the proofs are easy enough to carry.\n\nRepresenting the FA as a 2-stack-symbols PDA is probably the simplest representation for it. In the non-regular case, the finite control part of the PDA is the same as that of the GSM in the proof sketch above. Instead of outputting $a$'s and $b$'s like the GSM, the PDA counts the difference in number with the stack.\n\n\u2022 I had a question about context-freeness in the case of three words. I deleted it when I realised it could be analyzed similarly. I had first thought that proving non-CFness would make an original exercise, but the GSM ruins it. \u2013\u00a0babou Jun 19 '13 at 21:12\n\u2022 It is not clear what do you mean by \"occur independently of each other\", \"come necessarily together\" etc. Please write formal definitions instead, and prove that they cover all cases. \u2013\u00a0sdcvvc Jun 19 '13 at 21:45\n\u2022 I am not sure what you are asking, and what level of formalization you need, for what purpose. I realized that analyzing by hand possible relations of the two words is not garanteed to be correct, and does not matter anyway. What matters is whether an occurence of one word can exist without creating at the same time an occurence (or several) of the other word. The details do not matter as it will always be localized and thus manageable finitely. The two ends do not matter either as tey are localized too. Even overlaps of occurrences do not matter since they can only be finitely many in 1 place \u2013\u00a0babou Jun 19 '13 at 22:53\n\u2022 I asked you about precise definitions of the terms mentioned in the comment. Thank you for writing them. Was I supposed to guess them previously? Anyway, you seem to claim that $0^i 1 \\triangleleft \\triangleright 1 0^i$. This does not satisfy condition 1. of the definition of \"$w_1$ always occurs with $w_2$\", since there is no occurrence of $1 0^i$ in $s=0^M 0^i 1 1^M$. \u2013\u00a0sdcvvc Jun 20 '13 at 14:10\n\u2022 Sorry, I did not mean to make you guess. It only took me time to understand what exactly you wanted. My failing only. Regarding your counter example, you are correct. But for me it only means that I have to be a little bit more careful about telomeres, in the definition of the relations. I defined them too quickly, but $0^M$ or $1^M$ do not convey much information in this context. This is really a boundary pathological example within a pathological case, that actually cannot occur when more than 2 symbols are used. I just do not believe it changes anything. \u2013\u00a0babou Jun 20 '13 at 15:08","date":"2019-12-08 17:01:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7180712819099426, \"perplexity\": 290.7476229188519}, \"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-2019-51\/segments\/1575540511946.30\/warc\/CC-MAIN-20191208150734-20191208174734-00279.warc.gz\"}"}
| null | null |
Q: find which types from given dll are used in my solution I've got sulution which uses external dll. What is the best way to list all types from that external dll used in my solution?
Maybe resharper has got any option which may suport this?
A: References are always used within a project. To find the types used from that assembly just open the context menu of that referenced assembly under References of your project and select Find Code Dependent on Module
If you want to see which types are used, you can group them in the result window by type:
If that DLL is referenced within multiple projects you would have to do that on every project. I don't think there is a way to do that an a solution level.
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{"url":"https:\/\/insignia3d.com\/tight-lines-axdny\/binary-operation-table-16091b","text":"Share\n\n# binary operation table\n\n## binary operation table\n\nBinary Operations Let S be any given set. Determine whether the following operation define a binary operation on the given set or not: (i) \u00e2\u0080\u0098*\u00e2\u0080\u0099 on N defined by a * b = a b for all a, b \u00e2\u0088\u0088 N. (ii) \u00e2\u0080\u0098O\u00e2\u0080\u0099 on Z defined by a O b = a b for all a, b \u00e2\u0088\u0088 Z. This table shows the operation * (\u00e2\u0080\u009cstar\u00e2\u0080\u009d). 11.2 Multiplication tables For small sets, we may record a binary operation using a table, called the multiplication table (whether or not the binary operation is multiplication). Oracle Database Lite SQL also supports set operators. Represent operation * as a table on A. This module implements general operation tables, which are very matrix-like. This table is known as a composition table. Addition + : R \u00d7 R \u00e2\u0086\u0092 R e is called identity of * if a * e = e * a = a i.e. 2. uniquely associates each pair of elements in to some element of .. Example 1. There are two general classes of operators: unary and binary. Here e is called identity element of binary operation. This table can be formed as follows: Addition, subtraction, multiplication are binary operations on Z. In other words, $$\\star$$ is a rule for any two elements in the set $$S$$. B. The result of AND operation in Octal Decimal Result:.. (vii) Let S = N, with de ned by a b = ab (e.g., 2 3 = 23 = 8). Here, the output result relies on the operation executed on the input or proposition values and the value can be either true or false. The result of AND operation in Decimal Hex Result:.. A binary operation on S is a correspondence that associates with each ordered pair (a, b) of elements of S a uniquely ... \u00e2\u0080\u0093 A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow.com - id: 47753e-NTE0N 0. Algebraic operations with binary numbers. This is a binary operation. A binary operation on a nite set is commutative the table is symmetric about the diagonal running from upper left to lower right. Apart from these differences, operations such as addition, subtraction, multiplication, and division are all computed following the same rules as the decimal system. Let $$S$$ be a non-empty set, and $$\\star$$ said to be a binary operation on $$S$$, if $$a \\star b$$ is defined for all $$a,b \\in S$$. A binary operation given by a table. Learn how to make a\u00e2\u0080\u00a6 A. For the operation on , every element has an inverse, namely .. For the operation on , the only element that has an inverse is ; is its own inverse.. For the operation on , the only invertible elements are and .Both of these elements are equal to their own inverses. Because of the many interesting examples of binary operations \u00e2\u0080\u00a6 A binary operation on a nonempty set Ais a function from A Ato A. Some of the examples of binary operations are AND, OR, NOR, XOR, XNOR, etc. A binary operation on a finite set (a set with a limited number of elements) is often displayed in a table that demonstrates how the operation is performed. The result of the operation on a and b is another element from the same set X. A binary operation, , is defined on the set {1, 2, 3, 4}. 1. is defined for every pair of elements in , and . Situation 2: Sometimes, a binary operation on a finite set (a set with a limited number of elements) is displayed in a table which shows how the operation is to be performed. De\u00ef\u00ac\u0081nition 3.2 D. 4. Operators listed on \u00e2\u0080\u00a6 The usual division \/ is not a binary operation on R since \/ The binary operations include two variables for input values. The operation \u00ce\u00a6 is not associative for real numbers. Example: Consider the set A = {1, 2, 3} and a binary operation * on the set A defined by a * b = 2a+2b. Example 13.1.4. Binary Logic Operations. Solution: QUESTION: 4. In order to do the binary calculations yourself most would prefer using a table for smaller numbers and a calculator for larger ones. Using our tool in binary calculator mode you can perform the four basic arithmetic operations on binary numbers: addition, subtraction, multiplication and division. The result of AND operation in Binary Octal Result:.. A binary operation is an operation that applies to two quantities or expressions and .. A binary operation on a nonempty set is a map such that . Examples of binary operation on from to include addition (), subtraction (), multiplication) and division (). As is the case for other functions, there are several ways of specifying a binary operation. For example, the following is the multiplication table of a binary operation \u00e2\u0088\u0097 : {a,b}\u00d7{a,b} \u00e2\u0088\u0092\u00e2\u0086\u0092 {a,b}. Pandas include a couple useful twists, however: for unary operations like negation and trigonometric functions, these ufuncs will preserve index and column labels in the output, and for binary operations such as addition and multiplication, Pandas will automatically align indices when passing the objects to the ufunc. Access RD Sharma Solutions For Class 12 Maths Chapter 3 \u00e2\u0080\u0093 Binary Operations. Hot Network Questions How did musicians acquire samples for tracker music (MOD, S3M, XM and the like)? Similarly, standard multiplication is associative on $\\mathbb{R}$ because the order of operations is not strict when it comes to multiplying out an expression that is solely multiplication, i.e.,: (2) Definition: Binary operation. The composition table helps us to verify most of the properties satisfied by the binary operations. * a b a a b b b a In studying binary operations on sets, we tend to be interested in those operations that have certain properties which we discuss next. Then is closed under the operation *, if a * b \u00e2\u0088\u0088 A, where a and b are elements of A. To perform this operation we need a minimum of 1 input variable that can take the values of binary numbers i.e., 0 or 1 to get an output with one binary value (0\/1). There are many properties of the binary operations which are as follows: 1. How many elements of this operation have an inverse?. Closure Property: Consider a non-empty set A and a binary operation * on A. 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Variables, and appropriate logical operations = 0 or else A\u00e2\u0080\u0099 = 1, 2, 3, }...\n\n++","date":"2021-04-10 12:17:43","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\": 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\": 0.722771942615509, \"perplexity\": 979.6398887273393}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038056869.3\/warc\/CC-MAIN-20210410105831-20210410135831-00635.warc.gz\"}"}
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Lota James (#5) gets up at the wire over Diva Candy Girl (#6) in the $17,500 Dash In A Flash Stakes Saturday afternoon at Canterbury Park.
SHAKOPEE, MN�JULY 15, 2017�Lotta James, who two races back started for a $12,5000 tag at Remington Park, prevailed over Divas Candy Girl to win the $17,500 Dash In A Flash Stakes Saturday afternoon at Canterbury Park. Posttime-favorite Bye Byefreighttrain finished a neck back in the blanket finish to take third.
Ridden by jockey Nik Goodwin, Lotta James scooted over the 110 yards in a quick :6.987 clocking into a 15-mph headwind to earn a 98-speed index. It was the fifth career win in 17 starts for the son of multiple stakes sire IVORY JAMES.
Saddled by trainer R. Allen Hybsha, Lotta James picked up $8,850 to boost his career mark to $46,044. The brown gelding races for owner Whiting Ranch. Steve Holt bred Lota James in Oklahoma from the Shazoom mare Lotawatah.
Diva Candy Girl, a stakes winning sister to four-time champion Spit Curl Diva, picked up $3,540 for her second-place effort. Jason Olmstead conditions the filly for owner Thomas Scheckel. Carl Pevehouse and Ezra Lee bred Divas Candy Girl in Oklahoma. Danny Velazquez rode the earner of $83,075.
Kasey Willis saddled Bye Bye Freightrain to his third-place effort for owner/breeder L.P. Frank. Also, an Oklahoma-bred, the graded stakes winner was ridden by Cristian Esqueada. The $2,655 third-place check boosted Bye Bye Freightrain's earnings to $240,153.
Fast Eddys Eyeinyou, Dash For Number One, Ms Dynasty, Iowas Texan Tip and Fast Prize Janie completed the field.
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction and results}
\subsection{Main result} Throughout this article, a \emph{system} is a probability space $(X,\mathcal{X},\mu)$ together with invertible, measure preserving
transformations $T_1,\ldots, T_\ell\colon X\to X$ that commute. A multiple correlation sequence is a sequence of the form
$$
\int T_1^{n_1}f_1\cdot \ldots \cdot T_\ell^{n_\ell}f_\ell\, d\mu
$$
where $(X,\mathcal{X},\mu, T_1,\ldots, T_\ell)$ is a system, $f_1,\ldots, f_\ell\in L^\infty(\mu)$, and $n_1,\ldots, n_\ell\in {\mathbb Z}$. The study of the limiting behavior of averages of such sequences, where the iterates are restricted to certain subsets of ${\mathbb Z}^\ell$, has been an indispensable tool
in ergodic Ramsey theory and in particular in proving various far reaching extensions of Szemer\'edi's theorem on arithmetic progressions.
Although the precise structure
of the multiple correlation sequences
is unknown even when $n_1=\cdots=n_\ell=n$, there is a widespread belief that modulo negligible terms the building blocks are sequences with algebraic structure (see
\cite[Problem 1]{Fr11} for a related conjecture).
\begin{definition}[\cite{BHK05}]
For $\ell\in {\mathbb N}$, an \emph{$\ell$-step nilsequence} is a sequence of the form $(F(g^n\Gamma))$, where $F\in C(X)$, $X=G/\Gamma$, $G$ is an $\ell$-step nilpotent Lie group, $\Gamma$ is a discrete cocompact subgroup, and
$g\in G$. A \emph{$0$-step nilsequence}
is a constant sequence.
\end{definition}
When $T_i=T^i$, $i=1,\ldots, \ell,$ following the discovery of characteristic factors with algebraic structure for some closely related multiple ergodic averages,
V.~Bergelson, B.~Host, and B.~Kra proved the following beautiful result (see also \cite{Mei90} for related work for $\ell=3$):
\begin{theorem*}[\mbox{\cite[Theorem 1.9]{BHK05}}]\label{T:0}
For $\ell\in {\mathbb N}$, let $(X,\mathcal{X},\mu, T)$ be an ergodic system and $f_1,\ldots,f_\ell\in L^{\infty}(\mu)$ be functions with $\norm{f_i}_\infty\leq 1$. Then we have the
decomposition
$$
\int T^{n}f_1\cdot \ldots \cdot T^{\ell n}f_\ell\ d\mu=a_{st}(n)+a_{er}(n), \quad n\in {\mathbb N}, $$
where
\begin{enumerate}
\item $(a_{st}(n))$ is a uniform limit of $(\ell-1)$-step nilsequences
with $\norm{a_{st}}_{\infty}\leq 1$;
\item $\lim_{N-M\to \infty} \frac{1}{N-M}\sum_{n=M}^{N-1} |a_{er}(n)|^2=0$.
\end{enumerate}
\end{theorem*}
This result was extended by A.~Leibman to cover polynomial iterates in \cite{L10} and not necessarily ergodic transformations in \cite{L14}.
The proofs of these results depend in an essential way on the fact that characteristic factors for some suitable multiple ergodic averages
are inverse limits of nilsystems. This is no longer true for correlation sequences involving actions of commuting transformations, which is why efforts to prove decomposition results for such sequences did not bring any results so far.
In fact, characteristic factors for commuting actions
are known to be extremely complex (for related work see \cite{Au11a, Au11b}) which has raised suspicions that decomposition results in this more general setup
may involve sequences very different
from nilsequences. Our main result settles this rather elusive problem;
we show that
modulo error terms that are small in uniform density, correlation sequences of actions of commuting transformations are nilsequences.
\begin{theorem}\label{T:1}
For $\ell\in {\mathbb N}$ let $(X,\mathcal{X},\mu, T_1,\ldots, T_\ell)$ be a system and $f_1,\ldots,f_\ell\in L^{\infty}(\mu)$ be functions with $\norm{f_i}_\infty\leq 1$. Then for every $\varepsilon>0$ we have the
decomposition
\begin{equation}\label{E:mulcom}
\int T_1^{n}f_1\cdot \ldots \cdot T_\ell^{n}f_\ell\ d\mu=a_{st}(n)+a_{er}(n), \quad n\in {\mathbb N},
\end{equation}
where
\begin{enumerate}
\item $(a_{st}(n))$ is an $(\ell-1)$-step nilsequence with $\norm{a_{st}}_{\infty}\leq 1$;
\item \label{E:2} $\lim_{N-M\to \infty} \frac{1}{N-M}\sum_{n=M}^{N-1} |a_{er}(n)|^2\leq \varepsilon$.
\end{enumerate}
\end{theorem}
\begin{remark}
We do not know if a strengthening similar to the one in \cite[Theorem 1.9]{BHK05} holds where one
uses uniform limits of nilsequences in $(i)$ and
takes $\varepsilon=0$ in $(ii)$.
\end{remark}
Our argument is rather versatile and does not rely on the theory of characteristic factors; we rather focus on some distinctive properties correlation sequences as in \eqref{E:mulcom} satisfy (see Theorem~\ref{T:2}). The idea that starts the proof comes from answering the following natural question:
``Can a multiple correlation sequence as in \eqref{E:mulcom} be asymptotically orthogonal to all $(\ell-1)$-step nilsequences?''.
On the one hand, using an inverse theorem of B.~Host and B.~Kra
(see Theorem~\ref{T:inverse}), one gets that any such sequence has to be $U_{\ell}$-uniform.
On the other hand, by successively applying van der Corput's lemma one sees that
a sequence of the form \eqref{E:mulcom} is asymptotically orthogonal to all $U_{\ell}$-uniform sequences. Hence, any sequence that provides a positive answer to our question has to be asymptotically orthogonal to itself, that is, has to converge to $0$ in density.
With this idea in mind, we prove our main result as follows: Given a sequence $(a(n))$ as in \eqref{E:mulcom}, we consider the
$(\ell-1)$-step nilsequence, call it $a_{st}$, that lies ``closest'' to $(a(n))$ with respect to the semi-norm $\norm{\cdot}_2$ defined in \eqref{E:seminorm}. Then $a_{er}:=a-a_{st}$
is asymptotically orthogonal to all $(\ell-1)$-step nilsequences, and arguing as before, we
get that $a_{st}$ and $a_{er}$ have the asserted properties.
A slight complication appears because for $\ell\geq 2$ the space of
$(\ell-1)$-step nilsequences (or uniform limits of such sequences) is not $\norm{\cdot}_2$-complete; this is the reason why we are led to an error term $a_{er}$ that is small, but not zero, in uniform density. For our argument to work we also have to make sure that various limits of uniform Ces\`aro
averages exist; to guarantee this, we use a result of T.~Austin~\cite{Au09}.
Using a variant of the previous argument and a result of M.~Walsh~\cite{W12}
we get:
\begin{theorem}\label{T:1'}
Let $\ell,m\in {\mathbb N}$ and $p_{i,j}\in {\mathbb Z}[t]$, $i=1,\ldots, \ell, j=1,\ldots, m$, be polynomials.
Then there exists $k\in {\mathbb N}$, $k=k(\ell,m,\max{\deg(p_{i,j})})$,
such that for every system
$(X,\mathcal{X},\mu, T_1,\ldots, T_\ell)$, functions
$f_1,\ldots,f_m\in L^{\infty}(\mu)$ with $\norm{f_i}_\infty\leq 1$, and $\varepsilon>0$, we have
\begin{equation}\label{E:complicated}
\int (\prod_{i=1}^\ell T_i^{p_{i,1}(n)})
f_1\cdot \ldots \cdot (\prod_{i=1}^\ell T_i^{p_{i,m}(n)})
f_m\, d\mu =a_{st}(n)+a_{er}(n), \quad n\in {\mathbb N},
\end{equation}
where
\begin{enumerate}
\item $(a_{st}(n))$ is a $k$-step nilsequence with $\norm{a_{st}}_{\infty}\leq 1$;
\item \label{E:2} $\lim_{N-M\to \infty} \frac{1}{N-M}\sum_{n=M}^{N-1} |a_{er}(n)|^2\leq \varepsilon$.
\end{enumerate}
\end{theorem}
\subsection{A more general framework}
It turns out that Theorem~\ref{T:1} is a manifestation of a more general principle which
asserts that if a sequence is asymptotically orthogonal to all $U_{\ell}$-uniform sequences and
satisfies some necessary regularity conditions, then it admits a decomposition like the one in Theorem~\ref{T:1}. To make this more precise we introduce some notation (see Section~\ref{SS:seminorms} for the definition of the uniformity seminorms).
\begin{definition}
Let $\ell\in {\mathbb N}$. We say that the bounded sequence $a\colon {\mathbb N}\to {\mathbb C}$ is
\begin{enumerate}
\item
{\em $\ell$-anti-uniform} if
there exists
$C:=C(\ell,a)$ such that
$$
\limsup_{N-M\to
\infty} \Big|\frac{1}{N-M}\sum_{n=M}^{N-1} a(n)b(n)\Big|\leq C \norm{b}_{U_{\ell}(\mathbb{N})}$$ for every $b\in \ell^{\infty}$.
\item {\em $\ell$-regular} if
the limit
$$
\lim_{N-M\to \infty} \frac{1}{N-M}\sum_{n=M}^{N-1} a(n)\psi(n)
$$ exists for every $(\ell-1)$-step nilsequence $(\psi(n))$.
\end{enumerate}
\end{definition}
\begin{theorem}\label{T:2}
For $\ell\in {\mathbb N}$ let $a\colon {\mathbb N}\to {\mathbb C}$ be a sequence with $\norm{a}_\infty\leq 1$ that is $\ell$-anti-uniform and $\ell$-regular.
Then for every $\varepsilon>0$ we have the decomposition
$$
a(n)=a_{st}(n)+a_{er}(n), \quad n\in {\mathbb N},
$$
where
\begin{enumerate}
\item \label{E:1a} $(a_{st}(n))$ is an $(\ell-1)$-step nilsequence with $\norm{a_{st}}_{\infty}\leq 1$;
\item \label{E:1b} $\lim_{N-M\to \infty} \frac{1}{N-M}\sum_{n=M}^{N-1} |a_{er}(n)|^2\leq \varepsilon$.
\end{enumerate}
\end{theorem}
\begin{remarks}
For general $\ell$-regular sequences a similar result is proved in \cite[Theorem 2.19]{HK09} with an error term that is small with respect to the seminorm $\norm{\cdot}_{U_\ell({\mathbb N})}$.
A sequence $(a(n))$ that satisfies the asserted decomposition has to be $\ell$-regular. It also
has to satisfy the estimate defining the $\ell$-anti-uniformity property if one introduces an arbitrarily small error term $\varepsilon$ on the right hand side and allows $C$ to depend on $\varepsilon$ (this follows from
\cite[Theorem 2.14]{HK09}).
Theorem~\ref{T:2} fails if we use standard Ces\`aro averages to define the notions of anti-uniformity and regularity (and leave the definition of $\norm{\cdot}_{U_\ell({\mathbb N})}$ as is); the sequence
$(e^{ i \sqrt{n}})$, illustrates this. The same sequence shows that
anti-uniformity does not imply regularity ($(e^{ i \sqrt{n}})$ is $2$-anti-uniform but not $1$-regular).
\end{remarks}
\subsection{Applications}\label{SS:Applications}
On $\ell^\infty({\mathbb N})$ we define the seminorm $\norm{\cdot}_2$ by
\begin{equation}\label{E:seminorm}
\norm{a}_2^2:=\limsup_{N-M\to \infty} \frac{1}{N-M}\sum_{n=M}^{N-1}|a(n)|^2.
\end{equation}
For $\ell\in {\mathbb N}$ we consider the following subspaces of $\ell^\infty({\mathbb N})$:
$$
\mathcal{A}_\ell:=\Big\{(\psi(n)) \colon \psi \text{ is an } (\ell-1)\text{-step nilsequence}\Big\};
$$
$$
\mathcal{B}_\ell:=\Big\{\Big(\int T^{k_1n}f_1\cdot \ldots \cdot T^{k_{\ell} n}f_\ell \, d\mu\Big)\colon (X,\mathcal{X}, \mu,T) \text{ is a system}, f_i\in L^\infty(\mu), k_i=\ell!/i \Big\};
$$
$$
\mathcal{C}_\ell:=\Big\{\Big(\int T_1^nf_1\cdot \ldots \cdot T_{\ell}^{n}f_\ell \, d\mu\Big)\colon (X,\mathcal{X}, \mu, T_1,\ldots, T_\ell) \text{ is a system and } f_i\in L^\infty(\mu)\Big\}.
$$
After Proposition~\ref{P:nilkey} we explain why in the definition of $\mathcal{B}_\ell$ we use the exponents $k_1,\ldots, k_\ell$ instead of $1,\ldots, \ell$.
The space $\mathcal{A}_\ell$ is linear since if for $i=1,2$, $(F_i(g_i^n\Gamma_i))$ are $(\ell-1)$-step nilsequences on $G_i/\Gamma_i$, then their sum is the $(\ell-1)$-step nilsequence $(F(g^n\Gamma))$ on $G/\Gamma$, where $G=G_1\times G_2$,
$\Gamma:=\Gamma_1\times \Gamma_2$, $g:=(g_1,g_2)$, $F(g\Gamma):=F_1(g_1\Gamma_1)+F_2(g_2\Gamma_2)$.
To see that the space
$\mathcal{C}_\ell$ is linear (similarly for $\mathcal{B}_\ell$), let $a,b\in \mathcal{C}_\ell$ be defined by the systems $(X_i, \mathcal{X}_i, \mu_i, T_i)$ and the functions $f^i_1,\ldots, f^i_\ell$, $i=1,2$. Then $c:=(a+b)/2$ is also a multiple correlation sequence defined by the system $(X, \mathcal{X}, \mu, T)$, where $X=X_1\cup X_2$ (considered as disjoint subsets) with the corresponding $\sigma$-algebra $\mathcal{X}$, $\mu:=(\mu_1+\mu_2)/2$, $T$ equals $T_1$ on $X_1$ and $T_2$ on $X_2$, and $f_i:=f^1_i{\bf 1}_{X_1}+f^2_i{\bf 1}_{X_2}$, $i=1,\ldots, \ell$
It is a rather striking fact that, modulo sequences that are small in uniform density, the three subspaces
$\mathcal{A}_\ell$, $\mathcal{B}_\ell$, $\mathcal{C}_\ell$
coincide.
\begin{theorem} \label{T:3}
For every $\ell\in {\mathbb N}$ we have
$$
\overline{\mathcal{A}_{\ell}}=\overline{\mathcal{B}_\ell}=\overline{\mathcal{C}_\ell}
$$
where the closure is taken with respect to the seminorm $\norm{\cdot}_2$ defined in \eqref{E:seminorm}.
\end{theorem}
It is not hard to see that the first equality fails if we consider closures with respect to the $\norm{\cdot}_\infty$ norm. The second equality may still hold under such circumstances but this is not something we can prove with the methods developed so far.
The next two results illustrate some rather surprising principles:
$(i)$ convergence results for actions of a single transformation automatically imply stronger convergence results for actions of commuting transformations; and
$(ii)$ convergence results involving linear iterates
automatically imply stronger convergence results involving polynomial iterates.
\begin{theorem} \label{T:4}
Let $(r_n)$ be a strictly increasing sequence of integers such that $r_n=O(n)$. Then for every $\ell\in {\mathbb N}$ the following statements are equivalent:
\begin{enumerate}
\item \label{E:41} For every $(\ell-1)$-step nilsequence $(\psi(n))$ the limit $\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N}\psi(r_n)$ exists.
\item \label{E:42} For every system
$(X,\mathcal{X},\mu, T)$, functions $f_1,\ldots, f_\ell\in L^\infty(\mu)$, and for $k_i=\ell!/i$, $i=1,\ldots, \ell$,
the following limit exists
$$
\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N} \int T^{k_1r_n}f_1 \cdot\ldots\cdot T^{k_\ell r_n}f_\ell \, d\mu.
$$
\item \label{E:43}For every system
$(X,\mathcal{X},\mu, T_1,\ldots, T_\ell)$ and functions $f_1,\ldots, f_\ell\in L^\infty(\mu)$
the following limit exists
$$
\lim_{N\to \infty} \frac{1}{N}\sum_{n=1}^{N} \int T_1^{r_n}f_1\cdot \ldots\cdot T_{\ell}^{r_n} f_\ell \, d\mu.
$$
\end{enumerate}
\end{theorem}
\begin{remark}
Equivalently, the growth condition $r_n=O(n)$ holds if the set $R:=\{r_1,r_2,\ldots\}$ has positive lower natural density.
\end{remark}
In the previous result we have established an equivalence for every fixed $\ell\in {\mathbb N}$, in the next result
we have to assume that a certain property is known for every $\ell\in {\mathbb N}$ in order to establish an equivalence (this is needed for the equivalence of (ii) and (iii)).
\begin{theorem} \label{T:4'}
Let $(r_n)$ be a strictly increasing sequence of integers such that $r_n=O(n)$. Then the following statements are equivalent:
\begin{enumerate}
\item \label{E:4'1} For every
$\ell\in {\mathbb N}$ and $\ell$-step
nilsequence $(\psi(n))$ the limit $\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N}\psi(r_n)$ exists.
\item \label{E:4'2} For every $\ell\in {\mathbb N}$, system
$(X,\mathcal{X},\mu, T)$, and functions $f_1,\ldots, f_\ell\in L^\infty(\mu)$,
the following limit exists
$$
\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N} \int T^{r_n}f_1\cdot \ldots \cdot T^{\ell r_n}f_\ell \, d\mu.
$$
\item \label{E:4'3} For every $\ell\in {\mathbb N}$, polynomials $p_1,\ldots, p_\ell\in {\mathbb Z}[t]$, system
$(X,\mathcal{X},\mu, T_1,\ldots, T_\ell)$, and functions $f_1,\ldots, f_\ell\in L^\infty(\mu)$,
the following limit exists
$$
\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N} \int T_1^{p_1(r_n)}f_1\cdot \ldots\cdot T_{\ell}^{p_\ell(r_n)} f_\ell \, d\mu.
$$
\end{enumerate}
\end{theorem}
Similar results hold if in \eqref{E:4'1}-\eqref{E:4'3} of Theorems~\ref{T:4} and \ref{T:4'} one replaces the limit
$\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N}$ with the limit $\lim_{N-M\to \infty}\frac{1}{N-M}\sum_{n=M}^{N-1}$ and the growth assumption on $(r_n)$ with the assumption that the range of this sequence has positive lower Banach density. Furthermore, the same method can be used to prove convergence criteria for weighted averages where for a given
bounded sequence of complex numbers $(w_n)$ one
replaces in \eqref{E:4'1}-\eqref{E:4'3} of Theorems~\ref{T:4} and \ref{T:4'} the averaging operation $\frac{1}{N}\sum_{n=1}^{N}$
with the averaging operation $\frac{1}{N}\sum_{n=1}^{N} w_n$.
\subsection{Conjectures}
The growth assumption on $(r_n)$ in Theorems~\ref{T:4} and \ref{T:4'} is crucial for our argument to work as the proofs use
Theorem~\ref{T:1} which is not helpful for sequences that grow faster than linearly. Nevertheless, we
believe that the following is true:
\begin{conjecture}
In Theorems~\ref{T:4} and \ref{T:4'} the growth assumption on $(r_n)$ is superfluous.
\end{conjecture}
We also believe in the following strengthening of the second identity in Theorem~\ref{T:3}:
\begin{conjecture}
For every $\ell\in \mathbb{N}$ we have
$\overline{\mathcal{B}_\ell}=\overline{\mathcal{C}_\ell}$ where the closure is taken with respect to the norm $\norm{\cdot}_\infty$.
\end{conjecture}
\subsection{Notation}
We denote by ${\mathbb N}$ the set of positive integers.
\noindent
If $(a(n))$ is a bounded sequence we denote by $\limsup_{N-M\to \infty} |\frac{1}{N-M}\sum_{n=M}^{N-1}a(n)|$ the limit (it exists by subadditivity)
$
\lim_{N\to \infty} \sup_{M\in {\mathbb N}} \Big|\frac{1}{N}\sum_{n=M}^{M+N-1}a(n)\Big|.
$
\subsection{ Acknowledgements.} I would like to thank B.~Host, B.~Kra, M.~Wierdl, and the referee for helpful remarks.
\section{Proofs of results}
\subsection{Uniformity seminorms and the Host-Kra inverse theorem}\label{SS:seminorms}
We give a slight variant of the uniformity seminorms defined by B.~Host and B.~Kra in \cite{HK09}.
\begin{definition}
Let $\ell\in {\mathbb N}$ and $a\colon {\mathbb N}\to {\mathbb C}$ be a bounded sequence.
\begin{enumerate}
\item
Given a sequence of intervals ${\bf I}=(I_N)$ with lengths tending to infinity,
we say that the sequence $(a(n))$ is {\em distributed
regularly along ${\bf I}$} if the limit
$$
\lim_{N\to \infty} \frac{1}{|I_N|}\sum_{n\in I_N} a_1(n+h_1)\cdot\ldots\cdot a_r(n+h_r)
$$
exists for every $r\in {\mathbb N}$ and $h_1,\ldots, h_r\in {\mathbb N}$, where $a_i$ is either $a$ or $\bar{a}$.
\item If ${\bf I}$ is as in (i) and $(a(n))$ is distributed
regularly along ${\bf I}, $ we define inductively
$$\norm{a}_{{\bf I}, 1}:=
\lim_{N\to \infty} \Big|\frac{1}{|I_N|}\sum_{n\in I_N} a(n)\Big|;
$$
and for $\ell\geq 2$ (one can show as in \mbox{\cite[Proposition 4.3]{HK09}} that the next limit exists)
$$
\norm{a}_{{\bf I}, \ell}^{2^{\ell}} :=\lim_{H\to \infty} \frac{1}{H}\sum_{h=1}^H \norm{\sigma_ha\cdot \bar{a}}^{2^\ell-1}_{{\bf I}, \ell-1}
$$
where $\sigma_h$ is the shift transformation defined by $(\sigma_ha)(n):=a(n+h)$.
\item If $(a(n))$ is a bounded sequence we let
$$
\norm{a}_{U_\ell({\mathbb N})}:=\sup_{{\bf I}}\norm{a}_{{\bf I}, \ell}
$$
where the sup is taken over all sequences of intervals ${\bf I}$ with lengths tending to infinity along which
the sequence $(a(n))$ is distributed
regularly.
\end{enumerate}
\end{definition}
An application of Lemma~\ref{L:VDC}
shows that $\norm{a}_{{\bf I}, 1}$, as defined here, is smaller than the corresponding quantity
defined in \cite{HK09} (they can be different though). Furthermore, the inductive formula is identical in both cases
(see \cite[Proposition 4.4]{HK09}), hence $\norm{\cdot}_{U_\ell({\mathbb N})}$, as defined here, is a seminorm that is smaller than the corresponding seminorm
defined in \cite{HK09}. In fact, it can be shown that the two seminorms coincide but we will not need this.
Using the main structural result in \cite{HK05}, B.~Host and B.~Kra proved an inverse theorem
that will be a key ingredient in the proof of Theorem~\ref{T:2}. We state a slight variant of it next (\cite[Theorem 2.16]{HK09}
gives a stronger lower bound but it does not allow to assume that $\norm{b}_\infty\leq 1$). Its proof
amounts to a simple modification of the argument given in \cite[Theorem 2.16]{HK09}; we give the details for completeness.
\begin{theorem}[\mbox{\cite[Theorem 2.16]{HK09}}]\label{T:inverse}
Let $a\colon {\mathbb N}\to {\mathbb C}$ be a sequence of complex numbers with $\norm{a}_\infty\leq 1$ and $\ell\in {\mathbb N}$. Then for every $\varepsilon>0$ there exists an $(\ell-1)$-step nilsequence $(b(n))$ with $\norm{b}_\infty\leq 1$
such that
$$
\limsup_{N-M\to \infty}\Big|\frac{1}{N-M}\sum_{n=M}^{N-1} a(n)b(n)\Big|\geq \norm{a}_{U_{\ell}({\mathbb N})}^{2^\ell}-\varepsilon.
$$
\end{theorem}
\begin{remark}
It is crucial that the seminorms were defined using uniform and not standard Ces\`aro averages as in the latter case it is shown in \mbox{\cite[Paragraph 2.4.3]{HK09}} that the corresponding inverse theorem fails.
For standard Ces\`aro averages
a finitary inverse theorem was proved in~\cite{GTZ12c} but it is not clear whether it has an infinitary variant that is useful for our purposes.
\end{remark}
\begin{proof}
We refer the reader to \cite{HK09} for notation used in this argument. In what follows
we assume that the seminorms
$\norm{a}_{{\bf I}, \ell}$ are defined as in \cite{HK09}.
Let $0<\varepsilon<1$. By \cite[Proposition 6.2]{HK09} there exists a sequence of intervals ${\bf I}=(I_N)$ with lengths tending to infinity and an $(\ell-1)$-step nilsequence $(c(n))$ of the form $c(n)=F(g^n\Gamma)$, where $F$ is a continuous function on an $(\ell-1)$-step nilmanifold
$X=G/\Gamma$ and $g\in G$ is an element that acts ergodically on $X$, such that the sequences $a-c$ and $a$ satisfy property $\mathcal{P}(\ell)$ on ${\bf I}$ and moreover we have the estimates
\begin{equation}\label{E:2est}
\norm{a-c}_{{\bf I}, \ell}\leq \varepsilon, \quad \norm{a}_{{\bf I}, \ell}\geq \norm{a}_{U_\ell({\mathbb N})}-\varepsilon.
\end{equation}
Furthermore,
we have $\norm{F}_\infty\leq 1$, this is because in the proof of \cite[Proposition 6.2]{HK09} the function $F$ is defined as a conditional expectation of a function bounded by $1$. We let $b(n):=H(g^n\Gamma)$, where $H:=\mathcal{D}_\ell F$, and check that the asserted properties are satisfied.
First note that $(b(n))$ is an $(\ell-1)$-step nilsequence and since $\norm{F}_\infty\leq 1$ we have $\norm{H}_\infty\leq 1$, hence $\norm{b}_\infty\leq 1$. Furthermore, by \cite[Corollary 5.3]{HK09} we have $H\in C(X)$, hence $F\cdot H\in C(X)$, and since $g$ acts ergodically on $X$ we have
$$
\lim_{N\to \infty}\frac{1}{|I_N|}\sum_{n\in I_N} c(n)b(n)=
\int F\cdot H\, dm_X=
\norm{F}_\ell^{2^\ell}=\norm{c}_{{\bf I}, \ell}^{2^\ell}
$$
where we used the identity $\int F\cdot \mathcal{D}_\ell F\, dm_X=\norm{F}_\ell^{2^\ell}$ and
\cite[Corollary 3.11]{HK09} to justify the last two identities.
By \eqref{E:2est} and the triangle inequality this is greater or equal than
$$
(\norm{a}_{{\bf I}, \ell}-\varepsilon)^\ell\geq
(\norm{a}_{U_\ell({\mathbb N})}-2\varepsilon)^\ell
\geq \norm{a}_{U_\ell({\mathbb N})}^\ell-k_\ell\varepsilon
$$
for some positive integer $k_\ell$.
On the other hand, by \cite[Theorem 2.13]{HK09} we have
$$
\limsup_{N\to \infty}\Big|\frac{1}{|I_N|}\sum_{n\in I_N}(a(n)-c(n))b(n) \Big|\leq
\norm{a-c}_{{\bf I}, \ell} \norm{b}^*_{\ell}\leq \varepsilon
$$
where we used \eqref{E:2est} and that $\norm{b}^*_{\ell}=\norm{\mathcal{D}_\ell F}_\ell^*=\norm{F}_\ell^{2^\ell-1}\leq 1$
(the second identity follows from \cite[Equation (14)]{HK09}). Combining the previous bounds we get the asserted result.
\end{proof}
\subsection{Proof of Theorem~\ref{T:2}}
Let $\ell\in {\mathbb N}$ and $(a(n))$ be an $\ell$-regular and $\ell$-anti-uniform sequence with $\norm{a}_\infty\leq 1$.
We first remark that the limit
\begin{equation}\label{E:exists}
\lim_{N-M\to \infty} \frac{1}{N-M}\sum_{n=M}^{N-1} |a(n)|^2 \quad \text{ exists}.
\end{equation}
This follows from our anti-uniformity assumption
and \cite[Theorem 2.19]{HK09} (it applies since $(a(n))$ is $\ell$-regular) which states
that for every $\epsilon>0$ we have a decomposition $a=a_1+a_2$ where $a_1$ is an $(\ell-1)$-step nilsequence
and $\norm{a_2}_{U_\ell({\mathbb N})}\leq \epsilon$. Writing $|a(n)|^2=a\bar{a}_1+a\bar{a}_2$ one checks the asserted convergence at once.
We let
$$
Y:=\Big\{(\psi(n)) \colon \psi \text{ is an } (\ell-1)\text{-step nilsequence}\Big\}
$$
and
$$
X:=\text{span}\{{Y,a\}}.
$$
On $X\times X$ we define the bilinear form
$$
\langle f, g \rangle:=\lim_{N-M\to \infty} \frac{1}{N-M}\sum_{n=M}^{N-1} f(n)\overline{g}(n).
$$
Note that the limit exists for $f,g\in X$. This is the case if $f$ or $g$ is equal to $a$ because of our regularity assumption and \eqref{E:exists}, and when both $f$ and $g$ are in $Y$ because limits of
uniform Ces\`aro averages of nilsequences exist \cite{L05, Les91}. This bilinear form induces the seminorm
$$
\norm{f}_2:=\sqrt{\langle f, f \rangle}.
$$
This is the restriction on $X$ of the seminorm \eqref{E:seminorm} defined on $\ell^\infty({\mathbb N})$.
Let $\varepsilon>0$.
There exists $y_0\in Y$ such that
\begin{equation}\label{E:distance}
\norm{a-y_0}_2^2\leq d^2+\delta^2
\end{equation}
where
\begin{equation}\label{E:ded}
d:=\inf\{ \norm{a-y}_2\colon y\in Y\}, \quad \delta:=(\varepsilon/(4C))^{2^\ell},
\end{equation}
and $C:=C(\ell,a)$ is the constant determined by our $\ell$-anti-uniformity assumption on $a$. We can assume that $C\geq 1$.
Furthermore, we can assume without loss of generality that
\begin{equation}\label{E:y0}
\norm{y_0}_\infty\leq 1.
\end{equation}
Indeed, let $y_0:=(F(g^n\Gamma))$ where $X=G/\Gamma$ is a nilmanifold, $g\in G$, and $F\in C(X)$.
Then the sequence $\tilde{y}_0:=(\tilde{F}(g^n\Gamma))$, where $\tilde{F}:=F\cdot \mathbf{1}_{|F|\leq 1}+e^{2\pi i \arg(F)}\cdot \mathbf{1}_{|F|\geq 1} \in C(X)$, is a nilsequence, $\norm{\tilde{y}_0}_\infty\leq 1$, and
as $\norm{a}_\infty\leq 1$ we get that $|a(n)-\tilde{y}_0(n)|\leq |a(n)-y_0(n)|$ for every $n\in {\mathbb N}$, hence $\norm{a-\tilde{y}_0}_2\leq \norm{a-y_0}_2$.
It follows from \eqref{E:distance} that for every $y\in Y$ we have
$$
-\delta^2\leq \norm{a-(y_0+\delta y)}_2^2-\norm{a-y_0}_2^2=-2\delta\text{Re}(\langle a-y_0,y\rangle)+\delta^2\norm{y}_2^2.
$$
Hence,
$$
\text{Re}(\langle a-y_0,y\rangle) \leq \delta \ \text{ for every } y\in Y \text{ with } \norm{y}_2\leq 1.
$$
Inserting $-y$ and $\pm i y$ in place of $y$ we deduce that
\begin{equation}\label{E:ay0}
\sup_{y \in Y\colon \norm{y}_2\leq 1} |\langle a-y_0,y\rangle| \leq 2\delta.
\end{equation}
Since the set $\{y\in Y\colon \norm{y}_2\leq 1\}$ contains all $(\ell-1)$-step nilsequences that are bounded by $1$, we deduce from Theorem~\ref{T:inverse}
that
\begin{equation}\label{E:uniform}
\norm{a-y_0}_{U_{\ell}({\mathbb N})}\leq (2\delta)^{2^{-\ell}}.
\end{equation}
We let
$$
a_{st}:=y_0, \quad a_{er}:=a-y_0.
$$
Then
$$
a=a_{st}+a_{er}
$$
and $(a_{st}(n))$ is an $(\ell-1)$-step nilsequence with $\norm{a_{st}}_\infty\leq 1$ by \eqref{E:y0}.
Since $a$ is $\ell$-anti-uniform we get using \eqref{E:uniform} and the definition of $\delta$ in \eqref{E:ded} that
$$
|\langle a,a_{er}\rangle|\leq C\norm{a_{er}}_{U_{\ell}({\mathbb N})}\leq \varepsilon/2.
$$
Furthermore, \eqref{E:ay0} gives
$$
|\langle a_{st},a_{er}\rangle|\leq \varepsilon/2.
$$
Combining the last two estimates we deduce that
$$
\norm{a_{er}}_2^2 =\langle a_{er},a_{er}\rangle\leq
|\langle a,a_{er}\rangle|+ |\langle a_{st},a_{er}\rangle|
\leq \varepsilon.
$$
This completes the proof of Theorem~\ref{T:2}.
\subsection{Proof of Theorem~\ref{T:1}} In view of Theorem~\ref{T:2}, it suffices to prove that for every $\ell\in {\mathbb N}$ the sequence
$a\colon {\mathbb N}\to {\mathbb C}$ defined by
\begin{equation}\label{E:an}
a(n):=\int T_1^{n}f_1\cdot \ldots \cdot T_\ell^{n}f_\ell\ d\mu, \quad n\in {\mathbb N},
\end{equation}
is $\ell$-anti-uniform and $\ell$-regular.
\subsubsection{Anti-uniformity} Throughout, we can and will assume that $\norm{f_i}_\infty\leq 1$ for $i=1,\ldots, \ell$.
The $\ell$-anti-uniformity follows by successive applications of the following Hilbert space variant of van der Corput's estimate
(for a proof see \cite{Be87a}).
\begin{lemma}\label{L:VDC}
Let $(v_n)$ be a bounded sequence of vectors in an inner product
space and $(I_N)$ be a sequence of intervals with lengths tending to infinity.
Then
$$
\limsup_{N\to\infty}
\norm{\frac{1}{|I_N|}\sum_{n\in I_N} v_n}^2\leq 4 \ \!
\limsup_{H\to\infty} \frac{1}{H}\sum_{h=1}^H
\limsup_{N\to\infty}\Big|
\frac{1}{|I_N|}\sum_{n\in I_N} \langle v_{n+h},v_{n}\rangle \Big|.
$$
\end{lemma}
It suffices to show that for every $\ell\in {\mathbb N}$ and every sequence of intervals ${\bf I}:=(I_N)$ with lengths tending to infinity,
any sequence $(a(n))$ given by \eqref{E:an} satisfies the estimate
$$
\limsup_{N\to
\infty} \Big|\frac{1}{|I_N|}\sum_{n\in I_N} a(n)b(n)\Big|\leq 4 \norm{b}_{U_{\ell}({\mathbb N})}
$$
for every $b\in \ell^{\infty}({\mathbb N})$. Using a diagonal argument and passing to a subsequence of $(I_N)$
(if necessary) we can and will assume that the sequence $(b(n))$ is distributed regularly along the sequence
${\bf I}$.
It suffices to establish that for any sequence $(a(n))$ as in \eqref{E:an} which is bounded by $1$ and any $b\in \ell^\infty({\mathbb N})$ which is distributed regularly along a sequence of intervals ${\bf I}$, we have
\begin{equation}\label{E:anbn}
\limsup_{N\to
\infty} \Big|\frac{1}{|I_N|}\sum_{n\in I_N} a(n)b(n)\Big|\leq 4 \norm{b}_{{\bf I}, \ell}.
\end{equation}
We prove this by induction on $\ell$. For $\ell=1$ the result holds trivially.
Suppose that $\ell\geq 2$ and the statement holds for $\ell-1$.
We compose with $T_\ell^{-n}$, use the Cauchy-Schwarz inequality, and then Lemma~\ref{L:VDC} (on the space $L^2(\mu)$) for the sequence
$$
v_n := b(n)\cdot \tilde{T}_1^nf_1\cdot \tilde{T}_2^nf_2\cdot\ldots\cdot \tilde{T}_{\ell-1}^nf_{\ell-1}, \quad n\in {\mathbb N},
$$
where $\tilde{T}_i:=T_iT_\ell^{-1}$ for $i=1,\ldots,\ell-1$.
We deduce that the square of the left hand side in \eqref{E:anbn} is bounded by
\begin{equation}\label{E:u_n}
\limsup_{N\to\infty}\Bigl\lVert \frac{1}{|I_N|}\sum_{n\in I_N}
v_{n}\Bigr\lVert_{L^2(\mu)}^2\leq 4 \limsup_{H\to
\infty}\frac{1}{H}\sum_{h=1}^{H} \limsup_{N\to\infty}\Big|
\frac{1}{|I_N|}\sum_{n\in I_N} \langle{ v_{n+h}, v_n \rangle }\Big|.
\end{equation}
A simple computation gives that
$$
\frac{1}{|I_N|}\sum_{n\in I_N} \langle{ v_{n+h}, v_n \rangle }
= \frac{1}{|I_N|}\sum_{n\in I_N} b(n+h)\cdot \bar{b}(n) \int
\tilde{T}_1^n \tilde{f}_{1,h} \cdot\ldots\cdot \tilde{T}_{\ell-1}^n
\tilde{f}_{\ell-1,h}\,d\mu
$$
where
$\tilde{f}_{j,h}=\tilde{T}_{j}^hf_{j}\cdot\bar{f}_{j}$ for $j=1,\ldots, \ell-1$.
Note that the maps
$\tilde{T}_1, \ldots, \tilde{T}_{\ell-1}$ commute, for $h\in {\mathbb N}$ the sequence $(b(n+h) \bar{b}(n))$ is distributed regularly along ${\bf I}$,
and $\norm{\tilde{f}_{j,h}}_\infty\leq 1$ for $j=1,\ldots, \ell-1$. Using the induction hypothesis and the defining property of the seminorms we can bound the right hand
side in \eqref{E:u_n} by $16$ times
$$
\lim_{H\to \infty} \frac{1}{H}\sum_{h=1}^{H}\norm
{\sigma_hb\cdot b}_{{\bf I}, \ell-1 } \leq \lim_{H\to \infty}
\Big(\frac{1}{H} \sum_{h=1}^{H}\norm
{\sigma_hb\cdot b}_{{\bf I}, \ell-1 }^{2^{\ell-1}}\Big)^{1/2^{\ell-1}} =\norm {b}_{{\bf I}, \ell}^2
$$
where $(\sigma_h b)(n):=b(n+h)$.
Taking square roots we get the asserted estimate.
\subsubsection{Regularity} Let $\ell\in {\mathbb N}$.
To prove that $(a(n))$ is $\ell$-regular we will use
a known mean convergence result for multiple ergodic averages and Proposition~\ref{P:nilkey} below.
We start with the following
result of B.~Green and T.~Tao:
\begin{lemma}[\mbox{\cite[Lemma~14.2]{GT08}}]\label{L:nilkey}
For $\ell\in {\mathbb N}$ let $X=G/\Gamma$ be an $(\ell-1)$-step nilmanifold. Then there exists
a continuous map $P\colon X^{\ell}\to X$ such that
\begin{equation}\label{E:asd}
P(hg\Gamma, h^2g\Gamma, \ldots, h^{\ell} g\Gamma)=g\Gamma, \quad \text{ for every } g,h\in G.
\end{equation}
\end{lemma}
The result in \mbox{\cite[Lemma~14.2]{GT08}} gives $P(g\Gamma,hg\Gamma, h^2g\Gamma, \ldots, h^{\ell-1} g\Gamma)=h^{\ell} g\Gamma$.
Inserting $h^{-\ell}g $ in place of $g$, then $h^{-1}$ in place of $h$, and rearranging coordinates, we get \eqref{E:asd}.
\begin{proposition}\label{P:nilkey}
For $\ell\in {\mathbb N}$ let $(\psi(n))$ be an $(\ell-1)$-step nilsequence.
Then for every $\varepsilon>0$ there exists a
system $(X,\mathcal{X},\mu,T)$ and functions $f_1,\ldots, f_{\ell}\in L^\infty(\mu)$,
such that the
sequence $(b(n))$, defined by
\begin{equation}\label{E:bn}
b(n):=\int T^{k_1n}f_1 \cdot \ldots \cdot T^{k_{\ell} n}f_{\ell}\ d\mu, \quad n\in {\mathbb N},
\end{equation}
where $k_i:=\ell!/i$ for $i=1,\ldots, \ell$, satisfies
$$
\norm{\psi-b}_\infty\leq \varepsilon.
$$
\end{proposition}
\begin{remarks}
To prove a variant of this result that uses the integers $1, \ldots, \ell$ in place of $k_1,\ldots, k_\ell$,
one would have to prove a non-trivial variant of Lemma~\ref{L:nilkey} that establishes in place of \eqref{E:asd} the identity
$P(h^{k_1}g\Gamma, h^{k_2}g\Gamma, \ldots, h^{k_\ell} g\Gamma)=g\Gamma
$ for every $g,h\in G$.
Combining \cite[Theorem A (ii)]{BL07} with Proposition~\ref{P:nilkey} one deduces that for every bounded generalized polynomial $p\colon {\mathbb N}\to {\mathbb R}$ (see definition in \cite{BL07}) the sequences $(p(n))$
and $(e^{ip(n)})$ can be approximated arbitrarily well in $\norm{\cdot}_2$ by a sequence of the form \eqref{E:bn}.
\end{remarks}
\begin{proof}
Let $\varepsilon>0$ and
$$
\psi(n):=F(g^n\Gamma)
$$ where $F\in C(X)$, $X=G/\Gamma$ is an $(\ell-1)$-step nilmanifold, and $g\in G$.
By \cite[Paragraph 1.11]{L05} we have that $X$ is isomorphic to a subnilmanifold of a nilmanifold $\tilde{X}=\tilde{G}/\tilde{\Gamma}$, where $\tilde{G}$ is a connected and simply connected $(\ell-1)$-step nilpotent Lie group, $\tilde{\Gamma}$ is a discrete cocompact subgroup of $\tilde{G}$, and all elements of $G$ are represented in $\tilde{G}$. Then $\psi(n)=\tilde{F}(\tilde{b}^n\tilde{\Gamma})$ for some
$\tilde{b}\in \tilde{G}$ and $\tilde{F}\in C(\tilde{X})$. Hence, in what follows we can and will assume that the group $G$ is connected.
Using Lemma~\ref{L:nilkey} with $g^n$ in place of $g$ and $h:=g^m$, $m,n\in {\mathbb N}$, we get that there exists a continuous map $P\colon X^{\ell}\to X$ such that
\begin{equation}\label{E:gn}
g^n\Gamma=P(g^{m+n}\Gamma, g^{2m+n}\Gamma,\ldots, g^{\ell m+n}\Gamma) \quad \text{ for every } m,n\in {\mathbb N}.
\end{equation}
Let $g_0\in G$ be such that $g_0^{\ell!}=g$ (such a $g_0$ exists since $G$ is connected, hence divisible) and for $i=1,\ldots, \ell$ let $g_i:=g_0^i$.
Applying \eqref{E:gn} with $g_0$ in place of $g$ and $\ell! n$ (a multiple of $n$ is needed that is divisible by all the coefficients of $m$ that appear in \eqref{E:gn}) in place of $n$ we get
$$
\psi(n)=F(g_0^{\ell! n}\Gamma)=\tilde{F}(g_1^{m+k_1n}\Gamma, g_2^{m+k_2n}\Gamma,\ldots, g_{\ell}^{ m+k_{\ell} n}\Gamma) \quad \text{ for every } m,n\in {\mathbb N},
$$
where $\tilde{F}:=F\circ P\in C(X^{\ell})$.
Averaging over $m\in {\mathbb N}$ we get
$$
\psi(n)=\lim_{M\to \infty} \frac{1}{M}\sum_{m=1}^M \tilde{F}(g_1^{m+k_1n}\Gamma, g_2^{m+k_2n}\Gamma,\ldots, g_{\ell}^{ m+k_{\ell} n}\Gamma) \quad \text{ for every } n\in {\mathbb N}.
$$
Since $\tilde{F}$ can be approximated uniformly
by linear combinations of functions of the form $\tilde{f}_1\otimes\cdots\otimes \tilde{f}_{\ell}$, where for $i=1,\ldots, \ell$ the function $\tilde{f}_i \in C(X^\ell)$ depends on the coordinate $x_i$ only, we get that $(\psi(n))$ can be approximated in the $\norm{\cdot}_\infty$ norm within $\varepsilon$ by a finite linear combination of sequences $(a(n))$ of the form
\begin{equation}\label{E:limform}
a(n):=\lim_{M\to \infty} \frac{1}{M}\sum_{m=1}^M \tilde{f}_1(\tilde{g}^{m+k_1n}\tilde{\Gamma})\cdot \tilde{f}_2(\tilde{g}^{m+k_2n}\tilde{\Gamma})\cdot \ldots \cdot \tilde{f}_{\ell}(\tilde{g}^{ m+k_{\ell} n}\tilde{\Gamma}), \quad n\in {\mathbb N},
\end{equation}
where $\tilde{X}:=X^{\ell}$ , $\tilde{\Gamma}:=\Gamma\times\cdots \times \Gamma$, $\tilde{f}_i\in C(\tilde{X})$, and $\tilde{g}:=(g_1, \ldots, g_{\ell})$.
It is known (see \cite{L05} for example) that the limit in \eqref{E:limform} is equal to
$$
\int_{\tilde{Y}} \tilde{f}_1(\tilde{g}^{k_1n}\tilde{y})\cdot \tilde{f}_2(\tilde{g}^{k_2n}\tilde{y})\cdot \ldots \cdot \tilde{f}_{\ell}(\tilde{g}^{k_{\ell} n}\tilde{y}) \, dm_{\tilde{Y}}, \quad n\in {\mathbb N},
$$
where $\tilde{Y}$ is the subnilmanifold of $\tilde{X}$ defined by the closure of the set
$\{\tilde{g}^m \tilde{\Gamma}\colon m\in {\mathbb N}\}$. This proves that the sequence $(a(n))$ has the form \eqref{E:bn}.
Since finite linear combinations of sequences of the form
\eqref{E:bn} still have the form \eqref{E:bn} (see Section~\ref{SS:Applications}) the proof is complete.
\end{proof}
We are now ready to verify that if $(a(n))$ is as in \eqref{E:an}, then it is $\ell$-regular for every $\ell\in {\mathbb N}$.
By Proposition~\ref{P:nilkey}, in order to check that the limit
$\lim_{N-M\to \infty} \frac{1}{N-M}\sum_{n=M}^{N-1}a(n)\psi(n) $ exists for every $(\ell-1)$-step nilsequence $(\psi(n))$,
it suffices to check that the limit
\begin{equation}\label{E:abnn}
\lim_{N-M\to \infty} \frac{1}{N-M}\sum_{n=M}^{N-1} a(n)b(n)
\end{equation}
exists for every sequence
$(b(n))$ of the form $\int S^{k_1n}g_1\cdot \ldots \cdot S^{k_{\ell} n}g_{\ell}\ d\nu$ , where $k_1,\ldots, k_\ell\in {\mathbb N}$, $(Y,\mathcal{Y},\nu, S)$ is a system, and $g_1,\ldots, g_{\ell}\in L^\infty(\nu)$. This follows from the mean convergence result of T.~Austin~\cite{Au09} (which strengthens the convergence result of T.~Tao
\cite{Ta08} to uniform averages)
applied to
the transformations
$\tilde{T}_i:=T_i\times S^{k_i}$ acting on $X\times Y$ with the measure $\tilde{\mu}:=\mu\times \nu$ and the functions $\tilde{f_i}:=f_i\otimes g_i \in L^\infty(\tilde{\mu})$, $i=1,\ldots, \ell$.
\subsection{Proof of Theorem~\ref{T:1'}}
Modulo a known convergence result of M.~Walsh~\cite{W12}
the argument is similar to the one used to prove Theorem~\ref{T:1}, we explain the minor modifications needed next.
To verify $k$-anti-uniformity for some $k\in {\mathbb N}$ that depends only on $\ell, m$ and the maximum degree of the polynomials $p_{i,j}$,
one has to make successive uses of Lemma~\ref{L:VDC}
and apply
an inductive argument, often called PET induction, introduced by V.~Bergelson in \cite{Be87a}. The details are very similar to those in the proof of \mbox{\cite[Lemma 3.5]{FrHK11}} and so we omit them.
To verify regularity, we can argue as in the case of linear iterates, using the convergence result of M.~Walsh \cite{W12} for averages of expressions of the form \eqref{E:complicated}. At the very last step one needs to verify that if $(a(n))$ is as in \eqref{E:complicated}, then the limit
\eqref{E:abnn}
exists for every sequence
$(b(n))$ of the form $\int S^{k_1n}g_1\cdot \ldots \cdot S^{k_r n}g_r\ d\nu$,
where $r\in {\mathbb N}$ is arbitrary, $k_1,\ldots, k_r\in {\mathbb N}$, $(Y,\mathcal{Y},\nu, S)$ is a system, and $g_1,\ldots, g_r\in L^\infty(\nu)$.
The only change needed is to use Walsh's convergence result for
the $\ell+r$ commuting measure preserving transformations
$T_i\times \text{id}$, $i=1,\ldots, \ell$, and $\text{id}\times S^{k_j}$, $j=1,\ldots, r$, acting on $X\times Y$ with the measure $\tilde{\mu}:=\mu\times \nu$, and the functions $f_i\otimes 1$, $i=1,\ldots, \ell$ and $1\otimes g_j$, $j=1 \ldots, r$. If the polynomial iterates are chosen appropriately, one verifies that $a(n)b(n)$ is also a multiple correlation sequence with polynomial iterates, hence, by Walsh's convergence result \cite{W12}, the limit \eqref{E:abnn} exists.
\subsection{Extension to nilpotent groups} Essentially the same argument can be used when the transformations $T_1,\ldots, T_\ell$ generate a nilpotent group; the only extra difficulty occurs in proving $k$-anti-uniformity for some $k\in {\mathbb N}$ that depends also on the degree of nilpotency of the group generated by $T_1,\ldots, T_\ell$. In this case, the PET induction is somewhat more complicated, but can be handled by modifying the PET induction used in
\mbox{\cite[Lemma 3.5]{FrHK11}} along the lines of the argument used to prove \mbox{\cite[Theorem 4.2]{W12}}.
\subsection{Proof of Theorem~\ref{T:3}}
The inclusion $\overline{\mathcal{A}_{\ell}}\subset \overline{\mathcal{B}_\ell}$
follows from Proposition~\ref{P:nilkey}.
The inclusion $\overline{\mathcal{B}_{\ell}}\subset \overline{\mathcal{C}_\ell}$
is obvious.
The inclusion $\overline{\mathcal{C}_{\ell}}\subset \overline{\mathcal{A}_\ell}$
follows from Theorem~\ref{T:1}.
\subsection{Proof of Theorems~\ref{T:4} and \ref{T:4'}}
The implication $\eqref{E:42}\Rightarrow \eqref{E:41}$ follows from Proposition~\ref{P:nilkey}.
(for Theorem~\ref{T:4'} in order to get property (i) for some fixed $\ell\in {\mathbb N}$ we use property (ii) for $\ell!$).
The implication $\eqref{E:41}\Rightarrow \eqref{E:43}$ follows from Theorems~\ref{T:1} and \ref{T:1'}.
The implication $\eqref{E:43}\Rightarrow \eqref{E:42}$ is obvious.
The same argument applies
for the extensions mentioned after Theorem~\ref{T:4'} related to uniform and weighted Ces\`aro averages.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 3,060
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Why Re-signing Mike Muscala Should be One of the Hawks' Biggest Priorities this Summer
Muscala heads into the summer as an unrestricted free agent...
By Graham Chapple May 12, 2017, 10:00am EDT
Share All sharing options for: Why Re-signing Mike Muscala Should be One of the Hawks' Biggest Priorities this Summer
Mike Muscala's NBA journey has certainly been an interesting one up until this point...
From various assignments to the NBA D-League to being buried at the end of the bench, from making surprise contributions in the 2015 playoffs to being buried at the end of the bench again next season, from becoming the Hawks' primary back-up big to finding himself receiving DNP-CD's after the acquisition of Ersan llysaova to seeing valuable time in the NBA playoffs…
It really has been a roller coaster ride for the man affectionately known as "Moose" but while his role hasn't always consistent there was alway one thing that was: Mike Muscala has improved every season of his career.
After the best season of his career, Muscala now reaches an interesting crossroads. For the first time in his young NBA career, he can test the market as an unrestricted free agent.
Moose — among league circles — is one of those players who's considered to be a 'sneaky-good' type of player: a player who is good but not a lot of people really know it. The general fan certainly wouldn't assume so, but those who know, know. He's a player that'll be on a lot of team's radars this summer, his name amongst those that appeared on the Orlando Magic's infamous whiteboard...
The Hawks have quite a number of players hitting the market this summer: Paul Millsap, Thabo Sefolosha, Tim Hardaway Jr. (restricted), Ersan Ilyasova, Mike Muscala, José Calderón and Kris Humphries.
Paul Millsap is the Hawks' biggest priority this summer (and rightly so) but Mike Muscala should be the Hawks' second biggest priority and ahead of the likes of THJ, Ilyasova and Calderón.
Let me make the case for why this should be the case...
He's a modern stretch four/five
One of Muscala's strengths is shooting, he has a great shooting touch for someone his size at 6"11. His ability to hit the three-pointer/stretch the defense instantly renders him a very valuable commodity in this league.
Just as a visual example, Gorgui Dieng doesn't really want to cover Muscala in the corner on this possession and Moose makes him pay for it.
His ability to shoot the three-pointer makes life uncomfortable for bigs like Dieng, players like that with his size don't really want to go out and guard that shot — it leaves the paint more vulnerable.
Muscala has worked hard at improving his stroke and it's paid off. In fact, fewer people — at his position (power forward/center) — shoot the ball better than Muscala.
Amongst centers who played 60 or more games and attempted at least one three-pointer per game, only Pau Gasol (53.8%) and Jason Smith (47.4%) shot better percentages from behind the arc than Mike Muscala (41.8%). That's a higher percentage than the likes of some the league's most notable three-point shooters from the center spot in Marc Gasol, Al Horford and DeMarcus Cousins, though those particular three players average over 3.6 three-point attempts per game compared to Muscala's 1.7 so...take that as you will.
From the forward spot, Muscala is still amongst the top company from three. Under the same criteria as above, only Otto Porter (43.4%), Joe Ingles (44.1%) and Jason Smith (47.4%) shot higher percentages from three than Muscala this season. Porter and Ingles are small forwards, so the only power forward/center to shoot a higher percentage from three was Jason Smith.
I really want to hammer this home: that stat is not pulled from players only in the Southeast Division or the Eastern Conference...it's the entire league. Mike Muscala is right up there as one of the best three-point shooter from the power forward/center spot... Believe it.
Muscala's ability to play both the 4 and 5 gives coach Bud freedom to experiment with different lineups. His versatility means that coach Bud is not restricted to playing him in certain lineups like he has to with Dwight Howard or Tiago Splitter in the past.
Players with Muscala's skill-set at his size aren't common in this league and they're players you want to bring to your club, not let go...
Muscala plays an important offensive role
With Al Horford and Mike Scott no longer with the team, Muscala's ability to stretch the floor has become extremely important for the Hawks. It's become important because it meant (at least before the acquisition of Ersan Ilyasova) that Moose was the only big — other than Paul Millsap — who could stretch the floor.
In the past, Bud has been spoiled for choice when it comes to bigs who can stretch the floor, the likes of Millsap, Horford, Scott, and Pero Antić all available at Bud's disposal as floor spacers. However, with different roster moves/decisions made, it's just been Millsap and Muscala who have been able to stretch the floor from the forward/center spots.
At times, Bud likes to play both Millsap and Muscala together, and the two-man numbers for that lineup are pretty decent: plus-68 in 472 minutes. Were it not for their respective injuries, those numbers would probably be higher. When those two are on the floor, there's good spacing, good pace, good defense and good ball movement. In short, the Hawks look the most like their older selves when Millsap and Muscala are on the floor.
Bud — as we've seen more of as the season has come to an end — has also leant toward sitting Dwight in fourth quarters/down the stretch in the fourth quarters in favor of spacing the floor, valuing Muscala's ability to space the floor (as well as his ability to get up and down the court much quicker than Dwight) which opens opportunities up for other players.
"When we need to score and we need to catch up, we need to play a little more of a spread offense," said Budenholzer after Atlanta's Game 6 loss against the Wizards when asked about the decision not to play Dwight Howard in the fourth quarter of a must win game.
Though Muscala didn't play much in that particular fourth quarter, it's still his type of skill-set that Bud values and a skill-set he can depend upon when he feels the Hawks need to play with spacing and shooting — similar to how they played when Al Horford was with the team
"He can play the 4 or the 5," Budenholzer said of Muscala in December. "He can score in the paint. He can space the court, and shoot and make threes. He can pass as well. He protects the rim. He's a really good player."
Bud relied heavily on Muscala in April 9th's home game against the Cleveland Cavaliers, a game the Hawks trailed by 26 points heading into the fourth quarter. Muscala played 11:59 (according to NBA.com) of that fourth quarter.
With Muscala's ability to make passes like this:
And stretch the floor and hit huge shots like this:
And this (in overtime):
...The Hawks were able to pull off one of the most unlikeliest comebacks in NBA history. Without Muscala's shooting and the spacing/opportunities for others he provides (because of his spacing), the Hawks probably don't make this comeback.
When Al Horford and Mike Scott were with the team last year (and their ability to stretch the floor), the Hawks didn't urgently need Moose's skill-set and that was reflected in Muscala only playing 9.4 minutes a game in 60 games. However, with those two players no longer in the mix and with Dwight Howard — a completely different type of center than Horford — now in the fold in Atlanta, and with Muscala's continued development, Moose's skill-set is as valuable for the Hawks now than it ever has been. It's one the Hawks need to keep around.
A versatile defender
When we looked at some of the shooting company Muscala was hanging with from the forward/center spot there was one name that Muscala was tied with: Jason Smith. Offensively, the two do a great job stretching the floor while also being able to score inside. Defensively, however, it's a very different story...
In the playoffs, the Hawks looked to exploit Smith on the offensive end — especially with Millsap — because Smith isn't a great defender. You could argue that Smith is a better shooter than Muscala but Moose is certainly the better defender.
Muscala's ability to stretch the floor is what most people talk about when it comes to his overall game, but his defense is no joke either.
One of the greatest aspects of Muscala's defensive game is his vertical defense, his ability to go straight up without fouling.
On opening night, Kelly Oubre operates in transitio, and as he rises to lay the ball up he finds there's a wall in his way: Mike Muscala. Moose does a good job going straight up (not bringing those arms down) and deflects the ball out of bounds.
Against the much more experienced David West in transition, Moose proves to be the last line of defense between West and an easy layup. He sticks with the play and keeps his arms straight up and challenges the shot cleanly. As a result, West's shot misses.
I know what you may be thinking at this point. "Yes, yes this is all well and good, but Kelly Oubre is not exactly an offensive juggernaut and David West is 1,000 years old" and that's a fair point, but Muscala is capable of effectively challenging shots from much more advanced and athletic offensive players. Like, say, James Harden. Yeah, bet you weren't expecting that, but it's true.
Moose's verticality makes life for Harden difficult on this possession and his shot rolls off the rim.
An excellent challenge against one of the league's best offensive players.
Muscala is also an underrated shot blocker and does a good job in 'help' situations.
After doing a good job raising his arms and forcing Greg Monroe to pass the ball out of the post, Jabari Parker attacks Taurean Prince off of the dribble and gets a shot up but Muscala — arriving as the help defender — is on hand to block the shot.
This isn't a development of his game that has only emerged in the last year or two, Muscala could always protect the rim when called upon. In his four years in college with Bucknell, he averaged 2 blocks per game. Shot blocking has been a facet of Muscala's game at every level, even as a young NBA player.
From the 2014-15 season:
Again, Muscala goes straight up and doesn't just challenge Waiters' shot but blocks it outright.
Muscala is a very capable shot blocker, both in help situations and in one-on-one situations.
Part of what makes Moose a good defender are his physical attributes. He's quick on his feet which helps him get back in transition in a hurry, he can move his feet well defensively and he utilizes his size and length to challenge shots at the rim.
The only area you could say Muscala is lacking is, perhaps, his strength which is sometimes taken advantage of in the post by some of the stronger bigs in the league. He hasn't got the body of a Dwight Howard or Hassan Whiteside, sure, but he's not exactly a twig either. He can absorb contact.
(Moose also showcasing those quick feet which helps keep Jabari in front of him on this possession)
Alright, so Jabari might not be the best example of showcasing Muscala's ability to absorb contact, I understand. But how about this example with Hassan Whiteside? Sure, he's is knocked back a bit but Moose takes the hit and the shot from Whiteside misses.
Though he's no loose root in a sapling, Muscala is targeting on getting stronger in the off season.
"I think strength for me, defensively, it can help a lot and that is going to be a focus for me too," Muscala said recently.
We've talked about offense and defense separately, but Moose is also capable of combining the two. Off of the block, Muscala rebounds the ball, passes it to Jeff Teague, runs the floor, gets the ball back, takes Nerlens Noel off of the dribble and scores at the rim plus the foul.
A combination of speed, quickness, technique, size and length help make Muscala a versatile defender who can just about everything — at his position — defensively. His defense is part of the reason why Bud can call upon him when the Hawks go away from Dwight because, unlike some, Muscala isn't a give-give back type of player...
He's one of the Hawks' own
'Hawks University' is a term used by some to describe how the Hawks and their developmental coaches take a player who has usually been disregarded/underrated by other teams, or a player who has maybe been forgotten/given up on, and develop him into a serviceable NBA player.
In the past the Hawks have turned DeMarre Carroll, Kent Bazemore and more recently Tim Hardaway Jr. into serviceable NBA players and they have earned/are in line for big-money contracts as a result of enrolling at 'Hawks University' (as well as, of course, putting in huge work themselves).
Mike Muscala is another such example and everyone who has watched the Hawks over the years can see how far Mike has come as a player thanks to all the hard work both from him and the coaching staff. You can see the progress he's made on both sides of the floor.
In fact, they've done such a good job that I can honestly say that Mike Muscala is the one player (other than Paul Millsap) who runs the Mike Budenholzer system almost every time he's on the floor.
When I say the Mike Budenholzer system, I'm talking about ball movement, man movement, cutting, good shot selection etc. And Muscala does all of these things.
He moves the ball, moves off the ball (sets screens off and on the ball which keeps offense moving) never holds the ball for too long, never stays in one spot for too long, takes good shots and makes smart basketball decisions. And he's doing these things every time he's on the floor.
When you watch a team for long enough, you can see who is and isn't running what the coach wants him to run. Dennis Schröder is an example of a player who doesn't always (and that's being kind) run the Budenholzer offense when he's on the floor. No disrespect to Dennis (he's a very talented player), but he's not running the Hawks system as often as he should, being the point guard of an what's supposed to be a selfless offense and all... I couldn't look at Dennis Schröder and say he's the extension of Mike Budenholzer on the floor. I could with Mike Muscala, I could say "I can see the Mike Budenholzer in him on the floor".
That sounds wild, I understand, but if you've watched this team play you might understand what I mean. Maybe I can explain this another way...
Take a look at this sequence from Muscala in a game against the Minnesota Timberwolves earlier this season and we'll talk afterwards.
Let's go over what happened in that incredible sequence.
Muscala stretched the defense with his perimeter game and hit the open three after Millsap collapsed the defense after a switch onto Ricky Rubio occurred.
On the defensive end, Moose did a good job covering for Dennis Schröder (who was beaten by Rubio) and did a good job challenging Gorgui Dieng at the rim following the pass from Rubio.
Following the miss, Muscala was hustling the other way, beating every other Hawk up the floor.
After relocating behind the three-point line, Mike received the ball.
Instead of forcing something that isn't there, Muscala realised quickly there was nothing on for him and instead of using valuable clock time or chucking up a shot (as Dennis, THJ and Baze are all prone to do in the same situation at times) he gave the ball up quickly and immediately went to to set a screen for the man he passed the ball to. Never standing still, always moving and looking to make something happen whether he has the ball or not.
After receiving the ball back from Dennis, Muscala had the presence of mind to know that Bazemore was open behind the line. Moose could've taken the ball inside and try shoot over the smaller Andrew Wiggins but he didn't. Instead he made the better basketball play and found the open man.
Baze declined to shoot and drove inside.
Muscala helped facilitate the offense by setting a good screen on Wiggins that removed Wiggins from the play.
Bazemore missed the shot over the outstretched Karl-Anthony Towns but Muscala was there to crash the offensive glass and tip the ball home.
Ball movement, man movement, screening, defense, outside shooting, smart decision making, hustle...Mike Muscala did it all in 57 glorious seconds, though in reality it took hundreds of hours of practice and hard work.
Mike Muscala is one of the Hawks' own. It would be terrible for the club (and the fans, he is a fan favorite) if they allowed him to slip away as they did with Edy Tavares.
They say the point guard is the extension of the head coach, but Mike Muscala is the extension of the entire Hawks' coaching staff when he's on the floor. What kind of message would it send if the Hawks allowed him to leave?
Though Muscala has said he would like to stay in Atlanta, the fact of the matter is this: Muscala has earned less than $3 million in his four year career and given his unique skill-set and his overall game, he's in line for a sizeable increase in salary, possibly more than the Hawks might be willing to pay him.
Smart teams know about him and some of those teams — unlike the Hawks — have cap space to splash. Atlanta is probably the best place for Muscala to be at this stage of his career, but will the thought of a lucrative deal lure him away from Atlanta? The Hawks own Muscala's Bird Rights, meaning they can go above the cap to re-sign him, but will other their free agents such as Paul Millsap, Tim Hardaway Jr. and possibly Ersan Ilysava distract the Hawks from Muscala?
Only time will tell...
Hawks fourth-quarter run not enough in 117-108 loss to Knicks 10 straight losses at home.
Hawks fall to Heat 124-118 as offense falters late
Hawks trade Cam Reddish, Solomon Hill to Knicks for Kevin Knox, protected first-round pick The Cam Reddish era appears to be over in Atlanta.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 3,131
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Q: Selecting proper button element id using JQuery ID Selector ('#id') from a datatable in which each row has its own commandButton I have a .xhtml page in which there is a <p:datatable> element. Inside this table, there could be many rows, and each row has its own <h:commandButton>. It is a requirement that each row have its own button (for submitting payments). The table looks something like this:
<p:panel id="payment_method_panel" >
<f:facet name="header">
<h:outputText value="MyMethods" style="font-size:15px; font-family:Arial; color:#006666;" />
</f:facet>
<h:outputText>
Payment methods associated with this account.
</h:outputText>
<br/><br/>
<h:commandButton value="Add/Make New Payment" />
<p:dataTable id="methods" var="method" value="#{insuredPaymentView.paymentInstruments}" >
<p:column style="white-space:normal;" >
<h:commandButton id="makePaymentButton" value="Pay Now" />
</p:column>
<!-- other columns -->
</p:datatable>
</p:panel>
What I hope to accomplish is to have a user click the payment button, and this action will invoke some javascript that looks like:
<script type="text/javascript">
$(function () {
$('#myContainer').containerOne();
$('#makePaymentButton').click(function () {
var container= $('#myContainer').data('container');
/*start a payment proceedure*/
container.makePayment({
paymentUuid: guid(),
paymentContext: 'NewBusiness',
referenceNumber: '1',
paymentAmount: '10.00',
policyholderName: 'Name',
billingAddress: 'address',
billingZip: 'zip'
});
});
})
</script>
Obviously in a real scenario, those values would be dynamic based off the chosen row.
I think the issue is that the jquery #id selector for the makePaymentButton is not unique, so I suppose I need to get the unique button's ID from the row the user intended. Since there could be numerous rows, how do I reference the proper button/id?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 92
|
{"url":"https:\/\/robscode.onl\/d365-form-adaptors-in-separate-model\/","text":"# [D365] Form Adaptors in separate Model\n\nWhen task recordings are used to generate automatic tests, form adaptors are needed in order to make the controls available to the test code.\n\nOne way to generate or update them is to set the property Generate Form Adaptors on the project settings to True:\n\nThose classes should not be shipped with the deployable package later. A good way to prevent this is to keep them in a separate model.\nAnd a smart thing to do is to set everything up to automatically generate those form adaptors in the dedicated model.\n\nHere is a step by step rundown:\n\n#### Create a new model for the form adaptors\n\nVS Dynamics 365 > Model Management > Create model\nGive it a name like MyModuleFormAdaptors and the same layer as your working model.\nBecause this model holds forms from the Application Platform\/Foundation, add references to following models:\n- Application Platform\n- Application Foundation\n- Electronic Reporting App Suite Integration\nReference: D365 docs\n\n#### Update descriptor file\n\nOpen the descriptor file of the new model and add the following node:\n\n<FormAdaptorSourceModel>MyModule<\/FormAdaptorSourceModel>\n\n#### Refresh and build models\n\nRefresh the models and build your main model and the new one to make sure the changes are detected.\n\n#### Check references from test models\n\nBe sure to check the referenced packages of your test models and add the new one to them if needed so the test code can actually use the form adaptors.\n\nFrom now on form adaptors will be generated in your separate model.","date":"2022-09-26 22:04:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.2735563814640045, \"perplexity\": 2482.4192175780563}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030334942.88\/warc\/CC-MAIN-20220926211042-20220927001042-00120.warc.gz\"}"}
| null | null |
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