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02 Academy Birmingham
Emily's Army on the march to Brum
Rising US band coming to Birmingham 02 Academy 3 next month
Emily's Army
Get the biggest What's On stories by email
Emily's Army will be embarking on a UK tour this summer, to coincide with the release of their second album 'Lost at Seventeen' - and will be playing Birmingham 02 Academy 3 on Sunday, July 21.
Emily's Army - Cole and Max Becker, Travis Newman, and Joey Armstrong - have already received acclaim in the US for their debut album Don't Be A D**k with alternative press calling them "overwhelmingly tight and poppy" and Entertainment Weekly applauding the album's "raggedly sharp guitar licks."
A spokesman said: "The band formed in 2004 and soon found inspiration in Max and Cole's teenage cousin Emily, who was diagnosed with Cystic Fibrosis in 1998 and has suffered from the disease her entire life.
"In tribute, the band adopted the name Emily's Army, which is also the moniker for a fundraising organization in their cousin's honour. At a time when young bands are homogenized and tailor made for consumption and eventual regurgitation, Emily's Army is a breath of fresh air."
Apart from their acclaim and inspiration this teenage pop-punk band's latest record 'Lost At Seventeen' has been produced by Green Day's Billie Joe Armstrong.
The album focuses on the inescapable uncertainty we all feel as teenagers as best displayed in the title track "Lost At 17" and single "Avenue" which induce immediate nostalgia of what it feels like to have your whole life ahead of you.
Influenced by the rich East Bay music scene, Gilman Street, classic power pop, and garage rock, Emily's Army craft heartfelt teenage anthems dealing in cautionary tales, politics, and the pressures of adolescence. The band's UK dates are proceeded by their appearance on this year's Van Warped Tour across America.
Tickets are £6 and are available at www.gigsandtours.com/www.ticketmaster.co.uk or from the 24-hour hotline 0844 811 0051 and 0844 826 2826.
What's On NewsletterPrivacy notice
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Subscribe to our What's On newsletterPrivacy noticeEnter email Subscribe
Days OutWe had breakfast with 140 monkeys at this West Midland attractionTrentham Monkey Forest in Staffordshire is a unique day out for the family
Things To Do In BirminghamBest outdoor water parks and aqua parks near BirminghamThere are giant floating obstacle courses in Bromsgrove, Henley-in-Arden, Gloucestershire and more
Space EventsMuslims will say special prayers during lunar eclipse - and this is whyWhat the lunar eclipse means for followers of Islam
Tom WatsonGeorge Galloway in bid to replace Labour Deputy Leader Tom Watson as MP for West Bromwich EastLeft-wing politician George Galloway says Tom Watson is failing to back Brexit and is wrecking Jeremy Corbyn's campaign to become Prime Minister
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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En la mitología griega Turímaco era el séptimo rey de Sición.
Sucedió a su padre Egiro unos cuatrocientos años antes de la Guerra de Troya. Tuvo un hijo llamado Leucipo, que le sucedió.
Referencias
Reyes de Sición
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{
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{"url":"https:\/\/hal.archives-ouvertes.fr\/hal-00083222","text":"# Wall laws for fluid flows at a boundary with random roughness\n\nAbstract : The general concern of this paper is the effect of rough boundaries on fluids. We consider a stationary flow, governed by incompressible Navier-Stokes equations, in an infinite domain bounded by two horizontal rough plates. The roughness is modeled by a spatially homogeneous random field, with characteristic size $\\eps$. A mathematical analysis of the flow for small $\\eps$ is performed. The Navier's wall law is rigorously deduced from this analysis.\nKeywords :\nDocument type :\nPreprints, Working Papers, ...\nDomain :\n\nCited literature [23 references]\n\nhttps:\/\/hal.archives-ouvertes.fr\/hal-00083222\nContributor : David Gerard-Varet Connect in order to contact the contributor\nSubmitted on : Thursday, June 29, 2006 - 7:31:10 PM\nLast modification on : Thursday, March 17, 2022 - 10:08:16 AM\nLong-term archiving on: : Monday, April 5, 2010 - 11:37:55 PM\n\n### Citation\n\nArnaud Basson, David Gerard-Varet. Wall laws for fluid flows at a boundary with random roughness. 2006. \u27e8hal-00083222\u27e9\n\nRecord views","date":"2022-10-03 18:49:46","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\": 0.5887278318405151, \"perplexity\": 3949.633500889255}, \"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\/1664030337428.0\/warc\/CC-MAIN-20221003164901-20221003194901-00012.warc.gz\"}"}
| null | null |
{"url":"http:\/\/mathhelpforum.com\/advanced-algebra\/139205-write-inverse-terms-matrix-print.html","text":"# Write the inverse of A in terms of the matrix A\n\n\u2022 April 14th 2010, 03:56 PM\nDarK\nWrite the inverse of A in terms of the matrix A\nI have no idea where to begin, if someone could help me get started.\n\nAlso, I'm a little unsure about what to do for the first question as well (part a).\n\u2022 April 14th 2010, 04:04 PM\ndwsmith\nMatrix multiplication general isn't commutative.\n\n$(A-B)(A+B)=A^2+AB-BA-B^2$\n\nIf $AB \\neq BA$, then $AB-BA$ isn't guaranteed to equal to 0.\n\u2022 April 15th 2010, 06:00 AM\nHallsofIvy\nYou titled this \"write the inverse of A in terms of the matrix A\" but then didn't ask about (b)!\n\nIf $A^2+ A- I_n= 0$ then $A(A+ I_n)= (A+ I_n)A= I_n$.","date":"2015-05-29 14:41:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 5, \"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\": 0.8123429417610168, \"perplexity\": 683.6750715196096}, \"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-2015-22\/segments\/1432207930143.90\/warc\/CC-MAIN-20150521113210-00135-ip-10-180-206-219.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"http:\/\/pywonderland.com\/modular\/","text":"# The modular group\n\nExample:\n\nRequirements: cairocffi.\n\nThis program draws the hyperbolic tiling of the upper plane by fundamental domains of the modular group.\n\n# Math behind this image\n\nThe modular group $\\mathrm{PSL}_2(\\mathbb{Z})$ is an infinite group acts discretely on the upper plane by fractional linear transformations:\n\n$\\mathrm{PSL}_2(\\mathbb{Z}) = \\{\\frac{az+b}{cz+d},\\ a,b,c,d\\in\\mathbb{Z}, ad-bc=1\\}.$\n\nIt can be generated by two elements $S: z\\to -1\/z$ and $T: z\\to z+1$: $\\mathrm{PSL}_2(\\mathbb{Z}) = \\langle S,T \\,|\\, S^2=(ST)^3=1\\rangle,$ using $S$ and $ST$ as generators instead shows this group is isomorphic to the free product of the two cyclic groups $\\mathbb{Z}_2$ and $\\mathbb{Z}_3$: $\\mathrm{PSL}_2(\\mathbb{Z})\\cong \\mathbb{Z}_2\\ast\\mathbb{Z}_3.$\n\nAll the math above can be found on the wiki page, and these are almost enough for drawing our image: to draw the hyperbolic tiling, one just choose any fundamental domain $D$ of $\\mathrm{PSL}_2(\\mathbb{Z})$ (the classical choice is the region colored gray in the example image), and for each element $g$ in $\\mathrm{PSL}_2(\\mathbb{Z})$ (in practice we only iterate $g$ up to a given length.), represent $g$ as a word in $\\{A,B\\}$ where $A=S$ and $B=ST$, then draw the transformed domain $gD$.\n\nNote that the word representation of $g$ by $A,B$ is generally not unique, so we have to be careful to make sure each element is traversed exactly once. This is easy in the case of using the free product representation: each element $g$ can be uniquely expressed in the form $g=x_1x_2\\cdots x_n$ where each $x_i$ is either $A$ or $B$ and no two successive $A$'s and no three successive $B$'s occur in this representation.\n\nThe problem of this approach is that the resulting $gD$'s are not symmetrically distributed on the two sides of the $y$ axis, some parts of the upper plane are densely tiled by the $gD$'s whereas other parts got very sparse, which is not aesthetically pleasing. I learned a similar approach from Bill Casselman's expository essay which used another presentation of $\\mathrm{PSL}_2(\\mathbb{Z})$: $\\mathrm{PSL}_2(\\mathbb{Z})=\\langle A,B,C \\,|\\, C^2=AB=(CA)^3=1\\rangle.$ Where $C: z\\to -1\/z$, $A: z\\to z+1$, and $B: z\\to z-1$ (the only difference is we have included $B=A^{-1}$ as a generator).\n\nBut then how to traverse the modular group using this representation? Here is some deep math got involved: the modular group is an automatic group, which means there exists a finite state automata that recognize the language of the modular group. This automata is shown below (image credits to Casselman also):\n\nSo traversing the words in the modular group up to a given length is equivalent to traversing this automata by breadth-first search up to a given number of steps, that's much easier now.","date":"2018-05-27 03:05:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.8882052898406982, \"perplexity\": 177.8592477923904}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-22\/segments\/1526794867995.55\/warc\/CC-MAIN-20180527024953-20180527044953-00001.warc.gz\"}"}
| null | null |
Zinédine Ould Khaled (born 14 January 2000) is a French professional footballer who plays as a midfielder for Ligue 1 club Angers SCO.
Club career
Ould Khaled made his professional debut with Angers in a 2–0 Ligue 1 win over Nantes on 7 March 2020.
Personal life
Born in France, Ould Khaled is of Algerian descent. He was named after the French footballer Zinedine Zidane.
References
External links
FFF Profile
Living people
2000 births
Footballers from Val-de-Marne
French footballers
French sportspeople of Algerian descent
Association football midfielders
Angers SCO players
Ligue 1 players
Championnat National 2 players
Championnat National 3 players
People from Alfortville
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,011
|
Q: Swift MKMapView sometimes becomes nil and app crashes I am displaying a map like this:
//Map
@IBOutlet var mapView: MKMapView!
var location: String?
override func viewDidLoad() {
super.viewDidLoad()
self.mapView.delegate = self
if let location = location {
let address = location
let geocoder = CLGeocoder()
geocoder.geocodeAddressString(address) { (placemarks, error) in
if let placemarks = placemarks {
if placemarks.count != 0 {
let annotation = MKPlacemark(placemark: placemarks.first!)
self.mapView.addAnnotation(annotation)
let span = MKCoordinateSpanMake(0.1, 0.1)
let region = MKCoordinateRegionMake(annotation.coordinate, span)
self.mapView.setRegion(region, animated: false)
}
}
}
}
}
//Fixing map memory issue
override func viewWillDisappear(_ animated:Bool){
super.viewWillDisappear(animated)
self.applyMapViewMemoryFix()
}
//Fixing map memory issue
func applyMapViewMemoryFix(){
switch (self.mapView.mapType) {
case MKMapType.hybrid:
self.mapView.mapType = MKMapType.standard
break;
case MKMapType.standard:
self.mapView.mapType = MKMapType.hybrid
break;
default:
break;
}
self.mapView.showsUserLocation = false
self.mapView.delegate = nil
self.mapView.removeFromSuperview()
self.mapView = nil
}
If I exit out from the VC and go back again a few times fast the app will eventually crash because MKMapView becomes nil.
The crash happens here self.mapView.addAnnotation(annotation)
I am not sure why but my guess is that geocoder.geocodeAddressString has not finished loading/searching and if I exit out from the VC fast enough mapView becomes nil
A: You set self.mapView to nil in your viewWillDisappear method. So your app will crash anytime you leave the view controller before the geocoder is done.
Simply add a proper check for nil in the geocoder block.
override func viewDidLoad() {
super.viewDidLoad()
self.mapView.delegate = self
if let location = location {
let address = location
let geocoder = CLGeocoder()
geocoder.geocodeAddressString(address) { (placemarks, error) in
if let placemarks = placemarks {
if placemarks.count != 0 {
if let mapView = self.mapView {
let annotation = MKPlacemark(placemark: placemarks.first!)
mapView.addAnnotation(annotation)
let span = MKCoordinateSpanMake(0.1, 0.1)
let region = MKCoordinateRegionMake(annotation.coordinate, span)
mapView.setRegion(region, animated: false)
}
}
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,737
|
Q: Input multiple values separated by Comma as Input parameter in Jenkins I am new to Jenkins world.I am trying to create a job in Jenkins where i need to input multiple file names separated by comma as input parameter.And, catch these file names in Unix command inside Execute shell area to delete the same using rm -rf command(these files are available in some specific path).How can we use Jenkins to do the same?
Files names are like:
18_07_2015.log
22_07_2015.log
29_07_2015.log
Thanks in advance.
A: You may use String parameters in your jenkins job:
Then add the param to your shell script:
rm -rf ${FILE_PARAM}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,540
|
Q: want to get the selected java project packges using contextmenu in eclipse plugin development i am developing an eclipse plugin i want to fetch only packages of the selected javaproject using contect menu. i have got the selection using
in my handler and i have passed it to mywizard page. i am unable to set the selection as treeviewersetinput.
if i use treeViewer.setInput(selection).
i have got the selection like this in my wizard page.
code Snippet:
public void init(IStructuredSelection selection)
{
Object firstElement = selection.getFirstElement();
if (firstElement instanceof IJavaProject) {
IJavaProject javaProject = (IJavaProject) firstElement;
IFolder folder = javaProject.getProject().getFolder("src");
srcFolder = javaProject
.getPackageFragmentRoot(folder);
System.out.println(srcFolder.getPath().isRoot());
System.out.println(firstElement);
treeViewer.setInput(srcFolder);
}
}
**i am getting error as:**
src
<default> (...)
com (...)
com.hcl (not open)
com.hcl.example (not open)
com.hcl.example.menu (not open)
com.hcl.example.menu.handlers (not open)
!ENTRY org.eclipse.ui 4 0 2013-07-28 15:50:07.870
!MESSAGE Unhandled event loop exception
!STACK 0
org.eclipse.e4.core.di.InjectionException: java.lang.NullPointerException
at org.eclipse.e4.core.internal.di.MethodRequestor.execute(MethodRequestor.java:63)
at org.eclipse.e4.core.internal.di.InjectorImpl.invokeUsingClass(InjectorImpl.java:229)
at org.eclipse.e4.core.internal.di.InjectorImpl.invoke(InjectorImpl.java:210)
at org.eclipse.e4.core.contexts.ContextInjectionFactory.invoke(ContextInjectionFactory.java:131)
at org.eclipse.e4.core.commands.internal.HandlerServiceImpl.executeHandler(HandlerServiceImpl.java:171)
at org.eclipse.e4.ui.workbench.renderers.swt.HandledContributionItem.executeItem(HandledContributionItem.java:814)
at org.eclipse.e4.ui.workbench.renderers.swt.HandledContributionItem.handleWidgetSelection(HandledContributionItem.java:707)
at org.eclipse.e4.ui.workbench.renderers.swt.HandledContributionItem.access$7(HandledContributionItem.java:691)
at org.eclipse.e4.ui.workbench.renderers.swt.HandledContributionItem$4.handleEvent(HandledContributionItem.java:630)
at org.eclipse.swt.widgets.EventTable.sendEvent(EventTable.java:84)
at org.eclipse.swt.widgets.Widget.sendEvent(Widget.java:1053)
at org.eclipse.swt.widgets.Display.runDeferredEvents(Display.java:4169)
at org.eclipse.swt.widgets.Display.readAndDispatch(Display.java:3758)
at org.eclipse.e4.ui.internal.workbench.swt.PartRenderingEngine$9.run(PartRenderingEngine.java:1029)
at org.eclipse.core.databinding.observable.Realm.runWithDefault(Realm.java:332)
at org.eclipse.e4.ui.internal.workbench.swt.PartRenderingEngine.run(PartRenderingEngine.java:923)
at org.eclipse.e4.ui.internal.workbench.E4Workbench.createAndRunUI(E4Workbench.java:86)
at org.eclipse.ui.internal.Workbench$5.run(Workbench.java:588)
at org.eclipse.core.databinding.observable.Realm.runWithDefault(Realm.java:332)
at org.eclipse.ui.internal.Workbench.createAndRunWorkbench(Workbench.java:543)
at org.eclipse.ui.PlatformUI.createAndRunWorkbench(PlatformUI.java:149)
at org.eclipse.ui.internal.ide.application.IDEApplication.start(IDEApplication.java:124)
at org.eclipse.equinox.internal.app.EclipseAppHandle.run(EclipseAppHandle.java:196)
at org.eclipse.core.runtime.internal.adaptor.EclipseAppLauncher.runApplication(EclipseAppLauncher.java:110)
at org.eclipse.core.runtime.internal.adaptor.EclipseAppLauncher.start(EclipseAppLauncher.java:79)
at org.eclipse.core.runtime.adaptor.EclipseStarter.run(EclipseStarter.java:353)
at org.eclipse.core.runtime.adaptor.EclipseStarter.run(EclipseStarter.java:180)
at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
at sun.reflect.NativeMethodAccessorImpl.invoke(Unknown Source)
at sun.reflect.DelegatingMethodAccessorImpl.invoke(Unknown Source)
at java.lang.reflect.Method.invoke(Unknown Source)
at org.eclipse.equinox.launcher.Main.invokeFramework(Main.java:629)
at org.eclipse.equinox.launcher.Main.basicRun(Main.java:584)
at org.eclipse.equinox.launcher.Main.run(Main.java:1438)
at org.eclipse.equinox.launcher.Main.main(Main.java:1414)
Caused by: java.lang.NullPointerException
at com.hcl.green.type.handler.GenerateGreenfieldLayer.init(GenerateGreenfieldLayer.java:333)
at com.hcl.green.type.handler.GreenWizard.addPages(GreenWizard.java:53)
at org.eclipse.jface.wizard.WizardDialog.createContents(WizardDialog.java:605)
at org.eclipse.jface.window.Window.create(Window.java:431)
at org.eclipse.jface.dialogs.Dialog.create(Dialog.java:1089)
at org.eclipse.jface.window.Window.open(Window.java:790)
at com.hcl.green.type.handler.GreenSelectionHandler.execute(GreenSelectionHandler.java:70)
at org.eclipse.ui.internal.handlers.HandlerProxy.execute(HandlerProxy.java:290)
at org.eclipse.ui.internal.handlers.E4HandlerProxy.execute(E4HandlerProxy.java:76)
at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
at sun.reflect.NativeMethodAccessorImpl.invoke(Unknown Source)
at sun.reflect.DelegatingMethodAccessorImpl.invoke(Unknown Source)
at java.lang.reflect.Method.invoke(Unknown Source)
at org.eclipse.e4.core.internal.di.MethodRequestor.execute(MethodRequestor.java:56)
... 34 more
can any one please let me know how to proceed.
A: Looks like a missing null check at
com.hcl.green.type.handler.GenerateGreenfieldLayer.init(GenerateGreenfieldLayer.java:333)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,226
|
The Hasena Woodline Tida Modern Bed is available in all hasena solid wood finishes. The legs are chunky for a modern look, 10 cm high. The Bed is matched with the Litto headboard, which would come in the same finish selected for the frame.
If you would like to view a selection of alternative headboards or for any other queries and bespoke quotations, please contact us.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,265
|
Reading: Participation in a mobile health intervention to improve retention in early HIV care in an i...
Participation in a mobile health intervention to improve retention in early HIV care in an informal urban settlement in Nairobi, Kenya: a gender analysis
M. van der Kop ,
University of British Columbia/Karolinska Institutet, Vancouver, BC/CA
D. Ojakaa,
Amref Health Africa, Nairobi, KE
A. Ekström,
Karolinska Instituet, Stockholm, Sweden
J. Kimani,
University of Nairobi, Nairobi, KE
L. Thabane,
McMaster Unviersity, Hamilton, ON/CA
O. Awiti-Ujiji,
Karolinska Institutet, Stockholm, Sweden
R. Lester
University of British Columbia, Vancouver, BC/CA
How to Cite: Kop, M. van . der ., Ojakaa, D., Ekström, A., Kimani, J., Thabane, L., Awiti-Ujiji, O. and Lester, R., 2015. Participation in a mobile health intervention to improve retention in early HIV care in an informal urban settlement in Nairobi, Kenya: a gender analysis. Annals of Global Health, 81(1), pp.9–10. DOI: http://doi.org/10.1016/j.aogh.2015.02.536
Kop, M. van . der ., Ojakaa, D., Ekström, A., Kimani, J., Thabane, L., Awiti-Ujiji, O. and Lester, R., 2015. Participation in a mobile health intervention to improve retention in early HIV care in an informal urban settlement in Nairobi, Kenya: a gender analysis. Annals of Global Health, 81(1), pp.9–10. DOI: http://doi.org/10.1016/j.aogh.2015.02.536
Kop M van der, Ojakaa D, Ekström A, Kimani J, Thabane L, Awiti-Ujiji O, et al.. Participation in a mobile health intervention to improve retention in early HIV care in an informal urban settlement in Nairobi, Kenya: a gender analysis. Annals of Global Health. 2015;81(1):9–10. DOI: http://doi.org/10.1016/j.aogh.2015.02.536
Kop, M. van . der ., Ojakaa, D., Ekström, A., Kimani, J., Thabane, L., Awiti-Ujiji, O., & Lester, R. (2015). Participation in a mobile health intervention to improve retention in early HIV care in an informal urban settlement in Nairobi, Kenya: a gender analysis. Annals of Global Health, 81(1), 9–10. DOI: http://doi.org/10.1016/j.aogh.2015.02.536
Kop M van der and others, 'Participation in a Mobile Health Intervention to Improve Retention in Early HIV Care in an Informal Urban Settlement in Nairobi, Kenya: A Gender Analysis' (2015) 81 Annals of Global Health 9 DOI: http://doi.org/10.1016/j.aogh.2015.02.536
Kop, M. van der, D. Ojakaa, A. Ekström, J. Kimani, L. Thabane, O. Awiti-Ujiji, and R. Lester. 2015. "Participation in a Mobile Health Intervention to Improve Retention in Early HIV Care in an Informal Urban Settlement in Nairobi, Kenya: A Gender Analysis". Annals of Global Health 81 (1): 9–10. DOI: http://doi.org/10.1016/j.aogh.2015.02.536
Kop, M. van der, D. Ojakaa, A. Ekström, J. Kimani, L. Thabane, O. Awiti-Ujiji, and R. Lester. "Participation in a Mobile Health Intervention to Improve Retention in Early HIV Care in an Informal Urban Settlement in Nairobi, Kenya: A Gender Analysis". Annals of Global Health 81, no. 1 (2015): 9–10. DOI: http://doi.org/10.1016/j.aogh.2015.02.536
Kop, M. van . der ., et al.. "Participation in a Mobile Health Intervention to Improve Retention in Early HIV Care in an Informal Urban Settlement in Nairobi, Kenya: A Gender Analysis". Annals of Global Health, vol. 81, no. 1, 2015, pp. 9–0. DOI: http://doi.org/10.1016/j.aogh.2015.02.536
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,516
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{"url":"http:\/\/physics.stackexchange.com\/questions\/92776\/k%c3%a4hler-and-complex-manifolds","text":"K\u00e4hler and complex manifolds\n\nI was woundering if anyone knows any good references about K\u00e4hler and complex manifolds? I'm studying supergravity theories and for the simplest $\\mathcal{N}=1$ supergravity we'll get these. Now in the course-notes they're quite short about these complex manifolds. I was hoping someone of you guys might know a good (quite complete book) about the subject?\n\nTo get a rigorous mathematician's point of view, I've also posted this topic in on the math-stackexchange.\n\n-\nMaybe these lectures (Chapter 4). \u2013\u00a0 Trimok Jan 8 at 12:11\nWould Mathematics be a better home for this question? \u2013\u00a0 Emilio Pisanty Jan 8 at 13:18\n@EmilioPisanty I also have a copy of this question in the mathematics-part of the forum (math.stackexchange.com\/q\/630838). But I figured that maybe a physicist point of view might also be helpful ? \u2013\u00a0 Nick Jan 8 at 15:05\nIn that case, you should always indicate the fact that you've cross-posted, in both posts. \u2013\u00a0 Emilio Pisanty Jan 8 at 16:00\n@EmilioPisanty Edited it! :) \u2013\u00a0 Nick Jan 8 at 17:16\n\nI strongly suggest Nakahara. Geometry, Topology and Physics.\n\nThere is a whole chapter in complex differential geometry and the Kahler case is treated well.\n\nIt is a good and clear introduction, written from a physicist and for physicists. However, it is not complete. With this I mean that if you want to have a strong knowledge of the subject (for example to work on it) you need some more than Nakahara.\n\nBut I'd give it a shot.\n\n-\n\nChapter 0 of Griffiths and Harris, principles of algebraic geometry, gives a very good introduction in some 120 pages. In the remainder of the book the main focus is on complex algebraic varieties, which is a special, though still very broad, subclass.\n\n-\n120 pages, seems like quite a long way to go. Are you perhaps familiar with some works that might give a quicker acces (given that I have had e \"first encounter\" with differential geometry ?) \u2013\u00a0 Nick Jan 8 at 11:45\nI think it could still be useful. It is broken up in 7 sections that can probably be skipped entirely when they treat something you know already. Kaehler manifolds are only introduced in the last section, which is just over 20 pages. If you already know complex geometry including sheaf cohomology you can start there, otherwise you'll have to go through more of them. With your differential geometry background you can probably go very fast through at least two of the other 6. \u2013\u00a0 doetoe Jan 10 at 23:35\nthanks :), seems indeed a good (readable) source ! \u2013\u00a0 Nick Jan 13 at 13:05\n\nI guess your needs are related to compactifications of supergravity theories. if this is true, then the book \"Compact manifolds with special holonomy\" by Joyce will be very useful. It has a section devoted to Kahler manifold since they indeed are of great importance for compactifications.\n\nThen I'll suggest to look at review on flux compactifications, e.g. by M.Grana https:\/\/inspirehep.net\/record\/691224 . This describes geometry of manifolds with special geometry in application to physics (supergravity and phenomenology) while the book by Joyce contains more differential geometry.\n\nFinally, recently I found this old paper https:\/\/inspirehep.net\/record\/16270 very useful. It has some discussion on Kahler manifolds as well.\n\n-\n\nYou might find this excellent book entitled \"Mirror Symmetry\" by Hori et al, available online http:\/\/www2.maths.ox.ac.uk\/cmi\/library\/monographs\/cmim01.pdf, useful. Chapter 5, in particular, is a nice summary.\n\n-","date":"2014-08-29 05:28:41","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\": 0.6768133640289307, \"perplexity\": 758.8224023489142}, \"config\": {\"markdown_headings\": false, \"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-2014-35\/segments\/1408500831903.50\/warc\/CC-MAIN-20140820021351-00361-ip-10-180-136-8.ec2.internal.warc.gz\"}"}
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{"url":"http:\/\/www.math.uri.edu\/~grove\/","text":"# Edward A. Grove\n\nDepartment of Mathematics URI\n\n## Professional Information\n\n\u2022 Title: Professor of Mathemathics\n\n\u2022 Educational Background: Ph.D, Brown University, 1969\n\n## Contact Information\n\n\u2022 Office: 245 Tyler Hall\n\u2022 Telephone: (401) 874-5299\n\u2022 E-mail: grove@math.uri.edu\n\n## Research Information\n\n\u2022 Basic Theory of Nonlinear Difference Equations of Order Greater than One.\n\n\u2022 Global Stability and Periodic Behavior of Solutions.\n\n## Books\n\n1. \"An Introduction to Complex Variables\" (with G. Ladas), Houghton Mifflin, Boston, 1974.\n\n2. \"Periodicities in Nonlinear Difference Equations\" (with G. Ladas), CRC Press, 2005.\n\n## Publications\n\n1. SU(n) actions on manifolds with vanishing first and second integral Pontrjagin classes, Springer-Verlag. Lecture Notes in Mathematics, 298(1972) 324-333.\n\n2. N-person games for systems of hyperbolic linear partial differential equations (with J. Papadakis), Differential Games and Control Theory, Marcel Dekker, New York, (1974) 27-35.\n\n3. An Introduction to Complex Variables (with G. Ladas), Houghton Mifflin, Boston, 1974. (Book) See above.\n\n4. SU(n) actions on manifolds with vanishing first and second integral Pontrjagin classes, Transactions of the American Mathematical Society, 19(1974) 331-350.\n\n5. Classification of compact homogeneous spaces with non-zero Euler class and with zero Pontrjagin classes (with W.Y. Hsiang), Bull. Inst. Math. Acad. Sinica, Vol. 8, No. 2\/3, Part II (July, 1980) 365-382. (Special issue in memory of H.C. Wang.)\n\n6. Oscillations of first order neutral delay differential equations (with M.K. Grammatikopoulos and G. Ladas), J. Math. Anal. Appl., 120(1986) 510-520.\n\n7. Oscillations and asymptotic behavior of neutral differential equations with deviating arguments (with M.K. Grammatikopoulos and G. Ladas), Applicable Anal., 22(1986) 1-19.\n\n8. Sturm comparison theorems for neutral differential equations (with M.R.S. Kulenovic and G. Ladas), Canadian Mathematical Society Conference Proceedings, 8(1987), 163-169.\n\n9. Oscillation and asymptotic behavior of second order neutral differential equations with deviating arguments (with M.K. Grammatikopoulos and G. Ladas), Canadian Mathematical Society Conference Proceedings, 8(1987) 153-161.\n\n10. Sufficient conditions for oscillation and non-oscillation of neutral equations (with M.R.S. Kulenovic and G. Ladas), J. Diff. Eqns., 68(1987), 373-382.\n\n11. A necessary and sufficient condition for the oscillation of neutral equations (with G. Ladas and A. Meimaridou), J. Math. Anal. Appl., 126(1987) 341-354.\n\n12. Oscillations and asymptotic behavior of first-order neutral delay differential equations (with G. Ladas and S.W. Schultz), Applicable Analysis, 27(1988), 67-78.\n\n13. Neutral delay differential equations with positive and negative coefficients (with K. Farrell and G. Ladas), Applicable Analysis, 27(1988), 181-197.\n\n14. Sufficient conditions for the oscillation of delay and neutral delay equations (with G. Ladas and J. Schinas), Canad. Math. Bull., 31(1988), 459-466.\n\n15. Neutral delay differential equations with variable delays (with K. Gopalsamy and G. Ladas), Proceedings of The International Conference on Theory and Applications of Differential Equations, March 21-25, 1988, Ohio University; Differential Equations and Applications, Ohio University Press, 1988, 343-347.\n\n16. On the characteristic equation for equations with continuous and piecewise constant arguments (with I. Gyori and G. Ladas), Radovi Matematicki 5(1989), 271-281.\n\n17. Oscillations of neutral difference equations (with D.A. Georgiou and G. Ladas), Applicable Analysis, 33(1989), 243-253.\n\n18. A Myskis-type comparison result for neutral equations (with M.R.S. Kulenovic and G. Ladas), Mathematische Nachricten, 146(1990), 195-206.\n\n19. Oscillation of neutral difference equations with variable coefficients (with D.A. Georgiou and G. Ladas), Differential Equations: Stability and Control'', Marcel Dekker, 1990, 165-173.\n\n20. A simple model for price fluctuations in a single commodity market (with A.M. Farahani), Oscillation and Dynamics in Delay Equations, American Mathematical Society, 1992, 97-103.\n\n21. Global attractivity in a Food-Limited'' population model (with G. Ladas and Q. Qian), Dynamic Systems and Applications, 2(1993), 243-250.\n\n22. Periodicity in a simple genotype selection model (with V.Lj. Kocic, G. Ladas and R. Levins), Differential Equations and Dynamical Systems, 1(1993) 35-50.\n\n23. Oscillation and stability in a simple genotype model (with V.Lj. Kocic, G. Ladas and R. Levins), Quarterly of Applied Mathematics, 52(1994) 499-508.\n\n24. Oscillation of a difference equation with periodic coefficients (with E. Camouzis and G. Ladas), Applicable Analysis, 53(1994), 143-148.\n\n25. Oscillation and stability in models of a perennial grass (with R. DeVault, G. Ladas, R. Levins and C. Puccia), The Proceedings of Dynamic Systems and Applications - 1, 87-93, Dynamic Publishers Inc., U.S.A., 1994.\n\n26. On a rational recursive sequence (with E.J. Janowski, C.M. Kent and G. Ladas), Communications on Applied Nonlinear Analysis, 1(1994), 61-72.\n\n27. Monotone unstable solutions of difference equations and conditions for boundedness (with E. Camouzis, G. Ladas and V.Lj. Kocic), Journal of Difference Equations and Applications, 1(1995), 17-44.\n\n28. Bounds on the lengths of semi-cycles of difference equations (with R. DeVault, G. Ladas. R. Levins and C. Puccia), Proceedings of the First International Conference on Difference Equations, May 25-28, 1994, San Antonio, Texas, USA, Gordon and Breach Science Publishers (1995), 143-161.\n\n29. Oscillation and stability in a delay model of a perennial grass (with R. DeVault, G. Ladas, R. Levins and C. Puccia), Journal of Difference Equations and Applications, 1(1995), 173-185.\n\n30. Oscillation and stability in a genotype selection model with several delays (with V.Lj. Kocic, G. Ladas and R. Levins), Journal of Difference Equations and Applications, 2(1996), 205-217.\n\n31. Lyness equations with variable coefficients (with C.M. Kent and G. Ladas), Proceedings of the Second International Conference on Difference Equations and Applications, August 7-11, 1995, Vesprem, Hungary, Gordon and Breach Science Publishers, 1997, 281-288.\n\n32. Classification of invariants for certain difference equations (with V.Lj. Kocic and G. Ladas), Proceedings of the Second International Conference on Difference Equations and Applications, August 7-11, 1995, Vesprem, Hungary, Gordon and Breach Science Publishers, 1997, 289-294.\n\n33. On a nonlinear equation with piecewise constant argument (with G. Ladas and S. Zhang), Commun. Appl. Nonlinear Anal., 4(1997), 67-79.\n\n34. Boundedness and persistence of the non-autonomous Lyness and Max equations, (with G. Ladas and C.M. Kent), Journal of Difference Equations and Applications, 3(1998), 241-258.\n\n35. On some difference equations with eventually periodic solutions (with A.M. Amleh, G. Ladas, and C.M. Kent), Journal of Math. Anal. and Applications, 223(1998), 196-215.\n\n36. On the non-autonomous equation x_{n+1} = max { A_{n}\/x_{n},B_{n}\/x_{n-1}}, (with W.J. Briden, G. Ladas, and L.C. McGrath), Proceedings of the Third International Conference on Difference Equations and Applications, September 1-5, 1997, Taipei, Taiwan, Gordon and Breach Science Publishers, (1999), 49-73.\n\n37. Eventually periodic solutions of x_{n+1} = max { 1\/x_{n} , A_{n} \/x_{n-1}} (with W.J. Briden, C.M. Kent, and G. Ladas), Commun. Appl. Nonlinear Anal., 6(1999), 31-34.\n\n38. On the recursive sequence x_{n+1} = alpha + x_{n-1}\/x_{n} (with A.M. Amleh, D.A. Georgiou, and G. Ladas), Journal of Math. Anal. and Applications, 233(1999), 790-798.\n\n39. On the global behavior of solutions of a biological model (with G. Ladas, R. Levins, and N. Prokup), Commun. Appl. Nonlinear Anal., 7(2000), 33-46.\n\n40. A global convergence result with applications to periodic solutions (with H. El-Metwally and G. Ladas), Journal of Math. Anal. and Applications, 245(2000), 161-170.\n\n41. Global Stability in Some Population Models (with C.M. Kent, G. Ladas, R. Levins and S. Valicenti), Proceedings of the Fourth International Conference on Difference Equations and Applications}, August 27-31, 1998, Poznan, Poland, Gordon and Breach Science Publishers, (2000), 149-176.\n\n42. Existence and Behavior of Solutions of a Rational System (with G. Ladas, L.C. McGrath and C.T. Teixeira), Commun. Appl. Nonlinear Anal., 8(2001), 1-25.\n\n43. On the Global Attractivity and Periodic Character of Some Difference Equations (with H. El-Metwalli, G. Ladas and H. Voulov), Journal of Difference Equations and Applications, 7(2001), 837-850.\n\n44. On the Difference Equation x_{n+1} = max { \\frac{1}{x_{n}},\\frac{A_{n} }{ x_{n-1}} } with a Period 3 Parameter (with C.M. Kent, G. Ladas and M.A. Radin), Fields Institute Communications Series, Volume 29, 2001.\n\n45. On the difference equation x_{n+1} = a + b x_{n-1}e^{-x_{n}} (with H. El-Metwalli, G. Ladas, R. Levins, and M. Radin), Nonlinear Analysis, 47(2001) 4623-4634.\n\n46. On the global character of y_{n+1} = \\frac{ py_{n-1} + y_{n-2} }{ q + y_{n-2} } (with G. Ladas, M. Predescu, and M. Radin), MATH. SCI. RES. HOT-LINE 5(2001), 25-39.\n\n47. Periodicity in Nonlinear Difference Equations (with G. Ladas), Revista Cubo., 4(2002), 195-230.\n\n48. On the Periodic Character of the Difference Equation x_{n+1} = \\frac{ p + x_{n-2k} }{ 1 + x_{n-2l} } (with G. Ladas and M. Predescu), Math. Sci. Res. Journal, 6(2002), 221-233.\n\n49. On the Global Character of the Difference Equation x_{n+1} = \\frac{ alpha + gamma x_{n-(2k+1)} + \\delta x_{n-2l} }{ A + x_{n-2l} } (with G. Ladas, M. Predescu, and M. Radin), Journal of Difference Equations and Applications 9(2003), 171-199.\n\n50. On the Dynamics of y_{n+1} = \\frac{ p + y_{n-2} }{ qy_{n-1} + y_{n-2} } (with G. Ladas and L.C. McGrath), Proceedings of the 6th ICDEA, Augsburg, Germany, Taylor and Francis, London, 2003.\n\n51. On the Trichotomy Character of x_{n+1} = \\frac{ \\alpha + \\gamma x_{n-1} }{ A + B x_{n} + x_{n-2} } (with E. Chatterjee, G. Ladas, and Y. Kostrov), Journal of Difference Equations and Applications 9(2003), 1113-1128.\n\n52. On the Difference Equation y_{n+1} = \\frac{ y_{n-(2k+1)} + p }{ y_{n-(2k+1)} + qy_{n-2l} } (with H.A. El- Metwally, G. Ladas and L.C. McGrath), New Progress in Difference Equations , CRC, 2004, p.433-453.\n\n53. On Period-two solutions of x_{n+1} = \\frac{ \\alpha + \\beta x_{n} + \\gamma x_{n-1} }{ A + B x_{n} + C x_{n-1} } (with G. Ladas), Proceedings of the 7th ICDEA, Taylor and Francis (2004).\n\n54. Progress Report on Rational Difference Equations (with M.R.S Kulenovic and G. Ladas), Journal of Difference Equations (2004).\n\n55. Periodicities in Nonlinear Difference Equations (with G. Ladas), CRC Press, 2005. (Book.) See above.\n\n56. On Third-Order Rational Difference Equations, Part 4 (with Y. Kostrov, G. Ladas, and M. Predescu), Journal of Difference Equations and Applications (2005).\n\n57. Riccati Difference Equations With Real Period-2 Coefficients (with Y. Kostrov, G. Ladas, and S.W. Schultz), Communications on Applied Nonlinear Analysis Volume 14(2007), Number 2, p.33-56\n\n## Personal Interests and Hobbies\n\n\u2022 Walking\n\u2022 Card Games\n\u2022 My puppy dog Phoenix","date":"2013-05-24 03:14:55","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\": 0.6787608861923218, \"perplexity\": 8343.349899243538}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368704133142\/warc\/CC-MAIN-20130516113533-00033-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
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Filipino food is the best comfort food!
Top Recovery Filipino Food
2 years ago 2 years ago Food & Drinks, Where to eat
Filipinos love to eat! On a regular day, a typical pinoy eats three heavy meals and two snacks in between—there's no doubt food gives us comfort. We all have our ups and downs and one thing that keeps us sane and going is food. Whether you've been left by the love of your life, went…
Mark Aldrin HipolitoFood & Drinks, Where to eatFilipino, food, merienda, recovery, snacks 0 0 0
South of Manila Food Trip
2 years ago 2 years ago Food & Drinks, Travel, Where to eat
I've always wanted to drive down around the South of Manila and just eat my way through the different towns there. Three must-haves when visiting the south are bulalo, lomi, and longganisa. All of these taste so much more delicious after driving around in the new Countryman, shout out to Mini Cooper for letting me…
Erwan HeussaffFood & Drinks, Travel, Where to eatbatangas, bulalo, cavite, food, imus, lomi, longganisa, manila, metro manila, south, tagaytay 0 0 0
P10 vs P800 Lugaw Crawl in Manila, Philippines
Despite being an insanely hot country, the Philippines cannot get enough of their bubbling hot dishes. From saucy stews like kare-kare to bowls of sinigang or tinola, Filipinos find comfort in the warm and the soupy. Enter a Filipino classic that has its roots in both the Spanish and Chinese influences that have permeated the…
Erwan HeussaffFood & Drinks, Travel, Where to eatbreakfast, congee, dinner, Filipino, food, goto, hot, lugaw, lunch, Philippines, soup 0 0 0
Sweet Potato Snack Hunt in Japan
I recently went on a Sweet Potato Hunt in tokyo. In Japan, come winter time, you will see different types of Satsumaimo (sweet potato) flavoured snacks. For about 3 centuries, ishi yaki imo (roasted sweet potatoes on stones) were sold street side from vendors and then eventually trucks. This tuber has been popular in japan,…
Erwan HeussaffFood & Drinks, Travel, Where to eatdessert, food, ice cream, japan, soft serve, sweet, sweet potato, tokyo, winter 0 0 0
Best Things to Do in Tokyo, Japan (Tokyo Metro Guide)
Making an Overnight Tokyo guide was something I've always wanted to do ever since we started doing the series. But it always seemed like a gargantuan task because of how huge the city is. Thanks to Tokyo Metro, we were finally able to create Overnight Tokyo! Please watch and enjoy the video below: Here's a…
Erwan HeussaffFood & Drinks, Travel, Where to eatbars, dining, drinks, food, japan, ramen, tokyo, where to eat 0 0 0
Tokyo Street Food: The Tsukiji Outer Market
I was recently in Tokyo to shoot a city guide (coming soon, make sure you subscribe to my channel!) and honestly, a trip to this city will never be complete without a visit to the Tsukiji Market. I ventured the outer area of the market which is full of amazing things to try. Most places…
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Affordable Treats and Bars in Tokyo
Food in Tokyo is known to be expensive—it will definitely eat a large part of your budget especially if you're aiming for high end dinners and classy bars. But like with any city, expenses are a choice. Don't think that all the local Tokyo citizens spend hundreds of dollars every meal every day. Of course…
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Must Eats in Paris
I was recently on a trip to France with my sister Solenn for a wedding, and we only had two full days in Paris before heading to the south. I decided to shoot a vlog and show you guys the Parisian food scene and what I usually look for when I'm in the city. Here's…
Erwan HeussaffFood & Drinks, Travel, Where to eatdining, france, paris, recommendations, travel 0 0 0
The Best Korean Restaurants in Manila
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Where is the Best Halo Halo in Manila, Philippines?
Halo halo, literally translated to "mix-mix", doesn't actually originate from the Philippines. Rather, inspiration may have come from the Japanese dessert called "kakigori", as seen in other Asian icy desserts like Singapore's "ice kachang", Malaysia's "cendol", and Vietnam's "che ba mau". The idea of putting sweetened toppings and creamy syrups or liquids may have come…
Erwan HeussaffFood & Drinks, Where to eatdessert, Filipino, halo halo 1 0 0
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
The acceleration of the Universe is one of the most elusive problems in modern
cosmology. Since its discovery in the last decade of the twentieth century by
Supernovae (SNIa) observations \citep{Riess:1998,Perlmutter:1999}, and its confirmation
by the acoustic peaks of the cosmic microwave background (CMB) radiation
\citep{WMAP:2003elm}, it has been a theoretical and observational challenge to
construct a model that combines all of its characteristics. From a theoretical
point of view, and assuming homogeneous and isotropic symmetries (cosmological
principle), the need for a component
with features able to reproduce the Universe acceleration is vital to obtain
accurate values for the observable Universe age and size. Recently, the
confidence in the detection of this acceleration at late times has been
increased with precise observations of the large scale structure
\citep{Nadathur:2020kvq}.
The best candidate to explain the observed acceleration
is the well-known Cosmological Constant (CC), interpreted under the assumption that quantum
vacuum fluctuations generate the constant energy density observed and, with this,
a late-time acceleration. However, when we apply the Quantum Field Theory to
assess the energy density, the result is in total discrepancy with
observations, giving rise to the so-called {\it fine-tuning problem}
\citep{Zeldovich:1968ehl, Weinberg}. In addition, recent observations developed by the collaboration \emph{Supernova $H_0$ for the Equation of State} (SH0ES) \citep{Riess:2020fzl} show a
discrepancy for the obtained value of $H_0$ when compared to Planck
observations based on the $\Lambda$ Cold Dark Matter ($\Lambda$CDM) model
\citep{Aghanim:2018}. This generates a tension of $4.2\sigma$ between the
mentioned experiments, bringing a new crisis and the need for new ways to tackle
the problem \citep{DiValentino:2020zio}, as long as this discrepancy is not related to unknown
systematic errors affecting the measurements
\citep{DES:2019fny,birrer2021, Efstathiou:2021ocp, Freedman:2021ahq,
Shah:2021onj}. Is in this vein that the community has been
proposing other alternatives to address the problem of the Universe
acceleration. In general, there are two main directions that one could follow.
The first is to maintain general relativity an introduce new peculiar forms of
matter, such as scalar fields \citep{Copeland:2006wr,Cai:2009zp,review:universe},
Chaplygin gas \citep{Chaplygin,Villanueva_2015,Hernandez-Almada:2018osh},
viscous fluids \citep{Cruz, MCruz:2017, CruzyHernandez,AlmadaViscoso,
Hernandez-Almada:2020ulm, Almada:2020}, etc, collectively known as dark-energy
sector. The second way is to construct modified gravitational theories
\citep{CANTATA:2021ktz,Capozziello:2011et} such as braneworlds models
\citep{Maartens:2010ar,Garcia-Aspeitia:2016kak,Garcia-Aspeitia:2018fvw},
emergent gravity
\citep{PEDE:2019ApJ,Pan:2019hac,PEDE:2020,Hernandez-Almada:2020uyr,
Garcia-Aspeitia:2019yni,Garcia-Aspeitia:2019yod}, Einstein-Gauss-Bonet
\citep{Glavan:2019inb,Garcia-Aspeitia:2020uwq}, thermodynamical models
\citep{Saridakis:2020cqq,Leon:2021wyx}, torsional gravity \citep{Cai:2015emx}, $f(R)$ theories \citep{Dainotti:2021},
etc.
On the other hand, there is an increasing interest in dark energy
alternative models with the holographic principle. This is inspired by the
relation between entropy and the area of a black hole. It states that the observable
degree of freedom of a physical system in a volume can be encoded in a
lower-dimensional description on its boundary \citep{Hooft:1993, Susskind:1995}.
The holographic principle imposes a connection between the infrared (IR)
cutoff, related to large-scale of the Universe, with the ultraviolet (UV) one,
related to the vacuum energy. Application of the holographic principle to the
Universe horizon gives rise to a vacuum energy of holographic origin, namely holographic dark energy \citep{Li:2004rb,Wang:2016och}. Holographic dark
energy proves to lead to interesting phenomenology and, thus, it has been
studied in detailed
\citep{Li:2004rb,Wang:2016och,Horvat:2004vn, Pavon:2005yx,
Wang:2005jx,
Nojiri:2005pu,Kim:2005at, Setare:2008pc,Setare:2008hm}, confronted to observations
\citep{Zhang:2005hs,Li:2009bn,Feng:2007wn,Zhang:2009un,Lu:2009iv,
Micheletti:2009jy} and extended to various frameworks
\citep{Gong:2004fq,Saridakis:2007cy,
Cai:2007us,Setare:2008bb,Saridakis:2007ns,
Suwa:2009gm, BouhmadiLopez:2011qvd,
Khurshudyan:2014axa,
Saridakis:2017rdo,Nojiri:2017opc,
Saridakis:2018unr,
Kritpetch:2020vea,Saridakis:2020zol,Dabrowski:2020atl,
daSilva:2020bdc, Mamon:2020spa,
Bhattacharjee:2020ixg,
Huang:2021zgj,Lin:2021bxv,Colgain:2021beg,
Nojiri:2021iko,Shekh:2021ule}.
Recently, an extension of the holographic dark energy scenario was constructed
in \citep{Drepanou:2021jiv}, based on Kaniadakis entropy. The latter is an
extended entropy arising from the relativistic extension of standard
statistical theory, quantified by one new parameter
\citep{Kaniadakis:2002zz,Kaniadakis:2005zk}. In the case where this Kaniadakis
parameter becomes zero, i.e. when Kaniadakis entropy becomes the standard
Bekenstein-Hawking
entropy, Kaniadakis-holographic dark energy recovers
standard-holographic dark energy, however, in the general case, it exhibits a range of behaviors with interesting cosmological implications.
In this work, we investigate Kaniadakis-holographic dark energy, in
order to tackle the late time universe acceleration problem. The outline of the paper is
as follows. In Section \ref{MB} the mathematical background of the model is
considered, presenting the master equations. Section \ref{sec:data} presents
the observational confrontation analysis that includes three data samples and
the results from the corresponding constraints. Section \ref{sec:SA} is
dedicated to the dynamical system investigation and the stability analysis.
Finally, in Section \ref{sec:Con} we give a brief summary and a discussion of the
results. Throughout the manuscript we use natural units where
$\tilde{c}=\hbar=k_{B}=1$ (unless stated otherwise).
\section{ Kaniadakis holographic dark energy} \label{MB}
In this section we briefly review Kaniadakis holographic dark energy and we
elaborate the corresponding equations in order to bring them to a form suitable
for observational confrontation.
The essence of holographic dark energy is the
inequality $\rho_{DE} L^4\leq S$, with $\rho_{DE}$ being the holographic dark energy density, $L$ the largest distance
(typically a horizon), and $S$ the entropy expression in the case of a black hole with a horizon $L$ \citep{Li:2004rb,Wang:2016och}. In the standard
application using Bekenstein-Hawking entropy
$S_{BH}\propto A/(4G)=\pi L^2/G$, where $A$ is the area and $G$ the Newton's
constant, one obtains standard-holographic dark energy,
i.e. $\rho_{DE}=3c^2 M_p^2 L^{-2}$, where $M_p^2=(8\pi G)^{-1}$ is the Planck mass
and $c$ is the model parameter arising from the saturation of the above
inequality.
On the other hand, one can construct the one-parameter
generalization of the classical entropy, namely Kaniadakis entropy
$S_{K}=- k_{_B} \sum_i n_i\, \ln_{_{\{{\scriptstyle
K}\}}}\!n_i $ \citep{Kaniadakis:2002zz,Kaniadakis:2005zk},
where $k_{_B}$ is the Boltzmann constant and
with $\ln_{_{\{{\scriptstyle
K}\}}}\!x=(x^{K}-x^{-K})/2K$. This is characterized by the
dimensionless parameter $-1<K<1$, which accounts for the relativistic
deviations from standard statistical mechanics, and
in the limit $K\rightarrow0$ it recovers standard entropy.
Kaniadakis entropy can be re-expressed as
\citep{Abreu:2016avj,Abreu:2017hiy,Abreu:2021avp}
\begin{equation}
\label{kstat}
S_{K} =-k_{_B}\sum^{W}_{i=1}\frac{P^{1+K}_{i}-P^{1-K}_{i}}{2K},
\end{equation}
where $P_i$ is the probability of a specific microstate of the system and
$W$ the total number of possible configurations. Applied in the
black-hole framework, it results into
\citep{Drepanou:2021jiv,Moradpour:2020dfm,Lymperis:2021qty}
\begin{equation} \label{kentropy}
S_{K} = \frac{1}{K}\sinh{(K S_{BH})},
\end{equation}
which gives standard
Bekenstein-Hawking entropy
in the limit $K\rightarrow 0$.
Finally, since any deviations from standard thermodynamics are expected to be
small, one can approximate (\ref{kentropy}) for $K\ll1$,
acquiring \citep{Drepanou:2021jiv}
\begin{equation}\label{kentropy2}
S_{K} = S_{BH}+ \frac{K^2}{6} S_{BH}^3+ {\cal{O}}(K^4).
\end{equation}
In order to analyze the dynamics of the universe, we consider the homogeneous and isotropic cosmology based on the
Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) line element
$ds^2=-dt^2+a(t)(dr^2+r^2d\Omega^2)$, where $d\Omega^2\equiv
d\theta^2+\sin^2\theta d\varphi^2$, $a(t)$ is the scale factor and we consider
null spatial curvature $k=0$. Furthermore, as usual we use $L$
as the future event horizon $R_h\equiv a \int_t^{\infty }
\frac{1}{a(s)} \, ds$.
Inserting these into the above formulation, and using Kaniadakis entropy
instead of Bekenstein-Hawking one, we extract the energy density of
Kaniadakis holographic dark energy as \citep{Drepanou:2021jiv}
\begin{eqnarray}
&& \rho_{DE}= \frac{3c^2M_p^2}{R_h^2}+K^2 M_{p}^6
R_h^2, \label{rhoDE}
\end{eqnarray}
with $c>0$ and $K$ being the two parameters of the model.
Hence, we can write the Friedmann and Raychaudhuri equations as
\begin{eqnarray}
&&H^2=\frac{1}{3M_p^2}(\rho_m+\rho_{DE}), \label{Frie}\\
&&\dot{H}=-\frac{1}{2M_p^2}(\rho_m+p_m+\rho_{DE}+p_{DE}), \label{Ray}
\end{eqnarray}
where $H\equiv \dot{a}/a$ is the Hubble parameter, $\rho_m$ and $p_m$ are the energy density and
pressure of matter perfect fluid, while the matter conservation leads to dark
energy conservation and, in turn, to the dark energy pressure
\begin{eqnarray}
&& p_{DE}= -\frac{2c^2 M_p^2}{R_h^3 H}-\frac{
c^2M_p^2}{R_h^2}+K^2 M_{p}^6\left[\frac{2R_h}{3
H}-\frac{5}{3} R_h^2\right]. \label{pDE}
\end{eqnarray}
The combination of Raychaudhuri equation \eqref{Ray}
and \eqref{rhoDE}, \eqref{pDE} gives
\begin{eqnarray}
\dot{H}&=&\frac{c^2}{R_h^3 H}+\frac{c^2(3
w_m+1)}{2R_h^2}-\frac{3}{2}(w_m+1) H^2 \nonumber \\
&&-K^2 M_p^4\left[\frac{R_h}{3
H}-\frac{1}{6}
R_h^2 (3 w_m+5)\right],
\end{eqnarray}
where $w_m\equiv p_m/\rho_m$ is the equation of state (EoS) parameter for
matter, considered from now on as dust ($w_m=0$). From this expression we can
construct the deceleration and jerk parameters, which give us information about
the transition to an accelerated Universe. Thus, using the definition
of $R_h$,
we obtain that the energy density is
\begin{align}
\rho_{DE} = & \frac{3 c^2 M_p^2}{a^2 \left(\int_t^{\infty } \frac{1}{a(s)} \,
ds\right)^2}+K^2 M_{p}^6 a^2 \left(\int_t^{\infty }
\frac{1}{a(s)} \, ds\right)^2, \label{2rhoDE}
\end{align}
and the pressure
\begin{align}
p_{DE}= & -\frac{2 c^2 M_p^2}{a^3 H
\left(\int_t^{\infty } \frac{1}{a(s)} \, ds\right)^3}-\frac{c^2
M_p^2}{a^2 \left(\int_t^{\infty } \frac{1}{a(s)} \,
ds\right)^2}\nonumber \\
& +K^2 M_{p}^6\left[\frac{2a \left(\int_t^{\infty }
\frac{1}{a(s)} \, ds\right)}{3 H} -\frac{5}{3}
a^2 \left(\int_t^{\infty } \frac{1}{a(s)} \,
ds\right){}^2\right]. \label{2pDE}
\end{align}
Moreover, the fractional energy density of DE is defined as
\begin{equation}
\Omega_{DE}:=\frac{ \rho_{DE}}{3 M_p^2 H^2}= \frac{K^2 M_p^6 a^4 \left(\int_t^{\infty } \frac{1}{a(s)} \,
ds\right)^4+3 c^2 M_p^2}{3 M_p^2 a^2 H^2
\left(\int_t^{\infty } \frac{1}{a(s)} \, ds\right)^2}. \label{defODE1}
\end{equation}
From definition
\eqref{defODE1} we have four branches for
\begin{equation}
\mathcal{I}(t):= \int_t^{\infty } {a(s)}^{-1} \, ds,
\end{equation}
which give four possible expressions for the particle horizon
\begin{align}
{R_h}_{1,2}(t)&= \mp \frac{\left[3H^2\Omega_{DE}-\sqrt{9 H^4\Omega_{DE}^2-12 c^2 K^2
M_p^4}\right]^{ {1}/{2}}}{\sqrt{2} |K| M_p^2}, \label{1y2}\\
{R_h}_{3,4}(t)&= \mp \frac{\left[3 H^2\Omega_{DE}+\sqrt{9 H^4\Omega_{DE}^2-12 c^2 K^2
M_p^4}\right]^{ {1}/{2}}}{\sqrt{2} |K| M_p^2}. \label{3y4}
\end{align}
${R_h}_1(t)$ and ${R_h}_3(t)$ are both discarded since they lead to negative particle horizon.
To decide between the choices ${R_h}_2(t)$ and ${R_h}_4(t)$, which are both non negative, we calculate the limit $K\rightarrow 0$ and obtain
\begin{equation}
\lim_{K\rightarrow 0} {R_{h}}_2 = \frac{c}{H
\sqrt{\Omega_{DE}}}, \quad \lim_{K\rightarrow 0} {R_{h}}_4 = \infty.
\end{equation}
That is, $ {R_{h}}_2(t)$ defined by \eqref{1y2} is the only physical solution.
We proceed by introducing the usual dimensionless variable
\begin{equation}
E\equiv \frac{H}{H_0},
\label{Edefin}
\end{equation}
where $H_0$ is the Hubble constant at present time and, for convenience, we define the dimensionless constant $\beta\equiv \frac{K M_p^2}{H_0^2}$.
Differentiating
(\ref{defODE1}) and (\ref{Edefin}), and using the Friedmann equations we obtain
the master equations
\begin{align}
{\Omega_{DE}^{\prime}} = (1-\Omega_{DE}) \left[3 (w_m+1) \Omega_{DE} +2 \mathcal{X} \right], \label{eq2.15}
\\
E^{\prime} =E \left[ -\frac{3}{2} (w_m+1) (1-\Omega_{DE})+ \mathcal{X}\right], \label{eq2.16}
\end{align}
where
\begin{eqnarray}
\mathcal{X} &\equiv & \frac{1}{E^3}\left[\frac{2 \beta ^2 E^4 \Omega_{DE}^2-8 \beta ^4 c^2/3}{3 E^2 \Omega_{DE}-\sqrt{9 E^4 \Omega_{DE}^2-12 \beta ^2
c^2}}\right]^{1/2}\nonumber \\
&&-\frac{1}{E^2}[E^4 \Omega_{DE}^2-4 \beta ^2 c^2/3]^{1/2}. \label{eq2.17}
\end{eqnarray}
We use initial conditions $\Omega_{DE}(0)\equiv\Omega_{DE}^{(0)}=1-
\Omega_m^{(0)}, E(0)=1$,
where primes denote derivatives with respect to e-foldings number
$N=\ln(a/a_0)$, and $N=0$ marks the current time (from now on, the index
``0'' marks the value of a quantity at present).
The physical region of the phase space is
\begin{equation}
3 E^4 \Omega_{DE}^2 -4 \beta ^2 c^2\geq 0.
\end{equation}
Notice that $\mathcal{X}\rightarrow \frac{\Omega_{DE}^{ {3}/{2}}}{c}- \Omega_{DE}$
as $\beta\rightarrow 0$.
From the matter conservation equation, we arrive at
\begin{equation}
\rho_m^{\prime}(N)= -3 (1+w_m)\rho_m, \quad \rho_m(0)= 3 M_p^2 H_0^2 \Omega_{m}^{(0)},
\end{equation}
and, therefore, we have $\rho_{m}(N)= 3 H_0^2M_p^2 \Omega_m^{(0)} e^{-3 N (w_m+1)} $ which then leads to
\begin{align}
& \Omega_{DE} (N)=1- \Omega_{m} (N)= 1- \frac{\Omega_m^{(0)}
e^{-3 N (w_m+1)}}{E^2}.\label{defODE}
\end{align}
Defining $Z=E^2$, we obtain the equation
\begin{equation}
Z^{\prime} = - 3 (w_m+1) \Omega_m^{(0)} e^{-3 N (w_m+1)} + 2 \mathcal{X} Z, \quad Z(0)=1, \label{final1}
\end{equation}
where
\begingroup\makeatletter\def\f@size{7.5}\check@mathfonts
\begin{align}
&\mathcal{X} Z= -\left[ \left({{\Omega_m^{(0)}} e^{-3 N (w_m+1)}}-{Z}\right)^2-\frac{4 \beta ^2 c^2}{3}\right]^{1/2} \nonumber \\
& + \left[\frac{2 \beta ^2 \left(Z-{\Omega_m^{(0)}} e^{-3 N (w_m+1)}\right)^2-\frac{8 \beta ^4 c^2}{3}}{3 Z^2-3 Z
{\Omega_m^{(0)}} e^{-3 N (w_m+1)} -Z\sqrt{9 \left(Z-{\Omega_m^{(0)}} e^{-3 N (w_m+1)}\right)^2-12 \beta ^2 c^2}}\right]^{1/2}. \label{final2}
\end{align}
\endgroup
Thus, the evolution of $E^2(z)$ can be obtained by substituting \eqref{final2} into \eqref{final1}.
More precisely, substituting \eqref{final2} into \eqref{final1}, integrating, and imposing
the initial condition $Z(0)=1$, gives $E^2(N)$. In order to express it as
$E^2(z)$, we use the relation $N= \ln (a/a_0)=-\ln(1+z)$, which is a relation
between the e-folding ($N$), the scale factor ($a$), and the redshift ($z$).
Additionally, we can now write the deceleration parameter $q(z)$, and a cosmographic parameter which is
related to the third-order derivative of the scale factor, i.e. the cosmographic jerk parameter $j(z)$, which are given by the formulas
\begin{align}
q :=& -1- \frac{E^{\prime}}{E}, \label{q}\\
j:= & q(2q+1)-q', \label{j}
\end{align}
where $j=1$ corresponds to the case of a cosmological constant.
Hence, equation \eqref{q} becomes
\begin{eqnarray}
q&=&-1+\frac{3}{2} (w_m+1) (1-\Omega_{DE})- \mathcal{X},
\end{eqnarray}
with $ \mathcal{X}$ defined by \eqref{eq2.17}.
$j$ is found by direct evaluation of \eqref{j}.
We have mentioned before that taking the limit $\beta \rightarrow 0$ in \eqref{eq2.15}
and \eqref{eq2.16}, and neglecting error terms $O\left(\beta ^2\right)$, we acquire the approximated differential
equations
\begin{align}
\Omega_{DE}^{\prime}= \frac{\Omega_{DE} (1-\Omega_{DE}) \left(3 w_m c+c+2 \sqrt{\Omega_{DE}}\right)}{c}, \label{ODEK0}
\end{align}
\begin{align}
E^{\prime}= \frac{E \left\{2 \Omega_{DE}^{ {3}/{2}}+c [3 w_m
(\Omega_{DE}-1)+\Omega_{DE}-3]\right\}}{2 c}. \label{HK0}
\end{align}
Equations \eqref{ODEK0} and \eqref{HK0} characterize standard holographic cosmology.
Imposing the conditions
\begin{equation}
E(\Omega_{DE}^{(0)})=1, \;\; \ln \left(\frac{a}{a_0}\right)\Big|_{\Omega_{DE}^{(0)}}=0,
\end{equation}
we obtain the implicit solutions
\begin{eqnarray}
E= & \left(\frac{{\Omega_{DE}}}{{\Omega_{DE}^{(0)}}}\right)^{-\frac{3 (w_m+1)}{6 w_m+2}} \left(\frac{1-\sqrt{{\Omega_{DE}}}}{1-\sqrt{{\Omega_{DE}^{(0)}}}}\right)^{\frac{c-1}{3 c w_m+c+2}}
\left(\frac{\sqrt{{\Omega_{DE}}}+1}{\sqrt{{\Omega_{DE}^{(0)}}}+1}\right)^{\frac{c+1}{3 c w_m+c-2}} \nonumber \\
& \times \left(\frac{3 c w_m+c+2 \sqrt{{\Omega_{DE}}}}{3 c w_m+c+2 \sqrt{{\Omega_{DE}^{(0)}}}}\right)^{-\frac{12 (w_m+1)}{(3 w_m+1) \left((3 c w_m+c)^2-4\right)}},
\end{eqnarray}
and
\begin{eqnarray}
&(1+z)^{-1}:= \left(\frac{a}{a_0}\right) \nonumber \\
&= \left(\frac{\Omega_{DE}}{{\Omega_{DE}^{(0)}}}\right)^{\frac{1}{3 w_m+1}} \left(\frac{1-\sqrt{\Omega_{DE}}}{1-\sqrt{{\Omega_{DE}^{(0)}}}}\right)^{-\frac{c}{3 c w_m+c+2}} \left(\frac{\sqrt{\Omega_{DE}}+1}{\sqrt{{\Omega_{DE}^{(0)}}}+1}\right)^{-\frac{c}{3 c w_m+c-2}} \nonumber \\
& \times \left(\frac{3 c w_m+c+2 \sqrt{\Omega_{DE}}}{3 c w_m+c+2 \sqrt{{\Omega_{DE}^{(0)}}}}\right)^{\frac{8}{(3 w_m+1) \left((3 c
w_m+c)^2-4\right)}}.
\end{eqnarray}
Lastly, expanding around $\beta=0$ and $\Omega_{DE}=1$ and removing second order terms, the deceleration parameter \eqref{q} and the cosmographic jerk parameter \eqref{j} (in the dark-energy dominated epoch) are given by
\begin{align}
&q =-\frac{1}{c}+\frac{(1-\Omega_{DE}) (3 c w_m+c+3)}{2 c}, \\
&j= \frac{2-c}{c^2}+\frac{(1- \Omega_{DE}) (3 c w_m+c+3) [c (3 w_m+2)-2]}{2 c^2}.
\end{align}
Furthermore, expanding around $\beta=0$ and $\Omega_{DE}=0$ and removing second order terms, the deceleration parameter \eqref{q} and the cosmographic jerk parameter \eqref{j} (in the matter dominated epoch) are given by
\begin{align}
&q =\frac{1}{2} (3 w_m+1) (1- \Omega_{DE}),\\
&j=\frac{1}{2} [9 w_m (w_m+1)+2](1- \Omega _{DE}).
\end{align}
\section{Observational analysis} \label{sec:data}
One of the goals of this work is to provide observational bounds on the
parameter of Kaniadakis entropy $K$ or, more conveniently $\beta$, however we
are also interested in the behavior of all cosmological parameters, namely on
the vector
${\bm\Theta} = \{h, \Omega_m^{(0)}, \beta, c\}$. For
the parameter estimation we use the recent measurements of the observational
Hubble data as well as data from type Ia supernovae, and baryon acoustic
oscillations observations. In what follows, we first briefly introduce these
datasets and the Bayesian methodology, and then we apply it in the scenario of
Kaniadakis-holographic dark energy, providing the resulting observational
constraints.
\subsection{Data and methodology } \label{sec:Data}
\subsubsection{Cosmic chronometer data}
The Hubble parameter $H(z)$ describes the expansion rate of the Universe as a
function of redshift $z$. Currently, this parameter can be estimated from baryon
acoustic oscillations measurements and differential age in passive galaxies
(dubbed as cosmic chronometers). While the former could be biased due to
the assumption of a fiducial cosmology, the samples from cosmic
chronometers are independent from the underlying cosmological model. Thus, in
this work we only consider the $31$ points from cosmic chronometer sample
presented in \citet{Moresco:2016mzx,Magana:2018} in the redshift range
$0.07<z<1.965$. We assume a Gaussian likelihood function for this observation
as $\mathcal{L}_{\mathrm {CC}}\propto \exp{(-\chi_{\mathrm{CC}}^{2}/2)}$, where
the figure-of-merit is
\begin{equation} \label{eq:chi2_CC}
\chi^2_{\mathrm{CC}}=\sum_i^{31}\left[\frac{H_{mod}({\bm\Theta},z_i)-H_{dat}(z_i)}{\sigma^i_{dat} }\right]^2,
\end{equation}
where $H_{dat}(z_i)$ and $\sigma^i_{obs}$ are the measured Hubble parameter and its observational uncertainty
at the redshift $z_i$,
respectively. The
predicted Hubble parameter by the Kaniadakis-holographic dark energy is denoted
by $H_{mod}({\bm\Theta})$, and it can be obtained by solving the system
of equations \eqref{eq2.15}-\eqref{eq2.17}.
\subsubsection{Pantheon SNIa sample}
Since the discovery of the late cosmic acceleration with the observations of
high redshift type Ia supernovae (SNIa) by \citet{Riess:1998, Perlmutter:1999},
the
observation of these distant objects is a crucial test to determine if a
cosmological scenario is a viable candidate for the description of the
late-time Universe. The probe consists of confronting the observed luminosity
distance (or distance module) of SNIa with the theoretical prediction of any
model. Up
to now, the Pantheon sample \citep{Scolnic:2018} is the largest
collection of high-redshift SNIa, with $1048$ data points with measured redshifts
in the range $0.001<z<2.3$. The authors also provide a binned sample
containing 40 points of binned distances $\mu_{dat, bin}$ in the redshift range
$0.014<z<1.61$. In this work, we use the binned set and we consider a
Gaussian likelihood $\mathcal{L}_{SNIa} \propto \exp{(-\chi_{SNIa}^2/2)}$. By
marginalizing the nuisance parameters, the figure-of-merit function
$\chi_{SNIa}^2$ is given by
\begin{equation}
\chi_{SNIa}^{2}=a +\log \left( \frac{e}{2\pi} \right)-\frac{b^{2}}{e}, \label{fPan}
\end{equation}
where
$a=\Delta\boldsymbol{\tilde{\mu}}^{T}\cdot{\bm C_{P}^{-1}}
\cdot\Delta\boldsymbol{\tilde{\mu}},\,
b=\Delta\boldsymbol{\tilde{\mu}}^{T}\cdot\bm{C_{P}^{-1}}\cdot\Delta{\bm 1}$, $e=\Delta{\bm 1}^{T}\cdot{\bm C_{P}^{-1}}\cdot\Delta{\bm 1}$, and
$\Delta\boldsymbol{\tilde{\mu}}$ is the vector of residuals between the model
distance modulus and the observed (binned) one. The covariance matrix $\bm{C_{P}}$
takes into account systematic and statistical uncertainties \citep{Scolnic:2018}.
Moreover, the theoretical counterpart of the distance modulus for any
cosmological model is given by $\mu_{mod}({\bm\Theta},z) = 5 \log_{10} \left(
d_L({\bm\Theta},z) / 10 {\rm pc} \right)$, where $d_{L}$ is the luminosity
distance given by
\begin{equation}
d_{L}({\bm\Theta},z)=\frac{\tilde{c}}{H_{0}}(1+z) \int^{z}_{0}\frac{{\rm dz}^{\prime}}{E(z^{\prime})},
\label{eq:dl}
\end{equation}
where $\tilde{c}$ is the light speed.
\subsubsection{Baryon Acoustic Oscillations}
Baryon Acoustic Oscillations (BAO) are fluctuation patterns in the matter density field as result of internal interactions in the hot primordial plasma during the pre-recombination stage. Based on luminous
red galaxies, a sample of 15 transversal BAO scale measurements within the
redshift $0.110<z<2.225$ were collected by \citet{Nunes_2020}.
Assuming a
Gaussian likelihood, $\mathcal{L}_{\mathrm {BAO}}\propto
\exp{(-\chi_{\mathrm{BAO}}^{2}/2)}$, we build the figure of merit function as
\begin{equation} \label{eq:chi2_BAO}
\chi^2_{\rm BAO} = \sum_{i=1}^{15} \left[ \frac{\theta_{dat}^i -
\theta_{mod}(\Theta,z_i) }{\sigma_{\theta_{dat}^i}}\right]^2\,,
\end{equation}
where $\theta_{dat}^i \pm \sigma_{\theta_{dat}^i}$ is the BAO angular scale and
its uncertainty at $68\%$ measured at $z_i$. The theoretical BAO angular scale
counterpart, denoted as $\theta_{mod}$, is estimated by
\begin{equation}
\theta_{mod}(z) = \frac{r_{drag}}{(1+z)D_A(z)}\,,
\end{equation}
where $D_A=d_L(z)/(1+z)^2$ is the angular diameter distance at $z$ which
depends on the dimensionless luminosity distance $d_L(z)$, and $r_{drag}$ is
the sound horizon at the baryon drag epoch, considered to be $r_{drag}=137.7 \pm
3.6\,$Mpc \citep{Aylor:2019}.
\subsubsection{Bayesian analysis}
A Bayesian statistical analysis based on Markov Chain Monte Carlo (MCMC)
algorithm is performed to bound the free parameters of the Kaniadakis-holographic dark energy. The MCMC approach is implemented through the \texttt{emcee} python
module \citep{Emcee:2013} in which we generate $1000$ chains with $250$ steps,
each one after a burn-in phase. The latter is stopped when the chains have
converged based on the auto-correlation time criteria. Thus, the inference of
the parameter space is obtained by minimizing a Gaussian log-likelihood,
$-2\ln(\mathcal{L}_{\rm data})\varpropto \chi^2_{\rm data}$, considering flat
priors in the intervals: $h\in[0.2,1]$, $\Omega_m^{(0)}\in[0,1]$,
$\beta\in[-1,1]$, $c\in[0,2]$ for each dataset.
Additionally, a combined analysis is performed by assuming no correlation
between the datasets, hence the figure of merit is
\begin{equation} \label{eq:chi2_joint}
\chi^2_{\rm Joint} = \chi^2_{\rm CC}+\chi^2_{\rm SNIa}+\chi^2_{\rm BAO}\,,
\end{equation}
namely, the sum of the $\chi^2$ corresponding to each sample as previously defined.
\subsection{Results from observational constraints}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{contour1_hkan_cc0.pdf}
\caption{
Two-dimensional likelihood contours at $68\%$ and $99.7\%$ confidence level
(CL), alongside the corresponding 1D posterior distribution of the free
parameters, in Kaniadakis-holographic dark energy case. The stars
denote the mean values using the joint analysis, and the dashed lines represent
the best-fit values for $\Lambda$CDM cosmology \citep{Planck:2020}. }
\label{fig:contours}
\end{figure}
We perform the full confrontation described above for the scenario of
Kaniadakis holographic dark energy, and in
Fig. \ref{fig:contours} we present the 2D parameter likelihood contours at
$68\%$ ($1\sigma$) and $99.7\%$ ($3\sigma$) confidence level (CL) respectively,
alongside the corresponding 1D posterior distribution of the parameters.
Additionally, Table \ref{tab:bestfits} shows the mean values of the parameters
and their uncertainties at $1\sigma$.
\begin{table*}
\caption{Mean values of various parameters and their $68\%$ CL
uncertainties for Kaniadakis-holographic dark energy. The quantities $\Delta$AICc ($\Delta$BIC) are the
differences with respect to $\Lambda$CDM paradigm.}
\centering
\begin{tabular}{|lccccccc|}
\hline
Sample & $\chi^2$ & $h$ & $\Omega_m^{(0)}$ & $\beta$ & $c$ &
$\Delta$AICc & $\Delta$BIC \\
\hline
CC & $14.69$ & $0.690^{+0.072}_{-0.043}$ & $0.284^{+0.066}_{-0.055}$ & $0.012^{+0.486}_{-0.489}$ & $0.729^{+0.665}_{-0.350}$ & $5.2$ & $7.0$ \\ [0.9ex]
SNIa & $48.52$ & $0.597^{+0.279}_{-0.271}$ & $0.259^{+0.059}_{-0.069}$ & $0.013^{+0.480}_{-0.482}$ & $0.932^{+0.492}_{-0.302}$ & $5.1$ & $7.5$ \\ [0.9ex]
BAO & $13.01$ & $0.758^{+0.041}_{-0.035}$ & $0.403^{+0.167}_{-0.151}$ & $-0.006^{+0.433}_{-0.418}$ & $0.756^{+0.759}_{-0.463}$ & $8.2$ & $5.6$ \\ [0.9ex]
CC+SNIa+BAO & $98.07$ & $0.761^{+0.011}_{-0.010}$ & $0.211^{+0.043}_{-0.044}$ & $-0.003^{+0.412}_{-0.420}$ & $1.151^{+0.401}_{-0.287}$ & $21.7$ & $26.1$ \\ [0.9ex]
\hline
\end{tabular}
\label{tab:bestfits}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{plot_Hz_hkan_cc0_joint.pdf}\\
\includegraphics[width=0.5\textwidth]{plot_qz_hkan_cc0_joint.pdf}\\
\includegraphics[width=0.5\textwidth]{plot_jz_hkan_cc0_joint.pdf}
\caption{Reconstruction of
the Hubble function ($H(z)$, upper panel), the deceleration
parameter ($q(z)$, middle panel), and the jerk parameter ($j(z)$, bottom panel)
for the
Kanadiakis-holographic dark energy using the combined (CC+SNIa+BAO) analysis in the redshift range $0<z<2$.
The shaded regions
represent the $68\%$ confidence level,
and the square points
depict the results of the
$\Lambda$CDM scenario with $h=0.723$ and $\Omega_m^{(0)}=0.290$, namely the values obtained through observational confrontation using the same datasets with the analysis of Kaniadakis holographic dark energy.}
\label{fig:cosmography}
\end{figure}
In order to statistically compare these results
with
$\Lambda$CDM cosmology, we apply the corrected
Akaike information criterion (AICc) \citep{AIC:1974, Sugiura:1978, AICc:1989} and
the Bayesian information criterion (BIC) \citep{schwarz1978}.
They give a
penalty according to size of data sample ($N$) and the number of degrees of freedom
($k$) defined as
${\rm AICc}= \chi^2_{min}+2k +(2k^2+2k)/(N-k-1)$
and ${\rm BIC}=\chi^2_{min}+k\log(N)$ respectively, where $\chi^2_{min}$ is the
minimum value of the $\chi^2$. Thus, a model with lower values of AICc and BIC
is preferred by the data. According to the difference between a given model and
the reference one, denoted as $\Delta\rm{AICc}$, one has the following:
if $\Delta \rm{AICc}<4$, both models are supported by the data equally, i.e
they are statistically equivalent.
If $4<\Delta\rm{AICc}<10$, the data still support the given model but less than the preferred one.
If $\Delta \rm{AICc}>10$,
it indicates that the data does not support the
given model.
Similarly, the difference between a candidate model and the reference model,
denoted as $\Delta \rm{BIC}$,
is interpreted
in this way:
if $\Delta \rm{BIC}<2$,
there is no evidence against the candidate
model,
if $2<\Delta\rm{BIC}<6$,
there is modest evidence against the candidate model,
if $6<\Delta \rm{BIC}<10$, there is strong evidence against the candidate model, and
$\Delta \rm{BIC}>10$ gives the strongest evidence against it.
Hence, we have performed the above comparison, taking $\Lambda$CDM scenario as
the reference model, and
we display the results in the last two columns of
Table \ref{tab:bestfits}.
A first observation is that the Kaniadakis parameter $\beta$ is constrained
around 0 as expected, namely around the value in which Kaniadakis entropy
recovers the standard Bekenstein-Hawking one. A second observation is that
the scenario at hand gives a slightly smaller value for $\Omega_m^{(0)}$
comparing to $\Lambda$CDM cosmology, however it
estimates
a higher value for the
present Hubble constant $h$, closer to its direct measurements through long-period Cepheids.
In particular, it is consistent within
$1\sigma$ with the value reported by \citet{Riess:2019} and it exhibits a
deviation of $4.18\sigma$ from the one obtained by Planck \citet{Planck:2020}.
On the other hand, based on our mean value of $c=1.151^{+0.401}_{-0.287}$ it is interesting that we do not observe a turning point in the $H(z)$ reconstruction shown in Fig. \ref{fig:cosmography}, a feature from which the usual holographic dark energy suffers when $c<1$ \citep{Colgain:2021beg}. Hence, we deduce that Kaniadakis holographic dark energy
can also solve such a problem and thus avoid to violate the Null Energy Condition (NEC).
Concerning the comparison with $\Lambda$CDM scenario,
for the combined dataset analysis we find that $\Delta\rm{AICc}$
implies that $\Lambda$CDM is strongly favored
over Kaniadakis-holographic dark energy.
This result is also supported by BIC, for which $\Delta \rm{BIC}$ gives a strong
evidence against it. Notice that these comparisons were performed by using the same datasets for both models $\Lambda$CDM and Kaniadakis cosmology.
Finally, based on the combined (CC+SNIa+BAO) analysis, in Fig.
\ref{fig:cosmography} we present the reconstruction of the Hubble parameter
$H(z)$,
the deceleration parameter $q(z)$ (equation \eqref{q}), and
the cosmographic jerk parameter
$j(z)$ (equation \eqref{j}), in the redshift range $0<z<2$. For comparison, we also depict the
corresponding curves for $\Lambda$CDM scenario.
Concerning the current values, our analysis leads to
$H_0 = 76.09^{+1.06}_{-1.02}\, \rm{km/s/Mpc}$,
$q_0 = -0.537^{+0.064}_{-0.064}$,
$j_0 = 0.815^{+0.315}_{-0.274}$, where the uncertainties correspond to
$1\sigma$ CL. Additionally, using the joint analysis we find the redshift
for the deceleration-acceleration transition as $z_T =
0.860^{+0.213}_{-0.138}$, and the Universe age as $t_U =
13.000^{+0.406}_{-0.350} \,\rm{Gyrs}$.
Notice that $z_T$ value is
in agreement within $1\sigma$ with the value reported in
\citet{Herrera-Zamorano:2020} for $\Lambda$CDM paradigm ($z_T=0.642^{+0.014}_{-0.014}$).
\section{Dynamical system and stability analysis} \label{sec:SA}
In this section we apply the powerful method of phase-space and stability
analysis, which allows us to obtain a qualitative description of the local and
global dynamics of cosmological scenarios, independently of the initial
conditions and the
specific evolution of the universe.
The extraction of asymptotic solutions give theoretical values that can be compared with the observed ones, such as the dark-energy and total equation-of-state parameters, the deceleration parameter, the density parameters of the different sectors, etc., and also allows the classification of the cosmological solutions \citep{Ellis}.
In order to perform the stability analysis of a given cosmological scenario,
one transforms it to its autonomous form $\label{eomscol}
\textbf{X}'=\textbf{f(X)}$
\citep{Ellis,Ferreira:1997au,Copeland:1997et,Perko,Coley:2003mj,Copeland:2006wr,Chen:2008ft,Cotsakis:2013zha,Giambo:2009byn},
where $\textbf{X}$ is the column vector containing the auxiliary variables and
primes denote derivative
with respect to a conveniently chosen time variable. Then, one extracts the critical points
$\bm{X_c}$ by imposing the condition $\bm{X}'=0$ and,
to determine
their stability properties, one expands around them with $\textbf{U}$ the
column vector of the perturbations of the variables. Therefore,
for each critical point the perturbation equations are expanded to first
order as $\label{perturbation} \bm{U}'={\bm{Q}}\cdot
\bm{U}$, with the matrix ${\bm {Q}}$ containing the coefficients of the
perturbation equations. Finally, the eigenvalues of ${\bm {Q}}$ determine the
type and stability of the critical point under consideration.
\subsection{Local dynamical system formulation}
In this subsection we study the stability of system
\eqref{eq2.15}-\eqref{eq2.16} with $\mathcal{X}$ defined in \eqref{eq2.17}, in
the phase space
\begin{equation}
\left\{(E, \Omega_{DE})\in \mathbb{R}^2: 3 E^4 \Omega_{DE}^2 -4 \beta ^2 c^2\geq 0 \right\}.\label{Phase35}
\end{equation}
For generality, we keep the matter equation-of-state parameter $w_m$ in the
calculations, and it can be set to zero in the
final result if needed.
Since $\beta$ and $c$ appear quadratic in
\eqref{eq2.15}, \eqref{eq2.16}
\eqref{eq2.17} and \eqref{Phase35}, these equations are invariant under the changes $c\mapsto -c$ and $\beta\mapsto -\beta$. Therefore, in this section we focus on $\beta>0$ and $c>0$. When $\beta<0$ we change $\beta$ by $-\beta$ and $c$ by $-c$ on the next discussion.
The equilibrium points dominated by dark energy (namely possessing
$\Omega_{DE}=1$) with finite $H$ are:
\begin{itemize}
\item $L_1: (E, \Omega_{DE})=\left(\frac{\sqrt{2 \beta c }}{\sqrt[4]{3}},1\right)$. This point always satisfies
$-12 c^2\beta^2 +
9 E^4\Omega_{DE}^2 =0$. The eigenvalues are $\left\{-3 (w_m+1),\infty \;
\text{sgn}\left(\left(\sqrt{2}-2 c\right)\right)\right\}$. It is a stable point
for $c>\frac{\sqrt{2}}{2}$ and $w_m>-1$, and a saddle for
$c<\frac{\sqrt{2}}{2}$ and $w_m>-1$ .
\item $L_2: (E, \Omega_{DE})=\left(\frac{\sqrt{ \beta }}{\sqrt[4]{3(1-c^2)}}, 1\right)$. This point satisfies the reality condition if
$\frac{3 \beta ^2 \left(1-2 c^2\right)^2}{1-c^2}\geq 0$, namely $\beta =0, c^2>
1$ or $\beta\neq 0, c^2<1$. For $ c^2\leq \frac{1}{2}$ the eigenvalues are
\begin{eqnarray*}
{\lambda_1, \lambda_2}= \left\{\frac{ \left(4 c^4-4 c^2-1\right) |c|+\left(-8
c^4+6 c^2+1\right)
\sqrt{ 1-c^2}}{ \left|c-2 c^3\right|}, \right.\nonumber \\
\left.
2
\left(\sqrt{\frac{1}{c^2}-1}-1\right) \left(2 c^2-1\right)-3
(w_m+1)\right\}.
\end{eqnarray*}
This is a saddle point, as it can be
verified numerically in Fig. \ref{fig:StabilityL2}.
Moreover, for $\frac{1}{2}<c^2<1$, the eigenvalues are $\left\{2-2 c^2,-3
(w_m+1)\right\}$, and thus for $w_m>-1$ it is also a
saddle point.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{StabilityL2.pdf}
\caption{{\it{The eigenvalues corresponding to the point $L_2$, for
$w_m\in[-1,1]$, $c\in [0, \sqrt{2}/2]$. }}}
\label{fig:StabilityL2}
\end{figure}
\end{itemize}
Since $\Omega_{DE}^2 \geq \frac{4 \beta ^2 c^2}{3E^4}\geq 0$, we deduce that
the only possibility to have matter domination, namely $\Omega_{DE}=0$, is when
$E\rightarrow \infty$, due to the reality condition $c^2 \beta^2 \geq 0$.
It is convenient to define the dimensionless compact variable $T=(1+E)^{-1}$
such that $T\rightarrow 0$ as $E\rightarrow \infty$ and $T\rightarrow 1$ as $E\rightarrow 0$. Then, we obtain
\begin{small}
\begin{eqnarray}
&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
T'= \frac{3}{2} (T-1) T (w_m+1)
(\Omega_{DE}-1) -\frac{T^3
\sqrt{\frac{(T-1)^4 \Omega_{DE}^2}{T^4}-\frac{4 \beta ^2 c^2}{3}}}{T-1} \nonumber \\
& \!\!\!\!\!\!\!\! -\frac{ T^5 \sqrt{\frac{2\beta^2 (T-1)^4
\Omega_{DE}^2}{T^4}-\frac{8 \beta ^4 c^2}{3}}}{(T-1)^2 \sqrt{3
(T-1)^2 \Omega_{DE}-\sqrt{9 (T-1)^4 \Omega_{DE}^2-12 \beta ^2 c^2 T^4}}}, \label{syst1}
\end{eqnarray}
\begin{eqnarray}
&
\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\! \! \! \! \! \!
\! \! \! \!
\Omega_{DE}'= (\Omega_{DE}-1) \left[ -3 (w_m+1) \Omega_{DE} + \frac{2 T^2
\sqrt{\frac{(T-1)^4 \Omega_{DE}^2}{T^4}-\frac{4 \beta ^2 c^2}{3}}}{(T-1)^2}
\right. \nonumber \\
&
\ \ \ \ \ \ \ \ \ \ \ \
\left. +\frac{2 T^2 \sqrt{2 \beta^2 (T-1)^4 \Omega_{DE}^2-\frac{8}{3} \beta
^4 c^2 T^4}}{(T-1)^3 \sqrt{3 (T-1)^2 \Omega_{DE}-\sqrt{9 (T-1)^4
\Omega_{DE}^2-12 \beta ^2 c^2 T^4}}}\right], \label{syst2}
\end{eqnarray}
\end{small}
defined on the physical region
\begin{equation}
9 (T-1)^4 \Omega_{DE}^2-12 \beta ^2 c^2 T^4\geq 0.
\end{equation}
In summary the sources/sinks are:
\begin{itemize}
\item $L_1: (E, \Omega_{DE})=\left(\frac{\sqrt{2 \beta c }}{\sqrt[4]{3}},1\right)$ is a stable point
for $c>\frac{\sqrt{2}}{2}$ and $w_m>-1$, and a saddle for
$c<\frac{\sqrt{2}}{2}$ and $w_m>-1$.
\item For the dark-energy dominated solution $L_3: (T, \Omega_{DE})=(0,1)$, the eigenvalues are
$\left\{\frac{c-1}{c},-\frac{3 c w_m+c+2}{c}\right\}$, thus it is a stable
point for $-1<w_m<1$ and $ 0<c<1$ or a saddle for $-1<w_m<1$ and $ c>1$.
\item The past attractor is the matter dominated solution $L_4: (T, \Omega_{DE})=(0,0)$, for which the eigenvalues are
$\left\{3 (w_m+1),\frac{3 (w_m+1)}{2}\right\}$, and since they are always
positive for $-1<w_m<1$ it is an unstable point.
\end{itemize}
We remark here that $E=E_c$ finite corresponds to the de Sitter solution with $H=
E_c H_0$, and $a(t)\propto e^{ E_c H_0 t}$. That is, point $L_1$ satisfies $a(t)\propto e^{ \frac{\sqrt{2 |\beta c|}}{\sqrt[4]{3}} H_0 t}$ and
it is a late-time attractor providing the accelerated regime. Additionally, for $\beta\neq 0, c^2<1$, the point $L_2$ exists and satisfies $a(t)\propto e^{\frac{\sqrt{|\beta| }}{\sqrt[4]{3 (1-c^2)}} H_0 t}$, and since it is a saddle it can provide a transient accelerated phase that can be related to inflation.
\begin{figure}
\centering
\includegraphics[width=8cm,scale=0.6]{PHASE-PLOT1.pdf}
\caption{{\it{Phase-space plot of the dynamical system
\eqref{syst1}-\eqref{syst2}, for the best fit values $\beta = -0.003$ and $c = 1.151$ of Kaniadakis-holographic dark energy, and for dust matter ($w_m = 0$).
The red curve represents the solution for the initial data
$\Omega_{DE}|_{z=0}=0.71$, corresponding to the mean value from the joint analysis CC+SNIa+BAO, and for $T|_{z=0}=0.5$. The
dashed blue region is the physical region $9 (T-1)^4 \Omega_{DE}^2-12 \beta ^2
c^2 T^4\geq 0$, where the equations are real-valued. }}}
\label{DS1}
\end{figure}
In order to present the results in a more transparent way, in Fig. \ref{DS1} we show
a phase-space plot of the system \eqref{syst1}-\eqref{syst2} for the
best fit values $\beta = -0.003$ and $c = 1.151$ and for dust matter ($w_m =
0$). The red curve represents the solution for the initial data
$\Omega_{DE}|_{z=0}=0.71$, corresponding to the mean value from the joint analysis CC+SNIa+BAO, and for $T|_{z=0}=0.5$. The dashed blue region is
the physical region $9 (T-1)^4 \Omega_{DE}^2-12 \beta ^2 c^2 T^4\geq 0$, where
the equations are real-valued.
From this figure it is confirmed that the late-time attractor is the
dark-energy dominated solution $\Omega_{DE}=1$ with $T=0$. The past attractor
is the matter-dominated solution $\Omega_{DE}=0$ with $T=0$. At the finite region, point
$L_1$ is
the stable one.
Setting $\Omega_{DE}=1$, the system \eqref{syst1}-\eqref{syst2} becomes a one-dimensional dynamical system:
\begin{align}
T' &=\frac{T^3 \sqrt{\frac{(T-1)^4}{T^4}-\frac{4 \beta ^2 c^2}{3}}}{1-T} \nonumber \\
& -\frac{T^3 \sqrt{2 \beta ^2 (T-1)^4-\frac{8}{3} \beta ^4 c^2 T^4}}{(T-1)^2 \sqrt{3 (T-1)^2-\sqrt{9 (T-1)^4-12 \beta ^2 c^2
T^4}}}.\label{1DDS}
\end{align}
The origin $T=0$ has eigenvalue $\lambda=1-\frac{1}{| c| }$. Moreover, the system admits, at most, four additional equilibrium points $T_c$, with $T_c\in\{T_1, T_2, T_3, T_4\}$ satisfying
$\frac{(T-1)^4}{T^4}-\frac{4 \beta ^2 c^2}{3}=0$. Explicitly, we have that
\begin{subequations}\label{valuesofT}
\begin{align}
T_{1,2}&= \frac{3}{3-4 \beta ^2 c^2} -\frac{2 \sqrt{3} | c \beta | }{\left| 3-4 c^2 \beta ^2\right| } \nonumber \\
& \mp \frac{\sqrt{2} \sqrt{12 | c \beta | \left| 3-4 c^2 \beta ^2\right| +\sqrt{3} \left(16 \beta ^4
c^4-9\right)} \sqrt{| c \beta | }}{\left| 3-4 c^2 \beta ^2\right| ^{3/2}}, \nonumber
\end{align}
\begin{align}
T_{3,4}&= \frac{3}{3-4 \beta ^2 c^2} + \frac{2 \sqrt{3} | c \beta | }{\left| 3-4 c^2 \beta ^2\right| }
\\
& \mp \frac{\sqrt{2} \sqrt{12 | c \beta | \left| 3-4 c^2 \beta ^2\right| +\sqrt{3} \left(9-16
\beta ^4 c^4\right)} \sqrt{| c \beta | }}{\left| 3-4 c^2 \beta ^2\right| ^{3/2}}.
\end{align}
\end{subequations}
Such points with $0<T_c<1$, corresponding to de Sitter solution $a(t)\propto e^{H_0 t \left(\frac{1}{T_c}-1\right)}$, are stable for $c\geq 1$ and otherwise are saddle.
For the
best-fit values $\beta = -0.003$ and $c = 1.151$, the origin has eigenvalue $ \lambda \approx 0.13$, and therefore it is a source. In this case the only real value is $T_3 \approx 0.941$. The exact eigenvalue is negative infinity (for $c\geq 1$) at the exact value of
$T_3$, and therefore it is stable.
In Fig. \ref{DS1D} we draw a phase-space plot of the one-dimensional dynamical system \eqref{1DDS}, for the best fit values $\beta = -0.003$ and $c = 1.151$ of Kaniadakis holographic dark energy. The equilibrium point $T=0$ is unstable,
while the de Sitter equilibrium point
$T=T_c\approx 0.941$ is stable.
\begin{figure}
\centering
\includegraphics[width=8.5cm,scale=0.7]{PHASE-PLOT1D.pdf}
\caption{{\it{Phase-space plot of the one-dimensional dynamical system
\eqref{1DDS}, for the best fit values $\beta = -0.003$ and $c = 1.151$ of Kaniadakis-holographic dark energy. The equilibrium point $T=0$ is unstable,
while the de Sitter equilibrium point
$T=T_c\approx 0.941$ is stable.}}}
\label{DS1D}
\end{figure}
\subsection{Global dynamical systems formulation}
In the previous subsection we performed the local analysis
of the scenario. However, due to the presence of rational functions
that are not analytic in the whole domain, it becomes necessary to investigate
the full global dynamics.
We start by defining the dimensionless variables $\theta, T$ as
\begin{align}
T= \frac{H_0}{H+ H_0}= \frac{1}{1+E}, \quad \theta= \arcsin \left(\sqrt{1-
\frac{\rho_{DE}}{3 M_p^2 H^2 }}\right),
\end{align}
such that
\begin{equation}
\sin^2 (\theta)= \frac{\rho_m}{3 M_p^2 H^2 }, \quad \cos^2 (\theta)= \frac{\rho_{DE}}{3 M_p^2 H^2 }.
\end{equation}
For an expanding universe ($H>0$), we have that $T\in[0,1]$, while $\theta$
is a periodic coordinate and, thus, we can set $\theta\in[-\pi, \pi]$.
Therefore, we obtain a global phase-space formulation.
\subsubsection{Standard holographic dark energy ($\beta=0$)}
\label{sect4.3.1}
In order to present the features of Kaniadakis-holographic dark energy in
comparison with standard-holographic dark energy, we first analyze the latter
case for completeness, namely we consider the system
\eqref{ODEK0}-\eqref{HK0} for $\beta=0$. In this case, we obtain
\begin{align}
& T^{\prime}=\frac{(T-1) T \left\{\cos ^2(\theta ) [(3 w_m+1)c +2 \cos (\theta
)]-3 c (w_m+1)\right\}}{2 c}, \label{Case1_a}\\
& \theta^{\prime}=-\frac{[(3 w_m+1)c+2 \cos (\theta )] \sin (2 \theta )}{4 c}.
\label{Case1_b}
\end{align}
The critical points of the above system, alongside their associated
eigenvalues, are presented in Table \ref{tab:my_label1}.
Note that $\theta$ is unique modulo $2\pi$, and focus on
$\cos \theta\geq 0$. In the following list $\arctan[x,y]$
gives the arc tangent of $y/x$, taking into account on which quadrant the point $(x,y)$ is in. When $x^2+y^2=1$, $\arctan[x,y]$ gives the number $\theta$ such that $x=\cos\theta$ and $y=\sin\theta$.
\begin{table*}
\centering
\caption{ \label{tab:my_label1}
The critical points and their associated eigenvalues of the system
\eqref{Case1_a}-\eqref{Case1_b} for $\beta=0$ in
\eqref{ODEK0}-\eqref{HK0}, namely for the case of standard holographic dark
energy. We use the notation $x=\frac{1}{2}c (3 w_m+1)$, while $c_1\in \mathbb{Z}$.}
\begin{tabular}{|c|c|c|}\hline
Label& $ (T,\theta)$ & Eigenvalues\\\hline
$P_1$ & $\left( 0, 2 \pi c_1\right)$ & $\left\{\frac{c-1}{c},-\frac{3
w_m c+c+2}{2 c}\right\}$ \\
$P_2$ & $\left( 0, \frac{1}{2} \pi \left(4 c_1-1\right)\right)$ &
$\left\{\frac{3 (w_m+1)}{2},\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_3$ & $\left(0, \frac{1}{2} \pi \left(4 c_1+1\right)\right)$ &
$\left\{\frac{3 (w_m+1)}{2},\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_4^\pm$ & $ \left(0, 2 \pi c_1 \pm \pi \right)$ & $\left\{1+\frac{1}{c},-\frac{3
w_m}{2}+\frac{1}{c}-\frac{1}{2}\right\} $\\
$P_5$ & $\left(0, \arctan\left[-x,-\sqrt{1-x^2}\right]+2 \pi c_1\right)$ & $\left\{\frac{3 (w_m+1)}{2},\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\
$P_6$ & $ \left(0, \arctan\left[-x,\sqrt{1-x^2}\right]+2 \pi c_1\right)$ & $\left\{\frac{3 (w_m+1)}{2},\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\
$P_7$ & $ \left(1, 2 \pi c_1\right)$ & $\left\{\frac{1}{c}-1,-\frac{3
w_m c+c+2}{2 c}\right\}$ \\
$P_8$ & $\left(1, \frac{1}{2} \pi \left(4 c_1-1\right)\right)$ &
$\left\{-\frac{3}{2} (w_m+1),\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_9$ & $\left(1, \frac{1}{2} \pi \left(4 c_1+1\right)\right)$ &
$\left\{-\frac{3}{2} (w_m+1),\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_{10}^\pm$ & $ \left(1, 2 \pi c_1\pm\pi \right)$ & $\left\{-\frac{c+1}{c},-\frac{3
w_m}{2}+\frac{1}{c}-\frac{1}{2}\right\}$ \\
$P_{11}$ & $ \left(1, \arctan\left[-x,-\sqrt{1-x^2}\right]+2 \pi c_1\right)$ & $\left\{-\frac{3}{2} (w_m+1),\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\
$P_{12}$ & $ \left(1, \arctan\left[-x,\sqrt{1-x^2}\right]+2 \pi c_1\right) $& $\left\{-\frac{3}{2} (w_m+1),\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\\hline
\end{tabular}
\end{table*}
In summary, in the case $\beta=0$,
the critical points can be completely characterized. In particular:
\begin{itemize}
\item Point $P_1$ always exists. It corresponds to a dark-energy
dominated solution, i.e. $\Omega_{DE}=1$ with $T=0$. It is a stable point for
$-1<w_m<1, \quad 0<c<1$.
\item Points $P_2$ and $P_3$ exist always. They are two representations
of the matter-dominated solution $\Omega_{DE}=0$ with $T=0$. They are past
attractors, i.e. unstable points, for $-\frac{1}{3}<w_m\leq 1$, while they are
saddle for $-1<w_m<-\frac{1}{3}$.
\item Points $P_4^\pm$ exist always. They correspond to the dark-energy
dominated solution with $\Omega_{DE}=1$ with $T=0$. They are unstable points
for $0<c<\frac{1}{2}, -1\leq w_m\leq
1$, or $ c\geq \frac{1}{2}, \quad -1\leq w_m<\frac{2-c}{3 c}$, while they are
saddle for $c>\frac{1}{2}, \quad \frac{2-c}{3 c}<w_m\leq
1$.
\item Points $P_5$ and $P_6$ exist for $-1\leq \frac{1}{2}c (3 w_m+1)\leq 1$. They are sources for $0\leq c\leq 1, \quad -1<w_m<-\frac{1}{3}$ or $c>1, \quad -\frac{c+2}{3
c}<w_m<-\frac{1}{3}$.
For $0\leq c<\frac{1}{2},\quad -\frac{1}{3}<w_m\leq 1$, or $c\geq \frac{1}{2}, \quad
-\frac{1}{3}<w_m<\frac{2-c}{3 c}$, they are saddle.
\item Point $P_7$ exists always. It corresponds to a dark-energy dominated
solution $\Omega_{DE}=1$ with $T=1$. It is a stable point for $c>1, \quad -\frac{c+2}{3 c}<w_m\leq 1$.
\item Points $P_8$ and $P_9$ exist always. They are two representations
of the matter-dominated solution $\Omega_{DE}=0$ with $T=1$. They are stable
points for $-1<w_m<-\frac{1}{3}$, while they are saddle points for $-\frac{1}{3}<w_m\leq 1$.
\item Points $P_{10}^\pm$ are two representations of the matter-dominated
solution $\Omega_{DE}=0$ with $T=1$. They are stable points for $c>\frac{1}{2},
\frac{2-c}{3 c}<w_m\leq 1$, while they are saddle for $0<c<\frac{1}{2}, \quad
-1\leq w_m\leq 1$, or $c\geq \frac{1}{2}, \quad -1\leq w_m<\frac{2-c}{3
c}$.
\item Points $P_{11}$ and $P_{12}$ exist for
$-1\leq \frac{1}{2}c (3 w_m+1)\leq 1$.
They are saddle for $0\leq c\leq 1, \quad -1<w_m<-\frac{1}{3}$ or $c>1, \quad -\frac{c+2}{3
c}<w_m<-\frac{1}{3}$, while for $0\leq c<\frac{1}{2},\quad -\frac{1}{3}<w_m\leq 1$, or $c\geq \frac{1}{2}, \quad
-\frac{1}{3}<w_m<\frac{2-c}{3 c}$, they are stable.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=8cm,scale=0.6]{PHASE-PLOT3.pdf}
\caption{ {\it{Phase-space plot of the dynamical system
\eqref{Case1_a}-\eqref{Case1_b} for
$\beta=0$ in
\eqref{ODEK0}-\eqref{HK0}, namely for standard holographic dark energy, for the value $c = 1.151$, and for dust
matter $w_m = 0$. The red curve represents the solution for the initial data
$\Omega_{DE}|_{z=0}=0.71$ (i.e., $\theta
(0)=\arccos\left(\frac{1}{10}\sqrt{71}\right)\approx 0.569$), corresponding to the mean value
obtained with
the joint analysis CC+SNIa+BAO, and for $T|_{z=0}=0.5$. The
dashed blue region is the physical region where the equations are real-valued.
}}}
\label{DS3}
\end{figure}
In order to give a better picture of the system behavior,
Fig. \ref{DS3}
display a phase-space plot of the system \eqref{Case1_a}-\eqref{Case1_b} for
$\beta=0$ in
\eqref{ODEK0}-\eqref{HK0},
and
dust matter. The red curve
corresponds to the universe evolution according to parameter mean values from the joint analysis.
From this figure we deduce that the late-time attractor is the dark-energy dominated solution with $\Omega_{DE}=1$ and $T=1$ (point $P_7$), while the past attractor
is the matter-dominated solution with $\Omega_{DE}=0$ and $T=0$ (point $P_3$).
For other initial conditions there are other late-time attractors, such as points $P_{11}$ and $P_{12}$
which are stable
for the best-fit parameters
since they satisfy $c\geq \frac{1}{2}, \quad
-\frac{1}{3}<w_m<\frac{2-c}{3 c}$.
These points are scaling solutions since they have
$\Omega_{DE}= x^2$ and $\Omega_{DM}= 1-x^2$, with $x=\frac{c}{2} (3 w_m+1)= \frac{c}{2} $ for $w_m=0$.
Additionally, points $P_2$, $P_3$, which are matter-dominated solutions, and points $P_{4}^\pm$, which are dark-energy dominated solutions, are also past attractors.
\subsubsection{Kaniadakis holographic dark energy ($\beta\neq 0$)}
Let us now investigate the full extended model of Kaniadakis holographic
dark energy, namely the general case where $\beta\neq 0$.
The full system \eqref{eq2.15}-\eqref{eq2.16} becomes
\begin{eqnarray}
T^{\prime}= & \frac{3}{2} (1-T) T (w_m+1) \sin ^2(\theta ) +\frac{T^3 \sqrt{\frac{(1-T)^4 \cos
^4(\theta )}{T^4}-\frac{4 \beta ^2 c^2}{3}}}{1-T}\nonumber \\
& -\frac{ T^5 \sqrt{\frac{18 (1-T)^4 \cos ^4(\theta )\beta^2}{T^4}-24 \beta ^4 c^2}}{3 (T-1)^2 \sqrt{3 (T-1)^2 \cos ^2(\theta )-\sqrt{9 (1-T)^4 \cos ^4(\theta )-12 \beta ^2 c^2 T^4}}}, \label{FullT}
\end{eqnarray}
\begin{eqnarray}
\theta^{\prime}=& -\frac{3}{4} (w_m+1) \sin (2 \theta )+\frac{T^2 \tan (\theta ) \sqrt{\frac{(1-T)^4 \cos ^4(\theta )}{T^4}-\frac{4 \beta ^2
c^2}{3}}}{(T-1)^2} \nonumber \\
& -\frac{\sqrt{\frac{2}{3}} T^2 \tan (\theta ) \sqrt{-\beta ^2 \left(4 \beta ^2 c^2 T^4-3 (1-T)^4 \cos ^4(\theta )\right)}}{(1-T)^3 \sqrt{3 (T-1)^2 \cos ^2(\theta )-\sqrt{9 (1-T)^4 \cos ^4(\theta )-12 \beta ^2
c^2 T^4}}}. \label{Fulltheta}
\end{eqnarray}
Moreover, the physical region of the phase space is
\begin{equation}
3 (1-T)^4 \cos ^4(\theta )-4 \beta ^2
c^2 T^4\geq 0. \label{region}
\end{equation}
We proceed by studying the critical points of the system
\eqref{FullT}-\eqref{Fulltheta} in the physical region
\eqref{region} and their stability. We mention that for $\beta \neq 0$ the
invariant set
$T=1$ is not physical.
Near the invariant set $T=0$ the system \eqref{FullT}-\eqref{Fulltheta} becomes
\begin{align}
& T'=\left[-\frac{\cos ^3(\theta )}{c}+\cos ^2(\theta )+\frac{3}{2} (w_m+1)
\sin ^2(\theta )\right] T+O\left(T^2\right),\\
& \theta'=-\frac{[(3 w_m+1)c+2 \cos
(\theta )] \sin (2 \theta )}{4 c}+O\left(T^2\right).
\end{align}
In Table \ref{tab:my_label2} we summarize the critical points $P_1$ to $P_6$, alongside their
associated eigenvalues. Furthermore, the stability conditions are the same as
discussed in subsection \ref{sect4.3.1}.
In summary, in the invariant set $T=0$, the critical points are:
\begin{itemize}
\item Point $P_1$ exists always. It corresponds to a dark-energy
dominated solution, i.e. $\Omega_{DE}=1$ with $T=0$. It is a stable point for
$-1<w_m<1, \quad 0<c<1$.
\item Points $P_2$ and $P_3$ exist always. They are two representations
of the matter-dominated solution $\Omega_{DE}=0$ with $T=0$. They are past
attractors, i.e. unstable points, for $-\frac{1}{3}<w_m\leq 1$, while they are
saddle for $-1<w_m<-\frac{1}{3}$.
\item Points $P_4^\pm$ exist always. They correspond to the dark-energy
dominated solution with $\Omega_{DE}=1$ with $T=0$. They are unstable points
for $0<c<\frac{1}{2}, \quad -1\leq w_m\leq
1$, or $ c\geq \frac{1}{2}, \quad -1\leq w_m<\frac{2-c}{3 c}$, while they are
saddle for $c>\frac{1}{2}, \frac{2-c}{3 c}<w_m\leq
1$.
\item Points $P_5$ and $P_6$ exist for $-1\leq \frac{1}{2}c (3 w_m+1)\leq 1$. They are unstable
for $0\leq c\leq 1, \quad -1<w_m<-\frac{1}{3}$ or $c>1, \quad -\frac{c+2}{3
c}<w_m<-\frac{1}{3}$, while for $0\leq c<\frac{1}{2},\quad -\frac{1}{3}<w_m\leq 1$, or $c\geq \frac{1}{2}, \quad
-\frac{1}{3}<w_m<\frac{2-c}{3 c}$, they are saddle.
\end{itemize}
Moreover, the system admits, at most, twelve additional equilibrium points $(\theta, T)$, with $\theta\in
\{\theta_1, \theta_2, \theta_3\}$ satisfying
$\cos^2(\theta)=1$, and $T\in\{T_1, T_2, T_3, T_4\}$ satisfying
$\frac{(T-1)^4}{T^4}-\frac{4 \beta ^2 c^2}{3}=0$, explicitly given by \eqref{valuesofT}.
Such points with $0<T_c<1$, corresponding to de Sitter solution $a(t)\propto e^{H_0 t \left(\frac{1}{T_c}-1\right)}$, are stable for $c\geq 1$ or saddle otherwise.
Notice that the physical values are the real values of $T_i$ satisfying
$0\leq T_i\leq 1$, $i=1,2,3,4$. One eigenvalue is always $-\frac{3}{2} (1 +
w_m)$, while the other one is infinite. The stability conditions are found
numerically and, moreover, for $\beta=0$ we find $T_i=0$.
Hence, we re-obtain points $P_7$ and $P_{10}^{\pm}$ in Table \ref{tab:my_label1}. Indeed, for $\beta=0$ all the results of section \ref{sect4.3.1} are recovered.
\begin{table*}
\centering
\begin{tabular}{|c|c|c|}\hline
Label& $ (T,\theta)$ & Eigenvalues\\\hline
$P_1$ & $\left( 0, 2 \pi c_1\right)$ & $\left\{\frac{c-1}{c},-\frac{3
w_m c+c+2}{2 c}\right\}$ \\
$P_2$ & $\left( 0, \frac{1}{2} \pi \left(4 c_1-1\right)\right)$ &
$\left\{\frac{3 (w_m+1)}{2},\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_3$ & $\left(0, \frac{1}{2} \pi \left(4 c_1+1\right)\right)$ &
$\left\{\frac{3 (w_m+1)}{2},\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_4^\pm$ & $ \left(0, 2 \pi c_1 \pm \pi \right)$ & $\left\{1+\frac{1}{c},-\frac{3
w_m}{2}+\frac{1}{c}-\frac{1}{2}\right\} $\\
$P_5$ & $\left(0,
\arctan\left[-x,-\sqrt{1-x^2}\right]+2 \pi c_1\right)$ & $\left\{\frac{3
(w_m+1)}{2},\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\
$P_6$ & $ \left(0,
\arctan
\left[-x,\sqrt{1-x^2}\right]+2 \pi c_1\right)$ & $\left\{\frac{3
(w_m+1)}{2},\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\\hline
\end{tabular}
\caption{ \label{tab:my_label2}
The critical points and their associated eigenvalues of the system
\eqref{FullT}-\eqref{Fulltheta} in the invariant set $T=0$.
We use the notation $x=\frac{1}{2}c (3 w_m+1)$, $c_1\in \mathbb{Z}$.}
\end{table*}
The solutions of physical interest are those with
$T=0$. Point $P_1$, which corresponds to a dark-energy dominated solution
$\Omega_{DE}=1$ with $T=0$, is stable for $-1<w_m<1, \quad 0<c<1$. Points $P_2$
and $P_3$, which are two representations of the matter-dominated solution
$\Omega_{DE}=0$ with $T=0$, are past attractors for $-\frac{1}{3}<w_m\leq 1$ or
saddle for $-1<w_m<-\frac{1}{3}$.
Points $P_4^\pm$, which correspond to a dark-energy dominated solution
are unstable for $0<c<\frac{1}{2}, \quad -1\leq w_m\leq
1$, or $ c\geq \frac{1}{2}, \quad -1\leq w_m<\frac{2-c}{3 c}$, while they are
saddle points for $c>\frac{1}{2}, \quad \frac{2-c}{3 c}<w_m\leq
1$. Finally, points $P_5$ and $P_6$ exist for $-1\leq \frac{1}{2}c (3 w_m+1)\leq 1$. They are sources for $0\leq c\leq 1, \quad -1<w_m<-\frac{1}{3}$ or $c>1, \quad -\frac{c+2}{3
c}<w_m<-\frac{1}{3}$, while for $0\leq c<\frac{1}{2},\quad -\frac{1}{3}<w_m\leq 1$, or $c\geq \frac{1}{2}, \quad
-\frac{1}{3}<w_m<\frac{2-c}{3 c}$, they are saddle.
Finally, note that the region where $T\rightarrow 1$ is contained in the
complex-valued domain. This forbids solutions with $H=0$, which appear in the
standard-holographic dark energy scenario of \eqref{Case1_a}-\eqref{Case1_b}.
\begin{figure}
\centering
\includegraphics[width=8cm,scale=0.6]{PHASE-PLOT2.pdf}
\caption{{\it{Phase-space plot of the dynamical system
\eqref{FullT}-\eqref{Fulltheta}
for the best fit values
$\beta =-0.003$ and $c = 1.151$, and for dust matter ($w_m = 0$).
The red curve represents the solution for the initial data
$\Omega_{DE}|_{z=0}=0.71$ (i.e., $\theta
(0)=\arccos\left(\frac{1}{10}\sqrt{71}\right)\approx 0.569$), corresponding to the mean value from the joint analysis CC+SNIa+BAO, and for $T|_{z=0}=0.5$. The
dashed-blue region is the physical region where the equations are real-valued.
}}}
\label{DS2}
\end{figure}
In Fig. \ref{DS2} we
show a phase-space plot of the system \eqref{FullT}-\eqref{Fulltheta} for the
best-fit values $\beta = -0.003$
and $c = 1.151$ and for dust matter ($w_m = 0$).
In this case the only real value is $T_3 \approx 0.941$. At points $(-\pi,
T_3)$, $(0, T_3)$, and $(\pi, T_3)$, the eigenvalues are $-\frac{3}{2}$ and one eigenvalue is negative infinity at the exact value of
$T_3$, therefore they are sink. For comparison, we have added
the red curve, corresponding to the solution for the initial data
$\Omega_{DE}|_{z=0}=0.71$ (i.e., $\theta
(0)=\arccos\left(\frac{1}{10}\sqrt{71}\right)\approx 0.569$), which is the mean value from the joint analysis CC+SNIa+BAO, and for $T|_{z=0}=0.5$. From this figure it is confirmed that the late-time attractor is the
dark-energy dominated solution (de Sitter solution with $a(t)\propto e^{H_0 t \left(\frac{1}{T_c}-1\right)}, H_0= h \times 100\,\mathrm{km\, s}^{-1} \mathrm{Mpc}^{-1}, T_c\approx 0.941, h= 0.761$), while the past
attractor
is the matter-dominated solution.
\section{Summary and discussion} \label{sec:Con}
We investigated the scenario of Kaniadakis-holographic dark energy scenario by
confronting it with observational data.
This is an extension of the usual holographic dark-energy model which
arises
from
the use of the generalized Kaniadakis entropy instead of the
standard Boltzmann-Gibbs one, which in turn
appear
from the relativistic
extension of standard statistical
theory.
We applied the Bayesian approach
to extract the likelihood bounds
of the Kaniadakis parameter, as well as
the other free model parameters. In
particular, we performed a Markov Chain Monte Carlo analysis using
data from cosmic chronometers, supernovae type Ia, and
Baryon Acoustic Oscillations observations.
Concerning the Kaniadakis parameter, we
found
that it is constrained
around 0, namely, around the value in which Kaniadakis entropy
recovers the standard Bekenstein-Hawking one, as expected. Additionally, for
$\Omega_m^{(0)}$ we obtained a slightly smaller value
compared
to $\Lambda$CDM scenario.
Furthermore, we reconstructed the evolution of the Hubble,
deceleration and jerk parameters in the redshift range $0<z<2$.
We find that,
within one
sigma confidence level with those reported in \citet{Herrera-Zamorano:2020}, the deceleration-acceleration transition redshift is
$z_T = 0.86^{+0.21}_{-0.14}$,
and
the age of the Universe
is $t_U = 13.000^{+0.406}_{-0.350}\,\rm{Gyrs}$.
Lastly, we applied the usual information criteria in order to compare the
statistical significance of the fittings with $\Lambda$CDM cosmology. Both criteria AICc and BIC conclude that the $\Lambda$CDM scenario is strongly favored
in comparison
to Kaniadakis-holographic dark energy.
Finally, we performed a detailed dynamical-system analysis
to extract
the local and global features of the evolution in the scenario of Kaniadakis-holographic dark energy. We extracted the critical points as well as their
stability properties and found
that
the past
attractor of the Universe is the matter-dominated solution, while the late-time
stable solution is the dark-energy-dominated one with $H\rightarrow 0$.
In summary, Kaniadakis-holographic dark energy presents interesting
cosmological behavior and is in agreement with observations. We remark that the scenario may solve the turning point in the Hubble parameter reconstruction of standard holographic dark energy \citep{Colgain:2021beg}, which violates the NEC, and thus it is an interesting improvement in this context.
\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgements}
We thank the anonymous referee for thoughtful remarks and suggestions. Authors acknowledge Eoin O. Colgain for fruitful comments. G.L. was funded by Agencia Nacional de Investigaci\'on y Desarrollo - ANID for financial support through the program FONDECYT Iniciaci\'on grant no. 11180126 and by Vicerrectoría de Investigación y Desarrollo Tecnológico at UCN. J.M. acknowledges the support from ANID project Basal AFB-170002 and ANID REDES 190147. M.A.G.-A. acknowledges support from SNI-M\'exico, CONACyT research fellow, ANID REDES (190147), C\'atedra Marcos Moshinsky and Instituto Avanzado de Cosmolog\'ia (IAC). A.H.A. thanks to the PRODEP project, Mexico for resources and financial support and thanks also to the support from Luis Aguilar, Alejandro de Le\'on, Carlos Flores, and Jair Garc\'ia of the Laboratorio Nacional de Visualizaci\'on Cient\'ifica Avanzada. V.M. acknowledges support from Centro de Astrof\'{\i}sica de Valpara\'{i}so and ANID REDES 190147. This work is partially supported by the Ministry of Education and Science of the Republic of Kazakhstan, Grant AP08856912.
\section*{Data Availability}
The data underlying this article were cited in Section \ref{sec:Data}.
\bibliographystyle{mnras}
\section{Introduction}
The acceleration of the Universe is one of the most elusive problems in modern
cosmology. Since its discovery in the last decade of the twentieth century by
Supernovae (SNIa) observations \citep{Riess:1998,Perlmutter:1999}, and its confirmation
by the acoustic peaks of the cosmic microwave background (CMB) radiation
\citep{WMAP:2003elm}, it has been a theoretical and observational challenge to
construct a model that combines all of its characteristics. From a theoretical
point of view, and assuming homogeneous and isotropic symmetries (cosmological
principle), the need for a component
with features able to reproduce the Universe acceleration is vital to obtain
accurate values for the observable Universe age and size. Recently, the
confidence in the detection of this acceleration at late times has been
increased with precise observations of the large scale structure
\citep{Nadathur:2020kvq}.
The best candidate to explain the observed acceleration
is the well-known Cosmological Constant (CC), interpreted under the assumption that quantum
vacuum fluctuations generate the constant energy density observed and, with this,
a late-time acceleration. However, when we apply the Quantum Field Theory to
assess the energy density, the result is in total discrepancy with
observations, giving rise to the so-called {\it fine-tuning problem}
\citep{Zeldovich:1968ehl, Weinberg}. In addition, recent observations developed by the collaboration \emph{Supernova $H_0$ for the Equation of State} (SH0ES) \citep{Riess:2020fzl} show a
discrepancy for the obtained value of $H_0$ when compared to Planck
observations based on the $\Lambda$ Cold Dark Matter ($\Lambda$CDM) model
\citep{Aghanim:2018}. This generates a tension of $4.2\sigma$ between the
mentioned experiments, bringing a new crisis and the need for new ways to tackle
the problem \citep{DiValentino:2020zio}, as long as this discrepancy is not related to unknown
systematic errors affecting the measurements
\citep{DES:2019fny,birrer2021, Efstathiou:2021ocp, Freedman:2021ahq,
Shah:2021onj}. Is in this vein that the community has been
proposing other alternatives to address the problem of the Universe
acceleration. In general, there are two main directions that one could follow.
The first is to maintain general relativity an introduce new peculiar forms of
matter, such as scalar fields \citep{Copeland:2006wr,Cai:2009zp,review:universe},
Chaplygin gas \citep{Chaplygin,Villanueva_2015,Hernandez-Almada:2018osh},
viscous fluids \citep{Cruz, MCruz:2017, CruzyHernandez,AlmadaViscoso,
Hernandez-Almada:2020ulm, Almada:2020}, etc, collectively known as dark-energy
sector. The second way is to construct modified gravitational theories
\citep{CANTATA:2021ktz,Capozziello:2011et} such as braneworlds models
\citep{Maartens:2010ar,Garcia-Aspeitia:2016kak,Garcia-Aspeitia:2018fvw},
emergent gravity
\citep{PEDE:2019ApJ,Pan:2019hac,PEDE:2020,Hernandez-Almada:2020uyr,
Garcia-Aspeitia:2019yni,Garcia-Aspeitia:2019yod}, Einstein-Gauss-Bonet
\citep{Glavan:2019inb,Garcia-Aspeitia:2020uwq}, thermodynamical models
\citep{Saridakis:2020cqq,Leon:2021wyx}, torsional gravity \citep{Cai:2015emx}, $f(R)$ theories \citep{Dainotti:2021},
etc.
On the other hand, there is an increasing interest in dark energy
alternative models with the holographic principle. This is inspired by the
relation between entropy and the area of a black hole. It states that the observable
degree of freedom of a physical system in a volume can be encoded in a
lower-dimensional description on its boundary \citep{Hooft:1993, Susskind:1995}.
The holographic principle imposes a connection between the infrared (IR)
cutoff, related to large-scale of the Universe, with the ultraviolet (UV) one,
related to the vacuum energy. Application of the holographic principle to the
Universe horizon gives rise to a vacuum energy of holographic origin, namely holographic dark energy \citep{Li:2004rb,Wang:2016och}. Holographic dark
energy proves to lead to interesting phenomenology and, thus, it has been
studied in detailed
\citep{Li:2004rb,Wang:2016och,Horvat:2004vn, Pavon:2005yx,
Wang:2005jx,
Nojiri:2005pu,Kim:2005at, Setare:2008pc,Setare:2008hm}, confronted to observations
\citep{Zhang:2005hs,Li:2009bn,Feng:2007wn,Zhang:2009un,Lu:2009iv,
Micheletti:2009jy} and extended to various frameworks
\citep{Gong:2004fq,Saridakis:2007cy,
Cai:2007us,Setare:2008bb,Saridakis:2007ns,
Suwa:2009gm, BouhmadiLopez:2011qvd,
Khurshudyan:2014axa,
Saridakis:2017rdo,Nojiri:2017opc,
Saridakis:2018unr,
Kritpetch:2020vea,Saridakis:2020zol,Dabrowski:2020atl,
daSilva:2020bdc, Mamon:2020spa,
Bhattacharjee:2020ixg,
Huang:2021zgj,Lin:2021bxv,Colgain:2021beg,
Nojiri:2021iko,Shekh:2021ule}.
Recently, an extension of the holographic dark energy scenario was constructed
in \citep{Drepanou:2021jiv}, based on Kaniadakis entropy. The latter is an
extended entropy arising from the relativistic extension of standard
statistical theory, quantified by one new parameter
\citep{Kaniadakis:2002zz,Kaniadakis:2005zk}. In the case where this Kaniadakis
parameter becomes zero, i.e. when Kaniadakis entropy becomes the standard
Bekenstein-Hawking
entropy, Kaniadakis-holographic dark energy recovers
standard-holographic dark energy, however, in the general case, it exhibits a range of behaviors with interesting cosmological implications.
In this work, we investigate Kaniadakis-holographic dark energy, in
order to tackle the late time universe acceleration problem. The outline of the paper is
as follows. In Section \ref{MB} the mathematical background of the model is
considered, presenting the master equations. Section \ref{sec:data} presents
the observational confrontation analysis that includes three data samples and
the results from the corresponding constraints. Section \ref{sec:SA} is
dedicated to the dynamical system investigation and the stability analysis.
Finally, in Section \ref{sec:Con} we give a brief summary and a discussion of the
results. Throughout the manuscript we use natural units where
$\tilde{c}=\hbar=k_{B}=1$ (unless stated otherwise).
\section{ Kaniadakis holographic dark energy} \label{MB}
In this section we briefly review Kaniadakis holographic dark energy and we
elaborate the corresponding equations in order to bring them to a form suitable
for observational confrontation.
The essence of holographic dark energy is the
inequality $\rho_{DE} L^4\leq S$, with $\rho_{DE}$ being the holographic dark energy density, $L$ the largest distance
(typically a horizon), and $S$ the entropy expression in the case of a black hole with a horizon $L$ \citep{Li:2004rb,Wang:2016och}. In the standard
application using Bekenstein-Hawking entropy
$S_{BH}\propto A/(4G)=\pi L^2/G$, where $A$ is the area and $G$ the Newton's
constant, one obtains standard-holographic dark energy,
i.e. $\rho_{DE}=3c^2 M_p^2 L^{-2}$, where $M_p^2=(8\pi G)^{-1}$ is the Planck mass
and $c$ is the model parameter arising from the saturation of the above
inequality.
On the other hand, one can construct the one-parameter
generalization of the classical entropy, namely Kaniadakis entropy
$S_{K}=- k_{_B} \sum_i n_i\, \ln_{_{\{{\scriptstyle
K}\}}}\!n_i $ \citep{Kaniadakis:2002zz,Kaniadakis:2005zk},
where $k_{_B}$ is the Boltzmann constant and
with $\ln_{_{\{{\scriptstyle
K}\}}}\!x=(x^{K}-x^{-K})/2K$. This is characterized by the
dimensionless parameter $-1<K<1$, which accounts for the relativistic
deviations from standard statistical mechanics, and
in the limit $K\rightarrow0$ it recovers standard entropy.
Kaniadakis entropy can be re-expressed as
\citep{Abreu:2016avj,Abreu:2017hiy,Abreu:2021avp}
\begin{equation}
\label{kstat}
S_{K} =-k_{_B}\sum^{W}_{i=1}\frac{P^{1+K}_{i}-P^{1-K}_{i}}{2K},
\end{equation}
where $P_i$ is the probability of a specific microstate of the system and
$W$ the total number of possible configurations. Applied in the
black-hole framework, it results into
\citep{Drepanou:2021jiv,Moradpour:2020dfm,Lymperis:2021qty}
\begin{equation} \label{kentropy}
S_{K} = \frac{1}{K}\sinh{(K S_{BH})},
\end{equation}
which gives standard
Bekenstein-Hawking entropy
in the limit $K\rightarrow 0$.
Finally, since any deviations from standard thermodynamics are expected to be
small, one can approximate (\ref{kentropy}) for $K\ll1$,
acquiring \citep{Drepanou:2021jiv}
\begin{equation}\label{kentropy2}
S_{K} = S_{BH}+ \frac{K^2}{6} S_{BH}^3+ {\cal{O}}(K^4).
\end{equation}
In order to analyze the dynamics of the universe, we consider the homogeneous and isotropic cosmology based on the
Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) line element
$ds^2=-dt^2+a(t)(dr^2+r^2d\Omega^2)$, where $d\Omega^2\equiv
d\theta^2+\sin^2\theta d\varphi^2$, $a(t)$ is the scale factor and we consider
null spatial curvature $k=0$. Furthermore, as usual we use $L$
as the future event horizon $R_h\equiv a \int_t^{\infty }
\frac{1}{a(s)} \, ds$.
Inserting these into the above formulation, and using Kaniadakis entropy
instead of Bekenstein-Hawking one, we extract the energy density of
Kaniadakis holographic dark energy as \citep{Drepanou:2021jiv}
\begin{eqnarray}
&& \rho_{DE}= \frac{3c^2M_p^2}{R_h^2}+K^2 M_{p}^6
R_h^2, \label{rhoDE}
\end{eqnarray}
with $c>0$ and $K$ being the two parameters of the model.
Hence, we can write the Friedmann and Raychaudhuri equations as
\begin{eqnarray}
&&H^2=\frac{1}{3M_p^2}(\rho_m+\rho_{DE}), \label{Frie}\\
&&\dot{H}=-\frac{1}{2M_p^2}(\rho_m+p_m+\rho_{DE}+p_{DE}), \label{Ray}
\end{eqnarray}
where $H\equiv \dot{a}/a$ is the Hubble parameter, $\rho_m$ and $p_m$ are the energy density and
pressure of matter perfect fluid, while the matter conservation leads to dark
energy conservation and, in turn, to the dark energy pressure
\begin{eqnarray}
&& p_{DE}= -\frac{2c^2 M_p^2}{R_h^3 H}-\frac{
c^2M_p^2}{R_h^2}+K^2 M_{p}^6\left[\frac{2R_h}{3
H}-\frac{5}{3} R_h^2\right]. \label{pDE}
\end{eqnarray}
The combination of Raychaudhuri equation \eqref{Ray}
and \eqref{rhoDE}, \eqref{pDE} gives
\begin{eqnarray}
\dot{H}&=&\frac{c^2}{R_h^3 H}+\frac{c^2(3
w_m+1)}{2R_h^2}-\frac{3}{2}(w_m+1) H^2 \nonumber \\
&&-K^2 M_p^4\left[\frac{R_h}{3
H}-\frac{1}{6}
R_h^2 (3 w_m+5)\right],
\end{eqnarray}
where $w_m\equiv p_m/\rho_m$ is the equation of state (EoS) parameter for
matter, considered from now on as dust ($w_m=0$). From this expression we can
construct the deceleration and jerk parameters, which give us information about
the transition to an accelerated Universe. Thus, using the definition
of $R_h$,
we obtain that the energy density is
\begin{align}
\rho_{DE} = & \frac{3 c^2 M_p^2}{a^2 \left(\int_t^{\infty } \frac{1}{a(s)} \,
ds\right)^2}+K^2 M_{p}^6 a^2 \left(\int_t^{\infty }
\frac{1}{a(s)} \, ds\right)^2, \label{2rhoDE}
\end{align}
and the pressure
\begin{align}
p_{DE}= & -\frac{2 c^2 M_p^2}{a^3 H
\left(\int_t^{\infty } \frac{1}{a(s)} \, ds\right)^3}-\frac{c^2
M_p^2}{a^2 \left(\int_t^{\infty } \frac{1}{a(s)} \,
ds\right)^2}\nonumber \\
& +K^2 M_{p}^6\left[\frac{2a \left(\int_t^{\infty }
\frac{1}{a(s)} \, ds\right)}{3 H} -\frac{5}{3}
a^2 \left(\int_t^{\infty } \frac{1}{a(s)} \,
ds\right){}^2\right]. \label{2pDE}
\end{align}
Moreover, the fractional energy density of DE is defined as
\begin{equation}
\Omega_{DE}:=\frac{ \rho_{DE}}{3 M_p^2 H^2}= \frac{K^2 M_p^6 a^4 \left(\int_t^{\infty } \frac{1}{a(s)} \,
ds\right)^4+3 c^2 M_p^2}{3 M_p^2 a^2 H^2
\left(\int_t^{\infty } \frac{1}{a(s)} \, ds\right)^2}. \label{defODE1}
\end{equation}
From definition
\eqref{defODE1} we have four branches for
\begin{equation}
\mathcal{I}(t):= \int_t^{\infty } {a(s)}^{-1} \, ds,
\end{equation}
which give four possible expressions for the particle horizon
\begin{align}
{R_h}_{1,2}(t)&= \mp \frac{\left[3H^2\Omega_{DE}-\sqrt{9 H^4\Omega_{DE}^2-12 c^2 K^2
M_p^4}\right]^{ {1}/{2}}}{\sqrt{2} |K| M_p^2}, \label{1y2}\\
{R_h}_{3,4}(t)&= \mp \frac{\left[3 H^2\Omega_{DE}+\sqrt{9 H^4\Omega_{DE}^2-12 c^2 K^2
M_p^4}\right]^{ {1}/{2}}}{\sqrt{2} |K| M_p^2}. \label{3y4}
\end{align}
${R_h}_1(t)$ and ${R_h}_3(t)$ are both discarded since they lead to negative particle horizon.
To decide between the choices ${R_h}_2(t)$ and ${R_h}_4(t)$, which are both non negative, we calculate the limit $K\rightarrow 0$ and obtain
\begin{equation}
\lim_{K\rightarrow 0} {R_{h}}_2 = \frac{c}{H
\sqrt{\Omega_{DE}}}, \quad \lim_{K\rightarrow 0} {R_{h}}_4 = \infty.
\end{equation}
That is, $ {R_{h}}_2(t)$ defined by \eqref{1y2} is the only physical solution.
We proceed by introducing the usual dimensionless variable
\begin{equation}
E\equiv \frac{H}{H_0},
\label{Edefin}
\end{equation}
where $H_0$ is the Hubble constant at present time and, for convenience, we define the dimensionless constant $\beta\equiv \frac{K M_p^2}{H_0^2}$.
Differentiating
(\ref{defODE1}) and (\ref{Edefin}), and using the Friedmann equations we obtain
the master equations
\begin{align}
{\Omega_{DE}^{\prime}} = (1-\Omega_{DE}) \left[3 (w_m+1) \Omega_{DE} +2 \mathcal{X} \right], \label{eq2.15}
\\
E^{\prime} =E \left[ -\frac{3}{2} (w_m+1) (1-\Omega_{DE})+ \mathcal{X}\right], \label{eq2.16}
\end{align}
where
\begin{eqnarray}
\mathcal{X} &\equiv & \frac{1}{E^3}\left[\frac{2 \beta ^2 E^4 \Omega_{DE}^2-8 \beta ^4 c^2/3}{3 E^2 \Omega_{DE}-\sqrt{9 E^4 \Omega_{DE}^2-12 \beta ^2
c^2}}\right]^{1/2}\nonumber \\
&&-\frac{1}{E^2}[E^4 \Omega_{DE}^2-4 \beta ^2 c^2/3]^{1/2}. \label{eq2.17}
\end{eqnarray}
We use initial conditions $\Omega_{DE}(0)\equiv\Omega_{DE}^{(0)}=1-
\Omega_m^{(0)}, E(0)=1$,
where primes denote derivatives with respect to e-foldings number
$N=\ln(a/a_0)$, and $N=0$ marks the current time (from now on, the index
``0'' marks the value of a quantity at present).
The physical region of the phase space is
\begin{equation}
3 E^4 \Omega_{DE}^2 -4 \beta ^2 c^2\geq 0.
\end{equation}
Notice that $\mathcal{X}\rightarrow \frac{\Omega_{DE}^{ {3}/{2}}}{c}- \Omega_{DE}$
as $\beta\rightarrow 0$.
From the matter conservation equation, we arrive at
\begin{equation}
\rho_m^{\prime}(N)= -3 (1+w_m)\rho_m, \quad \rho_m(0)= 3 M_p^2 H_0^2 \Omega_{m}^{(0)},
\end{equation}
and, therefore, we have $\rho_{m}(N)= 3 H_0^2M_p^2 \Omega_m^{(0)} e^{-3 N (w_m+1)} $ which then leads to
\begin{align}
& \Omega_{DE} (N)=1- \Omega_{m} (N)= 1- \frac{\Omega_m^{(0)}
e^{-3 N (w_m+1)}}{E^2}.\label{defODE}
\end{align}
Defining $Z=E^2$, we obtain the equation
\begin{equation}
Z^{\prime} = - 3 (w_m+1) \Omega_m^{(0)} e^{-3 N (w_m+1)} + 2 \mathcal{X} Z, \quad Z(0)=1, \label{final1}
\end{equation}
where
\begingroup\makeatletter\def\f@size{7.5}\check@mathfonts
\begin{align}
&\mathcal{X} Z= -\left[ \left({{\Omega_m^{(0)}} e^{-3 N (w_m+1)}}-{Z}\right)^2-\frac{4 \beta ^2 c^2}{3}\right]^{1/2} \nonumber \\
& + \left[\frac{2 \beta ^2 \left(Z-{\Omega_m^{(0)}} e^{-3 N (w_m+1)}\right)^2-\frac{8 \beta ^4 c^2}{3}}{3 Z^2-3 Z
{\Omega_m^{(0)}} e^{-3 N (w_m+1)} -Z\sqrt{9 \left(Z-{\Omega_m^{(0)}} e^{-3 N (w_m+1)}\right)^2-12 \beta ^2 c^2}}\right]^{1/2}. \label{final2}
\end{align}
\endgroup
Thus, the evolution of $E^2(z)$ can be obtained by substituting \eqref{final2} into \eqref{final1}.
More precisely, substituting \eqref{final2} into \eqref{final1}, integrating, and imposing
the initial condition $Z(0)=1$, gives $E^2(N)$. In order to express it as
$E^2(z)$, we use the relation $N= \ln (a/a_0)=-\ln(1+z)$, which is a relation
between the e-folding ($N$), the scale factor ($a$), and the redshift ($z$).
Additionally, we can now write the deceleration parameter $q(z)$, and a cosmographic parameter which is
related to the third-order derivative of the scale factor, i.e. the cosmographic jerk parameter $j(z)$, which are given by the formulas
\begin{align}
q :=& -1- \frac{E^{\prime}}{E}, \label{q}\\
j:= & q(2q+1)-q', \label{j}
\end{align}
where $j=1$ corresponds to the case of a cosmological constant.
Hence, equation \eqref{q} becomes
\begin{eqnarray}
q&=&-1+\frac{3}{2} (w_m+1) (1-\Omega_{DE})- \mathcal{X},
\end{eqnarray}
with $ \mathcal{X}$ defined by \eqref{eq2.17}.
$j$ is found by direct evaluation of \eqref{j}.
We have mentioned before that taking the limit $\beta \rightarrow 0$ in \eqref{eq2.15}
and \eqref{eq2.16}, and neglecting error terms $O\left(\beta ^2\right)$, we acquire the approximated differential
equations
\begin{align}
\Omega_{DE}^{\prime}= \frac{\Omega_{DE} (1-\Omega_{DE}) \left(3 w_m c+c+2 \sqrt{\Omega_{DE}}\right)}{c}, \label{ODEK0}
\end{align}
\begin{align}
E^{\prime}= \frac{E \left\{2 \Omega_{DE}^{ {3}/{2}}+c [3 w_m
(\Omega_{DE}-1)+\Omega_{DE}-3]\right\}}{2 c}. \label{HK0}
\end{align}
Equations \eqref{ODEK0} and \eqref{HK0} characterize standard holographic cosmology.
Imposing the conditions
\begin{equation}
E(\Omega_{DE}^{(0)})=1, \;\; \ln \left(\frac{a}{a_0}\right)\Big|_{\Omega_{DE}^{(0)}}=0,
\end{equation}
we obtain the implicit solutions
\begin{eqnarray}
E= & \left(\frac{{\Omega_{DE}}}{{\Omega_{DE}^{(0)}}}\right)^{-\frac{3 (w_m+1)}{6 w_m+2}} \left(\frac{1-\sqrt{{\Omega_{DE}}}}{1-\sqrt{{\Omega_{DE}^{(0)}}}}\right)^{\frac{c-1}{3 c w_m+c+2}}
\left(\frac{\sqrt{{\Omega_{DE}}}+1}{\sqrt{{\Omega_{DE}^{(0)}}}+1}\right)^{\frac{c+1}{3 c w_m+c-2}} \nonumber \\
& \times \left(\frac{3 c w_m+c+2 \sqrt{{\Omega_{DE}}}}{3 c w_m+c+2 \sqrt{{\Omega_{DE}^{(0)}}}}\right)^{-\frac{12 (w_m+1)}{(3 w_m+1) \left((3 c w_m+c)^2-4\right)}},
\end{eqnarray}
and
\begin{eqnarray}
&(1+z)^{-1}:= \left(\frac{a}{a_0}\right) \nonumber \\
&= \left(\frac{\Omega_{DE}}{{\Omega_{DE}^{(0)}}}\right)^{\frac{1}{3 w_m+1}} \left(\frac{1-\sqrt{\Omega_{DE}}}{1-\sqrt{{\Omega_{DE}^{(0)}}}}\right)^{-\frac{c}{3 c w_m+c+2}} \left(\frac{\sqrt{\Omega_{DE}}+1}{\sqrt{{\Omega_{DE}^{(0)}}}+1}\right)^{-\frac{c}{3 c w_m+c-2}} \nonumber \\
& \times \left(\frac{3 c w_m+c+2 \sqrt{\Omega_{DE}}}{3 c w_m+c+2 \sqrt{{\Omega_{DE}^{(0)}}}}\right)^{\frac{8}{(3 w_m+1) \left((3 c
w_m+c)^2-4\right)}}.
\end{eqnarray}
Lastly, expanding around $\beta=0$ and $\Omega_{DE}=1$ and removing second order terms, the deceleration parameter \eqref{q} and the cosmographic jerk parameter \eqref{j} (in the dark-energy dominated epoch) are given by
\begin{align}
&q =-\frac{1}{c}+\frac{(1-\Omega_{DE}) (3 c w_m+c+3)}{2 c}, \\
&j= \frac{2-c}{c^2}+\frac{(1- \Omega_{DE}) (3 c w_m+c+3) [c (3 w_m+2)-2]}{2 c^2}.
\end{align}
Furthermore, expanding around $\beta=0$ and $\Omega_{DE}=0$ and removing second order terms, the deceleration parameter \eqref{q} and the cosmographic jerk parameter \eqref{j} (in the matter dominated epoch) are given by
\begin{align}
&q =\frac{1}{2} (3 w_m+1) (1- \Omega_{DE}),\\
&j=\frac{1}{2} [9 w_m (w_m+1)+2](1- \Omega _{DE}).
\end{align}
\section{Observational analysis} \label{sec:data}
One of the goals of this work is to provide observational bounds on the
parameter of Kaniadakis entropy $K$ or, more conveniently $\beta$, however we
are also interested in the behavior of all cosmological parameters, namely on
the vector
${\bm\Theta} = \{h, \Omega_m^{(0)}, \beta, c\}$. For
the parameter estimation we use the recent measurements of the observational
Hubble data as well as data from type Ia supernovae, and baryon acoustic
oscillations observations. In what follows, we first briefly introduce these
datasets and the Bayesian methodology, and then we apply it in the scenario of
Kaniadakis-holographic dark energy, providing the resulting observational
constraints.
\subsection{Data and methodology } \label{sec:Data}
\subsubsection{Cosmic chronometer data}
The Hubble parameter $H(z)$ describes the expansion rate of the Universe as a
function of redshift $z$. Currently, this parameter can be estimated from baryon
acoustic oscillations measurements and differential age in passive galaxies
(dubbed as cosmic chronometers). While the former could be biased due to
the assumption of a fiducial cosmology, the samples from cosmic
chronometers are independent from the underlying cosmological model. Thus, in
this work we only consider the $31$ points from cosmic chronometer sample
presented in \citet{Moresco:2016mzx,Magana:2018} in the redshift range
$0.07<z<1.965$. We assume a Gaussian likelihood function for this observation
as $\mathcal{L}_{\mathrm {CC}}\propto \exp{(-\chi_{\mathrm{CC}}^{2}/2)}$, where
the figure-of-merit is
\begin{equation} \label{eq:chi2_CC}
\chi^2_{\mathrm{CC}}=\sum_i^{31}\left[\frac{H_{mod}({\bm\Theta},z_i)-H_{dat}(z_i)}{\sigma^i_{dat} }\right]^2,
\end{equation}
where $H_{dat}(z_i)$ and $\sigma^i_{obs}$ are the measured Hubble parameter and its observational uncertainty
at the redshift $z_i$,
respectively. The
predicted Hubble parameter by the Kaniadakis-holographic dark energy is denoted
by $H_{mod}({\bm\Theta})$, and it can be obtained by solving the system
of equations \eqref{eq2.15}-\eqref{eq2.17}.
\subsubsection{Pantheon SNIa sample}
Since the discovery of the late cosmic acceleration with the observations of
high redshift type Ia supernovae (SNIa) by \citet{Riess:1998, Perlmutter:1999},
the
observation of these distant objects is a crucial test to determine if a
cosmological scenario is a viable candidate for the description of the
late-time Universe. The probe consists of confronting the observed luminosity
distance (or distance module) of SNIa with the theoretical prediction of any
model. Up
to now, the Pantheon sample \citep{Scolnic:2018} is the largest
collection of high-redshift SNIa, with $1048$ data points with measured redshifts
in the range $0.001<z<2.3$. The authors also provide a binned sample
containing 40 points of binned distances $\mu_{dat, bin}$ in the redshift range
$0.014<z<1.61$. In this work, we use the binned set and we consider a
Gaussian likelihood $\mathcal{L}_{SNIa} \propto \exp{(-\chi_{SNIa}^2/2)}$. By
marginalizing the nuisance parameters, the figure-of-merit function
$\chi_{SNIa}^2$ is given by
\begin{equation}
\chi_{SNIa}^{2}=a +\log \left( \frac{e}{2\pi} \right)-\frac{b^{2}}{e}, \label{fPan}
\end{equation}
where
$a=\Delta\boldsymbol{\tilde{\mu}}^{T}\cdot{\bm C_{P}^{-1}}
\cdot\Delta\boldsymbol{\tilde{\mu}},\,
b=\Delta\boldsymbol{\tilde{\mu}}^{T}\cdot\bm{C_{P}^{-1}}\cdot\Delta{\bm 1}$, $e=\Delta{\bm 1}^{T}\cdot{\bm C_{P}^{-1}}\cdot\Delta{\bm 1}$, and
$\Delta\boldsymbol{\tilde{\mu}}$ is the vector of residuals between the model
distance modulus and the observed (binned) one. The covariance matrix $\bm{C_{P}}$
takes into account systematic and statistical uncertainties \citep{Scolnic:2018}.
Moreover, the theoretical counterpart of the distance modulus for any
cosmological model is given by $\mu_{mod}({\bm\Theta},z) = 5 \log_{10} \left(
d_L({\bm\Theta},z) / 10 {\rm pc} \right)$, where $d_{L}$ is the luminosity
distance given by
\begin{equation}
d_{L}({\bm\Theta},z)=\frac{\tilde{c}}{H_{0}}(1+z) \int^{z}_{0}\frac{{\rm dz}^{\prime}}{E(z^{\prime})},
\label{eq:dl}
\end{equation}
where $\tilde{c}$ is the light speed.
\subsubsection{Baryon Acoustic Oscillations}
Baryon Acoustic Oscillations (BAO) are fluctuation patterns in the matter density field as result of internal interactions in the hot primordial plasma during the pre-recombination stage. Based on luminous
red galaxies, a sample of 15 transversal BAO scale measurements within the
redshift $0.110<z<2.225$ were collected by \citet{Nunes_2020}.
Assuming a
Gaussian likelihood, $\mathcal{L}_{\mathrm {BAO}}\propto
\exp{(-\chi_{\mathrm{BAO}}^{2}/2)}$, we build the figure of merit function as
\begin{equation} \label{eq:chi2_BAO}
\chi^2_{\rm BAO} = \sum_{i=1}^{15} \left[ \frac{\theta_{dat}^i -
\theta_{mod}(\Theta,z_i) }{\sigma_{\theta_{dat}^i}}\right]^2\,,
\end{equation}
where $\theta_{dat}^i \pm \sigma_{\theta_{dat}^i}$ is the BAO angular scale and
its uncertainty at $68\%$ measured at $z_i$. The theoretical BAO angular scale
counterpart, denoted as $\theta_{mod}$, is estimated by
\begin{equation}
\theta_{mod}(z) = \frac{r_{drag}}{(1+z)D_A(z)}\,,
\end{equation}
where $D_A=d_L(z)/(1+z)^2$ is the angular diameter distance at $z$ which
depends on the dimensionless luminosity distance $d_L(z)$, and $r_{drag}$ is
the sound horizon at the baryon drag epoch, considered to be $r_{drag}=137.7 \pm
3.6\,$Mpc \citep{Aylor:2019}.
\subsubsection{Bayesian analysis}
A Bayesian statistical analysis based on Markov Chain Monte Carlo (MCMC)
algorithm is performed to bound the free parameters of the Kaniadakis-holographic dark energy. The MCMC approach is implemented through the \texttt{emcee} python
module \citep{Emcee:2013} in which we generate $1000$ chains with $250$ steps,
each one after a burn-in phase. The latter is stopped when the chains have
converged based on the auto-correlation time criteria. Thus, the inference of
the parameter space is obtained by minimizing a Gaussian log-likelihood,
$-2\ln(\mathcal{L}_{\rm data})\varpropto \chi^2_{\rm data}$, considering flat
priors in the intervals: $h\in[0.2,1]$, $\Omega_m^{(0)}\in[0,1]$,
$\beta\in[-1,1]$, $c\in[0,2]$ for each dataset.
Additionally, a combined analysis is performed by assuming no correlation
between the datasets, hence the figure of merit is
\begin{equation} \label{eq:chi2_joint}
\chi^2_{\rm Joint} = \chi^2_{\rm CC}+\chi^2_{\rm SNIa}+\chi^2_{\rm BAO}\,,
\end{equation}
namely, the sum of the $\chi^2$ corresponding to each sample as previously defined.
\subsection{Results from observational constraints}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{contour1_hkan_cc0.pdf}
\caption{
Two-dimensional likelihood contours at $68\%$ and $99.7\%$ confidence level
(CL), alongside the corresponding 1D posterior distribution of the free
parameters, in Kaniadakis-holographic dark energy case. The stars
denote the mean values using the joint analysis, and the dashed lines represent
the best-fit values for $\Lambda$CDM cosmology \citep{Planck:2020}. }
\label{fig:contours}
\end{figure}
We perform the full confrontation described above for the scenario of
Kaniadakis holographic dark energy, and in
Fig. \ref{fig:contours} we present the 2D parameter likelihood contours at
$68\%$ ($1\sigma$) and $99.7\%$ ($3\sigma$) confidence level (CL) respectively,
alongside the corresponding 1D posterior distribution of the parameters.
Additionally, Table \ref{tab:bestfits} shows the mean values of the parameters
and their uncertainties at $1\sigma$.
\begin{table*}
\caption{Mean values of various parameters and their $68\%$ CL
uncertainties for Kaniadakis-holographic dark energy. The quantities $\Delta$AICc ($\Delta$BIC) are the
differences with respect to $\Lambda$CDM paradigm.}
\centering
\begin{tabular}{|lccccccc|}
\hline
Sample & $\chi^2$ & $h$ & $\Omega_m^{(0)}$ & $\beta$ & $c$ &
$\Delta$AICc & $\Delta$BIC \\
\hline
CC & $14.69$ & $0.690^{+0.072}_{-0.043}$ & $0.284^{+0.066}_{-0.055}$ & $0.012^{+0.486}_{-0.489}$ & $0.729^{+0.665}_{-0.350}$ & $5.2$ & $7.0$ \\ [0.9ex]
SNIa & $48.52$ & $0.597^{+0.279}_{-0.271}$ & $0.259^{+0.059}_{-0.069}$ & $0.013^{+0.480}_{-0.482}$ & $0.932^{+0.492}_{-0.302}$ & $5.1$ & $7.5$ \\ [0.9ex]
BAO & $13.01$ & $0.758^{+0.041}_{-0.035}$ & $0.403^{+0.167}_{-0.151}$ & $-0.006^{+0.433}_{-0.418}$ & $0.756^{+0.759}_{-0.463}$ & $8.2$ & $5.6$ \\ [0.9ex]
CC+SNIa+BAO & $98.07$ & $0.761^{+0.011}_{-0.010}$ & $0.211^{+0.043}_{-0.044}$ & $-0.003^{+0.412}_{-0.420}$ & $1.151^{+0.401}_{-0.287}$ & $21.7$ & $26.1$ \\ [0.9ex]
\hline
\end{tabular}
\label{tab:bestfits}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{plot_Hz_hkan_cc0_joint.pdf}\\
\includegraphics[width=0.5\textwidth]{plot_qz_hkan_cc0_joint.pdf}\\
\includegraphics[width=0.5\textwidth]{plot_jz_hkan_cc0_joint.pdf}
\caption{Reconstruction of
the Hubble function ($H(z)$, upper panel), the deceleration
parameter ($q(z)$, middle panel), and the jerk parameter ($j(z)$, bottom panel)
for the
Kanadiakis-holographic dark energy using the combined (CC+SNIa+BAO) analysis in the redshift range $0<z<2$.
The shaded regions
represent the $68\%$ confidence level,
and the square points
depict the results of the
$\Lambda$CDM scenario with $h=0.723$ and $\Omega_m^{(0)}=0.290$, namely the values obtained through observational confrontation using the same datasets with the analysis of Kaniadakis holographic dark energy.}
\label{fig:cosmography}
\end{figure}
In order to statistically compare these results
with
$\Lambda$CDM cosmology, we apply the corrected
Akaike information criterion (AICc) \citep{AIC:1974, Sugiura:1978, AICc:1989} and
the Bayesian information criterion (BIC) \citep{schwarz1978}.
They give a
penalty according to size of data sample ($N$) and the number of degrees of freedom
($k$) defined as
${\rm AICc}= \chi^2_{min}+2k +(2k^2+2k)/(N-k-1)$
and ${\rm BIC}=\chi^2_{min}+k\log(N)$ respectively, where $\chi^2_{min}$ is the
minimum value of the $\chi^2$. Thus, a model with lower values of AICc and BIC
is preferred by the data. According to the difference between a given model and
the reference one, denoted as $\Delta\rm{AICc}$, one has the following:
if $\Delta \rm{AICc}<4$, both models are supported by the data equally, i.e
they are statistically equivalent.
If $4<\Delta\rm{AICc}<10$, the data still support the given model but less than the preferred one.
If $\Delta \rm{AICc}>10$,
it indicates that the data does not support the
given model.
Similarly, the difference between a candidate model and the reference model,
denoted as $\Delta \rm{BIC}$,
is interpreted
in this way:
if $\Delta \rm{BIC}<2$,
there is no evidence against the candidate
model,
if $2<\Delta\rm{BIC}<6$,
there is modest evidence against the candidate model,
if $6<\Delta \rm{BIC}<10$, there is strong evidence against the candidate model, and
$\Delta \rm{BIC}>10$ gives the strongest evidence against it.
Hence, we have performed the above comparison, taking $\Lambda$CDM scenario as
the reference model, and
we display the results in the last two columns of
Table \ref{tab:bestfits}.
A first observation is that the Kaniadakis parameter $\beta$ is constrained
around 0 as expected, namely around the value in which Kaniadakis entropy
recovers the standard Bekenstein-Hawking one. A second observation is that
the scenario at hand gives a slightly smaller value for $\Omega_m^{(0)}$
comparing to $\Lambda$CDM cosmology, however it
estimates
a higher value for the
present Hubble constant $h$, closer to its direct measurements through long-period Cepheids.
In particular, it is consistent within
$1\sigma$ with the value reported by \citet{Riess:2019} and it exhibits a
deviation of $4.18\sigma$ from the one obtained by Planck \citet{Planck:2020}.
On the other hand, based on our mean value of $c=1.151^{+0.401}_{-0.287}$ it is interesting that we do not observe a turning point in the $H(z)$ reconstruction shown in Fig. \ref{fig:cosmography}, a feature from which the usual holographic dark energy suffers when $c<1$ \citep{Colgain:2021beg}. Hence, we deduce that Kaniadakis holographic dark energy
can also solve such a problem and thus avoid to violate the Null Energy Condition (NEC).
Concerning the comparison with $\Lambda$CDM scenario,
for the combined dataset analysis we find that $\Delta\rm{AICc}$
implies that $\Lambda$CDM is strongly favored
over Kaniadakis-holographic dark energy.
This result is also supported by BIC, for which $\Delta \rm{BIC}$ gives a strong
evidence against it. Notice that these comparisons were performed by using the same datasets for both models $\Lambda$CDM and Kaniadakis cosmology.
Finally, based on the combined (CC+SNIa+BAO) analysis, in Fig.
\ref{fig:cosmography} we present the reconstruction of the Hubble parameter
$H(z)$,
the deceleration parameter $q(z)$ (equation \eqref{q}), and
the cosmographic jerk parameter
$j(z)$ (equation \eqref{j}), in the redshift range $0<z<2$. For comparison, we also depict the
corresponding curves for $\Lambda$CDM scenario.
Concerning the current values, our analysis leads to
$H_0 = 76.09^{+1.06}_{-1.02}\, \rm{km/s/Mpc}$,
$q_0 = -0.537^{+0.064}_{-0.064}$,
$j_0 = 0.815^{+0.315}_{-0.274}$, where the uncertainties correspond to
$1\sigma$ CL. Additionally, using the joint analysis we find the redshift
for the deceleration-acceleration transition as $z_T =
0.860^{+0.213}_{-0.138}$, and the Universe age as $t_U =
13.000^{+0.406}_{-0.350} \,\rm{Gyrs}$.
Notice that $z_T$ value is
in agreement within $1\sigma$ with the value reported in
\citet{Herrera-Zamorano:2020} for $\Lambda$CDM paradigm ($z_T=0.642^{+0.014}_{-0.014}$).
\section{Dynamical system and stability analysis} \label{sec:SA}
In this section we apply the powerful method of phase-space and stability
analysis, which allows us to obtain a qualitative description of the local and
global dynamics of cosmological scenarios, independently of the initial
conditions and the
specific evolution of the universe.
The extraction of asymptotic solutions give theoretical values that can be compared with the observed ones, such as the dark-energy and total equation-of-state parameters, the deceleration parameter, the density parameters of the different sectors, etc., and also allows the classification of the cosmological solutions \citep{Ellis}.
In order to perform the stability analysis of a given cosmological scenario,
one transforms it to its autonomous form $\label{eomscol}
\textbf{X}'=\textbf{f(X)}$
\citep{Ellis,Ferreira:1997au,Copeland:1997et,Perko,Coley:2003mj,Copeland:2006wr,Chen:2008ft,Cotsakis:2013zha,Giambo:2009byn},
where $\textbf{X}$ is the column vector containing the auxiliary variables and
primes denote derivative
with respect to a conveniently chosen time variable. Then, one extracts the critical points
$\bm{X_c}$ by imposing the condition $\bm{X}'=0$ and,
to determine
their stability properties, one expands around them with $\textbf{U}$ the
column vector of the perturbations of the variables. Therefore,
for each critical point the perturbation equations are expanded to first
order as $\label{perturbation} \bm{U}'={\bm{Q}}\cdot
\bm{U}$, with the matrix ${\bm {Q}}$ containing the coefficients of the
perturbation equations. Finally, the eigenvalues of ${\bm {Q}}$ determine the
type and stability of the critical point under consideration.
\subsection{Local dynamical system formulation}
In this subsection we study the stability of system
\eqref{eq2.15}-\eqref{eq2.16} with $\mathcal{X}$ defined in \eqref{eq2.17}, in
the phase space
\begin{equation}
\left\{(E, \Omega_{DE})\in \mathbb{R}^2: 3 E^4 \Omega_{DE}^2 -4 \beta ^2 c^2\geq 0 \right\}.\label{Phase35}
\end{equation}
For generality, we keep the matter equation-of-state parameter $w_m$ in the
calculations, and it can be set to zero in the
final result if needed.
Since $\beta$ and $c$ appear quadratic in
\eqref{eq2.15}, \eqref{eq2.16}
\eqref{eq2.17} and \eqref{Phase35}, these equations are invariant under the changes $c\mapsto -c$ and $\beta\mapsto -\beta$. Therefore, in this section we focus on $\beta>0$ and $c>0$. When $\beta<0$ we change $\beta$ by $-\beta$ and $c$ by $-c$ on the next discussion.
The equilibrium points dominated by dark energy (namely possessing
$\Omega_{DE}=1$) with finite $H$ are:
\begin{itemize}
\item $L_1: (E, \Omega_{DE})=\left(\frac{\sqrt{2 \beta c }}{\sqrt[4]{3}},1\right)$. This point always satisfies
$-12 c^2\beta^2 +
9 E^4\Omega_{DE}^2 =0$. The eigenvalues are $\left\{-3 (w_m+1),\infty \;
\text{sgn}\left(\left(\sqrt{2}-2 c\right)\right)\right\}$. It is a stable point
for $c>\frac{\sqrt{2}}{2}$ and $w_m>-1$, and a saddle for
$c<\frac{\sqrt{2}}{2}$ and $w_m>-1$ .
\item $L_2: (E, \Omega_{DE})=\left(\frac{\sqrt{ \beta }}{\sqrt[4]{3(1-c^2)}}, 1\right)$. This point satisfies the reality condition if
$\frac{3 \beta ^2 \left(1-2 c^2\right)^2}{1-c^2}\geq 0$, namely $\beta =0, c^2>
1$ or $\beta\neq 0, c^2<1$. For $ c^2\leq \frac{1}{2}$ the eigenvalues are
\begin{eqnarray*}
{\lambda_1, \lambda_2}= \left\{\frac{ \left(4 c^4-4 c^2-1\right) |c|+\left(-8
c^4+6 c^2+1\right)
\sqrt{ 1-c^2}}{ \left|c-2 c^3\right|}, \right.\nonumber \\
\left.
2
\left(\sqrt{\frac{1}{c^2}-1}-1\right) \left(2 c^2-1\right)-3
(w_m+1)\right\}.
\end{eqnarray*}
This is a saddle point, as it can be
verified numerically in Fig. \ref{fig:StabilityL2}.
Moreover, for $\frac{1}{2}<c^2<1$, the eigenvalues are $\left\{2-2 c^2,-3
(w_m+1)\right\}$, and thus for $w_m>-1$ it is also a
saddle point.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{StabilityL2.pdf}
\caption{{\it{The eigenvalues corresponding to the point $L_2$, for
$w_m\in[-1,1]$, $c\in [0, \sqrt{2}/2]$. }}}
\label{fig:StabilityL2}
\end{figure}
\end{itemize}
Since $\Omega_{DE}^2 \geq \frac{4 \beta ^2 c^2}{3E^4}\geq 0$, we deduce that
the only possibility to have matter domination, namely $\Omega_{DE}=0$, is when
$E\rightarrow \infty$, due to the reality condition $c^2 \beta^2 \geq 0$.
It is convenient to define the dimensionless compact variable $T=(1+E)^{-1}$
such that $T\rightarrow 0$ as $E\rightarrow \infty$ and $T\rightarrow 1$ as $E\rightarrow 0$. Then, we obtain
\begin{small}
\begin{eqnarray}
&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
T'= \frac{3}{2} (T-1) T (w_m+1)
(\Omega_{DE}-1) -\frac{T^3
\sqrt{\frac{(T-1)^4 \Omega_{DE}^2}{T^4}-\frac{4 \beta ^2 c^2}{3}}}{T-1} \nonumber \\
& \!\!\!\!\!\!\!\! -\frac{ T^5 \sqrt{\frac{2\beta^2 (T-1)^4
\Omega_{DE}^2}{T^4}-\frac{8 \beta ^4 c^2}{3}}}{(T-1)^2 \sqrt{3
(T-1)^2 \Omega_{DE}-\sqrt{9 (T-1)^4 \Omega_{DE}^2-12 \beta ^2 c^2 T^4}}}, \label{syst1}
\end{eqnarray}
\begin{eqnarray}
&
\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\! \! \! \! \! \!
\! \! \! \!
\Omega_{DE}'= (\Omega_{DE}-1) \left[ -3 (w_m+1) \Omega_{DE} + \frac{2 T^2
\sqrt{\frac{(T-1)^4 \Omega_{DE}^2}{T^4}-\frac{4 \beta ^2 c^2}{3}}}{(T-1)^2}
\right. \nonumber \\
&
\ \ \ \ \ \ \ \ \ \ \ \
\left. +\frac{2 T^2 \sqrt{2 \beta^2 (T-1)^4 \Omega_{DE}^2-\frac{8}{3} \beta
^4 c^2 T^4}}{(T-1)^3 \sqrt{3 (T-1)^2 \Omega_{DE}-\sqrt{9 (T-1)^4
\Omega_{DE}^2-12 \beta ^2 c^2 T^4}}}\right], \label{syst2}
\end{eqnarray}
\end{small}
defined on the physical region
\begin{equation}
9 (T-1)^4 \Omega_{DE}^2-12 \beta ^2 c^2 T^4\geq 0.
\end{equation}
In summary the sources/sinks are:
\begin{itemize}
\item $L_1: (E, \Omega_{DE})=\left(\frac{\sqrt{2 \beta c }}{\sqrt[4]{3}},1\right)$ is a stable point
for $c>\frac{\sqrt{2}}{2}$ and $w_m>-1$, and a saddle for
$c<\frac{\sqrt{2}}{2}$ and $w_m>-1$.
\item For the dark-energy dominated solution $L_3: (T, \Omega_{DE})=(0,1)$, the eigenvalues are
$\left\{\frac{c-1}{c},-\frac{3 c w_m+c+2}{c}\right\}$, thus it is a stable
point for $-1<w_m<1$ and $ 0<c<1$ or a saddle for $-1<w_m<1$ and $ c>1$.
\item The past attractor is the matter dominated solution $L_4: (T, \Omega_{DE})=(0,0)$, for which the eigenvalues are
$\left\{3 (w_m+1),\frac{3 (w_m+1)}{2}\right\}$, and since they are always
positive for $-1<w_m<1$ it is an unstable point.
\end{itemize}
We remark here that $E=E_c$ finite corresponds to the de Sitter solution with $H=
E_c H_0$, and $a(t)\propto e^{ E_c H_0 t}$. That is, point $L_1$ satisfies $a(t)\propto e^{ \frac{\sqrt{2 |\beta c|}}{\sqrt[4]{3}} H_0 t}$ and
it is a late-time attractor providing the accelerated regime. Additionally, for $\beta\neq 0, c^2<1$, the point $L_2$ exists and satisfies $a(t)\propto e^{\frac{\sqrt{|\beta| }}{\sqrt[4]{3 (1-c^2)}} H_0 t}$, and since it is a saddle it can provide a transient accelerated phase that can be related to inflation.
\begin{figure}
\centering
\includegraphics[width=8cm,scale=0.6]{PHASE-PLOT1.pdf}
\caption{{\it{Phase-space plot of the dynamical system
\eqref{syst1}-\eqref{syst2}, for the best fit values $\beta = -0.003$ and $c = 1.151$ of Kaniadakis-holographic dark energy, and for dust matter ($w_m = 0$).
The red curve represents the solution for the initial data
$\Omega_{DE}|_{z=0}=0.71$, corresponding to the mean value from the joint analysis CC+SNIa+BAO, and for $T|_{z=0}=0.5$. The
dashed blue region is the physical region $9 (T-1)^4 \Omega_{DE}^2-12 \beta ^2
c^2 T^4\geq 0$, where the equations are real-valued. }}}
\label{DS1}
\end{figure}
In order to present the results in a more transparent way, in Fig. \ref{DS1} we show
a phase-space plot of the system \eqref{syst1}-\eqref{syst2} for the
best fit values $\beta = -0.003$ and $c = 1.151$ and for dust matter ($w_m =
0$). The red curve represents the solution for the initial data
$\Omega_{DE}|_{z=0}=0.71$, corresponding to the mean value from the joint analysis CC+SNIa+BAO, and for $T|_{z=0}=0.5$. The dashed blue region is
the physical region $9 (T-1)^4 \Omega_{DE}^2-12 \beta ^2 c^2 T^4\geq 0$, where
the equations are real-valued.
From this figure it is confirmed that the late-time attractor is the
dark-energy dominated solution $\Omega_{DE}=1$ with $T=0$. The past attractor
is the matter-dominated solution $\Omega_{DE}=0$ with $T=0$. At the finite region, point
$L_1$ is
the stable one.
Setting $\Omega_{DE}=1$, the system \eqref{syst1}-\eqref{syst2} becomes a one-dimensional dynamical system:
\begin{align}
T' &=\frac{T^3 \sqrt{\frac{(T-1)^4}{T^4}-\frac{4 \beta ^2 c^2}{3}}}{1-T} \nonumber \\
& -\frac{T^3 \sqrt{2 \beta ^2 (T-1)^4-\frac{8}{3} \beta ^4 c^2 T^4}}{(T-1)^2 \sqrt{3 (T-1)^2-\sqrt{9 (T-1)^4-12 \beta ^2 c^2
T^4}}}.\label{1DDS}
\end{align}
The origin $T=0$ has eigenvalue $\lambda=1-\frac{1}{| c| }$. Moreover, the system admits, at most, four additional equilibrium points $T_c$, with $T_c\in\{T_1, T_2, T_3, T_4\}$ satisfying
$\frac{(T-1)^4}{T^4}-\frac{4 \beta ^2 c^2}{3}=0$. Explicitly, we have that
\begin{subequations}\label{valuesofT}
\begin{align}
T_{1,2}&= \frac{3}{3-4 \beta ^2 c^2} -\frac{2 \sqrt{3} | c \beta | }{\left| 3-4 c^2 \beta ^2\right| } \nonumber \\
& \mp \frac{\sqrt{2} \sqrt{12 | c \beta | \left| 3-4 c^2 \beta ^2\right| +\sqrt{3} \left(16 \beta ^4
c^4-9\right)} \sqrt{| c \beta | }}{\left| 3-4 c^2 \beta ^2\right| ^{3/2}}, \nonumber
\end{align}
\begin{align}
T_{3,4}&= \frac{3}{3-4 \beta ^2 c^2} + \frac{2 \sqrt{3} | c \beta | }{\left| 3-4 c^2 \beta ^2\right| }
\\
& \mp \frac{\sqrt{2} \sqrt{12 | c \beta | \left| 3-4 c^2 \beta ^2\right| +\sqrt{3} \left(9-16
\beta ^4 c^4\right)} \sqrt{| c \beta | }}{\left| 3-4 c^2 \beta ^2\right| ^{3/2}}.
\end{align}
\end{subequations}
Such points with $0<T_c<1$, corresponding to de Sitter solution $a(t)\propto e^{H_0 t \left(\frac{1}{T_c}-1\right)}$, are stable for $c\geq 1$ and otherwise are saddle.
For the
best-fit values $\beta = -0.003$ and $c = 1.151$, the origin has eigenvalue $ \lambda \approx 0.13$, and therefore it is a source. In this case the only real value is $T_3 \approx 0.941$. The exact eigenvalue is negative infinity (for $c\geq 1$) at the exact value of
$T_3$, and therefore it is stable.
In Fig. \ref{DS1D} we draw a phase-space plot of the one-dimensional dynamical system \eqref{1DDS}, for the best fit values $\beta = -0.003$ and $c = 1.151$ of Kaniadakis holographic dark energy. The equilibrium point $T=0$ is unstable,
while the de Sitter equilibrium point
$T=T_c\approx 0.941$ is stable.
\begin{figure}
\centering
\includegraphics[width=8.5cm,scale=0.7]{PHASE-PLOT1D.pdf}
\caption{{\it{Phase-space plot of the one-dimensional dynamical system
\eqref{1DDS}, for the best fit values $\beta = -0.003$ and $c = 1.151$ of Kaniadakis-holographic dark energy. The equilibrium point $T=0$ is unstable,
while the de Sitter equilibrium point
$T=T_c\approx 0.941$ is stable.}}}
\label{DS1D}
\end{figure}
\subsection{Global dynamical systems formulation}
In the previous subsection we performed the local analysis
of the scenario. However, due to the presence of rational functions
that are not analytic in the whole domain, it becomes necessary to investigate
the full global dynamics.
We start by defining the dimensionless variables $\theta, T$ as
\begin{align}
T= \frac{H_0}{H+ H_0}= \frac{1}{1+E}, \quad \theta= \arcsin \left(\sqrt{1-
\frac{\rho_{DE}}{3 M_p^2 H^2 }}\right),
\end{align}
such that
\begin{equation}
\sin^2 (\theta)= \frac{\rho_m}{3 M_p^2 H^2 }, \quad \cos^2 (\theta)= \frac{\rho_{DE}}{3 M_p^2 H^2 }.
\end{equation}
For an expanding universe ($H>0$), we have that $T\in[0,1]$, while $\theta$
is a periodic coordinate and, thus, we can set $\theta\in[-\pi, \pi]$.
Therefore, we obtain a global phase-space formulation.
\subsubsection{Standard holographic dark energy ($\beta=0$)}
\label{sect4.3.1}
In order to present the features of Kaniadakis-holographic dark energy in
comparison with standard-holographic dark energy, we first analyze the latter
case for completeness, namely we consider the system
\eqref{ODEK0}-\eqref{HK0} for $\beta=0$. In this case, we obtain
\begin{align}
& T^{\prime}=\frac{(T-1) T \left\{\cos ^2(\theta ) [(3 w_m+1)c +2 \cos (\theta
)]-3 c (w_m+1)\right\}}{2 c}, \label{Case1_a}\\
& \theta^{\prime}=-\frac{[(3 w_m+1)c+2 \cos (\theta )] \sin (2 \theta )}{4 c}.
\label{Case1_b}
\end{align}
The critical points of the above system, alongside their associated
eigenvalues, are presented in Table \ref{tab:my_label1}.
Note that $\theta$ is unique modulo $2\pi$, and focus on
$\cos \theta\geq 0$. In the following list $\arctan[x,y]$
gives the arc tangent of $y/x$, taking into account on which quadrant the point $(x,y)$ is in. When $x^2+y^2=1$, $\arctan[x,y]$ gives the number $\theta$ such that $x=\cos\theta$ and $y=\sin\theta$.
\begin{table*}
\centering
\caption{ \label{tab:my_label1}
The critical points and their associated eigenvalues of the system
\eqref{Case1_a}-\eqref{Case1_b} for $\beta=0$ in
\eqref{ODEK0}-\eqref{HK0}, namely for the case of standard holographic dark
energy. We use the notation $x=\frac{1}{2}c (3 w_m+1)$, while $c_1\in \mathbb{Z}$.}
\begin{tabular}{|c|c|c|}\hline
Label& $ (T,\theta)$ & Eigenvalues\\\hline
$P_1$ & $\left( 0, 2 \pi c_1\right)$ & $\left\{\frac{c-1}{c},-\frac{3
w_m c+c+2}{2 c}\right\}$ \\
$P_2$ & $\left( 0, \frac{1}{2} \pi \left(4 c_1-1\right)\right)$ &
$\left\{\frac{3 (w_m+1)}{2},\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_3$ & $\left(0, \frac{1}{2} \pi \left(4 c_1+1\right)\right)$ &
$\left\{\frac{3 (w_m+1)}{2},\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_4^\pm$ & $ \left(0, 2 \pi c_1 \pm \pi \right)$ & $\left\{1+\frac{1}{c},-\frac{3
w_m}{2}+\frac{1}{c}-\frac{1}{2}\right\} $\\
$P_5$ & $\left(0, \arctan\left[-x,-\sqrt{1-x^2}\right]+2 \pi c_1\right)$ & $\left\{\frac{3 (w_m+1)}{2},\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\
$P_6$ & $ \left(0, \arctan\left[-x,\sqrt{1-x^2}\right]+2 \pi c_1\right)$ & $\left\{\frac{3 (w_m+1)}{2},\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\
$P_7$ & $ \left(1, 2 \pi c_1\right)$ & $\left\{\frac{1}{c}-1,-\frac{3
w_m c+c+2}{2 c}\right\}$ \\
$P_8$ & $\left(1, \frac{1}{2} \pi \left(4 c_1-1\right)\right)$ &
$\left\{-\frac{3}{2} (w_m+1),\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_9$ & $\left(1, \frac{1}{2} \pi \left(4 c_1+1\right)\right)$ &
$\left\{-\frac{3}{2} (w_m+1),\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_{10}^\pm$ & $ \left(1, 2 \pi c_1\pm\pi \right)$ & $\left\{-\frac{c+1}{c},-\frac{3
w_m}{2}+\frac{1}{c}-\frac{1}{2}\right\}$ \\
$P_{11}$ & $ \left(1, \arctan\left[-x,-\sqrt{1-x^2}\right]+2 \pi c_1\right)$ & $\left\{-\frac{3}{2} (w_m+1),\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\
$P_{12}$ & $ \left(1, \arctan\left[-x,\sqrt{1-x^2}\right]+2 \pi c_1\right) $& $\left\{-\frac{3}{2} (w_m+1),\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\\hline
\end{tabular}
\end{table*}
In summary, in the case $\beta=0$,
the critical points can be completely characterized. In particular:
\begin{itemize}
\item Point $P_1$ always exists. It corresponds to a dark-energy
dominated solution, i.e. $\Omega_{DE}=1$ with $T=0$. It is a stable point for
$-1<w_m<1, \quad 0<c<1$.
\item Points $P_2$ and $P_3$ exist always. They are two representations
of the matter-dominated solution $\Omega_{DE}=0$ with $T=0$. They are past
attractors, i.e. unstable points, for $-\frac{1}{3}<w_m\leq 1$, while they are
saddle for $-1<w_m<-\frac{1}{3}$.
\item Points $P_4^\pm$ exist always. They correspond to the dark-energy
dominated solution with $\Omega_{DE}=1$ with $T=0$. They are unstable points
for $0<c<\frac{1}{2}, -1\leq w_m\leq
1$, or $ c\geq \frac{1}{2}, \quad -1\leq w_m<\frac{2-c}{3 c}$, while they are
saddle for $c>\frac{1}{2}, \quad \frac{2-c}{3 c}<w_m\leq
1$.
\item Points $P_5$ and $P_6$ exist for $-1\leq \frac{1}{2}c (3 w_m+1)\leq 1$. They are sources for $0\leq c\leq 1, \quad -1<w_m<-\frac{1}{3}$ or $c>1, \quad -\frac{c+2}{3
c}<w_m<-\frac{1}{3}$.
For $0\leq c<\frac{1}{2},\quad -\frac{1}{3}<w_m\leq 1$, or $c\geq \frac{1}{2}, \quad
-\frac{1}{3}<w_m<\frac{2-c}{3 c}$, they are saddle.
\item Point $P_7$ exists always. It corresponds to a dark-energy dominated
solution $\Omega_{DE}=1$ with $T=1$. It is a stable point for $c>1, \quad -\frac{c+2}{3 c}<w_m\leq 1$.
\item Points $P_8$ and $P_9$ exist always. They are two representations
of the matter-dominated solution $\Omega_{DE}=0$ with $T=1$. They are stable
points for $-1<w_m<-\frac{1}{3}$, while they are saddle points for $-\frac{1}{3}<w_m\leq 1$.
\item Points $P_{10}^\pm$ are two representations of the matter-dominated
solution $\Omega_{DE}=0$ with $T=1$. They are stable points for $c>\frac{1}{2},
\frac{2-c}{3 c}<w_m\leq 1$, while they are saddle for $0<c<\frac{1}{2}, \quad
-1\leq w_m\leq 1$, or $c\geq \frac{1}{2}, \quad -1\leq w_m<\frac{2-c}{3
c}$.
\item Points $P_{11}$ and $P_{12}$ exist for
$-1\leq \frac{1}{2}c (3 w_m+1)\leq 1$.
They are saddle for $0\leq c\leq 1, \quad -1<w_m<-\frac{1}{3}$ or $c>1, \quad -\frac{c+2}{3
c}<w_m<-\frac{1}{3}$, while for $0\leq c<\frac{1}{2},\quad -\frac{1}{3}<w_m\leq 1$, or $c\geq \frac{1}{2}, \quad
-\frac{1}{3}<w_m<\frac{2-c}{3 c}$, they are stable.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=8cm,scale=0.6]{PHASE-PLOT3.pdf}
\caption{ {\it{Phase-space plot of the dynamical system
\eqref{Case1_a}-\eqref{Case1_b} for
$\beta=0$ in
\eqref{ODEK0}-\eqref{HK0}, namely for standard holographic dark energy, for the value $c = 1.151$, and for dust
matter $w_m = 0$. The red curve represents the solution for the initial data
$\Omega_{DE}|_{z=0}=0.71$ (i.e., $\theta
(0)=\arccos\left(\frac{1}{10}\sqrt{71}\right)\approx 0.569$), corresponding to the mean value
obtained with
the joint analysis CC+SNIa+BAO, and for $T|_{z=0}=0.5$. The
dashed blue region is the physical region where the equations are real-valued.
}}}
\label{DS3}
\end{figure}
In order to give a better picture of the system behavior,
Fig. \ref{DS3}
display a phase-space plot of the system \eqref{Case1_a}-\eqref{Case1_b} for
$\beta=0$ in
\eqref{ODEK0}-\eqref{HK0},
and
dust matter. The red curve
corresponds to the universe evolution according to parameter mean values from the joint analysis.
From this figure we deduce that the late-time attractor is the dark-energy dominated solution with $\Omega_{DE}=1$ and $T=1$ (point $P_7$), while the past attractor
is the matter-dominated solution with $\Omega_{DE}=0$ and $T=0$ (point $P_3$).
For other initial conditions there are other late-time attractors, such as points $P_{11}$ and $P_{12}$
which are stable
for the best-fit parameters
since they satisfy $c\geq \frac{1}{2}, \quad
-\frac{1}{3}<w_m<\frac{2-c}{3 c}$.
These points are scaling solutions since they have
$\Omega_{DE}= x^2$ and $\Omega_{DM}= 1-x^2$, with $x=\frac{c}{2} (3 w_m+1)= \frac{c}{2} $ for $w_m=0$.
Additionally, points $P_2$, $P_3$, which are matter-dominated solutions, and points $P_{4}^\pm$, which are dark-energy dominated solutions, are also past attractors.
\subsubsection{Kaniadakis holographic dark energy ($\beta\neq 0$)}
Let us now investigate the full extended model of Kaniadakis holographic
dark energy, namely the general case where $\beta\neq 0$.
The full system \eqref{eq2.15}-\eqref{eq2.16} becomes
\begin{eqnarray}
T^{\prime}= & \frac{3}{2} (1-T) T (w_m+1) \sin ^2(\theta ) +\frac{T^3 \sqrt{\frac{(1-T)^4 \cos
^4(\theta )}{T^4}-\frac{4 \beta ^2 c^2}{3}}}{1-T}\nonumber \\
& -\frac{ T^5 \sqrt{\frac{18 (1-T)^4 \cos ^4(\theta )\beta^2}{T^4}-24 \beta ^4 c^2}}{3 (T-1)^2 \sqrt{3 (T-1)^2 \cos ^2(\theta )-\sqrt{9 (1-T)^4 \cos ^4(\theta )-12 \beta ^2 c^2 T^4}}}, \label{FullT}
\end{eqnarray}
\begin{eqnarray}
\theta^{\prime}=& -\frac{3}{4} (w_m+1) \sin (2 \theta )+\frac{T^2 \tan (\theta ) \sqrt{\frac{(1-T)^4 \cos ^4(\theta )}{T^4}-\frac{4 \beta ^2
c^2}{3}}}{(T-1)^2} \nonumber \\
& -\frac{\sqrt{\frac{2}{3}} T^2 \tan (\theta ) \sqrt{-\beta ^2 \left(4 \beta ^2 c^2 T^4-3 (1-T)^4 \cos ^4(\theta )\right)}}{(1-T)^3 \sqrt{3 (T-1)^2 \cos ^2(\theta )-\sqrt{9 (1-T)^4 \cos ^4(\theta )-12 \beta ^2
c^2 T^4}}}. \label{Fulltheta}
\end{eqnarray}
Moreover, the physical region of the phase space is
\begin{equation}
3 (1-T)^4 \cos ^4(\theta )-4 \beta ^2
c^2 T^4\geq 0. \label{region}
\end{equation}
We proceed by studying the critical points of the system
\eqref{FullT}-\eqref{Fulltheta} in the physical region
\eqref{region} and their stability. We mention that for $\beta \neq 0$ the
invariant set
$T=1$ is not physical.
Near the invariant set $T=0$ the system \eqref{FullT}-\eqref{Fulltheta} becomes
\begin{align}
& T'=\left[-\frac{\cos ^3(\theta )}{c}+\cos ^2(\theta )+\frac{3}{2} (w_m+1)
\sin ^2(\theta )\right] T+O\left(T^2\right),\\
& \theta'=-\frac{[(3 w_m+1)c+2 \cos
(\theta )] \sin (2 \theta )}{4 c}+O\left(T^2\right).
\end{align}
In Table \ref{tab:my_label2} we summarize the critical points $P_1$ to $P_6$, alongside their
associated eigenvalues. Furthermore, the stability conditions are the same as
discussed in subsection \ref{sect4.3.1}.
In summary, in the invariant set $T=0$, the critical points are:
\begin{itemize}
\item Point $P_1$ exists always. It corresponds to a dark-energy
dominated solution, i.e. $\Omega_{DE}=1$ with $T=0$. It is a stable point for
$-1<w_m<1, \quad 0<c<1$.
\item Points $P_2$ and $P_3$ exist always. They are two representations
of the matter-dominated solution $\Omega_{DE}=0$ with $T=0$. They are past
attractors, i.e. unstable points, for $-\frac{1}{3}<w_m\leq 1$, while they are
saddle for $-1<w_m<-\frac{1}{3}$.
\item Points $P_4^\pm$ exist always. They correspond to the dark-energy
dominated solution with $\Omega_{DE}=1$ with $T=0$. They are unstable points
for $0<c<\frac{1}{2}, \quad -1\leq w_m\leq
1$, or $ c\geq \frac{1}{2}, \quad -1\leq w_m<\frac{2-c}{3 c}$, while they are
saddle for $c>\frac{1}{2}, \frac{2-c}{3 c}<w_m\leq
1$.
\item Points $P_5$ and $P_6$ exist for $-1\leq \frac{1}{2}c (3 w_m+1)\leq 1$. They are unstable
for $0\leq c\leq 1, \quad -1<w_m<-\frac{1}{3}$ or $c>1, \quad -\frac{c+2}{3
c}<w_m<-\frac{1}{3}$, while for $0\leq c<\frac{1}{2},\quad -\frac{1}{3}<w_m\leq 1$, or $c\geq \frac{1}{2}, \quad
-\frac{1}{3}<w_m<\frac{2-c}{3 c}$, they are saddle.
\end{itemize}
Moreover, the system admits, at most, twelve additional equilibrium points $(\theta, T)$, with $\theta\in
\{\theta_1, \theta_2, \theta_3\}$ satisfying
$\cos^2(\theta)=1$, and $T\in\{T_1, T_2, T_3, T_4\}$ satisfying
$\frac{(T-1)^4}{T^4}-\frac{4 \beta ^2 c^2}{3}=0$, explicitly given by \eqref{valuesofT}.
Such points with $0<T_c<1$, corresponding to de Sitter solution $a(t)\propto e^{H_0 t \left(\frac{1}{T_c}-1\right)}$, are stable for $c\geq 1$ or saddle otherwise.
Notice that the physical values are the real values of $T_i$ satisfying
$0\leq T_i\leq 1$, $i=1,2,3,4$. One eigenvalue is always $-\frac{3}{2} (1 +
w_m)$, while the other one is infinite. The stability conditions are found
numerically and, moreover, for $\beta=0$ we find $T_i=0$.
Hence, we re-obtain points $P_7$ and $P_{10}^{\pm}$ in Table \ref{tab:my_label1}. Indeed, for $\beta=0$ all the results of section \ref{sect4.3.1} are recovered.
\begin{table*}
\centering
\begin{tabular}{|c|c|c|}\hline
Label& $ (T,\theta)$ & Eigenvalues\\\hline
$P_1$ & $\left( 0, 2 \pi c_1\right)$ & $\left\{\frac{c-1}{c},-\frac{3
w_m c+c+2}{2 c}\right\}$ \\
$P_2$ & $\left( 0, \frac{1}{2} \pi \left(4 c_1-1\right)\right)$ &
$\left\{\frac{3 (w_m+1)}{2},\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_3$ & $\left(0, \frac{1}{2} \pi \left(4 c_1+1\right)\right)$ &
$\left\{\frac{3 (w_m+1)}{2},\frac{1}{2} (3 w_m+1)\right\}$ \\
$P_4^\pm$ & $ \left(0, 2 \pi c_1 \pm \pi \right)$ & $\left\{1+\frac{1}{c},-\frac{3
w_m}{2}+\frac{1}{c}-\frac{1}{2}\right\} $\\
$P_5$ & $\left(0,
\arctan\left[-x,-\sqrt{1-x^2}\right]+2 \pi c_1\right)$ & $\left\{\frac{3
(w_m+1)}{2},\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\
$P_6$ & $ \left(0,
\arctan
\left[-x,\sqrt{1-x^2}\right]+2 \pi c_1\right)$ & $\left\{\frac{3
(w_m+1)}{2},\frac{1}{8}
(3 w_m+1) \left(c^2(1+3 w_m)^2-4\right)\right\}$ \\\hline
\end{tabular}
\caption{ \label{tab:my_label2}
The critical points and their associated eigenvalues of the system
\eqref{FullT}-\eqref{Fulltheta} in the invariant set $T=0$.
We use the notation $x=\frac{1}{2}c (3 w_m+1)$, $c_1\in \mathbb{Z}$.}
\end{table*}
The solutions of physical interest are those with
$T=0$. Point $P_1$, which corresponds to a dark-energy dominated solution
$\Omega_{DE}=1$ with $T=0$, is stable for $-1<w_m<1, \quad 0<c<1$. Points $P_2$
and $P_3$, which are two representations of the matter-dominated solution
$\Omega_{DE}=0$ with $T=0$, are past attractors for $-\frac{1}{3}<w_m\leq 1$ or
saddle for $-1<w_m<-\frac{1}{3}$.
Points $P_4^\pm$, which correspond to a dark-energy dominated solution
are unstable for $0<c<\frac{1}{2}, \quad -1\leq w_m\leq
1$, or $ c\geq \frac{1}{2}, \quad -1\leq w_m<\frac{2-c}{3 c}$, while they are
saddle points for $c>\frac{1}{2}, \quad \frac{2-c}{3 c}<w_m\leq
1$. Finally, points $P_5$ and $P_6$ exist for $-1\leq \frac{1}{2}c (3 w_m+1)\leq 1$. They are sources for $0\leq c\leq 1, \quad -1<w_m<-\frac{1}{3}$ or $c>1, \quad -\frac{c+2}{3
c}<w_m<-\frac{1}{3}$, while for $0\leq c<\frac{1}{2},\quad -\frac{1}{3}<w_m\leq 1$, or $c\geq \frac{1}{2}, \quad
-\frac{1}{3}<w_m<\frac{2-c}{3 c}$, they are saddle.
Finally, note that the region where $T\rightarrow 1$ is contained in the
complex-valued domain. This forbids solutions with $H=0$, which appear in the
standard-holographic dark energy scenario of \eqref{Case1_a}-\eqref{Case1_b}.
\begin{figure}
\centering
\includegraphics[width=8cm,scale=0.6]{PHASE-PLOT2.pdf}
\caption{{\it{Phase-space plot of the dynamical system
\eqref{FullT}-\eqref{Fulltheta}
for the best fit values
$\beta =-0.003$ and $c = 1.151$, and for dust matter ($w_m = 0$).
The red curve represents the solution for the initial data
$\Omega_{DE}|_{z=0}=0.71$ (i.e., $\theta
(0)=\arccos\left(\frac{1}{10}\sqrt{71}\right)\approx 0.569$), corresponding to the mean value from the joint analysis CC+SNIa+BAO, and for $T|_{z=0}=0.5$. The
dashed-blue region is the physical region where the equations are real-valued.
}}}
\label{DS2}
\end{figure}
In Fig. \ref{DS2} we
show a phase-space plot of the system \eqref{FullT}-\eqref{Fulltheta} for the
best-fit values $\beta = -0.003$
and $c = 1.151$ and for dust matter ($w_m = 0$).
In this case the only real value is $T_3 \approx 0.941$. At points $(-\pi,
T_3)$, $(0, T_3)$, and $(\pi, T_3)$, the eigenvalues are $-\frac{3}{2}$ and one eigenvalue is negative infinity at the exact value of
$T_3$, therefore they are sink. For comparison, we have added
the red curve, corresponding to the solution for the initial data
$\Omega_{DE}|_{z=0}=0.71$ (i.e., $\theta
(0)=\arccos\left(\frac{1}{10}\sqrt{71}\right)\approx 0.569$), which is the mean value from the joint analysis CC+SNIa+BAO, and for $T|_{z=0}=0.5$. From this figure it is confirmed that the late-time attractor is the
dark-energy dominated solution (de Sitter solution with $a(t)\propto e^{H_0 t \left(\frac{1}{T_c}-1\right)}, H_0= h \times 100\,\mathrm{km\, s}^{-1} \mathrm{Mpc}^{-1}, T_c\approx 0.941, h= 0.761$), while the past
attractor
is the matter-dominated solution.
\section{Summary and discussion} \label{sec:Con}
We investigated the scenario of Kaniadakis-holographic dark energy scenario by
confronting it with observational data.
This is an extension of the usual holographic dark-energy model which
arises
from
the use of the generalized Kaniadakis entropy instead of the
standard Boltzmann-Gibbs one, which in turn
appear
from the relativistic
extension of standard statistical
theory.
We applied the Bayesian approach
to extract the likelihood bounds
of the Kaniadakis parameter, as well as
the other free model parameters. In
particular, we performed a Markov Chain Monte Carlo analysis using
data from cosmic chronometers, supernovae type Ia, and
Baryon Acoustic Oscillations observations.
Concerning the Kaniadakis parameter, we
found
that it is constrained
around 0, namely, around the value in which Kaniadakis entropy
recovers the standard Bekenstein-Hawking one, as expected. Additionally, for
$\Omega_m^{(0)}$ we obtained a slightly smaller value
compared
to $\Lambda$CDM scenario.
Furthermore, we reconstructed the evolution of the Hubble,
deceleration and jerk parameters in the redshift range $0<z<2$.
We find that,
within one
sigma confidence level with those reported in \citet{Herrera-Zamorano:2020}, the deceleration-acceleration transition redshift is
$z_T = 0.86^{+0.21}_{-0.14}$,
and
the age of the Universe
is $t_U = 13.000^{+0.406}_{-0.350}\,\rm{Gyrs}$.
Lastly, we applied the usual information criteria in order to compare the
statistical significance of the fittings with $\Lambda$CDM cosmology. Both criteria AICc and BIC conclude that the $\Lambda$CDM scenario is strongly favored
in comparison
to Kaniadakis-holographic dark energy.
Finally, we performed a detailed dynamical-system analysis
to extract
the local and global features of the evolution in the scenario of Kaniadakis-holographic dark energy. We extracted the critical points as well as their
stability properties and found
that
the past
attractor of the Universe is the matter-dominated solution, while the late-time
stable solution is the dark-energy-dominated one with $H\rightarrow 0$.
In summary, Kaniadakis-holographic dark energy presents interesting
cosmological behavior and is in agreement with observations. We remark that the scenario may solve the turning point in the Hubble parameter reconstruction of standard holographic dark energy \citep{Colgain:2021beg}, which violates the NEC, and thus it is an interesting improvement in this context.
\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgements}
We thank the anonymous referee for thoughtful remarks and suggestions. Authors acknowledge Eoin O. Colgain for fruitful comments. G.L. was funded by Agencia Nacional de Investigaci\'on y Desarrollo - ANID for financial support through the program FONDECYT Iniciaci\'on grant no. 11180126 and by Vicerrectoría de Investigación y Desarrollo Tecnológico at UCN. J.M. acknowledges the support from ANID project Basal AFB-170002 and ANID REDES 190147. M.A.G.-A. acknowledges support from SNI-M\'exico, CONACyT research fellow, ANID REDES (190147), C\'atedra Marcos Moshinsky and Instituto Avanzado de Cosmolog\'ia (IAC). A.H.A. thanks to the PRODEP project, Mexico for resources and financial support and thanks also to the support from Luis Aguilar, Alejandro de Le\'on, Carlos Flores, and Jair Garc\'ia of the Laboratorio Nacional de Visualizaci\'on Cient\'ifica Avanzada. V.M. acknowledges support from Centro de Astrof\'{\i}sica de Valpara\'{i}so and ANID REDES 190147. This work is partially supported by the Ministry of Education and Science of the Republic of Kazakhstan, Grant AP08856912.
\section*{Data Availability}
The data underlying this article were cited in Section \ref{sec:Data}.
\bibliographystyle{mnras}
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Q: (Why) Do N trials of a B.P. reduce variance by a factor of N? This question arrises from the lecture of a book called "Data mining" by Witten and Frank. I try to understand a statement and there is one point I just do not understand nor do I find any explanation:
We are considering a Bernoulli process which has a success rate p and thus mean p and variance p(1-p). So far so good. Now the text says
If N trials are taken from a Bernoulli process, the expected success
rate f=S/N is a random variable with the same mean p; the variance is
reduced by a factor of N to p(1-p)/N.
Now why is that?
If this question is trivial please excuse. I am not a mathematician but a physicist trying to get some knowledge in Data Science.
Thanks in advance!
Btw: I found this question here already but it makes me even more confused: Variance of n Bernoulli Trials
A: Variance and expected value are additive. So the expected value of $n$ trials where each has expected value $p$ is $np$, and the variance of $n$ trials where each has variance $p(1-p)$ is $np(1-p)$. Consequently, the standard deviation goes from $\sqrt{p(1-p)}$ to $\sqrt{np(1-p)}$. But in your setting, you do not consider the sum, but the average ("rate") of $n$ Bernoulli trials. Hence we divide the expected value by $n$, arriving back at $p$; and we divide the standard deviation by $n$ - or equivalently, we divide the variance by $n^2$, which gives us a variance of $\frac{np(1-p)}{n^2}=\frac{p(1-p)}n$.
Intuitively, the more often we repeat a Bernoulli trial, the less does the observed rate deviate from the expected rate.
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E-text prepared by Al Haines
BACK TO THE WOODS
The Story of a Fall from Grace
BY HUGH McHUGH
AUTHOR OF
"JOHN HENRY," "DOWN THE LINE WITH JOHN
HENRY," "IT'S UP TO YOU," ETC.
ILLUSTRATED
1902
To all the boys in the Hammer Club:--Greetings
and gesundheit! Get together now and hit
hard--for the Devil loveth a Cheerful Knocker.
CONTENTS.
JOHN HENRY'S LUCKY DAYS
JOHN HENRY'S GHOST STORY
JOHN HENRY'S BURGLAR
JOHN HENRY'S COUNTRY COP
JOHN HENRY'S TELEGRAM
JOHN HENRY'S TWO QUEENS
JOHN HENRY'S HAPPY HOME
LIST OF ILLUSTRATIONS
Yours till the last whistle blows, believe me! John Henry
Clara J.--A Dream of Peaches--Please Pass the Cream
Uncle Peter--the Original Trust Tamer
Aunt Martha--a Short, Stout Bundle of Good Nature
Tacks--the Boy Disaster
Bunch Jefferson--All to the Good and Two to Carry
CHAPTER I.
JOHN HENRY'S LUCKY DAYS.
Seven, come eleven!
After promising Clara J. that I would never again light a pipe at
the race track, there I stood, one of the busiest puff-puff laddies
on the circuit.
Well, the truth of the matter is just this: I fell asleep at the
switch and somebody put the white lights all over me.
Just how I happened to join the Dream Builders' Association I don't
know, but for several weeks I was Willie the Wild Boy at the race
track and I kept all the Bookmakers busy trying not to laugh when
they took my money.
Every day when I showed up at the gate the Pipers played "Darling,
Dream of Me!" and every time I picked a skate the Smokers' Society
went into executive session and elected me a life member.
Every horse that finished last gave me the trembling lip as he
crawled home, well aware of the fact that I had caught him with the
goods.
I blame Bunch Jefferson for putting the bug in my Central.
Bunch went down to the skating pond one day with $18 and picked
four live wires at an average of 8 to 1. Then he began to talk
about himself.
After that event whenever I happened to meet Bunch he would raise
his megaphone and fill the neighborhood with hot ozone, fresh from
the oven.
It was pitiful to see that boy swell.
Just to cure Bunch and drive him out of the balloon business I made
up my mind one day I'd run down to the Flatfish Factory and drag a
few honest dollars away from the Bookmakers.
Splash!
That's where I fell overboard.
One bright Saturday P. M. found me clinging to a wad the size of
a fountain pen and trying to decide whether I'd better play
Dinkalorum at 40 to 1 or Hysterics at 9 to 5.
I finally decided that a ten-spot on Dinkalorum would net me enough
to give Bunch a line of sad talk, so I stepped up to the poor-box
and contributed.
Dinkalorum started off in the lead like a pale streak and I
immediately bought an entirely new set of furniture for the flat.
About half way around a locomotive whistle happened to blow near
by. Dinkalorum, being a Union horse, thought it was six o'clock
and refused absolutely to work a minute overtime.
I had to put the furniture back in the store.
In the next race I decided to play a system of my own invention so
I took my program, counted seven up, four down and two up, all of
which resulted in Pink Slob at 60 to 1.
It looked good and I handed Isadore Longfinger $10 for the purpose
of tearing $600 away from him a little later on.
Pink Slob got away in the lead but he made the mistake of walking
fast instead of running, with the result that when the other horses
were back in the stable Pinkie was still giving a heel and toe
exhibition around near third base.
It wasn't my day, so I squeezed into the thirst parlor and bathed
my injured feelings with sarsaparilla.
Just before the last race I ran across Bunch. He was over $300 to
the good and he wanted to treat me to a lot of kind words he felt
like saying about himself.
Oh! but maybe he wasn't the City Boy with the Head in the Suburbs!
When I reached home that night I felt like a sock that needs
darning.
Clara J. had invited Uncle Peter to take dinner with us and he
began to give me the nervous look-over as soon as I answered roll
call.
Uncle Peter is a very stout, old gentleman. When he squeezes into
our little flat the walls act like they are bow-legged.
Uncle Peter always goes through the folding doors sideways and
every time he sits down the man in the flat below kicks because we
move the piano so often.
Tacks was also present.
Tacks is my youthful brother-in-law with a mind like a walking
delegate because he's always looking for trouble and when he finds
it he passes it up to somebody who doesn't need it.
"Evening, John!" gurgled Uncle Peter; "late, aren't you?"
"Cars blocked, delayed me," I sighed.
"New York will be a nice place when they get it finished, won't
it?" chirped Tacks.
Just then Aunt Martha squeezed in from a shopping excursion and I
went out in the hall while she counted up and dragged out the day's
spoils for Clara J. to look at.
Aunt Martha is Uncle Peter's wife only she weighs more and breathes
oftener.
When the two of them visit our bird cage at the same time the
janitor has to go out and stand in front of the building with a
view to catching it if it falls.
That night I waded into all the sporting papers and burned dream
pipes till the smoke made me dizzy.
The next day I hit the track with three sure-fires and a couple of
perhapses.
There was nothing to it. All I had to do was to keep my nerve and
not get side-tracked and I'd have enough coin to make Andrew
Carnegie's check book look like a punched meal ticket.
I played them--and when the Angelus was ringing Moses O'Brien and
three other Bookbinders were out buying meal tickets with my money.
Things went along this way for about a week and I was all to the
bad.
One evening Clara J. said to me, "John, I looked through your check
book to-day and I've had a cold on my chest ever since. At first I
thought I had opened the refrigerator by mistake."
At last the blow had fallen!
I had promised her faithfully before we were married that I'd never
play the ponies again and I fell and broke my word.
The accident was painful, and I'd be a sad scamp to put her wise at
this late day, especially after being fried to a finish.
I simply didn't dare confess that my money had gone into a fund to
furnish a home for Incurable Bookmakers--what to do? What to do?
She had me lashed to the mast.
"May I inquire," my wife continued with the breath of winter in her
tones, "why it's all going out and nothing coming in? Have you
begun so soon to lead a double life?"
Mother, call your baby boy back home! If Uncle Peter would only
drop in, or Tacks or Aunt Martha or even the janitor!
Suddenly it occurred to me:
"Dearie," I said, "you have surprised my secret, and now nothing
remains but the pleasure of telling you everything."
A thaw set in.
"As you have stated, not incorrectly, my dear, large bundles of
Green Fellows have severed their home ties and tiptoed into the
elsewhere," I continued, gradually getting my nerve back.
The thermometer continued to go up.
"Clara J., on several occasions you have expressed a desire to
leave this torn-up city and retire to the woodlands, haven't you?"
I asked.
She nodded and the weather grew warmer.
"Once you said to me, 'Oh, John, if they'd only take New York off
the operating table and give the poor city a chance to get well,
how nice it would be!'--didn't you?"
Another nod.
"Well," I said, backing Munchausen in a corner and dragging his
medals away from him, "that's the answer, You for the Burbs! You
for the chateau up the track! Henceforth, you for the cage in the
country where the daffydowndillys sing in the treetops and
buttercups chirp from bough to bough!"
"Oh, John!" she exclaimed, faint with delight; "do you really mean
you've bought a home in the country? How perfectly lovely! You,
dear, dear, old John! And that's what you've been doing with all
your money, just to surprise me! Bless your dear good heart! Oh!
I'm so glad, and so delighted. Won't it be simply grand?"
I could feel the cold, spectral form of Sapphira leaning over my
left shoulder, urging me on.
"What is it like? How many rooms? Where is it?" she inquired, all
in one breath.
Where was the blamed thing? What did it look like? How did I
know? She could search me. I could feel my ears getting red.
Presently I braced and mumbled, "No more details till the castle is
completed, then I'll coax you out there and let you revel."
"How soon will that be?" she asked, "To-morrow? Yes, John,
to-morrow?"
"No," I whispered croupily, "in--in about a week."
I wanted time to arrange my earthly affairs.
"Oh! lovely!" she said, and kissing me rushed away to break the
news to mother.
I felt like a rain check after the sun comes out.
Suddenly Hope tugged at my heart strings and I remembered that I
had a week in which to beat the ponies to a pulp and win out enough
coin to buy six Swiss Cheese cottages in the country.
Day after day I waded in among the jelly fish at the track but the
best I ever got was an $8 win.
Eight dollars wouldn't buy a dog house.
I was desperate. Every evening I had to sit around and listen
while Clara J. told Tacks or Uncle Peter or Aunt Martha or Mother
what she intended doing when we moved to the country.
They had it all cooked up. Uncle Peter and Aunt Martha were coming
to live with us and Tacks would be there to let us live with him.
Uncle Peter intended starting a garden truck farm in the back yard
and Tacks figured on building a chicken coop somewhere between the
front gate and the parlor.
Aunt Martha and Clara J. almost came to blows over the question of
milking the cow. Aunt Martha insisted that cows are milked by
machinery and Clara J. was equally positive that moral suasion is
the only means by which a cow can be brought to a show down.
In the meantime I was dying every half hour.
Finally the day preceding the long-talked of country excursion
arrived and I began to figure on the safest and least inexpensive
methods of suicide.
I went to the track in the afternoon and threw out enough gold dust
to paint our country home from cellar to attic--but never a sardine
showed.
Frostbitten and suffocated by the odor of burning money I crept
into a seat in the car and began to plan my finale.
Presently an elbow poked me in the ribs and I looked into the
smiling face of Bunch Jefferson.
"Still piking, eh?" he chuckled; "you wouldn't trail along after
Your Uncle Bunch and get next to the candy man, would you? Only
$400 to the good to-day. Am I the picker from Picklesburg, son of
the old man Pickwick?--well, I guess yes!"
Then in that desperate moment I broke down and confessed all to
Bunch. I told him how my haughty spirit disdained a tip and how in
the pride of my heart I doped the cards myself and fell in the
well. I told him of my feverish desire to beat the Bookmakers down
through the earth till they yelled for mercy, and I told him of my
pitiful dilemma and how I had to build a home in the country before
noon to-morrow or do a dog trot to the Bad lands.
Then Bunch began to laugh--a long, loud, discordant laugh which
ended in, "John, I'll help you make good!" and then I began to sit
up and notice things.
"I'm away head of this pitty-pat game at the Merry-go-Round," Bunch
went on, "and it so happens that recently I peeled the wrapper off
my roll and swapped it for a country home for my sister and her
daughter. She's a young widow, my sister is, and one of the
loveliest little ladies that ever came over the hill. And she has
a daughter that's a regular plate of peaches and cream."
Still I sat in darkness, and he went on:
"Now, my sister won't move out there for a day or two, so
to-morrow, promptly on schedule time, you lead your domestic fleet
over the sandbars to that house and point with pride to its various
beauties--are you wise?"
"But, Great Scott, man! it's not mine!" I gasped.
"Roll a small pill and get together," admonished Bunch, with a
seraphic smile. "Can't you figure the trick to win? All you have
to do is to coax your gang out there and then break the painful
news to them that you've suddenly discovered the place is haunted
and that you're going to sell it and buy a better bandbox--getting
wise?"
"Bunch," I murmured, weakly, "you've saved my life, temporarily,
at least. Where is this palace?"
"Only forty minutes from the City Hall--any old City Hall," he
answered, "It's at Jiggersville, on the Sitfast & Chewsmoke R.R.,
eighteen miles from Anywhere, hot and cold sidewalks and no
mosquitoes in the winter. Here you are, full particulars," and
with this Bunch handed me a printed card which let me into all the
secrets of that haven of rest in the tall grass.
Bless good old Bunch!
I offered to buy him a quart of Ruinart but he said his thirst
wasn't working, so I had to paddle off home.
That evening for the first time in several weeks I felt like
speaking to myself.
I was the life of the party and I even beamed approvingly when
Uncle Peter tuned up his mezzo contralto voice and began to write a
book about the delights of a country home.
It was a cinch, I assured myself, that the ghost story I had
broiled up to tell on the morrow would send my suburban-mad family
scurrying back to town.
Many times mentally I went over the blood curdling details and I
flattered myself that I surely had a lot of shivery goods for sale.
I couldn't see myself losing at all, at all.
So me for Jiggersville in the morning.
CHAPTER II.
JOHN HENRY'S GHOST STORY.
When the alarm clock went to work the next morning Clara J. turned
around and gave it a look that made its teeth chatter.
She had been up and doing an hour before that clock grew nervous
enough to crow.
Her enthusiasm was so great that she was a Busy-Lizzie long before
7 o'clock and we were not booked to leave the Choo-Choo House till
10:30.
About 8 o'clock she dragged me away from a dream and I reluctantly
awoke to a realization of the fact that I was due to deliver some
goods which I had never seen and didn't want to see.
"Get up, John!" Clara J. suggested, with a degree of excitement in
her voice; "it's getting dreadfully late and you know I'm all
impatience to see that lovely home you've bought for me in the
country!"
[Illustration: Clara J.--A Dream of Peaches--Please Pass the
Cream.]
Me under the covers, gnawing holes in the pillow to keep from
swearing.
"Oh, dear me!" she sighed, "I'm afraid I'm just a bit sorry to
leave this sweet little apartment. We've been so happy here,
haven't we?"
I grabbed the ball and broke through the center for 10 yards.
"Sorry," I echoed, tearfully; "why, it's breaking my heart to leave
this cozy little collar box of a home and go into a great large
country house full of--of--of rooms, and--er--and windows,
and--er--and--er--piazzas, and--and--and cows and things like that."
"Of course we wouldn't have to keep the cow in the house," she
said, thoughtfully.
"Oh, no," I said, "that's the point. There would be a barn, and
you haven't any idea how dangerous barns are. They are the curse
of country life, barns are."
"Well, then, John, why did you buy the cow?" she inquired, and I
went up and punched a hole in the plaster.
Why did I buy the cow? Was there a cow? Had Bunch ever mentioned
a cow to me? Come to think of it he hadn't and there I was cooking
trouble over a slow fire.
When I came to she was saying quietly, "Besides, I think I'd rather
have a milkman than a cow. Milkmen swear a lot and cheat sometimes
but as a rule they are more trustworthy than cows, and they very
seldom chase anybody. Couldn't you turn the barn into a gymnasium
or something?"
"Dearie," I said, trying my level best to get a mist over my lamps
so as to give her the teardrop gaze, "something keeps whispering to
me, 'Sidestep that cave in the wilderness!' Something keeps
telling me that a month on the farm will put a crimp in our
happiness, and that the moment we move into a home in the tall
grass ill luck will get up and put the boots to our wedded bliss."
Then I gave an imitation of a choking sob which sounded for all the
world like the last dying shriek of a bathtub when the water is
busy leaving it.
"Nonsense, John!" laughed Clara J.; "it's only natural that you
regret leaving our first home, but after one day in the country
you'll be happy as a king."
"Make it a deuce," I muttered; "a dirty deuce at that."
"Now," she said, joyfully; "I'm going to cook your breakfast. This
may be your very last breakfast in a city apartment for months,
maybe years, so I'm going to cook it myself. I've got every trunk
packed--haven't I worked hard? Get up, you lazy boy!" and with
this she danced out of the room.
Every trunk packed! Did she intend taking them with her, and if
she did how could I stop her?
Back to the woods!
I began to feel like a street just before they put the asphalt down.
For some time I lay there with my brain huddled up in one corner of
my head, fluttering and frightened.
Presently an insistent scratch-r-r-r-r aroused me and I began to
sit up and notice things.
The things I noticed consisted chiefly of Tacks and the kitchen
carving knife. The former was seated on the floor laboriously
engineering the latter in an endeavor to produce a large
arrow-pierced heart on the polished panel of the bedroom door.
"What's the idea?" I inquired.
"I'm farewelling the place," he answered, mournfully. "They's only
two more doors to farewell after I get this one finished. Ain't
hearts awful hard to drawr just right, 'specially when the knife
slips!"
"You little imp!" I yelled; "do you mean to tell me you've been
doing a Swinnerton all over this man's house? S'cat!" and I
reached for a shoe.
"Cut it!" cried Tacks, indignantly. "Didn't the janitor say he'd
miss me dreadful, and how can he miss me 'less'n he sees my loving
rememberments all over the place every time he shows this
compartment to somebody else? And it is impolite to go 'way
forever and ever amen without farewelling the janitor!"
"Where do you think you're going?" I inquired, trying hard to be
calm.
"To the country to live, sister told me," Tacks bubbled; "and we
ain't never coming back to this horrid city, sister told me; and
you bought the house for a surprise, sister told me; and it has a
pizzazus all around it, sister told me; and a cow that gives
condensed milk, sister told me; and they's hens and chickens and
turkey goblins and a garden to plant potato salad, and they's a
barn with pigeons in the attic, and they's a lawn with a barbers
wire fence all around it, sister told me; and our trunks are all
packed, and we ain't never coming back here no more, sister told
me; and I must hurry and farewell them two doors!"
Tacks was slightly in the lead when my shoe reached the door, so he
won.
At breakfast we were joined by Uncle Peter and Aunt Martha, both of
whom fairly oozed enthusiasm and Clara J.'s pulse began to climb
with excitement and anticipation.
I was on the bargain counter, marked down from 30 cents.
Every time Uncle Peter sprang a new idea in reference to his
garden, and they came so fast they almost choked him, I felt a
burning bead of perspiration start out to explore my forehead.
Presently to put the froth of fear upon my cup of sorrow there came
a telegram from "Bunch" which read as follows:
New York ----
John Henry
No. 301 W. 109th St.
Sister and family will move in country house tomorrow be sure to
play your game to-day good luck.
Bunch.
"Poor John! you look so worried," said Clara J., anxiously; "I
really hope it is nothing that will call you back to town for a
week at least. It will take us fully a week to get settled, don't
you think so, Aunt Martha?"
I dove into my coffee cup and stayed under a long time. When I
came to the surface again Uncle Peter was explaining to Tacks that
baked beans grew only in a very hot climate, and in the general
confusion the telegram was forgotten by all except my harpooned
self.
Clara J. and Aunt Martha were both tearful when we left the flat to
ride to the station, but to my intense relief no mention was made
of the trunks, consequently I began to lift the mortgage from my
life and breathe easier.
On the way out Tacks left a small parcel with one of the hall boys
with instructions to hand it to the janitor as soon as possible.
"It's a little present for the janitor in loving remembrance of his
memory," Tacks explained with something that sounded like a catch
in his voice.
"Hasn't that boy a lovely disposition?" Aunt Martha beamed on
Tacks; "to be so forgiving to the janitor after the horrid man had
sworn at him and blamed him for putting a cat in the dumb waiter
and sending it up to the nervous lady on the seventh floor who
abominated cats and who screamed and fell over in a tub of suds
when she opened the dumb-waiter door to get her groceries and the
cat jumped at her. Mercy! how can the boy be so generous!"
Tacks bore up bravely under this panegyric of praise and his face
wore a rapt expression which amounted almost to religious fervor.
"What did you give the janitor, Angel-Face?" I asked.
"Only just another remembrance," Tacks answered, solemnly. "I
happened to find a poor, little dead mouse under the gas range and
I thought I'd farewell the janitor with it."
Aunt Martha sighed painfully and Uncle Peter chuckled inwardly like
a mechanical toy hen.
On the train out to Jiggersville Clara J. was a picture entitled,
"The Joy of Living"--kind regards to Mrs. Pat Campbell; Ibsen
please write.
As for me with every revolution of the wheels I grew more and more
like a half portion of chipped beef.
"Oh, John!" said Clara J., her voice shrill with excitement; "I
forgot to tell you! I left my key with Mother, and she's going to
superintend the packing of the furniture this afternoon. By
evening she expects to have everything loaded in the van and we
won't have to wait any time for our trunks and things!"
"Great Scott!" I yelled; "maybe you won't like the house! Maybe
it's only a shanty with holes in the roof--er, I mean, maybe you'll
be disappointed with the lay-out! What's the blithering sense of
being in such a consuming fever about moving the fiendish
furniture? I'm certain you'll hate the very sight of this
corn-crib out among the ant hills. Can't you back-pedal on the
furniture gag and give yourself a chance to hear the answer to what
you ask yourself?"
Clara J. looked tearfully at me for a moment; then she went over
and sat with Aunt Martha and told her how glad she was we were
moving to the country where the pure air would no doubt have a
soothing effect on my nerves because I certainly had grown
irritable of late.
At last we reached the little old log cabin down the lane and after
the first glimpse I knew it was all off.
The place I had borrowed from Bunch for a few minutes was a dream,
all right, all right.
With its beautiful lawns and its glistening gravelled walks; with a
modern house perfect in every detail; with its murmuring brooklet
rushing away into a perspective of nodding green trees and with the
bright sunshine smiling a welcome over all it made a picture
calculated to charm the most hardened city crab that ever crawled
away from the cover of the skyscrapers.
As for Clara J. she simply threw up both hands and screamed for
help. She danced and yelled with delight. Then she hugged and
kissed me with a thousand reiterated thanks for my glorious present.
I felt as joyous as a jelly fish. Ten-legged microbes began to
climb into my pores. Everything I had in my system rushed to my
head. I could see myself in the giggle-giggle ward in a bat house,
playing I was the king of England.
I was a joke turned upside down.
After they had examined every nook and cranny of the place and had
talked themselves hoarse with delight I called them all up on the
front piazza for the purpose of putting out their lights with my
ghost story.
I figured on driving them all back to the depot with about four
paragraphs of creepy talk, so when I had them huddled I began in a
hoarse whisper to raise their hair.
I told them that no doubt they had noticed the worried expression
on my face and explained that it was due chiefly to the fact that I
had learned quite by accident that this beautiful place was haunted.
Tacks grew so excited that he dropped a garden spade off the piazza
and into a hot house below, breaking seven panes of glass, but the
others only smiled indulgently and I went on.
I jumped head first into my most blood-curdling story and related
in detail how a murder had been committed on the very site the
house was built on and how a fierce bewhiskered spirit roamed the
premises at night and demanded vengeance. I described in awful
words the harrowing spectacle and all I got at the finish was the
hoot from Uncle Peter.
"Poor John," said Clara J., "I had no idea you were so run down.
Why, you're almost on the verge of nervous prostration. And how
thoughtful you were to pick out a haunted house, for I do love
ghosts. Didn't you know that? I'll tell you what let's do. I'll
give a prize for the first one who sees and speaks to this unhappy
spirit--won't it be jolly? Where are you going, John?"
"Me, to the undertakers--I mean I must run back to town. That
telegram this morning--important business--forgot all about it--see
you later--don't breathe till I get back--I mean, don't live till
I--Oh! the devil!"
Just then I fell over the lawn mower, picked myself up hastily and
rushed off to town to find Bunch for I was certainly up against it
good and hard.
CHAPTER III.
JOHN HENRY'S BURGLAR.
When finally I located Bunch and told him the bitter truth he acted
like a zee-zee boy in a Wheel House.
Laugh! Say! he just threw out his chest and cackled a solo that
fairly bit its way through my anatomy.
Every once in a white he'd give me the red-faced glare and
snicker, "Oh, you mark! You Cincherine! You to the seltzer
bottle--fizz!--fizz! The only and original Wheeze Puller, not!
You're all right--backwards!"
Then he'd throw his ears back and let a chortle out of his
thirst-teaser that made the neighborhood jump sideways and rubber
for a cop.
"What are you going to do?" he asked me when presently his face
grew too tired to hold any more wrinkles.
[Illustration: Uncle Peter--the Original Trust Tamer.]
"Give me the count," I sighed; 'I'm down and out."
"Have you no plan at all?" inquired Bunch.
"Plan, nothing," I said; "every time I try to think of a plan my
brain gets bashful and hides. There's nothing in my noddle now but
a headache."
"Well," said Bunch, "I'll throw a wire at my sister and tell her
not to move out to Jiggersville until day after to-morrow. In the
mean time we'll have to get a crowbar and pry your family circle
loose from my premises. Nothing doing in the ghost business, eh?"
"Nothing," I answered, mournfully; "I couldn't coax a shiver."
"A fire wouldn't do, would it?" Bunch suggested, thoughtfully.
"It wouldn't do for you, unless you are aces with the insurance
Indians," I answered.
"We-o-o-u-w!" yelled Bunch, "I have it--burglars!"
"Burglars!" I repeated, mechanically.
"Sure! it's a pipe!" Bunch went on with enthusiasm. "You will play
Spike Hennessy and I'll be Gumshoe Charlie. We'll disguise
ourselves with whiskers and break into the house about 2 o'clock in
the morning. We'll arouse the sleeping inmates, shoot our
bullet-holders in the ceiling once or twice and hand them enough
excitement to make them gallop back to town on the first train. Do
you follow me, eh, what?"
"Not me, Bunch," I shook my head sadly. "Nix on the burgle for
yours truly. I must take the next train back to the woods.
Otherwise wee wifey may suspect something and begin to pass me out
the zero language. But I like the burglar idea. Couldn't you do
it as a monologue?"
"What! all by my lonesome?" cried Bunch. "Say! John, doesn't that
sound like making me work a trifle too hard to get my own goods
back ?"
I sighed and looked as helpless as a nut under the hammer.
Bunch laughed again. "Oh, very well," he said, "I see I'm the only
life-saver on duty so I'll do a single specialty and pull you out
of the pickle bottle."
I grasped my rescuer's hand and shook it warmly in silence.
"Leave a front window open," Bunch directed, "and somewhere around
two o'clock I'll squeeze through."
"I'll have it worked up good and proper," I said, eagerly. "I'll
throw out dark hints all the evening and have the bunch ready to
quiver when the crash comes. As soon as I hear your signal I'll
rush bravely down stairs and you shoot the ceiling. I'll give you
a struggle and chase you outside. Then I'll run you down behind
the barn. There, free from observation, you can shoot a couple of
holes in my coat so that I can produce evidence of a fierce fight,
and then you to the tall timber. I'll crawl breathlessly back to
my palpitating household, and, displaying my wounded coat, declare
everything off. I'll refuse to live any longer in a house where
murder and sudden death occupy the spare room. It looks to me like
a cinchalorum, Bunch, a regular cinchalorum!"
"It sounds good," Bunch acquiesced, "and I'll give you an imitation
of the best little amateur cracksman that ever swung a jimmy. I'll
take a late train out and hang around till it's time to ring the
curtain up. By the way, are there any revolvers on the premises?"
"Not a gun," I answered, "not even an ice-pick. Uncle Peter won't
show fight. All he'll show will be a blonde night gown cutting
across lots to beat the breeze. Aunt Martha will climb to the
attic, Clara J. will be busy doing a scream solo, and Tacks will
crawl under the bed and pull the bed after him. There'll be no
interference, Bunch; it's easy money!"
With this complete understanding we parted and I hustled back to
Jiggersville.
I found the family still delirious with delight with the exception
of Clara J. whose enthusiasm had been dampened by my sudden
departure.
My reappearance brought her back to earth, however, and in the
presence of so many new excitements she didn't even question me
with regard to my City trip.
As the evening wore on my nervousness increased and I began to
wonder if Bunch would really turn the trick or give me the loud
snicker and leave me flat.
I had gone too far now to confess everything to Clara J. She'd
never forgive me.
If I told her the facts in the case the long Arctic Winter Night
would set in, and I'd be playing an icicle on the window frame.
I felt as lonely as a coal scuttle during the strike.
About six o'clock Uncle Peter waded into the sitting room, flushed
and happy as a school boy. "I've just left the garden," he
chuckled.
"No, you haven't," I said, glancing at his shoes; "you've brought
most of it in here with you."
I never touched him. The old gentleman sat down in a loud rocker
and began to tell me a lot of things I didn't want to hear. Uncle
Peter always intersperses his remarks on current topics with bits
of parboiled philosophy that make one want to get up and drive him
through the carpet with a tack hammer. When it comes to wise saws
and proverbial stunts Uncle Peter has Solomon backed up in the
corner.
"John," he said, "this country life is great. Early to bed and
early to rise makes a man's stomach digest mince pies--how's that?
Notice the air out here? How pure and fresh and bracing! You
ought to go out and run a mile, John!"
"I'd like to run ten miles," I answered, truthfully.
"Exercise, that's the essence of life, my boy!" he continued. "I
firmly believe I could run five miles to-day without straining a
muscle."
I laughed internally and thought of the glorious opportunity he'd
have before the morning broke.
"You may or may not know, John," the old gentleman kept on, "that I
was a remarkably fine swordsman in my younger days. Parry, thrust,
cut, slash--heigho! those were the times. And, to tell you the
truth, I'm still able to hold my own with the sword or pistol. I
found a sword hanging on the wall in the hall to-day and I've been
practising a few swings."
A vision of Uncle Peter running a rusty sword into the interior
department of the disguised and disgusted Bunch rose before me, but
I blew it away with a laugh.
"He laughs best who laughs in his sleeve," chuckled the old party.
"Now that we're out in the country all of us should learn to handle
a sword or a pistol. It gives us self reliance. It's very
different from living in the city, I tell you. A tramp in the
lock-up is worth two in the kitchen. I shot at a mark for an hour
to-day."
"What with?" I gasped.
"With a bow and arrow I bought for Tacks yesterday directly I
learned we were coming to the country. I hit the bull's eye five
out of six times. An ounce of prevention is worth two hundred
pounds of policemen, you know. Tacks practised, too, and drove an
arrow through a strange man's overalls and was chased half a mile
for his skill in marksmanship, but, as I said before, the exercise
will do him good."
"Where do you keep this bow and arrow?" I inquired, with a studied
assumption of carelessness.
"To-night I'll keep it under my pillow. _Honi soit qui oncle
Pierre_, which means, evil be to him who monkeys with Uncle Peter,"
he said, solemnly. "To-morrow I'm going to town to buy a bull dog
revolver, maybe a bull dog _and_ a revolver, for a dog in the
manger is the noblest Roman of them all."
I could see poor Bunch scooting across the lawn with a bunch of
arrows in his ramparts and Uncle Peter behind, prodding his citadel
with a carving knife.
I began to get a hunch that our plan of campaign was threatened
with an attack of busy Uncle Peter, and I had just about decided to
remove his door key and lock the old man up in his room when Clara
J. came in to announce dinner.
Aunt Martha and Clara J. had collaborated on the dinner and it was
a success. Uncle Peter said so, and his appetite is one of those
brave fighting machines that never says die till every plate is
clean.
I was so nervous I couldn't eat a bite, but I pleaded a toothache,
so they all gave me the sympathetic stare and passed me up.
We went to bed early and I rehearsed mentally the stage business
for the drama about to be enacted when Bunch crept through the
picket lines.
About midnight a dog in the neighborhood began to hurl forth a
series of the most distressing bow-bows I ever heard. I arose, put
up the window and looked out.
I saw a tall man with a bunch of whiskers on his face flying across
the lot pursued by a black-and-tan pup, which snapped eagerly at
the man's heels and seemed determined to eat him up if ever the
runner stopped long enough.
I felt in my bones that the one in the lead was Bunch, and I sighed
deeply and went back to bed.
I must have dropped into an uneasy sleep for Clara J. was tapping
me on the arm when I started up and asked the answer.
"There's somebody in the house," she whispered, not a bit
frightened, to my surprise and dismay, "Maybe it's only the ghost
you told us about--what a lark!"
"Somebody in the house," I muttered, going on the stage blindly to
play my part; "and there isn't a gun in the castle."
"Yes there is," she answered, joyfully, I fancied; "mother brought
father's revolver over yesterday and made me put it in my satchel.
She said we would feel safer at night with it in the house. Do let
me shoot him; I can aim straight, indeed I can! Why, John, what
makes you tremble so?"
"I'm not trembling, you goose!" I snarled; "I can't find my shoes,
that's all. Doggone if I'm going to live in a joint like this with
ghosts and burglars all over the place."
Just then an alarming yell ascended from the regions below,
followed by a crash and a series of the most picturesque,
sulphur-lined oaths that mortal man ever gave vent to.
It was Bunch. His trademark was on every word. I could recognize
his brimstone vocabulary with my eyes shut.
But what dire fate had befallen him? Surely, not even an amateur
cracksman would give himself and the whole snap away unless the
provocation was great.
Lights began to appear all over the house. Aunt Martha in a weird
makeup came out of her room screaming, "What is it? What is it?"
followed by Uncle Peter and his trusty bow and arrow.
I began to pray. It was all over. A rosewood casket for Bunch.
Me for the Morgue.
Just as I was ready to rush down to investigate, Tacks came
bounding up the stairs, two steps at a time, clad only in his
nightie.
_Up the stairs_, mind you! The nerve of that kid!
"Gi'me the prize, sister!" he yelled; "I caught the ghost! I
caught him!"
"What do you mean?" I said, shaking him.
Tacks grinned from ear to ear. "You know they's a trap door in the
hall so's to get down in the cellar and it ain't finished yet, so
this evening I took the door up and laid heavy paper on it so's if
the ghost walked on it he'd go through and he did, and I get the
prize, don't I, sister?"
I rushed down to the scene of the explosion, followed by my excited
household.
Leaning over the yawning cellar trap door I yelled, "Who's down
there?"
"Oh! you go to hell!" came back the voice of the disgusted Bunch,
whereupon Aunt Martha almost fainted, while Uncle Peter loaded his
bow and arrow and prepared to sell his life dearly.
Great Scott! what a situation! The man who owned the house nursing
his bruises in the muddy cellar while the bunch of interlopers
above him clamored for his life.
While I puzzled my dizzy think-factory for a way out of the dilemma
there came a terrific knock at the door and Tacks promptly opened
it.
"Have you got him? Have you got him?" inquired the elongated and
cadaverous specimen of humanity who burst into the hall and stared
at us.
"I seen him early this evening a'hangin' around these here premises
and I ups and chases him twicet, but the skunk outrun me," the
newcomer gurgled, as he excitedly swung a policeman's billy the
size of a fence rail.
"Then I seen the lights here and says I, 'they has him'! Perduce
the maleyfactor till I trot him to the lock-up!" and with this the
minion of the law rolled up his sleeves and prepared for action.
"I presume you are the chief of police?" inquired Uncle Peter, with
an affable smile.
"I'm all the police they is and my name is Harmony Diggs, and
they's no buggular livin' can get out'n my clutches oncet I gits
these boys on him," the visitor shouted, waving an antiquated pair
of handcuffs excitedly in the air.
Tacks watched him open-mouthed. That boy was having the time of
his life and it would have pleased me immeasurably to paddle him to
sleep with Harmony's night stick.
"I caught him!" Tacks cried in exultant tones when the village
copper looked his way; "he's down there."
"Down there, eh?" snorted the country Sherlock, getting on his
knees and peering into the depths, but just then Bunch handed him a
handful of hard mud which located temporarily over Harmony's left
eye and put his optic on the blink.
With the other eye, however, Mr. Diggs caught a glimpse of a step
ladder, which he immediately lowered through the trap, and drawing
a murderous looking revolver from his pocket, commanded Bunch to
come up or be shot.
Bunch decided to come up. I didn't hold the watch on him, but I
figure it took him about seven-sixteenths of a second to make the
decision.
As the criminal slowly emerged from the cellar the spectators stood
back, spellbound and breathless; Aunt Martha with a long tin dipper
raised in an attitude of defense, and Uncle Peter with the bow and
arrow ready for instant use.
These war-like precautions were unnecessary, however. Bunch was a
sight. His clothing had accumulated all the mud in the unfinished
cellar and his false whiskers were skewed around, giving his face
the expression of a prize gorilla.
Bunch looked at me reproachfully, but never opened his head. Say!
if ever there was a dead game sport, Bunch Jefferson is the answer.
He didn't even whimper when the village Hawkshaw snapped the
bracelets on his wrist and said, "Come on, Mr. Buggular! This
here's a fine night's work for everybody in this neighborhood
because you've been a source of pesterment around here for six
months. If you don't get ten years, Mr. Buggular, then I ain't no
guess maker. Come along; goodnight to you, one and all; that there
boy that catched this buggular ought to get rewarded nice!"
"He will be," I said mentally, as Mr. Diggs led the suffering Bunch
away to the Bastile.
"I've got to see that villain landed in a cell," I said to Clara J.
as the door closed on the victor and vanquished.
"Do, John!" she answered; "but don't be too hard on the poor
fellow. You can't tell what temptations may have led him astray.
I certainly am disappointed for I was sure it was the ghost.
Anyway, the burglar had whiskers like the ghost's, didn't he?"
I didn't stop to reply, but grabbing my coat rushed away to
formulate some plan to get Bunch out of hock.
CHAPTER IV.
JOHN HENRY'S COUNTRY COP.
Ahead of me, plodding along the pike under the moonlight, were
Bunch and his cadaverous captor, the former bowed in sorrow or
anger, probably both, and the latter with head erect, haughty as a
Roman conqueror.
Bunch's make-up was a troubled dream. Over a pair of hand-me-down
trousers, eight sizes too large for him, he wore a three-dollar
ulster. On his head was an automobile cap, and his face was
covered with a bunch of eelgrass three feet deep. He was surely
all the money.
As I drew near I could hear Mr. Diggs expatiating on crime in
general and housebreaking in particular, and I fancied I could also
hear Bunch boiling and seething within.
[Illustration: Aunt Martha--a Short, Stout Bundle of Good Nature.]
"Mr. Buggular," Diggs was saying, "I don't know just what your home
trainin' was as a child, but they's a screw loose somewhere or
you'd a'never been brought to this here harrowful perdickyment,
nohow. I s'pose you jest started in nat'rally to be a heenyus
maleyfactor early in life, huh? You needn't to answer if you're
afeared it'll incrimigate you, but I s'pose you took to it when a
boy, pickin' pockets or suthin' like that, huh?"
"Oh, cut it out, you old goat, and don't bother me!" snapped Bunch,
just as I joined them.
"A dangerous maleyfactor," said Diggs to me, as he tightened his
grip on Bunch's arm; "but they ain't no call for you to assist the
course of justice, because if the dern critter starts to run I'll
pump him chuck full of lead. He's been a'tellin' me he started on
the downward path to predition as a child-stealer."
"I told you nothing, you old tadpole," shrieked Bunch, unable to
contain himself longer.
"Very well," said Harmony, soothingly, "they ain't no call for you
to say nothin' more that'll incrimigate you before the bar of
Justice. Steady, now, or I'll tap you with this here cane!"
"Brace up, good old sport; I'll get you out of this in a jiffy," I
whispered to Bunch at the first opportunity, and he gave me a
cold-storage look that chased the chills all over me.
Presently we arrived at the little brick structure which
Jiggersville proudly called its calaboose, and after much fumbling
of keys, Mr. Diggs opened the jackpot and we all stayed.
The yap policeman was for taking Bunch right back to the donjon
cell in the rear, but with a $5 bill I secured a stay of
proceedings.
My forehead was damp with perspiration so I took off my hat and
laid it on the bench in the little court room where Bunch sat
moodily and with bowed head.
Then I coaxed the rural Vidocq over in the corner and gave him a
game of talk that I thought would warm his heart, but he listened
in dumbness and couldn't see "no sense in believing the maleyfactor
was anythin' more'n a derned cuss, nohow!"
"I have every reason to believe that we have made a mistake," I
said to Harmony in a hoarse whisper. "From an envelope dropped by
this party in my house I am lead to believe that he's a respectable
gentleman who entered my premises quite by mistake."
The chin whiskers owned and engineered by Diggs bobbed up and down
as he chewed a reflective cud, but he couldn't see the matter in my
light at all.
I had used all kinds of arguments and was just about to give up in
despair when a voice in the doorway caused us both to turn.
There stood Bunch Jefferson, the real fellow, looking as fresh as a
daisy.
"What's the trouble, John?" he asked, smiling benignly on Diggs.
While I was talking to the representative of the law, Mr. Slick saw
his opportunity and grabbed it by the hind leg. He had quietly
reached the door, and once outside the sledding was excellent.
Bunch had his business suit on under the burglar make-up. It
didn't take him two minutes to work the shine darbies over his
hands. He then peeled off the ulster and the tuppeny trousers,
and throwing these and the Svengalis over the fence, he was home
again from the Bad Lands.
The transformation scene was made complete by the fact that Bunch
was now wearing my hat.
In answer to Bunch's question, the redoubtable Diggs smiled
indulgently and said with pride-choked tones, "A maleyfactor, sir,
caught in the meshes of the law and hauled before this here trybune
of Justice by these hands!"
The eagle eye of Diggs was now triumphantly sighted along the arm
and over the bony hand to where the criminal was supposed to be,
but when the gaze finally rested on an empty bench the expression
of pained surprise on the old man-hunter's map was calculated to
make a hen cackle.
Diggs rushed over to the bench, turned it upside down, looked
behind the chairs, and then, emitting a roar that rattled the
rafters, he hustled back to see if by any chance the prisoner had
locked himself up in a cell.
Bunch gave the old geezer the minnehaha and yelled, "Say! you with
the me-ya-ya's on the chin! Did somebody give you the hot-foot and
make a quick exit?"
Diggs was now in full eruption and heavy showers of Reub lava rose
from his vocal organs and fell all over the place, while he
thrashed around the calaboose in a frenzy of excitement.
"Maybe you're sending out a general alarm about that human meteor
that passed me on the pike a few minutes ago?" Bunch suggested.
Diggs turned and eyed him in open-mouthed silence.
"A mutt with a pink ulster and one of those pancakes on his head
like the drivers of the gasoline carts wear," Bunch suggested.
"It's him! it's the maleyfactor!" exclaimed Harmony, tightening his
grip on the night stick; "which way did the derned cuss go?"
Bunch pointed due south-east, and with a howl of rage Diggs sprang
forward and bounced down the pike like a hungry kangaroo on its way
to a lunch counter.
I began to wrap up my enjoyment and send it forth in short gurgles
of merriment until Bunch pressed the button and the scene was
changed to Greenland's Icy Mountains.
"Funny, isn't it?" he sneered; "regular circus, with yours in
haste, Bunch Jefferson, to do the grand and lofty tumbling! I'm
the Patsy, oh, maybe! It was a fine play, all right, but I didn't
expect you to stack the cards!"
"On the level, Bunch, believe me, it wasn't my fault," I spluttered.
"Not your fault," he snapped back; "then I suppose it was mine! I
suppose I fell down the elevator shaft just to please mother, eh?
Maybe you think I dropped into the excavation just to pass the time
away? Have you an idea that I dove down into the earth because I
wanted to get back to the mines? Wasn't your fault, indeed! Maybe
you think I fell in the well simply because I wanted to give an
imitation of the old oaken bucket, yes?"
I tried to tell him all about Tacks and the ghost story, but he
wouldn't stand for it.
"You should have been waiting for me on the stairs," he argued,
unreasonably, rubbing one of the bruises in his choice collection,
"Didn't you catch me early in the evening being chased from pillar
to post by everything in the neighborhood that had legs long enough
to run? When I tried to hide in the corner of a farm over there, a
bull dog came up on rubber shoes and bit his initials on some of my
personal property before I could crawl through the fence. Every
time I showed up on the pike that human accident that breathes like
a man and talks like a rabbit chased me eight miles there and back.
The first time I tried to approach the infernal house I fell over a
grindstone and signed checks in the gravel with my nose.
Hereafter, when you want a burglar, pick somebody your own size.
I'm going to hunt a hospital and get sewed together again."
I put on all steam and tried to square myself, but Bunch only shook
his head and said I was outlawed.
"You can't run on my race track," he exclaimed as he started for
the depot; "that last race was crooked and you stood in with the
dope mixer."
I watched him down the hill until he disappeared in the station,
then, sad at heart, I trudged back to the old homestead that had
caused all my trouble.
It was now broad daylight, but nowhere within my line of vision
could I get a peep of the doughty Diggs.
No doubt he was still cutting across lots trying to head off the
"maleyfactor."
CHAPTER V.
JOHN HENRY'S TELEGRAM.
When I reached the cottage I found all the members of my household
dressed for the day, and lined up on the piazza, eager for news
from the battlefield.
"Gee whiz!" exclaimed Uncle Peter, "the boy is bareheaded! Where's
your hat, John?"
"Mercy! I hope you're not scalped!" Aunt Martha cried,
sympathetically.
I explained that the desperado put up a stiff fight against Diggs
and myself and, warming up to the subject, I went into the details
of a hand to hand struggle that made them all shiver and blink
their lanterns.
When finally I finished with the statement that the robber knocked
us both down and had made a successful break for liberty. Uncle
Peter gave expression to a yell of dismay, and once again he and
his bow and arrow held a reunion.
Tacks suggested that we burn the house down so the burglar wouldn't
be able to find it if he came around after dark. I thought
extremely well of the suggestion, but didn't dare say so.
Aunt Martha had just about decided to untie a fit of hysterics,
when Clara J. reached for the kerosene bucket and threw oil on the
troubled waters.
"Let's drop all this nonsense about burglars and ghosts and go to
breakfast," she suggested. "I don't believe there ever was a ghost
within sixty miles of this house, and to save my soul I couldn't be
afraid of a burglar whose specialty consisted of falling in the
cellar and swearing till help came!"
After breakfast I was dragged away to the brook to fish for lamb
chops or whatever kind of an animal it was that Uncle Peter and
Tacks decided would bite. Aunt Martha posted off to the city on
urgent business, the nature of which she carefully concealed from
everybody.
Clara J. said she'd be delighted to have the house all to herself
for an hour or two, there were so many rooms to look through and so
many plans to make.
Uncle Peter gave her his bow and arrow with full instructions how
to shoot if danger threatened, and Tacks carefully rubbed the steps
leading up to the piazza with soap so the burglar would fall and
break his neck. Then the little shrimp called my attention to his
handiwork and demonstrated its availability by slipping thereon
himself and going the whole distance on his face. He didn't break
his neck, however, so to my mind his burglar alarm failed to make
good.
As time wore on I felt more and more like a mock turtle being led
to the soup house.
The fact that Bunch was sore worried me, and I began to realize
that it was now only a question of a few hours when I'd have to
crawl up to Clara J. and hand in my resignation.
Every time I drew a picture of that scene and heard myself telling
her I was nothing but a fawn-colored four-flush I could see my
future putting on the mitts and getting ready to hand me one.
And when I thought of the dish of fairy tales I had cooked for that
girl I could feel something running around in my head and trying to
hide. I suppose it was my conscience.
At the brook, Uncle Peter began to throw out hints that he was the
original lone fisherman. The lobster never lived that could back
away from him, and as for fly-casting, well, he was Piscatorial
Peter, the Fancy Fish Charmer from Fishkill.
The old gentleman is very rich, but he loves to live around with
his relatives, not because he's stingy, but simply because he likes
them and knows they are good listeners.
Uncle Peter is a reformed money-maker. He wrote the first Monopoly
that ever made faces at a defenceless public. He was the owner of
the first Trust ever captured alive, and he fed it on government
bonds and small dealers till it grew tame enough to eat out of a
pocketbook.
Uncle Peter sat down on a rock overhanging the clay bank which
sloped up about four feet above the lazy brooklet. He carefully
arranged his expensive rod, placed his fish basket near by and
entered into a dissertation on angling that would make old Ike
Walton get up and leave the aquarium.
In the meantime Tacks decided to do some bait fishing, so with an
old case knife he sat down behind Uncle Peter and began to dig
under the rock for worms.
"Fishing is the sport of kings," the old man chuckled; "an it's a
long eel that won't turn when trodden upon. If you're not going to
fish, John, do sit down! You're throwing a shadow over the water
and that scares the finny monsters. A fish diet is great for the
brain, John! You should eat more fish."
"There's many a true word spoken from the chest," I sighed, just as
Uncle Peter made his first cast and cleverly wound about eight feet
of line around a spruce tree on the opposite bank.
The old man began to boil with excitement as he pulled and tugged
in an effort to untangle his line, and just about this time Tacks
became the author of another spectacular drama.
In the search for the elusive worm that feverish youth known as
Tacks the Human Catastrophe, had finally succeeded in prying the
rock loose and immediately thereafter Uncle Peter dropped his rod
with a yell of terror and proceeded to follow the man from Cook's.
[Illustration: Tacks--the Boy Disaster.]
The rock reached the brook first, but the old gentleman gave it a
warm hustle down the bank and finished a close second. He was in
the money, all right.
Tacks also ran--but in an opposite direction.
For some little time my spluttering relative sat dumfounded in
about two feet of dirty water, and when finally I dipped him out of
the drink he looked like a busy wash-day. Everything was damp hut
his ardor.
However, with characteristic good nature he squeezed the water out
of his pockets and declared that it was just the kind of exercise
he needed. He made me promise not to tell Aunt Martha, because she
was very much opposed to his going in bathing on account of the
undertow. Then I sneaked him up to his room and left him to change
his clothes.
On the piazza I found Clara J., her face shrouded in the after-glow
of a wintry sunset.
She handed me a telegram minus the envelope and asked me, with a
voice that was intended to be cuttingly sarcastic, "Is there any
answer?"
I opened the message and read:
New York.
John Henry,
Jiggersville, N. Y.
The two queens will be out this afternoon. They are good girls so
treat them white.
Bunch.
The unspeakable idiot, to send me a wire worded like that! No
wonder Clara J. was sitting on the ice cream freezer! Of course
it only meant that Bunch's sister and her daughter were coming out
to look at their property, but--suffering mackerel! what an eye
Clara J. was giving me!
"And who are the two queens?" she queried, bitterly.
My face grew redder and redder. Every minute I expected to turn
into a complete boiled lobster. I could see somebody reaching for
the mayonaise to sprinkle me.
"Well," she continued, "is there no answer? Of course, they are
good girls, and you'll treat them white, but--" Then the heavens
opened and the floods descended.
"Oh, John!" she sobbed; "how could you be so unkind, so cruel!
Think of it, a scandal on the very first day in my new home, and I
was so happy!"
I would confess everything. There was no other way out of it. I
was on my knees by her side just about to blurt forth the awful
truth when my courage failed and suddenly I switched my bet and
gave the cards another cut.
"It's all a mistake," I whispered; "it's only Bunch Jefferson doing
a comedy scene. Don't you understand, dear; when Bunch tries to
get funny all the undertakers have a busy season. I simply don't
know who he means by the two queens, and as for scandal, well, you
know me, Pete!"
I threw out my chest and gave an imitation of St. Anthony.
"You must know who he means," she insisted, brightening a bit,
however.
"Ah, I have it!" I cried, brave-hearted liar that I was; "he means
my Aunt Eliza and her daughter, Julia! You remember Aunt Eliza,
and Julia?"
"I never heard you speak of them before," she said, still
unconvinced.
Good reason, too, for up to this awful moment I never had an Aunt
Eliza or a cousin Julia, but relatives must be found to fit the
emergency.
"Oh, you've forgotten, my dear," I said, soothingly. "Aunt Eliza
and Julia are two of the best Aunts I ever had--er, I mean Aunt
Eliza is the best cousin--well, let it go at that! Bunch may have
met them on the street, you see, and they inquired for my address.
Yes, that's it. Dear, old Aunt Eliza!"
"Is she very old?" Clara J. asked, willing to be convinced if I
could deliver the goods.
"Old," I echoed, then suddenly remembering Bunch's description;
"oh, no; she's a young widow, about 28 or 41, somewhere along in
there. You'll like her immensely, but I hope she doesn't come out
until we get settled in a year or two."
Clara J. dried her eyes, but I could see that she hadn't restored
me to her confidence as a member in good standing.
She pleaded a headache and went away to her room, while I sat down
with Bunch's telegram in my hands and tried to find even a cowpath
through the woods.
Uncle Peter came out, none the worse for his cold plunge, and sat
down near me.
"Ah, my boy, isn't this delightful!" he cried, drinking in the air.
"There's nothing like the country, I tell you! Look at that view!
Isn't it grand? John, to be frank with you, up until I saw this
place I didn't have much faith in your ability as a business man,
but now I certainly admire your wisdom in selecting a spot like
this--what did it cost you?"
Cost me! so far it had cost me an attack of nervous prostration,
but I couldn't tell him that. I hesitated for the simple reason
that I hadn't the faintest idea what the place had cost Bunch. I
had been too busy to ask him.
"It's all right, John," the old fellow went on; "don't think me
inquisitive. A rubberneck is the root of all evil. It's only
because I've been watching you rather closely since we came out
here and you seem to be nervous about something. I had an idea
maybe it took all your ready money to buy the place, and possibly
you regret spending so much--but don't you do it! The best day's
work you ever did was when you bought this place!"
"Yes, I believe you!" I sighed, wearily, as I turned to look down
the road.
I stiffened in the chair for I saw my finish in the outward form of
two women rapidly approaching the house,
"It's Bunch's sister and her daughter," I moaned to myself. "Well,
I'll be generous and let the blow fall first on Uncle Peter!"
Accordingly, I made a quick exit,
In the kitchen I found Clara J., her headache forgotten, busily
preparing to cook the dinner.
She's a foxy little bundle of peaches, that girl is; and I was wise
to the fact that her suspicion factory was still working over-time,
turning out material for the undersigned.
I felt it in my bones that the steer I gave her about Aunt Eliza
had been placed in cold storage for safe keeping.
Her brain was busy running to the depot to meet the scandal Bunch's
telegram hinted at, but she pretended to catch step and walk along
with me.
"John," she said, "I certainly do hope your relatives won't come
out for some little time, because we really aren't ready for
visitors, now are we, dear?"
"Indeed we are not," I groaned.
"I can't help thinking it awfully strange that you should be
notified of their coming by Mr. Jefferson, and in such peculiar
language," she said, after a pause.
"Didn't I tell you Bunch is a low comedian," I said, weakly.
"Besides, he knows them very well. Aunt Fanny is very fond of
Bunch."
"Aunt Fanny," she repeated, dropping a tin pan to the floor with a
crash; "I thought you said her name was Eliza?"
"Sure thing!" I chortled; while my heart fell off its perch and
dropped in my shoes. "Her name is Eliza Fanny; some of us call her
Aunt Eliza, some Aunt Fanny--see?"
She hadn't time to see, for at that moment Tacks rushed in,
exclaiming, "Say, sister, they's two strange women on the piazza
talking to Uncle Peter, and maybe when they go one of them will
fall down the steps if I put some more soap there!"
Like a whirlwind he was gone again. Clara J. simply looked at me
queerly and said, "The queens are here; treat them white, John!"
I felt as happy as a piece of cheese.
CHAPTER VI.
JOHN HENRY^S TWO QUEENS.
"Well!" said Clara J., after a painful pause, "why don't you go and
welcome your Aunt Eliza?"
Aunt Lize would be the central figure in a hot old time if she went
where I wished her at that moment.
Somebody had tied both my feet to the floor.
I had visions of two excited females lambasting me with umbrellas
and demanding their property back.
Completely at a loss I sank into a chair, feeling as bright and
chipper as a poached egg.
I felt that I belonged just about as much as a knothole does in a
barb-wire fence.
In that few minutes Bunch was more than revenged.
I was on the pickle boat for sure.
Sailing! sailing! over the griddle, me!
Scientists tell us that when a man is drowning every detail of his
lifetime passes before him in the fraction of a second.
Well, that moving picture gag was worked on me, without the aid of
a bathing suit.
When I awoke, Clara J. was saying, "Possibly it would look better
if I went with you. Wait just a moment, till I get this apron
off--there! come along!"
I arose, and with delightful unanimity the chair arose also,
clinging like a passionate porusplaster to my pantaloons.
"Mercy'" exclaimed Clara J., "that little villain, Tacks, has been
making molasses candy!"
"It strikes me," I said, trying hard to be calm, "that after making
the candy he decided to make a monkey of me. Darn the blame thing,
it won't let go! I suppose I've got to be a perpetual furniture
mover the rest of my life!"
Just then Uncle Peter came bubbling into the kitchen, talking in
short explosions like a bottle of vichy, and I collaborated with
the chair in a hasty squatty-vous!
"Two women on the piazza," he fizzed; "been talking to them an hour
and all I could get out of them was 'yes' and 'no.' Not bad
looking, but profoundly dumb."
"Hush!" said Clara J., glancing uneasily at me and then back at
Uncle Peter, as she raised a warning finger to her lips.
"Oh, they can't hear me," the old gentleman went on; "John, you
better go out and see them. They have a card with your name
written on it. I'm no lady's man, anyhow."
"Do they look like queens?" Clara J. asked, uneasily.
"Well, they aren't exactly Cleopatras, but not bad, not bad!" he
gurgled.
"Is one older than the other?" Clara J. cross-questioned.
"Might be mother and daughter," Uncle Peter fancied.
"It's surely Bunch's bunch," I groaned inwardly, wondering how I'd
look galloping across the country with a kitchen chair trailing
along behind.
"Uncle Peter, it must be John Henry's Aunt Eliza and cousin Julia.
He expects them, don't you, John?" Clara J. explained. "We shall
be ready to welcome them in just a little while;" here she glanced
cautiously at the chair. "In the meantime you show them into the
spare room and say that John will see them very soon."
The old gentleman eyed me suspiciously and retired without a word.
I'm afraid Uncle Peter found it hard to take.
With the kind assistance of the carving knife Clara J. removed all
of me from the chair, with the exception of a few feet of trousers,
and I made a quick change of costume.
A few minutes later I joined her in the parlor, where the scene was
set for my finish. I picked out a quiet spot near the piano to die.
Uncle Peter was enjoying every minute of it.
He hurried off to escort the visitors to the parlor and a moment
later Aunt Martha bustled in.
"Are they here?" she asked breathlessly.
"How did you know they were coming?" inquired Clara J. in surprised
tones.
"How did I know!" exclaimed Auntie; "why I sent them!"
Every hand was against me. The parachute had failed to work and I
was dropping on the rocks.
Faintly and far away I could hear the ambulance coming at a gallop.
Sweet spirits of ammonia, but I was up against it!
It was plainly evident to me that Aunt Martha knew the awful
relatives of Bunch, and that the old lady was camping on my trial.
Yes; there she stood, old Aunt Nemesis, glaring at me from behind
her spectacles.
I decided to die without going over near the piano.
"Where are they?" I could hear Aunt Martha asking in the same tone
of voice I was certain the Roman Emperor used when just about to
frame up a finale for a few Christians from over the Tiber.
"Uncle Peter has gone for them; we put them in the spare room,"
answered Clara J.
"What! _in the spare room_!" gasped Aunt Martha, collapsing in a
chair just as Uncle Peter appeared in the doorway, bowing low
before the visitors, who stalked clumsily into the parlor.
For some reason or other Clara J. omitted the formality of
springing forward and greeting my relatives effusively, so she
simply said, "You are very welcome, Aunt Eliza and cousin Julia!"
"Great heavens! what does this mean?" shrieked Aunt Martha. "It
cannot be possible that these two women are relatives of yours,
John! Why, I engaged them both in an intelligence office; one for
the kitchen, the other as parlor maid!"
"Sure not," I chirped, in joy-freighted accents, as I grasped the
glorious situation. "They aren't my relatives and never were. The
more I look at them the more convinced I am that there's no room
for them to perch on my family tree. I disown them both. Back to
the woods with the Swede imposters!"
I win by an eyelash.
I was so happy I went over to the mantel and began to bite the
bric-a-brac.
Clara J. didn't know whether to laugh or cry, so she compromised by
giggling at Uncle Peter, who sat on the piano stool whirling
himself around rapidly and muttering, "any kind of exercise is good
exercise."
Aunt Martha stared around the room from one to another in
speechless amazement, while the two innocent causes of all the
trouble stood motionless, with their noses tip-tilted to the
ceiling.
Presently Aunt Martha broke the spell just as I was about to eat a
cut-glass vase in the gladness of my heart.
"Go to the kitchen!" she said sharply to the newcomers, whereupon
they both turned in unison and looked the old lady all over.
Finally they decided to discharge Aunt Martha, for the oldest
member of the troupe folded her arms decisively and said, "Sure, it
ain't in any lunatic asylum I'll be afther livin', bless th'
Saints! If yez have a sinsible moment left in your head will yez
give us th' car fare back to th' city, and it'll be a blessed hour
for me whin I plants me feet on th' ferryboat, so it will!"
Uncle Peter checked the fiery course of the piano stool and began
to make his double chin do a gurgle, whereupon the youngest of the
two female impersonators handed him a glare that put out his
chuckle and he started the piano stool again at the rate of 45
revolutions per minute.
"Th' ould buffalo over there showed us up to th' spare room,
thinkin' to be funny," she who was fated never to be our cook, went
on, "and if I wasn't in a daffy house and him nothin' but a bug
it's the weight of that chair he'd feel over his bald spot. Th'
ould goosehead, to set us down on th' porch and talk to us for an
hour about th' landshcape and th' atmusphere, and to ask me, a
respectable lady, what kind of exercise I was partial to! It's a
Hiven's own blessin' I didn't hand him a poke in th' slats, so it
is!"
Uncle Peter, with palpably assumed indifference, slid off the piano
stool and faded behind the furthermost window curtain, while I went
up to the belligerent visitor and said, "On your way, Gismonda; the
referee gives the fight to you; here's the gate receipts!"
With this I handed her a ten-spot which she looked at suspiciously
and said, "If ever I get that ould potato pounder over in New York
it's exercise I'll give him! Sure, I'll run him from th' Bat'hry
to Harlem widout a shtop for meals, bad cess to him!"
Having delivered this parting knock at Uncle Peter, the queen of
the kitchen flounced out of the house, followed by the younger one
who had played only a thinking part in the strenuous scene.
Aunt Martha still sat motionless in the chair, quite on the verge
of tears, when Clara J. went over to her and said, "Why didn't you
tell me you were going after servants, Auntie?"
"I wanted to surprise you," the old lady replied, plaintively.
"They were to be my contribution to the household."
"You handed us a surprise, all right; didn't she, Uncle Peter?" I
chirped in with a view to laughing off the whole affair, but just
then a series of startling shrieks caused us all to rush for the
piazza.
At the gate we beheld a kicking, struggling mass of lingerie and
bad dialect, which presently resolved itself into the forms of my
temporary relatives who were now busily engaged in macadamizing the
roadway with their heads.
Then Tacks came yelling on the scene: "I thought maybe they was
female burglars so I stretched a wire acrost the gate and they was
in such a hurry getting away that they never noticed it till it was
too everlastingly late!"
Before we could remonstrate with the Boy-Disaster he let another
whoop out of him and darted off in the direction of the barn.
That whoop brought the two wire-tappers to their feet and after
they both shook their fists eagerly in our direction they started
in frenzied haste for the depot.
As they scurried frantically out of our neighborhood Uncle Peter
smiled blandly and murmured, "For lecturers, female reformers and
all those who lead a sedentary life there's nothing like exercise!"
Putting my arm around Clara J.'s waist I whispered, "Didn't I tell
you it was one of Bunch's put-up jobs? He's jealous because I'm so
happy out here with you, that's all! As for the telegram, forget
it!"
"All right, John," said Clara J., "but nevertheless that same
telegram gave you a busy day, didn't it?"
"It surely did, but it was only because I hated to have you
worried," I answered as she went in the house to console Aunt
Martha.
I sat down in a chair expecting every moment to have the Prince of
Liars come up and congratulate me.
Humming a tune quietly to himself Uncle Peter watched the flying
squadron disappear in a bend of the road, then he sat down near me
and said, "John, you're worried about something and I've a pretty
fair idea what it is. This property is too big a load for you to
carry, eh?"
From the depths of my heart I replied, "It certainly is!"
"Well," said the old gentleman, "it surely has made a hit with me.
I never struck a place I liked half as well as this. How would you
like to sell it to me, then you and Clara J. could live with us,
eh? Come on, now, what d'ye say?"
I sat there utterly unable to say anything.
"What did it cost you; come on, now, John?" the old fellow urged.
"Oh, about $14,000," I whispered, picking out the first figure I
could think of.
"It's worth it and more, too," he said. "I'll give you $20,000 for
it--say the word!"
"Well, if you insist!" I replied, weakly; and the next minute he
danced off to write me a check.
In the tar barrel every time I opened my mouth! Hard luck was
certainly putting the wrapping paper all over me.
Well, the only thing to do now was to hustle up to town in the
morning and inform Bunch that I had sold his property.
I felt sure he'd be tickled to a stand-still--not!
CHAPTER VII.
JOHN HENRYS HAPPY HOME.
Early the next morning I broke camp and took the trail to town,
determined never to come back alive unless Bunch agreed to sell the
plantation to Uncle Peter.
The old gentleman had crowded his check for $20,000 into my
trembling hands the night before with instructions to deposit it in
my bank, and at my convenience I was to let him have the deed to
the place.
Well, if Bunch should refuse to play ball I could send the check
back to Uncle Peter, and a telegram to Clara J., telling her that I
was back in the flat, laid up with a spavined fetlock or something.
Uncle Peter was out in the garden planting puree of split peas or
some other spring vegetable when I started for the train, so all
the Recording Angel had to put down against me was the new batch of
Ochiltrees I told Clara J.
I soon located Bunch, and to my surprise found him more inclined to
josh than to jolt.
[Illustration: Bunch Jefferson--All to the Good and Two to Carry.]
"Ah! my friend from the bush!" he exclaimed; "are you in town to
buy imitation coal, or is it to get a derrick and hoist your home
affairs away from my property? Why don't you take a tumble, John,
and let go?"
"Bunch," I said, "believe me, this is the crudest game of
freeze-out I ever sat in. My throat is sore from singing, 'Father,
dear father, come home with me now!' and every move I make nets me
a new ornamentation on my neck. Why didn't I tell the good wife
that the ponies put the crimp in my pocketbook instead of crawling
into this chasm of prevarication and trouble?"
"You can search me!" Bunch answered, thoughtfully.
"And that phony wire you sent me yesterday almost gave me a
plexus," I said bitterly. "Why did you frame up one of those
when-we-were-twenty-one dispatches from the front? It sounded like
a love song from Willie Hayface of Cohoes, after his first day on
Broadway. Didn't you know that my wife was liable to open that
queer fellow and put me on the toasting fork?"
Bunch blinked his eyes solemnly, but when I told him all about the
trouble his telegram had caused he simply rose up on his hind legs
and laughed me to a sit down.
"Well," he gasped after a long fit of cackling; "sister did intend
going out to Jiggersville and the only way I could stop her was to
suddenly discover that her health wasn't any too good, so I chased
her off to Virginia Hot Springs for a couple of weeks."
After all, Bunch had his redeeming qualities.
"I sent you that wire before I took sister's temperature," Bunch
explained, "and I quite forgot to send another which would put a
copper on the queens."
Once more he laughed uproariously and chortled between the
outbursts, "Now--ha, ha, ha!--I'm even for--ha, ha, ha!--for that
shoot the chute I did in your--ha, ha, ha--in your cellar--oh! ha,
ha, ha, ha!"
"Oh, quit your kidding!" I begged, and then, suddenly, "Say, Bunch,
will you sell the old homestead?"
Bunch stopped laughing and looked me over from head to foot. "Is
this on the level or simply another low tackle?"
"It's the goods," I answered: "I simply can't frighten, coax,
scare, drive or push my home companions away from your property, so
I'd like to buy it if you're game to cut the cards?"
"Been playing the lottery?" he snickered.
"No, but I have the Pierponts, all right, all right," I replied;
"will you put $14,000 in your kick and pass me over the baronial
estate?"
"Fourteen thousand!" Bunch repeated slowly. "Sure, I will. If you
can Morgan that amount I'll make good with the necessary documents,
and then you and your family troubles may sit around on fly paper
in Jiggersville for the rest of your natural lives for all I care."
I explained to Bunch that I wanted the deed made out in the name of
Peter Grant for the reason that Uncle Peter was a bigger farmer
than I, and in short order the preliminary arrangements were
completed to the satisfaction and relief of both parties concerned.
That evening I went back to Jiggersville feeling as light as a pin
feather on a young duck.
Uncle Peter could have the property; Bunch could buy his sister
another castle, and I was ahead of the game just $6,000, more than
enough to square me for all the green paper I had torn up at the
track.
Of course, it did look as though Uncle Peter had been whipsawed,
but when I considered the bundles the old gentleman had stored away
in the vaults, and when I remembered his eagerness to cough, I
simply couldn't produce one pang of conscience.
Two days later Bunch had a certified check for $14,000 and Uncle
Peter was the happy owner of the country estate.
"We will live with you and Aunt Martha a little while," I said to
him; "but if you have no objection I'd like to buy a small lot down
near the brook from you and build a bit of a cage there for
ourselves."
Uncle Peter chuckled affirmatively, but seemed unwilling to
continue the subject further. "Isn't it glorious out here," he
smiled. "Pure air, fresh from the bakery of Heaven! I have
younged myself ten years since we came out here. Yesterday I fell
in a bear trap which Tacks had dug and carefully concealed with
brush and leaves. It took me four hours to get out because I'm
rather stout, but the exercise surely did me good."
Can you beat him?
A week later the second anniversary of our wedding would roll
around, and although Clara J. was a trifle hard to win over, I
finally coaxed her to let me have Bunch out to spend a few hours
with us on that occasion.
At the appointed hour Bunch arrived and Clara J. greeted him with
every word of that telegram darting forth darkly from her eyes.
"Mrs. John," said Bunch, "I'm simply delighted to know you. I've
often heard your husband speak well of you."
She had to smile in spite of herself.
"Mrs. John," Bunch went on, with splendid assurance; "you should be
proud of this matinee idol husband of yours, for, to tell you the
truth, he's all the goods--he certainly is."
Clara J. looked somewhat embarrassed, and as for me, I was away out
to sea in an open boat. I hadn't the faintest idea what Bunch was
driving at.
"You surely have a wonderful influence over him," the lad with the
blarney continued. "A week or so ago I threw some bait at him just
to test him and he didn't even nibble. You know, in the old days
John and I often trotted in double harness to the track--bad place
for young men--sure!"
Bunch surveyed the property with a quick glance and said, "Yes, I
sent John a telegram. 'The two queens will be out this afternoon,'
I wired, meaning two horses that simply couldn't lose. 'They are
good girls, so treat them white,' I told him, meaning that he
should put up his roll on them and win a hatfull; but, Mrs. John, I
never touched him. He simply ignored my telegram and sat around in
the hammock all day, reading a novel, I suppose. I apologize to
you, Mrs. John, for trying to drag him away from the path of
rectitude, but, believe me, I didn't know when I sent the message
that he had promised you to give the ponies the long farewell!"
Clara J. laughed with happiness, all her doubts dispersed, and
said, "Oh, don't mention it, Mr. Bunch! I'm simply delighted to
welcome you to our new home. You have never been out here before,
have you?"
Bunch glanced at me, then through the open front door in the
direction of the scene of his downfall, and said, hesitatingly,
"Never before, thank you, kindly!"
Good old Bunch. He had squared me with my wife and the world--oh,
well, some day, perhaps, I'd get a chance to even up.
"John," he said, a few minutes later, when we took a short stroll
around the place. "Now that I've started in to tell the whole
truth I musn't skip a paragraph. This is a pleasant bit of
property, but the solemn fact remains that I put the boots to you.
I gave you the gaff for $6,000, old friend, and it breaks my heart
to tell you that I'm not sorry. Bunch for Number One, always!"
"What do you mean?" I asked.
"This farm only cost me $8,000," he said, giving me the pitying
grin.
"It cost me $14,000 and I sold it for $20,000," I said, slowly.
We stopped and shook hands.
"Who's the come-on?" he asked, presently.
"Uncle Peter," I answered, "but the old boy has so much he has to
kick a lot of it out of the house every once in a while, so it's
all right."
After dinner we were all sitting on the piazza listening to a
treatise from Uncle Peter on the subject of the growth and proper
care of wheat cakes, or asparagus, I forget which, when suddenly
the cadaverous form of the Sherlock Holmes of Jiggersville appeared
before us.
"Evenin' all!" bowed Harmony Diggs, clinging tightly to a bundle
which he held under his arm.
"Find that robber yet?" inquired Bunch, winking at me.
"That's just what I dropped around for to tell you, thinkin' maybe
you'd be kinder interested in knowin' the facts in the case,"
Harmony went on, carefully placing the precious bundle on the steps.
"I got a clue from this here gent," he said, pointing a bony finger
at Bunch, "and I ups and chases that there maleyfactor for four
miles, well knowin' that the cause of justice would suffer and the
reward of fifty dollars be nil and voidless if the critter got
away. But I got him, by crickey, I got him!"
He looked from one to the other, seeking a sign of applause, and
Bunch said, "Where did you catch him?"
"About four miles yonder," Diggs explained, indefinitely. "It was
a fierce fight while it lasted, but they ain't no maleyfactor
livin' can escape the clutches of these here hands oncet they
entwines him. I pulled the dem cuss out of his clothes!"
With this thrilling announcement he opened the bundle and proudly
displayed the burglar harness which Bunch had worn on that
memorable night.
"And the burglar himself?" Bunch questioned.
Diggs raised his head slowly, and with theatrical effect answered,
"I give the cussed scoun'rel the doggonest drubbin' a mortal
maleyfactor ever got and let him go. That was nearly two weeks
ago, and he ain't showed up since, dag him!"
"You win, Mr. Ananias!" said Bunch, handing Diggs a ten dollar
bill, as he whispered to me, "That story is worth the money."
"What's that for?" inquired Diggs, somewhat taken aback.
"That's my contribution to the reward for the robber," Bunch told
him.
"Well," spluttered Diggs; "it don't seem zactly right, seein' as
how I on'y pulled the cuss out of his clothes and then let him go
with a lambastin'."
"The ten-spot is for the clothes you pulled him out of," Bunch
said, picking up the garments and handing them to me. "Keep them,
John, as a souvenir of your first burglar--and true friend, Bunch!"
I took them reverently, and said, "For your sake, Bunch, they'll be
handed down from generation to generation."
Clara J. blushed and said, "Oh, John!" and I thought Uncle Peter
would chuckle himself into a delirium.
"Good-night, Mr. Ananias!" Bunch called, as Diggs made a farewell
bow and turned to go.
"Good-night, one and all," replied Diggs, then a thought struck him
and he turned with, "Say, who's this here Mr. Annienias? Seems
like the name's familiar, but it ain't mine."
"Mr. Ananias is the first detective mentioned in history," Bunch
explained, and Mr. Diggs beamed over us all.
"Wait a moment, Mr. Officer," Aunt Martha piped in; "have a drop of
refreshment before you go. Tacks, run in and pour Mr. Officer a
drink from that bottle on the sideboard!"
Diggs stood there swallowing his palate in delightful anticipation
until Tacks handed him a brimming glass from which the brave
thief-taker took one eager mouthful, whereupon he emitted a shriek
of terror that could be heard for miles.
"Water! water! quick! I'm a'burnin' up!" cried the astonished
Diggs.
Uncle Peter in his eagerness to quench the flames poured half a
pitcher full of ice water down the back of Diggs' neck.
"It ain't there, it's down my throat!" yelled the unfortunate
Harmony, whereupon Uncle Peter poured the rest of the ice water
over the constable's head.
When, finally, the old fellow was revived he faintly declined any
more refreshment, and with a sad "good-night," faded away in the
twilight.
"Gee!" exclaimed Tacks, as he watched the retreating form, "I'm
afraid I upset some tobascum sauce in that glass by mistake."
Presently, Bunch went off to the depot to take a train back to the
city, and for some little time we sat in silence on the piazza.
"Grand, isn't it?" Uncle Peter said, breaking the spell. "Couldn't
be any nicer, now, could it?" Then he went over and stood near
Clara J.
"Little woman," he said; "ever since we first talked of moving out
here I noticed how worried John was."
"So did I," she answered, taking my hand in hers.
"A day or two ago I found out what the trouble was," the old
gentleman continued; "this property was too heavy a load for a
young man to carry, especially when he's just married, so I bought
it from him!"
Before Clara J. could express a word Uncle Peter put his arm around
Aunt Martha's waist and continued, "Aunt Martha and I talked it all
over last night and in celebration of your second anniversary we
want you to accept this little present," and with this he placed a
document in Clara J.'s hands.
"It's the deed to the property," Aunt Martha said, "all for you,
Clara J., but if you don't mind, we'd like to live here!"
"Yes," said Uncle Peter; "that garden certainly needs someone to
look after it!"
Clara J. was crying softly and hugging Aunt Martha,
My own eyes were damp and I yearned to have somebody run the lawn
mower over me.
"I'll race you down to the gate and back," I suggested.
"You're on," laughed Uncle Peter; "I believe I do need a little
exercise!"
***
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 2,611
|
import sys
import os
try:
import biplist
except ImportError:
sys.stdout.write("You must install the biplist module: pip install biplist\n")
sys.exit(1)
def extplist(plistfile):
#print "Processing " + plistfile
try:
plistelements=biplist.readPlist(plistfile)
except biplist.InvalidPlistException:
print "Invalid plist file: " + plistfile
return
except OverflowError:
print "Invalid plist data: " + plistfile
return
except IOError:
print "Bad file name: " + plistfile
return
# Sometimes we get a list or string, not a dictionary
if not isinstance(plistelements, dict):
return
for key in plistelements:
if isinstance(plistelements[key], biplist.Data):
# Try to extract plist
try:
newp=biplist.readPlistFromString(plistelements[key])
except biplist.InvalidPlistException:
# Not valid plist data
continue
newfilename=os.path.basename(plistfile[:-6]) + "-" + key + ".plist"
i=1
while True:
if not os.path.exists(newfilename):
break
else:
newfilename=os.path.basename(plistfile[:-6]) + "-" + key + "-" + str(i) + ".plist"
i+=1
print "Writing " + newfilename
try:
biplist.writePlist(newp,newfilename)
except:
print "Could not write plist file " + newfilename
print sys.exc_info()[0]
extplist(newfilename)
if __name__ == "__main__":
if len(sys.argv) == 1:
print "Usage: %s [plist files]"%sys.argv[0]
sys.exit(0)
for pfile in sys.argv[1:]:
if pfile[-6:] != ".plist":
print "Skipping file '%s'. File names must end in '.plist'"%pfile
continue
extplist(pfile)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,785
|
Archive for the tag "Cooperstown"
Ten Facts About Cooperstown, New York
Virtually every baseball fan knows that the Baseball Hall of Fame is in Cooperstown, New York. But what do we know about Cooperstown, N.Y.? I've been to Cooperstown a couple of times, though it's been nearly twenty years since I've had the opportunity to visit the Hall of Fame. I thought I might take a few minutes to see what kind of information I could uncover about Cooperstown. Here are some facts I've decided to share with you:
Cooperstown, New York (Photo credit: Dougtone)
1) Cooperstown is not named for the writer, James Fenimore Cooper (although the author did live and pen some of his stories, such as "The Last of the Mohicans," in Cooperstown. It is actually named for his father, William Cooper, who founded this town in the late 1780's (though it first became officially incorporated in 1812.)
2) Cooperstown Dreams Park was established in 1996, and the Youth Baseball League it serves features up to 1,350 teams competing per season. The season lasts from the end of May until the end of August.
3) The population of Cooperstown is 1,833, down nearly ten percent since the year 2000. The population of Cooperstown is 91% white. There are six black families and one resident of full-blooded Native-American ancestry. Females outnumber males 55% to 45%. There is one registered sex offender in town limits.
4) About one-quarter of the people of Cooperstown walk to work. That's very cool, except in the winter.
5) Approximately 35% of the population are affiliated with a religious congregation. Nationally, about 51% of Americans are affiliated with a particular religious congregation. A plurality in Cooperstown are Catholics (43%.)
6) The most common first name among deceased individuals in Cooperstown is Mary. The most common last name among deceased individuals is Smith. I would suggest that if your name is Mary Smith, you might want to avoid Cooperstown. On the other hand, you would have a life expectancy of 81.5 years old.
7) The first speeding ticket issued in Cooperstown was given out in 1906.
8) No one born in Cooperstown has ever played Major League baseball.
9) Company G of the 176th Infantry Regiment of New York was recruited from Otsego County (in which Cooperstown is located), as well as a few of the other surrounding counties. They saw action in Virginia, North Carolina and Louisiana. The majority of casualties this regiment suffered occurred at the Battle of Cedar Creek in Virginia in 1864.
10) The last public hanging in Cooperstown took place in December 1827. The man condemned to death was a first cousin of James Fenimore Cooper named Levi Kelley, convicted of killing his tenant, Abraham Spafard. While the hangman was putting the noose around Kelley's neck, the grandstand collapsed under the weight of the crowd of onlookers, killing one person and mortally wounding another. The execution, however, went on as scheduled.
By The Numbers: Cooperstown – Celebrating Baseball And Culture
Upstate NY DA: cooperstown gunman captured in Va.
Fenimore Art Museum – Cooperstown, NY
Posted in American History, baseball, Hall of Fame and tagged Cooperstown, Cooperstown New York, James Fenimore Cooper, Last of the Mohicans, National Baseball Hall of Fame & Museum, New York, North Carolina, William Cooper
My Hall of Fame Ballot, and a Cautionary Tale
Are you familiar with the Hall of Fame for Great Americans, located on the campus of Bronx Community College in New York City? Not many people are. It was formally dedicated in May, 1901, as place to honor prominent Americans who had a significant impact on U.S. history and culture. Modeled on the Pantheon in Rome, its 630 foot open-air colonnade was conceived as a place where marble busts of America's most significant writers, presidents, inventors, and the like would be commemorated for all time. A very serious blue ribbon panel of 100 men was cobbled together to make initial nominations, and for several decades, the landmark was taken quite seriously.
As you have probably guessed by now, the existence of this Hall of Fame put the seed of an idea into the head of Ford Frick, who passed this idea along to Stephen Clark (of the Cooperstown Clarks), whose very wealthy local family connections paved the way for this unlikely caper to come to fruition. Stephen saw this as an idea to bring business to Cooperstown, suffering from the ravages of the Great Depression, and nearly overnight, this quaint little village was dedicated as hallowed ground where the Abner Doubleday legend also conveniently took root. That there was no easy way to transport people to Cooperstown to visit the proposed new shrine doesn't seem to have fazed Clark.
Meanwhile, while the National Baseball Hall of Fame in Cooperstown was just getting off the ground, the more established, high-brow Hall down in the Bronx (on what was then the campus of New York University) was in its heyday. The New York Bar Association went so far as advocating for certain of its members, and newspapers breathlessly covered the annual inductions.
In a fantastic little article I recently discovered, Baltimore Sun columnist Joe Mathews (August 1, 1997), wrote, in a sentence that could serve as a cautionary tale for the institution up in Cooperstown, "The 97-year old monument is a shrine not only to [them], but to an ideal of fame that, like the hall itself, is dusty and decaying."
Apropos to nothing, my favorite sentence in the article is, "The first hall of fame was the brainchild of a Presbyterian minister who was influenced by his concern for prostitution, democracy, and the Roman Empire." (emphasis added.) Mets brass, take note. Want to put asses in the seats at Citi Field next season? Why not go with "Prostitution, Democracy and the Roman Empire" as next season's slogan? It's certainly much more compelling than "Show up at Shea" (1998), or "Experience It" (2003).
Now, back to our story.
Hardly anyone ever visits The Hall of Fame for Great Americans these days anymore, even though it sits on an easily accessible college campus. Its committee of electors made its final official inductions in 1976. Among the four final inductees were a horticulturist and a judge. None of the final four have yet had a bronze bust built in their honor. Its Board of Trustees formally dissolved in 1979. Since then, the colonnade has been far more popular with pigeons than with people. You may still visit the 98 bronze busts in existence. Self-guided tours are available daily from 10:00-5:00, with a suggested donation of $2.00 per person.
Attendance to the Baseball Hall of Fame has steadily declined over the past twenty years, from a high of over 400,000 in the early 1990's to around 260,000 last year. Although the Hall of Fame is a non-profit institution, and is, in effect, a ward of the State of New York, it appears that its operating budget was over two million dollars in the red in its last fiscal year. Over the past decade, the HOF has more often than not lost money.
Outwardly, the Baseball Hall of Fame appears to be a healthy, thriving entity. It has a modern website, a Board of Directors featuring such luminaries as Tom Seaver and Joe Morgan, and disproportionate influence on how the game itself is remembered from one generation to the next. Its solid brick exterior and its pastoral location connote classical American values such as fortitude, temperance and diligence. And it contains part of the original facade of Ebbet's Field. What can go wrong?
By all means, consider the official Hall of Fame ballot a sacred totem of a mystical shrine, if you will, but consider this: Will our choices result in a stronger institution, more relevant to modern American sensibilities of entertainment and utility, or will they further contribute to the atrophy that apparently is slowly setting in?
Having said that, and while chafing at the ten-player limit arbitrarily imposed on actual BBWAA voters, here are my choices, in no particular order, for induction into the Baseball Hall of Fame:
1) Greg Maddux
2) Mike Piazza
3) Craig Biggio
4) Jeff Bagwell
5) Tim Raines
6) Tom Glavine
7) Mike Mussina
8) Alan Trammell
9) Frank Thomas
10) Don Mattingly
I'm sure the most controversial pick on this list will be Don Mattingly. Fine. Up until I set about typing this post, I would not have included him among this group, either. But in light of all the previous paragraphs I've written about The Hall in this article, the relevant question is, would the enshrinement of Donnie Baseball be a good thing for the future viability of The Hall, or would it somehow be a "bad" thing.
Three questions:
1) Was Don Mattingly ever the best player in the game during his career?
2) Did Don Mattingly represent the game, his team, and himself with nothing but respect both on the field and off?
3) Did he meet the 10-year minimum length career criteria for Hall eligibility?
The answer to each of these questions is yes. From 1984-87, there was no better player in the American League than Don Mattingly. He was always nothing but professional. He played for 14 seasons. At various times in his career, he led his league in hits, doubles, RBI, batting average, slugging percentage, OPS, OPS+, and total bases. From 1984-89, he averaged 330 total bases per season. Perhaps most impressively, however, he never struck out more than 43 times in any single full season in his career.
In his only playoff appearance, in 1995, vs. Seattle, he batted .417 in 25 plate appearances. He was a six time All Star, won three Silver Sluggers, nine Gold Gloves, and his .996 Fielding Percentage is among the ten best all-time at his position. He won an MVP award, and finished runner-up once as well. If he picked up a bat today, at age 52, he would probably still outhit Ike Davis.
Perhaps more to the point, Mattingly has legions of loyal fans who might just possibly trek all the way up to Cooperstown to see their hero enshrined, and to listen to his acceptance speech. Years from now, dads might still be taking their kids to see Mattingly's plaque at The Hall. How many parents do you think bring their kids all the way up to Cooperstown each year to stand in awe of the plaques of HOF "immortals" such as Herb Pennock, Rick Ferrell, Lloyd Waner, or Dave Bancroft?
Explain to me, then, how inducting Don Mattingly into the Baseball Hall of Fame would be bad for baseball, or for The Hall itself?
In the final analysis, the Hall of Fame is an idea as much as it is a place. All baseball fans, in their heart of hearts, have their own idea as to what constitutes fame in this context. When the chasm between what fans believe in their hearts is legitimate fame relative to the actual composition of the institution itself grows too wide, then the fans, faced with an untenable choice, will always follow one and ignore the other. Should that happen, The Baseball Hall of Fame may one day bear an uncanny resemblance to that other unfortunately failed Hall of Fame further downstate on a bluff overlooking the indifferent Harlem River.
Posted in baseball, Baseball Commentary, Hall of Fame and tagged Cooperstown, Don Mattingly, Greg Maddux, Hall of Fame, Hall of Fame for Great Americans, Mike Mussina, National Baseball Hall of Fame and Museum, Tom Glavine
Who Belongs In the Hall of Fame? (Almost Anyone)
Quick, tell me three things you know about Dave Bancroft.
O.K., tell me two things you know about Dave Bancroft. No, he is not the U.S. Ambassador to Great Britain. No, he is not the twenty-something lead vampire in the "Twilight" franchise.
Alright, so using deductive reasoning, you figured out he was a baseball player. This is, after all, a website dedicated to baseball.
Well, did you know that he played from 1915-1930? Did you know that he was a career .279 hitter who played good defense, amassed 2,004 career hits and scored just over a thousand runs, primarily as a shortstop for the Phillies, Giants and Braves?
Did you know his nickname was "Beauty?"
His career OPS (On-Base plus Slugging Percentage) barely topped .700, pretty low for a hitter in any era at any position, other than pitcher.
In other words, he had a substantial, if unremarkable career as a baseball player. A career not unlike those of Tony Fernandez, Alvin Dark, Dick Groat and Jay Bell.
With one substantial difference. Dave Bancroft was selected by the Veterans Committee to be enshrined as a member of the Baseball Hall of Fame in 1971, forty years after he retired.
Dave Bancroft is just one of several members of the Hall of Fame who are, at best, questionable choices to represent baseball in what is essentially baseball's equivalent to Mt. Olympus or Mt. Rushmore. After all, would we carve a huge profile of Grover Cleveland on Mt. Rushmore just because he was a two-term president?
Grover Cleveland Alexander, on the other hand, might make a pretty good choice.
So why do the Baseball Writers of America, who are tasked with the assignment of choosing the newest inductees on an annual basis, struggle so much with their respective choices? Doing so indicates that they truly believe that the HOF standard are players such as Ty Cobb, Willie Mays, and Tom Seaver.
The reality has always been dramatically different. As Bill James wrote in "The New Bill James Historical Baseball Abstract" published in 2001, "The Ted Williams…standard for Hall of Fame selection has never existed anywhere except in the imaginations of people who don't know anything about the subject."
This is not to say the writers shouldn't take their responsibility seriously. But how much should a writer agonize over whether Andre Dawson was significantly better than Dale Murphy (who despite two MVP Awards, is not in the Hall.)
Voting for or against Andre Dawson can be justified using the the lowest common denominator logic of "Well, Player A is in, and Player B was just about as good as Player A, so Player B should be in as well."
This logic has permeated the selection of Hall of Famers for many years, and will probably continue to do so for the foreseeable future. It is the logic that resulted in the induction of players such as Tony Perez and Orlando Cepeda, and probably Jim Rice.
Sometimes, it helps to have a brother who is already in the HOF. Lloyd Waner was voted in by the Veterans Committee. It must have helped his cause that his brother, Paul, was already in The Hall.
If you are fortunate enough to have been part of a cool sounding trio of infielders, your chances of being inducted into The Hall also apparently increase. Thus Tinkers-to-Evers-to-Chance resulted in at least two, if not three, questionable inductions (Frank Chance might have been the only deserving member of that trio.)
In my last post, I stated that the two essential questions regarding the issue of baseball immortality are 1) Who deserves to be remembered? and 2) How do they deserve to be remembered?
The Veterans Committee seems to have been primarily motivated by the fear that certain players that they hold dear, former friends, peers and colleagues, might simply vanish into obscurity.
The irony is that even induction into The Hall of Fame doesn't necessarily mean a ticket to immortality. After all, do people really make the pilgrimage to the tiny little village of Cooperstown, smack dab in the middle of nowhere, so that they could stand in awe in front of Fred Clarke's plaque?
So what is a sportswriter, ballot in hand, to do?
Here's another question? Why has the task been given to sportswriters? Do we allow news journalists to select our senators for us? Writers and journalists are generally competent at reporting on the world around them. This does not automatically indicate a degree of wisdom superior to that of the average baseball fan.
And if they are so competent, then why are they given 15 chances per retired player to get it right? (or wrong, depending on how you view the induction of Jim Rice.)
In my next blog, I will suggest a few alterations to the current induction system. Ultimately, perhaps, the Dave Bancrofts of the world can receive their due as significant contributors to the game in a more reasonable fashion, if only to leave more elbow room for Mays, Williams and Ruth.
Posted in Baseball History, Baseball Today and tagged Baseball Hall of Fame, Bill James, Cooperstown, Dave Bancroft, sportswriters, Veterans Committee
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Q: Looping and concatenating in Keras Sequential API Lets say I have a model defined like this:
from tensorflow.keras.models import Sequential, Model
from tensorflow.keras.layers import (BatchNormalization, concatenate,
Conv2D, Conv2DTranspose, DepthwiseConv2D,
Dropout, Input, MaxPooling2D,
ReLU, ZeroPadding2D)
input_layer = Input((64, 64, 3))
conv1 = Conv2D(16, (3, 3), padding="same")(input_layer)
conv1 = BatchNormalization()(conv1)
conv1 = ReLU()(conv1)
pool1 = MaxPooling2D((2,2))(conv1)
conv2 = Conv2D(32, (3, 3), padding="same")(pool1)
conv2 = BatchNormalization()(conv2)
conv2 = ReLU()(conv2)
pool2 = MaxPooling2D((2,2))(conv2)
conv3 = Conv2D(64, (3, 3), padding="same")(pool2)
conv3 = BatchNormalization()(conv3)
conv3 = ReLU()(conv3)
pool3 = MaxPooling2D((2,2))(conv3)
mid = Conv2D(128, (3, 3), padding="same")(pool3)
mid = BatchNormalization()(mid)
mid = ReLU()(mid)
dconv3 = Conv2DTranspose(64, (3, 3), strides=(2, 2), padding="same")(mid)
cat3 = concatenate([dconv3, conv3])
dconv2 = Conv2DTranspose(32, (3, 3), strides=(2, 2), padding="same")(dconv3)
cat2 = concatenate([dconv2, conv2])
dconv1 = Conv2DTranspose(16, (3, 3), strides=(2, 2), padding="same")(dconv2)
cat1 = concatenate([dconv1, conv1])
output_layer = Conv2D(1, (1,1), padding="same", activation="sigmoid")(dconv1)
model = Model(input_layer, output_layer)
The model is a very simple UNET which requires that the down sample blocks be concatenated with the upsample blocks. Lets now imagine that I want to define this exact model but with some arbitrary depth aka 2, 3, 4, 5 etc.. downsample and upsample blocks. Instead of having to go in and manually modify the parameters, I would like to automate the model building.
I am very close to accomplishing this, but I fail during concatenation. See below.
class configurable_model():
def __init__(self, csize, channels, start_neurons, depth):
self.csize = csize
self.channels = channels
self.start_neurons = start_neurons
self.depth = depth
def _convblock(self, factor, name):
layer = Sequential(name=name)
layer.add(Conv2D(self.start_neurons * factor, (3, 3), padding="same"))
layer.add(BatchNormalization())
layer.add(ReLU())
return layer
def build_model(self):
model = Sequential()
model.add(Input((self.csize, self.csize, self.channels), name='input'))
factor = 1
for idx in range(self.depth):
model.add(self._convblock(factor, f'downblock{idx}'))
model.add(MaxPooling2D((2,2)))
factor *= 2
model.add(self._convblock(factor, name='middle'))
for idx in reversed(range(self.depth)):
factor //= 2
model.add(Conv2DTranspose(self.start_neurons * factor, (3, 3),
strides=(2, 2), padding="same",
name=f'upblock{idx}'))
#how do I do the concatenation??
model.add(concatenate([model.get_layer(f'upblock{idx}'),
model.get_layer(f'downblock{idx}')]))
model.add(Conv2D(1, (1,1), padding="same",
activation="sigmoid", name='output'))
return model
test = configurable_model(64, 3, 16, 3)
model = test.build_model()
I have tried converting to the functional API, but run into the problem of 'naming' the layers and keeping track of them in the for loops. I tried Concatenate instead of concatenate. I tried model.get_layer('layername').output and model.get_layer('layername').output() in the concatenate statement, etc... nothing is working. The code above gives the error: ValueError: A Concatenate layer should be called on a list of at least 2 inputs.
A: I was able to get the functional version working by storing the downblocks in a dictionary that I reference later during concatenation. See below:
class configurable_model():
def __init__(self, csize, channels, start_neurons, depth):
self.csize = csize
self.channels = channels
self.start_neurons = start_neurons
self.depth = depth
def _convblock(self, factor, name=None):
block = Sequential(name=name)
block.add(Conv2D(self.start_neurons * factor, (3, 3), padding="same"))
block.add(BatchNormalization())
block.add(ReLU())
block.add(Conv2D(self.start_neurons * factor, (3, 3), padding="same"))
block.add(BatchNormalization())
block.add(ReLU())
return block
def build_model(self):
input_layer = Input((self.csize, self.csize, self.channels), name='input')
x = input_layer
factor = 1
downblocks = {}
for idx in range(self.depth):
x = self._convblock(factor, f'downblock{idx}')(x)
downblocks[f'downblock{idx}'] = x
x = MaxPooling2D((2, 2), name=f'maxpool{idx}')(x)
factor *= 2
x = self._convblock(factor, 'Middle')(x)
for idx in reversed(range(self.depth)):
factor //= 2
x = Conv2DTranspose(self.start_neurons * factor, (3, 3),
strides=(2, 2), padding="same",
name=f'upsample{idx}')(x)
cat = concatenate([x, downblocks[f'downblock{idx}']])
x = self._convblock(factor, f'upblock{idx}')(cat)
output_layer = Conv2D(1, (1, 1), padding="same",
activation="sigmoid", name='output')(x)
return Model(input_layer, output_layer)
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
Most stars form in clusters and smaller groups in the densest parts of giant
molecular clouds (GMCs). The fragmentation of molecular
clouds results in dense filaments which contain still denser cores.
These cold star-forming cores are best detected using far-infrared (FIR) and
submillimetre (submm) dust continuum. By studying their physical and
chemical characteristics one hopes to understand conditions leading to
protostellar collapse and the timescale related to this process. Furthermore,
the distribution and spacing of dense cores place constraints on the
fragmentation mechanisms (e.g., turbulent fragmentation and ambipolar
diffusion) and the possible interaction between newly born stars and their
surroundings (e.g., \cite{megeath2008}).
The estimates of the core masses resulting from dust continuum data can also be
used to examine the possible connection between the mass distribution of dense
cores and the stellar initial mass function (IMF), a question of great current
interest (e.g., \cite{simpson2008}; \cite{swift2008}; \cite{goodwin2008};
Rathborne et al. 2009, submitted).
Parameters affecting the cloud dynamics, such as the
degree of ionization and the abundances of various positive ions are
chemically related to deuterium fractionation and depletion of heavy
species (e.g., CO). Besides being important for the core
dynamics through the magnetic support and molecular line cooling,
these parameters depend on the core
history and characterise its present evolutionary stage.
For example, substantial CO depletion and deuterium
enrichment are supposed to be a characteristic
of prestellar cores in the pivotal stage before collapse
(\cite{caselli1999}; \cite{bacmann2002}; \cite{lee2003}).
Observations (\cite{tafalla2002}, 2004, 2006) and some theoretical models
(\cite{bergin1997}; \cite{aikawa2005}) suggest, however, that N-containing
species such as the chemically closely related nitrogen
species N$_2$H$^+$ and NH$_3$ (and their deuterated isotopologues),
remain in the gas phase at densities for which CO and other C-containing
molecules are already depleted (e.g., \cite{flower2005}; 2006b and references
therein).
They are therefore considered as useful spectroscopic tracers of prestellar
cores and the envelopes of protostellar cores.
There is, however, some evidence that N$_2$H$^+$ finally
freezes out at densities $n({\rm H_2})=$ several $\times10^5$ to $\gtrsim10^6$
cm$^{-3}$ (e.g., \cite{bergin2002}; \cite{pagani2005}, 2007). Contrary
to this, NH$_3$ abundance appears to \textit{increase} toward the centres of
e.g., L1498 and L1517B (\cite{tafalla2002}, 2004). Similar result was found
by Crapsi et al. (2007) using interferometric observations toward L1544.
\subsection{Ori B9}
Most of molecular material in the Orion complex is concentrated
on the Orion A and B clouds. Star formation in Orion B (L1630) takes mainly
place in four clusters, NGC 2023, NGC 2024, NGC 2068, and NGC 2071
(e.g., \cite{lada1992}; \cite{launhardt1996}).
The Orion B South cloud, which encompasses
the star-forming regions NGC 2023/2024 is the only site of O and B star
formation in Orion B (see, e.g., \cite{nutter2007}, hereafter NW07).
Apart of the above mentioned four regions, only single stars or small groups of
low- to intermediate mass stars are currently forming in Orion B
(\cite{launhardt1996}).
Ori B9 lies in the central part of Orion B and is a relatively isolated cloud
at a projected distance of $\sim40\arcmin$ (5.2 pc at 450 pc\footnote{We assume
a distance to the Orion star-forming regions of 450 pc.}) northeast from
the closest star cluster NGC 2024 (\cite{caselli1995}) which is the most
prominent region of current star formation in Orion B.
Ori B9 has avoided previous (sub)mm mappings which have
concentrated on the well-known active regions in the northern and
southern part of the GMC (NW07 and references therein).
In this paper we present results from the submm continuum
mapping of the Ori B9 cloud with LABOCA on APEX, and from spectral line
observations towards three N$_2$H$^+$ peaks found by Caselli \& Myers
(1994) in the clump associated with the low-luminosity FIR source
IRAS 05405-0117
(see Fig.~2 in Caselli \& Myers (1994))\footnote{The N$_2$H$^+(1-0)$
map of Caselli and Myers (1994) shows two separated gas condensations of
$\sim0.1$ pc in size. The southern condensation has a weak subcomponent.
The positions of our molecular line observations are given in Table~1 of
Harju et al. (2006).}.
This source has the narrowest CS linewidth ($0.48$ km s$^{-1}$) in the Lada
et al. (1991) survey, and the narrowest NH$_3$ linewidth (average linewidth
is $0.29$ km s$^{-1}$) in the survey by Harju et al. (1993).
A kinetic temperature of 10 K was derived from ammonia in this region.
We have previously detected the H$_2$D$^+$ ion towards two of the N$_2$H$^+$
peaks (\cite{harju2006}). These detections suggest that 1) the degree of
molecular depletion is high and 2) the ortho:para ratio of H$_2$ is low,
and thus the cores should have reached an evolved chemical stage.
The high density and low temperature may have resulted in CO
depletion. This possibility is supported by the fact that
the clump associated with IRAS 05405-0117 does not stand out in the CO map
of Caselli \& Myers (1995).
In the present study we determine the properties and
spatial distribution of dense cores in the Ori B9 cloud. We also
derive the degree of deuteration and ionization degree within the clump
associated with IRAS 05405-0117.
\vspace{1cm}
The observations and the data reduction procedures are described in Sect. 2.
The observational results are presented in Sect. 3. In Sect. 4 we describe the
methods used to derive the physical and chemical properties of the observed
sources. In Sect. 5 we discuss the results of our study, and in Sect. 6
we summarise our main conclusions.
\section{Observations and data reduction}
\subsection{Molecular lines: N$_2$H$^+$ and N$_2$D$^+$}
The spectral line observations towards the three above mentioned N$_2$H$^+$
peaks were performed with the IRAM 30 m telescope on Pico Veleta, Spain, on
May 18--20, 2007.
The spectra were centred at the frequencies of the strongest N$_2$H$^+(1-0)$
and N$_2$D$^+(2-1)$ hyperfine components. We used the following rest
frequencies: 93173.777 MHz (N$_2$H$^+$($JF_1F=123\rightarrow012$),
\cite{caselli1995b}) and 154217.154 MHz (N$_2$D$^+$($234\rightarrow123$),
\cite{gerin2001}). Dore et al. (2004), and very recently
Pagani et al. (2009b), have refined the N$_2$H$^+$ and N$_2$D$^+$ line
frequencies. The slight differences between the
"new" and "old" frequencies have, however, no practical effect on
to the radial velocities or other parameters derived here.
The observations were performed in the
frequency switching mode with the frequency throw set to 7.9 MHz for the 3 mm
lines and 15.8 MHz for the 2 mm lines.
As the spectral backend we used the VESPA
(Versatile Spectrometer Assembly) facility autocorrelator which has
a bandwidth of 20 MHz and a channel width of 10 kHz.
The lines were observed in two polarisations using the (AB)
100 GHz and the (CD) 150 GHz receivers. The horizontal polarisation
at higher frequency (D150) turned out to be very noisy and was thus excluded
from the reduction.
The channel width used corresponds to 0.032 km s$^{-1}$ and 0.019 km s$^{-1}$
at the observed frequencies of N$_2$H$^+(1-0)$ and N$_2$D$^+(2-1)$,
respectively. The half-power beamwidth (HPBW) and the main beam efficiency,
$\eta_{\rm MB}$, are $26\farcs4$ and 0.80 at 93 GHz, and $16\arcsec$ and 0.73
at 154 GHz.
Calibration was achieved by the chopper wheel method. The pointing and focus
were checked regularly towards Venus and several quasars.
Pointing accuracy is estimated to be better than $4-6\arcsec$.
The single-sideband (SSB) system temperatures were $\sim150-190$ K at 93 GHz
and $\sim290-340$ K at 154 GHz.
We reached an rms sensitivity in antenna temperature units of about
0.03 K in N$_2$H$^+(1-0)$ and about $0.05-0.07$ K in N$_2$D$^+(2-1)$.
The observational parameters are listed in Table~\ref{table:obs}.
The CLASS programme, which is part of the GAG software developed at the
IRAM and the Observatoire de
Grenoble\footnote{{\tt http://www.iram.fr/IRAMFR/GILDAS}},
was used for the reductions.
Third order polynomial baselines were subtracted from the individual
N$_2$H$^+(1-0)$ spectra before and after folding them.
From each individual N$_2$D$^+(2-1)$ spectra the
fourth order polynomial baselines were subtracted before folding.
Finally, the summed spectra were Hanning smoothed yielding the velocity
resolutions of 0.064 km s$^{-1}$ for N$_2$H$^+(1-0)$ and 0.038 km s$^{-1}$ for
N$_2$D$^+(2-1)$.
We fitted the lines using the hyperfine structure fitting method
of the CLASS programme. This method assumes that all the hyperfine components
have the same excitation temperature and width
($T_{\rm ex}$ and $\Delta {\rm v}$, respectively),
and that their separations and relative line strengths are fixed to the values
given in Caselli et al. (1995), Gerin et al. (2001), and Daniel et al. (2006).
Besides $T_{\rm ex}$ and $\Delta {\rm v}$ this method gives an estimate of the
total optical depth, $\tau_{\rm tot}$, i.e, the sum of the peak optical depths
of the hyperfine components. These parameters can be used to estimate
the column density of the molecule.
\begin{table*}
\caption{Observational parameters.}
\begin{minipage}{2\columnwidth}
\centering
\renewcommand{\footnoterule}{}
\label{table:obs}
\begin{tabular}{c c c c c c c c c c c}
\hline\hline
Molecule & Transition & Frequency & Instrument & $F_{\rm eff}$ & $\eta_{\rm MB}$\footnote{$\eta_{\rm MB}=B_{\rm eff}/F_{\rm eff}$, where $B_{\rm eff}$ and $F_{\rm eff}$ are the beam and forward efficiencies, respectively.} & \multicolumn{2}{c}{Resolution} & $F_{\rm throw}$ & $T_{\rm sys}$ & Obs. date\\
& & [GHz] & & & & [\arcsec] & [km s$^{-1}$] & [MHz] & [K] & \\
\hline
N$_2$H$^+$ & $J=1-0$ & 93.173777\footnote{Frequency taken from Caselli et al. (1995).} & 30 m/AB100 & 0.95 & 0.80 & 26.4 & 0.064 & 7.9 & 150--190 & 18--20 May 2007\\
N$_2$D$^+$ & $J=2-1$ & 154.217154\footnote{Frequency taken from Gerin et al. (2001).} & 30 m/CD150 & 0.93 & 0.73 & 16 & 0.038 & 15.8 & 290--340 & 18--20 May 2007\\
\hline
\multicolumn{3}{c}{Continuum at 870 $\mu$m} & APEX/LABOCA & 0.97 & 0.73 & 18.6 & - & - & - & 4 Aug. 2007\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\subsection{Submillimetre continuum}
The 870 $\mu$m continuum observations toward the Ori B9 cloud were carried out
on 4 August 2007 with the 295 channel bolometer array LABOCA
(Large APEX Bolometer Camera) on APEX.
The LABOCA central frequency is about 345 GHz and the bandwidth is about
60 GHz. The HPBW of the telescope is $18\farcs6$ (0.04 pc at 450 pc)
at the frequency used. The total field of view (FoV) for the LABOCA is
$11\farcm4$.
The telescope focus and pointing were checked using the planet Mars and
the quasar J0423-013.
The submm zenith opacity was determined using the sky-dip method and
the values varied from 0.16 to 0.20, with a median value of 0.18.
The uncertainty due to flux calibration is estimated to be $\sim10$\%.
The observations were done using the on-the-fly (OTF) mapping mode with
a scanning speed of $3\arcmin$ s$^{-1}$. A single map consisted of 200 scans
of $30\arcmin$ in length in right ascension and spaced by $6\arcsec$ in
declination.
The area was observed three times, with a final sensitivity of about
0.03 Jy beam$^{-1}$ (0.1 M$_{\sun}$ beam$^{-1}$ assuming a dust temperature
of 10 K).
The data reduction was performed using the BoA (BOlometer Array Analysis
Software) software package according to guidelines in the BoA User and
Reference
Manual (2007)\footnote{{\tt http://www.astro.uni-bonn.de/boawiki/Boa}}.
This included flat fielding, flagging bad/dark channels and data
according to telescope speed and acceleration, correcting for the atmospheric
opacity, division into subscans, baseline subtractions and median noise
removal, despiking, and filtering out the low frequencies of the $1/f$-noise.
Finally, the three individual maps were coadded.
\subsection{Spitzer/MIPS archival data}
Pipeline (version S16.1.0) reduced ``post-BCD (Basic Calibrated Data)''
Spitzer/MIPS images at 24 and 70 $\mu$m were downloaded from the Spitzer
data archive using the Leopard
software package\footnote{{\tt http://ssc.spitzer.caltech.edu/propkit/spot/}}.
We used the software package MOPEX
(MOsaicker and Point source EXtractor)\footnote{
{\tt http://ssc.spitzer.caltech.edu/postbcd/mopex.html}} to perform aperture
and point-spread function (PSF) fitted photometry on the sources.
The point sources were extracted using the APEX package (distributed as part
of MOPEX).
At 24 $\mu$m a 5.31 pixel aperture with a sky annulus between 8.16 and 13.06
pixels for background subtraction were used. At 70 $\mu$m the pixel aperture
was 8.75 pixels and the sky annulus ranged from 9.75 to 16.25 pixels.
The pixel scale is $2\farcs45$/pixel for 24 $\mu$m and $4\farcs0$/pixel
for 70 $\mu$m. The MIPS resolution is $\sim6\arcsec$ and $\sim18\arcsec$
at 24 and 70 $\mu$m, respectively. These values correspond to 0.01 pc and
0.04 pc at the cloud distance of 450 pc.
The aperture correction coefficients used with these settings
are 1.167 and 1.211 at 24 and 70 $\mu$m, respectively, as given on the Spitzer
Science Center (SSC)
website\footnote{{\tt http://ssc.spitzer.caltech.edu/mips/apercorr}}.
The absolute calibration uncertainties are about 4\% for 24 $\mu$m, and about
10\% for 70 $\mu$m (\cite{engelbracht2007}; \cite{gordon2007}).
\section{Observational results}
\subsection{Dust emission}
The obtained LABOCA map is presented in Fig.~\ref{figure:map}.
Altogether 12 compact sources can be identified on this map.
A source was deemed real if it had a peak flux density $>5\sigma$
(i.e., $>0.15$ Jy beam$^{-1}$) relative to the local background.
The coordinates, peak and integrated flux densities, deconvolved angular
FWHM diameters, and axis ratios of the detected sources are listed in
Table~\ref{table:cores}.
The coordinates listed relate to the dust emission peaks.
The integrated flux densities have been derived
by summing pixel by pixel the flux density in the source area.
The uncertainty on flux density is derived from
$\sqrt{\sigma_{\rm cal}^2+\sigma_{\rm S}^2}$, where $\sigma_{\rm cal}$ is the
uncertainty from calibration, i.e., $\sim10$\% of flux density,
and $\sigma_{\rm S}$ is the uncertainty from flux density determination
based on the rms noise level near the source area.
We have computed the deconvolved source angular diameters, $\theta_{\rm s}$,
assuming that the brightness distribution is Gaussian.
The values of $\theta_{\rm s}$ correspond to the geometric mean
of the major and minor axes FWHM obtained from two-dimensional Gaussian fits to
the observed emission which has been corrected for the beam size.
The uncertainty on $\theta_{\rm s}$ has been calculated by
propagating the uncertainties on the major and minor axes FWHM, which are
formal errors from the Gaussian fit. The axis ratio is defined as
the ratio of deconvolved major axis FWHM to minor axis FWHM.
Both the flux density determination and Gaussian fitting to the sources were
done using the Miriad software package (\cite{sault1995}).
Four of the detected sources have IRAS (Infrared Astronomical Satellite) point
source counterparts, whereas eight are new submm sources.
We designate the eight new sources as SMM 1, SMM 2,
etc. The locations of three N$_2$H$^+(1-0)$ line emission peaks from Caselli
\& Myers (1994) are indicated on the map with plus signs
(see Figs.~\ref{figure:map} and ~\ref{figure:positions}).
The N$_2$H$^+$ peak Ori B9 E which lies $40\arcsec$ east of IRAS 05405-0117
does not correspond to any submm peak (see Fig.~\ref{figure:positions}).
The N$_2$H$^+$ peak Ori B9 N lies about $39\arcsec$ southeast of the closest
dust continuum peak (see Sect. 5.5). One can see that our pointed
N$_2$H$^+$/ N$_2$D$^+$ observations, made before the LABOCA mapping, missed
the strongest submm peak SMM 4 located near IRAS 05405-0117.
\begin{figure*}
\centering
\includegraphics[width=18cm]{laboca_map_test.eps}
\caption{LABOCA 870 $\mu$m map of the Ori B9 cloud. The three large
plus signs in the centre of the field mark the positions of our molecular line
observations (see Fig.~\ref{figure:positions}). The small plus sign in
the upper left shows the dust peak position of IRAS 05412-0105.
The beam HPBW ($18\farcs6$) is shown in the bottom left.}
\label{figure:map}
\end{figure*}
\begin{table*}
\caption{Submillimetre sources in the Ori B9 cloud.}
\begin{minipage}{2\columnwidth}
\centering
\renewcommand{\footnoterule}{}
\label{table:cores}
\begin{tabular}{c c c c c c c}
\hline\hline
& \multicolumn{2}{c}{Peak position} & $S_{870}^{\rm peak}$ & $S_{870}$ & $\theta_{\rm s}$ & \\
Name & $\alpha_{2000.0}$ [h:m:s] & $\delta_{2000.0}$ [$\degr$:$\arcmin$:$\arcsec$] & [Jy beam$^{-1}$] & [Jy] & [\arcsec] & Axis ratio\\
\hline
IRAS 05399-0121 & 05 42 27.4 & -01 19 50 & 0.81 & $2.7\pm0.3$ & $30\pm4$ & 1.3\\
SMM 1 & 05 42 30.5 & -01 20 45 & 0.41 & $3.0\pm0.3$ & $57\pm7$ & 2.5\\
SMM 2 & 05 42 32.9 & -01 25 28 & 0.21 & $0.7\pm0.1$ & $26\pm5$ & 3.8\\
SMM 3 & 05 42 44.4 & -01 16 03 & 1.14 & $2.5\pm0.4$ & $19\pm3$ & 1.5\\
IRAS 05405-0117 & 05 43 02.7 & -01 16 21 & 0.19 & $0.9\pm0.1$\footnote{These values include both the IRAS 05405-0117 and Ori B9 E (see text and Fig.~\ref{figure:positions}).} & $45\pm8$ $^a$ & 1.2\\
SMM 4 & 05 43 03.9 & -01 15 44 & 0.28 & $1.3\pm0.1$ & $34\pm6$ & 2.3\\
SMM 5 & 05 43 04.5 & -01 17 06 & 0.16 & $0.6\pm0.1$ & $38\pm4$ & 1.8\\
SMM 6 & 05 43 05.1 & -01 18 38 & 0.26 & $2.5\pm0.3$ & $92\pm35$ & 4.1\\
Ori B9 N & 05 43 05.7 & -01 14 41 & 0.16 & $1.0\pm0.1$ & $47\pm5$ & 1.2\\
SMM 7 & 05 43 22.1 & -01 13 46 & 0.32 & $0.8\pm0.1$ & $24\pm4$ & 1.8\\
IRAS 05412-0105 & 05 43 46.4 & -01 04 30 & 0.17 & $0.5\pm0.1$ & - & -\\
IRAS 05413-0104 & 05 43 51.3 & -01 02 50 & 0.66 & $0.9\pm0.2$ & $25\pm8$ & 1.6\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\subsection{Spitzer/MIPS images}
The retrieved Spitzer/MIPS images at 24 and 70 $\mu$m are presented in
Fig.~\ref{figure:spitzer}. All four IRAS sources that were detected by LABOCA
are visible at both 24 and 70 $\mu$m (IRAS 05412-0105 and 05413-0104 northeast
from the central region are outside the regions shown in
Fig.~\ref{figure:spitzer}). From the new submm sources, SMM 3 and
SMM 4 are also visible at both 24 and 70 $\mu$m, while there is 24 $\mu$m
source near SMM 5 which is not detected at 70 $\mu$m.
The rest of the submm sources are visible at neither of the wavelenghts.
In Table~\ref{table:spitzer} we list the sources detected at both 24 and 70
$\mu$m. In this table we give the 24 and 70 $\mu$m peak positions of the
sources and their flux densities at both wavelengths obtained from the
aperture photometry. The $1\sigma$ uncertainties on flux densities were
derived as described in Sect. 3.1, i.e. as a quadratic sum of the
calibration and photometric uncertainties.
\begin{figure}[!h]
\resizebox{\hsize}{!}{\includegraphics[angle=270]{24um.eps}}
\resizebox{\hsize}{!}{\includegraphics{70um.eps}}
\caption{Spitzer/MIPS 24 $\mu$m (top) and 70 $\mu$m (bottom) images of the
central part of the Ori B9 cloud. The logarithmic colour scale range
from 33.4 to 729.5 MJy sr$^{-1}$ and 27.9 to 1357.4 MJy sr$^{-1}$ in the 24
and 70 $\mu$m images, respectively.}
\label{figure:spitzer}
\end{figure}
\begin{table*}
\caption{Spitzer 24/70 $\mu$m sources in Ori B9.}
\begin{minipage}{2\columnwidth}
\centering
\renewcommand{\footnoterule}{}
\label{table:spitzer}
\begin{tabular}{c c c c c c c c c}
\hline\hline
& \multicolumn{2}{c}{24 $\mu$m peak position} & \multicolumn{2}{c}{70 $\mu$m peak position} & $S_{24}$ & $S_{70}$ \\
Name & $\alpha_{2000.0}$ [h:m:s] & $\delta_{2000.0}$ [$\degr$:$\arcmin$:$\arcsec$] & $\alpha_{2000.0}$ [h:m:s] & $\delta_{2000.0}$ [$\degr$:$\arcmin$:$\arcsec$] & [Jy] & [Jy]\\
\hline
IRAS 05399-0121 & 05 42 27.6 & -01 20 01 & 05 42 27.7 & -01 19 57 & $1.3\pm0.05$ & $24.4\pm2.4$\\
SMM 3 & 05 42 45.3 & -01 16 14 & 05 42 45.1 & -01 16 13 & $0.005\pm0.0002$ & $3.6\pm0.4$\\
IRAS 05405-0117 & 05 43 03.1 & -01 16 29 & 05 43 03.0 & -01 16 30 & $1.3\pm0.05$ & $2.7\pm0.3$\\
SMM 4 & 05 43 05.7 & -01 15 55 & 05 43 05.6 & -01 15 52 & $0.036\pm0.001$ & $1.3\pm0.1$\\
IRAS 05412-0105 & 05 43 46.3 & -01 04 44 & 05 43 46.1 & -01 04 43 & $0.6\pm0.02$ & $2.5\pm0.3$\\
IRAS 05413-0104 & 05 43 51.4 & -01 02 53 & 05 43 51.3 & -01 02 51 & $0.2\pm0.01$ & $17.9\pm1.8$\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\subsection{N$_2$H$^+$ and N$_2$D$^+$}
The three positions of our molecular line observations are indicated
in Figs.~\ref{figure:map} and ~\ref{figure:positions}.
These positions correspond to the three
N$_2$H$^+(1-0)$ peaks found by Caselli \& Myers (1994) (see also Sect. 1.1).
The Hanning smoothed N$_2$H$^+(1-0)$ and N$_2$D$^+(2-1)$ spectra are
shown in Figs.~\ref{figure:n2h+} and \ref{figure:n2d+}, respectively.
The seven hyperfine components of N$_2$H$^+(1-0)$ are clearly resolved
towards all three positions. The N$_2$H$^+(1-0)$ spectra
towards Ori B9 E and Ori B9 N show additional lines, which can be explained
by N$_2$H$^+(1-0)$ emission originating at a different radial velocity
(see Fig.~\ref{figure:n2h+}). In the case of Ori B9 E, the additional
N$_2$H$^+(1-0)$ lines originate in gas at a radial velocity of 1.3 km s$^{-1}$,
whereas towards Ori B9 N the additional gas component has a ${\rm v}_{\rm LSR}$
of 2.2 km s$^{-1}$. These velocities are $\sim7-8$ km s$^{-1}$ lower than the
average velocity of the Ori B9 cloud, suggesting that they are produced by
a totally different gas component.
We checked that the additional components are not caused by, e.g., a
phase-lock failure by summing randomly selected subsets of the spectra.
All sums constructed in this manner showed the same features with
equal intensity ratios. The ``absorption''-like
feature at $\sim20$ km s$^{-1}$ in the N$_2$H$^+(1-0)$ spectrum of Ori B9 N
is an arfefact caused by the frequency switching folding process.
Only the strongest hyperfine group of N$_2$D$^+(2-1)$ was detected.
The relatively poor S/N ratio hampers the hyperfine component
fitting. Towards Ori B9 N, the additional velocity component at
$\sim2.2$ km s$^{-1}$ was also detected in N$_2$D$^+(2-1)$.
In Table~\ref{table:line_parameters} we give the N$_2$H$^+(1-0)$ and
N$_2$D$^+(2-1)$ line parameters derived from Hanning smoothed spectra.
The LSR velocities and linewidths (FWHM) are
listed in columns (2) and (3), respectively. The total optical depth and
excitation temperature for the lines are given in columns (6) and (7),
respectively. The excitation temperatures, $T_{\rm ex}$, of the
N$_2$H$^+(1-0)$ transition were derived from the antenna equation
\begin{equation}
\label{eq:ant_1}
T_{\rm A}^{*}= \eta \frac{h\nu}{k_{\rm B}}\left[F(T_{\rm ex})-F(T_{\rm bg})\right]\left(1-e^{-\tau}\right) \, ,
\end{equation}
where $\eta$ is the beam-source coupling efficiency, $h$ is the
Planck constant, $\nu$ is the transition frequency, $k_{\rm B}$ is the
Boltzmann constant, $T_{\rm bg}=2.725$ K is the cosmic microwave background
(CMB) temperature, and the function $F(T)$ is defined by
$F(T)\equiv\left(e^{h\nu/k_{\rm B}T}-1\right)^{-1}$.
We assumed that $\eta=\eta_{\rm MB}$, and we used the main beam
brightness temperature, $T_{\rm MB}=\eta_{\rm MB}^{-1}T_{\rm A}^*$,
and the optical thickness, $\tau$, of the brightest hyperfine component.
The uncertainty of $T_{\rm ex}$ has been calculated by propagating
the uncertainties on $T_{\rm MB}$ and $\tau$.
The total optical depth of the N$_2$D$^+(2-1)$ line cannot be calculated
directly because the hyperfine components are not resolved in the spectra.
We estimated the total optical depth in the following manner. First, we
calculated the optical depth of the main hyperfine group of N$_2$D$^+(2-1)$
from the antenna equation using the $T_{\rm MB}$ obtained from a Gaussian fit
to the group of 4 strongest hyperfines. In the calculation, we
adopted the excitation temperature of the N$_2$H$^+(1-0)$ lines.
Second, the total optical depths of N$_2$D$^+(2-1)$ were
calculated taking into account that the main group correspond to 54.3\% of
$\tau_{\rm tot}$.
The uncertainty on $\tau_{\rm tot}$ has been calculated by propagating
the uncertainties on $T_{\rm MB}$ and $T_{\rm ex}$.
\begin{table*}
\begin{minipage}{2\columnwidth}
\caption{N$_2$H$^+(1-0)$ and N$_2$D$^+(2-1)$ line parameters derived from
Hanning smoothed spectra. The integrated line intensity
($\int T_{\rm A}^*({\rm v}){\rm dv}$) includes all the
hyperfine components in the case of N$_2$H$^+$, whereas for N$_2$D$^+$ only
the main group is included.}
\centering
\renewcommand{\footnoterule}{}
\label{table:line_parameters}
\begin{tabular}{c c c c c c c}
\hline\hline
Line/Position & ${\rm v}_{\rm LSR}$ [km s$^{-1}$] & $\Delta {\rm v}$ [km s$^{-1}$] & $T_{\rm A}^*$ [K] & $\int T_{\rm A}^*({\rm v}){\rm dv}$ [K km s$^{-1}$] & $\tau_{\rm tot}$ & $T_{\rm ex}$ [K]\\
\hline
{\bf N$_2$H$^+$}$(1-0)$\\
IRAS 05405-0117\footnote{Caselli \& Myers (1994) derived ${\rm v}_{\rm LSR}=9.209\pm0.003$ km s$^{-1}$, $\Delta {\rm v}=0.313\pm0.008$ km s$^{-1}$, and $\tau_{\rm tot}=4.594\pm0.825$.} & $9.228\pm0.001$ & $0.290\pm0.002$ & $2.37\pm0.04$ & $3.97\pm0.05$ & $6.1\pm0.03$ & $6.8\pm0.07$\\
Ori B9 E\footnote{For the other velocity component hyperfine fit yields ${\rm v}_{\rm LSR}=1.310\pm0.013$ km s$^{-1}$, $\Delta {\rm v}=0.436\pm0.035$ km s$^{-1}$, and $\tau_{\rm tot}=6.6\pm2.0$.} & $9.163\pm0.002$ & $0.298\pm0.005$ & $1.46\pm0.04$ & $2.26\pm0.03$ & $3.5\pm0.5$ & $6.1\pm0.3$\\
Ori B9 N\footnote{For the other velocity component ${\rm v}_{\rm LSR}=2.219\pm0.006$ km s$^{-1}$, $\Delta {\rm v}=0.378\pm0.021$ km s$^{-1}$, $T_{\rm A}^*=0.52\pm0.03$ K, $\tau_{\rm tot}=1.0$, and $T_{\rm ex}=5.9\pm0.17$ K.} & $9.149\pm0.003$ & $0.261\pm0.008$ & $1.82\pm0.09$ & $2.29\pm0.04$ & $2.2\pm0.8$ & $8.3\pm1.4$\\
{\bf N$_2$D$^+$}$(2-1)$\\
IRAS 05405-0117 & $9.414\pm0.012$ & $0.319\pm0.027$ & $0.31\pm0.04$ & $0.13\pm0.01$ & $0.26\pm0.01$\footnote{$\tau_{\rm tot}$ calculated by taking into account
that the main hyperfine group corresponds to 54.3\% of the total optical
depth. $\tau_{\rm main \, group}$ is calculated using $T_{\rm MB}$ from Gaussian fit to the main group and $T_{\rm ex}$ from N$_2$H$^+(1-0)$.} & $6.8\pm0.07$\\Ori B9 E & $9.285\pm0.013$ & $0.194\pm0.029$ & $0.27\pm0.03$ & $0.09\pm0.01$ & $0.28\pm0.03^d$ & $6.1\pm0.3$\\
Ori B9 N\footnote{For the other velocity component ${\rm v}_{\rm LSR}=2.290\pm0.044$ km s$^{-1}$, $\Delta {\rm v}=0.187\pm0.090$ km s$^{-1}$, $T_{\rm A}^*=0.22\pm0.06$ K, and $\tau_{\rm tot}=0.12\pm0.01$ ($\tau_{\rm tot}$ is calculated as described in footnote $d$).} & $9.255\pm0.009$ & $0.136\pm0.021$ & $0.28\pm0.07$ & $0.09\pm0.01$ & $0.16\pm0.05^d$ & $8.3\pm1.4$\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\begin{figure}[!h]
\resizebox{\hsize}{!}{\includegraphics[angle=270]{clump_new.ps}}
\caption{Blow-up of Fig.~\ref{figure:map} showing the IRAS 05405-0117 clump
region. The large plus signs indicate the positions of our H$_2$D$^+$,
N$_2$H$^+$, and N$_2$D$^+$ observations towards three condensations shown in
Fig.~2 of Caselli \& Myers (1994). Also shown are the 24 $\mu$m peak positions
of SMM 4 and IRAS 05405-0117, and the 24 $\mu$m peak near SMM 5 (small green
plus signs, cf. Fig.~\ref{figure:spitzer}). The beam HPBW ($18\farcs6$) is
shown in the bottom left.}
\label{figure:positions}
\end{figure}
\section{Physical and chemical parameters of the sources}
\subsection{Spectral energy distributions}
The 24 and 70 $\mu$m flux densities together with the integrated flux
densities at 870 $\mu$m were used to fit the spectral energy
distribution (SED) of SMM 3 and SMM 4. For the IRAS sources also the archival
IRAS data were included.
The flux densities in the 12, 25, 60, and
100 $\mu$m IRAS bands are listed in Table~\ref{table:iras}.
The derived SEDs for SMM 3, SMM 4, and IRAS 05405-0117 are shown in
Fig.~\ref{figure:sed}.
For all six sources detected at three or more wavelengths, the data were
fitted with a two-temperature composite model.
The parameters resulting from the fitting are given in Table~\ref{table:sed}.
We adopted a gas-to-dust mass ratio of 100 and dust opacities
corresponding to a MRN size distribution with thick ice mantles at a gas
density of $n_{\rm H}=10^5$ cm$^{-3}$ (\cite{ossenkopf1994}).
The total (cold+warm) mass and the integrated bolometric luminosity are given
in columns (2) and (3) of Table~\ref{table:sed}, respectively.
The temperatures of the two components are listed
in columns (4) and (5). In columns (6) and (7) we give the mass and luminosity
fractions of the cold component vs. the total mass and luminosity, and in
column (8) we list the ratio of submm luminosity
(numerically integrated longward of 350 $\mu$m) to total bolometric luminosity
($L_{\rm submm}/L_{\rm bol}$). Column (9) list the normalised
envelope mass, $M_{\rm tot}/L_{\rm bol}^{0.6}$, which is an
evolutionary indicator in the sense that it correlates with the protostellar
outflow strength (i.e., with the mass accretion rate), and thus
decreases with time (\cite{bontemps1996}).
In column (10) we give the source SED classification (see Sect. 5.1).
In all cases the mass of the warm component is negligible
($\sim10^{-7}-10^{-4}$ M$_{\sun}$) and thus the bulk of the material is
cold ($M_{\rm cold}/M_{\rm tot}\sim1$).
\begin{table}
\caption{IRAS flux densities in Jy.}
\centering
\renewcommand{\footnoterule}{}
\label{table:iras}
\begin{tabular}{c c c c c}
\hline\hline
Name & $S_{12}$ & $S_{25}$ & $S_{60}$ & $S_{100}$\\
\hline
IRAS 05399-0121 & 0.25 & 1.59 & 22.94 & 45.93\\
IRAS 05405-0117 & 0.40 & 1.55 & 3.75 & 19.67\\
IRAS 05412-0105 & 0.26 & 0.65 & 1.66 & 73.58\\
IRAS 05413-0104 & 0.25 & 0.31 & 17.33 & 59.46\\
\hline
\end{tabular}
\end{table}
\begin{figure}[!h]
\resizebox{\hsize}{!}{\includegraphics[angle=270]{n2h+_IRAS.eps}}
\resizebox{\hsize}{!}{\includegraphics[angle=270]{n2h+_E.eps}}
\resizebox{\hsize}{!}{\includegraphics[angle=270]{n2h+_N.eps}}
\caption{N$_2$H$^+(1-0)$ spectra toward IRAS 05405-0117 (top), Ori B9 E
(middle), and Ori B9 N (bottom) after Hanning smoothing.
Hyperfine fits to the spectra are indicated with green lines.
The residuals of the fits are shown below the spectra. Hyperfine fits to the
other velocity component are indicated with red lines (see text). The
small ``absorption''-like feature at $\sim20$ km s$^{-1}$ in the bottom panel
is arfefact caused by frequency switching.}
\label{figure:n2h+}
\end{figure}
\begin{figure}[!h]
\resizebox{\hsize}{!}{\includegraphics[angle=270]{n2d+_IRAS.eps}}
\resizebox{\hsize}{!}{\includegraphics[angle=270]{n2d+_E.eps}}
\resizebox{\hsize}{!}{\includegraphics[angle=270]{n2d+_N.eps}}
\caption{N$_2$D$^+(2-1)$ spectra toward IRAS 05405-0117 (top), Ori B9 E
(middle), and Ori B9 N (bottom) after Hanning smoothing. Hyperfine fits to the
spectra are indicated with green lines. The lines under the spectra indicate
the positions and relative intensities of the hyperfine components
(see Table~2 in Gerin et al. (2001)). Undermost are plotted the residuals of
the fits. Hyperfine fit to the other velocity component in the bottom
panel is indicated with red line (see text).}
\label{figure:n2d+}
\end{figure}
\begin{figure}[!h]
\resizebox{\hsize}{!}{\includegraphics{SMM3_sed.ps}}
\resizebox{\hsize}{!}{\includegraphics{SMM4_sed.ps}}
\resizebox{\hsize}{!}{\includegraphics{IRAS05405_sed.ps}}
\caption{SEDs of the sources SMM 3 (top), SMM 4 (middle), and IRAS 05045-0117
(bottom). 24 and 70 $\mu$m data points are derived from archival Spitzer/MIPS
data, while the 870 $\mu$m measurement is performed with LABOCA.
For IRAS 05045-0117 we include also IRAS archival data
(see Table~\ref{table:iras}). $1\sigma$ error bars are indicated for Spitzer
and LABOCA data points. The solid lines in all plots represent the sum
of two (cold$+$warm) components (see columns (4) and (5) of
Table~\ref{table:sed}).
The ``ripple'' between $\sim30-60$ $\mu$m in the SED of IRAS 05045-0117 is due
to simple logarithmic interpolation used to derive the luminosity.}
\label{figure:sed}
\end{figure}
\begin{table*}
\caption{Results of the SED fits.}
\begin{minipage}{2\columnwidth}
\centering
\renewcommand{\footnoterule}{}
\label{table:sed}
\begin{tabular}{c c c c c c c c c c c}
\hline\hline
& $M_{\rm tot}$ & $L_{\rm bol}$ & $T_{\rm cold}$ & $T_{\rm warm}$ & & & & $M_{\rm tot}/L_{\rm bol}^{0.6}$ & \\
Source & [M$_{\sun}$] & [L$_{\sun}$] & [K] & [K] & $M_{\rm cold}/M_{\rm tot}$ & $L_{\rm cold}/L_{\rm bol}$ & $L_{\rm submm}/L_{\rm bol}$ & [M$_{\sun}$/L$_{\sun}^{0.6}$] & Class\\
\hline
IRAS 05399-0121 & $2.8\pm0.3$ & $21\pm1.2$ & $18.5\pm0.1$ & $103.9\pm0.2$ & $\sim1$ & 0.90 & 0.02 & 0.45 & 0/I\\
SMM 3 & $7.2\pm2.2$ & $3.5\pm0.2$ & $11.8\pm0.9$ & $36.9\pm0.2$ & $\sim1$ & 0.74 & 0.11 & 5.57 & 0\\
IRAS 05405-0117 & $1.6\pm0.2$ & $6.4\pm0.4$ & $16.1\pm0.1$ & $112.4\pm0.4$ & $\sim1$ & 0.69 & 0.03 & 0.53 & 0\\
SMM 4 & $3.8\pm0.2$ & $1.7\pm0.2$ & $11.6\pm0.2$ & $51.4\pm4.9$ & $\sim1$ & 0.76 & 0.11 & 2.76 & 0\\
IRAS 05412-0105 & $1.3\pm0.3$ & $5.8\pm0.6$ & $17.0\pm0.3$ & $127.8\pm0.5$ & $\sim1$ & 0.86 & 0.03 & 0.45 & 0\\
IRAS 05413-0104 & $1.0\pm0.2$ & $13.2\pm1.2$ & $20.3\pm0.4$ & $152.3\pm1.1$ & $\sim1$ & 0.97 & 0.02 & 0.21 & 0\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\subsection{Linear sizes, mass estimates and densities}
The linear sizes (radii $R=\theta_{\rm s}d/2$) were computed from the
angular FWHM sizes listed in Table~\ref{table:cores}.
The masses of the cores (gas+dust mass, $M_{\rm cont}$) are calculated
from their integrated 870 $\mu$m continuum flux density, $S_{870}$, assuming
that the thermal dust emission is optically thin:
\begin{equation}
\label{eq:mass}
M_{\rm cont}=\frac{S_{870}d^2}{B_{870}(T_{\rm d})\kappa_{870}R_{\rm d}} \; ,
\end{equation}
where $d$ is the distance, and $B_{870}(T_{\rm d})$ is the Planck function
with dust temperature $T_{\rm d}$.
For all IRAS sources as well as for SMM 3 and SMM 4 we adopted the dust
temperatures resulting from the SED fitting (see Table~\ref{table:sed},
column (4)).
For all the other sources it was assumed that $T_{\rm d}=10$ K.
The assumed dust temperature of 10 K is justified by
the estimates obtained from NH$_3$ (\cite{HWW1993}) and is commonly adopted
for starless cores. The assumption that $T_{\rm d}=T_{\rm kin}$, where
$T_{\rm kin}$ is the gas kinetic temperature, is likely to
be valid at densities $n({\rm H_2})>10^5$ cm$^{-3}$ (\cite{burke1983}).
The opacity per unit mass column density at
$\lambda=870$ $\mu{\rm m}$ is assumed to be
$\kappa_{870}\simeq0.17$ m$^2$ kg$^{-1}$. This value is interpolated from
Ossenkopf \& Henning (1994, see Sect. 4.1). The value $1/100$ is adopted
for the dust-to-gas mass ratio, $R_{\rm d}$.
The virial masses, $M_{\rm vir}$, of IRAS 05405-0117 and Ori B9 N have been
estimated by approximating the mass distribution by a homogenous, isothermal
sphere without magnetic support and external pressure (see, e.g., Eqs.~(1) and
(2) in Chen et al. (2008) where the linewidth of N$_2$H$^+$ is used).
The resulting virial masses are about 4.3
M$_{\sun}$ for IRAS 05405-0117 and 2.8 M$_{\sun}$ for Ori B9 N.
The corresponding $M_{\rm cont}/M_{\rm vir}$ ratios are about 0.3 and 1.4.
Note that it is usual that protostellar cores, like IRAS 05405-0117, appears
to be below the self-gravitating limit ($M_{\rm cont}/M_{\rm vir}=0.5$),
though they are forming stars (e.g., \cite{enoch2008}).
Since Ori B9 N appears to be
gravitationally bound, it is probably prestellar.
There are several factors that would lead to virial masses being
overestimated. For example, using the radial density profile with power-law
indices $p=1-1.5$ (see Sect. 5.3), would lead to $M_{\rm vir}$ being reduced
by factors of $1.1-1.25$.
The volume-averaged H$_2$ number densities, $\langle n({\rm H_2}) \rangle$,
were calculated assuming a spherical geometry for the sources and using masses,
$M_{\rm cont}$, and radii, $R$, estimated for them from the dust continuum map.
The obtained radii, masses, and volume-averaged H$_2$ number densities are
given in colums (2), (3), and (5) of Table~\ref{table:parameters},
respectively.
\begin{table*}
\caption{Linear radii, masses, and H$_2$ column and volume-averaged number
densities of all detected submm sources.}
\begin{minipage}{2\columnwidth}
\centering
\renewcommand{\footnoterule}{}
\label{table:parameters}
\begin{tabular}{c c c c c}
\hline\hline
& $R$ & $M_{\rm cont}$ & $N({\rm H_2})$ & $ \langle n({\rm H_2}) \rangle $\\
Source & [pc] & [M$_{\sun}$] & [$10^{22}$ cm$^{-2}$] & [$10^5$ cm$^{-3}$]\\
\hline
IRAS 05399-0121 & 0.03 & 3.6 & 2.61 & 7.5\\
SMM 1 & 0.06 & 11.8 & 3.89 & 3.1\\
SMM 2 & 0.03 & 2.8 & 1.99 & 5.8\\
SMM 3 & 0.02 & 7.2 & 7.80 & 51.1\\
IRAS 05405-0117 & 0.05\footnote{These values include both the IRAS 05405-0117 and Ori B9 E.} & 1.5$^a$ & 0.76 & 0.7$^a$\\
SMM 4 & 0.04 & 3.8 & 1.98 & 3.4\\
SMM 5 & 0.04 & 2.3 & 1.51 & 2.1\\
SMM 6 & 0.10 & 9.9 & 2.45 & 0.6\\
Ori B9 N & 0.05 & 3.9 & 1.51 & 1.7\\
SMM 7 & 0.03 & 3.1 & 3.01 & 6.5\\
IRAS 05412-0105 & -\footnote{Deconvolving the angular size was not possible, and thus the radius and number density could not be estimated.} & 0.8 & 0.63 & -$^b$\\
IRAS 05413-0104 & 0.03 & 1.0 & 1.85 & 2.2\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\subsubsection{Total mass of the Ori B9 region}
We made an estimate of the total mass in the region by using the
near-infrared extinction mapping technique
(NICER, \cite{lombardi2001}).
In this technique, the near-infrared colors, namely $H-K$ and
$J-H$, of the stars shining through the dust cloud are compared to the colours
of stars in a nearby field that is free from dust.
The reddened colours of the stars behind the dust cloud can then be interpreted
in terms of extinction due to the relatively well-known ratios of optical
depths at $JHK$ wavelengths (for further details of the method, we refer to
Lombardi \& Alves (2001)). To implement the method, we retrieved
$JHK$ photometric data from the 2MASS archive, covering a
$30\arcmin \times 19\arcmin$
region centred at $(\alpha, \delta)_\mathrm{J2000}$ = (5:43:00, -01:16:20).
Applying NICER to these data yielded an extinction map with the
resolution of FWHM=$2\farcm5$, indicating extinction values of
$A_\mathrm{V}=8\dots 12$ mag at the positions of the detected sources.
The total mass of the region was calculated from the derived extinction map by
summing up the extinction values of all pixels assuming the gas-to-dust ratio
of $N(\mathrm{H}) = 2\cdot 10^{21}$ cm$^{-2}$ mag$^{-1}$
(\cite{bohlin1978}), and the mean molecular weight per H$_2$ molecule of 2.8.
The total mass of the region resulting from the calculation is 1400 M$_{\sun}$.
The total mass of the cores within the region implied by the submm dust
emission data is only $\sim50$ M$_{\sun}$, about $3.6$\% of the total mass in
the region.
\subsection{Column densities, fractional abundances, and the degree of
deuterium fractionation}
The H$_2$ column densities, $N({\rm H_2})$, towards the submm peaks and the
positions selected for the line observations were calculated using the
following equation:
\begin{equation}
\label{eq:N_H2}
N({\rm H_2})=\frac{I_{870}^{\rm dust}}{B_{870}(T_{\rm d})\mu_{\rm H_2} m_{\rm H}\kappa_{870}R_{\rm d}} \, .
\end{equation}
$I_{870}^{\rm dust}$ is the observed dust peak surface brightness, which
is related to the peak flux density via 1 Jy/18\farcs6 beam
$=1.085\cdot10^{-18}$ W m$^{-2}$ Hz$^{-1}$ sr$^{-1}$.
$\mu_{\rm H_2}=2.8$ is the mean molecular weight per H$_2$ molecule, and
$m_{\rm H}$ is the mass of the hydrogen atom.
The same dust temperature values were used as in the mass estimates
(Eq.~(\ref{eq:mass})).
The N$_2$H$^+$ column densities were calculated using the
equation
\begin{equation}
\label{eq:N_tot}
N_{\rm tot}=\frac{3\epsilon_0 h}{2\pi^2 \mu_{\rm el}^2}\frac{1}{S_{\rm ul}}e^{E_{\rm u}/k_{\rm B}T_{\rm ex}}F(T_{\rm ex})Z(T_{\rm ex})\int \tau ({\rm v}) {\rm dv} \, ,
\end{equation}
where $\epsilon_0$ is the vacuum permittivity, $\mu_{\rm el}$ is the permanent
electric dipole moment, $S_{\rm ul}$ is the line strength, $E_{\rm u}$ is the
upper state energy, $Z$ is the rotational partition function, and
$\int \tau {\rm dv}$ is the integrated optical thickness.
We assumed a dipole moment of 3.4 D for both N$_2$H$^+$ and N$_2$D$^+$
(\cite{havenith1990}). For the rotational transition
$J_{\rm u} \rightarrow J_{\rm u}-1$ of a linear molecule (like N$_2$H$^+$),
$S_{\rm ul}=J_{\rm u}$.
For the N$_2$H$^+$ lines the optical thicknesses
were derived from the Gaussian fits to the hyperfine components, and thus
the integral $\int \tau {\rm dv}$ can be replaced by
$\frac{\sqrt{\pi}}{2\sqrt{\ln 2}}\Delta {\rm v}\tau_0$. Here
$\Delta {\rm v}$ is the linewidth of an individual hyperfine component, and
$\tau_0$ is the sum of the peak optical thicknesses of all the seven
components.
The N$_2$D$^+$ column densities were calculated in two different ways: 1) as
in the case of N$_2$H$^+$, and 2) using Eq.~(1) with the approximation
of optically thin line ($\tau \ll 1$):
\begin{equation}
\label{eq:ant_2}
T_{\rm A}^{*}\approx \eta \frac{h\nu}{k_{\rm B}}\left[F(T_{\rm ex})-F(T_{\rm bg})\right]\tau \,.
\end{equation}
Again, we assumed that $\eta=\eta_{\rm MB}$.
The integrated opacity was estimated from the integrated $T_{\rm MB}$ of the
main hyperfine group (54.3\% of the total integrated intensity):
\begin{equation}
\label{eq:tau}
\int \tau {\rm dv}=\frac{\int T_{\rm MB}{\rm dv}}{\frac{h\nu}{k_{\rm B}}\left[F(T_{\rm ex})-F(T_{\rm bg})\right]} \,.
\end{equation}
A comparison of the column density determination
via the two methods shows that $N({\rm N_2D^+})$, when using the first method,
is 1.3 (IRAS 05405-0117), 0.7 (Ori B9 E), and 0.6 (Ori B9 N) times the value
obtained using the second one.
As the N$_2$D$^+$ line areas are somewhat uncertain,
the N$_2$D$^+$ column densities determined by using the first method have been
adopted in this paper.
The fractional N$_2$H$^+$ and N$_2$D$^+$ abundances, $x({\rm N_2H^+})$ and
$x({\rm N_2D^+})$, were calculated by dividing the corresponding column
densities by $N({\rm H_2})$ from the dust continuum. For $x({\rm N_2H^+})$
the dust map was smoothed to 26\farcs4, the resolution of the N$_2$H$^+$
observations. No smoothing was done in the case of N$_2$D$^+$, as the
resolutions of the N$_2$D$^+$ and dust continuum observations are
similar (16\farcs0 and 18\farcs6, respectively).
The degree of deuterium fractionation in N$_2$H$^+$ is defined as the
column density ratio $R_{\rm deut} \equiv N({\rm N_2D^+})/N({\rm N_2H^+})$.
The obtained H$_2$ column densities are given in column (4) of
Table~\ref{table:parameters}. The N$_2$H$^+$ and N$_2$D$^+$
column densities, fractional abundances, and the values of
$R_{\rm deut}$ are listed in Table~\ref{table:d_frac}.
The uncertainties on $N({\rm N_2H^+})$ and
$N({\rm N_2D^+})$ have been calculated by propagating the uncertainties on
$T_{\rm ex}$, $\tau_{\rm tot}$, and $\Delta {\rm v}$, and the uncertainties on
$N({\rm N_2D^+})/N({\rm N_2H^+})$ ratios are propagated from
$N({\rm N_2H^+})$ and $N({\rm N_2D^+})$.
\begin{table*}
\caption{N$_2$H$^+$ and N$_2$D$^+$ column densities, fractional abundances,
and the column density ratio.}
\begin{minipage}{2\columnwidth}
\centering
\renewcommand{\footnoterule}{}
\label{table:d_frac}
\begin{tabular}{c c c c c c}
\hline\hline
& $N({\rm N_2H^+})$ & $N({\rm N_2D^+})$ & $x({\rm N_2H^+})$ & $x({\rm N_2D^+})$ & \\
Position & [$10^{12}$ cm$^{-2}$] & [$10^{11}$ cm$^{-2}$] & [$10^{-10}$] & [$10^{-11}$] & $R_{\rm deut} \equiv N({\rm N_2D^+})/N({\rm N_2H^+})$\\
\hline
IRAS 05405-0117 & $9.14\pm0.08$\footnote{Harju et al. (2006) estimated
slightly lower N$_2$H$^+$ column density, $\sim6-8\cdot10^{12}$ cm$^{-2}$, toward IRAS 05405-0117 from
the N$_2$H$^+(1-0)$ data of Caselli \& Myers (1994).} & $3.19\pm0.37$ & 11.1 & 4.9 & $0.03\pm0.004$\\
Ori B9 E & $4.54\pm0.65$ & $1.90\pm1.39$ & 6.9 & 5.0 & $0.04\pm0.03$\\
Ori B9 N\footnote{For the other velocity component $N({\rm N_2H^+})=1.56\pm0.09\cdot10^{12}$ cm$^{-2}$, $N({\rm N_2D^+})=7.64\pm5.44\cdot10^{10}$ cm$^{-2}$, and $R_{\rm deut}=0.05\pm0.03$.} & $4.11\pm1.51$ & $1.46\pm0.65$ & 3.9 & 1.9 & $0.04\pm0.02$\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\begin{table*}
\caption{Parameters derived in Sect. 4.4.}
\begin{minipage}{2\columnwidth}
\centering
\renewcommand{\footnoterule}{}
\label{table:ion}
\begin{tabular}{c c c c c c c c}
\hline\hline
Source & $x({\rm H_2D^+})$ & $x({\rm H_3^+})$ & $x({\rm e})_l$\footnote{The first value is calculated from Eq.~(\ref{eq:low_x(e)}), whereas the second value is calculated from Eq.~(\ref{eq:lowxe2}).} & $x({\rm e})_u$ & $\langle x({\rm e}) \rangle$\footnote{This is the mean value between the lower and upper limit, where $x({\rm e})_l$ is calculated from Eq.~(\ref{eq:lowxe2}).} & $\zeta_{\rm H_2}$\footnote{The second value is derived by including HCO$^+$ in the analysis (see text).} & $C_i$\\
& [$10^{-9}$] & [$10^{-8}$] & [$10^{-8}$] & [$10^{-7}$] & [$10^{-7}$] & [$10^{-16}$ s$^{-1}$] & [$10^3$ cm$^{-3/2}$ s$^{1/2}$]\\
\hline
Ori B9 E & 2.0 & 1.8 & 2.0/3.0 & 6.4 & 3.4 & 2.0/1.0 & 2.6\\
Ori B9 N & 1.0 & 0.9 & 1.0/2.0 & 6.3 & 3.3 & 2.5/1.3 & 4.1\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\subsection{Ionization degree and cosmic ray ionization rate}
The charge quasi-neutrality of plasma dictates that the number of
positive and negative charges are equal. Since electrons are the
dominant negative species, their fractional abundance nearly equals
the sum of the abundances of positive ions, $x({\rm cations})\simeq
x({\rm e})$. Thus, one may obtain a \textit{lower limit} for the
ionization fraction by summing the abundances of several molecular
ions:
\begin{equation}
\label{eq:low_x(e)}
x(\mathrm{e}) >
x(\mathrm{N_2H^+})+x(\mathrm{N_2D^+})+x(\mathrm{H_3^+})+x(\mathrm{H_2D^+}) \; .
\end{equation}
In the following we attemp to derive estimates for the cosmic ray
ionization rate and the fractional electron abundance using the
abundances of $\mathrm{N_2H^+}$, $\mathrm{N_2D^+}$, and
$\mathrm{H_2D^+}$ together with the reaction schemes and formulae
presented in Crapsi et al. (2004) and Caselli et al. (2008). The rate
coefficients for the ${\rm H_3^+ + H_2}$ isotopic system have been
newly calculated by Hugo et al. (2009). We use these for the
deuteration sequence ${\rm H_3^+}$ $\leftrightarrow$ ${\rm H_2D^+}$
$\leftrightarrow$ ${\rm D_2H^+}$ $\leftrightarrow$ ${\rm D_3^+}$ (see
their Table VIII) . For other reactions the rate coefficients have
been adopted from the UMIST database which is available at {\tt www.udfa.net}.
The main difference between the Hugo et al. coefficients and those of
Roberts et al. (2004) is that in the former, the effective backward
rate coefficient, $k_{-1}$, of the reaction ${\rm H_3^+}+{\rm HD}
\mathop{\rightleftharpoons}\limits^{k_1}_{k_{-1}}{\rm H_2D^+}+{\rm H_2}$, and the corresponding
coefficients for multiply deuterated forms of H$_3^+$ are higher if
the non-thermal ortho/para ratio of H$_2$ (hereafter o/p-H$_2$) is
taken into account (\cite{pagani1992}; \cite{gerlich2002}; \cite{flower2006a};
\cite{pagani2009a}; \cite{hugo2009}).
The ortho-H$_2$D$^+$ column density was derived towards Ori B9 E and N
by Harju et al. (2006). Using their value, $N({\rm o-H_2D^+}) \sim
3.0\cdot10^{12}$ cm$^{-2}$, and the H$_2$ column densities derived
here, we get $x({\rm o-H_2D^+})\approx8.0\cdot10^{-10}$ and
$4.1\cdot10^{-10}$ towards Ori B9 E and N,
respectively\footnote{Caselli et al. (2008) derived $N({\rm
o-H_2D^+})= 2.0/9.0\cdot10^{12}$ cm$^{-2}$ toward position which is
only 12\farcs7 southeast of our line observations position Ori B9 N,
assuming a critical density $n_{\rm cr}=10^5$ and $10^6$ cm$^{-3}$,
respectively.}. The ortho/para ratio of H$_2$D$^+$ (hereafter
o/p-${\rm H_2D^+}$) depends heavily on o/p-H$_2$. According to the
model of Walmsley et al. (2004, see their Fig.~3), the characteristic
steady-state value of o/p-H$_2$ is $\sim 10^{-4}$ in the density range
$n({\rm H_2}) \sim 10^5 - 10^6$ cm$^{-3}$ appropriate for the objects
of this study. This model deals with the situation of ``complete
depletion'' and it is not clear how valid the quoted o/p-H$_2$ is in
less depleted gas. The recent results of Pagani et al. (2009a) suggest
high values of o/p-H$_2$ ($\sim 4\cdot10^{-3} - 6\cdot10^{-2}$) in
L183.
For the moment we adopt the value o/p-${\rm H_2}=10^{-4}$. The
effect of increasing this ratio will be examined briefly
below. Assuming that o/p-${\rm H_2D^+}$ is mainly determined by
nuclear spin changing collisions with ortho- and para-H$_2$, the
quoted o/p-${\rm H_2}$ ratio implies an o/p-${\rm H_2D^+}$ of $\sim
0.7$ at $T=10$ K. The total (ortho$+$para) H$_2$D$^+$ abundances
corresponding to this o/p ratio are $x({\rm
H_2D^+})\approx2.0\cdot10^{-9}$ and $1.0\cdot10^{-9}$ towards Ori B9 E
and N, respectively.
The N$_2$D$^+$/N$_2$H$^+$ column density ratio which we denote by
$R_{\rm deut}$, gives a rough estimate for the H$_2$D$^+$/H$_3^+$
abundance ratio, denoted here by $r$. According to the relation
$R_{\rm deut}\approx(r+2r^2)/(3+2r+r^2)$ derived by Crapsi et al. (2004; a
more accurate formula is given in Eq.~(13) of Caselli et al. (2008)),
the values of $R_{\rm deut}$ given in Table~\ref{table:d_frac} imply
$r\approx0.08$ for IRAS 05405-0117, and $r\approx0.11$ for Ori B9 E and
N. Using these $r$ values we obtain the following fractional
abundances $x({\rm H_3^+})=x({\rm H_2D^+})/r$ $\approx1.8\cdot10^{-8}$
(Ori B9 E) and $\approx9.1\cdot10^{-9}$ (Ori B9 N). Substituting all
the derived abundances into Eq.~(\ref{eq:low_x(e)}) we get the
following lower limits for the degree of ionization: $x({\rm
e})>2.0\cdot10^{-8}$ in Ori B9 E, and $>1.0\cdot10^{-8}$ in Ori B9 N.
Despite the fact that the clump associated with IRAS 05405-0117 does
not stand out in the $^{13}$CO and C$^{18}$O maps, a moderate CO
depletion factor of 3.6 near Ori B9 N has been derived (\cite{caselli1995};
\cite{caselli2008}). This estimate is based on
$^{13}{\rm CO}(1-0)$ observations made with the FCRAO 14-m telescope (HPBW
$50\arcsec$), and a total H$_2$ column density, $N({\rm H_2})$,
derived from ammonia. By smoothing the LABOCA map to the resolution of
$50\arcsec$, we obtain an average H$_2$ column density of
$1.3\cdot 10^{22}$ cm$^{-2}$ around Ori B9 N. This is 2.6 times lower
than the value adopted by Caselli et al. (2008).
With this $N({\rm H_2})$ the fractional CO abundance, $x({\rm CO})$,
becomes $6.8\cdot10^{-5}$. The corresponding CO depletion factor, $f_{\rm D}$,
is only 1.4 with respect to the often adopted fractional abundance from
Frerking et al. (1982)\footnote{The depletion factor is used in the
text to express the fractional CO abundance with respect to the value
$9.5\cdot10^{-5}$. Adopting a higher reference abundance (see \cite{lacy1994})
would not change the results of the calculations.}.
The small value of $R_{\rm deut}$ is consistent with a low CO depletion
factor (e.g., \cite{crapsi2004}).
In chemical equilibrium the fractional ${\mathrm H_3^+}$ abundance is
\begin{equation}
x(\mathrm{H_3^+})=\frac{\zeta_{\mathrm{H_2}}/n(\mathrm{H_2})+
k_{-1} x(\mathrm{H_2D^+})}{D_0} \; ,
\label{eq:h3+}
\end{equation}
where $\zeta_{\rm H_2}$ is the cosmic ray ionization rate of H$_2$,
and
\begin{displaymath}
D_0 \equiv k_1 x(\mathrm{HD}) + k_{\rm CO} x({\rm CO}) +k_{\rm rec0} x({\rm e}) + k_{\rm g} x({\rm g^-}) +k_{\rm N_2} x({\rm N_2}) + ...
\end{displaymath}
The notation of Caselli et al. (2008) has been used here, i.e., $k_1$
and $k_{-1}$ are the forward and backward rate coefficients of the
reaction mentioned above, and the other terms in $D_0$ refer to the
destruction of H$_3^+$ in reactions with neutral molecules (e.g., CO
and N$_2$) and in recombination with electrons and on negatively
charged dust grains.
By solving numerically Eqs.~(8)-(10), and (13) of Caselli et
al. (2008) together with our Eq.~(\ref{eq:h3+}) we obtain the following
estimates for the fractional electron abundance and cosmic ray ionizations
rate: $x({\rm e}) = 6.4\cdot 10^{-7}$, $\zeta_{\rm H_2} = 2.0\cdot10^{-16}$
s$^{-1}$ in Ori B9 E, and $x({\rm e}) = 6.3 \cdot 10^{-7}$, $\zeta_{\rm H_2} =
2.5\cdot10^{-16}$ s$^{-1}$ in Ori B9 N. Here we have used CO
depletion factor 1.4, and the dust parameters ($k_{\rm g^-}$, $x({\rm
g^-}$) quoted in Eqs.~(11) and (12) of Caselli et al. which are based
on a MRN dust grain size distribution (\cite{mathis1977}) and
effective grain recombination coefficients derived by Draine \& Sutin
(1987). The average number densities, $\langle n({\rm H_2}) \rangle$,
derived in Sect. 4.2. have been used for the cosmic ray ionization rates.
The obtained values of $\zeta_{\rm H_2}$ are very similar to each other as
is expected for such a nearby cores (\cite{williams1998};
\cite{bergin1999} and references therein).
For comparison, Bergin et al. (1999) found that adopting $\zeta_{\rm
H_2}=5\cdot10^{-17}$ s$^{-1}$ in their chemical model best reproduced
their observations of massive cores in Orion. Note that the
``standard'' value often quoted in the literature is
$\zeta_{\rm H_2}=1.3\cdot10^{-17}$ s$^{-1}$.
Also the fractional electron abundances are clearly larger than those
calculated from the standard relation $x({\rm
e})\sim1.3\cdot10^{-5}n({\rm H_2})^{-1/2}$ (cf. \cite{mckee1989};
\cite{mckee1993}), where the electron fraction is due to cosmic ray ionization
only and $\zeta_{\rm H_2}$ has its above mentioned standard value.
The corresponding values would be $x({\rm e})\sim 5\cdot10^{-8}$
(Ori B9 E) and $x({\rm e}) \sim 3\cdot10^{-8}$ (Ori B9 N).
The mean value of the ionization degree found by Bergin et
al. (1999) for the massive cores in Orion is $\sim 8 \cdot 10^{-8}$.
The parameters derived above depend on the adopted o/p-H$_2$ which
affects the backward rate coefficient $k_{-1}$, $k_{-2}$, and $k_{-3}$
(see \cite{caselli2008}), and the CO depletion factor $f_{\rm D}$ which
affects the destruction of H$_3^+$. The fractional electron
abundance can be decreased to $\sim 8\cdot10^{-8}$ by increasing
o/p-H$_2$ to $2.4\cdot10^{-3}$ (this yields a $\zeta_{\rm H_2}$ of
$1.3\cdot10^{-16}$ s$^{-1}$). On the other hand, an increase in
$f_{\rm D}$ will lead to a higher $x({\rm e})$, but also to a lower
$\zeta_{\rm H_2}$. A solution where both $x({\rm e})$ and
$\zeta_{\rm H_2}$ obtain the average values derived by Bergin et al.
(1999) can be found by setting $f_{\rm D}$ to 4.4 and o/p-H$_2$ to
$3.4\cdot10^{-3}$.
However, the available observational data do not give grounds for
abandoning the present bona fide $f_{\rm D}$ value 1.4. So the
main uncertainty seems to be related to the unknown o/p-H$_2$.
Additional uncertainties to the $\zeta_{\rm H_2}$ values are caused by the
rough density estimates, and by the fact that densities in the positions
observed in molecular lines are probably lower than the average
densities adopted in the analysis. The electron abundance obtained assuming an
o/p-${\rm H_2}$ of $1.0\cdot10^{-4}$ is likely to be an upper limit. This o/p
ratio corresponds to steady state in highly depleted dense gas with
large abundances of H$^+$ and H$_3^+$ capable of efficient proton
exchange with H$_2$ (\cite{flower2007}). Their replacement by other
ions in less extreme situations can sustain higher o/p-H$_2$ ratios.
A substantial amount of CO implies the presence of HCO$^+$ in the gas.
By including the dissociative electron recombination of HCO$^+$, and
the proton exchange reaction between N$_2$H$^+$ (or N$_2$D$^+$) and CO
in the reaction scheme, the fractional HCO$^+$ abundance can be
solved. Through this estimate we get a slightly more stringent lower
limit on the electron abundance than that imposed by
Eq.~(\ref{eq:low_x(e)}) by demanding that
\begin{align}
x({\rm e}) & \geq x({\rm H_3^+})(1+r) + x({\rm N_2H^+})(1+R_{\rm deut}) \nonumber \\
&\quad+ x({\rm HCO^+})(1+R_{\rm deut}) \; .
\label{eq:lowxe2}
\end{align}
Here it has been assumed that N$_2$H$^+$ and HCO$^+$ have
similar degrees of deuterium fractionation. By varying o/p-H$_2$
until electron and the ``known'' cations are in balance we obtain with
o/p-${\rm H_2} = 2.7\,10^{-3}$ the lower limits $x({\rm e}) \geq
3\cdot10^{-8}$ and $x({\rm e}) \geq 2\cdot10^{-8}$ in Ori B9 E and N,
respectively. The corresponding values of $\zeta_{\rm H_2}$ are
$1.0\cdot10^{-16}$ s$^{-1}$ and $1.3\cdot10^{-16}$ s$^{-1}$. In both
solutions HCO$^+$ is more abundant than H$_3^+$, whereas in the case
of a large $x({\rm e})$ (small o/p-${\rm H_2}$) the HCO$^+$ abundance
lies between those of N$_2$H$^+$ and H$_2$D$^+$.
The obtained values of the cosmic ray ionization rate vary smoothly
with o/p-${\rm H_2}$, and all viable solutions point towards $\zeta_{\rm
H_2} \sim 1-2\cdot10^{-16}$ s$^{-1}$. In the model of McKee (1989) these
levels imply factional ionizations of $\sim 1.1-1.6 \cdot10^{-7}$ at the
density $10^5$ cm$^{-3}$. These values lie between the lower and upper
limits derived above. In what follows we assume that
$x({\rm e}) \sim 1\cdot10^{-7}$, keeping in mind that true electron abundance
is likely to be found within a factor of few from this value.
There is also uncertainty about the most abundant ion. According to
our calculation the electron abundance is an order of magnitude higher than
the summed abundances of the positive ions
H$_3^+$, HCO$^+$, and N$_2$H$^+$ for ${\rm o/p-H}_2 =
1.0\cdot10^{-4}$, whereas for the higher o/p-H$_2$ ratio $x({\rm
HCO^+})$ is comparable to $x({\rm e})$. This suggests that in the
first case the reaction scheme misses the most abundant cation(s). In
depleted regions with densities below $10^6$ cm$^{-3}$ protons, H$^+$,
are likely to be the dominant ions (\cite{walmsley2004}; \cite{pagani2009a}).
On the other hand, as discussed in Crapsi et al. (2004)
and references therein, if atomic oxygen is abundant in the gas phase
the major ion may be H$_3$O$^+$. To our knowledge this ion has not yet been
found in cold clouds (see also Caselli et al. 2008).
When discussing the ambipolar diffusion timescale in Sect. 5.6 we will assume
that the most abundant ion is either H$^+$ or HCO$^+$.
We furthermore estimate the value of a constant, $C_i$, that
describes the relative contributions of molecular ions and metal ions
to the ionization balance (\cite{williams1998}, their Eq.~(4);
\cite{bergin1999};
\cite{padoan2004}). The value of $C_i$ can be used to
estimate the strength of the ion-neutral coupling in terms of the wave
coupling parameter, $W \propto C_i$ (see Sect. 5.7).
In this analysis it is assumed that the electron abundance is
determined by cosmic ray ionization balanced by recombination and
it is appropriate for cores where $A_{\rm V}>4$ mag (i.e., ionization due to
cosmic rays dominates that resulting from UV radiation).
Adopting the electron abundance $1\cdot10^{-7}$ and
$\zeta_{\rm H_2}\sim10^{-16}$ s$^{-1}$ we find values of
$C_i\simeq2.6-4.1\cdot10^3$ cm$^{-3/2}$ s$^{1/2}$ in our cores.
These are similar to the value found by Bergin et al. (1999)
for the massive cores in Orion ($3.6\cdot10^3$ cm$^{-3/2}$ s$^{1/2}$).
McKee (1989) derives $C_i=3.2\cdot10^3$
cm$^{-3/2}$ s$^{1/2}$ for an idealised model of cosmic ray ionization and
Williams et al. (1998) obtained $C_i=2.0\cdot10^3$ cm$^{-3/2}$ s$^{1/2}$
for low-mass cores.
All the parameters derived in this Section are summarised in
Table~\ref{table:ion}.
\section{Discussion}
\subsection{Nature of submm sources in Ori B9}
By combining the submm LABOCA and far-infrared Spitzer data, we can distinguish
starless cores from protostellar cores. In addition to the four IRAS sources
in the region, two of the new submm sources, namely SMM
3 and SMM 4, are clearly associated with Spitzer point sources and are
protostellar. The remaining six submm cores are starless.
IRAS 05399-0121 was previously classified as a Class I protostar
(\cite{bally2002} and references therein). However, taking into account the
rather low bolometric (18.5 K) and kinetic temperatures
(13.7 K, \cite{HWW1993}), and high values of $L_{\rm submm}/L_{\rm bol}$
(2\%) and $M_{\rm tot}/L_{\rm bol}^{0.6}$ (0.45 M$_{\sun}$/L$_{\sun}^{0.6}$),
we suggest the source is in a transition phase from Class 0 to Class I
(see \cite{bontemps1996}; \cite{froebrich2005}).
This source is associated with the highly collimated jet HH 92
(\cite{bally2002}).
The SED of IRAS 05413-0104 derived here is consistent with its previous
classification as a Class 0 object (e.g., \cite{cabrit2007} and references
therein). The source is associated with the highly symmetric jet HH 212
(\cite{lee2006}, 2007; \cite{codella2007}; \cite{smith2007};
\cite{cabrit2007}). IRAS 05412-0105 and IRAS 05405-0117 have very similar SEDs,
and they, too, are likely to represent the Class 0 stage. The weak line wings
in the N$_2$H$^+(1-0)$ hyperfine lines of IRAS 05405-0117
(see Fig.~\ref{figure:n2h+}, top) could indicate the presence of outflow from
an embedded protostellar object.
$L_{\rm submm}/L_{\rm bol}$ ratio for both SMM 3 and SMM 4 is 11\%.
This together with low values of $T_{\rm bol}$ make these new submm sources
Class 0 candidates (e.g., \cite{froebrich2005} and references therein) that
are deeply embedded in a massive, cold envelope.
On a bolometric luminosity vs. temperature diagram these objects lie on the
evolutionary track for a Class 0 source with initially massive envelope
(see Fig.~12 in Myers et al. (1998)).
The starless cores SMM 1, 2, 5, 6, 7, and Ori B9 N, are
likely to be prestellar as their densities are relatively high
($0.6-5.8\cdot10^5$ cm$^{-3}$; see also Sect. 4.2).
The 24 $\mu$m Spitzer source near SMM 5 is probably not associated
with this core. It lies rather far form the core centre and
it is not detected at 70 $\mu$m (Fig.~\ref{figure:spitzer}).
There is an equal number of prestellar and protostellar cores
in Ori B9. This situation is similar to that recently found by
Enoch et al. (2008) in Perseus, Serpens, and Ophiuchus,
and suggests that the lifetimes of prestellar and protostellar
cores are comparable. Evolutionary timescales will be further discussed
in Sect. 5.6.
\subsection{Mass distribution and core separations}
The spatial and mass distribution of cores are both important parameters
concerning the cloud fragmentation mechanism.
Our core sample is, however, so small that it is not reasonable to study
the properties of these distributions directly. Therefore, we only compared
them with the distributions derived for another, larger core sample in Orion
GMC by NW07. We make this comparison particularly with Orion B North because
the SCUBA 850 $\mu$m map of Orion B North (see Fig.~2c in NW07) looks
qualitatively similar to Ori B9.
Orion B North also has deeper sensitivity and completeness limit than other
regions studied by NW07, and besides it contains large number of cores.
Fig.~\ref{figure:CMF} presents the observed cumulative mass functions, which
counts cores with mass less than $M$, i.e.,
$\mathcal{N}(M)=N(m<M)/N_{\rm tot}$, for both the core masses in Ori B9 and
masses of prestellar cores in Orion B North derived by NW07.
Note that the core mass function (CMF) studied by NW07 is constructed by
removing the Class I protostars from the sample, so that CMF includes only
cores which have all their mass initially available for star formation left.
Correspondingly, we have excluded IRAS 05399-0121 from our sample.
We have also multiplied the core masses by required factors to compare to NW07
values, due to differences in assumed values of $T_{\rm dust}$,
$\kappa_{\lambda}$, and distance (NW07 used the following values:
$T_{\rm dust}=20$ K, $\kappa_{850}=0.1$ m$^2$ kg$^{-1}$, and $d=400$ pc).
\begin{figure}[!h]
\resizebox{\hsize}{!}{\includegraphics{CMF.eps}}
\caption{Normalised cumulative mass functions, $\mathcal{N}(M)$,
for the prestellar cores in Ori B9 (solid line) and in Orion B North
(dashed line) studied by Nutter \& Ward-Thompson (2007).}
\label{figure:CMF}
\end{figure}
In order to determine if the two datasets are samples of the same core mass
distribution, we carried out the Kolmogorov-Smirnov (K-S) test.
The K-S test yields the maximum vertical difference between the cumulative
distributions of $D=0.166$, and probability of approximately 95\% that the
core mass distributions in Ori B9 and Orion B North are drawn from the same
parent distribution.
Fig.~\ref{figure:dist} (top) shows the observed core
separation distribution and the distribution expected for the same number of
randomly positioned cores over an identical area (0.22 deg$^2$).
The mean and median of the core separations in Ori B9
are $\log(r/{\rm AU})=5.47\pm0.09$ ($2.9\pm0.6\cdot10^5$ AU) and 5.42
($2.6\cdot10^5$ AU), respectively. The quoted error for the mean correspond
to the standard deviation.
These values are similar to those of randomly positioned cores,
for which the mean and median are $\log(r/{\rm AU})=5.59\pm0.05$ and
$5.59\pm0.06$, respectively. The quoted uncertainties are the standard
deviation of the sampling functions.
For core separation distribution in Orion B North studied by
NW07, the corresponding values are $5.67\pm0.03$ and 5.63, suggesting that the
fragmentation scale is similar in both Ori B9 and Orion B North.
Similar fragmentation scales and the fact that CMFs have resemblance to the
stellar IMF (\cite{goodwin2008}) suggest that the origin of cores in these two
regions is probably determined by turbulent fragmentation
(e.g., \cite{maclow2004}; \cite{ballesteros-paredes2007}).
The clustered mode of star formation in these two regions suggests that
turbulence is driven on large scales (e.g., \cite{klessen2001}).
Recently, Enoch et al. (2007) found the median separations
of $\log(r/{\rm AU})=3.79$, 4.41, and 4.36 in nearby molecular clouds
Ophiuchus, Perseus, and Serpens, respectively.
The spatial resolution of the Bolocam (31\arcsec) used by Enoch et al. at the
distance of Ophiuchus, Perseus, and Serpens, is 0.02, 0.04, and 0.04 pc.
The latter two are similar to our
resolution. The results suggest that the fragmentation scales in Perseus and
Serpens are different from that in Orion.
Fig.~\ref{figure:dist} (bottom) shows the comparison between the observed
nearest neighbour distribution and the distribution for randomly positioned
cores. The mean and median of the nearest neighbour distribution in Ori B9 are
$\log(r/{\rm AU})=4.75\pm0.09$ ($5.6\pm1.3\cdot10^4$ AU) and 4.62
($4.2\cdot10^4$ AU), respectively. These values are rather different
from those expected from random distributions, for which the mean and median
are $\log(r/{\rm AU})=5.08\pm0.08$ and $5.07\pm0.11$, respectively.
For core positions in Orion B North the mean and median are
$\log(r/{\rm AU})=4.46\pm0.03$ and 4.35, respectively (NW07).
Also this comparison supports the idea that the scale of fragmentation,
and the amount of clustering are similar in Ori B9 and Orion B North.
Note that the minimum observable separation is the beam size, i.e $18\farcs6$
or $\sim8.3\cdot10^3$ AU at 450 pc. We also note that the source sample is too
small to measure the significance of the clustering in Ori B9
based on the two-point correlation function.
\begin{figure}[!h]
\resizebox{\hsize}{!}{\includegraphics{separations.ps}}
\resizebox{\hsize}{!}{\includegraphics{nearest.ps}}
\caption{{\bf Top:} Observed core separation distribution (solid line) compared
with the expected distribution for random distribution of the same
number of sources as the observed sample over an identical area (dashed line).
{\bf Bottom:} Observed nearest neighbour distribution (solid line)
compared with the expected distribution for random distribution of the same
number of sources as the observed sample over the same area (dashed line).}
\label{figure:dist}
\end{figure}
\subsection{Sizes, shapes, and density structures of the cores}
Starless cores in Ori B9, for which the mean value of the deconvolved angular
size in units of the beam FWHM is
$\langle \theta_{\rm s}/\theta_{\rm beam} \rangle=2.5\pm0.8$, are larger on
average than protostellar cores
($\langle \theta_{\rm s}/\theta_{\rm beam} \rangle=1.6\pm0.2$).
These sizes are similar to those recently found by Enoch et al. (2008)
in Perseus ($\langle \theta_{\rm s}/\theta_{\rm beam} \rangle=2.2$ and 1.6 for
starless and protostellar cores, respectively).
The mean axis ratios at half-maximum contours of starless and protostellar
cores are also 2.5 and 1.6, respectively (see Table~\ref{table:cores}, column
(7)). This indicates that starless cores in Ori B9 are also more
elongated on average than protostellar cores (cf. \cite{offner2009}).
The values of $\theta_{\rm s}/\theta_{\rm beam}$ can be used to infer the
steepness of the core radial density profile (\cite{young2003};
\cite{enoch2008}). According to the correlation between
$\theta_{\rm s}/\theta_{\rm beam}$ and density-power-law index, $p$, found
by Young et al. (2003, see their Fig.~27) a mean
$\theta_{\rm s}/\theta_{\rm beam}$ values of 2.5 for starless and 1.6 for
protostellar cores imply an average index of $p\sim0.9-1.0$ and
$\sim1.4-1.5$, respectively.
Moreover, Fig.~25 of Young et al. (2003) suggest power-law indices $<1$ for
starless cores and $1.1-1.6$ for protostellar cores, consistent with those
inferred by the average deconvolved angular source sizes.
The low values of $p\sim1.0$ for starless cores suggest that they are
best modelled with shallower density profiles than the protostellar cores.
These results are in agreement with those found by Ward-Thompson et al.
(1999) using 1.3 mm dust continuum data, and Caselli et al. (2002b) using
N$_2$H$^+(1-0)$ maps.
\subsection{Deuterium fractionation and depletion in
the IRAS 05405-0117 region}
The N$_2$D$^+$/N$_2$H$^+$ column density ratio, $R_{\rm deut}$, is supposed to
increase strongly as the core evolves (\cite{caselli2002}; \cite{crapsi2005a},
their Fig. 5; \cite{fontani2006}; \cite{emprechtinger2009};
but see \cite{roberts2007}).
This can be understood so that the abundances of H$_3^+$, and its deuterated
forms which transfer deuterium to other molecules, increase with increasing
density due to molecular depletion and a lower degree of ionization.
Crapsi et al. (2005a) suggested that prestellar cores are characterised with
$R_{\rm deut}>0.1$, whereas starless cores with $R_{\rm deut}<0.1$ are not
necessarily so dense that CO would be heavily depleted (\cite{roberts2007}).
It should be noted that in cores without internal heating sources
the degree of deuterium fractionation is likely to increase inwards
as the density increases and temperature decreases due to attenuation
of starlight. This temporal and radial tendency is likely to
be reversed during the core collapse because of compressional heating
and the formation of a protostar (e.g., \cite{aikawa2008a}; see also Fig.~3
in Emprechtinger et al. (2009)).
The positions studied here have $R_{\rm deut}\sim0.03-0.04$.
This is $\sim2-3\cdot10^3$ times larger than the cosmic
D/H elemental abundance of $\sim1.5\cdot10^{-5}$ (\cite{linsky1995}; 2006;
\cite{oliveira2003}). Also, the H$_2$D$^+$/H$_3^+$ ratios we derived are
$\sim7\cdot10^3$ times larger than the cosmic D/H ratio.
Our $R_{\rm deut}$ values are similar to those found by Crapsi et al. (2005a)
toward several low-mass starless cores, and to those found by Emprechtinger et
al. (2009) toward Class 0 sources.
Like Emprechtinger et al. (2009), we find that the deuterium fractionation of
N$_2$H$^+$ in protostellar cores, which takes place in the cold extended
envelope, is similar to that in prestellar cores.
It has been found that the values of $R_{\rm deut}$ toward high-mass
star-forming cores are (usually) lower than those found in the present
study (\cite{fontani2006}; see also \cite{emprechtinger2009}).
This conforms with the fact that our sources are low- to intermediate
mass star-forming cores.
As discussed by Walmsley et al. (2004) and Flower et al. (2006a),
the H$_2$D$^+$ abundance depends (inversely) on the
ortho:para ratio of H$_2$, because the reaction
${\rm H_2D^+}+{\rm H_2}\overset{k_{-1}}{\rightarrow}{\rm H_3^+}+{\rm HD}$
is rapid between ortho forms. The ortho:para ratio of H$_2$ decreases
with time and gas density, and is therefore large at early stages of core
evolution. Consequently, a relatively high degree of deuterium fractionation
is a sign of matured chemistry characterised by a low ortho:para ratio of
H$_2$ and probably a high degree of molecular depletion.
Low CO depletion factor of 1.4 close to N$_2$H$^+$
peak Ori B9 N (see Sect. 4.4) is consistent with the $R_{\rm deut}$ value of
Ori B9 N (e.g., \cite{crapsi2004}; see also Fig.~4 in Emprechtinger et al.
(2009)).
The ortho-H$_2$D$^+$ detection towards Ori B9 E and N suggests an evolved
chemical stage and tells of a longlasting prestellar phase.
The non-detection toward IRAS 05405-0117 can be explained by a
lower ortho-H$_2$D$^+$ abundance due to the central heating by the protostar.
\subsection{Evidence for a N$_2$H$^+$ ``hole'' and chemical differentiation}
The N$_2$H$^+$ map of Caselli \& Myers (1994, see their Fig.~2) and submm dust
continuum map of the clump associated with IRAS 05405-0117
(Fig.~\ref{figure:positions}) are not very much alike.
The strongest dust continuum peak, SMM 4, does not stand out in N$_2$H$^+$.
Moreover, the northern N$_2$H$^+$ maximum, Ori B9 N, seem to be shifted with
respect to the northern dust peak, and the N$_2$H$^+$ peak Ori B9 E does not
correspond to any dust emission peak.
To determine whether the $\sim1.5$ times higher resolution of the LABOCA 870
$\mu$m map relative to the N$_2$H$^+$ map of Caselli \& Myers (1994)
contributed to the different appearance of the dust continuum and N$_2$H$^+$
maxima, we smoothed the LABOCA map to a resolution similar to that of the
N$_2$H$^+$ map (27\arcsec). The smoothed 870 $\mu$m map, however, still shows
the same differences between the dust continuum and N$_2$H$^+$.
The Class 0 candidate SMM 4 (see Sect. 4.1 and 5.1) can represent an extreme
case of depletion where also N$_2$H$^+$ has disappeared from the gas phase due
to freeze out on to the dust grain surfaces. There is some previous evidence
for N$_2$H$^+$ depletion in the centres of chemically evolved cores,
such as B68 (\cite{bergin2002}), L1544 (\cite{caselli2002a}), L1512
(\cite{lee2003}), and L1521F (\cite{crapsi2004}).
Example of the N$_2$H$^+$ depletion toward
Class 0 source is IRAM 04191+1522 in Taurus (\cite{belloche2004}).
Pagani et al. (2005) found clear signs of moderate N$_2$H$^+$ depletion in the
prestellar core L183 (see also \cite{pagani2007}).
Also, Schnee et al. (2007) found clear evidence
of N$_2$H$^+$ depletion toward the dust centre of TMC-1C.
Note that SMM 4 is probably not warm enough for CO to evaporate from the grain
mantles ($\sim20$ K, e.g., \cite{aikawa2008a}), so it is unlikely that CO,
which is the main destroyer of N$_2$H$^+$ (through reaction
${\rm N_2H^+} + {\rm CO}\rightarrow {\rm HCO^+} + {\rm N_2}$),
would have led to disappearance of N$_2$H$^+$ from the gas phase.
To study the chemical differentiation within the clump, we compare
our previously determined NH$_3$ column densities with the present
N$_2$H$^+$ column densities. The integrated NH$_3$ $(1,1)$ and
$(2,2)$ intensity maps of the clump (see Appendix A in Harju et al. (1993))
show roughly the same morphology as the submm map. The NH$_3$ column densities
toward IRAS 05405-0117, Ori B9 E, and Ori B9 N are
$11.9\pm1.3\cdot10^{14}$, $7.2\pm1.3\cdot10^{14}$, and $9.7\pm4.6\cdot10^{14}$
cm$^{-2}$, respectively.
The corresponding NH$_3$/N$_2$H$^+$ column density ratios
are about $130\pm14$, $159\pm37$, and $236\pm87$.
These values suggest that NH$_3$/N$_2$H$^+$ abundance ratio is higher towards
starless condensations than towards the IRAS source. Hotzel et al. (2004) found
a similar tendency in B217 and L1262: the NH$_3$/N$_2$H$^+$ abundance ratios
are at least twice as large in the dense starless parts of the cores
than in the regions closer to the YSO (see \cite{caselli2002b} for other
low-mass star-forming regions).
The same trend is also found in the high-mass star-forming region
IRAS 20293+3952 (\cite{palau2007}).
This is in accordance with chemistry models (\cite{aikawa2005}) and previous
observations (\cite{tafalla2004}) which suggest that NH$_3$ develops slightly
later than N$_2$H$^+$, and can resist depletion up to higher densities.
It should be noted that models by Aikawa et al. (2005) reproduce the
observed enhancement of the NH$_3$/N$_2$H$^+$ ratio by adopting
the branching ratio for the dissociative recombination of N$_2$H$^+$ as
measured by Geppert et al. (2004; i.e.,
${\rm N_2H}^++{\rm e}\rightarrow{\rm NH}+{\rm N}$ accounts for 64\% of the
total reaction). However, this branching ratio has been
retreated by the same authors\footnote{Their recent laboratory experiment
suggest that the above mentioned branching ratio is only 10\% (see
\cite{aikawa2008b}).}, and thus it is not clear at the moment what
is actually causing the increase of NH$_3$/N$_2$H$^+$ abundance ratio.
\subsection{Core evolution: quasi-static vs. dynamic}
The degree of ionization in dense cores determines the importance of magnetic
fields in the core dynamics.
The ionization fractions in low-mass cores are found to be
$10^{-8} < x({\rm e}) < 10^{-6}$ (\cite{caselli1998}; \cite{williams1998}).
The physical origin of the large variations in $x({\rm e})$ is not well
understood, though variations in $\zeta_{\rm H_2}$ or appropriate values
of metal depletion are assumed (\cite{padoan2004}). Padoan et al. (2004)
suggested that the observed variations in $x({\rm e})$ can be understood as the
combined effect of variations in core age, extinction, and density.
Fractional ionizations can be transformed into estimates of the
ambipolar diffusion (AD) timescale, $\tau_{\rm AD}$. We have used Eq.~(5)
of Walmsley et al. (2004) for this purpose. Assuming that H$^+$ is the
dominant ion (see Sect. 4.4), one obtains
$\tau_{\mathrm{AD}}\sim8\cdot10^{13}x({\rm e}) \ \mathrm{yr}$.
If the dominant ion is HCO$^+$,
$\tau_{\mathrm{AD}}\sim1.3\cdot10^{14}x({\rm e}) \ \mathrm{yr}$, i.e.,
$\sim60\%$ longer than in the former case.
Using the electron abundance $x({\rm e})=1\cdot10^{-7}$ we obtain that
$\tau_{\rm AD}\sim10^7$ yr.
This timescale is roughly 70 and 100 times longer than the free-fall time
($\tau_{\rm ff}\sim3.7\cdot10^7\left(n({\rm H_2}) [{\rm cm^{-3}}]\right)^{-1/2}$ yr) of Ori B9 E and N, respectively.
Since $\tau_{\rm AD}>\tau_{\rm ff}$, the cores may be supported against
gravitational collapse by magnetic fields and ion-neutral coupling.
The magnetic field that is needed to
support the cores can be estimated using the relation between the critical
mass required for collapse and the magnetic flux
(see Eq.~(2) in Mouschovias \& Spitzer (1976)). Using the masses and radii from
Table~\ref{table:parameters}, we obtain a critical magnetic field strength
of $\sim80$ $\mu$G for Ori B9 E/IRAS 05405-0117 and $\sim200$ $\mu$G for
Ori B9 N. These are rather high values compared to those that have been
observed (\cite{troland1996}; \cite{crutcher1999}; \cite{crutcher2000};
\cite{crutcher2004}; \cite{turner2006}; \cite{troland2008}).
According to the ``standard'' model of low-mass star formation,
$\tau_{\rm AD}/\tau_{\rm ff}\sim10$ (see, e.g., \cite{shu1987};
\cite{ciolek2001} and references therein).
Since AD is generally a slow process, the core evolution
toward star formation occur quasi-staticly.
The chemical abundances found in the present study
($x({\rm N_2H^+})\sim10^{-10}$, $x({\rm NH_3})\sim10^{-7}$, see Sect. 5.8)
are consistent with chemical models for a dynamically young, but chemically
evolved (age $>10^5$ yr) source (\cite{bergin1997}; \cite{roberts2004};
\cite{aikawa2005}; \cite{shirley2005}; see also \cite{kirk2007}
and references therein). This supports the idea that the sources have
been static or slowly contracting for more than $10^5$ yr, and conforms with
the estimated AD timescales.
On the other hand, the equal numbers of prestellar and protostellar cores
suggest that the prestellar core lifetime should be similar
to the lifetime of embedded protostars. Since the duration of the protostellar
stage is $\sim{\rm few}\cdot10^5$ yr
(e.g., \cite{ward-thompson2007}; \cite{hatchell2007};
\cite{galvan-madrid2007}; \cite{enoch2008}),
the prestellar core evolution should be rather dynamic and last for only
a few free-fall times, as is the case in star formation driven by supersonic
turbulence (e.g., \cite{maclow2004}; \cite{ballesteros-paredes2007}).
This seems to contradict with the above results of
AD timescales. However, in order to recognise the cores
in the submm map, they are presumed to be in the high-density stage of
their evolution. Thus, the short statistical lifetime deduced above is still
consistent with the quasi-static evolution driven by AD,
if we are only observing the densest stages of a longer scale core evolution
(e.g., \cite{enoch2008}; \cite{crutcher2009}). Also, the dynamic phase in the
core evolution with $\tau_{\rm AD}$ being only a few free-fall times might be
appropriate for magnetically near-critical (the mass-to-magnetic flux ratio
being $\sim80\%$ of the critical value) or already slightly supercritical cores
when rapid collapse ensues (\cite{ciolek2001}; see also \cite{tassis2004}).
\subsection{Linewidths and turbulence}
The N$_2$D$^+$ linewidths in Ori B9 E and N are significantly narrower
than the N$_2$H$^+$ linewidths (by factors of $\sim1.5$ and $\sim1.9$,
respectively). Crapsi et al. (2005a) found similar trend
in several low-mass starless cores (see their Table~4).
This is probably due to the fact that N$_2$D$^+$ traces the high density nuclei
of starless cores, where non-thermal turbulent motions are expected
to be insignificant (e.g., \cite{andre2007}; \cite{ward-thompson2007}).
The non-thermal component dominate the N$_2$H$^+$ linewidths in the observed
positions (the thermal linewidth of N$_2$H$^+$ is about 0.126 km s$^{-1}$ at
10 K, and thus $\Delta {\rm v}_{\rm NT}/\Delta {\rm v}_{\rm T}\sim2$).
However, the level of internal turbulence, as estimated from the ratio
between the non-thermal velocity dispersion and the isothermal
speed of sound (e.g., \cite{kirk2007}), is not dynamically significant.
Using Eq.~(7) of Williams et al (1998) and the derived values for the
molecular/metal ion-contribution constant $C_i$ and cosmic ray ionization rate
$\zeta_{\rm H_2}$ (see Table~\ref{table:ion}), we see that non-thermal
N$_2$H$^+$ line broadening in the observed positions can be
explained in part by magnetohydrodynamic (MHD) wave propagation.
The large wave coupling parameter in our sources ($W\gg1$), suggest that the
coupling between the field and gas is strong and the waves are not suppressed.
The derived values of $W$ ($\sim30-60$, in the case of minimum turbulence)
are in agreement with $\tau_{\rm AD}/\tau_{\rm ff}$ ratios
(\cite{williams1998}).
Also, the estimated degrees of coupling between the magnetic field and gas
conforms with the susceptibility to fragmentation (\cite{bergin1999}).
Caselli \& Myers (1995) analysed ammonia cores in the Orion B GMC
and found an inverse relationship between core linewidth and distance to the
nearest stellar cluster. The nearest stellar cluster to Ori B9 is NGC 2024
at the projected distance of 5.2 pc (see Sect. 1.1.), so its role as driving
external turbulence to the region is probably not significant.
\subsection{Formation of a small stellar group in Ori B9}
Internal turbulence or gravitational motions in the massive molecular cloud
core may promote fragmentation of the medium. This can easily generate
sheets and filaments (e.g., \cite{caselli1995}; \cite{andre2008}).
The collapse of these elongated clumps most probably results in the
formation of a small stellar group or a binary system rather than a single
star (e.g., \cite{launhardt1996}).
Only the densest parts of the filaments, the dense cores, are directly
involved in star formation.
It is unclear at the present time whether the collapse of an individual
prestellar core typically produces single stars or multiple protostellar
systems (see \cite{andre2008}).
The total mass of gas and dust of the clump associated with IRAS 05405-0117
as derived from the dust continuum
emission is $\sim14$ M$_{\sun}$, and it has elongated structure with multiple
cores (local maxima in the filament are separated by more than one beam size,
see Fig.~\ref{figure:positions}).
The previous mass estimates by Harju et al. (1993) based on NH$_3$ were much
higher: $\sim50$ M$_{\sun}$ derived from $N({\rm NH_3})$ distribution, and
$\sim310$ M$_{\sun}$ derived from peak local density.
The uncertainty in the abundance\footnote{Harju et al. assumed that
$x({\rm NH_3})\sim3\cdot10^{-8}$. Using the H$_2$ column densities from the
dust continuum we derive the values of $x({\rm NH_3})\sim1-2\cdot10^{-7}$ in
our line observation positions.} and lower resolution used are certainly
affecting the estimation of the mass from NH$_3$.
However, the clump has enough mass to form a small stellar group.
The kinetic temperature, velocity dispersion and the
fractional H$_2$D$^+$ abundance in the clump are similar to those
in the well-studied prestellar cores, e.g., L1544 and L183, where strong
emission of H$_2$D$^+$ line has been detected previously (see \cite{harju2006}
and references therein).
The masses, sizes, relatively high degree of deuteration and
the line parameters of the condensations indicate that they are low- to
intermediate-mass dense cores (cf. \cite{fontani2008}).
IRAS 05405-0117 and SMM 4 are likely to represent Class 0 protostellar cores
(see Sect. 5.1), whereas the subsidiary cores, e.g., Ori B9 N, are in an
earlier, prestellar phase.
\section{Summary and conclusions}
We mapped the Ori B9 cloud in the 870 $\mu$m dust continuum emission
with the APEX telescope. We also observed N$_2$H$^+(1-0)$ and N$_2$D$^+(2-1)$
spectral line emission towards selected positions in Ori B9 with the IRAM 30 m
telecope. These observations were used together with archival Spitzer/MIPS
data to derive the physical characteristics of the cores in Ori B9 and
the degree of deuterium fractionation and ionization degree within the
IRAS 05405-0117 clump region. The main results of this work are:
\hspace{0.5cm}
1. The LABOCA field contains 12 compact submm sources. Four of them are
previously known IRAS sources, and eight of them are new submm sources.
All the IRAS sources and two of the new submm sources are associated
with the Spitzer 24 and 70 $\mu$m sources. The previously unknown
sources, SMM 3 and SMM 4, are promising Class 0 candidates based on their SEDs
between 24 and 870 $\mu$m. There is equal number of starless and
protostellar cores in the cloud. We suggest that the majority of our starless
cores are likely to be prestellar because of their high densities.
2. The total mass of the cloud as estimated from the 2MASS
near-infrared extinction map is 1400 M$_{\sun}$.
The submm cores constitute about 3.6\% of
the total cloud mass. This percentage is in agreement with the observed low
values of star formation efficiency in nearby molecular clouds.
3. Mass distribution of the cores in Ori B9 and in Orion B North studied by
Nutter \& Ward-Thompson (2007) very likely represent the subsamples of
the same parent distribution. The CMF for the Orion B North
is well-matched to the stellar IMF (\cite{goodwin2008}).
Also the core separations in these two regions are similar, indicating that
the fragmentation length scale is similar. Since the fragmentation
length scales are alike, and the CMFs have resemblance to the IMF,
the origin of cores could be explained in terms of turbulent
fragmentation. The clustered mode of star formation in these two different
regions suggest that turbulence is driven on large scales.
4. On average, the starless cores are larger and more elongated than the
protostellar cores in Ori B9. The observed mean angular sizes and axis
ratios suggest average density-power-law indices $p\sim1$ and $\sim1.5$ for
starless and protostellar cores, respectively.
5. The fractional N$_2$H$^+$ and N$_2$D$^+$ abundances within the
clump associated with IRAS 05405-0117 are $\sim4-11\cdot10^{-10}$ and
$2-5\cdot10^{-11}$, respectively. The $N({\rm N_2D^+})/N({\rm N_2H^+})$ column
density ratio varies between 0.03-0.04. This is a typical degree of deuteration
in low-mass dense cores and conform with the earlier detection of H$_2$D$^+$.
There is evidence for a N$_2$H$^+$ ``hole'' in the protostellar Class 0
candidate SMM 4. The envelope of SMM 4 probably represents an extreme case of
depletion where also N$_2$H$^+$ has disappeared from the gas phase.
6. The ionization fraction (electron abundance) in the
positions studied is estimated to be $x({\rm e})\sim10^{-7}$. There is
uncertainty about the most abundant ionic species. The most likely candidates
are H$^+$ and HCO$^+$. The cosmic ray ionization rate in the
observed positions was found to be $\zeta_{\rm H_2}\sim1-2\cdot10^{-16}$
s$^{-1}$.
7. There seems to be a discrepancy between the chemical age
derived near IRAS 05405-0117 and the statistical age deduced from the
numbers of starless and protostellar cores which suggest that
the duration of the prestellar phase of core evolution is
comparable to the free-fall time.
The statistical age estimate is, however, likely to be biased by the fact that
the cores detected in this survey are rather dense ($n({\rm H_2})\gtrsim10^5$
cm$^{-3}$) and thus represent the most advanced stages.
\begin{acknowledgements}
We thank the referee, Paola Caselli, for her insightful comments and
suggestions that helped to improve the paper.
The authors are grateful to the sfaff of the IRAM 30 m telescope, for
their hospitality and help during the observations. We also thank the staff
at the APEX telescope site. We are very grateful to Edouard Hugo, Oskar Asvany,
and Stephan Schlemmer for making available their rate coefficients of the
reaction H$_3^+$ + H$_2$ with deuterated isotopologues.
O. M. acknowledges Martin Hennemann for providing
the SED fitting tool originally written by
J\"urgen Steinacker, and the Research Foundation of
the University of Helsinki. The team acknowledges support from the Academy of
Finland through grant 117206.
This work is based in part on observations made
with the Spitzer Space Telescope, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology under a contract with NASA.
\end{acknowledgements}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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Q: UINavigationController status bar color is transparent I am trying to figure out how to use UINavigationController. Whenever I create one, and set a UIViewController as rootViewController, It shows the background color of the UIViewController as the color of the status bar, like this:
The UIViewController .backgroundColor is set to System Teal Color, the Navigation Bar .background color is UIColor.gray. So the root view controller is showing through the navigation bar in the status bar area. How do I make the status bar the same color as the nav bar? Why is that area transparent?
Here's the main view controller class, which contains the UINavigationController as a child.
class ViewController: UIViewController {
let navVC: UINavigationController?
var mainVC: UIViewController?
required init?(coder: NSCoder) {
mainVC = MainVC(nibName: "MainVC", bundle: nil)
mainVC?.title = "Main"
navVC = UINavigationController(rootViewController: mainVC!)
navVC?.navigationBar.backgroundColor = UIColor.gray
super.init(coder: coder)
}
override func viewDidLoad() {
super.viewDidLoad()
// Do any additional setup after loading the view.
navVC?.willMove(toParent: self)
self.addChild(navVC!)
self.view.addSubview(navVC!.view)
navVC?.didMove(toParent: self)
navVC?.view.translatesAutoresizingMaskIntoConstraints = false
navVC?.view.topAnchor.constraint(equalTo: self.view.topAnchor).isActive = true
navVC?.view.leadingAnchor.constraint(equalTo: self.view.safeAreaLayoutGuide.leadingAnchor).isActive = true
navVC?.view.bottomAnchor.constraint(equalTo: self.view.safeAreaLayoutGuide.bottomAnchor).isActive = true
navVC?.view.trailingAnchor.constraint(equalTo: self.view.safeAreaLayoutGuide.trailingAnchor).isActive = true
}
}
Here's the project for this example.
Example Project
A: Thanks to lazarevzubov, who suggested this answer. Setting the navigationBar.standardAppearance and navigationBar.scrollEdgeAppearance fixed the problem.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Q: C++ repeating codes with a loops Im not sure if this is a stupid question, so shoot me if it is!
I am having this "dilemna" which I encounter very often. I have say two overloaded functions in C++
say we have this two overloads of F (just a pseudocode below)
void F(A a, .../*some other parameters*/)
{
//some code part
//loop Starts here
G1(a,.../* some other parameter*/)
//loop ends here
//some code part
}
void F(B b, .../*some other parameters*/)
{
//some code part
//loop Starts here
G2(b,.../* some other parameter*/)
//loop ends here
//some code part
}
where A and B are different types and G1 and G2 are different functions doing different things. The code part of the overloads except for G1 and G2 lines are the same and they are sometimes very long and extensive. Now the question is.. how can I write my code more efficiently. Naturally I want NOT to repeat the code (even if it's easy to do that, because its just a copy paste routine). A friend suggested macro... but that would look dirty. Is this simple, because if it is Im quite stupid to know right now. Would appreciate any suggestions/help.
Edit: Im sorry for those wanting a code example. The question was really meant to be abstract as I encounter different "similar" situation in which I ask myself how I am able to make the code shorter/cleaner. In most cases codes are long otherwise I wouldn't bother asking this in the first place. As KilianDS pointed out, it's also good to make sure that the function itself isn't very long. But sometimes this is just unavoidable. Many cases where I encounter this, the loop is even nested (i.e. several loops within each other) and the beginning of F we have the start of a loop and the end of F we end that loop.
Jose
A: A simple way to prevent the code duplication in this case would be to use a template. E.g:
void G(A a,.../* some other parameter*/)
{
G1(a,.../* some other parameter*/);
}
void G(B b,.../* some other parameter*/)
{
G2(b,.../* some other parameter*/);
}
template <typename T>
void F(T x, .../*some other parameters*/)
{
//some code part
//loop Starts here
G(x,.../* some other parameter*/)
//loop ends here
//some code part
}
Note how the overloaded G function is used to determine whether to call G1 or G2.
However, also note that this only prevents the code duplication, not the duplication in the compiled executable (because each template instantiation creates its own code).
Depending on the surrounding architecture there might be a number of other viable options (e.g. virtual methods instead of G1/G2 calls, function pointers, lambda functions if you have C++11...)
A: The most obvious solution is to put the common code parts into separate
functions, and call them:
void F( A a, ... )
{
commonPrefix(...);
G1( a, ... );
commonPostfix(...);
}
void F( B b, ... )
{
commonPrefix(...);
G2( a, ... );
commonPostfix(...);
}
if there is a lot of data shared between the prefix and the postfix, you
could create a class to hold it, and make the functions members.
Alternatively, you could forward to a template, possibly using traits:
template <typename T>
class GTraits;
template<>
class GTraits<A>
{
static void doG( A a, ... ) { G1( a, ... ); }
};
template <>
class GTraits<B>
{
static void doG( B b, ... ) { G2( b, ... ): }
};
template <typename T>
void doF( T t, ... )
{
// ...
GTraits<T>::doG( t, ... );
// ...
}
void F(A a, ...)
{
doF( a, ... );
}
void F(B b, ...)
{
doF( b, ... );
}
This could easily result in the common parts of the code being
duplicated, however. (Whether this is a problem or not depends. In
most cases, I suspect that the code bloat would be minimal.)
EDIT:
Since you say that the common code includes the loop logic: you can use
the template method pattern, something like:
class CommonBase
{
virtual void doG( other_params ) = 0;
public:
void doF()
{
// prefix
// loop start
doG( other_params );
// loop end
// suffix
}
};
You then define a separate derived class in each function:
void F( A a ,... )
{
class DoG : public CommonBase
{
A myA;
void doG( other_params ) { G1( myA, other_params ); }
public:
DoG( A a ) : myA( a ) {}
} do( a );
do.doF();
}
The fact that you need the forwarding constructor makes it a bit wordy,
but it does keep all of the common code common.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 8,187
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David Pérez Pallas (Vigo, Pontevedra, 24 d'agost de 1987) és un àrbitre de futbol espanyol que actua com a àrbitre VAR a la Segona Divisió d'Espanya. Pertany al Comitè Tècnic d'Àrbitres de Galícia.
Trajectòria
Durant les seves set temporades en el futbol espanyol ha arbitrat 156 partits, mostrant un total de 932 targetes grogues i 56 targetes vermelles. La seva temporada més prolífica en l'àmbit disciplinari va ser la temporada 2011-12 en la qual, militant en la Segona Divisió "B" d'Espanya va fer una mitjana de més de 7 targetes grogues i 1 vermella per partit.
Temporades
Referències
Àrbitres de futbol gallecs
Esportistes de Vigo
|
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"redpajama_set_name": "RedPajamaWikipedia"
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| 8,264
|
{"url":"https:\/\/ask.learncbse.in\/t\/if-earth-contracts-to-half-its-radius\/13967","text":"# If earth contracts to half its radius\n\nIf earth contracts to half its radius. What would be the length of the day?\n\nThe moment of inertia (If I = 2\/5 M${ R }_{ 2 }$) of the earth about its own axis will become one-fourth and so its angular velocity will become four times (L = I\u03c9= constant). Hence, the time period will reduce to one-fourth (T = 2$\\pi$\/\u03c9 to), i.e. 6 hours.","date":"2020-10-29 20:11:39","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\": 0.7849081158638, \"perplexity\": 1163.3566391929605}, \"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\/1603107905777.48\/warc\/CC-MAIN-20201029184716-20201029214716-00144.warc.gz\"}"}
| null | null |
Xu Tingting (; ur. 12 lipca 1989) – chińska lekkoatletka specjalizująca się w trójskoku.
Osiągnięcia
srebro mistrzostw Azji (Guangdong 2009)
medalistka mistrzostw kraju
Rekordy życiowe
trójskok – 14,15 (2008)
trójskok (hala) – 13,95 (2008)
Linki zewnętrzne
Chińscy trójskoczkowie
Urodzeni w 1989
|
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| 9,184
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\section{Introduction}
\begin{figure*}[ht]
\begin{center}
\includegraphics{ZM.pdf}
\caption{{\footnotesize Distribution in the $M$--$z$ plane of the \Planck\ SZ cluster catalogue
(open red circles) \citep{planck2013-p05a} compared with those from SPT
(black) \citep{rei13,ble14} and ACT (green)
\citep{mar11,has13}, MaDCoWS (yellow) \citep{bro14}, and
NORAS and REFLEX from the MCXC meta-catalogue
(blue) (\citet{pif11} and references therein). Some clusters may
appear several times as distinct points due to differences in the
mass estimate between surveys. The black dotted lines show the \Planck\ mass limit
for the medium-deep survey zone at 20\% completeness for a redshift limit of $z=0.5$. }}
\label{fig:mz}
\end{center}
\end{figure*}
It is only recently that cluster samples selected via their
Sunyaev--Zeldovich (SZ) signal have reached a significant sizes, e.g.,
the Early SZ (ESZ) catalogue from the
\textit{Planck}\ Satellite\footnote{\textit{Planck}\ (\url{http://www.esa.int/Planck})
is a project of the European Space Agency (ESA) with instruments
provided by two scientific consortia funded by ESA member states (in
particular the lead countries France and Italy), with contributions
from NASA (USA) and telescope reflectors provided by a collaboration
between ESA and a scientific consortium led and funded by Denmark.}
\citep{planck2011-5.1a,planck2013-p05a}, and catalogues from the South Pole Telescope
\citep[ SPT, ][]{rei13,ble14} and the Atacama Cosmology Telescope
\citep[ACT,][]{mar11,has13}. These are now considered as
new reference samples for cluster studies and associated cosmological analyses.
The present note describes updates to the construction and
properties of the \textit{Planck}\ catalogue of SZ sources PSZ1,
\citep[hereafter PXXIX2013, ][]{planck2013-p05a}, released in March 2013
as part of the first \textit{Planck}\ data delivery. The PSZ1 catalogue
contains 1227 entries, including 683 so-called {\it previously-known}
clusters. This category corresponds to the association of \Planck\ SZ
source detections with known clusters from the literature. The association is set to
the first identifier as defined in the hierarchy adopted by
PXXIX2013, namely: (i) identification with MCXC clusters
\citep{pif11}; (ii) identification with Abell and Zwicky objects; (iii) identification with clusters derived from SDSS-based
catalogues (primarily from \citet{wen12}); (iv) identification with clusters from SZ catalogues
\citep{has13,rei13}; (v) searches in the NED and SIMBAD
databases. Considerable added value, including consolidated
redshift and mass estimates (Fig.~\ref{fig:mz}), has been obtained through
compilation of this ancillary information.
Since its delivery March 2013, we have continued to update the PSZ1 catalogue by
focusing on the confirmation of newly-discovered clusters in PSZ1. This process
has first involved updating the redshifts of some previously-known clusters (Sect.~\ref{asso}).
We have also made use of recent results from
dedicated follow-up observations conducted by the {\it Planck}
Collaboration with the RTT150 \citep{rtt150} and ENO telescopes
(Planck Collaboration 2015, in prep.), which together have
allowed us to observe and measure redshifts for $\sim 150$ PSZ1 sources (Sect. ~\ref{rtt}).
We have also used published results from PanSTARRS \citep{liu14} and from the latest SPT catalogue \citep{ble14}, as described in Sects.~\ref{pans} and~\ref{spt}.
For all clusters with measured redshifts, we have computed the
estimated masses using the $Y_z$ mass proxy (\citealt{arn14} and PXXIX2013; Sec.~\ref{masses}).
Finally, we have revisited the cluster candidate classification scheme, which in PXXIX2013 was organised into three classes ({\it class-}1, 2, 3) in order of decreasing reliability.
As described in Sect. \ref{can}, we have now used the SZ spectral
energy distribution (SED) to refine the quality assessment of the
cluster candidates by adopting a new, novel quality flag derived from the Artificial
Neural Network analysis developed by \citet{agh14}.
\section{Redshift updates for \textit{previously-known} clusters}\label{asso}
In the external validation process performed in PXXIX2013, a total of
683 PSZ1 sources were associated with clusters published in X-ray,
optical, or SZ catalogues, or with clusters found in the NED or SIMBAD
databases. We refer to these as {\it previously-known clusters}. Their
redshifts, when available, were compiled from the literature and
a consolidated value was provided with the PSZ1 catalogue.
In the present update, we first re-examine the {\it
previously-known} clusters of the PSZ1 catalogue.
The dedicated follow-up of \textit{Planck}\ PSZ1 clusters with RTT150
described in \citet{rtt150} provided updates to the redshifts of 19
{\it previously-known} clusters. The follow-up of
\textit{Planck}\ PSZ1 clusters with ENO telescopes further updated the
redshifts of five
{\it previously-known} clusters. \\
We have updated the redshifts of ten
PSZ1 sources associated with SPT clusters provided in \citet{ble14}.
Finally, we have queried the NED and SIMBAD databases, and searched
in the cluster catalogues constructed from the SDSS data (namely
\citealt{wen12} and \citealt{roz14a}), for additional spectroscopic redshifts. When
these were available, we report them in the updated
version of the PSZ1 catalogue. The full process led us to change the
redshifts of 34 {\it previously-known} PSZ1 clusters. We have
also changed the published photometric redshift estimate of one ACT cluster (ACT-CL
J0559-5249) to a spectroscopic redshift value.
In summary, 69
sources from the PSZ1 catalogue associated with {\it
previously-known} clusters now have updated redshifts. Most of these consist of updates from photometric to spectroscopic values; however, eight redshifts were measured for the
first time for {\it previously-known} clusters.
\section{{\it Planck}-discovered clusters}
The PSZ1 catalogue contained 366 cluster candidates,
classified as {\it class-}1 to 3 in order of decreasing reliability,
and 178 {\it Planck}-discovered clusters confirmed mostly with
dedicated follow-up programmes undertaken by the {\it Planck}
Collaboration. Since the delivery of the PSZ1 catalogue in March 2013,
a number of additional confirmations, including results from the community, were
performed and redshifts were updated from photometric estimates to
spectroscopic values.
Combining the results from follow-up with the RTT150 \citet{rtt150}, ENO telescopes (Planck
collaboration 2015, in prep.), \citet{liu14}, \citet{roz14a}, and
\citet{ble14}, a total of 86 PSZ1 sources have been newly confirmed as {\it
Planck-}discovered clusters with measured redshifts.
\subsection{From RTT150 results}\label{rtt}
As part of the \textit{Planck}\ Collaboration optical follow-up programme, candidates were observed with the Russian Turkish Telescope
\citep[RTT150\footnote{\url{http://hea.iki.rssi.ru/rtt150/en/index.php}.},][]{rtt150} within the Russian quota of observational time,
provided by Kazan Federal University and Space Research Institute
(IKI, Moscow). Direct images and spectroscopic redshift measurements
were obtained using T\"UB\.ITAK Faint Object Spectrograph and Camera
(TFOSC\footnote{\url{http://hea.iki.rssi.ru/rtt150/en/}\\ \url{index.php?page=tfosc}.}).
For the highest-redshift clusters, complementary spectroscopic
observations were performed with the BTA 6-m telescope of the SAO RAS
using the SCORPIO focal reducer and spectrometer \citep{afa05}.
These observations have confirmed and measured redshifts for a total of 24 new candidates. Eleven of these have spectroscopic redshifts. We have updated the PSZ1 catalogue by including these newly-measured redshifts.
\subsection{From ENO telescopes}
Also as part of the \textit{Planck}\ Collaboration optical follow-up programme, candidates were observed
at European Northern Observatory (ENO\footnote{ENO:
\url{http://www.iac.es/eno.php?lang=en}. }) telescopes, both in
imaging (at IAC80, INT and WHT) and spectroscopy (at NOT, GTC, INT and
TNG). The observations were obtained as part of proposals for the
Spanish CAT time, and an {\it International Time Programme (ITP)},
accepted by the International Scientific Committee of the Roque de los
Muchachos and Teide observatories. We summarise here the main results
of these observing programmes. Further details will be presented in a
companion article (\textit{Planck}\ Collaboration 2015, in prep.).
These observations have confirmed and provided new redshifts
for a total of 26 candidates, that are reported in the updated
PSZ1 catalogue. These include the confirmation of 12 SZ sources as
newly-discovered clusters: two {\it class}-1, high reliability
candidates, five {\it class}-2, and five {\it class}-3 candidates.
\subsection{From PanSTARRS} \label{pans}
Based on the Panoramic Survey Telescope \& Rapid Response System
(PanSTARRS, \citealt{kai02}) data, \citet{liu14} have searched for
optical confirmation of the 237 \textit{Planck}\ SZ detections that overlap the
PanSTARRS footprint.
We only report here the redshifts for unambiguously confirmed clusters. Of these, 15 objects were included in the RTT150 follow-up, for which the redshifts are published in
\citet{rtt150}, and three objects were included in the ESO follow-up described above. In these cases, we report the \textit{Planck}\ Collaboration follow-up redshift values in the updated PSZ1 catalogue.
An additional two {\it Planck} clusters confirmed by
PanSTARRS have a counterpart in the \citet{roz14a} catalogue, with
spectroscopic redshifts that we update in the PSZ1 catalogue.
A total of 40 {\it Planck-discovered} clusters are confirmed, for the
first time, by \citet{liu14} in the PanSTARRS survey. All of these have
measured photometric redshifts that we have reported in the updated PSZ1 catalogue.
\begin{figure*}[!th]
\begin{center}
\includegraphics{histoZ_Planck.pdf}
\includegraphics{histoM_Planck.pdf}
\caption{{\footnotesize Distribution of redshifts (left panel) and masses (right
panel) for the \Planck\ SZ clusters. The black shaded area
represents the population of clusters above redshift of 0.5 (in the
right panel) and having a mass above $5 \times 10^{14} M_{\sun} $ (in the left
panel).}}
\label{fig:z_hist}
\end{center}
\end{figure*}
\subsection{From SPT} \label{spt}
A new catalogue of SZ clusters detected with the South Pole Telescope
(SPT) cluster catalogue was published in \citet{ble14}. It provides an
ensemble of spectroscopic and photometric redshifts. Four candidate {\it class-}1 and 2
clusters from the PSZ1 catalogue were confirmed
and have photometric redshifts in \citet{ble14}. These are included in the updated PSZ1 catalogue.
\subsection{From SDSS-RedMapper catalogue} \label{spt}
Comparison with the SDSS-based catalogue from \citet{roz14a} provided
confirmation and new redshift values for five {\it Planck-discovered} clusters. This
includes confirmation of two {\it Planck} cluster candidates (one {\it class-}2
and one {\it class-}3 candidate). We use the spectroscopic redshift values
available in the \citet{roz14a} in the updated PSZ1 catalogue.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.8cm]{zredshifts.pdf}
\caption{{\footnotesize Percentage of origin and type (photometric, spectroscopic) of
the redshifts reported in PSZ1. To date associations with MCXC
clusters provide 49.8\% of the redshifts, all spectroscopic. Follow
up observations by the \textit{Planck}\ collaboration (FUs) provide 24.6\% of the
redshifts, of which 64.73\% are spectroscopic. Associations with clusters
from SDSS-based catalogues result in 11.7\% of all redshifts, of which
58.9\% are spectroscopic. Redshifts from the NED and SIMBAD databases
represent 5.9\% of all redshifts, with 90.7\% of them spectroscopic. The confirmation
from PanSTARRS data provides 4.4\% of the total number of redshifts, all of them
photometric. Finally the association with SZ catalogues (SPT and
ACT) represents 3.5\% of all redshifts, of which 71.9\% are spectroscopic.
}}
\label{fig:zdist}
\end{center}
\end{figure}
\section{Mass estimate} \label{masses}
The size--flux degeneracy discussed in, e.g., \citet{planck2011-5.1a}
and PXXIX2013 can be broken when $z$ is known, using the
$\Mv$--$\da\YSZ$\ relation between $\theta_{500}$ and $Y_{500}$ see \citep{arn14}. The $Y_{500}$
parameter, denoted $Y_z$, is derived from the intersection of the
$\Mv$--$\da\YSZ$\ relation and the size--flux degeneracy curve. It is the SZ mass
proxy $Y_z$ that is equivalent to the X-ray mass proxy $\YX$.
For all the \Planck\ clusters with measured redshifts, $Y_z$ was
computed for our assumed cosmology, allowing us to derive an homogeneously-defined
SZ mass proxy, denoted $M_{500}^{Y_z}$, based on X-ray
calibration of the scaling relations (see discussion in PXXIX2013).
We show in Fig.~\ref{fig:z_hist}
(right panel, in red) the distribution of masses obtained from the SZ-based mass
proxy for all clusters with measured redshift. Note that since we use an X-ray calibration of the scaling relations, these masses are uncorrected for any bias due to the assumption of hydrostatic equilibrium in the X-ray mass analysis. The
shaded black area shows the distribution of masses for clusters with
redshifts higher than 0.5. They represent a total of 78 clusters.
\section{Cluster candidates}\label{can}
Since the delivery of the {\it Planck} catalogue and the confirmation in this \textit{addendum}
of 86 candidates as new clusters, the updated
PSZ1 catalogue now contains 280 cluster candidates. In the original PSZ1, these latter were
classified as {\it class-}1 to 3 in order of decreasing reliability;
the reliability being defined empirically from the combination
of internal {\it Planck} quality assessment and ancillary information
(e.g., searches in RASS, WISE, SDSS data). The updated PSZ1
catalogue contains 24 high quality ({\it class-}1) SZ detections
whereas lower reliability {\it class}-2 and 3 candidates represent
130 and 126 SZ sources, respectively.
With the updated PSZ1 catalogue, we now provide a new objective quality
assessment of the SZ sources derived from an artificial neural-network
analysis. The construction, training and validation of the network is
based on the analysis of the Spectral Energy Distribution (SED) of the
SZ signal in the {\it Planck} channels. The implementation is discussed in detail by
\citet{agh14}. The neural network was trained with an ensemble of three
samples: the confirmed clusters in the PSZ1 calatogue representing
good/high-quality SZ signal; the {\it Planck} Catalogue of Compact
Sources source, representing the IR and radio-source induced
detections; and random positions on the sky as examples of
noise-induced, very low reliability, detections.
In practice, we provide for each SZ source of the updated PSZ1 catalogue a
neural-network quality flag, $Q_N$, defined as in \citet{agh14}. This
flag separates the high quality SZ detections from the low quality
sources such that $Q_N< 0.4$ identifies low-reliability SZ sources
with a high degree of success. Figure \ref{fig:zqual} summarises for
each class of {\it Planck} cluster candidate the number of sources
below and above the threshold velue of $Q_N=0.4$. The {\it class-}1 cluster
candidates all have $Q_N>0.4$ except for one source for which
$Q_N=0.39$. The fraction of `good' $Q_N>0.4$ SZ detections in the
{\it class-}2 category is about 80\%, while the fraction of $Q_N>0.4$ candidates
drops to about 30\% for the {\it class-}3 cluster-candidates.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.8cm]{zqual.pdf}
\caption{{\footnotesize Number of {\it Planck} cluster-candidates below and above the
neural-network quality flag threshold $Q_N=0.4$, denoting a high-quality SZ detection, for each reliability class.}}
\label{fig:zqual}
\end{center}
\end{figure}
\section{Summary}
\begin{table*}[t]
\begingroup
\caption{{\footnotesize Summary of the updates of PSZ1v2 for each cluster or candidate type}}
\label{tab:summ}
\nointerlineskip
\vskip -3mm
\footnotesize
\setbox\tablebox=\vbox{
\newdimen\digitwidth
\setbox0=\hbox{\rm 0}
\digitwidth=\wd0
\catcode`*=\active
\def*{\kern\digitwidth}
\newdimen\signwidth
\setbox0=\hbox{+}
\signwidth=\wd0
\catcode`!=\active
\def!{\kern\signwidth}
\halign{$#$\hfil\tabskip 2em&
\hfil$#$\hfil\tabskip=0.8em&
$#$\hfil&
\hfil$#$\tabskip=2em&
\hfil$#$\hfil\tabskip=2em&
\hfil$#$\hfil\tabskip=2em&
$#$\hfil\tabskip=0.0em&
\hfil$#$&
\hfil$#$\tabskip=0pt\cr
\noalign{\vskip 3pt\hrule \vskip 1.5pt \hrule \vskip 5pt}
\hfil\hfil\hfil\hfil\hfil\hfil\hfil\hfil\hfil&\multispan3\hfil PSZ1 (2013)\hfil& \omit&\multispan3\hfil PSZ1v2 (2015)\hfil\cr
\noalign{\vskip -3pt}
\omit\hfil\hfil&\multispan3\hrulefill&\omit&\multispan3\hrulefill\cr
\omit\hfil&\hfil Number\hfil&\multispan2\hfil redshift\hfil&\mathrm{UPDATES}&Number&\multispan2\hfil redshift\hfil\cr
\omit\hfil\hfil&\omit&\multispan2\hrulefill&\omit&\omit&\multispan2\hrulefill\cr
\omit\hfil&\omit&type&number&\omit&\omit&type&number\cr
\noalign{\vskip 3pt\hrule\vskip 5pt}
\mathit{ `` Previously\; known'' } & \mathbf{683} & \omit&\omit& \mathbf{0} & \mathbf{683} &\omit&\omit\cr
\omit & \omit & \mathrm{undef}&29& -8 & \omit& \mathrm{undef}& 21\cr
\omit & \omit & \mathrm{estim}&5& -4 & \omit& \mathrm{estim}& 1\cr
\omit & \omit & \mathrm{phot}&97& -43 & \omit& \mathrm{phot}& 54\cr
\omit & \omit & \mathrm{spec}&552& +55 & \omit& \mathrm{spec}& 607\cr
\noalign{\vskip 6pt\hrule\vskip 5pt}
\mathit{ `` Planck\; discovered'' } & \mathbf{178} & \omit&\omit& \mathbf{+86} & \mathbf{264} &\omit&\omit\cr
\omit & \omit & \mathrm{undef}&19& -6 & \omit& \mathrm{undef}& 13\cr
\omit & \omit & \mathrm{phot}&72& +50 & \omit& \mathrm{phot}& 122\cr
\omit & \omit & \mathrm{spec}&87& +42 & \omit& \mathrm{spec}& 129\cr
\noalign{\vskip 6pt\hrule\vskip 5pt}
\mathit{Class-1 } & \mathbf{54} & \omit&\omit& \mathbf{-30} & \mathbf{24} &\omit&\omit\cr
\omit & \omit & \mathrm{undef}&54& -30 & \omit& \mathrm{undef}& 24\cr
\omit & \omit & \mathrm{phot}&\omit& \mathit{+22} & \omit& \mathrm{phot}& \omit\cr
\omit & \omit & \mathrm{spec}&\omit& \mathit{+8} & \omit& \mathrm{spec}& \omit\cr
\mathit{Class-2 } & \mathbf{170} & \omit&\omit& \mathbf{-40} & \mathbf{130} &\omit&\omit\cr
\omit & \omit & \mathrm{undef}&170& -40 & \omit& \mathrm{undef}& 130\cr
\omit & \omit & \mathrm{phot}&\omit& \mathit{+26} & \omit& \mathrm{phot}& \omit\cr
\omit & \omit & \mathrm{spec}&\omit& \mathit{+14} & \omit& \mathrm{spec}& \omit\cr
\mathit{Class-3 } & \mathbf{142} & \omit&\omit& \mathbf{-16} & \mathbf{126} &\omit&\omit\cr
\omit & \omit & \mathrm{undef}&142& -16 & \omit& \mathrm{undef}& 126\cr
\omit & \omit & \mathrm{phot}&\omit& \mathit{+10} & \omit& \mathrm{phot}& \omit\cr
\omit & \omit & \mathrm{spec}&\omit& \mathit{+6} & \omit& \mathrm{spec}& \omit\cr
\noalign{\vskip 6pt\hrule\vskip 3pt}}}
\endPlancktablewide
\endgroup
\end{table*}
We have updated the \textit{Planck}\ catalogue of SZ-selected sources detected in
the first 15.5 months of observations. The catalogue contains 1227
detections and was validated using external X-ray and optical/NIR data,
alongside a multi-frequency follow-up programme for confirmation.
The updated PSZ1 catalogue now contains 947 confirmed clusters, including
264 brand-new clusters, of which 214 have been confirmed by the
\Planck\ Collaboration's follow-up programme. The
remaining 280 cluster candidates have been divided into three classes
according to their reliability, i.e., the quality of evidence that
they are likely to be {\it bona fide} clusters. To date, high quality SZ
detections in PSZ1 represent 24 sources, all of which are classified as high-quality by our neural-network quality assessment procedure. Lower reliability, {\it class}-2
and 3 candidates represent 130 and 126 SZ sources respectively
(Table~\ref{tab:summ}). We find that $\sim 80$\% of the {\it class}-2
candidates are classified as high-quality by our neural-network quality assessment procedure, whereas only 35\% of the {\it class}-3 sources are considered as high-quality SZ
detections. Based on this assessement, the purity of the updated PSZ1
catalogue is $\sim 94\%$.
A total of 913 \Planck\ clusters (i.e., 74.2\% of all SZ detections) now have measured redshifts, of which 736 are spectroscopic values (i.e., 80.6\% of all
redshifts). The left-hand panel of Fig.~\ref{fig:z_hist} shows the distribution
in redshift of all clusters (red), and for the clusters with masses
above $5 \times 10^{14} M_{\sun}$ (shaded black). The median redshift of the
PSZ1 catalogue is about 0.23, and of order 35\% of the
\textit{Planck}\ clusters lie at redshifts above $z=0.3$.
The origins and types of redshifts are shown in
Fig.~\ref{fig:zdist}. Association with MCXC clusters
\citep{pif11} provides about $ 49.8\%$ of the redshifts, all of which are
spectroscopic. Follow up observations undertaken by the \textit{Planck}\ Collaboration
provide $ 24.6\%$ of the redshifts, about two thirds of them
being spectroscopic. SDSS-based catalogues yield $ 11.7\%$
of the redshifts, of which more than half of which are spectroscopic.
NED and SIMBAD database searches yield 5.9\% of the redshifts, the vast majority of which are spectroscopic. PanSTARRS data provide $ 4.4\%$ of the redshifts, all of which are photometric. Finally, association with the SPT and ACT SZ catalogues represent $\sim 3.5\%$ of all
redshifts, most of which are spectroscopic.
For the \Planck\ clusters with measured redshifts, we have provided a
homogeneously-defined mass estimated from the Compton $Y$
parameter. The $M$--$z$ distribution of the \textit{Planck}\ clusters is shown by open red circles
in Fig.~\ref{fig:mz}, where it is compared with
other large cluster surveys. Note that the masses are not homogenised
and some clusters may appear several times due to differences in the
mass estimation methods between surveys. We see that \Planck\ cluster
distribution probes a unique region in the $M$--$z$ space occupied by
massive, $M\ge5\times 10^{14}\,\mathrm{M}_{\odot}$, high-redshift ($z\ge 0.5$)
clusters. The \textit{Planck}\ detections almost double the number of massive
clusters above redshift 0.5 with respect to other surveys.
\begin{acknowledgements}
The development of \textit{Planck}\ has been supported by: ESA; CNES and
CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE
(USA); STFC and UKSA (UK); CSIC, MICINN, JA and RES (Spain); Tekes,
AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space
(Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland);
FCT/MCTES (Portugal); and PRACE (EU). The authors thank
N. Schartel, ESA {\it XMM-Newton} project scientist, for granting the
DDT used for confirmation of SZ
\textit{Planck}\ candidates. The authors thank TUBITAK, IKI, KFU and AST for
support in using RTT150; in particular we thank KFU and IKI for
providing significant amount of their observing time at RTT150. We
also thank BTA 6-m telescope TAC for support of optical follow-up
project. The authors acknowledge the use of the INT and WHT
telescopes operated on the island of La Palma by the Isaac Newton
Group of Telescopes at the Spanish Observatorio del Roque de los
Muchachos of the IAC; the NOT, operated on La Palma jointly by
Denmark, Finland, Iceland, Norway, and Sweden, at the Spanish
Observatorio del Roque de los Muchachos; the TNG, operated
on La Palma by the Fundacion Galileo Galilei of the INAF at the
Spanish Observatorio del Roque de los Muchachos; the GTC
telescope, operated on La Palma by the IAC at the Spanish Observatorio
del Roque de los Muchachos; and the IAC80 telescope
operated on the island of Tenerife by the IAC at the Spanish
Observatorio del Teide. Part of this research has been
carried out with telescope time awarded by the CCI International Time
Programme. The authors thank the TAC of the MPG/ESO-2.2m telescope for
support of optical follow-up with WFI under {\it Max Planck} time.
Observations were also conducted with ESO NTT at the La Silla Paranal
Observatory. This research has made use of SDSS-III data. Funding for
SDSS-III \url{http://www.sdss3.org/} has been provided by the Alfred
P. Sloan Foundation, the Participating Institutions, the National
Science Foundation, and DoE. SDSS-III is managed by the Astrophysical
Research Consortium for the Participating Institutions of the SDSS-III
Collaboration. \\ This research has made use of the following
databases: the NED and IRSA databases, operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with
the NASA; SIMBAD, operated at CDS, Strasbourg, France; SZ cluster
database \url{http://szcluster-db.ias.u-psud.fr} and SZ repository
operated by IDOC operated by IAS under contract with CNES and CNRS.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3
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"Doctor Who" Will Finally Have A Woman Play The Doctor
BY Seeta Charan
Doctor Who has finally chosen a woman to play the Doctor after 12 men! Yup, that's over 35 active years of women playing nothing more than sidekicks and love interests until now. Jodie Whittaker stars in the BBC hit show Broadchurch alongside two Doctor Who veterans. BBC announced her as the 13th Doctor through a video, and followed up with an interview so we could get to know our next Doctor a little better. In the interview, she addresses the question of a woman as the Doctor by saying, "I want to tell the fans not to be scared by my gender. Because this is a really exciting time, and Doctor Who represents everything that's exciting about change. The fans have lived through so many changes, and this is only a new, different one, not a fearful one."
Fans took to Twitter to express their joy over the announcement and loyal viewers defended the show's choice to cast Whittaker.
Past stars on Doctor Who made sure to show their support for Whittaker.
The previous Doctor discussed gender with the current companion, Bill, when she said, "So Time Lords are a bit flexible on the whole man/woman thing, yeah?" and he responded with, "We're billions of years beyond your petty obsession with gender and its associated stereotypes." This comment was probably foreshadowing the next Doctor. They had already been changing the image of the Doctor by making him younger with Tennant and Smith and then allowing Capaldi to keep his Scottish accent when all of the past Doctors have been British.
Doctor Who also welcomed its first openly gay, full-time companion, Bill, but the show has featured gay companions in the past. Madame Vastra and Jenny Flint were a lesbian couple who occasionally accompanied the Doctor in adventures. Casting Jodie Whittaker is a major step forward for Doctor Who, and we can only hope that Doctor Who will continue to cast women as their troubled, world-saving superheroes.
Photo Credit: Twitter, @bbcdoctorwho
Why We Need More Films About Women In Sports - And How We Can See Them
Rob Kardashian, Revenge Porn Is Never Okay
Dudes, Come Get Your Friends
Tags: Jodie Whittaker , Doctor Who , BBC , Peter Capaldi , Matt Smith , David Tennant , Broadchurch
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,689
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Christmas Glow is coming to Enjoy Centre and it's going to be the largest indoor family Christmas festival in the Edmonton area!
Best news of this is that it's INDOORS! No need to bundle your kids up for Christmas fun only to stay for 15 minutes because it's so cold!
Christmas Glow light tunnel from the Langley BC event.
It's the first time that Christmas Glow is in Alberta, with similar events happening in Langley, BC, and Barrie, ON.
Tickets are ON SALE now on their website and until September 25th pre-sale tickets are 40% off!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,703
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The Met One GT-321 is a Portable Handheld Particle Counter that can count particles all the way down to 0.3 microns which provides you with reliability and portability at the industry's lowest price.
The Met One GT-321 is totally self contained and has an internal battery, sample pump, and is easily programmable which allows you to pick the desired particle size from 5 different sizes of 5.0, 2.0, 1.0, 0..5, or 0.3 microns.
Other particle counters may take up to 1 minute to produce a result, but the Met One GT-321 delivers volumetric concentration readings in around 6 seconds! This results in quick facilitatation of site particle trending, tracing particle sources, and multi site sampling.
Only 2 buttons on the front panel control the sample cycle size and channels. The power switch is side mounted which allows for easy thumb operation.
The battery pack is self contained and provides power to the unit for 8 hrs. typical intermittent operation, & for a maximum of 5 hrs. continuous use. The AC adapter supplies the counter even while the battery pack has no charge, & recharges the drained battery in 16 hours when shut off.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 88
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Exclusive Video Premiere: 'Voyager,' Fur Trade
By Emily McDermott
ABOVE: FUR TRADE. PHOTO COURTESY OF BRENDAN MEDOWS.
We knew the union of Steve Bays from Hot Hot Heat and Parker Bossley from The Gay Nineties for their latest venture, Fur Trade, would produce a unique sound and aesthetic. Watching the duo's latest video "Voyager," which we're pleased to premiere here, makes viewers feel as though they are back in the '90s: an era of amateur and experimental editing, VHS, and prolific karaoke. Beginning slowly with a home-movie appeal, the video becomes more intricate—even psychedelic—as the song progresses, interlacing reality with fiction. The video blurs the lines between three awkward businessmen (one of whom happens to be Bossley's dad) in a karaoke bar, and a kimono-donning Japanese woman lip-synching the song's lyrics on the karaoke screen. "We were screaming the whole shoot, telling them to look more angry," Bays recalls.
Bays and Bossley juxtapose electronic synthesizers with live drums recorded using the built-in iPhone voice recorder. The video was filmed using an old-school VHS camera—a fitting medium for a song with the lyrics, "Borrow from the past and steal from the future."
"Despite the glitches and noise, it's much more flattering and intriguing than even the most expensive digital cam," Bays says. The trippy visuals and cast of characters were intentionally used to offset any potential irony resulting from the VHS and karaoke. As Bays says, "In the end, every moment of the video feels either over the top or totally uncomfortable."
FUR TRADE'S DEBUT ALBUM DON'T GET HEAVY IS OUT NOW. FOR MORE INFO ON THE BAND, VISIT THEIR SOUNDCLOUD.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,075
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Soledad O'Brien Becomes a New Anchor for CNN
by Observer Newsroom January 10, 2012 December 5, 2012
(NNPA) – On Jan. 2, Soledad O'Brien's new program aired on CNN entitled, "Starting Point." Her new show fills in the 7-9 a.m. slot left after the demise of the American Morning Block. CNN has referred to the shows as being a "conversational ensemble" with O'Brien at the center.
This change comes after CNN announced in late October that it was revamping its morning lineup, with O'Brien and former MSNBC anchor, Ashleigh Banfield were named to be among the anchors of a new early programming schedule.
Former "American Morning" anchor O'Brien, who co-hosted from 2003-2007, was recruited back to mornings for the second shift–just in time for the Jan. 3rd Iowa caucuses. According to Broadcasting & Cable, O'Brien will report live from Des Moines, Iowa on Jan. 2 and 3.
A graduate of Harvard University, O'Brien is a member of the National Association of Black Journalists and the National Association of Hispanic Journalists. She began her career as an associate producer and news writer at the then NBC affiliate WBZ-TV in Boston. She would work long hours as a local reporter and bureau chief for NBC affiliate KRON in San Francisco.
She later joined NBC in 1991 in New York where she worked as a field producer for Nightly News and TODAY. O'Brien came to CNN, where she anchored the network's Weekend Today since July 1999. At CNN, O'Brien would earn numerous awards and accolades for groundbreaking coverage and reports.
She became co-anchor of CNN's flagship morning program, American Morning in July 2003. There she covered world-changing events like Hurricane Katrina, Southeast and Thailand Tsunamis, and the 2005 London terrorist attacks. She earned the George Foster Peabody Award for her Katrina coverage and the Alfred I. DuPont Award for her coverage of the tsunami. Other accolades include the Gracie Allen Award in 2007 on the Israeli-Hezbollah conflict, a NAACP President's Award, also in 2007, for her humanitarian efforts and journalistic excellence. In 2008, she received the Johns Hopkins Bloomberg School of Public Health's Goodermote Humanitarian Award for her reports on Katrina and the Southeast Asia tsunami and was the first recipient of the Soledad O'Brien Freedom's Voice Award from Morehouse School of Medicine for promoting social change. O'Brien was also awarded the Brotherhood Crusade Pioneer African American Achievement Award by the Brotherhood Crusade in 2009.
One of her most recent projects was Black in America 2, which was a four-hour documentary that focused on successful community leaders who improved quality of life for African Americans.
O'Brien's Black in America in 2008 revealed that state of Blacks 40 years after the assassination of Martin Luther King Jr. She has also reported for the CNN documentary Words That Changed a Nation, which featured never-before-seen footage of Dr. King's private writings and notes, and her investigation of his assassination. Her project, Children of the Storm and One Crime at a Time documentaries have shown her dedication to stories coming out of New Orleans.
Tagged: CNN, Soledad O'Brien, Television
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 263
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Битва при Заальфельде произошла 10 октября 1806 между французской армией численностью 12 800 человек и прусской армией насчитывавшей 8300 солдат.
Сражение закончилось победой Франции.
Ход битвы
Принц Людвиг был одним из основных сторонников возобновления войны против Франции.
Основные силы Пруссии находились в соседней Йене, но принц Людвиг, неосведомленный, что подразделение маршала Ланна перед ним было всего лишь частью его сил (остальные подходили), думая, что превосходит его численностью, приготовился к бою. Принц поместил своих солдат на низину вне города, спиной к реке. Он приготовился к обороне от французов, которые атаковали с холма.
Ланн некоторое время вёл огонь артиллерией и когда противник показал признаки дезорганизации, он начал наступление своей пехотой, послав свои силы против фланга. Окружённый и превзойденный численностью французами, прусский фланг скоро начал ломаться и отступать под ударами французов.
Запоздало увидев свою ошибку, Людвиг лично повёл в атаку свою конницу и атаковал продвигающихся по флангу французов. Атака была отбита, и принц оказался в ближнем бою с противником и был убит сержантом-квартирмейстером 10-го гусарского полка Гуине.
Последствия
Спустя четыре дня после битвы при Заальфельде произошло двойное сражение при Йене и Ауэрштедте. Хотя война продолжалась в течение ещё семи месяцев, но тотальное поражение, понесённое прусской армией, привело к устранению Пруссии от антифранцузской коалиции вплоть до войны Шестой коалиции
Литература
Chandler, David G. (1994). The Campaigns of Napoleon. Weidenfeld & Nicolson.
Ссылки
Battle at Napoleonic Officers
The Memoirs of Baron de Marbot — Volume I
Order of Battle.
Сражения по алфавиту
Заальфельд
Сражения Франции
Сражения Пруссии
Сражения Саксонии
События 10 октября
Сражения 1806 года
Октябрь 1806 года
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,384
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\section{Introduction}
Lie point symmetries have been used in order to solve explicitly the
Einstein field equations, and to find new exact solutions in modified
theories of gravity (for instance see \cit
{LK,Gov,Cap,Dimakis,Yusuf,Maharaj1,Maharaj3} and references therein).
Furthermore, Lie point symmetries have also been used for the study of the
geodesic equations and for the determination of exact solutions of the wave
equation in various gravitation models \cite{FerozeT,GRG2,Camci,Bokh1,Azad}.
\ As far as concerns the wave equation in Riemannian spacetimes, a symmetry
analysis of wave equation in a power-law Bianchi III spacetime can be found
in \cite{Bokhari} and a symmetry analysis of the wave equation on static
spherically symmetric spacetimes, with higher symmetries, was carried out in
\cite{Tahir1}. Recently, in \cite{IJGMMP2}, we started a research program
where we performed the symmetry classification of the wave and the
Klein-Gordon equation in Bianchi I spacetimes. In this work we would like to
extend this analysis and perform a classification of the Lie and Noether
point symmetries for the wave equation and the Klein-Gordon equation in
pp-wave type N spacetimes.
The line element of a pp-wave spacetime is ($A,B=1,2)$ \cite{StephaniBook1
\begin{equation}
ds^{2}=-2dudv-2H\left( u,x^{A}\right) du^{2}+\delta _{AB}dx^{A}dx^{B}
\label{pp.01}
\end{equation
where $x^{A}=\left( y,z\right) $ are the Cartesian coordinates and $\delta
_{AB}=diag\left( 1,1\right) $ is the two dimensional Euclidian metric. The
non-zero connection coefficients of (\ref{pp.01}) ar
\begin{equation}
\Gamma _{uu}^{v}=H_{,u}~,~\Gamma _{uu}^{A}=H_{,A}~,~\Gamma _{Au}^{v}=H_{,A}.
\end{equation
The Laplace operator for the spacetime (\ref{pp.01}) is
\begin{equation}
\Delta \Psi \equiv -2\Psi _{,uv}+2H\left( u,x^{A}\right) \Psi _{,vv}+\Delta
_{\delta }\Psi ,
\end{equation
where $\Delta _{\delta }~$is the Laplacian of the two dimensional Euclidian
space. It follows that the Klein-Gordon equation in (\ref{pp.01}) has the
following for
\begin{equation}
-2\Psi _{,uv}+2H\left( u,x^{A}\right) \Psi _{,vv}+\Delta _{\delta }\Psi
+V\left( u,v,x^{A}\right) \Psi =0. \label{pp.02}
\end{equation}
The main property of a pp-wave spacetime (\ref{pp.01}) is that admits the
null Killing vector field $k=\partial _{v}$. However for special forms of
the function $H\left( u,x^{A}\right) $ (\ref{pp.01}) admits a greater
conformal algebra. The function $H\left( u,x^{A}\right) $ is computed from
the solution of Einstein field equations. For empty spacetime is given by
the equation $\Delta _{\delta }H=0$, which in Cartesian coordinates i
\[
H_{,yy}+H_{,zz}=0.
\
The classification of the Killing algebras of (\ref{pp.01}) has been done in
\cite{ppKV}, whereas in \cite{ppCKV} are given the conformal algebras of
\ref{pp.01}). In the following we use the classification of \cite{ppCKV},
which means that the results we find hold also for non empty spacetimes. \
The plan of the paper is as follows.
In Section \ref{collineations} we give basic definitions and properties of
Riemannian collineations and introduce the Lie and the Noether point
symmetries of differential equations. Furthermore, we discuss the relation
between the Lie and Noether symmetries of the Klein-Gordon equation with the
conformal algebra of the underlying Riemannian manifold. In Section \re
{isometryclass} we determine the functional form of the potential $V\left(
u,v,x^{A}\right) $ and the function $H\left( u,x^{A}\right) $ of (\ref{pp.01
), in order the Klein-Gordon equation (\ref{pp.02}) to admit Lie and Noether
point symmetries. The complete symmetry analysis for wave equation in the
pp-wave spacetime (\ref{pp.01}) is given in Section \ref{wave}. Finally, in
Section \ref{conclusion} we discuss our results and draw our conclusions.
\section{Collineations and symmetries of differential equations}
\label{collineations}
\subsection{Collineations of Riemannian manifolds}
Let a space with coordinates $\{x^{i}\}.$ Consider the one parameter point
transformatio
\begin{equation}
\bar{x}^{i}=x^{i}+\varepsilon \xi ^{i}\left( x^{k}\right) . \label{go.02}
\end{equation
in which $\xi ^{i}\left( x^{k}\right) $ are the components of a vector field
called the infinitesimal generator of (\ref{go.02}).
Let $\Omega $ be a geometric object in $V^{n}$ with transformation law
\Omega ^{a^{\prime }}=\Phi ^{a}\left( \Omega ^{k},x^{k},x^{k^{\prime
}}\right) $.$~\ $Under the action of the point transformation $\Omega $
changes to $\Phi \left( \Omega ^{k},x^{k},x^{k^{\prime }}\right) .$ We
define the Lie derivative $\mathcal{L}_{\xi }$ of $\Omega $ with respect to
the vector field $\xi $ as follows \cite{Yano}
\begin{equation}
\mathcal{L}_{\xi }\Omega =\lim_{\varepsilon \rightarrow 0}\frac{1}
\varepsilon }\left[ \Phi \left( \Omega ^{k},x^{k},x^{k^{\prime }}\right)
-\Omega \right] . \label{go.03}
\end{equation}
By definition the Lie derivative of a geometric object depends on its
transformation law. For functions, the transformation law is $F^{\prime
}\left( \bar{x}^{i}\right) =F\left( x^{i}\right) $, hence under the point
transformation (\ref{go.02}) we hav
\[
\bar{F}\left( \bar{x}^{i}\right) =F\left( x^{i}+\varepsilon \xi ^{i}\right)
=F\left( x^{i}\right) +\varepsilon F_{,i}\xi ^{i}+O\left( \varepsilon
^{2}\right) .
\]
Hence from (\ref{go.03}) it follows
\begin{equation}
\mathcal{L}_{\xi }F=F_{,i}\xi ^{i}. \label{go.04}
\end{equation
We say that the function $F\left( x^{i}\right) $ is invariant under the
action of (\ref{go.02}) if $\mathcal{L}_{\xi }F=0$; In this case $\xi $ is
called a symmetry of the function $F\left( x^{i}\right) .$
In general we hav
\begin{equation}
\mathcal{L}_{\xi }\Omega =\Lambda \label{go.05}
\end{equation
where $\Lambda $ is a tensor which has the same number and symmetries of the
indices with $\Omega $. We remark that $\Omega $ is not necessarily a
tensor. In this case we say that the vector field $\xi $ is a collineation
of $\Omega .$ The type of collineations depends on the tensor $\Lambda $.
In Riemannian Geometry (and in General Relativity) in general we are
interested on geometrical objects $\Omega $ which are defined in terms of
the metric\footnote
For the complete classification of the collineations of Riemannian manifold
see \cite{Katzin69,GSHall}.}. In particular in this work we shall consider
the geometric object $\Omega =$ $g_{ij},$ and $\Lambda =$ $2\psi \left(
x^{k}\right) g_{ij}$; that is, condition (\ref{go.05}) become
\begin{equation}
\mathcal{L}_{\xi }g_{ij}=2\psi \left( x^{k}\right) g_{ij}. \label{go.07}
\end{equation}
The vector field $\xi $ is called as conformal Killing vector (CKV)$.$ In
general $\psi \left( x^{k}\right) =\frac{1}{n}\xi _{;i}^{i}$, \ where $n$ is
the dimension of the Riemann space $V^{n}$ and $";"$ denotes the covariant
derivative with respect to the metric tensor $g_{ij}$. If $\psi _{;ij}=0$
the field $\xi $ is called special Conformal Killing vector (sp.CKV), when
\psi _{;i}=0$, i.e. $\psi \left( x^{k}\right) =\psi _{0}$, $\xi $ is called
Homothetic vector (HV) and when $\psi \left( x^{k}\right) =0$, $\xi $ is a
Killing vector (KV) of the metric tensor $g_{ij}$.
The CKVs\ of a metric form a Lie algebra and so do the KVs and the HV. If we
denote by $G_{CV},G_{HV}$, $G_{KV}$ these algebras we have the inclusion
relations
\begin{equation}
G_{KV}\subseteq G_{HV}\subseteq G_{CV} \label{go.08}
\end{equation
and~
\begin{equation}
0\leq \dim G_{H-K}\leq 1~ \label{go.09}
\end{equation
where $G_{H-K}=G_{HV}-G_{HV}\cap G_{KV};\ $the last relation means that a
Riemannian space admits at most one HV. Concerning the dimension of the
conformal algebra $G_{CV}$ we have $\dim G_{CV}\leq \frac{1}{2}\left(
n+1\right) \left( n+2\right) .$
Collineations constitute a strong constraint on the geometric structure of a
space. For example if a space admits $\frac{1}{2}n\left( n+1\right) $ then
it must be a space of constant curvature and there are only three types of
spaces with curvature $0,\pm K$ whose metric in Cartesian coordinates has
the general form
\begin{equation}
ds^{2}=\left( 1+\frac{K}{4}\eta _{ij}x^{i}x^{j}\right) ^{-1}\left( \eta
_{ij}dx^{i}dx^{j}\right) . \label{go.09a}
\end{equation}
\subsection{Lie point symmetries of differential equations}
Consider the second order partial differential equation $\Theta \left(
x^{k},\Psi ,\Psi _{,i},\Psi _{,ij}\right) =0$, where $x^{k}$ are the
independent variables, $\Psi =\Psi \left( x^{k}\right) $ is the dependent
variable and, $\Psi _{,i}=\frac{\partial \Psi }{\partial x^{i}}$. Latin
indices take the values $1,2,...,n$. \ Let $\left( x^{i},\Psi \right)
\rightarrow \left( \bar{x}^{i}\left( x^{k},\Psi ,\varepsilon \right) ,\bar
\Psi}\left( x^{k},\Psi ,\varepsilon \right) \right) ,$ be a one parameter
point transformation of the independent and dependent variables with
infinitesimal generator
\begin{equation}
X=\xi ^{i}\left( x^{k},\Psi \right) \partial _{i}+\eta \left( x^{k},\Psi
\right) \partial _{\Psi }. \label{go.10}
\end{equation}
The differential equation $\Theta $ can be seen as a geometric object on the
jet space $J=J\left( x^{k},\Psi ,\Psi _{,i},\Psi _{,ij}\right) .$ We say
that $X$ defines a Lie point symmetry of $\Theta \left( x^{k},\Psi ,\Psi
_{,i},\Psi _{,ij}\right) =0,$ if the following condition is satisfie
\begin{equation}
\mathcal{L}_{X^{\left[ 2\right] }}\Theta =\lambda \Theta , \label{go.12}
\end{equation
in which $X^{\left[ 2\right] },$ is the second extension/prolongation of $X$
in the space $J$, given by the formula
\begin{equation}
X^{\left[ 2\right] }=X+\eta _{i}\partial _{\Psi _{,i}}+\eta _{ij}\partial
_{\Psi _{,i}}, \label{go.13}
\end{equation
where,$~\eta _{i}=D_{i}\left( \eta \right) -\Psi _{,k}D_{i}\left( \xi
^{k}\right) $, $\eta _{ij}=D_{j}\left( \eta _{i}\right) -\Psi _{ki}D_{j}\xi
^{k}$, and $D_{i}$ is the operator of the total derivative, i.e. $D_{i}
\frac{\partial }{\partial x^{i}}+\Psi _{,i}\frac{\partial }{\partial \Psi
+\Psi _{,ij}\frac{\partial }{\partial \Psi _{,j}}+...~$\cite{Bluman,Ibrag}.
The existence of a Lie point symmetry for a partial differential equation
(PDE)\ means that there exist a "coordinate" system in which the
differential equation $\Theta $ is independent on one of the independent
variables. In addition, Lie point symmetries can be used in order to
transform solutions into solutions between different points of the space
\left( x^{i},\Psi \right) ~$\cite{Bluman3}.
If the differential equation follows form a Lagrangian $L=L\left( x^{k},\Psi
,\Psi _{,i}\right) $, that is $\Theta \equiv \mathbf{E}\left( L\right) =0$,
where $\mathbf{E}$ is the Euler-operator, then one defines a special type of
Lie symmetry by the condition
\begin{equation}
\mathcal{L}_{X^{\left[ 1\right] }}L+LD_{i}\xi ^{i}=D_{i}A^{i}, \label{go.14}
\end{equation
where $A^{i}$ is a vector field and $X^{\left[ 1\right] }$ \ is the first
prolongation of $X$. These Lie point symmetries are called Noether point
symmetries\footnote
In the literature the vector fields which satisfy condition (\ref{go.14})
have been called Noether Gauge symmetries. However, condition (\ref{go.14})
is that of the standard Noether's theorem \cite{Emmy} and the use of the
term Gauge is unnecessar, for instance see \cite{NG01,NG02,NG03}.} and have
the characteristic property that to the Lie symmetry $X$ there corresponds a
conserved current $I^{i}\left( x^{k},\Psi ,\Psi _{,i}\right) $, that is
D_{i}I^{i}=0$, where
\begin{equation}
I_{i}=\xi ^{j}\mathcal{H}_{ij}+\eta p_{i}-A_{i}. \label{go.15}
\end{equation
where $p^{i}=\frac{\partial L}{\partial \Psi _{,i}}$ and $\mathcal{H
_{~j}^{i}=\frac{\partial L}{\partial \Psi _{,i}}\Psi _{,j}-L$.
If (\ref{go.14}) holds, the Lie symmetry $X$ is called Noether symmetry and
the vector field $I^{i}$ Noether current. The Lie point symmetries of a PDE\
form a Lie algebra and the Noether point symmetries a subalgebra of this
algebra.
\subsection{Collineations of Riemannian spaces as point symmetries of the
Klein-Gordon equation}
One parameter point transformations in a Riemannian space define the
collineations in that space which \ characterize to a large extent the
geometry of the space. But one parameter point transformations define also
the Lie point symmetries of PDEs in that space. Therefore one should expect
that there exists a relation between the collineations of a space and the
Lie / Noether point symmetries of a PDE\ in that space. The reason for this
is twofold. First one may "see " the defining equation of a collineation as
a PDE\ in the space which remains invariant under the Lie point symmetry.
Second the dynamical field equations which describe the evolution of a
dynamical system the space should be affected by the geometry of the space.
This is most vividly seen in the case of the geodesic equations which on one
hand characterize the geometry of the space and on the other, in accordance
to the Principle of Equivalence, describe the equations of motion of a
"free" particle in the space.
Indeed it has been shown that Lie point symmetries of the geodesic equations
in a Riemannian space are elements of the special Projective algebra of the
space \cite{GRG2}, and that the Lie point symmetries form the projective
algebra of an extended manifold, for details see \cite{Aminova1,Aminova2}.
Similar results have been found for partial differential equations which
involve the metric tensor $g_{ij}$. \ Specifically, it has been shown that
the Lie point symmetries of the Schr\"{o}dinger equation are generated by
the Homothetic algebra of the space which defines the Laplace operator \cit
{IJGMMP}. Moreover, the Lie point symmetries of the wave and of the Poisson
equation are elements of the conformal algebra of the Riemannian manifold
\cite{Ibrag,Bozhkov}.
The Klein-Gordon equation in a general Riemannian space is defined as
follows
\begin{equation}
\Delta _{g}\Psi +V\left( x^{k}\right) \Psi =0, \label{kg.01}
\end{equation
where $\Delta _{g}=\frac{1}{\sqrt{\left\vert g\right\vert }}\frac{\partial }
\partial x^{i}}\left( \sqrt{\left\vert g\right\vert }g^{ij}\frac{\partial }
\partial x^{j}}\right) $, is the Laplace operator defined by the metric
tensor $g_{ij}$. Equation (\ref{kg.01}) arise from a variational principle
given by the following Lagrangian functio
\begin{equation}
L\left( x^{k},\Psi ,\Psi _{,k}\right) =\frac{1}{2}\sqrt{\left\vert
g\right\vert }g^{ij}\Psi _{,i}\Psi _{,j}-\frac{1}{2}\sqrt{\left\vert
g\right\vert }V\left( x^{k}\right) \Psi ^{2} \label{kg.02}
\end{equation}
In \cite{IJGMMP}, it has been shown that the Lie point symmetries for the
Klein-Gordon equation (\ref{kg.01}) ar
\begin{equation}
X=\xi ^{i}\left( x^{k}\right) \partial _{i}+\left( \frac{2-n}{2}\psi \left(
x^{k}\right) \Psi \right) \partial _{\Psi }~,~X_{\Psi }=\Psi \partial _{\Psi
}~,~X_{b}=b\left( x^{k}\right) \partial _{\Psi } \label{kg.03}
\end{equation
where $\xi ^{i}\left( x^{k}\right) $ is a CKV of $g_{ij}$ with conformal
factor $\psi \left( x^{k}\right) ,$ $b\left( x^{k}\right) $ is a solution of
the original equation (\ref{kg.01}) and the following condition holds
\begin{equation}
\xi ^{k}V_{,k}+2\psi V-\frac{2-n}{2}\Delta _{g}\psi =0 \label{kg.04}
\end{equation
where $n=\dim g_{ij}$. The two fields $X_{\Psi },X_{b}$ are called linear
symmetries, because they exist for a general linear partial equation. For
obvious reasons $X_{b}$ is called a solution symmetry.
Concerning the Noether point symmetries of (\ref{kg.01}) we have that for
the Lagrange function (\ref{kg.02}), the Lie point symmetries of (\ref{kg.01
) (except the two trivial ones) are also Noether point symmetries of (\re
{kg.02}) and that the field $A^{i}$ of condition (\ref{go.14}) has the
following form
\begin{equation}
A_{i}=\frac{2-n}{4}\sqrt{g}\psi _{,i}\left( x^{k}\right) u^{2}.
\label{kg.05}
\end{equation}
In the following we apply these results in order to classify the Lie and the
Noether point symmetries of the Klein-Gordon equation (\ref{kg.01}) and the
wave equation, $V\left( x^{k}\right) =0$, in pp-wave spacetimes.
\section{Lie and Noether point symmetries of the Klein-Gordon equation in
pp-wave spacetimes}
\label{isometryclass}
The pp-wave spacetimes have been classified according to the admitted
isometry algebra in \cite{ppKV}. The complete classification of the CKVs for
the pp-wave spacetimes has been done in \cite{ppCKV,ppnull}. However, as we
discussed above the Lie/Noether point symmetries of the Klein-Gordon
equation follow from the conformal algebra of the space which defines the
Laplace operator which means that in order to perform the classification
problem we follow the results of \cite{ppCKV,ppnull} in order to determine
all potentials $V\left( u,v,x^{k}\right) $ for which the resulting
Klein-Gordon equation (\ref{pp.02}) admits Lie and Noether point symmetries.
From \cite{ppKV} and \cite{ppCKV}, we have 14 isometry classes with some
subclasses, in which the spacetime (\ref{pp.01}) admits a greater conformal
algebra. We remark that equation (\ref{pp.02}) is a linear equation
therefore admits always the linear (trivial) symmetries $X_{\Psi },~X_{b}$.
\subsection{Isometry class 1}
This is the most general isometry class and $H=H\left( u,x^{A}\right) $ is
an arbitrary function. The spacetime (\ref{pp.01}) admits a one dimensional
conformal algebra given by the KV $k$. Hence we have that the Klein-Gordon
equation (\ref{pp.02}) admits the vector field $k$ as a Lie or Noether point
symmetry if and only if,
\begin{equation}
\mathcal{L}_{k}V=0,
\end{equation
from which follows that $V_{G}^{\left( 1\right) }=V\left( u,x,y\right) $.
\subsubsection{Subclass 1i}
If, $H=H\left( u,z\right) ,$ space (\ref{pp.01}) admits a three dimensional
conformal algebra spanned by the three KVs
\begin{equation}
k=\partial_{v}~,~X_{2}^{\left( 1i\right) }=\partial_{y}~,~X_{3}^{\left(
1i\right) }=y\partial _{v}+u\partial _{y}.
\end{equation
Therefore from conditions (\ref{kg.04}) we have that $X_{2}^{\left(
1i\right) },$ $X_{3}^{\left( 1i\right) }$ are Lie point symmetries whe
\begin{eqnarray}
X_{2}^{\left( 1i\right) } &:&V_{2}^{\left( 1i\right) }=V\left( u,v,z\right) ,
\\
X_{3}^{\left( 1i\right) } &:&V_{3}^{\left( 1i\right) }=V\left(
u,x,y^{2}-2uv\right) .
\end{eqnarray}
Furthermore from the linear combinations of the vector fields we have that
the vector $X_{\left( 1i\right) }=c_{1}k+c_{2}X_{2}^{\left( 1i\right)
}+c_{3}X_{3}^{\left( 1i\right) }$ is a Lie point symmetry of (\ref{pp.02})
whe
\begin{equation}
V_{G}^{\left( 1i\right) }=\left( u,x,\frac{2c_{1}y-2c_{2}v+c_{3}\left(
y^{2}-2uv\right) }{2\left( c_{2}+c_{3}u\right) }\right) .
\end{equation
The commutators of the Killing algebra are as follow
\begin{equation}
\left[ k,X_{2}^{\left( 1i\right) }\right] =0~,~\left[ k,X_{3}^{\left(
1i\right) }\right] =0~,~\left[ X_{2}^{\left( 1i\right) },X_{3}^{\left(
1i\right) }\right] =k
\end{equation}
\subsection{Isometry class 2}
When $H=H\left( u,r\right) ,$ the space admits a two dimensional conformal
algebra spanned by the KVs $k$ and $X_{2}^{\left( 2\right) }=\partial
_{\theta }$. Therefore we have that the field $X_{\left( 2\right)
}=c_{1}k+c_{2}X_{2}^{\left( 2\right) }$ is a Lie point symmetry of (\re
{pp.02}), if and only if,
\begin{equation}
V_{2}^{\left( 2\right) }=V\left( u,r,\theta -\frac{c_{2}}{c_{1}}v\right) .
\end{equation}
In this isometry class correspond four subclasses where the space (\re
{pp.01}) admits a greater conformal algebra. However in the fourth class the
exact form of the function $H\left( u,r\right) $ is not exact, hence we
study only the three subclasses.
\subsubsection{Subclass 2i}
Assume that $H=K\left( \alpha u+\beta \right) ^{q}\ln r,$ with $\beta \in
\mathbb{R}
$ and $K,\alpha \in
\mathbb{R}
^{\ast }$. In this case the space (\ref{pp.01}) admits an extra HV.
For $q\neq -1$ the HV vector is
\begin{equation}
H_{3}^{\left( 2i\right) }=\frac{2}{2\alpha +\alpha q}\left[ \left( \alpha
u+\beta \right) \partial _{u}+\left( \alpha \left( q+q\right) v-\frac{q+2}
2\left( q+1\right) }K\left( \alpha u+\beta \right) ^{q+1}\right) \partial
_{v}+\frac{2+q}{2}\alpha r\partial _{r}\right]
\end{equation
whereas for $q=-1$ the HV i
\begin{equation}
H_{3\left( -1\right) }^{\left( 2i\right) }=\frac{2}{\alpha }\left[ \left(
\alpha u+\beta \right) \partial _{u}-\frac{K}{2}\ln \left( \alpha u+\beta
\right) \partial _{v}+\alpha r\partial _{r}\right]
\end{equation
The corresponding factors $\psi _{H}^{\left( 2i\right) }~$and~$\psi
_{H}^{\left( 2i-1\right) }$ are equal to one.
We have that the vector fields $Y_{3}^{\left( 2i\right) }=H_{3}^{\left(
2i\right) }-\frac{1}{2}X_{\Psi }$, $Y_{3\left( -1\right) }^{\left( 2i\right)
}=H_{3\left( -1\right) }^{\left( 2i\right) }-\frac{1}{2}X_{\Psi }$ are Lie
point symmetries of (\ref{pp.02}) provided the potential has the for
\begin{equation}
Y_{3}^{\left( 2i\right) }:V_{3}^{\left( 2i\right) }=\left( \alpha u+\beta
\right) ^{-\left( q+2\right) }V\left( v\left( \alpha u+\beta \right)
^{-\left( q+1\right) }+\frac{q+2}{2\alpha \left( q+1\right) }K\ln \left(
\alpha u+\beta \right) ,r\left( \alpha u+\beta \right) ^{-1-\frac{q}{2
},\theta \right) ,
\end{equation
an
\begin{equation}
Y_{3\left( -1\right) }^{\left( 2i\right) }:V_{3\left( -1\right) }^{\left(
2i\right) }=\left( \alpha u+\beta \right) ^{-1}V\left( v+\frac{K}{4\alpha
\left( \ln \left( \alpha u+\beta \right) \right) ^{2},\frac{r}{\sqrt{\alpha
u+\beta }},\theta \right) .
\end{equation}
The generic fields $X_{\left( 2i\right) }=c_{1}k+c_{2}X_{2}^{\left( 2\right)
}+c_{3}H^{\left( 2i\right) }-\frac{1}{2}c_{3}X_{\Psi }$ and $X_{\left(
2i-1\right) }$\thinspace $=c_{1}k+c_{2}X_{2}^{\left( 2\right)
}+c_{3}H_{\left( -1\right) }^{\left( 2i\right) }-\frac{1}{2}c_{3}X_{\Psi }$
are Lie point symmetries when the potential is
\begin{equation}
X_{\left( 2i\right) }:V_{G}^{\left( 2i\right) }=\left( \alpha u+\beta
\right) ^{-\left( q+2\right) }V\left( f\left( u,v\right) ,r\left( \alpha
u+\beta \right) ^{-1-\frac{q}{2}},\theta -\frac{c_{2}\left( q+2\right) }
2c_{3}}\ln \left( \alpha u+\beta \right) \right) ,
\end{equation
o
\begin{equation}
X_{\left( 2i-1\right) }:V_{G\left( -1\right) }^{\left( 2i\right) }=\left(
\alpha u+\beta \right) ^{-1}V\left( v-\frac{c_{1}}{2c_{3}}\ln \left( \alpha
u+\beta \right) +\frac{K}{4\alpha }\left( \ln \left( \alpha u+\beta \right)
\right) ^{2},\frac{r}{\sqrt{\alpha u+\beta }},\theta -\frac{c_{2}}{2c_{3}
\ln \left( \alpha u+\beta \right) \right) ,
\end{equation
respectively; the function $f\left( u,v\right) $ is
\begin{equation}
f\left( u,v\right) =\left( v+\frac{c_{1}\left( q+2\right) }{2c_{3}\left(
q+1\right) }\right) \left( \alpha u+\beta \right) ^{-\left( q+1\right) }
\frac{q+2}{2\alpha \left( q+1\right) }K\ln \left( \alpha u+\beta \right) .
\end{equation}
\subsubsection{Subclass 2ii}
When $H=Ke^{-\frac{\Theta }{\beta }u}\ln r$, where$~\Theta \in
\mathbb{R}
$ and $\beta ,K\in
\mathbb{R}
^{\ast },$ the space (\ref{pp.01}) admits an extra HV. For $\Theta \neq 0$
the HV i
\begin{equation}
H_{3}^{\left( 2ii\right) }=-\frac{2\beta }{\Theta }\partial _{u}+\left( 2v
\frac{\beta }{\Theta }Ke^{-\frac{\Theta }{\beta }u}\right) \partial
_{v}+r\partial _{r}~,~\psi _{H}^{\left( 2ii\right) }=1,
\end{equation
and for $\Theta =0$ the HV i
\begin{equation}
H_{3\left( 0\right) }^{\left( 2ii\right) }=u\partial _{u}+\left( v-Ku\right)
\partial _{v}+r\partial _{r}~,~\psi _{H}^{\left( 2ii0\right) }=1.
\end{equation}
Hence, we have that $Y_{3}^{\left( 2ii\right) }=H^{\left( 2ii\right) }-\frac
1}{2}X_{\Psi },~Y_{3\left( 0\right) }^{\left( 2ii\right) }=H_{\left(
0\right) }^{\left( 2ii\right) }-\frac{1}{2}X_{\Psi }$ are Lie point
symmetries of (\ref{pp.02}) if and only if the potential is
\begin{eqnarray}
Y_{3}^{\left( 2ii\right) } &:&V_{3}^{\left( 2ii\right) }=V\left( ve^{\frac
\Theta }{\beta }u}+\frac{K}{2}u,re^{\frac{\Theta }{\beta }u},\theta \right)
e^{\frac{\Theta }{\beta }u}, \\
Y_{3\left( 0\right) }^{\left( 2ii\right) } &:&V_{3\left( 0\right) }^{\left(
2ii\right) }=V\left( vu^{-1}+K\ln u,ru^{-1},\theta \right) .
\end{eqnarray
The generic vector fields $X_{\left( 2ii\right) }=c_{1}k+c_{2}X_{2}^{\left(
2\right) }+c_{3}Y_{3}^{\left( 2ii\right) }$ and $X_{\left( 2ii0\right)
}=c_{1}k+c_{2}X_{2}^{\left( 2\right) }+c_{3}Y_{3\left( 0\right) }^{\left(
2ii\right) },$ are Lie point symmetries of the Klein Gordon equation if
\begin{eqnarray}
X_{\left( 2ii\right) } &:&V_{G}^{\left( 2ii\right) }=V\left( \left( v+\frac
c_{1}}{2c_{3}}\right) e^{\frac{\Theta }{\beta }u}+\frac{K}{2}u,re^{\frac
\Theta }{\beta }u},\theta -\frac{c_{2}}{2c_{3}}\frac{\Theta }{\beta
v\right) e^{\frac{\Theta }{\beta }u}, \\
X_{\left( 2ii0\right) } &:&V_{G}^{\left( 2ii0\right) }=u^{-2}V\left( \left(
v+\frac{c_{1}}{c_{3}}\right) u^{-1}+K\ln u,ru^{-1},\theta -\frac{c_{2}}{c_{3
}\ln u\right) .
\end{eqnarray}
\subsubsection{Subclass 2iii}
When $H=e^{g\left( u\right) }\ln r$ with $g\left( u\right) =-\ln \left( \rho
u^{2}+\alpha u+\beta \right) -\Theta \int \left( \rho u^{2}+\alpha u+\beta
\right) ^{-1}du$ and $\rho ,\alpha ,\beta \in
\mathbb{R}
^{\ast },~\Theta \in
\mathbb{R}
$ the space (\ref{pp.01}) admits a sp.CKV. For simplicity we study the case
\Theta =0$. The sp.CKV is
\begin{equation}
S_{3}^{\left( 2iii\right) }=2e^{-g\left( u\right) }\partial _{u}+\left( \rho
r^{2}+g\left( u\right) \right) \partial _{v}+\left( 2\rho u+\alpha \right)
r\partial _{r},
\end{equation
where the conformal factor is $\psi _{S}^{\left( 2iii\right) }=2\rho
u+\alpha ,$ with $\left( \psi _{S}^{\left( 2iii\right) }\right) _{;\mu \nu
}=0$.
Therefore from the sp.CKV $S_{3}^{\left( 2iii\right) },$ the generated
Lie/Noether point symmetry vector is $Y_{3}^{\left( 2iii\right)
}=S_{3}^{\left( 2iii\right) }-\frac{1}{2}\psi _{S}^{\left( 2iii\right)
}X_{\Psi }$ with corresponding potentia
\begin{equation}
V_{3}^{\left( 2iii\right) }=e^{g\left( u\right) }V\left( v-\frac{\rho }{2
r^{2}ue^{g\left( u\right) }-\int g\left( u\right) e^{g\left( u\right)
}du,re^{\frac{1}{2}g\left( u\right) },\theta \right) .
\end{equation}
Furthermore the vector field $Y_{\left( 2iii\right) }=c_{1}k+c_{2}X_{\left(
2\right) }^{2}+c_{3}Y_{3}^{\left( 2iii\right) }$ is a Lie point symmetry
vector of (\ref{pp.02}) i
\begin{equation}
V_{G}^{\left( 2iii\right) }=e^{-c_{3}g\left( u\right) }V\left( v-\frac{\rho
}{2}r^{2}ue^{g\left( u\right) }-\int e^{g\left( u\right) }\left( g\left(
u\right) +\frac{c_{1}}{c_{3}}\right) du,~re^{\frac{1}{2}g\left( u\right)
},~\theta -\frac{c_{2}}{c_{3}}\frac{\arctan \left( \frac{\psi _{S}^{\left(
2iii\right) }}{\sqrt{4\rho \beta -\alpha ^{2}}}\right) }{\sqrt{4\rho \beta
-\alpha ^{2}}}\right) .
\end{equation}
The commutators of the elements of the conformal algebras of the isometry
class 2 with the subclasses (2i)-(2iii) are given in table \ref{comClass2}.
\begin{table}[tbp] \centerin
\caption{Commutators of the conformal algebras of isometry class 2 of the
pp-wave spacetime (\ref{pp.01}) and of the subclasses (2i)-(2iii)
\begin{tabular}{c|ccccccc}
\hline\hline
$\left[ \mathbf{X}_{I},\mathbf{X}_{J}\right] $ & $k$ & $X_{2}^{\left(
2\right) }$ & $H_{3}^{\left( 2i\right) }$ & $H_{3\left( -1\right) }^{\left(
2i\right) }$ & $H_{3}^{\left( 2ii\right) }$ & $H_{3\left( 0\right) }^{\left(
2ii\right) }$ & $S_{3}^{\left( 2iii\right) }$ \\ \hline
$k$ & $0$ & $0$ & $2\frac{q+1}{q+2}k$ & $0$ & $2k$ & $k$ & $0$ \\
$X_{2}^{\left( 2\right) }$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
$H_{3}^{\left( 2i\right) }$ & $-2\frac{q+1}{q+2}k$ & $0$ & $0$ & & & &
\\
$H_{3\left( -1\right) }^{\left( 2i\right) }$ & $0$ & $0$ & & $0$ & & &
\\
$H_{3}^{\left( 2ii\right) }$ & $-2k$ & $0$ & & & $0$ & & \\
$H_{3\left( 0\right) }^{\left( 2ii\right) }$ & $-k$ & $0$ & & & & $0$ &
\\
$S_{3}^{\left( 2iii\right) }$ & $0$ & $0$ & & & & & $0$ \\ \hline\hline
\end{tabular
\label{comClass2
\end{table
\subsection{Isometry class 3}
In the isometry class 3, the function $H\left( u,x^{A}\right) $ is of the
form $H=u^{-2}W\left( s,t\right) $ where the coordinates $\left\{
s,t\right\} $ are
\begin{eqnarray}
s &=&y\sin \left( c\ln u\right) -z\cos \left( c\ln u\right) , \\
~t &=&y\cos \left( c\ln u\right) +z\sin \left( c\ln u\right) .
\end{eqnarray}
The pp-wave spacetime (\ref{pp.01}) admits two KVs, the fields $k$ and
X_{2}^{\left( 3\right) }=u\partial _{u}-v\partial _{v}$ with commutator
\left[ k,X_{2}^{\left( 3\right) }\right] =-k$. \ Hence the vector field
X_{3}=c_{1}k+c_{2}X_{2}^{\left( 3\right) }$ is a Lie point symmetry of (\re
{pp.02}) provided that
\begin{equation}
V_{G}^{\left( 3\right) }=V\left( uv-\frac{c_{1}}{c_{2}}u,s,t\right) .
\end{equation}
\subsection{Isometry class 4}
When $H=W\left( \bar{s},\bar{t}\right) $ wit
\begin{eqnarray}
\bar{s} &=&y\sin \left( cu\right) -z\cos \left( cu\right) \\
\bar{t} &=&y\cos \left( cu\right) +z\sin \left( cu\right)
\end{eqnarray
the pp-wave spacetime (\ref{pp.01}) admits a two dimensional conformal
algebra with elements the two KVs $k,~X_{2}^{\left( 4\right) }=\partial _{u}$
with commutator $\left[ k,X_{2}^{\left( 4\right) }\right] =0$. Therefore we
have that the field $X_{4}=c_{1}k+c_{2}X_{2}^{\left( 4\right) }$ is a Lie
point symmetry of (\ref{pp.02}), if and only if,
\begin{equation}
V_{G}^{\left( 4\right) }=V\left( v-\frac{c_{1}}{c_{2}}u,\bar{s},\bar{t
\right) .
\end{equation}
\subsection{Isometry class 5}
In this case the spacetime (\ref{pp.01}) admits a three dimensional Killing
algebra with commutator
\begin{equation}
\left[ k,X_{2}^{\left( 5\right) }\right] =0~,~\left[ X_{2}^{\left( 5\right)
},X_{3}^{\left( 5\right) }\right] =0~,~\left[ k,X_{3}^{\left( 5\right)
\right] =-k,
\end{equation
where $X_{2}^{\left( 5\right) }=\partial _{\theta }$, $X_{3}^{\left(
5\right) }=u\partial _{u}-v\partial _{v}$ and $H=u^{-2}W\left( r\right) $.
Therefore, the generic vector field $X_{5}=c_{1}k+c_{2}X_{2}^{\left(
5\right) }+c_{3}X_{3}^{\left( 5\right) }$ is a Lie point symmetry of (\re
{pp.02}), if and only i
\begin{equation}
V_{G}^{\left( 5\right) }=V\left( v-\frac{c_{1}}{c_{3}}u,~r~,~\theta -\frac
c_{2}}{c_{3}}\ln \left( u\right) \right) .
\end{equation}
Moreover, for the subclasses (5i) with $W\left( r\right) =\zeta \ln r,$ and
(5ii) with $W\left( r\right) =\left( \delta r^{-\sigma }-\sigma \left(
2-\sigma \right) ^{-2}r^{2}\right) ,~\left\vert \sigma \right\vert \neq 0,2
, the spacetime (\ref{pp.01}) admits a four dimensional conformal algebra.
\subsubsection{Subclass 5i}
When $H=\zeta u^{-2}\ln r$ the spacetime (\ref{pp.01}) admits the extra
sp.CKV
\begin{equation}
S_{4}^{\left( 5i\right) }=u^{2}\partial _{u}+\left( \frac{r^{2}}{2}-\zeta
\ln u\right) \partial _{v}+ur\partial _{r}~,~\psi _{4}^{\left( 5i\right) }=u.
\end{equation
Hence the field $Y_{4}^{\left( 5i\right) }=S_{4}^{\left( 5i\right) }-\frac{
}{2}uX_{\Psi }$ is a Lie point symmetry of (\ref{pp.02}) provide
\begin{equation}
V_{4}^{\left( 5i\right) }=u^{-2}V\left( ru^{-1},v-\frac{r^{2}+2\zeta \left(
1+\ln u\right) }{2u},\theta \right) .
\end{equation}
Similarly the field $X_{\left( 5i\right) }=c_{1}k+c_{2}X_{2}^{\left(
5\right) }+c_{3}X_{3}^{\left( 5\right) }+c_{4}Y_{4}^{\left( 5i\right) }$ is
a Lie point symmetry of (\ref{pp.02}) whe
\begin{equation}
V_{G}^{\left( 5i\right) }=\left( c_{3}+c_{4}u\right) ^{-2}V\left( \frac{r}
c_{3}+c_{4}u},F\left( u,r,v\right) ,\theta -\frac{c_{2}}{c_{3}}\ln \left(
\frac{u}{c_{3}+c_{4}u}\right) \right) ,
\end{equation
where
\begin{equation}
F\left( u,r,v\right) =\frac{c_{1}+c_{4}uv}{c_{4}\left( c_{3}+c_{4}u\right) }
\frac{c_{4}ur^{2}}{\left( c_{3}+c_{4}u\right) ^{2}}-\frac{\zeta }{c_{3}}\ln
\left( c_{3}+c_{4}u\right) +\frac{c_{4}}{c_{3}}\frac{\zeta \left( 1+u\ln
u\right) }{\left( c_{3}+c_{4}u\right) }.
\end{equation}
\subsubsection{Subclass 5ii}
Contrary to the subclass (5i), this subclass admits the proper CK
\begin{equation}
C_{4}^{\left( 5ii\right) }=u^{\frac{4}{2-\sigma }}\partial _{u}+\frac{\left(
\sigma +2\right) }{\left( \sigma -2\right) ^{2}}r^{2}u^{\frac{2\sigma }
2-\sigma }}\partial _{v}+\frac{2}{2-\sigma }ru^{\frac{\sigma +2}{2-\sigma
}\partial _{r},
\end{equation
with conformal factor $\psi _{4}^{\left( 5ii\right) }=\frac{2}{\sigma -2}u^
\frac{\sigma +2}{2-\sigma }}$. Since $C_{4}^{\left( 5ii\right) }$ is a not a
sp.CKV holds that $\left( \psi _{4}^{\left( 5ii\right) }\right) _{;ij}\neq 0
; however we note that $\Delta \left( \psi _{4}^{\left( 5ii\right) }\right)
=0$ which means that $\psi _{4}^{\left( 5ii\right) }$ is a solution of the
wave equation. Therefore, we have that the vector field $Y_{4}^{\left(
5ii\right) }=C_{4}^{\left( 5ii\right) }-\frac{1}{2}\psi _{4}^{\left(
5ii\right) }$ is a point symmetry of (\ref{pp.02}) when
\begin{equation}
V_{4}^{\left( 5ii\right) }=u^{\frac{4}{\sigma -2}}V\left( ru^{\frac{2}
\sigma -2}},v+\frac{r^{2}}{u\left( \sigma -2\right) },\theta \right) ,
\end{equation}
Furthermore, the field $Y_{\left( 5ii\right) }=c_{1}k+c_{2}X_{2}^{\left(
5\right) }+c_{3}X_{3}^{\left( 5\right) }+Y_{4}^{\left( 5ii\right) }$ is a
point symmetry of (\ref{pp.02}) whe
\begin{equation}
V_{G}^{\left( 5ii\right) }=\left( f_{1}\left( u\right) \right) ^{2}V\left(
rf_{1}\left( u\right) ,g\left( v,u,r\right) ,\theta -\frac{c_{2}}{c_{3}}\ln
f_{1}\left( u\right) \right) ,
\end{equation
wher
\begin{equation}
g\left( v,u,r\right) =v\left( f_{2}\left( u\right) \right) ^{\frac{\sigma -
}{\sigma +2}}-\frac{c_{1}}{c_{3}}\left( f_{2}\left( u\right) \right) ^{
\frac{4}{\sigma +2}}f_{2}\left( u\right) +c_{4}\frac{u^{\frac{4}{\sigma +2}
}{2-\sigma }\left( f_{1}\left( u\right) \right) ^{2}f_{3}\left( u\right)
r^{2},
\end{equation
an
\begin{equation}
f_{1}\left( u\right) =\left( c_{3}+c_{4}u^{\frac{\sigma +2}{2-\sigma
}\right) ^{-\frac{2}{\sigma +2}}~,~f_{2}\left( u\right) =\left(
c_{4}+c_{3}u^{\frac{\sigma +2}{2-\sigma }}\right) ,
\end{equation
\begin{equation}
f_{3}\left( u\right) =\int \left( f_{1}\left( u\right) \right) ^{2}\left(
f_{2}\left( u\right) \right) ^{-\frac{4}{2+\sigma }}u^{-\frac{2\left( \sigma
+4\right) }{2+\sigma }}du.
\end{equation}
The commutators of the elements of the conformal algebras of the isometry
class 5 with the subclasses (5i) and (5ii) are given in table \ref{comClass5
.
\begin{table}[tbp] \centerin
\caption{Commutators of the conformal algebras of isometry class 5 of
pp-wave spacetime (\ref{pp.01}) and of the subclasses (5i), (5ii)
\begin{tabular}{c|ccccc}
\hline\hline
$\left[ \mathbf{X}_{I},\mathbf{X}_{J}\right] $ & $k$ & $X_{2}^{\left(
5\right) }$ & $X_{3}^{\left( 5\right) }$ & $S_{4}^{\left( 5i\right) }$ &
C_{4}^{\left( 5ii\right) }$ \\ \hline
$k$ & $0$ & $0$ & $-k$ & $0$ & $0$ \\
$X_{2}^{\left( 5\right) }$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
$X_{3}^{\left( 5\right) }$ & $k$ & $0$ & $0$ & $S_{4}^{\left( 5i\right)
}-\zeta k$ & $-\frac{4}{\left( \sigma -2\right) }C_{4}^{\left( 5ii\right) }$
\\
$S_{4}^{\left( 5i\right) }$ & $0$ & $0$ & $-S_{4}^{\left( 5i\right) }+\zeta
k $ & $0$ & \\
$C_{4}^{\left( 5ii\right) }$ & $0$ & $0$ & $\frac{4}{\left( \sigma -2\right)
}C_{4}^{\left( 5ii\right) }$ & & $0$ \\ \hline\hline
\end{tabular
\label{comClass5
\end{table
\subsection{Isometry class 6}
In the isometry class 6, the spacetime (\ref{pp.01}) admits three KVs, the
field $k$ and the two vector fields $X_{2}^{\left( 6\right) }=\partial
_{\theta }~,~X_{3}^{\left( 6\right) }=\partial _{u},$ where $H\left(
u,x^{A}\right) =W\left( r\right) $.
Hence, we have that the generic field $X_{6}=c_{1}k+c_{2}X_{2}^{\left(
6\right) }+c_{3}X_{3}^{\left( 6\right) },$ is a Lie point symmetry of (\re
{pp.02}), i
\begin{equation}
V_{G}^{\left( 6\right) }=V\left( v-\frac{c_{1}}{c_{3}}u,r,\theta -\frac{c_{2
}{c_{3}}u\right) .
\end{equation}
For special functions $H\left( r\right) $ the spacetime (\ref{pp.01}) admits
a greater conformal algebra. There exist four possible subclasses in which
the pp-wave spacetime admits a greater conformal algebra.
\subsubsection{Subclass 6i}
When $W\left( r\right) =\frac{N}{4}r^{2}+\delta r^{-2}$, the spacetime (\re
{pp.01}) admits two proper non sp.CKVs, the field
\begin{eqnarray}
C_{4}^{\left( 6i\right) } &=&\sin \left( \sqrt{2N}u\right) \left( \partial
_{u}-\frac{Nr^{2}}{2}\partial _{v}\right) +\frac{\sqrt{2N}}{2}r\cos \left(
\sqrt{2N}u\right) \partial _{r}, \\
C_{5}^{\left( 6i\right) } &=&\cos \left( \sqrt{2N}u\right) \left( \partial
_{u}-\frac{Nr^{2}}{2}\partial _{v}\right) -\frac{\sqrt{2N}}{2}r\sin \left(
\sqrt{2N}u\right) \partial _{r},
\end{eqnarray
with conformal factors $\psi _{4}^{\left( 6i\right) }=\frac{\sqrt{2N}}{2
\cos \left( \sqrt{2N}u\right) $ and $\psi _{5}^{\left( 6i\right) }=-\frac
\sqrt{2N}}{2}\sin \left( \sqrt{2N}u\right) $ which satisfy the wave
equation, i.e. $\Delta \psi _{4-5}^{\left( 6i\right) }=0$.
Therefore, from the fields $C_{4}^{\left( 6i\right) },$ $C_{5}^{\left(
6i\right) }$ we have the possible point symmetries $Y_{4}^{\left( 6i\right)
}=C_{4}^{\left( 6i\right) }-\frac{1}{2}\psi _{4}^{\left( 6i\right) }X_{\Psi
} $ and $Y_{5}^{\left( 6i\right) }=C_{5}^{\left( 6i\right) }-\frac{1}{2}\psi
_{5}^{\left( 6i\right) }X_{\Psi }$ for the Klein-Gordon equation (\ref{pp.02
) provided the potential has the following form
\[
Y_{4}^{\left( 6i\right) }:V_{4}^{\left( 6i\right) }=\sin \left( \sqrt{2N
u\right) V\left( r^{2}\sin \left( \sqrt{2N}u\right) ,v-\frac{\sqrt{2N}}{4
r^{2}\cot \left( \sqrt{2N}u\right) ,\theta \right) ,
\
\[
Y_{5}^{\left( 6i\right) }:V_{5}^{\left( 6i\right) }=\cos \left( \sqrt{2N
u\right) V\left( r^{2}\cos \left( \sqrt{2N}u\right) ,v+\frac{\sqrt{2N}}{4
r^{2}\tan \left( \sqrt{2N}u\right) ,\theta \right) .
\]
Moreover, if the general vector field $X_{\left( 6i\right)
}=X_{6}+c_{4}Y_{4}^{\left( 6i\right) }+c_{5}Y_{5}^{\left( 6i\right) }$ is a
point symmetry of (\ref{pp.02}), the
\[
V_{G}^{\left( 6i\right) }=V\left( g_{1},g_{2},g_{3}\right) ,
\
wher
\begin{equation}
g_{1}=r^{2}f_{1}\left( u\right) ~,~g_{3}=\theta -c_{2}f_{2}\left( u\right) ,
\end{equation
\[
g_{2}=2v+\sqrt{2N}g_{1}\left( \frac{1}{2}c_{5}\sin \left( \sqrt{2N}u\right)
-c_{4}\cos ^{2}\left( \frac{\sqrt{2N}}{2}u\right) \right) -2c_{1}f_{2}\left(
u\right) ,
\
an
\begin{eqnarray}
f_{1}\left( u\right) &=&\left( c_{3}+c_{4}\sin \left( \sqrt{2N}u\right)
+c_{5}\cos \left( \sqrt{2N}u\right) \right) ^{-1} \\
f_{2}\left( u\right) &=&\frac{\sqrt{2}}{\sqrt{N}\sqrt{\left( c_{3}\right)
^{2}-\left( c_{4}\right) ^{2}-\left( c_{5}\right) ^{2}}}\arctan \left( \frac
\left( c_{3}-c_{5}\right) \tan \left( \frac{\sqrt{2N}}{2}u\right) +c_{4}}
\sqrt{\left( c_{3}\right) ^{2}-\left( c_{4}\right) ^{2}-\left( c_{5}\right)
^{2}}}\right) .
\end{eqnarray}
\subsubsection{Subclass 6ii}
When $N=0$, i.e. $W\left( r\right) =\delta r^{-2}$, the spacetime (\re
{pp.01}) admits a HV and a proper sp.CKV. These fields are respectively
\begin{equation}
H_{4}^{\left( 6ii\right) }=2u\partial _{u}+r\partial _{r}~,~\psi
_{4}^{\left( 6ii\right) }=1,
\end{equation
\begin{equation}
S_{5}^{\left( 6ii\right) }=u^{2}\partial _{u}+\frac{r^{2}}{2}\partial
_{v}+ru\partial _{r}~,~\psi _{5}^{\left( 6ii\right) }=\frac{u}{2}.
\end{equation
Note that $\left( \psi _{5}^{\left( 6ii\right) }\right) _{;\mu \nu }=0$.
Therefore from the fields $H_{4}^{\left( 6ii\right) }$ and $S_{5}^{\left(
6ii\right) }$ we have the possible Lie point symmetries of (\ref{pp.02})
Y_{4}^{\left( 6ii\right) }=H_{4}^{\left( 6ii\right) }-\frac{1}{2}X_{\Psi }$
and $Y_{5}^{\left( 6ii\right) }=S_{5}^{\left( 6ii\right) }-\frac{1}{2}\psi
_{5}^{\left( 6ii\right) }X_{\Psi }$ respectively.
This means that the generic point symmetry vector of (\ref{pp.02}) is
X_{\left( 6ii\right) }=X_{\left( 6\right) }+c_{4}Y_{4}^{\left( 6ii\right)
}+c_{5}Y_{5}^{\left( 6ii\right) }$ provided the potential has the for
\begin{equation}
V_{G}^{\left( 6ii\right) }=f_{1}\left( u\right) V\left( r^{2}f_{1}\left(
u\right) ,v-\frac{c_{5}}{2}r^{2}f_{1}\left( u\right) -c_{1}f_{2}\left(
u\right) ,\theta -c_{2}f_{2}\right) ,
\end{equation
where
\begin{equation}
f_{1}\left( u\right) =\left( c_{3}+2c_{4}u+c_{5}u^{2}\right)
^{-1}~~,~f_{2}\left( u\right) =\frac{\arctan \left( \frac{c_{4}+c_{5}u}
\sqrt{c_{3}c_{5}-\left( c_{4}\right) ^{2}}}\right) }{\sqrt{c_{3}c_{5}-\left(
c_{4}\right) ^{2}}}.
\end{equation}
\subsubsection{Subclass 6iii}
When $W\left( r\right) =\zeta \ln r$, the spacetime (\ref{pp.01}) admits the
extra H
\begin{equation}
H_{4}^{\left( 6iii\right) }=u\partial _{u}+\left( v-\zeta u\right) \partial
_{v}+r\partial _{r}~,~\psi _{4}^{\left( 6i\right) }=1.
\end{equation
It follows that $Y_{4}^{\left( 6iii\right) }=H_{4}^{\left( 6iii\right) }
\frac{1}{2}X_{\Psi }$ is a Lie point symmetry of (\ref{pp.02}) when
\begin{equation}
V=\frac{1}{u^{2}}V\left( vu^{-1}+\zeta \ln u,ru^{-1},\theta \right) .
\end{equation}
Moreover, the vector field $X_{\left( 6iii\right) }=X_{6}+c_{4}Y_{4}^{\left(
6iii\right) }$ is a Lie and Noether point symmetry of (\ref{pp.02}) if and
only if the potential has the following form
\begin{equation}
V_{G}^{\left( 6iii\right) }=V\left( \frac{c_{4}v+c_{1}+c_{3}\zeta }
c_{4}\left( c_{3}+c_{4}u\right) }-\frac{\zeta }{c_{4}}\ln \left(
c_{3}+c_{4}u\right) ,\frac{r}{c_{3}+c_{4}u},\theta -\frac{c_{2}}{c_{4}}\ln
\left( c_{3}+c_{4}u\right) \right) .
\end{equation}
\subsubsection{Subclass 6iv}
When $W\left( r\right) =\delta r^{-s}$ with $s\neq 2,0$, $\ $the spacetime
\ref{pp.01}) admits the extra H
\begin{equation}
H_{4}^{\left( 6iv\right) }=\frac{2+\sigma }{2}u\partial _{u}+\frac{2-\sigma
}{2}v\partial _{v}+r\partial _{r}~,~\psi _{4}^{\left( 6iv\right) }.
\end{equation}
Hence, the vector field $X_{\left( 6iv\right) }=X_{6}+c_{4}\left(
H_{5}^{\left( 6iv\right) }-\frac{1}{2}X_{\Psi }\right) $ is a Lie and
Noether point symmetry of (\ref{pp.02}) when
\begin{equation}
V_{G}^{\left( 6iv\right) }=\left( f_{1}\left( u\right) \right) \left( \left(
v+2\frac{c_{1}}{c_{3}\left( 2-\sigma \right) }\right) ^{2}f_{1}\left(
u\right) ,r^{2}f_{1}\left( u\right) ,2\theta +c_{2}\ln f_{1}\left( u\right)
\right) .
\end{equation
where $f_{1}\left( u\right) =\left( 2c_{3}+c_{4}u\left( 2+\sigma \right)
u\right) ^{-\frac{4}{2+\sigma }}$.
In table \ref{comClass6}, we give the commutators of the vector fields which
form the conformal algebras of the isometry class 6 and the subclasses
(6i)-(6iv).
\begin{table}[tbp] \centerin
\caption{Commutators of the conformal algebras of isometry class 6 of
spacetime (\ref{pp.01}) and of the subclasses (6i)-(6iv)
\begin{tabular}{c|ccccccccc}
\hline\hline
$\left[ \mathbf{X}_{I},\mathbf{X}_{J}\right] $ & $k$ & $X_{2}^{\left(
6\right) }$ & $X_{3}^{\left( 6\right) }$ & $C_{4}^{\left( 6i\right) }$ &
C_{5}^{\left( 6i\right) }$ & $H_{4}^{\left( 6ii\right) }$ & $S_{5}^{\left(
6ii\right) }$ & $H_{4}^{\left( 6iii\right) }$ & $H_{4}^{\left( 6iv\right) }$
\\ \hline
$k$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $k$ & $\left( 1-\frac{\sigma
}{2}\right) k$ \\
$X_{2}^{\left( 6\right) }$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ &
$0$ \\
$X_{3}^{\left( 6\right) }$ & $0$ & $0$ & $0$ & $C_{5}^{\left( 6i\right) }$ &
$-C_{4}^{\left( 6i\right) }$ & $2X_{3}^{\left( 6\right) }$ & $H_{4}^{\left(
6ii\right) }$ & $X_{3}^{\left( 6\right) }-\zeta k$ & $\left( 1+\frac{\sigma
}{2}\right) X_{3}^{\left( 6\right) }$ \\
$C_{4}^{\left( 6i\right) }$ & $0$ & $0$ & $-C_{5}^{\left( 6i\right) }$ & $0$
& $-X_{3}^{\left( 6\right) }$ & & & & \\
$C_{5}^{\left( 6i\right) }$ & $0$ & $0$ & $C_{4}^{\left( 6i\right) }$ &
X_{3}^{\left( 6\right) }$ & $0$ & & & & \\
$H_{4}^{\left( 6ii\right) }$ & $0$ & $0$ & $-2X_{3}^{\left( 6\right) }$ & &
& $0$ & $2S_{6}^{\left( 6ii\right) }$ & & \\
$S_{5}^{\left( 6ii\right) }$ & $0$ & $0$ & $-H_{4}^{\left( 6ii\right) }$ &
& & $2S_{6}^{\left( 6ii\right) }$ & $0$ & & \\
$H_{4}^{\left( 6iii\right) }$ & $-k$ & $0$ & $-X_{3}^{\left( 6\right)
}+\zeta k$ & & & & & $0$ & \\
$H_{4}^{\left( 6iv\right) }$ & $\left( \frac{\sigma }{2}-1\right) k$ & $0$ &
$-\left( 1+\frac{\sigma }{2}\right) X_{3}^{\left( 6\right) }$ & & & & &
& $0$ \\ \hline\hline
\end{tabular
\label{comClass6
\end{table
\subsection{Isometry class 7}
In the isometry class 7 $H\left( u,x^{A}\right) =e^{2\omega \theta }W\left(
r\right) .$ For this $H\left( u,x^{A}\right) $ spacetime (\ref{pp.01})
admits as KVs the fields $k$ and
\begin{equation}
X_{2}^{\left( 7\right) }=\omega u\partial _{u}-\omega v\partial
_{v}-\partial _{\theta }~,~X_{3}^{\left( 7\right) }=\partial _{u}.
\end{equation
The commutators of the Killing algebra ar
\[
\left[ k,X_{2}^{\left( 7\right) }\right] =-\omega k~,~\left[ k,X_{3}^{\left(
7\right) }\right] =0~,~\left[ X_{2}^{\left( 7\right) },X_{3}^{\left(
7\right) }\right] =-\omega X_{3}^{\left( 7\right) }.
\]
The field $X_{2}^{\left( 7\right) }$ is a point symmetry of (\ref{pp.02})
when
\begin{equation}
V_{2}^{\left( 7\right) }=V\left( vu,r,\omega \theta +\ln u\right) .
\end{equation}
Moreover, the generic KV $\ X_{7}=c_{1}k+c_{2}X_{2}^{\left( 7\right)
}+c_{3}X_{3}^{\left( 7\right) }$, is a \ Lie symmetry for equation (\re
{pp.02}) if and only if
\begin{equation}
V_{G}^{\left( 7\right) }=V\left( v\left( c_{3}+c_{2}\omega u\right)
-c_{1}u,r,\omega \theta +\ln \left( c_{2}\omega u+c_{3}\right) \right) .
\end{equation}
We note, that there exist subclasses of the isometry class 7 in which the
spacetime (\ref{pp.01}) admits a greater conformal algebra; however, these
subclasses are not vacuum pp-wave spacetimes. We continue with the isometry
class 8.
\subsection{Isometry class 8}
Consider $H=\exp \left( 2\delta t^{\prime }\right) W\left( s^{\prime
}\right) $ where
\begin{equation}
t^{\prime }=\eta y+\sigma z~,~s^{\prime }=\eta z-\sigma y,
\end{equation
$\delta =-\frac{c}{\eta ^{2}+\sigma ^{2}}$ and $c,\eta ,\sigma \in
\mathbb{R}
-\left\{ \zeta ^{2}\equiv \eta ^{2}+\sigma ^{2}=0\right\} $.
For $\delta \neq 0$, the spacetime (\ref{pp.01}) admits the three KV
\begin{equation}
k,~X_{2}^{\left( 8\right) }=u\partial _{u}~,~X_{3}^{\left( 8\right) }=\delta
\left( u\partial _{u}-v\partial _{v}\right) -\partial _{t^{\prime }},
\end{equation
and for $\delta =0$ admits four KVs, the extra KV i
\[
X_{4}^{\left( 8\right) }=t^{\prime }\partial _{v}+\left( \eta ^{2}+\sigma
^{2}\right) u\partial _{t^{\prime }}.
\]
Therefore, for $\delta \neq 0$, the general form of the potential $V\left(
u,v,s^{\prime },t^{\prime }\right) $ for which the Klein-Gordon equation
admits as a Lie point symmetry the vector field~$X_{8}=c_{1}k+c_{2}X_{2}^
\left( 8\right) }+c_{3}X_{3}^{\left( 8\right) },$ is as follows
\begin{equation}
V_{G}^{\left( 8\right) }=V\left( c_{1}u+v\left( \delta c_{3}u-c_{2}\right)
,s^{\prime },\delta t^{\prime }+\ln \left( \delta c_{3}u-c_{2}\right)
\right) .
\end{equation}
Finally, the commutators of the Killing algebra are as follow
\[
\left[ k,X_{2}^{\left( 8\right) }\right] =0~,~\left[ k,X_{3}^{\left(
8\right) }\right] =\delta k,~\left[ X_{2}^{\left( 8\right) },X_{3}^{\left(
8\right) }\right] =-X_{2}^{\left( 8\right) }.
\]
Similarly, for $\delta =0$ we have that the vector field $X_{8}^{\left(
0\right) }=X_{8}+c_{3}X_{4}^{\left( 8\right) }$ is a Lie point symmetry of
\ref{pp.02}), if and only if
\begin{equation}
V_{G\left( 0\right) }^{\left( 8\right) }=V\left(
\begin{array}{c}
s^{\prime },c_{2}t^{\prime }-\left[ c_{3}+\frac{c_{4}\zeta ^{2}}{2}u\right]
u, \\
c_{4}t^{\prime }u-c_{2}v+u\left( c_{1}u-\frac{c_{3}c_{4}}{2c_{2}}u^{2}-\frac
c_{4}^{2}\zeta ^{2}}{3c_{2}}u^{2}\right
\end{array
\right) .
\end{equation}
\subsubsection{Subclass 8i}
When $\delta =0$ and $W\left( s\right) =Ks^{\gamma }$, $K,\gamma \in
\mathbb{R}
^{\ast }$ and $\gamma \neq \pm 2,1$, the spacetime\ (\ref{pp.01}) admits a
five dimensional homothetic algebra. The extra HV i
\begin{equation}
H_{5}^{\left( 8i\right) }=\left( 1-\frac{\gamma }{2}\right) u\partial
_{u}+\left( 1+\frac{\gamma }{2}\right) v\partial _{v}+s^{\prime }\partial
_{s^{\prime }}+t^{\prime }\partial _{t^{\prime },}
\end{equation
where $\psi _{5}^{\left( 8i\right) }=1$. Hence, the field $Y_{5}^{\left(
8i\right) }=H_{5}^{\left( 8i\right) }-\frac{1}{2}X_{\Psi }$ is a Lie point
symmetry of (\ref{pp.02}) whe
\begin{equation}
V_{5}^{\left( 8i\right) }=u^{\frac{4}{\gamma -2}}V\left( vu^{\frac{\gamma +
}{\gamma -2}},s^{\prime }u^{\frac{2}{\gamma -2}},t^{\prime }u^{\frac{2}
\gamma -2}}\right) ,~\gamma \neq 2.
\end{equation}
Finally the generic field $X_{\left( 8i\right) }=X_{8}^{\left( 0\right)
}+c_{5}Y_{5}^{\left( 8i\right) }$, is a Lie point symmetry for equation (\re
{pp.02})), whe
\begin{equation}
V_{G}^{\left( 8i\right) }=V\left( s^{\prime }\left( g_{1}\right) ^{\frac{2}
\gamma -2}},\left( g_{1}\right) ^{\frac{2}{\gamma -2}}\left[ \gamma
c_{5}\left( c_{5}t^{\prime }+c_{3}\right) +2c_{4}\left( \eta ^{2}+\gamma
^{2}\right) \left( c_{2}+c_{5}u\right) \right] ,\frac{\left( g_{1}\right) ^
\frac{\gamma +2}{\gamma -2}}g_{2}}{\gamma ^{2}\left( \gamma +2\right) \left(
c_{5}\right) ^{3}}\right) ,
\end{equation
where functions $g_{1},g_{2}$ ar
\begin{equation}
g_{1}=c_{5}\left( \gamma -1\right) u-2c_{2},
\end{equation
\begin{eqnarray}
g_{2} &=&2c_{1}c_{5}^{2}\gamma ^{2}+4c_{2}\left( c_{4}\right) ^{4}\zeta
^{2}+4\gamma c_{3}c_{4}c_{5}+ \nonumber \\
&&+2\left( \gamma +2\right) c_{5}\left[ c_{4}\left( c_{5}t^{\prime }\gamma
+c_{4}\zeta ^{2}u\right) +\left( c_{5}\right) ^{2}\gamma ^{2}v\right] .
\end{eqnarray}
We continue with the subclasses (8ii) and (8iii), where $\gamma =2$ and
\gamma =-2$ respectively. As we have discussed, when $\gamma =1$ the space
\ref{pp.01}) is flat, and we do not consider that case.
\subsubsection{Subclass 8ii}
When $\gamma =2$, i.e. $W\left( s\right) =Ks^{2}$, the pp-wave spacetime
\ref{pp.01}) admits a seven dimensional conformal algebra. In particular
admits six KVs and a HV. The KVs are the fields $k,~X_{2}^{\left( 8\right)
},~X_{3}^{\left( 8\right) },X_{4}^{\left( 8\right) }~$and
\begin{eqnarray}
X_{5}^{\left( 8ii\right) } &=&\frac{\sqrt{2K}}{\zeta }s^{\prime }\cos \left(
\sqrt{2K}\zeta u\right) \partial _{v}+\sin \left( \sqrt{2K}\zeta u\right)
\partial _{s^{\prime }}, \\
X_{6}^{\left( 8ii\right) } &=&\frac{\sqrt{2K}}{\zeta }s^{\prime }\sin \left(
\sqrt{2K}\zeta u\right) \partial _{v}-\cos \left( \sqrt{2K}\zeta u\right)
\partial _{s^{\prime }},
\end{eqnarray
where the HV is the $H_{5}^{\left( 8i\right) }$. \ Therefore, the fields
X_{5}^{\left( 8ii\right) }$ and $X_{6}^{\left( 8ii\right) }$ are Lie point
symmetries of (\ref{pp.02}) when
\begin{eqnarray}
X_{5}^{\left( 8ii\right) } &:&V_{5}^{\left( 8ii\right) }=V\left( u,s^{\prime
2}-\frac{\sqrt{2}\zeta }{\sqrt{K}}v\tan \left( \sqrt{2K}\zeta u\right)
,t^{\prime }\right) , \\
X_{6}^{\left( 8ii\right) } &:&V_{6}^{\left( 8ii\right) }=V\left( u,s^{\prime
2}+\frac{\sqrt{2}\zeta }{\sqrt{K}}v\cot \left( \sqrt{2K}\zeta u\right)
,t^{\prime }\right) .
\end{eqnarray}
We continue with the next subclass, where $\gamma =-2.$
\subsubsection{Subclass 8iii}
When $W\left( s\right) =Ks^{-2}$, the pp-wave spacetime (\ref{pp.01}) admits
four KVs, one HV and the sp.CK
\begin{equation}
S_{6}^{\left( 8iii\right) }=u^{2}\partial _{u}+\frac{s^{\prime 2}+t^{\prime
2}}{2\zeta ^{2}}\partial _{v}+us^{\prime }\partial _{s^{\prime }}+ut^{\prime
}\partial _{t^{\prime }},~\psi _{6}^{\left( 8iii\right) }=u.
\end{equation
Therefore, from the sp.CKV we have that vector field $Y_{6}^{\left(
8iii\right) }=S_{6}^{\left( 8iii\right) }-\frac{1}{2}\psi _{6}^{\left(
8iii\right) }X_{\Psi }$ is a Lie point symmetry of the Klein-Gordon equation
whe
\begin{equation}
V_{6}^{\left( 8iii\right) }=u^{2}V\left( v-\frac{s^{\prime 2}+t^{\prime 2}}
2\zeta ^{2}u},su^{-1},tu^{-1}\right) .
\end{equation}
The commutators of the elements of the conformal algebras of the isometry
class 8 for $\delta =0,~$and of the subclasses (8i)-(8iii) are given in
table \ref{comClass8}. We would like to remark that for the subclasses (8ii)
and (8iii), we did not give the form of the potential for which the
Klein-Gordon equation (\ref{pp.02}) admits as Lie point symmetry the generic
symmetry vector, which follows from the linear combination of the CKVs,
because in this case the potential has a complex functional form.
\begin{table}[tbp] \centerin
\caption{Commutators of the conformal algebras for the isometry class 8 for $\delta=0$ and of the subclasses (8i)-(8iii) for
the pp-wave spacetime (\ref{pp.01}).
\begin{tabular}{c|cccccccc}
\hline\hline
$\left[ \mathbf{X}_{I},\mathbf{X}_{J}\right] $ & $k$ & $X_{2}^{\left(
8\right) }$ & $X_{3}^{\left( 8\right) }$ & $X_{4}^{\left( 8\right) }$ &
H_{5}^{\left( 8i\right) }$ & $X_{5}^{\left( 8ii\right) }$ & $X_{6}^{\left(
8ii\right) }$ & $S_{6}^{\left( 8iii\right) }$ \\ \hline
$k$ & $0$ & $0$ & $0$ & $0$ & $\left( 1+\frac{\gamma }{2}\right) k$ & $0$ &
0$ & $0$ \\
$X_{2}^{\left( 8\right) }$ & $0$ & $0$ & $0$ & $\zeta ^{2}X_{3}^{\left(
8\right) }$ & $\left( 1-\frac{\gamma }{2}\right) X_{2}^{\left( 8\right) }$ &
$-\sqrt{2K}\zeta X_{6}^{\left( 8ii\right) }$ & $\sqrt{2K}\zeta X_{5}^{\left(
8ii\right) }$ & $H_{5}^{\left( 8i\right) }$ \\
$X_{3}^{\left( 8\right) }$ & $0$ & $0$ & $0$ & $k$ & $X_{3}^{\left( 8\right)
}$ & $0$ & $0$ & $\frac{X_{4}^{\left( 8\right) }}{\zeta ^{2}}$ \\
$X_{4}^{\left( 8\right) }$ & $0$ & $-\zeta ^{2}X_{3}^{\left( 8\right) }$ &
-k$ & $0$ & $\frac{\gamma X_{4}^{\left( 8\right) }}{2}$ & $0$ & $0$ & $0$ \\
$H_{5}^{\left( 8i\right) }$ & $-\left( 1+\frac{\gamma }{2}\right) k$ &
\left( \frac{\gamma }{2}-1\right) X_{2}^{\left( 8\right) }$ &
-X_{3}^{\left( 8\right) }$ & $\frac{-\gamma X_{4}^{8}}{2}$ & $0$ &
-X_{5}^{\left( 8ii\right) }$ & $-X_{6}^{\left( 8ii\right) }$ &
2S_{6}^{\left( 8iii\right) }$ \\
$X_{5}^{\left( 8ii\right) }$ & $0$ & $\sqrt{2K}\zeta X_{6}^{\left(
8ii\right) }$ & $0$ & $0$ & $X_{5}^{\left( 8ii\right) }$ & $0$ & $\frac
\sqrt{2K}}{\zeta }k$ & \\
$X_{6}^{\left( 8ii\right) }$ & $0$ & $-\sqrt{2K}\zeta X_{5}^{\left(
8ii\right) }$ & $0$ & $0$ & $X_{6}^{\left( 8ii\right) }$ & $-\frac{\sqrt{2K
}{\zeta }k$ & $0$ & \\
$S_{6}^{\left( 8iii\right) }$ & $0$ & $H_{5}^{\left( 8i\right) }$ & $-\frac
X_{4}^{\left( 8\right) }}{\zeta ^{2}}$ & $0$ & $-2S_{6}^{\left( 8iii\right)
} $ & & & $0$ \\ \hline\hline
\end{tabular
\label{comClass8
\end{table
\subsection{Isometry class 9}
In isometry class 9, $H=Ke^{\eta y-\sigma z}$ with $K,\eta ,\sigma ~\in
\mathbb{R}
-\left\{ \zeta ^{2}\equiv \eta ^{2}+\sigma ^{2}=0\right\} .$ In this class,
spacetime (\ref{pp.01}) admits a five dimensional Killing algebra. The\ KVs
are the field $k$ and
\[
X_{2}^{\left( 9\right) }=u\partial _{u},~X_{3}^{\left( 9\right) }=u\partial
_{u}-v\partial _{v}+\frac{1}{\sigma }\partial _{z}~,
\
\[
X_{4}^{\left( 9\right) }=\left( y+\frac{\eta }{\sigma }z\right) \partial
_{v}+u\partial _{y}+\frac{\eta }{\sigma }u\partial _{z}~,~X_{5}^{\left(
9\right) }=\partial _{y}+\frac{\eta }{\sigma }\partial _{z}.
\
The commutators of the Lie algebra are given in table \ref{comClass9}. In
table \ref{pot9} we give the form of the potential $V\left( u,v,x,y\right) $
for which any of the elements of the Killing algebra of (\ref{pp.01}) is a
point symmetry of (\ref{pp.02}).
\begin{table}[tbp] \centerin
\caption{Commutators of the Killing algebra for the isometry class 9 of
pp-wave spacetime (\ref{pp.01}).
\begin{tabular}{c|ccccc}
\hline\hline
$\left[ \mathbf{X}_{I},\mathbf{X}_{J}\right] $ & $k$ & $X_{2}^{\left(
9\right) }$ & $X_{3}^{\left( 9\right) }$ & $X_{4}^{\left( 9\right) }$ &
X_{5}^{\left( 9\right) }$ \\ \hline
$k$ & $0$ & $0$ & $-k$ & $0$ & $0$ \\
$X_{2}^{\left( 9\right) }$ & $0$ & $0$ & $X_{2}^{\left( 9\right) }$ &
X_{5}^{\left( 9\right) }$ & $0$ \\
$X_{3}^{\left( 9\right) }$ & $k$ & $-X_{2}^{\left( 9\right) }$ & $0$ &
\frac{\eta }{\sigma ^{2}}k+X_{4}$ & $0$ \\
$X_{4}^{\left( 9\right) }$ & $0$ & $-X_{5}^{\left( 9\right) }$ & $-\frac
\eta }{\sigma ^{2}}k-X_{4}$ & $0$ & $-\frac{\eta ^{2}+\sigma ^{2}}{\sigma
^{2}}k$ \\
$X_{5}^{\left( 9\right) }$ & $0$ & $0$ & $0$ & $\frac{\eta ^{2}+\sigma ^{2}}
\sigma ^{2}}k$ & $0$ \\ \hline\hline
\end{tabular
\label{comClass9
\end{table
\begin{table}[tbp] \centerin
\caption{Lie symmetries and potentials for the Klein-Gordon (\ref{pp.02}) in
isometry class 9.
\begin{tabular}{cc}
\hline\hline
\textbf{Lie Sym.} & $\mathbf{V}\left( u,v,y,z\right) $ \\ \hline
$k$ & $V\left( u,y,z\right) $ \\
$X_{2}^{\left( 9\right) }$ & $V\left( v,y,z\right) $ \\
$X_{3}^{\left( 9\right) }$ & $V\left( vu,y,ue^{-\sigma z}\right) $ \\
$X_{4}^{\left( 9\right) }$ & $V\left( u,z-\frac{\eta }{\sigma }y,2v\sigma
^{2}-\frac{\zeta ^{2}}{u}y-\frac{2\eta \left( \sigma z-\eta y\right) y}
2\sigma ^{2}u}\right) $ \\
$X_{5}^{\left( 9\right) }$ & $V\left( u,v,z-\frac{\eta }{\sigma }y\right) $
\\ \hline\hline
\end{tabular
\label{pot9
\end{table
Furthermore, the generic vector field $X_{9}=c_{1}k+c_{2}X_{2}^{\left(
9\right) }+...+c_{5}X_{5}^{\left( 9\right) }$ is a Lie point symmetry of
\ref{pp.02}) if and only if
\begin{equation}
V_{G}^{\left( 9\right) }=V\left( g_{1},g_{2},\bar{g}\left(
c_{3}u+c_{2}\right) ^{C_{3}}\right) ,
\end{equation
where $C_{3}=\frac{c_{2}c_{4}}{c_{3}\sigma ^{2}}\left[ \zeta
^{2}c_{2}c_{4}-\eta c_{3}\left( c_{3}+\eta c_{5}\right) -\sigma
^{2}c_{3}c_{5}\right] $,
\begin{equation}
g_{1}\left( u,y\right) =\exp \left( c_{3}\left( c_{3}y-c_{4}u\right) \right)
\left( c_{3}u+c_{2}\right) ^{c_{2}c_{4}-c_{3}c_{5}},
\end{equation
\begin{equation}
g_{3}\left( u,z\right) =\exp \left( c_{3}\left( c_{3}\sigma z-c_{4}\eta
u\right) \right) \left( c_{3}u+c_{2}\right) ^{\eta \left(
c_{2}c_{4}-c_{3}c_{5}\right) -\left( c_{3}\right) ^{2}},
\end{equation
an
\begin{eqnarray}
\bar{g}\left( u,v,y,z\right) &=&c_{2}v+\left( c_{3}v-c_{4}y-c_{1}\right) u
\frac{c_{4}c_{5}}{c_{3}}\left( \frac{\zeta ^{2}}{\sigma ^{2}}\left( u+\frac
c_{2}}{c_{3}}\right) \right) + \nonumber \\
&&+\frac{c_{2}c_{4}}{\sigma ^{2}\left( c_{3}\right) ^{2}}\left[ \eta
c_{3}-c_{4}\zeta ^{2}\left( u+c_{2}\right) \right] + \nonumber \\
&&+\frac{\left( c_{4}\right) ^{2}}{2\sigma ^{2}c_{3}}\zeta ^{2}u^{2}+\frac
c_{4}\eta \left( 1-\sigma z\right) u}{\sigma ^{2}}.
\end{eqnarray}
\subsection{Plane wave spacetime: Isometry class 10}
When the function $H\left( u,x^{A}\right) $ has the form
\begin{equation}
H\left( u,x^{A}\right) =\frac{1}{2}\left( A\left( u\right) y^{2}+C\left(
u\right) z^{2}\right) +B\left( u\right) yz, \label{pw.01}
\end{equation
the spacetime (\ref{pp.01}) is a plane wave spacetime.
The spacetime (\ref{pp.01}) with (\ref{pw.01}) is vacuum when $A\left(
u\right) +C\left( u\right) =0$. Moreover, admits a six dimensional
homothetic algebra. The four KVs are given by the vector field \cit
{ppKV,ppCKV,ppnull
\begin{equation}
X_{a}^{\left( 10\right) }=\left( y\dot{d}_{a}\left( u\right) +z\dot{e
_{a}\left( u\right) \right) \partial _{v}+d_{a}\left( u\right) \partial
_{y}+e_{a}\left( u\right) \partial _{z},
\end{equation
where $\dot{d}_{a}=\frac{d}{du}\left( d_{a}\right) ,$ and the functions
d_{a}\left( u\right) ,e_{a}\left( u\right) $ satisfy the following system of
equation
\begin{eqnarray}
\ddot{d}_{a}+Cd_{a}+Be_{a} &=&0, \\
\ddot{e}_{a}+Ae_{a}+Bd_{a} &=&0.
\end{eqnarray}
The fifth KV is the field $k,$ and the proper HV i
\begin{equation}
H_{6}^{\left( 10\right) }=2v\partial _{v}+y\partial _{y}+z\partial
_{z}~,~\psi _{6}^{\left( 10\right) }=0.
\end{equation}
Hence, we have that the fields $X_{a}^{\left( 10\right) }$ are Lie point
symmetries of (\ref{pp.02}), if and only i
\begin{equation}
V_{a}^{\left( 10\right) }=V\left( u,v-\frac{\dot{e}_{a}}{d_{a}}yz-\frac{y^{2
}{2d_{a}^{2}}\left( \dot{d}_{A}d-\dot{e}_{a}e_{a}\right) ,z-\frac{e_{a}}
d_{Aa}}y\right) .
\end{equation}
Moreover, from the HV, we have that the field $Y_{6}^{\left( 10\right)
}=H_{6}^{\left( 10\right) }-\frac{1}{2}X_{\Psi }$ is a Lie point symmetry of
(\ref{pp.02}) provide
\begin{equation}
V_{6}^{\left( 10\right) }=v^{-1}V\left( u,y^{2}v^{-1},z^{2}v^{-1}\right) .
\end{equation}
Finally, the generic field $X_{10}=c_{1}k+c_{a}X_{a}^{\left( 10\right)
}+c_{6}Y_{6}^{\left( 10\right) }$ is a Lie point symmetry of (\ref{pp.02})
whe
\begin{equation}
V_{G}^{\left( 10\right) }=\left( c_{A}d_{a}+c_{5}y\right) ^{-2}V\left(
u,g\left( v,u,y,z\right) ,\frac{c_{5}z+c_{a}e_{a}}{c_{5}\left(
c_{a}d_{a}+c_{5}y\right) }\right) ,
\end{equation
wher
\begin{equation}
g\left( v,u,y,z\right) =\frac{2c_{5}v+c_{1}}{\left( c_{a}d_{a}+c_{5}y\right)
^{2}}+c_{2}\frac{\dot{d}_{a}\left( c_{a}d_{a}+c_{5}y\right) +\dot{e
_{A}\left( c_{a}e_{a}+2c_{5}z\right) }{\left( c_{a}d_{a}+c_{5}y\right) ^{2}}.
\end{equation}
The commutators of the homothetic algebra are \cite{ppCKV
\begin{eqnarray}
\left[ k,X_{a}^{\left( 10\right) }\right] &=&0~,~\left[ k,H_{6}^{\left(
10\right) }\right] =2k~,~ \\
\left[ X_{a}^{\left( 10\right) },H_{6}^{\left( 10\right) }\right] &=&X_{a}~,
\left[ X_{a}^{\left( 10\right) },X_{b}^{\left( 10\right) }\right] =2Q_{\left[
ab\right] }k,
\end{eqnarray
where $Q_{ab}\ $are constants. Furthermore, there are three subclasses, for
special form of the functions $A\left( u\right) ,B\left( u\right) ,C\left(
u\right) $ of (\ref{pw.01}), for which the plane symmetric spacetime admits
a greater conformal algebra. In the following we consider the two subclasses
for which the spacetime (\ref{pp.01}) admits extra sp.CKV.
\subsubsection{Subclass 10i}
When the functions $A\left( u\right) ,B\left( u\right) $ and $C\left(
u\right) $ of (\ref{pw.01}) ar
\begin{eqnarray}
A\left( u\right) &=&K\left( u^{2}+\beta \right) ^{-2}\left( \sin \left( \phi
\left( u\right) \right) +\lambda \right) , \\
B\left( u\right) &=&K\left( u^{2}+\beta \right) ^{-2}\cos \left( \phi \left(
u\right) \right) , \\
C\left( u\right) &=&-K\left( u^{2}+\beta \right) ^{-2}\left( \sin \left(
\phi \left( u\right) \right) -\lambda \right) ,
\end{eqnarray
where $\phi \left( u\right) =2\gamma \int \frac{du}{u^{2}+\beta },~$the
spacetime (\ref{pp.01}) admits the extra sp.CK
\begin{equation}
S_{7}^{\left( 10i\right) }=\left( u^{2}+\beta \right) \partial _{u}+\frac{1}
2}\left( y^{2}+z^{2}\right) \partial _{v}+\left( uy+\gamma z\right) \partial
_{y}+\left( uz-\gamma z\right) \partial _{z},
\end{equation
where $\psi _{7}^{\left( 10i\right) }=u$. Since $S_{7}^{\left( 10i\right) }$
is a sp.CKV the $\left( \psi _{7}^{\left( 10i\right) }\right) _{;\mu \nu
}=0. $ Therefore we from $S_{7}^{\left( 10i\right) }$ we have that the
vector field $Y_{7}^{\left( 10i\right) }=S_{7}^{\left( 10i\right) }-\frac{1}
2}\psi _{7}^{\left( 10i\right) }X_{\Psi }$, is a point symmetry of (\re
{pp.02}) whe
\begin{equation}
V_{7}^{\left( 10i\right) }=\left( u^{2}+\beta \right) ^{-1}V\left( v-\frac
r^{2}u}{2\left( u^{2}+\beta \right) },\frac{r^{2}}{u^{2}+\beta },\frac
e^{2\theta }}{\left( u^{2}+\beta \right) ^{\gamma }}\right) .
\end{equation}
\subsubsection{Subclass 10ii}
When $A\left( u\right) =-\alpha \left( u^{2}+\beta \right) ^{-2}$,~$B\left(
u\right) =-b\left( u^{2}+\beta \right) ^{-2}$ and $C\left( u\right)
=-c\left( u^{2}+\beta \right) ^{-2}$, the extra sp.CKV of (\ref{pp.02}) i
\begin{equation}
S_{7}^{\left( 10ii\right) }=\left( u^{2}+\beta \right) \partial _{u}+\frac{
}{2}\left( y^{2}+z^{2}\right) \partial _{v}+uy\partial _{y}+uz\partial
_{z}~,~\psi _{7}^{\left( 10ii\right) }=u,
\end{equation
which is the field $S_{7}^{\left( 10i\right) }$ for $\gamma =0$. Hence, the
field $Y_{7}^{\left( 10ii\right) }=S_{7}^{\left( 10ii\right) }-\frac{1}{2
\psi _{7}^{\left( 10ii\right) }X_{\Psi }$ is a Lie point symmetry of
integral when the potential has the for
\begin{equation}
V_{7}^{\left( 10ii\right) }=\left( u^{2}+\beta \right) ^{-1}V\left( v-\frac
r^{2}u}{2\left( u^{2}+\beta \right) },\frac{r^{2}}{u^{2}+\beta },\theta
\right) .
\end{equation}
There are also four more isometry classes in which the plane wave spacetime
\ref{pp.01}) admits a seven dimensional homothetic algebra \cite{ppCKV},
where there exists a six dimensional subalgebra and it is the homothetic
algebra of isometry class 10.
For these isometry classes, in table\footnote
In the isometry class 12, $\phi =2\delta \ln u.$} \ref{subclasses11}, we
give the functional form of $A,B,C$, the extra KV and the form of the
potential for which the corresponding KV is a Lie point symmetry of the
Klein-Gordon equation (\ref{pp.02}).
\begin{table}[tbp] \centerin
\caption{Point symmetries and potentials for the Klein-Gordon equation
(\ref{pp.02}) in the isometry classes 11-14
\begin{tabular}{cccccc}
\hline\hline
\textbf{Class} & $\mathbf{A}\left( u\right) $ & $\mathbf{B}\left( u\right) $
& $\mathbf{C}\left( u\right) $ & \textbf{Extra KV} & \textbf{Potential} \\
\hline
\textbf{11} & $\alpha u^{-2}$ & $\beta u^{-2}$ & $\gamma u^{-2}$ &
X_{7}^{\left( 11\right) }=u\partial _{u}-v\partial _{v}$ & $V\left(
vu,y,z\right) $ \\
\textbf{12} & $-cu^{-2}\left( \sin \phi +\lambda \right) $ & $cu^{-2}\left(
\cos \phi \right) $ & $cu^{-2}\left( \sin \phi -\lambda \right) $ &
X_{7}^{\left( 12\right) }=X_{7}^{\left( 11\right) }+\delta \partial _{\theta
}$ & $V\left( vu,r,e^{\theta }u^{-\delta }\right) $ \\
\textbf{13} & $\alpha $ & $\beta $ & $c$ & $X_{7}^{\left( 13\right)
}=\partial _{u}$ & $V\left( v,y,z\right) $ \\
\textbf{14} & $-c\sin \left( 2\delta u\right) +\lambda $ & $-c\cos \left(
2\delta u\right) $ & $c\sin \left( 2\delta u\right) +\lambda $ &
X_{7}^{\left( 14\right) }=\partial _{u}$ & $V\left( u,r,e^{\theta
}u^{-\delta }\right) $ \\ \hline\hline
\end{tabular
\label{subclasses11
\end{table
\section{Symmetry classification for the Wave equation}
\label{wave}
When the potential in (\ref{pp.02}) vanishes the Klein Gordon equation
becomes
\begin{equation}
-2\Psi _{,uv}+2H\left( u,x^{A}\right) \Psi _{,vv}+\Delta _{\delta }\Psi =0,
\label{wave.01}
\end{equation
which is the wave equation in spacetime (\ref{pp.01}).
Contrary to the Klein-Gordon equation, a CKV of the metric which defines the
Laplace operator generates a Lie/Noether symmetry for the wave equation only
when the conformal factor is a solution of the original equation (see
condition (\ref{kg.04})). Therefore, the KVs, the HV and the sp.CKVs
generate always point symmetries for the wave equation. Furthermore, in
section \ref{isometryclass} we showed that when the pp-wave spacetime admits
a proper CKV, then the conformal factor is a solution of the original
equation, which means that the proper CKVs, when there exist, generate
always Lie and Noether point symmetries for the wave equation (\ref{wave.01
).
The Lie and Noether point symmetries of the wave equation (except the
trivial ones) for the isometry classes 1 to 14, of section \re
{isometryclass} are given in table \ref{WaveSym1}.
\begin{table}[tbp] \centerin
\caption{Lie and Noether point symmetries for the wave equation in the
pp-wave spacetime (\ref{pp.01}), for the isometry classes of \cite{ppKV}
\begin{tabular}{|c|c|c|c|c|c|}
\hline\hline
\textbf{Class} & $\mathbf{\#}$\textbf{\ } & \textbf{Lie/Noether Sym.} &
\textbf{Class} & $\#$ & \textbf{Lie/Noether Sym.} \\ \hline
\textbf{1} & $1$ & $k$ & \textbf{6iv} & $4$ & $k,~X_{2}^{\left( 6\right)
},~X_{3}^{\left( 6\right) },~Y_{4}^{\left( 6iv\right) }$ \\
\textbf{1i} & $2$ & $k,~X_{2}^{\left( 1i\right) }$ & \textbf{7} & $3$ &
k,~X_{2}^{\left( 7\right) },~X_{3}^{\left( 7\right) }$ \\
\textbf{2} & $2$ & $k,~X_{2}^{\left( 2\right) }$ & \textbf{8} & $3$ &
k,~X_{2}^{\left( 8\right) },~X_{3}^{\left( 8\right) }$ \\
\textbf{2i} & $3$ & $k,~X_{2}^{\left( 2\right) },~Y_{3}^{\left( 2i\right)
}/Y_{3\left( -1\right) }^{\left( 2i\right) }$ & \textbf{8}$_{\left( 0\right)
}$ & $4$ & $k,~X_{2}^{\left( 8\right) },~X_{3}^{\left( 8\right)
},~X_{4}^{\left( 8\right) }$ \\
\textbf{2ii} & $3$ & $k,~X_{2}^{\left( 2\right) },~Y_{3}^{\left( 2ii\right)
}/Y_{3\left( 0\right) }^{\left( 2ii\right) }$ & \textbf{8i} & $5$ &
k,~X_{2}^{\left( 8\right) },~X_{3}^{\left( 8\right) },~X_{4}^{\left(
8\right) },~Y_{5}^{\left( 8i\right) }$ \\
\textbf{2iii} & $3$ & $k,~X_{2}^{\left( 2\right) },~Y_{3}^{\left(
2iii\right) }$ & \textbf{8ii} & $6$ & $k,~X_{2}^{\left( 8\right)
},~X_{3}^{\left( 8\right) },~X_{4}^{\left( 8\right) },~X_{5}^{\left(
8ii\right) },~X_{6}^{\left( 8ii\right) },Y_{5}^{\left( 8i\right) }~$ \\
\textbf{3} & $2$ & $k,~X_{2}^{\left( 3\right) }$ & \textbf{8iii} & $6$ &
k,~X_{2}^{\left( 8\right) },~X_{3}^{\left( 8\right) },~X_{4}^{\left(
8\right) },~Y_{5}^{\left( 8i\right) },~Y_{6}^{\left( 8iii\right) }$ \\
\textbf{4} & $2$ & $k,~X_{2}^{\left( 4\right) }$ & \textbf{9} & $5$ &
k,~X_{2}^{\left( 9\right) },~X_{3}^{\left( 9\right) },~X_{4}^{\left(
9\right) },~X_{5}^{\left( 9\right) }$ \\
\textbf{5} & $3$ & $k,~X_{2}^{\left( 5\right) },~X_{3}^{\left( 5\right) }$ &
\textbf{10} & $6$ & $k,~X_{a}^{\left( 10\right) },~Y_{6}^{\left( 10\right) }$
\\
\textbf{5i} & $4$ & $k,~X_{2}^{\left( 5\right) },~X_{3}^{\left( 5\right)
},~Y_{4}^{\left( 5i\right) }$ & \textbf{10i} & $7$ & $k,~X_{a}^{\left(
10\right) },~H_{6}^{\left( 10\right) },~Y_{7}^{\left( 10i\right) }$ \\
\textbf{5ii} & $4$ & $k,~X_{2}^{\left( 5\right) },~X_{3}^{\left( 5\right)
},~Y_{4}^{\left( 5ii\right) }$ & \textbf{10ii} & $7$ & $k,~X_{a}^{\left(
10\right) },~H_{6}^{\left( 10\right) },~Y_{7}^{\left( 10ii\right) }$ \\
\textbf{6} & $3$ & $k,~X_{2}^{\left( 6\right) },~X_{3}^{\left( 6\right) }$ &
\textbf{11} & $7$ & $k,~X_{a}^{\left( 10\right) },~H_{6}^{\left( 10\right)
},~X_{7}^{\left( 11\right) }$ \\
\textbf{6i} & $5$ & $k,~X_{2}^{\left( 6\right) },~X_{3}^{\left( 6\right)
},~Y_{4}^{\left( 6i\right) },~Y_{5}^{\left( 6i\right) }$ & \textbf{12} & $7$
& $k,~X_{a}^{\left( 10\right) },~H_{6}^{\left( 10\right) },~X_{7}^{\left(
12\right) }$ \\
\textbf{6ii} & $5$ & $k,~X_{2}^{\left( 6\right) },~X_{3}^{\left( 6\right)
},~Y_{4}^{\left( 6ii\right) },~Y_{5}^{\left( 6ii\right) }$ & \textbf{13} &
7 $ & $k,~X_{a}^{\left( 10\right) },~H_{6}^{\left( 10\right)
},~X_{7}^{\left( 13\right) }$ \\
\textbf{6iii} & $4$ & $k,~X_{2}^{\left( 6\right) },~X_{3}^{\left( 6\right)
},~Y_{4}^{\left( 6iii\right) }$ & \textbf{14} & $7$ & $k,~X_{a}^{\left(
10\right) },~H_{6}^{\left( 10\right) },~X_{7}^{\left( 14\right) }$ \\
\hline\hline
\end{tabular
\label{WaveSym1
\end{table
\section{Conclusions}
\label{conclusion}
In this work we performed a complete classification of the Lie /Noether
point symmetries for the Klein-Gordon and the wave equation in pp-wave
spacetimes using three results: (a) The general results of \cite{IJGMMP} and
\cite{IJGMMP2} concerning the relation between the Lie / Noether point
symmetries of the Klein-Gordon equation with the conformal algebra of the
underlying space; (b) the classification of the Klein Gordon equation based
on the isometries of (\ref{pp.01}) done in \cite{ppKV}, and (c) The
classification of the conformal algebra of the pp-wave spacetimes (\re
{pp.01}) done in \cite{ppCKV} and \cite{ppnull}.
In addition we used these results in order to calculate the Lie and the
Noether point symmetries of the wave equation (\ref{wave.01}) in a pp-wave
spacetime. \ We found that the Lie point symmetries form a Lie algebra
G_{W}~$(except the trivial symmetries), of dimension $\dim G_{W}\leq 7$
where the equality holds for the case where the space (\ref{pp.01}) is a
plane wave spacetime. In addition we noted that due to the fact that the
conformal factors of the CKVs of (\ref{pp.01}) are solutions of wave
equation (\ref{wave.01}) all CKVs of (\ref{pp.01}) give rise to a Lie point
symmetry give a Lie point symmetry of (\ref{wave.01}). Because we have
followed the classification of \cite{ppCKV}, the symmetry classification
holds and for non-empty spacetimes.\
A further use of the results obtained in this work is that they can be used
in order one to reduce and possibly to solve analytically the Klein-Gordon
equation (\ref{pp.02}) and wave equation (\ref{wave.01}) in a pp-wave
spacetime.
{\large {\textbf{Acknowledgements}}} \newline
The research of AP was supported by FONDECYT postdoctoral grant no. 3160121.
|
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"redpajama_set_name": "RedPajamaArXiv"
}
| 8,891
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\section{Introduction}\label{sec:introduction}
Different from the previous graph neural network (GNN) works \cite{zhang2020graph,Velickovic_Graph_ICLR_18,Kipf_Semi_CORR_16}, which mainly focus on the node embeddings in large-sized graph data, we will study the representation learning of the whole graph instances in this paper. Representative examples of graph instance data studied in research include the human brain graph, molecular graph, and real-estate community graph, whose nodes (usually only in tens or hundreds) represent the brain regions, atoms and POIs, respectively. Graph instance representation learning has been demonstrated to be an extremely difficult task. Besides the diverse input graph instance sizes, attributes and extensive connections, the inherent node-orderless property \cite{Meng_Isomorphic_NIPS_19} brings about more challenges on the model design.
Existing graph representation learning approaches are mainly based on the convolutional operator \cite{NIPS2012_4824} or the approximated graph convolutional operator \cite{Hammond_2011,NIPS2016_6081} for pattern extraction and information aggregation. By adapting CNN \cite{NIPS2012_4824} or GCN \cite{Kipf_Semi_CORR_16} to graph instance learning settings, \cite{Niepert_Learning_16,verma2018graph,Zhang2018AnED,xinyi2018capsule,Chen_Dual_19} introduce different strategies to handle the node orderless properties. However, due to the inherent learning problems with graph convolutional operator, such models have also been criticized for serious performance degradation on deep architectures \cite{zhang2019gresnet}. Deep sub-graph pattern learning \cite{Meng_Isomorphic_NIPS_19} is another emerging new-trend on graph instance representation learning. Different from these aforementioned methods, \cite{Meng_Isomorphic_NIPS_19} introduces the IsoNN (Isomorphic Neural Net) to learn the sub-graph patterns automatically for graph instance representation learning, which requires the identical-sized graph instance input and cannot handle node attributes.
In this paper, we introduce a new graph neural network, i.e., {\textsc{Seg-Bert}} (\underline{Se}gmented \textsc{\underline{G}raph-Bert}), for graph instance representation learning based on the recent {\textsc{Graph-Bert}} model. Originally, {\textsc{Graph-Bert}} is introduced for node representation learning \cite{zhang2020graph}, whose learning results have been demonstrated to be effective for various node classification/clustering tasks already. Meanwhile, to adapt {\textsc{Graph-Bert}} to the new problem settings, we re-design it in several major aspects: (a) {\textsc{Seg-Bert}} involves no node-order-variant inputs or functional components. {\textsc{Seg-Bert}} excludes the nodes' relative positional embeddings from the initial inputs; whereas both the self-attention \cite{Vaswani_Attention_17} and the representation fusion component in {\textsc{Seg-Bert}} can both handle the graph node orderless property naturally. (b) {\textsc{Seg-Bert}} unifies graph instance input to fit the model configuration. {\textsc{Seg-Bert}} has a segmented architecture and introduces three strategies to unify the graph instance sizes, i.e., \textit{full-input}, \textit{padding/pruning} and \textit{segment shifting}, which help feed {\textsc{Seg-Bert}} with the whole input graph instances (or graph segments) for representation learning. (c) {\textsc{Seg-Bert}} re-introduces graph links as initial inputs. Considering that some graph instances may have no node attributes, {\textsc{Seg-Bert}} re-introduces the graph links (in an artificial fixed node order) back as one part of the initial inputs for graph instance representation learning. (d) {\textsc{Seg-Bert}} uses the graph residual terms of the whole graph instance. Since we focus on learning the representations of the whole graph instances, the graph residual terms involved in {\textsc{Seg-Bert}} will be defined for all the nodes in the graph (or segments) instead of merely for the target nodes \cite{zhang2020graph}.
{\textsc{Seg-Bert}} is pre-trainable in an unsupervised manner, and the pre-trained model can be further transferred to new tasks directly or with necessary fine-tuning. To be more specific, in this paper, we will explore the pre-training and fine-tuning of {\textsc{Seg-Bert}} for several graph instance studies, e.g., \textit{node attribute reconstruction}, \textit{graph structure recovery}, and \textit{graph instance classification}. Such explorations will help construct the functional model pipelines for graph instance learning, and avoid the unnecessary redundant learning efforts and resource consumptions.
We summarize our contributions of this paper as follows:
\begin{itemize}
\item \textbf{Graph Representation Learning Unification}: We examine the effectiveness of {\textsc{Graph-Bert}} in graph instance representation learning task in this paper. The success of this paper will help unify the currently disconnected representation learning tasks on nodes and graph instances, as discussed in \cite{zhang2019graph}, with a shared methodology, which will even allow the future model transfer across graph datasets with totally different properties, e.g., from social networks to brain graphs.
\item \textbf{Node-Orderless Graph Instances}: We introduce a new graph neural network model, i.e., {\textsc{Seg-Bert}}, for graph instance representation learning by re-designing {\textsc{Graph-Bert}}. {\textsc{Seg-Bert}} only relies on the attention learning mechanisms and the node-order-invariant inputs and functional components in {\textsc{Seg-Bert}} allow it to handle the graph instance node orderless properties very well.
\item \textbf{Segmented Architecture}: We design {\textsc{Seg-Bert}} in a segmented architecture, which has a reasonable-sized input portal. To unify the diverse sizes of input graph instances to fit the input portals, {\textsc{Seg-Bert}} introduces three strategies, i.e., \textit{full-input}, \textit{padding/pruning} and \textit{segment shifting}, which can work well for different learning scenarios, respectively.
\item \textbf{Pre-Train \& Transfer \& Fine-Tune}: We study the unsupervised pre-training of {\textsc{Seg-Bert}} on graph instance studies, and explore to transfer such pre-trained models to the down-stream application tasks directly or with necessary fine-tuning. In this paper, we will pre-train {\textsc{Seg-Bert}} with unsupervised \textit{node attribute reconstruction} and \textit{graph structure recovery} tasks, and further fine-tune {\textsc{Seg-Bert}} on supervised \textit{graph classification} as the down-stream tasks.
\end{itemize}
The remaining parts of this paper are organized as follows. We will introduce the related work in Section~\ref{sec:related_work}. Detailed information about the {\textsc{Seg-Bert}} model will be introduced in Section~\ref{sec:method}, whereas the pre-training and fine-tuning of {\textsc{Seg-Bert}} will be introduced in Section~\ref{sec:analysis} in detail. The effectiveness of {\textsc{Seg-Bert}} will be tested in Section~\ref{sec:experiment}. Finally, we will conclude this paper in Section~\ref{sec:conclusion}.
\section{Related Work}\label{sec:related_work}
Several interesting research topics are related to this paper, which include \textit{graph neural networks}, \textit{graph representation learning} and \textit{{\textsc{Bert}}}.
\noindent \textbf{GNNs and Graph Representation Learning}: Different from the node representation learning \cite{Kipf_Semi_CORR_16,Velickovic_Graph_ICLR_18}, GNNs proposed for the graph representation learning aim at learning the representation for the entire graph instead \cite{Narayanan_Graph_17}. To handle the graph node permutation invariant challenge, solutions based various techniques, e.g., attention \cite{Chen_Dual_19,Meltzer_Permutation_19}, pooling \cite{Meltzer_Permutation_19,ranjan2019asap,Jiang_Gaussian_18}, capsule net \cite{Mallea_Capsule_19}, Weisfeiler-Lehman kernel \cite{NIPS2016_6166} and sub-graph pattern learning and matching \cite{Meng_Isomorphic_NIPS_19}, have been proposed.
For instance, \cite{Mallea_Capsule_19} studies the graph classification task with a capsule network; \cite{Chen_Dual_19} defines a dual attention mechanism for improving graph convolutional network on graph representation learning; and \cite{Meltzer_Permutation_19} introduces an end-to-end learning model for graph representation learning based on an attention pooling mechanism. On the other hand, \cite{ranjan2019asap} focuses on studying the sparse and differentiable pooling method to be adopted with graph convolutional network for graph representation learning; \cite{NIPS2016_6166} examines the optimal assignments of kernels for graph classification, which can out-perform the Weisfeiler-Lehman kernel on benchmark datasets; and \cite{Jiang_Gaussian_18} proposes to introduce the Gaussian mixture model into the graph neural network for representation learning. As an emerging new-trend, \cite{Meng_Isomorphic_NIPS_19} explores the deep sub-graph pattern learning and proposes to learn interpretable graph representations by involving sub-graph matching into a graph neural network.
\noindent \textbf{{\textsc{Bert}}}: {\textsc{Transformer}} \cite{Vaswani_Attention_17} and {\textsc{Bert}} \cite{Bert} based models have almost dominated NLP and related research areas in recent years due to their great representation learning power. Prior to that, the main-stream sequence transduction models in NLP are mostly based on complex recurrent \cite{Hochreiter_Long_Neural_97,DBLP:journals/corr/ChungGCB14} or convolutional neural networks \cite{kim-2014-convolutional}. However, as introduced in \cite{Vaswani_Attention_17}, the inherently sequential nature precludes parallelization within training examples. To address such a problem, a brand new representation learning model solely based on attention mechanisms, i.e., the {\textsc{Transformer}}, is introduced in \cite{Vaswani_Attention_17}, which dispense with recurrence and convolutions entirely. Based on {\textsc{Transformer}}, \cite{{Bert}} further introduces {\textsc{Bert}} for deep language understanding, which obtains new state-of-the-art results on eleven natural language processing tasks. By extending {\textsc{Transformer}} and {\textsc{Bert}}, many new {\textsc{Bert}} based models, e.g., T5 \cite{raffel2019exploring}, ERNIE \cite{Sun_ERNIE} and RoBERTa \cite{Liu_RoBERTa}, can even out-perform the human beings on almost all NLP benchmark datasets. Some extension trials of {\textsc{Bert}} on new areas have also been observed. In \cite{zhang2020graph}, the authors explore to extend {\textsc{Bert}} for graph representation learning, which discard the graph links and learns node representations merely based on the attention mechanism.
\section{Problem Formulation}\label{sec:formulate}
In this section, we will first introduce the notations used in this paper. After that, we will provide the definitions of several important terminologies, and then introduce the formal statement of the studied problem.
\subsection{Notations}
In the sequel of this paper, we will use the lower case letters (e.g., $x$) to represent scalars, lower case bold letters (e.g., $\mathbf{x}$) to denote column vectors, bold-face upper case letters (e.g., $\mathbf{X}$) to denote matrices, and upper case calligraphic letters (e.g., $\mathcal{X}$) to denote sets or high-order tensors. Given a matrix $\mathbf{X}$, we denote $\mathbf{X}(i,:)$ and $\mathbf{X}(:,j)$ as its $i_{th}$ row and $j_{th}$ column, respectively. The ($i_{th}$, $j_{th}$) entry of matrix $\mathbf{X}$ can be denoted as either $\mathbf{X}(i,j)$. We use $\mathbf{X}^\top$ and $\mathbf{x}^\top$ to represent the transpose of matrix $\mathbf{X}$ and vector $\mathbf{x}$. For vector $\mathbf{x}$, we represent its $L_p$-norm as $\left\| \mathbf{x} \right\|_p = (\sum_i |\mathbf{x}(i)|^p)^{\frac{1}{p}}$. The Frobenius-norm of matrix $\mathbf{X}$ is represented as $\left\| \mathbf{X} \right\|_F = (\sum_{i,j} |\mathbf{X}(i,j)|^2)^{\frac{1}{2}}$. The element-wise product of vectors $\mathbf{x}$ and $\mathbf{y}$ of the same dimension is represented as $\mathbf{x} \otimes \mathbf{y}$, whose concatenation is represented as $\mathbf{x} \sqcup \mathbf{y}$.
\subsection{Terminology Definitions}
Here, we will provide the definitions of several important terminologies used in this paper, which include \textit{graph instance} and \textit{graph instance set}.
\begin{definition}
(Graph Instance): Formally, a graph instance studied in this paper can be denoted as $G=(\mathcal{V}, \mathcal{E}, w, x)$, where $\mathcal{V}$ and $\mathcal{E}$ denote the sets of nodes and links in the graph, respectively. Mapping $w: \mathcal{E} \to \mathbbm{R}$ projects links in the graph to their corresponding weight. For unweighted graphs, we will have $w(e_{i,j}) = 1, \forall e_{i,j} \in \mathcal{E}$ and $w(e_{i,j}) = 0, \forall e_{i,j} \in \mathcal{V} \times \mathcal{V} \setminus \mathcal{E}$. For the nodes in the graph instance, they may also be associated with certain attributes, which can be represented by mapping $x: \mathcal{V} \to \mathcal{X}$ (here, $\mathcal{X} = \mathbbm{R}^{d_x}$ denotes the attribute vector space and $d_x$ is the space dimension).
\end{definition}
Based on the above definition, given a node $v_i$ in graph instance $G$, we can represent its connection weights with all the other nodes in the graph as $\mathbf{w}_i = [w(e_{i,j})]_{v_j \in \mathcal{V}} \in \mathbbm{R}^{|\mathcal{V}| \times 1}$. Meanwhile, the raw attribute vector representation of $v_i$ can also be simplified as $\mathbf{x}_i = x(v_i)$. The size of graph instance $G$ can be denoted as the number of involved nodes, i.e., $|\mathcal{V}|$. For the graph instances studied in this paper, they can be in different sizes actually, which together can be represented as a \textit{graph instance set}.
\begin{definition}
(Graph Instance Set): For each graph instance studied in this paper, it can be attached with some pre-defined class labels. Formally, we can represent $n$ labeled graph instances studied in this paper as set $\mathcal{G} = \left\{(G_i, \mathbf{y}_i) \right\}_{i = 1}^n$, where $\mathbf{y}_i \in \mathcal{Y}$ denotes the label vector of $G_i$ (here, $\mathcal{Y} = \mathbbm{R}^{d_y}$ is the class label vector space and $d_y$ is the space dimension).
\end{definition}
For representation simplicity, in reference to the graph instance set (without labels), we can also denote it as $\mathcal{G}$, which will be used in the following problem statement.
\begin{figure*}[t]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{./framework.pdf}
\caption{An Illustration of the {\textsc{Seg-Bert}} Model for Graph Instance Representation Learning.}
\label{fig:architecture}
\end{minipage}%
\vspace{-5pt}
\end{figure*}
\subsection{Problem Formulation}
Based on the notations and terminologies defined above, we can provide the problem statement as follows.
\noindent \textbf{Problem Statement}: Formally, given the labeled graph instance set $\mathcal{G}$, we aim at learning a mapping $f: \mathcal{G} \to \mathcal{R}^{d_h}$ to project the graph instances to their corresponding latent representations ($d_h$ denotes the hidden representation space dimension). What's more, we cast extra requirements on the mapping $f$ in this paper: (a) representations learned by $f$ should be invariant to node orders, (b) $f$ can accept graph instances in various sizes, as well as diverse categories of information inputs, (c) $f$ can be effectively pre-trained with the unsupervised learning tasks, and (d) representations learned by $f$ can also be transferred to the down-stream application tasks.
\section{The {\textsc{Seg-Bert}} Model}\label{sec:method}
In this section, we will provide the detailed information about the {\textsc{Seg-Bert}} model for graph instance representation learning. As illustrated in Figure~\ref{fig:architecture}, the {\textsc{Seg-Bert}} model has several key components: (1) \textit{graph instance serialization} to reshape the various-sized input graph instances into node list, where the node orders will not affect the learning results; (2) \textit{initial embedding extraction} to define the initial input feature vectors for all nodes in the graph instance; (3) input size unification to fit the input portal size of \textit{graph-transformer}; (4) \textit{graph-transformer} to learn the nodes' representations in the graph instance (or segments) with several layers; (5) \textit{representation fusion} to integrate the learned representations of all nodes in the graph instance; and (6) \textit{functional component} to compute and output the learning results. These components will all be introduced in this section in detail, whereas the learning detail of {\textsc{Seg-Bert}} will be discussed in the follow-up Section~\ref{sec:analysis} instead.
\subsection{Graph Serialization and Initial Embeddings}
Formally, given a graph instance $G \in \mathcal{G}$ from the graph set, we can denote its structure as $G = (\mathcal{V}, \mathcal{E}, w, x)$, involving node set $\mathcal{V}$ and link set $\mathcal{E}$, respectively. To simplify the notations, we will not indicate the graph instance index subscript in this section. Both the initial inputs and the functional components in {\textsc{Seg-Bert}} are node-order-invariant, i.e., the nodes' learned representations are not dependent on the node orders. Therefore, regardless of the nodes' orders, we can serialize node set $\mathcal{V}$ into a sequence $[v_1, v_2, \cdots, v_{|\mathcal{V}|}]$. For the same graph instance, if it is fed to train/tune {\textsc{Seg-Bert}} for multiple times, the node order in the list can change arbitrarily without affecting the learning representations.
For each node in the list, e.g., $v_i$, we can represent its raw features as a vector $\mathbf{x}_i = x(v_i) \in \mathbbm{R}^{d_x \times 1}$, which can cover various types of information, e.g., node tags, attributes, textual descriptions and even images. Via certain embedding mappings, we can denote the embedded feature representation of $v_i$'s raw features as
\begin{equation}
\mathbf{e}_i^{(x)} = \mbox{Embed} \left( \mathbf{x}_i \right) \in \mathbbm{R}^{d_h \times 1}.
\end{equation}
Depending on the input features, different approaches can be utilized to define the $\mbox{Embed}(\cdot)$ function, e.g., CNN for image features, LSTM for textual features, positional embedding for tags and MLP for real-number features. In the case when the graph instance nodes have no raw attributes on nodes, a dummy zero vector will be used to fill in the embedding vector $\mathbf{e}_i^{(x)}$ entries by default.
To handle the graph instances without node attributes, in this paper, we will also extend the original {\textsc{Graph-Bert}} model by defining the node adjacency neighborhood embeddings. Meanwhile, to ensure such an embedding is node-order invariant, we will cast an artificially fixed node order on this embedding vector (which is fixed forever for the graph instance). For simplicity, we will just follow the node subscript index as the node orders in this paper. Formally, for each node $v_i$, following the fixed artificial node order, we can denote its adjacency neighbors as a vector $\mathbf{w}_i = \left[w(i,j)\right]_{v_j \in \mathcal{V}} \in \mathbbm{R}^{|\mathcal{V}| \times 1}$. Via several fully connected (FC) layers based mappings, we can represent the embedded representation of $v_i$'s adjacency neighborhood information as
\begin{equation}
\mathbf{e}_i^{(w)} = \mbox{FC-Embed} \left( \mathbf{w}_i \right) \in \mathbbm{R}^{d_h \times 1}.
\end{equation}
The degrees of nodes can illustrate their basic properties \cite{Chung_Spectra_03,Bondy_Graph_76}, and according to \cite{Lovasz1996} for the Markov chain or random walk on graphs, their final stationary distribution will be proportional to the nodes' degrees. Node degree is also a node-order invariant actually. Formally, we can represent the degree of $v_i$ in the graph instance as $D(v_i) \in \mathbbm{N}$, and its embedding can be represented as
\begin{equation}
\begin{aligned}
\mathbf{e}_i^{(d)} &= \mbox{Position-Embed}\left( \mbox{D}(v_i) \right)\\
&= \left[sin\left (\frac{\mbox{D}(v_i)}{10000^{\frac{2 l}{d_{h}}}} \right), cos\left(\frac{\mbox{D}(v_j)}{10000^{\frac{2 l + 1}{d_{h}}}} \right) \right]_{l=0}^{\left \lfloor \frac{d_h}{2} \right \rfloor},
\end{aligned}
\end{equation}
where $\mathbf{e}_i^{(d)} \in \mathbbm{R}^{d_h \times 1}$ and the vector index $l$ will iterate through the vector to compute the entry values based on the $sin(\cdot)$ and $cos(\cdot)$ functions.
In addition to the node raw feature embedding, node adjacency neighborhood embedding and node degree embedding, we will also include the nodes' Weisfeiler-Lehman role embedding vector in this paper, which effectively denotes the nodes' global roles in the input graph. Nodes' Weisfeiler-Lehman code is node-order-invariant, which denotes a positional property of the nodes actually. Formally, given a node $v_i$ in the input graph instance, we can denote its pre-computed WL code as $\mbox{WL}(v_i) \in \mathbbm{N}$, whose corresponding embeddings can be denoted as
\begin{equation}
\begin{aligned}
\mathbf{e}_i^{(r)} &= \mbox{Position-Embed}\left( \mbox{WL}(v_i) \right) \in \mathbbm{R}^{d_h \times 1}.
\end{aligned}
\end{equation}
{\textsc{Seg-Bert}} doesn't include the \textit{relative positional embedding} and \textit{relative hop distance embedding} used in \cite{zhang2020graph}, as there exist no target node for the graph instances studied in this paper. Based on the above descriptions, we can represent the initially computed input embedding vectors of node $v_i$ in graph $G$ as
\begin{equation}\label{equ:initial_embedding}
\mathbf{h}^{(0)}_i = \mbox{sum} \left( \mathbf{e}_i^{(x)}, \mathbf{e}_i^{(w)}, \mathbf{e}_i^{(d)}, \mathbf{e}_i^{(r)} \right) \in \mathbbm{R}^{d_h \times 1}.
\end{equation}
\subsection{Graph Instance Size Unification Strategies}
Different from \cite{zhang2020graph}, where the sampled sub-graphs all have the identical size, the graph instance input usually have different number of nodes instead. To handle such a problem, we design {\textsc{Seg-Bert}} with an instance size unification component in this paper. Formally, we can denote the input portal size of {\textsc{Seg-Bert}} used in this paper as $k$, i.e., it can take the initial input embedding vectors of $k$ nodes at a time. Depending on the input graph instance sizes, {\textsc{Seg-Bert}} will define the parameter $k$ and handle the graph instances with different strategies:
\begin{itemize}
\item \textbf{Full-Input Strategy}: The input portal size $k$ of {\textsc{Seg-Bert}} is defined as the largest graph instance size in the dataset, and dummy node padding will be used to expand all graph instances to $k$ nodes (zero padding for the connections, raw attributes and other tags).
\item \textbf{Padding/Pruning Strategy}: The input portal size $k$ of {\textsc{Seg-Bert}} is assigned with a value slightly above the graph instance average size. For the graph instance input with less than $k$ nodes, dummy node padding (zero padding) is used to expand the graph to $k$ nodes; whereas for larger input graph instances, a $k$-node sub-graph (e.g., the first $k$ nodes) will be extracted from them and the remaining nodes will be pruned.
\item \textbf{Segment Shifting Strategy}: A fixed input portal size $k$ will be pre-specified, which can be a very small number. For the graph instance input with node set $\mathcal{V}$, the nodes will be divided into $\left \lceil \frac{|\mathcal{V}|}{k} \right \rceil$ segments, and dummy node padding will be used for the last segment if necessary. {\textsc{Seg-Bert}} will shift along the segments to learn all the nodes representations in the graph instances.
\end{itemize}
Generally, the \textit{full-input strategy} will use all the input graph nodes for representation learning, but for a small-graph set with a few number of extremely large graph instance(s), a very large $k$ will be used in {\textsc{Seg-Bert}}, which may introduce unnecessary high time costs. The \textit{padding/pruning strategy} balances the parameter $k$ among all the graph instances and can learn effective representations in an efficient way, but it may have information loss for the pruned parts of some graph instances. Meanwhile, the \textit{segment shifting strategy} can balance between the \textit{full-input strategy} and \textit{padding/pruning strategy}, which fuses the graph instance global information for representation learning with a small model input portal. More experimental tests of such different graph instance size unification strategies will also be explored with experiments on real-world benchmark datasets to be introduced in Section~\ref{sec:experiment}.
\begin{figure*}
\centering
\begin{subfigure}[b]{.23\textwidth}
\includegraphics[width=\linewidth]{./IMDBMulti_Train_Acc.png}
\caption{Train Acc (IMDB)}\label{fig:acc_train}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{.23\textwidth}
\includegraphics[width=\linewidth]{./IMDBMulti_Test_Acc.png}
\caption{Test Acc (IMDB)}\label{fig:acc_test}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{.23\textwidth}
\includegraphics[width=\linewidth]{./Proteins_Train_Acc.png}
\caption{Train Acc (Proteins)}\label{fig:acc_train}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{.23\textwidth}
\includegraphics[width=\linewidth]{./Proteins_Test_Acc.png}
\caption{Test Acc (Proteins)}\label{fig:acc_test}
\end{subfigure}%
\vspace{-5pt}
\caption{Learning records of {\textsc{Seg-Bert}} on the IMDB-Multi social-graph dataset and the Proteins bio-graph dataset. For the graph data with discrete structures, most of the comparison models studied in the experiments will overfit the training data easily and an early stop is usually necessary. The x axis: iteration, and the y axis: training/testing loss.}\label{fig:graph_bert_protein}
\vspace{-10pt}
\end{figure*}
\subsection{Segmented Graph-Transformer}
To learn the graph instance representations, we will introduce the segmented graph-transformer in this part to update such nodes' representations iteratively with $D$ layers. Formally, based on the above descriptions, we can denote the current segment as $s_j = [v_{j,1}, v_{j,2}, \cdots, v_{j,k}]$, whose initial input embeddings can be denoted as $\mathbf{H}_j^{(0)} = [ \mathbf{h}^{(0)}_{j,1}, \mathbf{h}^{(0)}_{j,2}, \cdots, \mathbf{h}^{(0)}_{j,k} ]^\top \in \mathbbm{R}^{k \times d_h}$ and $\mathbf{h}^{(0)}_{j,i} \in \mathbbm{R}^{d_h \times 1}$ is defined in Equation~(\ref{equ:initial_embedding}). Depending on the different unification strategies adopted, the segment can denote all the graph nodes, pruned/padded node subset or the node segments, respectively. For the segmented graph-transformer, it will update the segment representations iteratively for each layer $\forall l \in \{1, \cdots, D\}$ according to the following equation:
\begin{equation}
\begin{aligned}
\mathbf{H}_j^{(l)} &= \mbox{G-Transformer} ( {\mathbf{H}}_j^{(l-1)} ),\\
&= \mbox{Transformer} ({\mathbf{H}}_j^{(l-1)}) + \mbox{G-Res} ({\mathbf{H}}_j^{(l-1)}, \mathbf{X}_j),
\end{aligned}
\end{equation}
where $\mbox{Transformer} (\cdot)$ and $\mbox{G-Res} (\cdot)$ denote the transformer \cite{Vaswani_Attention_17} and graph residual terms \cite{zhang2019gresnet}, respectively. Here, we need to add some remarks on the graph residual terms used in the model. Different from \cite{zhang2020graph}, which integrates the target node residual terms to all the nodes in the batch, (1) for the graph instances without any node attributes, the nodes adjacency neighborhood vectors will be used to compute the residual terms, and (2) we compute the graph residual term with the whole graph instance instead in this paper (not merely for the target node).
Such a process will iterate through all the segments of the input graph instance nodes, and the finally learned node representations can be denoted as $\mathbf{H}^{(D)} = [\mathbf{h}_1^{(D)}, \mathbf{h}_2^{(D)}, \cdots, \mathbf{h}_{|\mathcal{V}|}^{(D)}]^\top \in \mathbbm{R}^{|\mathcal{V}| \times d_h}$. In this paper, for presentation simplicity, we just assume all the hidden layers are of the same dimension $d_h$. Considering that we focus on learning the representations of the entire graph instance in this paper, {\textsc{Seg-Bert}} will integrate such node representations together to define the fused graph instance representation vector as follows:\vspace{-5pt}
\begin{equation}
\begin{aligned}
\mathbf{z}& = \mbox{Fusion} \left( \mathbf{H}^{(D)} \right) = \frac{1}{|\mathcal{V}|} \sum_{i=1}^{|\mathcal{V}|} \mathbf{h}_1^{(D)}.\vspace{-5pt}
\end{aligned}
\end{equation}
Both vector $\mathbf{z}$ and matrix $\mathbf{H}^{(D)}$ will be outputted to the down-stream application tasks for the model training/tuning and graph instance representation learning, which will be introduced in the follow-up section in detail.
\section{{\textsc{Seg-Bert}} Learning}\label{sec:analysis}
In this part, we will focus on the pre-training and fine-tuning of {\textsc{Seg-Bert}} with several concrete graph instance learning tasks. To be more specific, we will introduce two unsupervised pre-training tasks to learn {\textsc{Seg-Bert}} based on \textit{node raw attribute reconstruction} and \textit{graph structure recovery}, which ensure the learned node and graph representations can capture both the raw attribute and structure information in the graph instances. After that, we will introduce one supervised tasks to fine-tune {\textsc{Seg-Bert}} for the \textit{graph instance classification}, which will cast extra refinements on the learned node and graph representations.
\subsection{Unsupervised Pre-Training}
The pre-training tasks used in this paper will enable {\textsc{Seg-Bert}} to effectively capture both the node raw attributes and graph instance structures in the learned nodes and graph representation vectors. In the case where the graph instances have no node raw attributes, only graph structure recovery will be used for the pre-training. Formally, based on the learned representations $\mathbf{H}^{(D)}$ of all the nodes in the graph instance, e.g., $G$, with several fully connected layers, we can project such latent representations to their raw features. Furthermore, by comparing the reconstructed feature matrix against the ground-truth raw features and minimizing their differences, we will be able to pre-train the {\textsc{Seg-Bert}} model with all the graph instances. Furthermore, based on the nodes learned representations, we can also compute the closeness (e.g., cosine similarity of the representation vectors) of the node pairs in the graph. Furthermore, compared against the ground-truth graph connection weight matrix, we can define the introduced loss term for graph structure recovery of all the graph instances to pre-train the {\textsc{Seg-Bert}} model.
\begin{table*}[t]
\caption{Experimental results of different comparison methods. For the results not reported in the previous works, we mark the corresponding entries with `$-$' in the table. The entries are the accuracy scores (mean$\pm$std) achieved by the baseline methods with the $10$ folds. For {\textsc{Seg-Bert}}(padding/pruning, none) and {\textsc{Seg-Bert}}(padding/pruning, raw), they denote {\textsc{Seg-Bert}} with the \textit{padding/pruning} strategy and different graph residual terms (raw vs none). At the last row on {\textsc{Seg-Bert}}*, we show the best results obtained by {\textsc{Seg-Bert}} with all these three unification strategies.}\label{tab:result}
\vspace{-5pt}
\centering
\small
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
\textbf{Datasets} &\textbf{IMDB-B} &\textbf{IMDB-M} &\textbf{COLLAB} &\textbf{MUTAG} &\textbf{PROTEINS} &\textbf{PTC} &\textbf{NCI1}\\
\hline
\textbf{\# Graphs} &1,000 &1,500 &5,000 &188 &1,113 &344 &4,110 \\
\textbf{\# Classes} &2 &3 &3 &2 &2 &3 &2\\
\textbf{Avg. \# Nodes} &19.8 &13.0 &74.5 &17.9 &39.1 &25.5 &29.8\\
\textbf{max \# Nodes} &136 &89 &492 &28 &620 &109 &111\\
\hline
\hline
\textbf{Methods} & \multicolumn{7}{c}{\textbf{Accuracy} (mean$\pm$std)}\\
\hline
WL {\scriptsize \cite{Shervashidze_WL_11}} &73.40$\pm$4.63 &49.33$\pm$4.75 &\textbf{79.02$\pm$1.77} &82.05$\pm$0.36 &74.68$\pm$0.49 &59.90$\pm$4.30 &\textbf{82.19$\pm$0.18} \\
GK {\scriptsize \cite{pmlr-v5-shervashidze09a}} &65.87$\pm$0.98&43.89$\pm$0.38 &72.84$\pm$0.28&81.58$\pm$2.11&71.67$\pm$0.55&57.26$\pm$1.41&62.28$\pm$0.29 \\
\hline
DGK {\scriptsize \cite{Yanardag_Deep_15}} &66.96$\pm$0.56&44.55$\pm$0.52 &73.09$\pm$0.25&87.44$\pm$2.72&75.68$\pm$0.54&60.08$\pm$2.55&\textbf{80.31$\pm$0.46} \\
AWE {\scriptsize \cite{Ivanov_Anonymous_18}} &\textbf{74.45$\pm$5.83}&\textbf{51.54$\pm$3.61} &73.93$\pm$1.94&87.87$\pm$9.76&$-$&$-$&$-$ \\
PSCN {\scriptsize \cite{Niepert_Learning_16}} &71.00$\pm$2.29&45.23$\pm$2.84 &72.60$\pm$2.15&\textbf{88.95$\pm$4.37}&75.00$\pm$2.51&62.29$\pm$5.68&76.34$\pm$1.68 \\
\hline
DAGCN {\scriptsize \cite{Chen_Dual_19}} &$-$&$-$&$-$&87.22$\pm$6.10 &76.33$\pm$4.3 &62.88$\pm$9.61 &\textbf{81.68$\pm$1.69}\\
SPI-GCN {\scriptsize \cite{SPIGCN}} &60.40$\pm$4.15 &44.13$\pm$4.61 &$-$ &84.40$\pm$8.14 &72.06$\pm$3.18 &56.41$\pm$5.71 &64.11$\pm$2.37\\
DGCNN {\scriptsize \cite{Zhang2018AnED}} &70.03$\pm$0.86&47.83$\pm$0.85 &73.76$\pm$0.49&85.83$\pm$1.66&75.54$\pm$0.94&58.59$\pm$2.47&74.44$\pm$0.47 \\
GCAPS-CNN {\scriptsize \cite{verma2018graph}} &71.69$\pm$3.40 &48.50$\pm$4.10 &77.71$\pm$2.51&$-$&\textbf{76.40$\pm$4.17}&\textbf{66.01$\pm$5.91}&\textbf{82.72$\pm$2.38} \\
CapsGNN {\scriptsize \cite{xinyi2018capsule}} &73.10$\pm$4.83 &50.27$\pm$2.65 &\textbf{79.62$\pm$0.91}&86.67$\pm$6.88&\textbf{76.28$\pm$3.63}&$-$&78.35$\pm$1.55 \\
\hline
{\textsc{Seg-Bert}}(padding/pruning, none) &\textbf{75.40$\pm$2.29}&\textbf{52.27$\pm$1.55}&\textbf{78.42$\pm$1.29}&\textbf{89.24$\pm$7.78}&\textbf{77.09$\pm$4.15}&\textbf{68.86$\pm$4.17}&70.15$\pm$1.84 \\
{\textsc{Seg-Bert}}(padding/pruning, raw) &\textbf{74.70$\pm$3.74}&\textbf{50.60$\pm$3.03}&74.90$\pm$1.78&\textbf{89.80$\pm$6.71}&\textbf{76.28$\pm$2.91}&\textbf{64.84$\pm$6.77}&68.10$\pm$2.55 \\
\hline
{\textsc{Seg-Bert}}* &\textbf{77.20$\pm$3.09}&\textbf{53.40$\pm$2.12}&\textbf{78.42$\pm$1.29}&\textbf{90.85$\pm$6.58}&\textbf{77.09$\pm$4.15}&\textbf{68.86$\pm$4.17}&70.15$\pm$1.84\\
\hline
\end{tabular}
\vspace{-15pt}
\end{table*}
\subsection{Transfer and Fine-Tuning}
When applying the pre-trained {\textsc{Seg-Bert}} and the learned representations in new application tasks, e.g., \textit{graph instance classification}, necessary fine-tuning can be needed. Formally, we can denote the batch of labeled graph instance as $\mathcal{T} = \left\{ (G_i, \mathbf{y}_i) \right\}_{i}$, where $\mathbf{y}_i$ denotes the label vector of graph instance $G_i$. Based on the fused representation vector $\mathbf{z}_i$ learned for graph instance $G_i \in \mathcal{G}$, we can further project it to the label vector with several fully connected layers together with the softmax normalization function, which can be denoted as
\begin{equation}
\hat{\mathbf{y}}_i = \mbox{softmax} \left( \mbox{FC} \left( \mathbf{z}_i \right) \right) \in \mathbbm{R}^{d_y \times 1}.
\end{equation}
Meanwhile, based on the known ground-truth label vector of graph instances in the training set, we can define the introduced loss term for the graph instance based on the corss-entropy term as follows:
\begin{equation}
\ell_{gc} = \sum_{(G_i, \mathbf{y}_i) \in \mathcal{T}} \sum_{j=1}^{d_y} - \mathbf{y}_i(j) \log \hat{\mathbf{y}}_i(j).
\end{equation}
By optimizing the above loss term, we will be able to re-fine {\textsc{Seg-Bert}} and the learned graph instance representations based on the application tasks specifically.
\section{Experiments}\label{sec:experiment}
To test the effectiveness of {\textsc{Seg-Bert}}, extensive experiments on real-world graph instance benchmark datasets will be done in this section. What's more, we will also compare {\textsc{Seg-Bert}} with both the classic and state-of-the-art graph instance representation learning baseline methods to demonstrate its advantages.
\noindent \textbf{Reproducibility}: Both the datasets and source code used can be accessed via the github page\footnote{https://github.com/jwzhanggy/SEG-BERT}. Detailed information about the server used to run the model can be found at the footnote\footnote{GPU Server: ASUS X99-E WS motherboard, Intel Core i7 CPU 6850K@3.6GHz (6 cores, 40 PCIe lanes), 3 Nvidia GeForce GTX 1080 Ti GPU (11 GB buffer each), 128 GB DDR4 memory and 128 GB SSD swap.}.
\begin{table*}[t]
\vspace{-7pt}
\caption{Evaluation results of different graph instance size unification strategies in {\textsc{Seg-Bert}}. No residual terms are used here, and the default epoch number is $500$. The time denotes the average time cost for mode training in the 10 folds. For {\textsc{Seg-Bert}} with the \textit{segment shifting} strategy, the default parameter $k$ is set as $20$.}\label{tab:strategy_result}
\vspace{-5pt}
\centering
\small
\setlength{\tabcolsep}{3.5pt}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c|c|c|c||c|c|c||c|c|c||c|c|c}
\hline
\multirow{2}{*}{\textbf{Strategies}} &\multicolumn{3}{c||}{\textbf{IMDB-B}} &\multicolumn{3}{c||}{\textbf{IMDB-M}} &\multicolumn{3}{c||}{\textbf{MUTAG}} &\multicolumn{3}{c}{\textbf{PTC}}\\
\cline{2-13}
&Accuracy&k&Time(s)&Accuracy&k&Time(s)&Accuracy&k&Time(s)&Accuracy&k&Time(s)\\
\hline
\hline
Full-Input &\textbf{76.90$\pm$1.76}&136&2808.70&\textbf{53.33$\pm$2.53}&89&2397.42&\textbf{90.85$\pm$6.58}&28&55.73&68.01$\pm$4.23&109&700.98\\
\hline
Padding/Pruning &{75.40$\pm$2.29} &50 &\textbf{1312.42} &{52.27$\pm$1.55} &50 &\textbf{1654.66} &{89.24$\pm$7.78} &25 &\textbf{55.19} &\textbf{68.86$\pm$4.17} &50 &\textbf{223.47}\\
\hline
Segment Shifting &\textbf{77.20$\pm$3.09}&20&1525.97&\textbf{53.40$\pm$2.12}&20&1730.08&90.29$\pm$7.74&20&88.40&66.54$\pm$4.18&20&295.83\\
\hline
\end{tabular}
\vspace{-10pt}
\end{table*}
\subsection{Dataset and Experimental Settings}
\noindent \textbf{Dataset Descriptions}: The graph instance datasets used in this experiments include IMDB-Binary, IMDB-Multi and COLLAB, as well as MUTAG, PROTEINS, PTC and NCI1, which are all the benchmark datasets as used in \cite{Yanardag_Deep_15} and all the follow-up graph classification papers \cite{xinyi2018capsule,verma2018graph,How_Xu_18}. Among them, IMDB-Binary, IMDB-Multi and COLLAB are the social graph datasets; whereas the remaining ones are the bio-graph datasets. Basic statistical information about the datasets is also available at the top of Table~\ref{tab:result}.
\noindent \textbf{Comparison Baselines}: The comparison baseline methods used in this paper include (1) \textbf{conventional graph kernel based methods}, e.g., Weisfeiler-Lehman subtree kernel (WL) \cite{Shervashidze_WL_11} and graphlet count kernel (GK) \cite{pmlr-v5-shervashidze09a}; (2) \textbf{existing deep learning based methods}, e.g., Deep Graph Kernel (DGK) \cite{Yanardag_Deep_15} and AWE \cite{Ivanov_Anonymous_18}, PATCHY-SAN (PSCN) \cite{Niepert_Learning_16}; and (3) \textbf{state-of-the-art deep learning methods}, e.g., Dual Attention Graph Convolutional Network (DAGCN) \cite{Chen_Dual_19}, Simple Permutation-Invariant Graph Convolutional Network (SPI-GCN) \cite{SPIGCN}, Graph Capsule CNN (GCAPS-CNN) \cite{verma2018graph} and Deep Graph CNN (DGCNN) \cite{Zhang2018AnED}, and Capsule Graph Neural Network (CapsGNN) \cite{xinyi2018capsule}. \textbf{Evaluation Metric}: The learning performance of these methods will be evaluated by Accuracy as the metric.
\noindent \textbf{Experimental Settings}: In the experiments, we will first compare {\textsc{Seg-Bert}} with the \textit{padding/pruning} strategy for input size unification against the baseline methods, which can help test if a sub-structures in graph instances can capture the characteristics of the whole graph instance or not. The other two strategies will be discussed at the end of this section in detail. For the input portal size $k$, it is assigned with a value slightly larger than the graph instance average sizes of each dataset. All the graph instances in the datasets will be partitioned into train, validate, testing sets according to the ratio 8:1:1 with the $10$-fold cross validation, where the validation set is used for parameter tuning and selection. Meanwhile, for fair comparisons (existing works use $9:1$ train/test partition without validation set), such $10\%$ validation set will also be used as training data as well to further fine-tune {\textsc{Seg-Bert}} after the parameter selection.
\noindent \textbf{Default Model Parameter Settings}: If not clearly specified, the results reported in this paper are based on the following parameter settings of {\textsc{Seg-Bert}}: \textit{input portal size}: $k=25$ (MUTAG), $k=50$ (IMDB-Binary, IMDB-Multi, NCI1, PTC) and $k=100$ (COLLAB, PROTEINS); \textit{hidden size}: 32; \textit{attention head number}: 2; \textit{hidden layer number}: $D=2$; \textit{learning rate}: 0.0005 (PTC) and 0.0001 (others); \textit{weight decay}: $5e^{-4}$; \textit{intermediate size}: 32; \textit{hidden dropout rate}: 0.5; \textit{attention dropout rate}: 0.3; \textit{graph residual term}: raw/none; \textit{training epoch}: 500 (early stop if necessary to avoid over-fitting).
\subsection{Graph Instance Classification Results}\label{subsec:main_result}
\noindent \textbf{Model Train Records}: As illustrated in Figure~\ref{fig:graph_bert_protein}, we show the learning records (training/testing accuracy) of {\textsc{Seg-Bert}} on both the IMDB-Multi social graph and the Protein bio-graph datasets. According to the plots, graph instance classification is very different from classification task on other data types, as the model can get over-fitting easily. Similar phenomena have been observed for most comparison methods on the other datasets as well. Even though the default training epoch is $500$ mentioned above, an early stop of training {\textsc{Seg-Bert}} is usually necessary. In this paper, we decide the early-stop learning epoch number based on the partitioned validation set.
\noindent \textbf{Main Results}: The main learning results of of all the comparison methods are provided in Table~\ref{tab:result}. The {\textsc{Seg-Bert}} method shown here adopts the \textit{padding/pruning} strategy to handle the graph data input. The residual terms adopted are indicated by the {\textsc{Seg-Bert}} method name in the parentheses. According to the evaluation results, {\textsc{Seg-Bert}} can greatly improve the learning performance on most of the benchmark datasets. For instance, the accuracy score of {\textsc{Seg-Bert}} (none) on IMDB-Binary is $75.40$, which is much higher than many of the state-of-the-art baseline methods, e.g., DGCNN, GCAPS-CNN and CapsGNN. Similarly learning performance advantages have also been observed on the other datasets, except NCI1.
For the raw residual term used in {\textsc{Seg-Bert}}, for some of the datasets as shown in Table~\ref{tab:result}, e.g., MUTAG, it can improve the learning performance; whereas for the remaining ones, its effectiveness is not very significant. The main reason can be for most of the graph instances studied here they don't have node attributes and the discrete nodes adjacency neighborhood embeddings can be very sparse, which renders the computed residual terms to be less effective for performance improvement.
\noindent \textbf{Graph Size Unification Strategy Analysis}: The analyses of the different graph size unification strategies is provided in Table~\ref{tab:strategy_result}. For {\textsc{Seg-Bert}} with the \textit{full-input strategy} on COLLAB, PROTEIN and NCI1 (with more instances and large max graph sizes), the training time costs are too high ($>$ 3 days to run the 10 folds). So, the results on these three datasets are not shown here. As illustrated in Table~\ref{tab:strategy_result}, the performance of \textit{full-input} is better than \textit{padding/pruning}, but it also consumes the highest time cost. The \textit{padding/pruning} strategy has the lowest time costs but its performance is slightly lower than \textit{full-input} and \textit{segment shifting}. Meanwhile, the learning performance of \textit{segment shifting} balances between \textit{full-input} and \textit{padding/pruning}, which can even out-perform \textit{full-input} as it introduce far less dummy paddings in the input.
\section{Conclusion}\label{sec:conclusion}
In this paper, we have introduce a new graph neural network model, namely {\textsc{Seg-Bert}}, for graph instance representation learning. With several significant modifications, the re-designed {\textsc{Seg-Bert}} has no node-order-variant inputs or function components, which can handle the node orderless property very well. {\textsc{Seg-Bert}} has an extendable architecture. For large-sized graph instance input, {\textsc{Seg-Bert}} will divide the nodes into segments, whereas all the nodes in the graph will be used for the representation learning of each segment. We have tested the effectiveness of {\textsc{Seg-Bert}} on several concrete application tasks, and the experimental results demonstrate that {\textsc{Seg-Bert}} can out-perform the state-of-the-art graph instance representation learning models effectively.
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"redpajama_set_name": "RedPajamaArXiv"
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Q: Swipe functionality working locally but fails on device farm I am trying to run a simple swipe function for an iOS app. The swipe functionality that I'm using is something like this:
public static void swipe (AppiumDriver<?> driver, WebElement element, String direction ) throws Exception {
int startX = element.getLocation().getX();
int startY = element.getLocation().getY();
int endX = element.getLocation().getX();
int endY=element.getLocation().getY();
switch (direction){
case "left":
System.out.println(startX);
startX += element.getSize().getWidth();
endX = -(element.getSize().getWidth());
break;
case "right":
startX = 0;
endX +=element.getSize().getWidth();
break;
case "up":
startY += element.getSize().getHeight();
endY = -element.getSize().getHeight();
break;
case "down":
endY += driver.manage().window().getSize().getHeight();
break;
default:
throw new Exception("invalid direction, must be left/right/up/down");
}
//driver.swipe(startX, startY, endX, endY, 1000);
new TouchAction(driver).press(startX, startY).waitAction(1000).moveTo(endX,
endY).release().perform();
System.out.println(startX +" " + startY+ " " + endX+ " " +endY);
}
The default capabilities I'm using locally are:
"platformName": "iOS"
"platformVersion": "10.3"
"automationName": "Appium"
"deviceName": "iPhone 7"
I have tried with the same desired capabilities locally and its working fine on the simulator. I am running it on the same device(i in the device farm.
I am also using the Appium v1.6.5 locally as well as in device farm.
A: Do not set your desired capabilities for device farm. It will not use them and in expected things happen when doing that.
https://github.com/awslabs/aws-device-farm-appium-tests-for-sample-app/blob/master/src/test/java/Tests/AbstractBaseTests/TestBase.java#L60
Try not setting them and see if that helps. Let me know what happens.
Best regards
James
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{"url":"http:\/\/mathoverflow.net\/questions\/83913\/no-non-trivial-homomorphism-to-a-group?sort=votes","text":"# No non-trivial homomorphism to a group\n\nHere is a question I posted some months ago in Math.SE, and t.b. mentioned to the following question by Florent MARTIN which is somehow related to my question;\n\nLet $G$ be a compact Hausdorff topological group, and let $H$ be a torsion-free group satisfying the ascending condition, i.e. there are no infinite strictly ascending chains $H_1 < H_2<...$ of subgroups of $H.$\n\nProve that there is no non-trivial homomorphism of $G$ into $H.$\n\nNote that, no topology is considered on $H$ and \"homomorphism\" simply means \"group homomorphism.\"\n\n-\nIf $G$ is finite, it's false. So I guess you should take $G$ infinite. Do you want to show that there is no injective homomorphism $G \\to H$? Because then you just need to show that $G$ fails to have the ascending condition. Hint: $G$ is uncountable. If that isn't what you mean, then take $G$ profinite and let $H$ be a finite quotient. Or am I missing something? \u2013\u00a0 Richard Kent Dec 20 '11 at 1:22\nDoesn't the answer to Florent's question answer yours? The acc implies finitely generated and by the answer to that question all fg images are finite. \u2013\u00a0 Benjamin Steinberg Dec 20 '11 at 1:23\nRichard, he said H is torsion-free so in particular not finite. \u2013\u00a0 Benjamin Steinberg Dec 20 '11 at 1:24\nOh, thanks. Missed it. \u2013\u00a0 Richard Kent Dec 20 '11 at 3:16\n\n## 1 Answer\n\nThere are no non-trivial homomorphisms from a compact group to a torsion-free finitely generated group by the theorem of Nikolov and Segal quoted in the answer by Andreas to Florent's question (mentioned by the OP above). Since ascending chain condition on subgroups implies finite generation, this answers the question.\n\nNb. Please upvote Andreas's answer if you like this one.\n\n-\nHmmm. BTW is there a more down to earth approach? \u2013\u00a0 Ehsan M. Kermani Dec 20 '11 at 3:03\nMost likely not. \u2013\u00a0 Benjamin Steinberg Dec 20 '11 at 3:30","date":"2015-08-28 07:39:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9278671741485596, \"perplexity\": 543.0241924634757}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-35\/segments\/1440644062327.4\/warc\/CC-MAIN-20150827025422-00263-ip-10-171-96-226.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/email.esm.psu.edu\/pipermail\/macosx-tex\/2008-June\/035651.html","text":"[OS X TeX] Using boldsymbol in figure caption with listoffigures\n\nPeter Lichtner lichtner at lanl.gov\nWed Jun 25 12:23:46 EDT 2008\n\nI am trying to create a caption to a figure using boldsymbol as in\n\n\\caption{$\\hat\\bzeta$}\n\nwhere \\bzeta is defined as\n\n\\newcommand{\\bzeta}{\\boldsymbol{\\zeta}}\n\nBut this gives an error in \\listoffigures. I assume I need a \\protect\nsomewhere but can't seem to get it to work. Any help appreciated.\n...Peter","date":"2020-07-14 07:16:17","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\": 0.9890215396881104, \"perplexity\": 4610.126484627267}, \"config\": {\"markdown_headings\": false, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-29\/segments\/1593657149205.56\/warc\/CC-MAIN-20200714051924-20200714081924-00261.warc.gz\"}"}
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Q: Scrapy Returns empty list when scraping from yahoo finance When I use this code on Yahoo Finance it returns an empty list but when used in another site it works fine. And its not an error in the xpath.
import pandas as pd
import requests
from scrapy import Selector
html = requests.get('https://finance.yahoo.com/quote/SM/key-statistics?p=SM').content
sel = Selector(text=html)
# Naming Sheet
ticker = sel.xpath('//*[@id="quote-header-info"]/div[2]/div[1]/div[1]/h1/text()').getall()
print(ticker)
A: In this way data is generating:
import pandas as pd
import requests
#from scrapy import Selector
headers = {'User-Agent':'Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/91.0.4472.124 Safari/537.36'}
html = requests.get('https://finance.yahoo.com/quote/SM/key-statistics?p=SM', headers = headers).content
sel = pd.read_html(html)
print(sel)
# Naming Sheet
# ticker = sel.xpath('//*[@id="quote-header-info"]/div[2]/div[1]/div[1]/h1/text()').get()
# print(ticker)
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What are the main areas to take note when setting up a new office?
Setting up a new office is an effective test of your business planning capacity. It's like a small project that requires from you to mobilize your budget management, design and visualization skills to incorporate all essentials into a smooth-running workspace.
The new office setup checklist won't work perfectly if you decide to leave it to chance. Solving problems as they arise can be an exception, but not the rule of a small office set up. Business plans make things easier. But, what if you don't have an inclusive business plan or just need to open the office immediately without waiting for the plan to be finalized and approved? Checklists save the day for many successful endeavors. New offices are no different.
Legal requirements are the first to consider when you are setting up a new office in a certain area. It's not bad to check out the area, such as its crime rates and law enforcement activity. Since health and safety are usually a priority, include them at the top of the new office setup checklist. While picking up a great location, don't forget to consider other aspects of your small business. Is this a startup that will work in a coworking space? Are your common partners located nearby? Are there any leased offices that could be appropriate without weighing on your startup budget? These factors will help you solidify the new office checklist. You can move the less critical stuff to the bottom of the list and avoid beginner's mistakes of space planning.
In the midst of hectic activity, many new managers forget that they need to work in a secure place. Choosing the most effective access control system is a basic business need. If you are renting a space or just sharing a coworking office, you might not be in charge of the access control. You need to get approvals and permissions. In contrast, you can equip your own or your rented space with electronic access control and manage it from your own central network. Choose the right doors and locks and organize the alarms, detectors and video surveillance to serve the needs of your team. At the same time, think of inside security. Do you need to use separate offices or will dividers do? Most startups share an office. In such space, your new office startup checklist should include a simple open layout plan. For security, use locked cabinets. The good thing about modern security is that it got smart, so you won't have to dig, drill, wire or install complex systems. You can rely on a mobile app.
When managing security, it is not sufficient to take into account the technology factor only. Nearly any camera or an alarm system can be deactivated, and a locking mechanism can be broken if used as a standalone safety measure. Think of the human factor that in many cases accounts for security breaches and information leaks.
Access control allows employers to find an all-round solution to their safety needs. It serves as an umbrella of multiple security devices, helps maintain effective control over premises, protects assets, and also presupposes user authentication and authorization. This means, you as an employer can identify your workers' rights and establish or deny access to certain offices or information based on employees' role within an organization.
Legacy access control requires investment in servers, wires, and in-house personnel who can supervise work of the system. As opposed to the traditional access control systems cloud-based access control is more cost-effective as it does not entail huge capital expenses. It is more scalable, flexible and spares you the hassle of hardwired components. Moreover, with IP access control you can entrust your security to a service provider who will overlook all the processes all year round, so there's no need to hire a security manager.
Think workstations, telecommunications, the Internet, and furnishings. These are the basics for a small or a large office. Consider whether you need to get all that stuck in the office space. The limited space should stock up the essentials. It might be wiser to outsource printing, copying or faxing services to an external provider. On the other hand, setting up the Internet connection is one of the first tasks on a new office setup checklist. Do you really need a big kitchen? Which team can share a desk? Who needs to work separately because of security reasons? If you are renting a space, your employees need to accommodate more to the space than vice versa. Visualizing how a day (or a week) at work would look like is a helpful exercise for a worry-free small office set up.
The small office set up plan is a challenging task. With a bit more attention to detail, you can turn the small office into a functional and pleasant space.
Purchasing vs renting or leasing equipment and services to save costs and space.
Design a smart office layout and place teams as neighbors, with easy-to-reach files at hand.
Maximize the floor plan by examining the separate areas. Include a reception desk instead of a visitor room, meeting cubicles instead of meeting rooms and kitchen tables that can replace a conference room when free.
Remove robust furniture and unnecessary IT equipment. Really think this part through.
Maintain safety and security standards by choosing anatomical furniture, proper lighting and workstations, as well as adequate toilets and smart access control.
Don't forget about interior design. If the office is plant or animal-friendly, think of how will this affect the daily work in such a small space.
You can cut down a long list for a small office set up by paying attention to what you can do by yourself and what it's best left to others. Small businesses and startups have a lot in common. Check out the startup checklist below for additional insight.
Space ownership. Do you own or lease the workspace? Are you working from home? Can you make alterations? Must you stick to the strict provisions of your lease? Do you need to meet specific security requirements? Have you thought of signing up for a coworking space membership? The answers to this questions will help you decide about the next items on this list.
IT and other communication systems. If the answer to the above question included sharing a workspace or renting an office, you will have a couple of fewer items to think of, because many coworking spaces or buildings with offices for rent have already taken care of this through their facility management departments or monthly membership packages. This can include web access, telephone and fax lines, but also air-conditioning, cleaning and food catering services. If you sign such a contract, cross off the IT and the additional services of your office equipment list.
Furniture, supplies and stationery. Purchase only the necessary; rely on the rented stuff for basic functioning. You'll need to think of who will deliver your daily necessities, which can include anything from batteries, paper, toner, cables to food and water. Do you need to buy whiteboards, projectors and copying machines? Can you find a per-hour conference room in the near vicinity that will cost less than using your own full time? Is your startup a busy place with a constant flow of new people? How will you identify, authenticate and grant access to clients, visitors and employees?
You can cut down a long list for a small office set up by paying attention to what you can do by yourself and what it's best left to others.
What is a good sample checklist for large offices / enterprises?
In a large firm, there are many departments and many people, which calls for space to be organized smartly and efficiently.
Should there be a separate area for each department?
Open space vs. private offices – what suits your business model more?
Open plan offices are more economical and are easier to supervise compared to private offices. However, you should consider that security can be hampered in such a working environment.
Create meeting rooms for customers and partners.
Common areas where employees can interact, such as a fully-fit kitchen or a lounge with comfortable seats.
Access control and how it can be implemented depending on the set up of the offices. The more complex hierarchy within your organization, the more layered access rights to assets and information should be given to your employees.
Security needs. Find a reliable access control supplier and make sure you get a scalable and flexible access control system that can be adjusted as your company grows and roles within an organization quickly change.
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The Swedish infantry musket, or the Swedish Land Pattern Musket, was a muzzle-loaded 0.63 (16.002 mm) to 0.81 (20.7mm)-inch calibre smoothbored long gun. These weapons were in service within the Royal Swedish Army from the mid-16th century until the mid-19th century.
History
At the end of the 16th century, the Swedish military musket became a style-setter. Its style remained the same until about 1660 in most armies. In Sweden, its basic style lasted for many years—until the end of the 1680s. The matchlock was the dominant mechanism on the Swedish Army soldiers' muskets as well as among other European armed forces, and remained so until the latter half of the 1600s when the snaphaunce mechanism increasingly took over. But it was not until the flintlock mechanism as well as the bayonet had taken hold in earnest—around the turn of the 17th–18th centuries—that the matchlock became completely obsolete among the various squadrons within the Swedish Empire. However, some weapons equipped with wheellock mechanism were primarily reserved for the cavalry. The Swedish, purely warlike musket design remained in its basic form from Model 1696 until Model 1775. Before that, long guns – military as well as civilian – were produced in a variety of designs.
Clear variants
Model 1673
Model 1688
Matchlock Musket M1688Snaphaunce Musket M1688'Model 1690
Model 1696
The flintlock carbine M1696 was the first bayonet-equipped.Swedish Army Museum
Model 1704
Model 1716
Model 1725
Model 1738
Model 1762Krävan with the krävan-fitting was abandoned in favour of a third scouring stick-pipe, where a ramrod (now made of iron) instead rested and a fourth scouring stick-pipe (all now in brass) next to the chamber. And the stock was equipped with a nose cap, also in brass.
Model 1775
With the manufacturing of the 1775 model, the pins holding the barrel in place were abandoned in favour of two scouring stick-pipe-bands with associated kräkor'' and a front barrel band nose cap with bow-shaped front sights in brass infused.
Model 1784
Model 1791
Model 1805
Model 1815
Model 1840
Model 1848
See also
List of wars involving Sweden
Military history of Sweden
Musket
External links
Swedish Army Museum
Forum for Living History
Swedish Ingermanland National Association (in Swedish)
References
Muskets
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Sanç I de Cerdanya, (1161 - 1223), príncep d'Aragó i comte de Cerdanya (1168-1223); de Provença (1181-1185); i de Rosselló (1185-1223).
Sanç I de Lleó (935 - 966), rei de Lleó (956 -958) i (960 -966).
Sanç I Llop duc de Gascunya, efectiu vers el 801 al 816
Sanç I de Mallorca o Sanç II de Cerdanya (1276 – Formiguera 1324), rei de Mallorca, comte de Rosselló i Cerdanya, vescomte de Carladés, baró d'Omelàs i senyor de Montpeller (1311–1324).
Sanç Garcés I de Pamplona (~865 - Resa, 925), rei de Navarra (905 - 925)
Sanç I de Portugal dit "el Poblador" (Coïmbra, 1154 - 1212), rei de Portugal (1185 -1212).
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Dell Pushes Hyperscale Boundaries with New PowerEdge
June 23, 2015 by Alison Diana
(Source: SDSC)
With today's introduction of the PowerEdge C6320, Dell continues its drive to propel high performance computing outside its traditional comfort zone of engineering or research and development and into general business enterprise applications.
"We recognize that beyond the hyperscale space there's an opportunity to offer this type of product," Brian Payne, executive director of product management and product strategy for Dell Server Solutions, told Enterprise Technology. "We took some of our mainstream products out of the datacenter solution and created the PowerEdge C6320."
The C6320 – which will ship in July – is packaged in a compact, 2U chassis designed for HPC and hyper-converged solutions and appliances, and runs on Intel Xeon E5-2600 v3 processors with up to 18 cores per socket (144 cores per 2U chassis). The latest PowerEdge family member has up to 512GB of DDR4 memory and up to 72TB of flexible local storage.
"It's a versatile platform. The C6320 is an appropriate piece of hardware for the enterprise space," said Rick Wagner, HPC systems manager at the San Diego Supercomputer Center, in an interview. "While they may not have 27 racks dedicated to data simulation and analysis and big data, there are a growing number of enterprises using high-performance computing for business processes."
The server was designed for high-end enterprise applications such as big data, and research and development, said Payne. "We're seeing hyperconverged appliances go into mainstream IT. Mainstream IT is very different from the way they operate in web tech and hyperconverged [organizations]," he said.
To address enterprise IT department needs, the C6320 is equipped with iDRAC8 with Lifecycle Controller. The integrated Dell Remote Access Controller allows IT staff to automate multiple routine management chores, reducing the time and eliminating processes required to deploy, monitor, and update servers. No operating system or hypervisor is required since it's embedded.
For more demanding workloads, the C6320 can be paired up with the PowerEdge C4130, a GPU dense and flexible rack serve designed to handle big data and other large loads.
While PowerEdge C6320 is appropriate for compute-hungry enterprise applications, it stays true to its HPC roots, as evidenced by one early customer. The San Diego Supercomputer Center (SDSC) at the University of California used 1,944 nodes or 46,656 cores in 27 racks of PowerEdge C6320 compute nodes to create Comet, its new petascale supercomputer.
"It's a big platform, but it's designed to support as many people as possible to get their science done as quickly as possible," said Wagner. "Comet has the capacity to run up to 50,000 jobs at once if they are utilizing only a single core."
Before preparing a proposal for the National Science Foundation's review, SDSC spoke to Intel, its storage partner, and several server vendors before opting to work with Dell, Wagner said.
"When we looked at features, price competitiveness, and scale, Dell came out well ahead. Dell is a well respect vendor. They've been in the enterprise for a long time," he said. "They delivered a very good cost so we could provide a competitive bid to NSF."
As the Dell Engineered Solutions portfolio evolves, the PowerEdge C6320 will make it the platform for appliances such as the Dell Engineered Solutions for VMware EVO:Rail, available next month; the Dell XC Series of Web-scale Converged Appliances, slated to ship with the C6320 in the fourth quarter, and future Dell HPC offerings.
Categories: Editor's Picks: Systems, Slider: Systems, Systems
Tags: Brian Payne,Comet,Dell,PowerEdge C6320,Rick Wagner,San Diego Supercomputer Center,SDSC
About the author: Alison Diana
Managing editor of Enterprise Technology. I've been covering tech and business for many years, for publications such as InformationWeek, Baseline Magazine, and Florida Today. A native Brit and longtime Yankees fan, I live with my husband, daughter, and two cats on the Space Coast in Florida.
FedEx Enlists Dell, Switch for Logistics Hub
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Nascent HCI Market Bestrides 'Multi-Modal' IT
Security Threats Soar Along with Data Volumes
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While it might not come as a shock to learn that Texas' wealthiest ZIP code lies within Houston, its location within the city is probably different than what you'd expect.
The Bayou City's priciest ZIP code isn't in River Oaks or Piney Point or Carlton Woods in The Woodlands.
The wealthiest ZIP code in the state of Texas is 77010, an area smaller than 30 square blocks in downtown Houston, according to a new report by information services group Experian, which created an extensive map that reveals the wealthiest ZIP codes based on individual income tax returns around the nation.
The area includes the George R. Brown Convention Center and Discovery Green, as well as a number of skyscrapers that house local branches of massive companies like the Hess Corporation and Ernst & Young. There are several luxury residential highrises, too, including One Park Place, located within the ZIP code area. Based on the 247 tax filings reported for the 77010 ZIP code, the adjusted income for the area is $670,198.
For each state, Experian lists the top three wealthiest zip codes based on adjusted income figures from the IRS. The other two in Texas are located in Dallas. The 75247 zip code — located near Love Field — ranked No. 2 in the state with an adjusted income of $548,640, and 75270 in downtown is No. 3 with an adjusted income of $382,037.
Nationally, three of the top five wealthiest zip codes are located in New York City, with the No. 1 highest adjusted income in the country falling in the 10104 zip code. It covers only one city block, between Fifth Avenue and Avenue of the Americas at West 51st and West 52nd Street, and has an adjusted income of $2,976,929.
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Il parco nazionale di Cabañeros (in spagnolo: Parque nacional de Cabañeros) è un parco nazionale situato in Castiglia-La Mancia, in Spagna.
Altri progetti
Collegamenti esterni
Cabañeros
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{"url":"https:\/\/dilequante.com\/an-alternative-way-to-visualize-stocks-correlations\/","text":"# An alternative way to visualize stocks\u2019 correlations\n\nIn the universe of Asset Management, and more precisely in the field of portfolio construction, two of the most used mathematical elements are covariance and correlation matrices. These elements are used to suggest a risk estimation on assets given a specific investment universe. They allow to bring two informations, one relates to the estimation of a standalone risk for each assets (a.k.a the volatility), the second tries to estimate how all the assets behave together (a.k.a the correlation). Here we will focus on the use of correlation matrix.\n\nBefore starting to elaborate a portfolio construction process to invest in a specific universe, one should investigate how the universe \u201clooks like\u201d. In other words, we would like to investigate if there are some groups of stocks that tend to behave similarly, that have similar features (a.k.a clusters).\n\nLet\u2019s take an example on major US equities, i.e. on a universe of around 600 stocks.\n\nThe basic visualization of a correlation matrix is simply a heatmap of all stocks correlation.\n\nWe can\u2019t see much information, as it is not very readable.\n\nRemark: By the way, here correlations are nearly all positive as we are looking in \u201cabsolute\u201d terms on stocks, i.e. keeping their beta market component. The purpose of this article is simply to show how to generate a good visualization of stocks\u2019 correlation structure, and not about the matrix generation which alone is a wide subject.\n\nOne improvement would be to cluster data thanks to a \u201clinkage\u201d function.\n\nThis is a bit better, as we can spot some clusters. But still, with a lot of data, we can\u2019t visually analyze it properly.\n\nNow, the method we are going to study is part of the Graph Theory field.\n\nGraphs are used to model pairwise relations between objects of a specified universe.\n\nExplained briefly, graphs are mathematical structures that possess two components, nodes and edges. Nodes represent the objects we are studying (here stocks), edges represent the links between all nodes (here correlations, or more precisely distances) and the strength of the links are reflected by a weight feature (here the correlation\/distance coefficient).\n\nLike for the clustered heatmap, we apply a linkage function (to compute a distance measure from the correlation matrix) and map data into a graph. Then we apply a minimization algorithm to chart what is called a \u201cminimum spanning tree\u201d (i.e. we minimize the total distance, i.e. the sum of edges\u2019 weights, between all stocks to generate a tree).\n\nHere each node represents a stock, and one \u201cedge\u201d (a link) relates each stock to at least another one which exhibits the most \u201csimilarity\u201d. The method is quite simple, starting from the correlation matrix, we compute a distance(1) matrix, which simply reflects the intensity of the correlation between stocks. When much correlated the distance is quite small and vice versa. Nodes are colored by their sectoral affiliation and size of nodes are dependent of the market capitalization of the stock.\n\nNow let\u2019s dig into the code to produce this chart.\n\nWe will be using the python library NetworkX which allows to generate and analyze networks, and what is our interest here, a minimum spanning tree.\n\nFirst, let\u2019s import all the librairies we will use.\n\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport networkx as nx \n\nThen let\u2019s define our data. We load our correlation matrix as well as our stocks data (i.e. name, sector, market cap), inside pandas dataframes.\n\ncorrel_matrix = pd.read_excel('correlation_data.xlsx')\ndata_labels = pd.read_excel('label_data.xlsx')\n\nNow that we have our data, we should create a graph object G from our correlation matrix.\n\nG = nx.from_numpy_matrix(np.asmatrix(correl_matrix)) # create a graph G from a numpy matrix\nG = nx.relabel_nodes(G,lambda x: correl_matrix.index[x]) # relabel nodes with our correlation labels\n\nFrom our graph G, we ask to generate the minimum spanning tree(2) (MST) T.\n\nT = nx.minimum_spanning_tree(G)\n\nIn order to chart our tree, we need to choose a \u201clayout\u201d, i.e. to position graphically our nodes. We will be using Fruchterman-Reingold force-directed algorithm(3) to create the nodes positions.\n\npos = nx.fruchterman_reingold_layout(T)\n\nNow we focus on aggregating our graph data into dataframes, the primary key being \u201cAsset ID\u201d.\n\nnodes = pd.DataFrame(sorted(T.nodes),columns=['Asset ID']).set_index('Asset ID')\nnodes = pd.merge(nodes,data_labels,how='left',left_index=True,right_index=True)\nedges = pd.DataFrame(sorted(T.edges(data=True)),columns=['Source','Target','Weight'])\n\nAs we would like to use sectors to define colors, we transform the \u201cGICS Sector\u201d serie to a pandas Categorical type and store the corresponding sectors codes.\n\nnodes_cat = nodes.copy()\nnodes_cat['GICS Sector Cat']=pd.Categorical(nodes_cat['GICS Sector'])\nnodes_cat['GICS Sector Cat Code'] = nodes_cat['GICS Sector Cat'].cat.codes\nnodes_cat = nodes_cat.reindex(T.nodes()) # keep the right index order\n\n\nThanks to the sector codes, we can map a \u201ccolormap\u201d to the sectors.\n\n# we create our color pallet\ncolor_df = pd.DataFrame(data=[plt.cm.Paired.colors]).transpose().rename(columns={0:'color'})\n\n# we map the color for each stock accordingly to their sector affiliation\nnodes_cat = pd.merge(nodes_cat,color_df, left_on=\"GICS Sector Cat Code\", right_index=True)\nnodes_cat = nodes_cat.reindex(T.nodes()) # keep the right index order\n\nFor the sake of a better visualization, we will only display the names of stocks that are the biggest in terms of market cap (here a weight in index > 1%). So we create a \u201clabels_to_draw\u201d vector that will store only the label name we want to display and an empty string otherwise.\n\nlabels_to_draw = {\nn: (nodes[nodes.index == n]['Asset Name'].values[0]\nif nodes[nodes.index == n]['Active Weight (%)'].values[0]*100 > 1.0\nelse '')\nfor n in T.nodes\n}\n\nSame for the size of nodes, we create a vector that will store the size for each node, depending on its index weight (i.e. market cap).\n\nnode_size_list = {\nn: ((nodes[nodes.index == n]['Active Weight (%)'].values[0]100+1)500)\nfor n in T.nodes\n}\n\nAs we have all our attributes ready, we can create our plot, with the nx.draw() function. To display a legend, we use empty scatter plots (as there is no such feature in networkx).\n\nplt.figure(figsize=(30, 30))\nnx.draw(T, pos, with_labels=True,\nlabels=labels_to_draw,\nedge_color = \"grey\",\nwidth = .1,\nnode_size=list(node_size_list.values()),\nfont_size=25,\nnode_color = nodes_cat['color'])\nfor v in range(len(nodes_cat['GICS Sector Cat'].cat.categories)):\nplt.scatter([],[], color=plt.cm.Paired.colors[v], label=nodes_cat['GICS Sector Cat'].cat.categories[v],s=10)\nplt.show()\n\nTo sum up, in this article we were able to produce an alternative way to visualize the correlation structure of an equity market universe. With this methodology, we can visually analyze more precisely on the existence of clusters and link it to some stocks characteristics (market cap, sectors, etc.).\n\nRemarks\n\n\u2022 (1) The distance measure usually used for correlations is the Euclidian distance:\nd = \\sqrt {2 \\times(1- \\rho)}\n\u2022 (2) The MST algorithm used here is Kruskal\u2019s algorithm. Another widely used algorithm is PRIM\u2019s algorithm.\n\u2022 (3) An alternative layout, that fits well in our use case, is the Force Atlas algorithm, but unfortunately it is not directly available on the networkx lib.\n\u2022 To manually produce MSTs in a more advanced way, Gephi software is really useful and well equiped.","date":"2021-06-19 00:57:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.6291337609291077, \"perplexity\": 2030.7986176198199}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487643354.47\/warc\/CC-MAIN-20210618230338-20210619020338-00487.warc.gz\"}"}
| null | null |
Q: How to rate limit the fetch API in React? I have a Vite.js App, React Template. In one of my components I want to fetch some data from https://wheretheiss.at/w/developer
API. I am using useEffect. Now, the documentation mentions that the rate limiting is 1 request per second. And here is my code:
import { useState, useEffect } from "react";
import axios from "axios";
export default function Info() {
const [info, setInfo] = useState({
longitude: 0,
latitude: 0,
altitude: 0,
velocity: 0,
visibility: "",
});
const getCoords = async () => {
const data = await axios({
method: "get",
url: "https://api.wheretheiss.at/v1/satellites/25544",
});
return data;
};
useEffect(() => {
fetch("https://api.wheretheiss.at/v1/satellites/25544")
.then((res) => res.json())
.then((data) => {
const lat = data.latitude;
const long = data.longitude;
const alt = data.altitude * 0.62137119;
const vel = data.velocity * 0.62137119;
const vis = data.visibility;
setInfo({
longitude: long.toFixed(4),
latitude: lat.toFixed(4),
altitude: alt.toFixed(4),
velocity: vel.toFixed(2),
visibility: vis === "daylight" ? "not visible" : "visible",
});
});
}, [info]);
return (
<section className="info">
<hr />
<ul>
<li>
<strong>Latitude: </strong>
<span>{info.latitude}°</span>
</li>
<li>
<strong>Longitude: </strong>
<span>{info.longitude}°</span>
</li>
<li>
<strong>Altitdude: </strong>
<span>{info.altitude} miles</span>
</li>
<li>
<strong>Velocity: </strong>
<span>{info.velocity} mph</span>
</li>
<li>
<strong>Visibility: </strong>
<span>{info.visibility}</span>
</li>
</ul>
</section>
);
}
I know that setting the info state as the dependencies array for my useEffect hook will determine React re-rendering the component in an infinite loop. My question is: How can I limit the fecthing for one time per second?
I cannot set the dependecie array as [] because I want to constantly fetch data, because the latitude, longitude, etc., are changing every second. I cannot leave it like that either because the app will throw this error:
Uncaught (in promise): Object {..., message: "Request failed with status code 429"}
which means my app is making to many requests above the rate limit. When I get this error, my coordinates are not being re-rendered anymore and they remain the same.
Do you have any idea?
A: You can use the 'axios-retry' npm package in your environment. It has the capabilities you need to use in your environment.
A: step 1:
import the react-debouncing lib
npm i loadash
now use the loadash's debounce to debounce the function so the
function will run only for 1sec so the request will also get limited
to 1req per sec
here is the modified code:
import { useState, useEffect } from "react";
import axios from "axios";
import { debounce } from 'loadash';
export default function Info() {
const [info, setInfo] = useState({
longitude: 0,
latitude: 0,
altitude: 0,
velocity: 0,
visibility: "",
});
const getCoords = async () => {
const data = await axios({
method: "get",
url: "https://api.wheretheiss.at/v1/satellites/25544",
});
return data;
};
let debouncedReq = debounce(()=>{
fetch("https://api.wheretheiss.at/v1/satellites/25544")
.then((res) => res.json())
.then((data) => {
const lat = data.latitude;
const long = data.longitude;
const alt = data.altitude * 0.62137119;
const vel = data.velocity * 0.62137119;
const vis = data.visibility;
setInfo({
longitude: long.toFixed(4),
latitude: lat.toFixed(4),
altitude: alt.toFixed(4),
velocity: vel.toFixed(2),
visibility: vis === "daylight" ? "not visible" : "visible",
});
});
}, 1000) // now the function is debounced for 1000ms
useEffect(() => {
debouncedReq()
}, [info]);
return (
<section className="info">
<hr />
<ul>
<li>
<strong>Latitude: </strong>
<span>{info.latitude}°</span>
</li>
<li>
<strong>Longitude: </strong>
<span>{info.longitude}°</span>
</li>
<li>
<strong>Altitdude: </strong>
<span>{info.altitude} miles</span>
</li>
<li>
<strong>Velocity: </strong>
<span>{info.velocity} mph</span>
</li>
<li>
<strong>Visibility: </strong>
<span>{info.visibility}</span>
</li>
</ul>
</section>
);
}
A: Your existing code relies on the fetch call succeeding to have another scheduled afterwards (triggered by setInfo being called with new data) and these are scheduled to run one after the other without any delay which quickly results in the rate limit quickly being hit.
wheretheiss.at rate limiting is roughly documented on their API page but they also indicate that you should check out the X-Rate-Limit-* headers returned by their server to know more info about it:
X-Rate-Limit-Limit: 350
X-Rate-Limit-Remaining: 193
X-Rate-Limit-Interval: 5 minutes
This means you can query slightly more often than every second so you can use setInterval to schedule your API calls:
useEffect(() => {
const interval = setInterval(() => {
fetch('https://api.wheretheiss.at/v1/satellites/25544')
.then((res) => res.json())
.then((data) => {
const lat = data.latitude;
const long = data.longitude;
const alt = data.altitude * 0.62137119;
const vel = data.velocity * 0.62137119;
const vis = data.visibility;
setInfo({
longitude: long.toFixed(4),
latitude: lat.toFixed(4),
altitude: alt.toFixed(4),
velocity: vel.toFixed(2),
visibility: vis === 'daylight' ? 'not visible' : 'visible',
});
});
}, 1_000);
return () => { clearInterval(interval) };
}, [setInfo])
Stackblitz link
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,408
|
Q: Python Tornado XSRF cookie issue with Chrome I am new to Tornado and I am trying to build a very simple login form with xsrf cookie.
Code as below:
class MainHandler(tornado.web.RequestHandler):
def get(self):
self.render("index.html")
def post(self):
self.write("You ahve submitted the form.!")
class App(tornado.web.Application):
def __init__(self):
handlers = [
(r"/", MainHandler),
]
settings = dict(xsrf_cookies=True)
tornado.web.Application.__init__(self, handlers, **settings)
if __name__ == "__main__":
app = App()
server = tornado.httpserver.HTTPServer(app)
server.listen(7001)
tornado.ioloop.IOLoop.current().start()
The form is also very simple:
<html>
<head><title>Text XSRF</title></head>
<body>
<form method="post">
{% module xsrf_form_html() %}
{% raw xsrf_form_html() %}
Username:<input type="text" name="username">
Password:<input type="password" name="pwd">
<input type="submit" value="submit">
</form>
</body>
</body>
</html>
I am adding to the host file a domain:
127.0.0.1 xsrf.test.com
And when I open Chrome and type in hxxp://xsrf.test.com:7001, I can see the login form, but when sending the POST request I get a
403 (XSRF cookie does not match POST argument)
However, if I hit hxxp://localhost:7001, I can submit the form as expected. IE and Safari works fine for both domains. So I am wondering is there anything I did wrong to make this work using "xsrf.test.com" in Chrome?
I did check Chrome->Settings->Content Settings->Cookies->Allow local data to be set is properly selected.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,275
|
HomeMagazine issuesApril 2021
The online edition of Share International magazine presents a selection of items from the printed edition. Each online edition includes a complete article by Benjamin Creme's Master. Most other articles reproduced here, covering a wide range of topics, are excerpts. The online edition usually also includes a selection of Questions and Answers, Readers' letters, and photographs of Signs of Maitreya's presence.
See the full table of contents of the printed edition at the foot of the page.
Questions and answers (a selection)
Readers' letters (a selection)
Compilation: The Law of Action and Reaction (excerpt)
Signs of the time
Lakhdar Brahimi calls for a spirit of solidarity (excerpt)
From our own correspondents
From the inception of Share International magazine, Benjamin Creme's Master provided an article every month for nearly 35 years. In 2010 He wrote: "The decisions made by men now will decide, in large measure, the whole future of this planet." Those decisions are now more urgent than ever and must be based on the acceptance of Brotherhood. Treasure Brotherhood, cherish Brotherhood, we are advised; it melts barriers and it transforms because it is Divine.
The Master —, through Benjamin Creme
Without doubt, this is a time of major importance to humanity. The decisions made by men now will decide, in large measure, the whole future of this planet. Future generations will marvel at the apparent ease with which so many today slough off concern for the world's ills: millions starve and die of want in a world blessed with a huge surplus of food; millions more are always hungry and undernourished.
Many know this to be true yet do nothing. How can this be? What prevents their action? The basis of this inaction is complacency, the source of all evil in the world. Complacency has its roots in the crime of separation which pulls men apart and prevents the flowering of Brotherhood.
Men soon must realize this truth or perish. Brotherhood is both an idea and the fact of our planetary life. Without the reality of Brotherhood as the basis for all action, man's every effort would come to nothing.
When men accept Brotherhood as the essential nature of life, every aspect of our daily living will change for the better. Every manifestation of Brotherhood melts the barriers which form themselves between men and lead to misunderstanding and distrust. Brotherhood assuages the pain of loss and misfortune. It is a precious gift to be cultivated and nourished. Treasure Brotherhood, it is the key which gives entry to the finest chambers of the heart. We, your Elder Brothers, cherish Brotherhood as Our highest nature, and strive to maintain and strengthen its reality. When men, too, grasp the beneficent truth of Brotherhood, they will realize the beauty which its nature displays, and grasp something of the beauty of divinity itself. Brotherhood is divine as men are divine. It could not be otherwise.
Men are about to experience a profound truth, an awareness of their essential Being. For most, it will come as an experience of rebirth to a state long lost in the distant past. Each, in his own way, will feel redeemed, made new, cleansed and purified. The joy and beauty of Brotherhood will thrill through their Being, and each will see themselves as a part of that beauty and love.
These articles are by a senior member of the Hierarchy of Masters of Wisdom. His name, well-known in esoteric circles, is not yet being revealed. Benjamin Creme, a principal spokesman about the emergence of Maitreya, was in constant telepathic contact with this Master who dictated his articles to him.
See further articles by this Master
At every lecture he gave around the world, and virtually every day of his life, Benjamin Creme was asked numerous questions covering a vast range of topics. We draw on this large recorded resource and publish answers provided by BC and his Master over the years, none of which have yet appeared in Share International.
Evolution — how it proceeds and the role of the Masters
You talk about evolution — but what is the motor for evolution? How does it work? You talk about evolution — but what is the motor for evolution? How does it work?
The Masters are the custodians of all the spiritual energies entering the planet. And Their task is to radiate some degree of these energies out into the world in such a way that the Plan of Evolution works out. Everything that humanity does is in response to energy. Generally speaking, we don't know that, but that's what happens. If the energy isn't there then nothing happens. Even just to lift a glass of water takes energy. Everything we do takes energy of one kind or another. We usually think of energy as a physical phenomenon; we think of it in physical terms, such as having enough energy to run. But I'm not talking about that kind of energy. This is about spiritual energy. And that spiritual energy embodies certain great ideas – like love, like unity, illumination, justice, right relationship, will, purpose. These are all ideas which embody certain energies and they become humanity's ideals; and as we put our ideals into effect we use the energies and so create our culture and our civilization. That's how culture and civilization grow – in response to the energies embodying the ideals which are a reflection of the ideas which themselves embody the energies in the first place.
The Masters know that, in order to bring out certain aspects of the Plan as They see it, They need to release certain energies which will produce a certain response or result, if we respond. They are experts at it, so in the end we usually respond; possibly not perfectly but to some degree.
Could you give a concrete example of the forces behind the evolutionary process? Could you give a concrete example of the forces behind the evolutionary process?
Suddenly the Berlin Wall comes down. Why? Because the people of Germany awakened to the pressure of this energy which was saying 'Freedom! Freedom!' or 'Justice!' and they pulled down the wall. It didn't happen the day before. It had to wait until there was that intensity of energy and a sufficient number of people responding to it for the event to take place. And that's how everything happens in the world. These energies reach a certain focus and intensity to which humanity willy-nilly responds — not everybody, but the more advanced and then the less sensitive, and so on. And when a kind of critical mass has been reached, then events happen — humanity acts, governments act. All these different things take energy.
The Masters are seeking all the time to direct, to guide humanity but we have freewill and They always respect that freewill. They never do and never will infringe our freewill. They cannot and will not force humanity to do anything. And so we do many destructive things. The Masters try to guide us not to do certain things — for example: to prevent war, to make war unthinkable — but if humanity thinks war and has the destructive tendency then they make war, despite the work of the Masters to prevent it. We, humanity, create wars, even though behind us stand the Masters doing everything They can to keep us on a peaceful constructive evolutionary path. The Masters know the way; They have been through evolution. They know the path but They can only teach us the way. They can't do it for us.
The letters published here describe encounters and experiences some of which were confirmed by Benjamin Creme's Master, while the more recent letters carry no such confirmation. It is the writers' and the editors' experience, based on familiarity with such occurrences, plus the writer's own intuitive response, which gives them the confidence to judge these encounters to be significant and meaningful — both personally and generally. Some experiences seem specific to the individual concerned while others speak for themselves in providing hope and inspiration to all. We present them for your consideration.
There is a small town in the Pennines, UK, called Hebden Bridge. Over the years I have posted Emergence copies into many letter boxes there.
A river runs through the middle of the town, and this year [2021], on a cold January Sunday morning, we walked the river side path into the deserted town.
Suddenly a tall young man jumped onto the iron handrail leading to the bridge and proceeded to 'tightrope' walk the three-inch iron rail, as if he was walking on air (in spite of the 30-foot drop into the river). He then jumped onto a parapet on the bridge and turned to look at us.
In my mind I was thinking 'Master Jesus' — no smile, no wave, just a gentle gaze which spoke volumes in the present situation.
Thank you Share International magazine for so many uplifting experiences you print. Much needed in these trying times.
Light and love
L.L., Yorkshire, UK
Flood of joy
Several years ago I had this experience immediately after giving a lecture in Los Angeles. Typically after a talk, people line up to ask me questions or share an experience and this was no exception. Suddenly a short man with dark hair stepped in front of the others to talk with me. He seemed very happy. We spoke briefly and I asked his name. He said, "I will tell you, but you have to repeat it after me". I said, "OK". He said, "U." I repeated, "U". He said, "R". I said, "R". He said, "Love". I said, "Love – U. R. Love". He beamed at me with a huge smile and said very excitedly, "So are you!" and threw his arms around me in a big hug. I was immediately filled with a flood of joy that is impossible to describe.
I watched as he walked away several steps to the water cooler to get a drink and I thought, 'The Water Carrier'. So I wrote to Benjamin Creme and he confirmed that the man was Maitreya using a familiar. What a wonderful experience of pure joy! And of course it seems the lesson is that, regardless of race, sex or sexual preference, belief system, financial status, level of education, country of origin, or personality type, we are all... "love".
D.L., USA
Angelic messenger
I have been doing Transmission Meditation for nearly six years now [written November 2004] and I also send Reiki healing to people who need it. But I don't really know if I'm doing it correctly — so one day I thought it would be quite nice just to have a sign to let me know if I'm being effective.
I realize it's no good asking for a sign without specifying what sort of sign! So, I came up with the idea of a cloud formation in the shape of an angel, which perhaps would not be too difficult to manifest and distinctive enough not to be imagination. I looked out of the window and saw puffs of clouds floating by, but no angel! I did this once or twice with no result, so I dismissed the thought out of my mind. It didn't really matter if I had a sign or not as I would go on meditating anyway to the best of my ability, in the hope of doing some good.
Two or three days later (15 September 2004), it was a lovely sunny day and I thought "I must get out in the sunshine". I went out and sat on the grass and, after a little while, I happened to look up. And there above me I saw a large angel, with great arching wings coming to a point at the feet. Its head in the North and feet in the South. It stayed there for five to ten minutes before slowly drifting out of shape.
I would very much like to know who or what manifested that angel for me and to say thank you for listening to me and giving me a sign.
J.C-C., Dorset, England
(Benjamin Creme's Master confirmed that the 'angel' was manifested by the Master Jesus.)
Rallying call!
It has come to our notice that more and more people are receiving our information from the website alone, and thus forgoing the small cost of subscribing to Share International magazine.
Not everyone can, or does, use the internet, and it is essential to have an outer, physical expression which can display, with photographs, the events of which we speak. That means there has to be a printed magazine, which of course requires a lot of work from volunteers, and a lot of money to produce and distribute. There may be a notion that Hierarchy dishes out money for this work, but that is not the case. Subscriptions to Share International are an essential part of maintaining our work and reaching the public. The cost of the subscription is kept as low as possible, the magazine is not subsidized by advertisements and printing and postal charges are increasing all the time.
Surely we would all agree that all those who seriously believe in this work would want to support the magazine, whether or not they read the information on the internet.
This selection of quotations is taken from Maitreya (Messages from Maitreya the Christ, Benjamin Creme's Master (A Master Speaks Volumes One and Two), and Benjamin Creme's writings.
Fundamentally, what Maitreya will say we already know — and accept to be true — which is that right human relationships are the basis of life. From moment to moment, by our thoughts and actions, we set into motion causes, the effects of which make our life what it is, for good or for ill. This is the great Law of Cause and Effect.
When we understand this Law and its relation to the Law of Rebirth, we will come to understand the need for harmlessness in all relationships. The rightness, the inevitability, the "commonsense-ness" of right relationship will be driven home to us. (Benjamin Creme, The Reappearance of the Christ and the Masters of Wisdom)
When men see the Christ in person, they will quickly assume a new attitude to life and its problems. They will understand that the problems are man-made, exist in man himself, and are not the fault of an uncaring God or the result of mindless chance. A new sense of responsibility will endow men with the impulse to act for the betterment of all. Co-operation, caring and trust will soon replace the present self-concern, and a new phase will open in the evolution of man. (Benjamin Creme's Master, from 'His name is Love')
My major need today is for those who share My vision to accept the responsibility of action. Many millions there are in the world who know the need of man, who see that vision, but know not the urgency of the time. I rely on all those with a knowledge of your brothers' needs, a sympathy for the sufferings of so many, and a will to change all that. May you be among those upon whom I may call, that together we can usher in a new and better world. (Maitreya, from Message No.46)
If we in the developed world are reluctant to make the sacrifices necessary for all people everywhere to live decent, civilized lives, then we will destroy ourselves. Not as punishment, but the direct result of the Law of Cause and Effect. If we do the right thing, we will transform the world; if we continue as we are, and do not see or accept the need for the changes, then they will not happen. Maitreya says: "Nothing happens by itself. Man must act and implement his will," (Message No.31); so it is up to us. Everything is up to us. (Benjamin Creme, Maitreya's Mission Volume Three)
In a very real sense the world has shrunk to village size and, as in village life, the actions of one affect the lives and interests of all. No longer can any nation stand aside and claim immunity from the results of its misdeeds. Power alone no longer confers this privilege. More and more, the nations are awakening to their mutual dependence and responsibility, and this fact augurs well for the world. (Benjamin Creme's Master, from 'Awakening to responsibility')
We present here phenomena which, to the editors, are "signs of hope" and "signs of the time". Fortunately, our current stock of phenomena confirmed as real and genuine by Benjamin Creme's Master is fairly large. However, in future we will also present material which has not been confirmed. We undertake to be as thorough as possible in our investigation of each 'miracle' or 'sign' and will present them for your consideration only, since we cannot now make use of the confirmation and additional information which in the past was always provided by Benjamin Creme's Master. Further details, when available, are given in the captions to the photographs.
Light Phenomena Worldwide
Light formations in Hannover, Germany. Photograph: B.G., Hannover.
NASA – On 22 February 2021, NASA's Solar Terrestrial Relations Observatory (STEREO) photographed a huge structured object emitting beams of energy close to the sun. (Source: helioviewer.org)
UK – On 15 November 2020, a resident of Weymouth, Dorset, took several photos of the dark, ominous-looking sky. When reviewing them later, a lighted, triangular aerial craft at low altitude could be seen in one of the photos — see detail, below. (Source: mufon.com)
Lakhdar Brahimi is an Algerian former freedom fighter, foreign minister, conflict mediator and UN diplomat, and has been a member of The Elders since the group was founded in 2007. In the following article, he reflects on the importance of working together to recover sustainably from the pandemic and combat other existential threats to humanity such as war and conflict, climate change and nuclear proliferation.
We have seen over the past 12 months how Covid-19 has exacerbated existing inequalities, especially for people already suffering multiple hardships in fragile states and conflict zones. It is vital now that the world does not succumb to 'vaccine nationalism' and other short-term measures that favour the privileged and wealthy — within and between countries and regions. If we have learned anything over the past year, it should be that, in a pandemic, none of us is safe until all of us are safe.
Poverty, inequality and injustice — these three ills of human society did probably not directly cause the pandemic, but they certainly helped it spread. In my personal experience, they were both the cause and the consequence of the conflicts which continue to spread and grow around the world, and I was delighted to discuss these issues with Mary Robinson, our Chair at The Elders, in the latest episode of the 'Finding Humanity' podcast.
The part of the world I come from — the Middle East and North Africa as it is called in 'UN-speak' — is beset with all sorts of problems, and in general, these problems are not being handled that well. 2021 marks the tenth anniversary of the 'Arab Spring', which raised huge hopes when it started in Tunisia in December 2010 and seemed to triumph less than one month later. Now, for most people, this is a bittersweet moment to recall both the bravery of protesters peacefully demanding greater freedom, and the extent to which early optimism has been replaced with deep doubts and even despair.
The region is receiving outside help and for that we must be grateful. But are our countries and people always receiving the right kind of help? The best help any external party can give in a conflict situation or peace process is to encourage the people at the heart of the situation to do what they think will improve the situation in their country.
Pursuing the narrow and selfish objectives of external players, ultimately, does not even achieve those narrow objectives and will damage rather than help the interests of the suffering people in the country in conflict, especially when those external players are working against, rather than with, each other: look at Libya, Lebanon, Syria, Iraq, Yemen and Afghanistan. (Source: theelders.org)
The Economics of Biodiversity: the Dasgupta Review, 2021
A recommended read (excerpt)
Phyllis Creme
"This comprehensive and immensely important report shows us how by bringing economics and ecology face to face we can help to save the natural world and in doing so save ourselves." (David Attenborough, Introduction to the Dasgupta Review, abridged version).
This is the urgent message to world leaders and communities worldwide of the Dasgupta Review. It was commissioned by the UK Chancellor of the Exchequer in 2019, as a 'comprehensive global review of the link between biodiversity and economic growth': to be led by Professor Sir Partha Dasgupta, Emeritus Professor of Economics at Cambridge University. Its brief was to 'assess the economic benefits of biodiversity globally; assess the economic costs and risks of biodiversity loss; and identify a range of actions that can simultaneously enhance biodiversity and deliver economic prosperity'. The Review sits alongside accounts of the climate emergency and is just as important.
This report is a thorough and far-reaching exploration of the relationship between economics and ecology. Although it is primarily focused on economics, it also displays a deep understanding of humanity's relation to nature overall; addressing the "failure of contemporary conceptions of economic possibilities to acknowledge that we are embedded within Nature; we are not external to it".
As the Review spells out, if economics were to treat Nature as a capital asset alongside material products and services, then we would take a different approach to the natural world. Sustainable economic growth requires a different measure from Gross Domestic Product. And this change is essential if we are to prevent the breakdown of this most important but sometimes 'invisible' and silent asset.
Since the 1950s, humanity has despoiled nature to a devastating extent — and has become financially richer in the process. But this is no longer sustainable. The Review argues that losses in biodiversity are undermining the productivity, resilience and adaptability of nature. In turn, this is putting economies, livelihoods and well-being at risk: "Our demands far exceed nature's capacity to supply us with the goods and services we all rely on. We would require 1.6 Earths to maintain the world's current living standards. … Humanity now faces a choice: we can continue down a path where our demands on Nature far exceed its capacity to meet them on a sustainable basis; or we can take a different path, one where our engagements with Nature are not only sustainable but also enhance our collective wellbeing and that of our descendants". Dasgupta has also pointed out that our toxic relationship with nature also gives rise to pandemics. …
In spite of the difficulties, and some apparently intransigent forces, the report is not pessimistic. …
Solutions are possible, but they rely on the will of individuals, governments and international bodies to act — and quickly. Fundamentally, we have to change our measures of economic success to guide us on a more sustainable path, for "Nature is our home. Good economics demands we manage it better. Our economies are embedded in nature, not external to it."
The Review concludes: "The fault is not in economics; it lies in the way we have chosen to practise it. Transformative change is possible — we and our descendants deserve nothing less."
A disciple's responsibility in crucial times – Part Two (excerpt)
Anne Marie Kvernevik
The Masters are still striving to teach the disciples and humanity the art of co-operation: "It is Our earnest desire that men learn the art of co-operation, and to this end shall We act as mentors, teaching through example. So liberating is co-operation it is surprising, is it not, that men have been so tardy in learning its joys."*
Benjamin Creme's Master stated that cooperation is the only way forward for humanity: "It is by co-operation alone that mankind will survive, by co-operation alone the new civilization will be built, by cooperation only that men can know and demonstrate the inner truth of their divinity."*
It is to be hoped that the Covid-19 pandemic will be a lesson for humanity, and especially for the world leaders, to recognize the oneness of humanity and that the way to solve our grave problems is by sharing and justice. It is to be hoped that the disciples in the world, through the support of the Masters — which they have — will also be teaching humanity (as embryonic models) the art of co-operation and cohesion by example.
Benjamin Creme's Master states: "Men must release themselves from the poison of competition, must realize it for the glamour which it is, and, seeing the Oneness of all men, embrace co-operation for the General Good. ('The Art of Co-operation', Share International, September 2000)
Benjamin Creme underlines the disciple's responsibility: "The soul aspect of a nation can only demonstrate through the disciples and initiates in the nation, because they are the ones who give expression to the soul aspect of any nation. It is up to the initiates and disciples to come forward with the ideas, the inspiring thought-form of co-operation on a global scale." …
[* From the article 'Co-operation', Share International, December 1984.]
Full contents of the printed edition
The Master —, through Benjamin Creme: Brotherhood
Key to pandemic prevention: heal 'broken relationship' with nature: Andrea Germanos
Phyllis Creme: The Economics of Biodiversity: the Dasgupta Review, 2021 (book recommendation)
Stephen Leahy: Making Peace With Nature — Humanity's to do list
Latino teenager sues an oil company and wins
Anne Marie Kvernevik: A disciple's responsibility in crucial times – part two
Pope Francis' visit to Iraq: "Landmark moment in modern religious history"
Lakhdar Brahimi calls for a spirit of solidarity
Signs of the time: Light phenomena worldwide
Andrea Germanos: 'Saving lives is never a crime': Aid groups reject charges over refugee rescue missions
The Law of Action and Reaction – a compilation
Aart Jurriaanse: Attributes of the disciple
Matteo Marchisio: What role for South-South cooperation in post-Covid recovery?
Thalif Deen: Money laundering: the darker side of the world's offshore financial system
Benjamin Creme: Questions and answers
Have you enjoyed reading Share International online?
You can also subscribe and have Share International magazine delivered to your home. The printed edition carries full versions of all the excerpts shown here, plus additional articles, interviews, book reviews, questions and answers, and more.
Copyright © 2021 Share International. All rights reserved.
The views expressed by authors other than Share International correspondents do not necessarily reflect those held by the editors of this magazine. By the same token, interviewees, and authors other than our own correspondents, do not necessarily subscribe to or support the information and approach which form the basis and context of this publication.
The reproduction of Share International content in printed or electronic form, including internet websites, requires written permission which will not be unreasonably withheld. Please apply via our Contact page. When Share International content is reproduced, please credit Share International as the source: © Share International. A clipping (in the case of print media), or a notification and link (in the case of online reproduction) would be appreciated.
Other issues in 2021
|
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\section{How to Use this Template}
\maketitle
\begin{abstract}
\noindent Traffic-light modelling is a complex task, because many factors have to be taken into account. In particular, capturing all traffic flows in one model can significantly complicate the model. Therefore, several realistic features are typically omitted from most models. We introduce a mechanism to include pedestrians and focus on situations where they may block vehicles that get a green light simultaneously. More specifically, we consider a generalization of the Fixed-Cycle Traffic-Light (FCTL) queue. Our framework allows us to model situations where (part of the) vehicles are blocked, e.g. by pedestrians that block turning traffic and where several vehicles might depart simultaneously, e.g. in case of multiple lanes receiving a green light simultaneously. We rely on probability generating function and complex analysis techniques which are also used to study the regular FCTL queue. We study the effect of several parameters on performance measures such as the mean delay and queue-length distribution.
\end{abstract}
\section{Introduction}
Traffic lights are currently omnipresent in urban areas and one of their aims is to let vehicles drive across an intersection in such a way that the delay is as small as possible. The modelling of queues in front of traffic lights therefore has always been and still is an important topic of study in road-traffic engineering. The overall aim is to create a model that is as realistic as possible, which poses to be a difficult task. There are many studies devoted to traffic control at intersections, ranging from simulation studies and the use of artificial intelligence to analytical and explicit calculations to find good control strategies. This study provides a more realistic extension of the so-called Fixed-Cycle Traffic-Light (FCTL) queue, see e.g.~\cite{darroch1964traffic}, which allows us to perform analytical computations. We call the model that we consider in this paper the blocked Fixed-Cycle Traffic-Light (bFCTL) queue with multiple lanes. Our main aim is to provide an exact computation of the steady-state queue length of the bFCTL queue with multiple lanes, although a transient analysis (possibly with time-varying parameters) is also possible.
The regular FCTL queue is a well-studied model in traffic engineering, see~\cite{boon2019pollaczek,boon2018networks,darroch1964traffic,hagen1989comparison,mcneil1968solution,newell1965approximation,oblakova2019exact,van2006delay,webster1958traffic}. The typical features of the FCTL queue are:
\begin{itemize}
\item A fixed cycle length, fixed green and red times;
\item A general arrival process;
\item Constant interdeparture times of queued vehicles;
\item Whenever the queue becomes empty during a green period, it remains empty since newly arriving vehicles pass the crossing at full speed without experiencing any delay.
\end{itemize}
Due to all the fixed settings, the model focuses on a single lane and does not capture any dependencies or interactions with other lanes.
Unfortunately, in many cases the FCTL queue cannot be applied as a realistic model to study the queue-length distribution in front of a traffic light. Take, for example, an intersection where vehicles from a single stream are spread onto two lanes which are both heading straight and where both lanes are governed by the same traffic light, see also Figure~\ref{fig:vis}(a). Indeed, since there are two parallel lanes in each direction, two vehicles can cross the intersection simultaneously and vehicles will in general switch lanes (if needed) to join the lane with the shorter queue. Moreover, it might be the case that the vehicles are blocked during the green period, e.g. because of a pedestrian crossing the intersection (receiving a green light at the same time as the stream of vehicles that we model), see Figure~\ref{fig:vis}(b) for a visualization. Such blockages might also occur in a multi-lane scenario (where all lanes are going in the same direction) as visualized in Figure~\ref{fig:vis}(c). It is apparent that these situations cannot be modeled by the standard FCTL queue. However, it is extremely relevant to understand such intersections better as is also indicated in e.g.~\cite{tageldin2019models,yan2018design} and more generally, it is e.g. important to investigate pedestrian behaviour at intersections as is done in e.g.~\cite{zhou2019simulation}.
The study in this paper provides an extension of the FCTL queue to account for such situations. They seem to be the most common in practice, see e.g.~\cite{shaoluen2020random} for another study on the case as in Figure~\ref{fig:vis}(b). For extensions and other scenarios, we refer the reader to Section~\ref{sec:discussion}. Note that the blocking mechanisms discussed in this paper give rise to more complicated model dynamics and dependencies, which make it impossible to use traditional methods (e.g. Webster's approximation for the mean delay \cite{webster1958traffic}).
\begin{figure}[h!]
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\\
\scriptsize (a)
& \hspace{1cm}
\scriptsize (b)
& \hspace{1cm}
\scriptsize (c)
\end{tabular}
\caption{A visualization of three intersections that can be modeled by the bFCTL queue with multiple lanes. In~(a), the blue rectangle indicates a combination of lanes which can be analyzed as a bFCTL queue with two lanes. The other lanes at the intersection, the complement of the blue rectangle, can be considered separately because of the fixed settings. In~(b), the blue rectangle indicates a lane that can be modeled as a bFCTL queue with a single lane with blockages. In~(c), the blue rectangle indicates two lanes that we can model as a bFCTL queue with \emph{two} lanes where vehicles are potentially blocked by pedestrians.}
\label{fig:vis}
\end{figure}
A shared right-turn lane as in Figure~\ref{fig:vis}(b), that is a lane with vehicles that are either turning right or are heading straight, has been studied before. However, to the best of our knowledge, there are no papers with a rigorous analysis taking stochastic effects into account while computing e.g. the mean queue length for such lanes. Shared right-turn lanes where vehicles are blocked by pedestrians crossing immediately after the right turn have been considered in e.g.~\cite{alhajyaseen2013left,chen2011saturation,chen2014investigation,chen2008influence,shaoluen2020random,milazzo1998effect,roshani2017effect,rouphail1997pedestrian}. Several case studies, such as~\cite{chen2014investigation} and \cite{roshani2017effect}, indicate that there is a potentially severe impact by pedestrians blocking vehicles. This is for example also reflected in the Highway Capacity Manual (HCM) as published by the Transportation Research Board~\cite{manual2010}, where the focus is on capacity estimation. Most papers have also focused on the estimation of the so-called saturation flow rate, or capacity, of shared lanes where turning vehicles are possibly blocked by pedestrians, see e.g.~\cite{chen2008influence,milazzo1998effect,rouphail1997pedestrian}. In~\cite{chen2011saturation}, it is stated that the used functions for the capacity estimation for turning lanes (such as those in the HCM) might have to be extended to account for stochastic behaviour. In a small case study, \cite{chen2011saturation} confirm that the capacity estimation by the HCM yields an overestimation in various cases. The overestimation of the capacity by the HCM is also observed in several other papers, such as in~\cite{chen2014investigation,chen2008influence} and \cite{shaoluen2020random}, and is probably due to random/stochastic effects. The bFCTL queue explicitly models such stochastic behaviour.
A potential application of the bFCTL queue with a single lane as depicted in Figure~\ref{fig:vis}(b) can be found in the model that is studied in~\cite{shaoluen2020random}, which has also been the source of inspiration for this paper. A description of the model in~\cite{shaoluen2020random} is as follows, where we replace the left-turn assumption for left-driving traffic to a right-turn assumption for the more standard case of right-driving traffic. We have a shared lane with straight-going and right-turning traffic controlled by a traffic light, where immediately after the right turn there is a crossing for pedestrians. The pedestrians may block the right-turning vehicles as the vehicles and pedestrians may receive a green light simultaneously. The right-turning vehicles that are blocked, immediately block all vehicles behind them.
Another potential application of the bFCTL queue is to account for bike lanes. Bikes might make use of a dedicated lane or mix with other traffic and in both cases a turning vehicle might be (temporarily) blocked by bicycles because the bicycles happen to be in between the vehicle and the direction that the vehicle is going. As such, blockages have an influence on the performance measures of the traffic light. It is important to take such influences into account in order to find good traffic-light settings. Several papers studying the impact of bikes can be found in~\cite{allen1998effect,chen2007influence,guo2012effect} and \cite{chen2018evaluating}. Also other types of blocking might occur, such as by a shared-left turn lane and opposing traffic receiving a green light simultaneously, see e.g.~\cite{chai2014traffic,levinson1989capacity,liu2011arterial,liu2008lane,ma2017two,wu2011modelling,yang2018analytical,yao2013optimal}. As such, the bFCTL queue (either with multiple lanes or not) is a relevant addition to the literature because it enables a more suitable modelling of traffic lights at intersections with crossing pedestrians and bikes, which leads to traffic-light control strategies for more realistic situations. In order to model a situation where two opposing streams of vehicles potentially block one another as in e.g.~\cite{yang2018analytical}, the bFCTL queue would have to be extended. For more references on the topics discussed in this paragraph see also the review paper by~\cite{cheng2016review}. Another related study is~\cite{oblakova2019exact} who introduce a model with ``distracted'' drivers, which can be considered as an FCTL queue with independent blockages, but this blocking mechanism is a special case of the one discussed in the present paper.
As mentioned before, we call the model that we consider in this paper the bFCTL queue with multiple lanes. On the one hand we thus allow for the modelling of vehicle streams that are spread over multiple lanes and on the other hand we allow for vehicles to be (temporarily) blocked during the green phase. The key observation to constructing the mathematical model is that we can model multiple parallel (say $m$) lanes as \emph{one} single queue where batches of (up to) $m$ delayed vehicles can depart in one time slot, for more details see Section~\ref{sec:materials}. The resulting queueing model is one-dimensional just like the standard FCTL queue, which allows us to obtain the probability generating function (PGF) of the steady-state queue-length distribution of the bFCTL queue with multiple lanes and to provide an exact characterization of the capacity.
In summary, our main contributions are as follows:
\begin{itemize}
\item[(i)] We extend the general applicability of the Fixed-Cycle Traffic-Light (FCTL) queue. We allow for traffic streams with multiple lanes and for vehicles to be blocked during the green phase. We refer to this model variation as the blocked Fixed-Cycle Traffic-Light (bFCTL) queue with multiple lanes.
\item[(ii)] We provide an exact capacity analysis for the bFCTL queue relieving the need for simulation studies.
\item[(iii)] We provide a way to compute the PGF of the steady-state queue-length distribution of the bFCTL queue and show that it can be used to obtain several performance measures of interest.
\item[(iv)] We provide a queueing-theoretic framework for the study of shared lanes with potential blockages by pedestrians. This e.g. allows for the study of several performance measures and allows us to model the impact of randomness on the performance measures.
\end{itemize}
\subsection*{Paper outline}
The remainder of this paper is organized as follows. In Section~\ref{sec:materials}, we give a detailed model description. This is followed by a capacity analysis, a derivation of the PGF of the steady-state queue-length distribution, and a derivation of some of the main performance measures in Section~\ref{sec:queue_length_derivation}. In Section~\ref{sec:results}, we provide an overview of relevant performance measures for some numerical examples and point out various interesting results. We wrap up with a conclusion and some suggestions for future research in Section~\ref{sec:discussion}.
\section{Detailed model description}\label{sec:materials}
In this section we provide a detailed model description of the bFCTL queue with multiple lanes.
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\draw[blue] (49,56.5) arc(-90:90:2.25);
\draw[red] (31,61.5) to (49,61.5);
\draw[red] (31,61.5) arc(270:90:2.25);
\draw[lightgray] (31,61.5) -- (31,66);
\draw[red] (31,66) to (49,66);
\draw[red] (49,61.5) arc(-90:90:2.25);
\end{tikzpicture} & \hspace{1cm}
\begin{tikzpicture}[scale=0.10]
\draw[black,fill=lightgray](0,7) rectangle (10,35);
\draw[black,fill=lightgray] (5,0) circle (140pt);
\draw[black,fill=black](1,8) rectangle (9,9);
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\draw[black,fill=black](1,13) rectangle (9,14);
\draw [decorate,decoration={brace,amplitude=5pt,mirror,raise=4ex},rotate=180]
(-5,-7) -- (-5,-14) node[midway,xshift=3em]{$m$};
\draw[black,thick] (0.5,7.5) rectangle (9.5,14.5);
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\draw [decorate,decoration={brace,amplitude=5pt,mirror,raise=4ex},rotate=180,blue]
(-5,-15) -- (-5,-22) node[midway,xshift=3em]{$m$};
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\end{tikzpicture}
\\
\scriptsize (a)
&
\hspace{0.2cm} \scriptsize (b)
\end{tabular}
\caption{Visualization of (a) the bFCTL model in terms of an intersection with a traffic stream spread over $m$ lanes and (b) the corresponding queueing model, where the server takes batches of $m$ vehicles into service simultaneously unless there are less than $m$ vehicles present; in that case all vehicles are taken into service.}
\label{fig:vis_queue}
\end{figure}
We assume that there are multiple lanes for a traffic stream, that is a group of vehicles coming from the same road and heading into one (or several) direction(s), governed by a \emph{single} traffic light. A visualization can be found in Figure~\ref{fig:vis_queue}(a). As can be seen in Figure~\ref{fig:vis_queue}(a), we assume that there are $m$ lanes and that vehicles spread themselves among the available lanes in such a way that $m$ vehicles can depart if there are at least $m$ vehicles. In practice, this assumption makes sense as drivers gladly minimize their delay by choosing free lanes. The traffic-light model is then turned into a queueing model with a \emph{single} queue with batch services of vehicles, see Figure~\ref{fig:vis_queue}(b). The batches generally consist of $m$ delayed vehicles (we consider delayed vehicles as is done in the study of the FCTL queue, see e.g.~\cite{boon2019pollaczek}), except if less than $m$ delayed vehicles are present at the moment that a batch is taken into service: then all vehicles are taken into service. We further assume that the time axis is divided into time intervals of constant length, where each interval corresponds to the time it takes for a batch of delayed vehicles to depart from the queue. We will refer to these intervals as slots.
We now turn to discuss two concrete, motivational examples that fit the framework of the bFCTL queue with multiple lanes. After that, we describe the assumptions of the bFCTL queue more formally.
\begin{example}[Shared right-turn lane]
In this example we consider the scenario as in Figure~\ref{fig:vis}(b). We have batches of vehicles of size $1$, i.e. batches are individual vehicles.
We distinguish between vehicles that are going straight ahead and vehicles that turn right. We do so because only right-turning vehicles can be blocked by crossing pedestrians. The probability that an arbitrary vehicle at the head of the queue is a turning vehicle is $p$. Such a turning vehicle is blocked by a pedestrian in slot $i$ with probability $q_i$, i.e.\,a pedestrian is present on the crossing with probability $q_i$. If a turning vehicle is blocked, all vehicles behind it are also blocked. Then, we proceed to the next slot, $i+1$, and check whether there are any pedestrians crossing (with probability $q_{i+1}$): if there are pedestrians crossing, all vehicles in the queue keep being blocked and otherwise, the turning vehicle at the head of the queue may depart and the blockage of all other vehicles is removed.
Moreover, if the queue becomes empty during the green period, it will in general not start building again (cf. the FCTL assumption for the regular FCTL queue, see e.g.~\cite{van2006delay}), \emph{except} if there arrives a turning vehicle and there is a crossing pedestrian. The turning vehicle is then blocked and any vehicles arriving in the same slot behind this vehicle are also blocked.
\end{example}
\begin{example}[Two turning lanes]\label{ex:two}
In this example we consider the scenario as in Figure~\ref{fig:vis}(c). We have batches of vehicles of size $2$.
In this example, there is no need to make a distinction between vehicles: each vehicle is a turning vehicle with probability $1$, i.e. $p=1$. During each slot $i$, there are pedestrians on the crossing with probability $q_i$ and if there is a pedestrian, all vehicles in the batch are blocked, as are all other vehicles in the queue: there are no vehicles that can complete the right turn. All vehicles in the queue keep being blocked until there are no pedestrians crossing anymore.
Also in this example, the queue of vehicles might dissolve entirely during the green period. If that happens, it only starts building again if there are vehicles arriving \emph{and} if there are pedestrians crossing. In such cases, all arriving vehicles get blocked and remain blocked until there are no pedestrians anymore.
\end{example}
We are now set to formalize the assumptions for the bFCTL queue with multiple lanes. We number them for clarity and provide additional remarks if necessary. We start with a standard assumption for FCTL queues and a standard assumption on the independence of arriving vehicles, see, e.g.~\cite{van2006delay}.
\begin{assumption} \label{ass:disctime} [Discrete-time assumption]
We divide time into discrete slots.
The red and green times, $r$ and $g$ respectively, are fixed multiples of those discrete slots and the total cycle length, $c=g+r$, thus consists of an integer number of slots. Each slot corresponds to the duration of the departure of a batch of maximally $m$ delayed vehicles, where $m$ is the maximum number of vehicles that can cross the intersection simultaneously. Any arriving vehicle that finds at least $m$ other vehicles waiting in front of the traffic light is delayed and joins the queue.
\end{assumption}
\begin{assumption}[Independence of arrivals]
All arrivals are assumed to be independent. In particular, the arrivals during slot $i$ do not affect the arrivals in slot $j$ when $i\neq j$.
\end{assumption}
The next three assumptions, Assumptions~\ref{as:division}, \ref{ass:removal}, and \ref{ass:adapted}, relate to the blockages of vehicles and that allow us to explicitly model such blockages.
\begin{assumption}[Green period division]\label{as:division}
For the green period we distinguish between two parts, $g_1$ and $g_2$, with $g=g_1+g_2$. During the first part of the green period, blockages might occur (see also Assumption~\ref{ass:removal} below). During the second part of the green period there are no blockages at all. We further assume that $g_2>0$ for technical reasons.
\end{assumption}
We make a division of the green period into two parts as is done in e.g.~\cite{shaoluen2020random}. Moreover, such a division is often present in reality and it slightly eases the computations later on. This e.g. means that during the second part of the green period there is a ``no walk'' sign flashing, during which pedestrians are not allowed to cross the intersection. We note that if $g_1=0$ (and $m=1$), we obtain the standard FCTL queue.
Further, we assume that the second part of the green period is strictly positive, mainly for technical reasons. This basically implies that at least one batch of vehicles can depart from the queue during each cycle and that there is \emph{no} batch of vehicles in the queue at the end of the cycle that has caused a blockage before. If $g_2$ would be zero and if a batch of vehicles is blocked at the end of slot $g_1$, this would allow for a blockage to carry over to the next cycle, leading to a slightly more complex model. Moreover, the red and green times could be taken random in the regular FCTL queue when the times are independent of one another, see e.g.~\cite{boon2021optimal}. At the expense of additional complexity, our framework for the bFCTL queue could be adjusted to account for such sources of randomness. This would allow one to model (to some extent) randomness in, for example, crossing times of pedestrians.
Next, we make an assumption about the blocking of batches of vehicles during the first part of the green period. We take into account that (i) not all batches of vehicles at the head of the queue are potentially blocked (e.g. because only turning batches of vehicles can be blocked); that (ii) if a batch of vehicles is blocked, all vehicles behind it are blocked as well; that (iii) once a blockage occurs, it carries over to the next slot; and that (iv) blockages occur only in the combined event of having a right-turning batch of vehicles at the head of the queue \emph{and} pedestrians crossing the road.
\begin{assumption}[Potential blocking of batches]\label{ass:removal}
A batch of vehicles, arriving at the head of the queue in time slot $i$, turns right with probability $p_i$. Independently, in time slot $j$, pedestrians cross the road with probability $q_j$, blocking right-turning traffic. As a consequence, whenever a new batch arrives at the head of the queue, this batch will be served in that particular time slot if (i) the batch goes straight ahead, \emph{or} (ii) the batch turns right but there are no crossing pedestrians. Once a batch (of right-turning vehicles) is blocked, it will remain blocked until the next time slot when no pedestrians cross the road. Note that this will be time slot $g_1+1$ at the latest. If the batch at the head of the queue is blocked, it will also block all the other batches in the queue, including those that would go straight. Both $p_i$ and $q_i$ are allowed to depend on the slot $i$.
\end{assumption}
\begin{remark
\label{rem:pi}
We make a couple of remarks on the values of the $p_i$.
First, we note that $p_i$ is not representing the probability that the batch at the head of the queue is a turning batch, but rather the probability that a \emph{newly arriving} batch that gets to the head of the queue in slot $i$, is a turning batch. In practice, this will usually \emph{not} depend on the slot in which the batch gets to the head of the queue. This would imply that $p_i=p$ (see, e.g., Example~\ref{ex:two}) and that we could drop the subscript $i$. However, we are able to let $p_i$ depend on the slot in the derivation of the formulas and opt to provide the general case where $p_i$ is allowed to depend on $i$.
Moreover, in the case that $m>1$, we will often assume that either $p_i=0$, as is the case in Figure~\ref{fig:vis}(a), or $p_i=1$, as is the case in Figure~\ref{fig:vis}(c). This is mainly due to the fact that \emph{all} vehicles in a batch have to be treated similarly: the framework of the bFCTL queue does not allow for batches consisting of one right-turning vehicle that is blocked and one straight-going vehicle that is allowed to depart because it is not blocked. I.e. a case with mixed traffic and \emph{multiple} lanes, such as the shared right-turn lane example in Figure~\ref{fig:vis}(b) but with $m>1$, is not modeled by the bFCTL queue. We do not consider this to be a severe restriction as it will often be the case in practice that $p_i=0$ or $p_i=1$ if $m>1$. We stress that the case with $m=1$ as depicted by the blue rectangle in Figure~\ref{fig:vis}(b) can be studied by the bFCTL queue.
\end{remark}
\begin{remark
\label{rem:blockages}
We would like to stress that the blockage of a batch of vehicles carries over to the next slot. E.g. if a vehicle is a right-turning vehicle in Figure~\ref{fig:vis}(b) and is blocked, it is still at the head of the queue in the next slot. So, as soon as a blockage actually takes place, we are essentially in a different state of the system than in the case where there is no blockage: if there is a blockage in time slot $i$ then we are sure that there is a right-turning batch at the head of the queue in time slot $i+1$. This is why we have two mechanisms for the blocking: on the one hand we have the $p_i$ to check whether \emph{new} batches that get to the head of the queue are right turning and on the other hand we have the pedestrians crossing in slot $i$ accounted for by the $q_i$.
\end{remark}
We need one final assumption which is a slightly adapted version of the standard FCTL assumption. We require a slight change because of the potential blocking of vehicles during the first part of the green phase and because of the possibility that there is more than one delayed vehicle departing in a single slot during the green period because of the batch-service structure.
\begin{assumption}[bFCTL assumption]\label{ass:adapted}
We assume that any vehicle arriving during a slot where $m-1$ or less vehicles are in the queue, may depart from the queue immediately together with the $m-1$ or less delayed vehicles. There are two exceptions: (i) if this batch of $m-1$ or less vehicles is blocked or (ii) if the queue was empty and there is an arriving vehicle that gets blocked, in which case that vehicle gets blocked together with any arriving vehicles after that vehicle. In the former case, all arriving vehicles together with the delayed vehicles remain at the queue. In the latter case, the first blocked vehicle is delayed and any arriving vehicles behind it (if any) are also delayed and blocked where we restrict ourselves to the situation where the queue is empty. If the queue was not empty, then we assume that either all arriving vehicles in that slot are blocked and delayed (because the batch at the head of the queue is blocked) or that all arriving vehicles are allowed to depart along with the batch of delayed vehicles (because the batch at the head of the queue is not blocked). Summarizing, if the queue length at the start of the slot is at least $1$ but at most $m-1$, we either have no departures (in case of a blockage) or \emph{all} vehicles are allowed to cross the intersection (including arriving vehicles). If the queue length is $0$, we only have a non-zero queue at the end of the slot if a vehicle gets blocked: then the blocked vehicle and any vehicles arriving behind it are queued.
\end{assumption}
\begin{remark
\label{rem:bFCTL}
The bFCTL assumption allows one to model a situation where arriving vehicles get blocked if the queue was already empty before the start of the slot. Although, in principle, one can use any distribution for the number of arriving vehicles that are blocked, there are only few logical choices in practice. For example, in the case of Figure~\ref{fig:vis}(b), the number of (potentially) blocked vehicles that arrive at the queue during slot $i$ would correspond to the number of vehicles counting from the first right-turning vehicle among all vehicles arriving in slot $i$: these vehicles will be blocked if there is a crossing pedestrian in slot $i$. In Figure~\ref{fig:vis}(c), any arriving vehicle is a turning vehicle. So, if there is a crossing pedestrian, all arriving vehicles in slot $i$ are blocked.
\end{remark}
The combination of all the above assumptions enables us to view the process as a discrete-time Markov chain, which in turn allows us to obtain the capacity and the PGF of the steady-state queue-length distribution of the bFCTL queue with multiple lanes. We do so in the next section.
\section{Capacity analysis, PGFs, and performance measures for the bFCTL queue}\label{sec:queue_length_derivation}
In this section we provide an exact analysis for the bFCTL queue. We start with an exact characterization of the capacity in Subsection~\ref{subsec:capacity}. In Subsection~\ref{subsec:pgf_derivation}, we obtain the steady-state queue-length distribution in terms of PGFs where we thus focus on the \emph{transforms} of the queue-length distribution, because we cannot directly obtain closed-form expressions for the probabilities. We can use the methods devised in e.g.~\cite{abate1992numerical} and~\cite{choudhury1996numerical} to obtain numerical values from the PGFs for the queue-length probabilities and moments respectively. Without giving details, we stress that our recursive approach in Subsection~\ref{subsec:pgf_derivation} also allows us to provide a transient analysis in which case we can also take time-varying parameters into account. In Subsection~\ref{sec:performance_measures}, we study several important performance measures of the bFCTL queue.
\subsection{Capacity analysis for the bFCTL queue}\label{subsec:capacity}
In this subsection we develop a computational algorithm to determine the capacity for the bFCTL queue. The capacity is defined as the maximum number of vehicles that can cross the intersection in the given lane group, per time unit. In the standard FCTL queue, the capacity can simply be determined by multiplying the saturation flow with the ratio of the green time and the cycle length. In the bFCTL model, however, there are subtle dependencies which carry over from one cycle to the next cycle. We will capture these dependencies by means of a Markov reward model. The Markov chain with the associated transition probabilities that we use is depicted in Figure~\ref{fig:MC}. We are interested in the number of departures of delayed vehicles in each time slot. For this reason, the Markov chain that we consider here only has states $(i,s)$ for $i=1,\dots,g_1$ representing the slots during the first part of the green period and $s=u,b$, representing the case where vehicles are not blocked ($s=u$) and the case where vehicles are blocked ($s=b$). We also have states $i$ for $i=g_1+1,\dots,g_1+g_2+r$ representing the slots during the second part of the green period and the red period. Finally, we create an artificial state $0$ to gather the rewards from states $(1,b)$ and $(1,u)$. The long-term mean number of departures of delayed vehicles can now be determined by means of a Markov reward analysis.
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[->,auto,node distance=2.2cm]
\tikzstyle{every state}=[fill=white,draw=black,text=black,minimum size=1.1cm]
\node[draw,circle,minimum height = 1cm] (A) {$0$};
\node[draw,circle,minimum height = 1cm] (B) [above right of=A] {$(1,u)$};
\node[draw,circle,minimum height = 1cm] (C) [below right of=A] {$(1,b)$};
\node[draw,circle,minimum height = 1cm] (D) [right = 1.5 cm of B] {$(2,u)$};
\node[draw,circle,minimum height = 1cm] (E) [right = 1.5 cm of C] {$(2,b)$};
\node[minimum size=1cm] (H) [right=1.5 cm of D] {$\dots$};
\node[minimum size=1cm] (I) [right=1.5 cm of E] {$\dots$};
\node[draw,circle,minimum height = 1cm] (J) [right=1.5 cm of H] {$(g_1,u)$};
\node[draw,circle,minimum height = 1cm] (K) [right=1.5 cm of I] {$(g_1,b)$};
\node[draw,circle,minimum height = 1cm] (L) [below right of=J] {$g_1+1$};
\node[draw,circle,minimum height = 1.75cm] (LL) [below=2.5 cm of L] {$g_1+2$};
\node[minimum size=1cm] (M) [left=0.5 cm of LL] {$\dots$};
\node[draw,circle,minimum height = 1.75cm] (N) [left=0.5 cm of M] {\begin{tabular}{c}$g_1+$\\$g_2$\end{tabular}};
\node[draw,circle,minimum height = 2cm] (O) [left=0.5 cm of N] {\begin{tabular}{c}$g_1+$\\$g_2+1$\end{tabular}};
\node[minimum size=1cm] (P) [left=0.5 cm of O] {$\dots$};
\node[draw,circle,minimum height = 1.75cm] (Q) [left=0.5 cm of P] {\begin{tabular}{c}$g_1+$\\$g_2+r$\end{tabular}};
\path (A) edge node {\footnotesize{$1-p_1q_1$}} (B);
\path (A) edge node {\footnotesize{$p_1q_1$}} (C);
\path (B) edge node {\footnotesize{$1-p_2q_2$}} (D);
\path (B) edge node[pos=0.78] {\footnotesize{$p_2q_2$}} (E);
\path (C) edge node[pos=0.6,xshift=1.5cm] {\footnotesize{$1-q_2$}} (D);
\path (C) edge node {\footnotesize{$q_2$}} (E);
\path (D) edge node {\footnotesize{$1-p_3q_3$}} (H);
\path (D) edge node[pos=0.78] {\footnotesize{$p_3q_3$}} (I);
\path (E) edge node[pos=0.6,xshift=1.5cm] {\footnotesize{$1-q_3$}} (H);
\path (E) edge node {\footnotesize{$q_3$}} (I);
\path (H) edge node {\footnotesize{$1-p_{g_1}q_{g_1}$}} (J);
\path (H) edge node[pos=0.78] {\footnotesize{$p_{g_1}q_{g_1}$}} (K);
\path (I) edge node[pos=0.6,xshift=1.7cm] {\footnotesize{$1-q_{g_1}$}} (J);
\path (I) edge node {\footnotesize{$q_{g_1}$}} (K);
\path (K) edge node[pos=0.1,xshift=0.8cm,yshift=-0.5cm] {\footnotesize{$1$}} (L);
\path (J) edge node {\footnotesize{$1$}} (L);
\path (L) edge node {\footnotesize{$1$}} (LL);
\path (LL) edge node {\footnotesize{$1$}} (M);
\path (M) edge node {\footnotesize{$1$}} (N);
\path (N) edge node {\footnotesize{$1$}} (O);
\path (O) edge node {\footnotesize{$1$}} (P);
\path (P) edge node {\footnotesize{$1$}} (Q);
\end{tikzpicture}
\caption{Markov chain used to study the capacity of the bFCTL queue.}\label{fig:MC}
\end{figure}
We use Markov reward theory to obtain the capacity of the bFCTL queue. In order to use Markov reward theory, we work backwards from state $g_1+g_2+r$ to obtain the reward in state $0$. Indeed, we get the mean number of vehicles that is able to depart from the queue in an arbitrary cycle when we compute the reward in state $0$. The rewards that we assign to each transition are as follows: if we make a transition to a state $(i,u)$ for $i=1,\dots,g_1$, we receive a reward $m$ reflecting the maximum of $m$ delayed vehicles departing from the queue. We also get a reward $m$ if we make a transition from state $g_1+i$ to state $g_1+i+1$ for $i=1,\dots,g_2-1$. For all other transitions, we receive no reward as there are no vehicles departing. We denote the received reward up to state $(i,s)$ with $r_{i,s}$ with $i=1,\dots,g_1$ and $s=u,b$ and the received reward up to state $i$ with $r_i$ for $i=0$ and $i=g_1+1,\dots,g_1+g_2+r$. Then we get the following relations between the rewards in the various states. We start with defining the total reward in state $g_1+g_2+r$ to be $0$ (there are no vehicle departures while being in state $g_1+g_2+r$), i.e.
\begin{equation} \label{eq:rc}
r_{g_1+g_2+r} = 0.
\end{equation}
For states $i=g_1+g_2,\dots,g_1+g_2+r-1$, we obtain
\begin{equation}
r_{i} = r_{i+1},
\end{equation}
as there are no departures during the red period. However, for states $i=g_1+1,\dots,g_1+g_2-1$, we have
\begin{equation}\label{eqnm1}
r_{i} = m + r_{i+1}
\end{equation}
as there are (potentially) $m$ delayed vehicles departing. For state $(g_1,b)$ we have that
\begin{equation}
r_{g_1,b} = r_{g_1+1},
\end{equation}
as there are no departures when the vehicles are blocked. For state $(g_1,u)$ we obtain
\begin{equation}
r_{g_1,u} = m + r_{g_1+1}\label{eqnm2}
\end{equation}
as there are, at most, $m$ delayed vehicles departing from the queue when we transition from state $(g_1,u)$ to $g_1+1$. Similarly, for states $(i,b)$ with $i=1,\dots,g_1-1$, we get
\begin{equation}
r_{i,b} = q_{i+1}r_{i+1,b}+(1-q_{i+1})r_{i+1,u}
\end{equation}
and for states $(i,u)$ with $i=1,\dots,g_1-1$, we get
\begin{equation}\label{eqnm3}
r_{i,u} = m+p_{i+1}q_{i+1}r_{i+1,b}+(1-p_{i+1}q_{i+1})r_{i+1,u}.
\end{equation}
Finally, for state $0$, we get
\begin{equation}\label{eq:r0}
r_{0} = p_1q_1r_{1,b}+(1-p_1q_1)r_{1,u}.
\end{equation}
Then we have that $r_0$ is the average reward received when traversing the Markov chain as depicted in Figure~\ref{fig:MC}. This average reward translates to the mean number of delayed vehicles that are able to depart from the queue during a cycle, which is exactly the capacity of this lane group. We can thus compute the capacity of the bFCTL queue for each set of input parameters. Along with the mean number arrivals per cycle, we can also check whether the bFCTL queue renders a stable queueing model. If we denote the mean number of arrivals in slot $i$ by $\mathbb{E}[Y_i]$, the mean number of arrivals per cycle is $\sum_{i=1}^c\mathbb{E}[Y_i]$ and the bFCTL queue is stable if $r_0>\sum_{i=1}^c\mathbb{E}[Y_i]$. The procedure to check for stability is summarized in Algorithm~\ref{alg:stability}.
\begin{algorithm}[H]
\caption{Algorithm to check for stability of the bFCTL queue.} \label{alg:stability}
\begin{algorithmic}[1]
\State Input: $\mathbb{E}[Y_i]$ for $i=1,\dots,c$, $g_1$, $g_2$, $c$, $p_{i}$ for $i=1,\dots,g_1$, and $q_{i}$ for $i=1,\dots,g_1$.
\State Use Equations~\eqref{eq:rc} up to \eqref{eq:r0} to determine $r_0$.
\If {$\sum_{i=1}^c\mathbb{E}[Y_i]< r_0$}
\State The bFCTL queue is stable.
\Else
\State The bFCTL queue is not stable.
\EndIf
\end{algorithmic}
\end{algorithm}
\begin{remark}\label{rem:capacity}
One of our model restrictions (Assumption~\ref{ass:disctime}) is that vehicles depart at the end of each time slot, meaning that we do not correct for the fact that turning vehicles might need more time to accelerate. A simple method to account for this effect, which reduces the capacity in practice, is to modify the reward structure of the Markov chain. One can modify the value of $m$ in Equations~\eqref{eqnm1}, \eqref{eqnm2}, and \eqref{eqnm3} to account for the lower departure rate of turning vehicles. For example, one can use
\begin{equation}\label{eqn:capacityCorrection}
m^* = p_i m_\textit{turn} + (1-p_i)m_\textit{through},
\end{equation}
where $m_\textit{through}$ and $m_\textit{turn}$ represent the average number of through-vehicles and turning vehicles, respectively, crossing the intersection per time unit. For this capacity calculation, these numbers do not need to be integers. See Section~\ref{subsec:capacity2} for a numerical example and a comparison to the HCM capacity formula.
\end{remark}
\subsection{Derivation of the PGFs for the bFCTL queue}\label{subsec:pgf_derivation}
First, we need to introduce some further concepts and notation before we continue our quest to obtain the relevant PGFs of the queue-length distribution. We introduce two states, one corresponding to a situation where the queue is blocked and one where this is not the case, cf. Assumption~\ref{ass:removal} and Remark~\ref{rem:blockages} and as is done in Subsection~\ref{subsec:capacity}. We denote the random variable of being in either of the two states with $S$ and $S$ takes the values $b$ (blocked) and $u$ (unblocked). By definition, blocked states only occur during the first part of the green period and if there are vehicles in the queue. We define $S$ to be equal to $u$ if the queue is empty. We denote the joint steady-state queue length (measured in number of vehicles) and the state $S$ at the end of slot $i=1,\dots,g_1$ with the tuple $(X_i,S)$ and we denote its PGF with $X_{i,j}(z)$ where $i=1,\dots,g_1$ and $j=u,b$. We note that $X_{i,b}(z)$ and $X_{i,u}(z)$ are partial generating functions: we e.g. have $X_{i,b}(z) = \mathbb{E}[z^{X_{i}}\mathds{1}\{S=b\}]$, where $\mathds{1}\{S=b\}=1$ if $S=b$ and $0$ otherwise. For the slots $i=1,\dots,c$ we denote the steady-state queue length with $X_i$ and its PGF with $X_i(z)$, so for $i=1,\dots,g_1$ we have that $X_i(z)=X_{i,u}(z)+X_{i,b}(z)$.
We note that, as we are looking at the steady-state distribution of the number of vehicles in the queue, we need to require stability of the queueing model. We can check whether or not the stability condition is satisfied by means of Algorithm~\ref{alg:stability} devised in Subsection~\ref{subsec:capacity}.
We further denote with $Y_i$ the number of arrivals during slot $i$ and with $Y_{i,b}$ we denote the total number of arrivals of potentially blocked vehicles during slot $i$, see also Assumption~\ref{ass:adapted}.
We denote their PGFs respectively with $Y_i(z)$ and $Y_{i,b}(z)$. Later in this subsection, we provide $Y_{i,b}(z)$ for several concrete examples.
In the next part of this subsection we provide the recursion between the $X_{i,j}(z)$, $i=1,\dots,g_1$ and $j=u,b$, and the $X_i(z)$, $i=g_1+1,\dots,c$. Afterwards, we wrap up with some technicalities that need to be overcome to obtain a full characterization of all the PGFs.
\subsubsection{Recursion for the $X_{i,j}(z)$}
We start with the relation between $X_{1,b}(z)$ and $X_c(z)$. We distinguish several cases while making a transition from slot $c$ to a blocked state in slot $1$. We get
\begin{equation}\label{eq:X1b}
\begin{aligned}
X_{1,b}(z) = & p_1q_1 \mathbb{E}[z^{X_c+Y_1}\mathds{1}\{X_c>0\}] + q_1 \mathbb{E}[z^{Y_{1,b}}\mathds{1}\{X_c=0\}\mathds{1}\{Y_{1,b}>0\}] +\\& 0\cdot\mathbb{E}[ \mathds{1}\{X_c=0\}\mathds{1}\{Y_{1,b}=0\}]\\
= & p_1q_1 X_{c}(z) Y_1(z) + q_1\mathbb{P}(X_{c}=0)\left(Y_{1,b}(z)-Y_{1,b}(0)-p_1Y_1(z)\right).
\end{aligned}
\end{equation}
We explain this relation as follows: if the queue is nonempty at the end of slot $c$, we need both a right-turning batch of vehicles and a crossing pedestrian in slot $1$ to get a blockage, which happens with probability $p_1q_1$. The queue length at the end of slot $1$ is then $X_c+Y_1$. The second term can be understood as follows: if $X_c=0$, the queue at the end of slot $c$ is empty and then we get to a blocked state if there is a pedestrian crossing (which happens with probability $q_1$) and if $Y_{1,b}>0$, in which case the queue length is $Y_{1,b}$. Note that we further have that the case $X_{1,b}=0$ cannot occur (by definition) as indicated by the term on the second line of Equation~\eqref{eq:X1b}.
Similarly, we derive $X_{1,u}(z)$:
\begin{align}\label{eq:X1u}
X_{1,u}(z) = &\nonumber (1-p_1q_1) \mathbb{E}[z^{X_c+Y_1-m}\mathds{1}\{X_c\geq m\}] + (1-p_1q_1)\mathbb{E}[z^0\mathds{1}\{1\leq X_c\leq m-1\}] + \\&(1-q_1)\mathbb{E}[z^0 \mathds{1}\{X_c=0\}]+q_1\mathbb{E}[z^0 \mathds{1}\{X_c=0\}\mathds{1}\{Y_{1,b}=0\}] \\
= &\nonumber(1-p_1q_1) X_{c}(z) \frac{Y_1(z)}{z^m} + (1-p_1q_1)\sum_{l=1}^{m-1}\mathbb{P}(X_{c}=l)\left(1-\frac{Y_{1}(z)}{z^{m-l}}\right)+\\&\mathbb{P}(X_c=0)\left(1-q_1+q_1Y_{1,b}(0)-(1-p_1q_1)\frac{Y_1(z)}{z^m}\right).\nonumber
\end{align}
This relation can be understood in the following way: first, if there are at least $m$ vehicles at the end of slot $c$ and if there is no blockage (which occurs with probability $1-p_1q_1$, i.e. the complement of a blockage occurring), then the queue length at the end of slot $1$ is $X_c+Y_1-m$. Secondly, if there is at least $1$ but at most $m-1$ vehicles at the end of slot $c$, we have an empty queue at the end of slot $1$ if there is no blockage (which is the case with probability $1-p_1q_1$). Thirdly, if the queue is empty at the end of slot $c$, then the queue remains empty if there are no pedestrians crossing (occurring with probability $1-q_1$) or if there is a pedestrian crossing (occurring with probability $q_1$) while $Y_{1,b}=0$. This fully explains Equation~\eqref{eq:X1u}.
In a similar way, we obtain the following relations for slots $i=2,\dots,g_1$:
\begin{equation}\label{eq:Xib}
\begin{aligned}
X_{i,b}(z) = & p_iq_i \mathbb{E}[z^{X_{i-1}+Y_i}\mathds{1}\{S=u\}] + q_i \mathbb{E}[z^{X_{i-1}+Y_i}\mathds{1}\{S=b\}] + \\& q_i\mathbb{E}[z^{Y_{i,b}}\mathds{1}\{X_{i-1}=0\}\mathds{1}\{S=u\}\mathds{1}\{Y_{i,b}>0\}]\\
= &p_iq_i X_{i-1,u}(z) Y_i(z) + q_iX_{i-1,b}(z)Y_i(z) +\\&q_i\mathbb{P}(X_{i-1}=0,S=u)\left(Y_{i,b}(z)-Y_{i,b}(0)-p_iY_{i}(z)\right),
\end{aligned}
\end{equation}
where we have to take both transitions from slot $i-1$ while being blocked (the case $S=b$) and not being blocked (the case $S=u$) into account, and
\begin{align}
X_{i,u}(z) = & (1-p_iq_i) \mathbb{E}[z^{X_{i-1}+Y_i-m}\mathds{1}\{X_{i-1}\geq m\}\mathds{1}\{S=u\}] +\nonumber\\& (1-q_i) \mathbb{E}[z^{X_{i-1}+Y_i-m}\mathds{1}\{X_{i-1}\geq m\}\mathds{1}\{S=b\}]+ \nonumber\\& (1-p_iq_i)\mathbb{E}[z^0\mathds{1}\{1\leq X_{i-1}\leq m-1\}\mathds{1}\{S=u\}] + \nonumber\\& (1-q_i)\mathbb{E}[z^0\mathds{1}\{1\leq X_{i-1}\leq m-1\}\mathds{1}\{S=b\}] +\nonumber\\& (1-q_i)\mathbb{E}[z^0\mathds{1}\{X_{i-1}=0\}\mathds{1}\{S=u\}]+\nonumber\\&q_i\mathbb{E}[z^0\mathds{1}\{X_{i-1}=0\}\mathds{1}\{S=u\}\mathds{1}\{Y_{i-1,b}=0\}] \\
= &(1-p_iq_i) X_{i-1,u}(z) \frac{Y_i(z)}{z^m} + (1-q_i)X_{i-1,b}(z)\frac{Y_i(z)}{z^m}+ \nonumber\\&(1-p_iq_i)\sum_{l=1}^{m-1}\mathbb{P}(X_{i-1}=l,S=u)\left(1-\frac{Y_{i}(z)}{z^{m-l}}\right)+\nonumber\\& (1-q_i)\sum_{l=1}^{m-1}\mathbb{P}(X_{i-1}=l,S=b)\left(1-\frac{Y_{i}(z)}{z^{m-l}}\right)+\nonumber\\&\mathbb{P}(X_{i-1}=0,S=u)\left(1-q_i+q_iY_{i,b}(0)-(1-p_iq_i)\frac{Y_i(z)}{z^m}\right).\nonumber
\end{align}
In order to derive $X_{g_1+1}(z)$, we note that we need to take the cases into account where the queue was blocked or not during slot $g_1$. We then get
\begin{align}
\begin{aligned}
X_{g_1+1}(z) = & \mathbb{E}[z^{X_{g_1}+Y_{g_1+1}-m}\mathds{1}\{X_{g_1}\geq m\}\mathds{1}\{S=u\}]+\\&\mathbb{E}[z^{X_{g_1}+Y_{g_1+1}-m}\mathds{1}\{X_{g_1}\geq m\}\mathds{1}\{S=b\}]+\\&\mathbb{E}[z^0\mathds{1}\{X_{g_1}\leq m-1\}\mathds{1}\{S=u\}]+\mathbb{E}[z^0\mathds{1}\{X_{g_1}\leq m-1\}\mathds{1}\{S=b\}] \\
= & \left(X_{g_1,u}(z) +X_{g_1,b}(z) \right)\frac{Y_{g_1+1}(z)}{z^m}+\\&\sum_{l=0}^{m-1}\left(\mathbb{P}(X_{g_1}=l,S=u\right)\left(1-\frac{Y_{g_1+1}(z)}{z^{m-l}}\right)+\\&\sum_{l=1}^{m-1}\left(\mathbb{P}(X_{g_1}=l,S=b\right)\left(1-\frac{Y_{g_1+1}(z)}{z^{m-l}}\right).
\end{aligned}
\end{align}
For $i=g_1+2,\dots,g_1+g_2$, we obtain the following
\begin{equation}
\begin{aligned}
X_{i}(z) = & \mathbb{E}[z^{X_{i-1}+Y_{i}-m}\mathds{1}\{X_{i-1}\geq m\}]+\mathbb{E}[z^0\mathds{1}\{X_{i-1}\leq m-1\}] \\
= & X_{i-1}(z) \frac{Y_{i}(z)}{z^m}+\sum_{l=0}^{m-1}\left(\mathbb{P}(X_{i-1}=l\right)\left(1-\frac{Y_{i}(z)}{z^{m-l}}\right),
\end{aligned}
\end{equation}
while for slots $i=g_1+g_2+1,\dots,c$ we get
\begin{equation}\label{eq:Xg1+g2+i}
\begin{aligned}
X_{i}(z) = \mathbb{E}[z^{X_{i-1}+Y_{i}}]= X_{i-1}(z)Y_i(z).
\end{aligned}
\end{equation}
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The combination of all equations above, provides us with a recursion with which we can express $X_{g_1+g_2}(z)$ in terms of $Y_i(z)$, $Y_{i,b}(z)$, $\mathbb{P}(X_{i}=l,S=u)$ and $\mathbb{P}(X_{i}=l,S=b)$ for $i=1,\dots,g_1$ and $l=0,\dots,m-1$, and $\mathbb{P}(X_{i} = l)$ for $i=g_1+1,\dots,g_1+g_2-1$, $i=c$, and $l=0,\dots,m-1$, with the following general form:
\begin{equation}\label{eq:generalbFCTL}
X_{g_1+g_2}(z) = \frac{X_n(z)}{X_d(z)},
\end{equation}
with known $X_n(z)$ and $X_d(z)$. We refrain from giving $X_n(z)$ and $X_d(z)$ in the general case because of their complexity and only provide them under simplifying assumptions later in this subsection. The $Y_{i}(z)$ are known, but we still need to obtain the $Y_{i,b}(z)$, the $\mathbb{P}(X_{i}=l,S=u)$ and $\mathbb{P}(X_{i}=l,S=b)$ for $i=1,\dots,g_1$ and $l=0,\dots,m-1$, and the $\mathbb{P}(X_{i} = l)$ for $i=g_1+1,\dots,g_1+g_2-1$, $i=c$, and $l=0,\dots,m-1$. We start with the $Y_{i,b}(z)$ and then come back to the unknown probabilities.
The occurrence of the PGF $Y_{i,b}(z)$ directly relates to Assumption~\ref{ass:adapted}. As mentioned before in Remark~\ref{rem:bFCTL}, one could, a priori, use any positively distributed, discrete random variable. However, when we have a specific example in mind, there is usually one logical definition, see also Remark~\ref{rem:Yib} below.
\begin{remark}\label{rem:Yib
In general, we define $Y_{i,b}$ to be the random variable of the total number of arrivals of potentially blocked vehicles during slot $i$, cf. Assumption~\ref{ass:adapted}. In case $m=1$, such as in Figure~\ref{fig:vis}(b), the interpretation of the $Y_{i,b}(z)$ is straightforward. We simply count the number of arriving vehicles starting from the first vehicle that is a turning vehicle. We get the following expression for $Y_{i,b}(z)$:
\begin{align*}
Y_{i,b}(z) & = \sum_{k=0}^\infty \mathbb{P}(Y_{i,b} = k)z^k\\
& = \sum_{j=0}^\infty \mathbb{P}(Y_i=j)(1-p_i)^j + \sum_{k=1}^\infty \sum_{j=k}^\infty \mathbb{P}(Y_i = j) (1-p_i)^{j-k} p_i z^k \\
& = Y_i(1-p_i) + \sum_{j=1}^\infty p_i \mathbb{P}(Y_i=j)(1-p_i)^j\sum_{k=1}^j \left(\frac{z}{1-p_i}\right)^k \\
& = Y_i(1-p_i) + \sum_{j=1}^\infty p_i \mathbb{P}(Y_i=j)(1-p_i)^j z\frac{1-\left(\frac{z}{1-p_i}\right)^j}{1-p_i-z}\\
& = Y_i(1-p_i) + \frac{p_i z}{1-p_i-z}\sum_{j=1}^\infty \mathbb{P}(Y_i=j)\left((1-p_i)^j-z^j\right)\\
& = Y_i(1-p_i) + \frac{p_iz}{1-p_i-z}\left(Y_i(1-p_i)-Y_i(z)\right),
\end{align*}
where in the second step we condition on the total number of arrivals and take into account how we can get to $k$ blocked vehicles; in the third step we interchange the order of the summation; and in the fourth step we compute a geometric series.
If $m>1$, the interpretation as above for the case $m=1$ is not necessarily meaningful. It is more difficult to compute the $Y_{i,b}$ in a logical and consistent way. This has to do with the fact that if $m>1$ we consider batches of vehicles that are either all blocked or not, whereas the $Y_{i,b}$'s are about individual vehicles. As mentioned before in Remark~\ref{rem:pi}, if $m>1$ we often have that either $p_i=0$ or $p_i=1$. If $p_i=0$, the general expression for $Y_{i,b}(z)$ reduces to:
\[
Y_{i,b}(z) = Y_{i}(1) + 0\cdot(Y_i(1)-Y_i(z)) = Y_{i}(1)=1,
\]
which makes sense as there are no turning vehicles in case $p_i=0$. If $p_i=1$, we have that:
\[
Y_{i,b}(z) = Y_{i}(0) - (Y_{i}(0)-Y_{i}(z))=Y_{i}(z),
\]
which is also logical: every arriving vehicle is a turning vehicle if $p_i=1$, so we have that $Y_{i,b}(z)=Y_{i}(z)$.
\end{remark}
Except for the constants $\mathbb{P}(X_{i}=l,S=u)$ and $\mathbb{P}(X_{i}=l,S=b)$ for $i=1,\dots,g_1$ and $l=0,\dots,m-1$, and $\mathbb{P}(X_{i} = l)$ for $i=g_1+1,\dots,g_1+g_2-1$, $i=c$, and $l=0,\dots,m-1$, we are now done. We explain how to find the (so far) unknown constants in the next part of this subsection.
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\subsubsection{Finding the unknowns in $X_{g_1+g_2}(z)$}\label{subsubsec:completion}
As mentioned before, we still need to find several unknowns before the expression for $X_{g_1+g_2}(z)$ is complete.
The standard framework for the FCTL queue as described in e.g.~\cite{van2006delay} is also applicable to the bFCTL queue with multiple lanes with some minor differences. Although we are dealing with more complex formulas, the key ideas are identical. We have $m(g_1+g_2)+(m-1)g_1$ unknowns in the numerator $X_n(z)$ of $X_{g_1+g_2}(z)$ in Equation~\eqref{eq:generalbFCTL} and we have $m(g_1+g_2)$ roots with $|z|\leq 1$ for the denominator $X_d(z)$ of $X_{g_1+g_2}(z)$, assuming stability of the queueing model. An application of Rouch\'{e}'s theorem, see e.g.~\cite{adan2006application}, shows that $X_d(z)$ indeed has $m(g_1+g_2)$ roots on or within the unit circle assuming stability. One root is $z=1$, which leads to a trivial equation and as a substitute for this root, we put in the additional requirement that $X_{g_1+g_2}(1)=1$. The remaining $(m-1)g_1$ equations are implicitly given in Equations~\eqref{eq:X1b} and \eqref{eq:Xib}. We give them here separately for completeness. We have for $k=1,...,m-1$
\begin{equation*}\label{eq:bFCTL_prob_blocked_first}
\mathbb{P}(X_{1} = k,S=b) =
p_1q_1 \sum_{l=1}^k \mathbb{P}(X_{c} = l)\mathbb{P}(Y_1 = k-l) + q_1\mathbb{P}(X_{c}=0)\mathbb{P}(Y_{1,b} = k),
\end{equation*} and for $i=2,\dots,g_1$ and $k=1,\dots,m-1$
\begin{equation*}\label{eq:bFCTL_prob_blocked}
\begin{aligned}
\mathbb{P}(X_{i} = k, S=b) = &
\sum_{l=1}^{k} \left\{p_iq_i\mathbb{P}(X_{i-1}=l,S=u)+q_i\mathbb{P}(X_{i-1}=l,S=b)\right\}\mathbb{P}(Y_i = k-l)\\&+ q_i\mathbb{P}(X_{i-1} = 0,S=u)\mathbb{P}(Y_{i,b} = k),
\end{aligned}
\end{equation*}
which provides us with the $(m-1)g_1$ additional equations. In total, we obtain a set of $m(g_1+g_2)+(m-1)g_1$ linear equations with $m(g_1+g_2)+(m-1)g_1$ unknowns, which we can solve to find the unknown $\mathbb{P}(X_{i}=l,S=u)$, for $i=1,\dots,g_1$ and $l=0,\dots,m-1$, the unknown $\mathbb{P}(X_{i}=l,S=b)$, for $i=1,\dots,g_1$ and $l=1,\dots,m-1$, and the unknown $\mathbb{P}(X_{i}=l)$, for $i=g_1+1,\dots,g_1+g_2-1$, $i=c$, and $l=0,\dots,m-1$. Due to the complicated structure of our formulas, we do not obtain a similar, easy-to-compute Vandermonde system as for the standard FCTL queue (see~\cite{van2006delay}), but a linear solver is in general able to find the unknowns (we did not encounter any numerical issues/problems in the examples that we studied).
There are several ways to obtain the roots of $X_d(z)$ in Equation~\eqref{eq:generalbFCTL}. Because those roots are subsequently used in solving a system of linear equations, we need to find the required roots with a sufficiently high precision, certainly if $m(g_1+g_2)+(m-1)g_1$ is large. In some cases, the roots can be found analytically, e.g. in case the number of arrivals per slot has a Poisson or geometric distribution. In other cases, the roots have to obtained numerically. There are several ways to do so. An algorithm to find roots is given in~\cite{boon2019pollaczek}, Algorithm~1, while two other methods, one based on a Fourier series representation and one based on a fixed point iteration, are described in~\cite{janssen2005analytic}.
\subsection{Performance measures}\label{sec:performance_measures}
Now that we have a complete characterization of $X_{g_1+g_2}(z)$, we can find the PGFs of the queue-length distribution at the end of the other slots by employing Equations~\eqref{eq:X1b} up to \eqref{eq:Xg1+g2+i}. This basically implies that we can find any type of performance measure related to the queue-length distribution. As an example we find the PGF of the queue-length distribution at the end of an arbitrary slot. We denote this PGF with $X(z)$ and obtain the following expression:
\begin{equation*}
X(z) = \frac{1}{c}\sum_{i=1}^c X_{i}(z) .
\end{equation*}
Another important performance measure is the delay distribution. The mean of the delay distribution, $\mathbb{E}[D]$, can easily be derived from the mean queue length at the end of an arbitrary slot by means of Little's law with a time-varying arrival rate (for a proof of Little's law in this setting see e.g.~\cite{stidham1972lambda}):
\begin{equation*}
\mathbb{E}[D] = \frac{X^\prime(1)}{\frac{1}{c}\sum_{i=1}^cY_{i}^\prime(1)}.
\end{equation*}
The PGF of the delay distribution can be derived (as is done for the FCTL queue in~\cite{van2006delay}), but such a derivation is more difficult. In the regular FCTL queue, the number of slots an arriving car has to wait is deterministic when conditioned on the number of vehicles in the queue and the time slot in which the car arrives. This is not the case for the bFCTL queue as the occurrence of blockages is random. By proper conditioning on the various blocked slots and queue lengths, one can obtain the delay distribution from the distribution of the queue length. We do not pursue this here.
If we want to obtain probabilities and moments from a PGF, we need to differentiate the PGF and respectively put $z=0$ or $z=1$. In our experience, this has not proven to be a problem. However, differentiation might become prohibitive in various settings, e.g. when $m(g_1+g_2)+(m-1)g_1$ becomes large or if we want to obtain tail probabilities. There are ways to circumvent such problems. If we are pursuing probabilities and do not want to rely on differentiation, we might use the algorithm developed by Abate and Whitt in~\cite{abate1992numerical} to numerically obtain probabilities from a PGF. For obtaining moments of random variables from a PGF, an algorithm was developed in~\cite{choudhury1996numerical} which finds the first $N$ moments of a PGF numerically. Essentially, this shows that, from the PGF, we can obtain any type of quantity related to the steady-state distribution of the queue length, in the form of a numerical approximation.
All formulas computed in this section have been verified by comparing the numerical results with a simulation which mimics our discrete-time queueing model. More information about this simulation is given in Appendix~\ref{a:simulation}.
\section{Examples}\label{sec:results}
We start in Subsection~\ref{subsec:special} with several special cases of the bFCTL queue for which we provide explicit expressions for the PGF of the overflow queue and relate those special cases to the existing literature. Subsequently, we make a comparison between the capacity obtained in the HCM~\cite{HCM} and the capacity in our model in Subsection~\ref{subsec:capacity2}. After that, we investigate the influence of several parameters on the performance measures in numerical examples. We consider performance measures like the mean and variance of the steady-state queue-length distribution, both at specific moments and at the end of an arbitrary slot, the mean delay, and several interesting queue-length probabilities. We study the influence of the $p_i$ and $q_i$ in Subsection~\ref{subsec:parameter}. In Subsection~\ref{subsec:layout}, we compare the case of turning and straight-going traffic on a single lane, as present in the bFCTL queue where blockages of all vehicles might occur, and cases where we have dedicated lanes for the right-turning and straight-going traffic where only turning vehicles are blocked. Note that we will consider each lane \emph{separately} in those examples, so there is no conflict with e.g. Remark~\ref{rem:pi}.
\subsection{Special cases of the bFCTL queue}\label{subsec:special}
We study several special cases of the bFCTL queue, e.g. cases where the bFCTL queue reduces to the FCTL queue.
If $q_i=1$, an explicit expression for the PGF of the distribution of the overflow queue, $X_{g_1+g_2}(z)$, can be written down relatively easily. When it is further assumed, for the ease of exposition, that all $p_i=p$, $Y_i\overset{d}= Y$, $Y_{i,b}\overset{d} = Y_b$ and $m=1$, the following expression for $X_{g_1+g_2}(z)$ is obtained:
\begin{equation}\label{eq:explicit_Xg_q=0}
X_{g_1+g_2}(z) = \frac{X_n(z)}{X_d(z)},
\end{equation} with
\begin{align}
\label{eq:Xn_q=0}
X_n&(z) =z^{g_1+g_2}\sum_{i=0}^{g_2-1}\left(\frac{Y(z)}{z}\right)^{g_2-i-1}\left(1-\frac{Y(z)}{z}\right)\mathbb{P}(X_{g_1+i}=0)+ \nonumber\\
& z^{g_1}Y(z)^{g_2}\sum_{i=0}^{g_1-1}\nonumber \Bigg\{ \mathbb{P}(X_{i}=0,S=u)\Bigg[\left(Y_b(0) - (1-p)\frac{Y(z)}{z}\right)\left((1-p)\frac{Y(z)}{z}\right)^{g_1-i-1}+\\&\left(Y_b(z)-Y_b(0)-p Y(z)\right)Y(z)^{g_1-i-1}\Bigg]+\nonumber\\
&p Y(z)^{g_1-i}\sum_{j=0}^{i-1} \mathbb{P}(X_{j}=0,S=u)\left(Y_b(0) - (1-p)\frac{Y(z)}{z}\right)\left((1-p)\frac{Y(z)}{z}\right)^{i-j-1}\Bigg\},
\end{align}
where $\mathbb{P}(X_{0}=0,S=u)$ is to be interpreted as $\mathbb{P}(X_{c}=0)$, and
\begin{equation}\label{eq:Xd_q=0}
X_d(z) = z^{g_1+g_2} - \left(\left(1-p\right)^{g_1}+p z^{g_1}\sum_{i=0}^{g_1-1}\left(\frac{1-p}{z}\right)^i\right) Y(z)^c.
\end{equation}
The reason that we provide an explicit formula for this particular case is that this formula is significantly easier than the formula in the case where $q_i< 1$ for one or more $i=1,\dots,g_1$. The stability condition (cf. Algorithm~\ref{alg:stability} in Subsection~\ref{subsec:capacity}) for this example is relatively easy to derive and reads as follows:
\begin{equation*}
\begin{cases}
\mu c < g_1+g_2, & \textrm{if } p = 0,\\
\mu c < g_2, & \textrm{if } p = 1,\\
\mu c < g_2+\left(1-(1-p)^{g_1}\right)\frac{1-p}{p}, & \textrm{otherwise},
\end{cases}
\end{equation*}
where $\mu$ is the mean arrival rate per slot, i.e. $\mu=\mathbb{E}[Y]$.
This can be understood as follows: if $p=0$ there are no turning vehicles and we obtain the regular FCTL queue with green period $g_1+g_2$. If $p=1$ all vehicles are turning vehicles and there are no departures during the first part of the green period because $q_i=1$, so we obtain the FCTL queue with green period $g_2$. The other case can be understood as follows: on the left-hand side we have the average number of arrivals per cycle whereas on the right-hand side we have the average number of slots available for delayed vehicles to depart. Indeed, on the right-hand side we have $g_2$, the number of green slots during the second part of the green period which are all available for vehicles to depart, and the number of green slots available for departures during the first green period:
\[
\sum_{i=1}^{g_1}(1-p)^i = \left(1-(1-p)^{g_1}\right)\frac{1-p}{p}.
\]
If $p_i=0$ for all $i$, i.e. there are no blockages occurring at all (regardless of the $q_i$), the FCTL queue with multiple lanes (with green period $g=g_1+g_2$) is obtained. Note that we do not have to include the state $S$, because there are no blockages of batches of vehicles. If $m=1$, we obtain the regular FCTL queue as studied in e.g.~\cite{van2006delay}. This can e.g. be observed when putting $p_i=0$ and $m=1$ in Equations~\eqref{eq:explicit_Xg_q=0}, \eqref{eq:Xn_q=0}, and \eqref{eq:Xd_q=0}. The expression for $X_{g_1+g_2}(z)$ or, alternatively, $X_g(z)$ is (after rewriting):
\begin{align}\label{eq:FCTL}
X_g(z) & = \frac{(z-Y(z))z^{g-1}\sum_{i=0}^{g-1}\mathbb{P}(X_i=0)\left(\frac{Y(z)}{z}\right)^{g-i-1}}{z^g-Y(z)^c},
\end{align}
where $\mathbb{P}(X_{0}=0)$ is to be interpreted as $\mathbb{P}(X_{c}=0)$. For general $m$, we have the following formula:
\begin{equation}
X_g(z) = \frac{z^{mg}\sum_{i=0}^{g-1}\sum_{l=0}^{m-1} \mathbb{P}(X_{i}=l)\left(1-\frac{Y(z)}{z^{m-l}}\right)\left(\frac{Y(z)}{z}\right)^{g-i-1}}{z^{mg}-Y(z)^c},
\end{equation}
where the $\mathbb{P}(X_{0}=l)$, $l=0,\dots,m-1$, are to be interpreted as $\mathbb{P}(X_{c}=l)$. The stability condition for this case can be verified to be
\[
\mu c < m g
\]
which is in accordance with Algorithm~\ref{alg:stability}.
It can also be verified that the bFCTL queue reduces to the regular FCTL queue with green time $g = g_2$ and red time $r+g_1$, if $p_i = 1$ and $q_i = 1$.
We note that for the FCTL queue with a single lane and no blockages (i.e. $p_i=0$ or $p_i=1$ and $q_i=1$) there is an alternative characterization of the PGF in terms of a complex contour integral, see~\cite{boon2019pollaczek}. It remains an open question whether such a contour-integral representation exists for the bFCTL with multiple lanes, as the polynomial structure in terms of $Y(z)/z$ as present in Equation~\eqref{eq:FCTL} is not present in the general bFCTL queue. This feature of the FCTL queue seems essential to obtain a contour-integral expression as is done in~\cite{boon2019pollaczek}.
In \cite{boon2019pollaczek}, a decomposition result is presented in Theorem~2. It shows that several related queueing processes can in fact be decomposed in the independent sum of the FCTL queue and some other queueing process. It is likely that the bFCTL queue with multiple lanes allows for some of those generalizations as well. We mention randomness in the green and red time distributions as a relevant potential extension.
\subsection{Capacity} \label{subsec:capacity2}
In order to compare our model and the existing literature (focusing on the HCM~\cite{HCM}), we provide several examples in this subsection.
The formula for the capacity of a permitted right-turn lane in a shared lane in the HCM is
\begin{equation*}
s_{sr} = \frac{s_{th}}{1+P_r\left(\frac{E_R}{f_{Rpb}}-1\right)},
\end{equation*}
cf.~\cite{HCM} equation (31-105). Here, $s_{sr}$ is the saturation flow of the shared lane, $s_{th}$ the saturation flow of an exclusive through lane, $P_r$ the right-turning portion of vehicles, $E_R$ the equivalent number of through vehicles for a protected right-turn vehicle and $f_{Rpb}$ is the bicycle-pedestrian adjustment factor for right-turn groups. The latter is defined as the average amount of time during the green period during which right-turning vehicles are not blocked, i.e., in our model, there are no pedestrians crossing. There is a procedure provided in the HCM to compute this factor, but in our model this simply corresponds to the $q_i$ and we will determine the $f_{Rpb}$ factor on the $q_i$. Further, in order to make a comparison with our model, we turn the saturation flow of the shared lane into a number of vehicles per cycle.
More concretely, we choose the green period to be $30$ seconds, split into the two phases as follows: $g_1=20$ and $g_2=10$. We pick the cycle length to be $90$ seconds, the time slots to have length $2$ seconds and we focus on a single shared lane, so we have at most $1$ vehicle departing per time slot. Further, we choose the right-turning portion vehicles to be $1$ or $0.9$ in our examples. Lastly, for the HCM formula, we assume that vehicles heading straight have a crossing time of $1$ second. To account for this effect in our bFCTL model, we use the correction discussed in Remark~\ref{rem:capacity}. In this example we have:
\[
m^* = p_i m_\textit{turn} + (1-p_i)m_\textit{through} = p_i\times 1 + (1-p_i)\times 2.
\]
This enables us to compute the capacity in our model and in the HCM up to the $q_i$.
We first focus on the cases with $p_i=1$ and we display the capacity according to the HCM in Figure~\ref{f:cap1}(a). Note that $f_{Rpb}$ is at least $1/3$ because $g_1=20$ and $g_2=10$, implying that during at least a part $1/3$ of the cycle, turning vehicles are not blocked. In Figure~\ref{f:cap1}(a) we also depict two capacities according to the bFCTL queue. In case $(1)$ we assume that all the $q_i$ are the same and are chosen in such a way that the $f_{Rpb}$ in the HCM formula is matched. E.g. in case $f_{Rpb}=1/3$, we choose $q_i=0$ as there are no pedestrians and in case $f_{Rpb}=2/3$, we choose $q_i=1/2$. In case $(2)$, we consider a step function for the $q_i$ such that
\begin{equation*}
q_i = \begin{cases}
1 & \textrm{if } i < k\\
0 & \textrm{if } i > k\\
k^* & \textrm{otherwise,}
\end{cases}
\end{equation*}
for some values of $k$ and $k^*$ such that the $q_i$ match with the value for $f_{Rpb}$ that is used in the formula for the HCM.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\textwidth]{capacity_p_1rev.pdf}
&
\includegraphics[width=0.45\textwidth]{capacity_p_2rev.pdf}\\
(a) & (b) \\[1ex]
\end{tabular}
\caption{Capacity in vehicles per cycle for the example according to the HCM and to the bFCTL queue with two different choices for the $q_i$ (case $(1)$ and case $(2)$) as detailed in the text. We have $p_i=1$ in (a) and $p_i=0.2$ in (b).}
\label{f:cap1}
\end{figure}
Figure~\ref{f:cap1}(a) makes sense: if $f_{Rpb}$ is for example equal to $1$, there are no pedestrians crossing (i.e. $q_i=0$), and then the number of vehicles departing per cycle is $(g_1+g_2)/2=15$. The capacity according to the bFCTL queue when $p_i=1$ is equal to (after simplification)
\begin{equation}\label{eq:cap}
\frac{g_1+g_2}{2} - \sum_{i=1}^{g_1} q_i.
\end{equation}
\textbf{This shows that when $\sum_{i=1}^{g_1} q_i$ is translated into the factor $f_{Rpb}$ in the HCM, we have an identical capacity.} E.g. if the $q_i=0$, then also in the bFCTL queue, the capacity is equal to $15$ vehicles per cycle. Equation~\eqref{eq:cap} also indicates that it does not matter in which slots the pedestrians are crossing if $p_i=1$ (when looking at the capacity). In this case, the $q_i$ only influence the capacity through their sum, however in general the individual $p_i$ and $q_i$ have an impact on the capacity (and the queue-length process). Similar observations hold if $p_i=0$, i.e. there are no turning vehicles.
\textbf{If the $p_i$ are not equal to $1$, there are differences between the capacity in the HCM and the bFCTL queue.} We study an example where $p_i=0.2$. The results are depicted in Figure~\ref{f:cap1}(b). The values for the capacity obtained with the function in the HCM are slightly lower than the values that we obtain in both cases of the bFCTL queue.
In contrast with the previous example, there are differences between all three choices which relate to various causes.
The main reason for the occurring difference between cases $(1)$ and $(2)$ in the bFCTL queue, is that the individual $q_i$ are determining the capacity rather than the total value of the $q_i$'s alone as was the case when $p_i=1$. \textbf{Here we thus see that our detailed description of the queueing model in terms of slots is necessary to fully understand the capacity (and, more generally, the queueing process).}
In this subsection we have been working under several assumptions. If one would, e.g., also incorporate start-up delays as is done in~\cite{shaoluen2020random}, we would see that the capacity in the HCM results in an overestimation of the capacity as is more generally observed~\cite{shaoluen2020random}. We also expect that the distribution of the $q_i$ over the different slots has a bigger impact on the capacity and queueing process if start-up delays are incorporated. Implementing such effects into our model is possible (probably in a similar way as including a departure variable as discussed above), but is beyond the scope of the present paper.
\subsection{The bFCTL queue with turning vehicles and pedestrians}\label{subsec:parameter}
In this subsection, we study the bFCTL queue with a single lane, so $m=1$. The setting in this subsection is as depicted in Figure~\ref{fig:vis}(b). We mainly focus on the distribution of $X_{g_1+g_2}$, to which we refer as the overflow queue, as this is the distribution from which some interesting performance measures can be derived.
This distribution reflects the probability distribution of the queue size at the moment that the green light switches to a red light. We also briefly consider some other performance measures.
\subsubsection{Influence of the number of turning vehicles}\label{sec:influenceofturning}
First, we vary the fraction of right-turning vehicles $p_i$ and study its influence on $X_{g_1+g_2}$. We choose the $p_i$ to be the same for each $i$, so we have $p_i=p$, and we vary $p$. We choose the value of the $q_i=q$ to be $1$, so there are always pedestrians on the pedestrian crossing during the first part of the green period with length $g_1$. In this way, we can effectuate the influence of the fraction of turning vehicles on the performance measures. Further, we choose $g_1$ to be either $2$ or $10$ and we choose $g_2=r=2g_1$. The arrival process is taken to be Poisson with mean $0.39$. Note that the lane is close to its point of saturation, because the capacity can be shown to be equal to $0.4$.
We display results for $\mathbb{P}(X_{g_1+g_2}\leq j)$ for $j=0,\dots,10$ in Figure~\ref{fig:ex3}.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[height=4.7cm]{linechartXg1g2p1_039.pdf}
&
\includegraphics[height=4.7cm]{linechartXg1g2p2_039.pdf}\\
(a) & (b) \\[1ex]
\end{tabular}
\caption{Cumulative Distribution Function (CDF) of the overflow queue for various values of $p_i=p$, $q_i=q=1$, and Poisson arrivals with mean $0.39$. In (a) we have $g_2=r=2g_1=4$ and in (b) we have $g_2=r=2g_1=20$.}
\label{fig:ex3}
\end{figure}
As can be observed from Figure~\ref{fig:ex3}, \textbf{the fraction of turning vehicles may dramatically influence the number of queueing vehicles}. There is virtually no queue at the end of the green period when there are no turning vehicles ($p=0$), whereas in \textbf{more than 50\%} of the cases there is a queue of at least $10$ vehicles at the end of the green period when all vehicles are turning vehicles ($p=1$). The blockages of the turning vehicles in the latter case effectively reduce the green period by a factor $1/3$ in our examples (as $q=1$), which causes the huge difference in performance. We note that the distribution of $X_{g_1+g_2}$ coincides with the overflow queue distribution in the FCTL queue when $p=0$ (when we take $g_1+g_2$ as the green period and $r$ as the red period in the FCTL queue) and when $p=1$ and $q=1$ (with $g_2$ the green period and $r+g_1$ the red period).
When comparing Figures~\ref{fig:ex3}(a) and ~\ref{fig:ex3}(b), we see that \textbf{the influence of $p$ is not uniform across the two examples}. In case $p=0$ or $p=1$, the probability of a large overflow queue is larger for the case where $g_1=2$. This might be clarified by noting that a larger cycle reduces the amount of within-cycle variance which reduces the probabilities of a large queue length. If $0<p<1$ this does not seem to be the case. This might be due to the fact that a relatively big part of the first green period is eaten away by turning vehicles that are blocked when $g_1=10$. For example, when $p>0$ and the first vehicle is a turning vehicle, immediately the entire period $g_1$ is wasted because $q=1$. This is of course also the case when $g_1=2$, but the blockage is resolved sooner and during the second part of the green period the blocked vehicle may depart relatively soon in comparison with the case where $g_1=10$.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.47\textwidth]{barChartXI0_039.pdf}
&
\includegraphics[width=0.47\textwidth]{barChartMeanXi_039.pdf}\\
(a) & (b) \\[1ex]
\end{tabular}
\caption{In~(a) $\mathbb{P}(X_{i}=0)$ for slot number $i=1,\dots,10$ is displayed for two different values of $p_i$, where orange corresponds to $p_i=p=0$ and blue to $p_i=p=0.6$, with $2g_1=g_2=r = 4$, $q_i=q=1$, and with Poisson arrivals with mean $0.39$. In~(b) the same two examples are studied, but the mean queue length $\mathbb{E}[X_{i}]$ at the end of slot $i$ is shown. }
\label{fig:ex3xk}
\end{figure}
In Figure~\ref{fig:ex3xk}(a), we see the probability of an empty queue after slot $i$, where $i=1,2,\dots,c$, for two different values of $p$. For the case $p=0$ (in orange) we have a monotone increasing sequence of probabilities during the green period as one would expect: this setup corresponds to a regular FCTL queue and once the queue empties during the green period, it stays empty. We see that for the case $p=0.6$ (in blue) the probabilities of an empty queue after slot $i$ are much lower (as there are more turning vehicles which might be blocked and hence cause the queue to be non-empty). In fact, the probability of an empty queue even decreases when going from slot $2$ to slot $3$. This can be clarified by the fact that the queue might start building again even when the queue is (almost) empty: e.g. if the queue is empty during the first green period and there is an arrival of a turning vehicle, that vehicle will be blocked as $q=1$ in which case the queue is no longer empty.
The same type of behaviour is reflected in the mean queue length at the end of a slot, as can be observed in Figure~\ref{fig:ex3xk}(b). Even though the green period already started, the queue in the example with $p=0.6$ still grows (in expected value) during the first part of the green period, see the first two blue bars. This is caused by the fact that vehicles might be blocked, \textbf{which demonstrates the possibly severe impact of blocked vehicles on the performance of the system}.
\subsubsection{Influence of the pedestrians}
Secondly, we investigate the influence of the presence of pedestrians by studying various values for the $q_i$. A high value of the $q_i$ corresponds to a high density of pedestrians as $q_i$ corresponds to the probability that a turning vehicle is not allowed to depart during the first green period. Conversely, a low value of the $q_i$ corresponds to a low density of pedestrians and a relatively high probability of a turning vehicle departing during the first green period. We choose $p_i=p=0.5$ and take $g_1=g_2=r=10$. We take Poisson arrivals with mean $0.36$. We study one set of examples where the $q_i$ are constant over the various slots, see Figure~\ref{fig:ex4xk}(a). We also study the influence of the dependence of the $q_i$ on $i$ by investigating two cases with all parameters as before in Figure~\ref{fig:ex4xk}(b). In one case we take $q_i=0.5$ for all $i$, but in the other case we take $q_i=1-(i-1)/g_1$. The latter case reflects a decreasing number of pedestrians blocking the turning flow of vehicles during the first part of the green period.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[height=4.5cm]{linechartXg1g2q1.pdf}
&
\includegraphics[height=4.5cm]{barChartMeanXIqi.pdf}\\
(a) & (b) \\[1ex]
\end{tabular}
\caption{In (a) the CDF of the overflow queue is displayed for various values of the $q_i$ with all $q_i=q$ the same, $p_i=p=0.5$, Poisson arrivals with mean $0.36$, and $g_1=g_2=r=10$. In (b) the $\mathbb{E}[X_{i}]$ are compared for slot number $i=1,\dots,30$ with in orange $q_i=0.5$ and in blue $q_i=1-(i-1)/g_1$ for $i=1,\dots,g_1$. Further, it is assumed that $p_i=p=0.5$, that the number of arrivals in each slot follows a Poisson distribution with mean $0.36$, and that $g_1=g_2=r=10$.}
\label{fig:ex4xk}
\end{figure}
We note that it is \textbf{important to estimate the correct blocking probabilities $q_i$ from data}, when applying our analysis to a real-life situation \textbf{as the $q_i$ have an impact on the performance measures}. In Figure~\ref{fig:ex4xk}(a), we clearly see that the more pedestrians, the longer the queue length at the end of the green period is. Indeed, if there are more pedestrians, there are relatively many blockages of vehicles which causes the queue to be relatively large.
Moreover, \textbf{it is important to capture the dependence of the $q_i$ on the slot $i$ in the right way}, see Figure~\ref{fig:ex4xk}(b). Even though, on average over all slots, the mean number of pedestrians present is similar in the two cases, we see a clear difference between the two examples. In the case with decreasing $q_i$ (in blue), we see an initial increase of the mean queue length during the first green slots of the cycle, caused by a relatively large fraction of turning vehicles ($p=0.5$) \emph{and} a high value of $q_i$. This is not the case in the other example where $q_i=0.5$ for all $i$. After some slots of the first green period, the decrease in the mean queue length is quicker for the example where the $q_i$ decrease when $i$ increases, which can (at least partly) be explained by the decreasing $q_i$. During the remaining part of the cycle, the queue in front of the traffic light behaves more or less the same in both examples and even the mean overflow queue, $\mathbb{E}[X_{g_1+g_2}]$, is not that much different for the two examples. This implies, as can also be observed in Figure~\ref{fig:ex4xk}(b), that the mean queue length during the red period is comparable as well for our setting. This does not hold for the mean queue length at the end of an arbitrary slot and the mean delay, because of the differences in the queue length during the first part of the green period.
\subsection{Shared right-turn lanes and dedicated lanes}\label{subsec:layout}
We continue with a study of several numerical examples that focus on the differences between shared right-turn lanes and dedicated lanes for turning traffic. We do so in order to provide relevant insights in the benefit of splitting the vehicles in different streams. Firstly, we study the difference between a single shared right-turn lane (as visualized in Figure~\ref{fig:vis2}(a)) and a case where the straight-going and turning vehicles are split into two different lanes. In the latter case, we thus have two lanes, one for the straight-going traffic and one for the turning traffic (as visualized in Figure~\ref{fig:vis2}(b)) which we can analyze as two separate bFCTL queues.
\begin{figure}[h!]
\centering
\begin{tabular}{ccc}
\begin{tikzpicture}[scale=0.1,rotate=90]
\draw[black,fill=lightgray](30,20) rectangle (40,50);
\draw[black,fill=lightgray](0,30) rectangle (50,40);
\draw[thick,white,dash pattern=on 7 off 4](0,35) to (50,35);
\draw[thick,white,dash pattern=on 7 off 4](35,0) to (35,60);
\draw[lightgray,fill=lightgray](30,30) rectangle (40,40);
\draw[white,->,thick](24,32.5) to [out = 0, in = 120] (28,30.5);
\draw[white,->,thick](24,32.5) to [out = 0, in = 180] (28.5,32.5);
\draw[white,fill=white](31,25.5) rectangle (32,28.5);
\draw[white,fill=white](33,25.5) rectangle (34,28.5);
\draw[white,fill=white](36,25.5) rectangle (37,28.5);
\draw[white,fill=white](38,25.5) rectangle (39,28.5);
\end{tikzpicture}
&\hspace{-1cm} \begin{tikzpicture}[scale=0.1,rotate=90]
\draw[black,fill=lightgray](30,20) rectangle (40,60);
\draw[black,fill=lightgray](0,30) rectangle (60,50);
\draw[thick,white,dash pattern=on 7 off 4](0,35) to (60,35);
\draw[thick,white,dash pattern=on 7 off 4](0,45) to (60,45);
\draw[thick,white](0,40) to (60,40);
\draw[thick,white,dash pattern=on 7 off 4](35,0) to (35,60);
\draw[lightgray,fill=lightgray](30,30) rectangle (40,50);
\draw[white,->,thick](24,32.5) to [out = 0, in = 120] (28,30.5);
\draw[white,->,thick](24,37.5) to [out = 0, in = 180] (28.5,37.5);
\draw[white,fill=white](31,25.5) rectangle (32,28.5);
\draw[white,fill=white](33,25.5) rectangle (34,28.5);
\draw[white,fill=white](36,25.5) rectangle (37,28.5);
\draw[white,fill=white](38,25.5) rectangle (39,28.5);
\end{tikzpicture}
&\hspace{-1cm} \begin{tikzpicture}[scale=0.1,rotate=90]
\draw[black,fill=lightgray](30,20) rectangle (40,60);
\draw[black,fill=lightgray](0,30) rectangle (60,50);
\draw[thick,white,dash pattern=on 7 off 4](0,35) to (60,35);
\draw[thick,white,dash pattern=on 7 off 4](0,45) to (60,45);
\draw[thick,white](0,40) to (60,40);
\draw[thick,white,dash pattern=on 7 off 4](35,0) to (35,60);
\draw[lightgray,fill=lightgray](30,30) rectangle (40,50);
\draw[white,->,thick](24,32.5) to [out = 0, in = 120] (28,30.5);
\draw[white,->,thick](24,32.5) to [out = 0, in = 180] (28.5,32.5);
\draw[white,->,thick](24,37.5) to [out = 0, in = 180] (28.5,37.5);
\draw[white,fill=white](31,25.5) rectangle (32,28.5);
\draw[white,fill=white](33,25.5) rectangle (34,28.5);
\draw[white,fill=white](36,25.5) rectangle (37,28.5);
\draw[white,fill=white](38,25.5) rectangle (39,28.5);
\end{tikzpicture}
\\
\hspace{-0.9cm}\scriptsize (a)
&
\hspace{-2.85cm}\scriptsize (b)
&
\hspace{-2.85cm}\scriptsize (c)
\end{tabular}
\caption{The various lane configurations considered in Subsection~\ref{subsec:layout}. In~(a) we have a single lane with a shared right-turn lane. In~(b) we have two dedicated lanes: one for straight-going vehicles and one for right-turning traffic, whereas in~(c) we have a two-lane setup with one lane for straight-going vehicles only and a shared right turn.}
\label{fig:vis2}
\end{figure}
Secondly, we compare two two-lane settings. The first is visualized in Figure~\ref{fig:vis2}(b), while the other is a two-lane scenario where one lane is a dedicated lane for straight-going traffic and the other is a shared right-turn lane as depicted in Figure~\ref{fig:vis2}(c). We thus allow for straight-going traffic to mix with some of the right-turning vehicles in the latter case. We do so in order to make sure that the shared right-turn lane together with the lane for vehicles heading straight has the same capacity as the two lanes where the two streams of vehicles are split (as opposed to the first example in this subsection). In both two-lane scenarios we, again, analyze the two lanes as two separate bFCTL queues.
\subsubsection{One lane for the shared right-turn}
We start with comparing the traffic performance of a single shared right-turn lane as in Figure~\ref{fig:vis2}(a), case ($1$), and a two-lane scenario where the turning vehicles and the straight-going vehicles are split as in Figure~\ref{fig:vis2}(b), case ($2$). We refer in the latter case to the lane which has right-turning vehicles as lane $1$ and to the other lane we refer as lane $2$. We assume that the arrival process is Poisson and that the arrival rate of turning vehicles, $\mu_1$, and straight-going vehicles, $\mu_2$, are the same in both cases. The total arrival rate of vehicles is $\mu=\mu_1+\mu_2$ in case~($1$). We choose $p_i=0.3$ for the shared right-turn lane, whereas in the two-lane case we have $p_i=1$ for lane $1$ and $p_i=0$ for lane $2$ and arrival rates $\mu_1=0.3\mu$ at lane $1$ and $\mu_2=0.7\mu$ at lane $2$. Further, we choose $q_i=1$, $g_1=8$, $g_2=20$, and $r=20$. We compute the mean queue length at the end of an arbitrary time slot for both lanes in case ($2$), denoted with $\mathbb{E}[X^{(i)}]$ for lane $i$, and the total mean queue length at the end of an arbitrary time slot, denoted with $\mathbb{E}[X^{t}]$, and which equals $\mathbb{E}[X^{(1)}]+\mathbb{E}[X^{(2)}]$. For case ($1$) we denote the mean queue length at the end of an arbitrary time slot with $\mathbb{E}[X^t]$. The delay of an arbitrary car is denoted with $\mathbb{E}[D]$ for both cases ($1$) and ($2$). We study an example with various values of $\mu$ in Figure~\ref{f:single}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.7\textwidth]{table_1_fig_cars.pdf}\\
(a)\\[1ex]
\includegraphics[width=0.7\textwidth]{table_1_fig_delaywebster.pdf}\\
(b) \\[1ex]
\caption{The total Poisson arrival rate, $\mu$, on the horizontal axis and in (a) the mean queue length at the end of an arbitrary time slot for the various cases and lanes where $\mathbb{E}[X^{t}]=\mathbb{E}[X^{(1)}]+\mathbb{E}[X^{(2)}]$ for case (2), and in (b) the mean delay for the various cases.}
\label{f:single}
\end{figure}
In Figure~\ref{f:single}, we can clearly see that the total mean queue length at the two lanes in case ($2$) is lower than the mean queue length at the single lane in case ($1$). This makes sense from various points of view: in case ($2$), we have twice as many lanes as in case ($1$), so we would expect a smaller total mean queue length in case ($2$). Moreover, in case ($1$), it might happen that straight-going vehicles are blocked. Such blockages cannot occur in case ($2$), as all turning traffic is on lane $1$ and all vehicles that go straight are on lane $2$. These two reasons are the main drivers for the performance difference in cases ($1$) and ($2$). From the point of view of the traffic performance, \textbf{it thus makes sense to split the traffic on a shared right-turn lane into two separate streams of vehicles on two lanes while assuming one lane available for departures in case ($1$) and two lanes in case ($2$)}. We observe similar results when looking at the mean delay and comparing cases ($1$) and ($2$).
\begin{remark}
We emphasized before that the blocking mechanism makes it impossible to use existing methods to analyze the queue lengths and delays. However, in this particular example we have chosen the parameter settings in such a way that case (2) \emph{can} be analyzed using existing methods. The reason is that we have two separate lanes, each with its own ``extreme'' blocking mechanism: lane 1 contains \emph{only} turning vehicles and \emph{all} of them are blocked during $g_1$. Essentially, this turns this lane into a regular FCTL queue with an extra long red period ($r + g_1$) and a shorter green period ($g_2$). Lane 2 contains only vehicles going straight, none of which are blocked. This means that this lane is essentially a regular FCTL queue as well. As a consequence, these two lanes can be analyzed separately using standard FCTL methods. When applying the method described in \cite{van2006delay}, the mean delay would be exactly the same as computed in Figure~\ref{f:single}(b). Moreover, this means that we can also use Webster's well-known approximation for the mean delay for case (2). This has also been visualized in Figure~\ref{f:single}(b) and, indeed, the approximation is remarkably accurate. Still, we stress that this is only possible because we have chosen an extreme blocking mechanism ($q_i=1$) in combination with Poisson arrivals (Webster's approximation only works for Poisson arrival processes).
\end{remark}
\subsubsection{Two lanes for the shared right-turn}
Now we turn to an example where we still have two dedicated lanes as in case ($2$) of the previous example, one for turning traffic and one for straight-going traffic, see Figure~\ref{fig:vis2}(b), but we compare it with a two-lane example where the vehicles mix, see Figure~\ref{fig:vis2}(c). All turning vehicles will be on lane $1$, but we allow some straight-going traffic to be present on lane $1$ too. Lane $1$ is thus a shared right-turn lane. On lane $2$, we only have vehicles that are heading straight. This could, e.g., model a scenario in which some straight-going vehicles desire to take a specific lane, strategically anticipating on an upcoming exit. Anticipation in lane changing behaviour is more generally investigated in e.g.~\cite{choudhury2013modelling} in urban scenarios. We could adapt the value of $p$ depending on this number of strategic vehicles. In order to make a comparison between the various cases that we study and that is as fair as possible, we assume the following: the total arrival rate and the fraction of turning vehicles are the same.
We assume that the probability that an arbitrary vehicle is a turning vehicle is $0.3$ and we vary the total Poisson arrival rate $\mu$ to study the influence of the strict splitting of the turning vehicles. In case ($1$), we thus have an arrival rate at the right-turning lane that satisfies $\mu_1=0.3\mu$, whereas on the other lane we have an arrival rate $\mu_2 = 0.7\mu$. At lane $1$ we have $p_i=1$ and at lane $2$ we have $p_i=0$. In case ($2$) we distinguish between two subcases. In subcase ($2$a) we assume that the total arrival rate at both lanes is the same and thus $\mu_1=\mu_2=0.5\mu$. In subcase ($2$b), we assume that the arrival rate is split in the ratio $2:3$, so $\mu_1= 0.4\mu$ and $\mu_2=0.6\mu$. This implies that in subcase ($2$a) we choose $p_i=0.6$ (the fraction of turning vehicles is then $p\mu_1=0.6\cdot 0.5\mu=0.3\mu$) and in subcase ($2$b) we choose $p_i=0.75$ (the fraction of turning vehicles is then $p\mu_1=0.75\cdot0.4\mu=0.3\mu$), to make sure that we match the number of turning vehicles in case ($1$). Further, we choose $q_i=1$, $g_1=8$, $g_2=16$ and $r=16$. Then, we study the mean queue length at the end of an arbitrary time slot of both lanes, $\mathbb{E}[X^{(1)}]$ and $\mathbb{E}[X^{(2)}]$, and the total average mean queue length at the end of an arbitrary time slot, denoted with $\mathbb{E}[X^t]$. We obtain Figure~\ref{f:split}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.65\textwidth]{table_2_fig_cars_1.pdf}\\
(a)\\[1ex]
\includegraphics[width=0.65\textwidth]{table_2_fig_cars_2.pdf}\\
(b)\\[1ex]
\includegraphics[width=0.65\textwidth]{table_2_fig_cars_t.pdf}\\
(c) \\[1ex]
\caption{The total Poisson arrival rate, $\mu$ and the mean queue length at the end of an arbitrary time slot for the various cases, split among lane $1$ (a), lane $2$ (b) and the total among the two lanes (c) where $\mathbb{E}[X^{t}]=\mathbb{E}[X^{(1)}]+\mathbb{E}[X^{(2)}]$.}
\label{f:split}
\end{figure}
In Figure~\ref{f:split}, we see only small differences in the total mean queue lengths at the end of an arbitrary time slot for low arrival rates. At both lanes, there are few vehicles in the queue. This is different for the examples in Figure~\ref{f:split} with a higher arrival rate. In all examples for case ($1$) we see that the mean queue length at lane $2$, the straight-going traffic lane, is higher than for lane $1$. This is due to the relatively high fraction of vehicles that \emph{have} to use lane $2$ due to the strict splitting between turning and straight-going vehicles. In some sense, lane $1$, which only has turning vehicles, has overcapacity that cannot be used for the busier lane $2$ with only straight-going traffic. This is different for the other two cases, where the traffic is split more evenly across the two lanes. As one would expect, the longest queue in subcase ($2$a) is present at lane $1$, as the arrival rate at both lanes is the same and because vehicles are only blocked at lane $1$, the shared right-turn lane. This points towards another potential improvement and this is found in subcase ($2$b) where we balance the arrival rate differently. \emph{The right balance leads to a more economic use of both lanes and, hence, also the best performance} in this example when looking at $\mathbb{E}[X^t]$.
The results in Figures~\ref{f:single} and \ref{f:split} might seem conflicting at a first glance, but they are not. In the case of a single, shared right-turn lane as in Figure~\ref{f:single}, we see a higher mean queue length than for the two dedicated lanes case in Figure~\ref{f:single}. This is the other way around in Figure~\ref{f:split} (considering case ($2$b)). This is mainly explained by the fact that in case ($2$b) in Figure~\ref{f:split}, we have two lanes and thus twice as many potential departures as in case ($1$) in Figure~\ref{f:single}. This is one of the main factors in the explanation of the differences in the mean performance between the examples studied in Figures~\ref{f:single} and \ref{f:split}.
The two examples in this subsection tell us that \textbf{a separate or dedicated lane for turning traffic does not necessarily improve the traffic flow}. The intuition behind this is that a dedicated lane might have overcapacity which is not employed (e.g. in the case of an asymmetric load on both lanes). This issue is less present when the two dedicated lanes are turned into two lanes, one exclusively for straight-going traffic and one shared lane. This is confirmed by our simulations. As such, \textbf{an in-depth study is needed to obtain the best layout of the intersection and the best traffic-light control}. As a side-remark, we surpass the possibility here that in Figure~\ref{f:split}, case ($1$), we might control the two lanes in a different way, e.g. by prolonging the green period for one of the lanes. This is not possible in cases ($2$a) and ($2$b).
\begin{comment}
\subsection{FCTL queue with multiple lanes}\label{subsec:multiFCTL}
The regular FCTL queue has only a single lane from which vehicles might depart, yet at bigger intersections, this is not realistic. There might be several lanes for, e.g., straight-going traffic which all receive green simultaneously. For a visualization, see Figure~\ref{fig:vis}(a).
\begin{table}[htb!]
\begin{center}
\caption{The bFCTL queue with $m$ lanes, $g=5$, $r=5$, Poisson arrivals, and no blockages. The load $\rho$, the number of lanes $m$, the mean arrival rate $\mu$, and several performance measures are displayed.}
\label{t:multiFCTLpois}
\[\begin{array}{|c|cc|ccHccHc|}\hline
\rho & m & \mu & \mathbb{E}[X_g] & \textrm{Var}[X_g] &
& \mathbb{P}(X_g\geq10) & \mathbb{E}[X^t]
&
& \mathbb{E}[D] \\\hline
0.2 & 1 & 0.1 & 0.000583 & 0.000788 & 1. & <0.00001 & 0.170 & 0.205 & 1.701 \\
& 2 & 0.2 & <0.00001 & 0.000010 & 1. & <0.00001 & 0.317 & 0.449 & 1.587 \\
& 5 & 0.5 & <0.00001 & <0.00001 & 1. & <0.00001 & 0.762 & 1.57 & 1.523 \\
& 10 & 1.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 1.505 & 4.75 & 1.505 \\
& 15 & 1.5 & <0.00001 & <0.00001 & 1. & <0.00001 & 2.252 & 9.56 & 1.502 \\
& 20 & 2.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 3.001 & 16.0 & 1.500 \\\hline
0.4 & 1 & 0.2 & 0.0217 & 0.0384 & 0.985 & <0.00001 & 0.404 & 0.565 & 2.021 \\
& 2 & 0.4 & 0.00324 & 0.00663 & 0.998 & <0.00001 & 0.711 & 1.25 & 1.778 \\
& 5 & 1.0 & 0.000013 & 0.000033 & 1. & <0.00001 & 1.661 & 4.86 & 1.661 \\
& 10 & 2.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 3.240 & 15.9 & 1.620 \\
& 15 & 3.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 4.816 & 33.2 & 1.605 \\
& 20 & 4.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 6.390 & 56.7 & 1.598 \\\hline
0.6 & 1 & 0.3 & 0.180 & 0.429 & 0.903 & 0.000029 & 0.817 & 1.4 & 2.724 \\
& 2 & 0.6 & 0.0770 & 0.215 & 0.963 & 0.000019 & 1.279 & 2.63 & 2.131 \\
& 5 & 1.5 & 0.00788 & 0.0298 & 0.997 & <0.00001 & 2.834 & 9.97 & 1.890 \\
& 10 & 3.0 & 0.00019 & 0.00101 & 1. & <0.00001 & 5.505 & 33.4 & 1.835 \\
& 15 & 4.5 & <0.00001 & 0.000030 & 1. & <0.00001 & 8.181 & 70.5 & 1.818 \\
& 20 & 6.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 10.85 & 121. & 1.809 \\\hline
0.8 & 1 & 0.4 & 1.097 & 4.181 & 0.645 & 0.00842 & 2.025 & 5.73 & 5.063 \\
& 2 & 0.8 & 0.795 & 3.465 & 0.763 & 0.00662 & 2.598 & 7.45 & 3.247 \\
& 5 & 2.0 & 0.359 & 2.038 & 0.917 & 0.00417 & 4.707 & 18.6 & 2.354 \\
& 10 & 4.0 & 0.109 & 0.836 & 0.982 & 0.00242 & 8.621 & 56.4 & 2.155 \\
& 15 & 6.0 & 0.0343 & 0.332 & 0.996 & 0.00130 & 12.68 & 118. & 2.113 \\
& 20 & 8.0 & 0.0109 & 0.127 & 0.999 & 0.00057 & 16.79 & 203. & 2.099 \\\hline
0.98 & 1 & 0.49 & 23.22 & 614.8 & 0.0918 & 0.638 & 24.44 & 617 & 49.88 \\
& 2 & 0.98 & 22.59 & 613.1 & 0.126 & 0.621 & 25.02 & 619 & 25.53 \\
& 5 & 2.45 & 21.02 & 606.9 & 0.219 & 0.580 & 27.06 & 631 & 11.04 \\
& 10 & 4.90 & 18.47 & 589.0 & 0.363 & 0.517 & 30.51 & 666 & 6.227 \\
& 15 & 7.35 & 15.90 & 558.9 & 0.489 & 0.451 & 33.93 & 718.3 & 4.616 \\
& 20 & 9.80 & 13.45 & 517.4 & 0.596 & 0.381 & 37.44 & 788.7 & 3.820 \\\hline
\end{array}\]
\end{center}
\end{table}
Our framework for the bFCTL queue with multiple lanes allows us to model such examples, which we demonstrate in this subsection. We study both the case of a Poisson distributed number of arrivals studied in~\cite{van2006delay}. We thus study a case where $g=g_1+g_2=5$, $p_i=0$ for all $i$, $r=5$, and with Poisson distributed arrivals in each slot with mean $\mu$. We study various cases of $\mu$ and analyze the overflow queue, denoted with $X_g$, the mean queue length at the end of an arbitrary time slot $\mathbb{E}[X^t]$, and the mean delay $\mathbb{E}[D]$. We also vary $m$ to study the influence of having multiple lanes in the FCTL queue. In order to make a comparison between the various cases with different $m$, we scale the arrival rate proportionally with $m$ so that the load or vehicle-to-capacity ratio, $\rho=(c\mu)/(mg)$, is fixed for different values of $m$. Then, we obtain Table~\ref{t:multiFCTLpois}.
We note that there is a difference between analyzing $m$ FCTL queues separately and the joint analysis of the $m$ lanes as presented here. \textbf{It is thus important to perform an analysis that accounts for the number of lanes that vehicles from a single stream can use.} This can most prominently be observed by fixing $\rho$ and considering various values of $m$: the mean and variance of the overflow queue (measured in number of vehicles) then decrease
if we have Poisson arrivals in each slot.
\textbf{This indicates that having more lanes at a single intersection while $\rho$ is fixed, is not necessarily beneficial when looking at the total number of vehicles in the queue}: a high variability in the number of arrivals per slot might result in an increase of the number of vehicles in the queue when the number of lanes is increased. However, in all cases the mean delay decreases if $\rho$ is fixed and $m$ increases.
\end{comment}
\section{Conclusion and discussion}\label{sec:discussion}
In this paper, we have established a recursion for the PGFs of the queue-length distribution at the end of each slot which can be used to provide a full queue-length analysis of the bFCTL queue with multiple lanes. This is an extension of the regular FCTL queue so that we can account for temporal blockages of vehicles receiving a green light, for example because of a crossing pedestrian at the turning lane or because of a (separate) bike lane, and to account for a vehicle stream that is spread over multiple lanes. These features might impact
the traffic-light performance as we have shown by means of various numerical examples. The blocking of turning vehicles and the number of lanes corresponding to a vehicle stream therefore has to be taken into account when choosing the settings for a traffic light.
We briefly touched upon how one should design the layout of an intersection. Interestingly, it might be suboptimal to have a dedicated lane for turning traffic. It seems that mixing turning and straight-going traffic has benefits over a strict separation of those two traffic streams when there are two lanes for this turning and straight-going traffic. We advocate a further investigation into the influence of separating or mixing different streams of vehicles in front of traffic lights. It might be possible to find the optimal division of straight-going and turning vehicles over the various lanes, e.g. by enumerating several possibilities. A more structured optimization seems difficult because of the intricate expressions involved, but would definitely be worthwhile to investigate. Some research on the splitting of different traffic streams has already been done in e.g.~\cite{kikuchi2007lengths,tian2006probabilistic,wu1999capacity} and \cite{zhang2008modeling} and the present study can be seen as an alternative way of modelling the situation at hand.
A possible extension of the results on the bFCTL queue is a study of (the PGF of) the delay distribution. We have refrained from deriving the delay distribution because of its (notational) complexity. Using proper conditioning, one can obtain (the PGF of) the delay distribution for the bFCTL queue.
The work in~\cite{shaoluen2020random}, in which a simulation study of a similar model is performed, has been a source of inspiration for the study in this paper.
There are some extensions possible when comparing our work with~\cite{shaoluen2020random}. We e.g. did not study the influence of start-up delays as is done in~\cite{shaoluen2020random}. Investigating such start-up delays at the beginning of the green period is easily done in our framework: we simply need to adjust the $Y_i$ for the first few slots. Another approach to deal with start-up delays is presented in~\cite{maesnetworks}. Start-up delays which depend on the blocking of vehicles and different slot lengths for different combinations of turning/straight going vehicles, are harder to tackle. One could e.g. introduce additional states (besides states $u$ and $b$) to deal with this. Although the developed recursion does not directly allow for such a generalization, it seems possible to account for this at the expense of a more complex recursion. For the ease of exposition, we have refrained from doing so and we leave a full study on this topic for future research.
A further possible extension of the bFCTL queue would be to consider different blocking behaviours: instead of e.g. a fixed probability $q_i$ for each slot $i$, a more general blocking process might be considered. For example, if there are no pedestrians during slot $i$ for the model depicted in Figure~\ref{fig:vis}(b), then the probability that there are also no pedestrians in slot $i+1$, might be relatively high. In other words, there might be \emph{dependence} between the various slots when considering the presence of pedestrians. We gave an example where there is dependence between the current and the next slot, but it is also possible to consider such dependencies among more than two slots. It is worthwhile to investigate generalizations of the blocking process in order to further increase the general applicability of the bFCTL queue with multiple lanes.
Another generalization for the blocking mechanism, is to block only a \emph{part} of the $m$ vehicles that are at the head of the queue. Indeed, we restrict ourselves to the cases where either all vehicles in a batch of size $m$ are blocked (or not). In various real-life examples, it might be the case that only part of the $m$ vehicles are blocked. It would be interesting to investigate whether such a model can be analyzed. Further, a situation with ``a right turn is always permitted'' scenario might be investigated. In such a case, right-turning vehicles are always free to turn, but might be blocked by straight-going vehicles in front them, which have to wait for a red traffic light, or are blocked by pedestrians. Straight-going vehicles might be blocked by turning traffic waiting for pedestrians. It seems that such a case, at the expense of additional complexity, can be tackled by a similar type of recursion as the one that is developed in this paper by extending and generalizing the blocking mechanism (and, thus, the recursion) to the red period.
\paragraph{Discussion.} We end this paper with a discussion on its practical applicability. Although we have extended the standard model for traffic signals with fixed settings, there are still quite some possible improvements, as discussed in the above paragraphs. Still, to the best of our knowledge, this paper is the first to present analytical results for traffic intersections with blocking mechanisms, based on a queueing theoretic approach. Note that standard formulas like Webster's approximation for the mean delay \cite{webster1958traffic} cannot be used in these situations. From a practical point of view, the most relevant extension to the current analysis would be to deal with start-up delays that depend on the blocking of vehicles. One way to do this, is by considering different slot lengths for different combinations of turning/straight going vehicles, inspired by an analysis in ~\cite{maesnetworks}. This would make it possible to compute a saturation flow adjustment factor due to the right-turning movements at shared lane conditions (see also Biswas et al. \cite{biswas2018}).
Finally, we also advocate an investigation whether the bFCTL queue with a vehicle-actuated mechanism (rather than the fixed green and red times that we consider) results in a tractable model.
\paragraph*{Acknowledgements} We would like to thank Onno Boxma for several interesting discussions that a.o. improved the readability of this manuscript. We are also thankful to Joris Walraevens who suggested the current exposition of the PGF recursion and to the reviewers who suggested several improvements of the paper.
\paragraph*{Funding} The work in this paper is supported by the Netherlands Organization for Scientific Research (NWO) under grant number 438-13-206.
\paragraph*{Disclosure statement} The authors report there are no competing interests to declare.
\bibliographystyle{tfcad}
\section{How to Use this Template}
\maketitle
\begin{abstract}
\noindent Traffic-light modelling is a complex task, because many factors have to be taken into account. In particular, capturing all traffic flows in one model can significantly complicate the model. Therefore, several realistic features are typically omitted from most models. We introduce a mechanism to include pedestrians and focus on situations where they may block vehicles that get a green light simultaneously. More specifically, we consider a generalization of the Fixed-Cycle Traffic-Light (FCTL) queue. Our framework allows us to model situations where (part of the) vehicles are blocked, e.g. by pedestrians that block turning traffic and where several vehicles might depart simultaneously, e.g. in case of multiple lanes receiving a green light simultaneously. We rely on probability generating function and complex analysis techniques which are also used to study the regular FCTL queue. We study the effect of several parameters on performance measures such as the mean delay and queue-length distribution.
\end{abstract}
\section{Introduction}
Traffic lights are currently omnipresent in urban areas and one of their aims is to let vehicles drive across an intersection in such a way that the delay is as small as possible. The modelling of queues in front of traffic lights therefore has always been and still is an important topic of study in road-traffic engineering. The overall aim is to create a model that is as realistic as possible, which poses to be a difficult task. There are many studies devoted to traffic control at intersections, ranging from simulation studies and the use of artificial intelligence to analytical and explicit calculations to find good control strategies. This study provides a more realistic extension of the so-called Fixed-Cycle Traffic-Light (FCTL) queue, see e.g.~\cite{darroch1964traffic}, which allows us to perform analytical computations. We call the model that we consider in this paper the blocked Fixed-Cycle Traffic-Light (bFCTL) queue with multiple lanes. Our main aim is to provide an exact computation of the steady-state queue length of the bFCTL queue with multiple lanes, although a transient analysis (possibly with time-varying parameters) is also possible.
The regular FCTL queue is a well-studied model in traffic engineering, see~\cite{boon2019pollaczek,boon2018networks,darroch1964traffic,hagen1989comparison,mcneil1968solution,newell1965approximation,oblakova2019exact,van2006delay,webster1958traffic}. The typical features of the FCTL queue are:
\begin{itemize}
\item A fixed cycle length, fixed green and red times;
\item A general arrival process;
\item Constant interdeparture times of queued vehicles;
\item Whenever the queue becomes empty during a green period, it remains empty since newly arriving vehicles pass the crossing at full speed without experiencing any delay.
\end{itemize}
Due to all the fixed settings, the model focuses on a single lane and does not capture any dependencies or interactions with other lanes.
Unfortunately, in many cases the FCTL queue cannot be applied as a realistic model to study the queue-length distribution in front of a traffic light. Take, for example, an intersection where vehicles from a single stream are spread onto two lanes which are both heading straight and where both lanes are governed by the same traffic light, see also Figure~\ref{fig:vis}(a). Indeed, since there are two parallel lanes in each direction, two vehicles can cross the intersection simultaneously and vehicles will in general switch lanes (if needed) to join the lane with the shorter queue. Moreover, it might be the case that the vehicles are blocked during the green period, e.g. because of a pedestrian crossing the intersection (receiving a green light at the same time as the stream of vehicles that we model), see Figure~\ref{fig:vis}(b) for a visualization. Such blockages might also occur in a multi-lane scenario (where all lanes are going in the same direction) as visualized in Figure~\ref{fig:vis}(c). It is apparent that these situations cannot be modeled by the standard FCTL queue. However, it is extremely relevant to understand such intersections better as is also indicated in e.g.~\cite{tageldin2019models,yan2018design} and more generally, it is e.g. important to investigate pedestrian behaviour at intersections as is done in e.g.~\cite{zhou2019simulation}.
The study in this paper provides an extension of the FCTL queue to account for such situations. They seem to be the most common in practice, see e.g.~\cite{shaoluen2020random} for another study on the case as in Figure~\ref{fig:vis}(b). For extensions and other scenarios, we refer the reader to Section~\ref{sec:discussion}. Note that the blocking mechanisms discussed in this paper give rise to more complicated model dynamics and dependencies, which make it impossible to use traditional methods (e.g. Webster's approximation for the mean delay \cite{webster1958traffic}).
\begin{figure}[h!]
\centering
\begin{tabular}{ccc}
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\scriptsize (a)
& \hspace{1cm}
\scriptsize (b)
& \hspace{1cm}
\scriptsize (c)
\end{tabular}
\caption{A visualization of three intersections that can be modeled by the bFCTL queue with multiple lanes. In~(a), the blue rectangle indicates a combination of lanes which can be analyzed as a bFCTL queue with two lanes. The other lanes at the intersection, the complement of the blue rectangle, can be considered separately because of the fixed settings. In~(b), the blue rectangle indicates a lane that can be modeled as a bFCTL queue with a single lane with blockages. In~(c), the blue rectangle indicates two lanes that we can model as a bFCTL queue with \emph{two} lanes where vehicles are potentially blocked by pedestrians.}
\label{fig:vis}
\end{figure}
A shared right-turn lane as in Figure~\ref{fig:vis}(b), that is a lane with vehicles that are either turning right or are heading straight, has been studied before. However, to the best of our knowledge, there are no papers with a rigorous analysis taking stochastic effects into account while computing e.g. the mean queue length for such lanes. Shared right-turn lanes where vehicles are blocked by pedestrians crossing immediately after the right turn have been considered in e.g.~\cite{alhajyaseen2013left,chen2011saturation,chen2014investigation,chen2008influence,shaoluen2020random,milazzo1998effect,roshani2017effect,rouphail1997pedestrian}. Several case studies, such as~\cite{chen2014investigation} and \cite{roshani2017effect}, indicate that there is a potentially severe impact by pedestrians blocking vehicles. This is for example also reflected in the Highway Capacity Manual (HCM) as published by the Transportation Research Board~\cite{manual2010}, where the focus is on capacity estimation. Most papers have also focused on the estimation of the so-called saturation flow rate, or capacity, of shared lanes where turning vehicles are possibly blocked by pedestrians, see e.g.~\cite{chen2008influence,milazzo1998effect,rouphail1997pedestrian}. In~\cite{chen2011saturation}, it is stated that the used functions for the capacity estimation for turning lanes (such as those in the HCM) might have to be extended to account for stochastic behaviour. In a small case study, \cite{chen2011saturation} confirm that the capacity estimation by the HCM yields an overestimation in various cases. The overestimation of the capacity by the HCM is also observed in several other papers, such as in~\cite{chen2014investigation,chen2008influence} and \cite{shaoluen2020random}, and is probably due to random/stochastic effects. The bFCTL queue explicitly models such stochastic behaviour.
A potential application of the bFCTL queue with a single lane as depicted in Figure~\ref{fig:vis}(b) can be found in the model that is studied in~\cite{shaoluen2020random}, which has also been the source of inspiration for this paper. A description of the model in~\cite{shaoluen2020random} is as follows, where we replace the left-turn assumption for left-driving traffic to a right-turn assumption for the more standard case of right-driving traffic. We have a shared lane with straight-going and right-turning traffic controlled by a traffic light, where immediately after the right turn there is a crossing for pedestrians. The pedestrians may block the right-turning vehicles as the vehicles and pedestrians may receive a green light simultaneously. The right-turning vehicles that are blocked, immediately block all vehicles behind them.
Another potential application of the bFCTL queue is to account for bike lanes. Bikes might make use of a dedicated lane or mix with other traffic and in both cases a turning vehicle might be (temporarily) blocked by bicycles because the bicycles happen to be in between the vehicle and the direction that the vehicle is going. As such, blockages have an influence on the performance measures of the traffic light. It is important to take such influences into account in order to find good traffic-light settings. Several papers studying the impact of bikes can be found in~\cite{allen1998effect,chen2007influence,guo2012effect} and \cite{chen2018evaluating}. Also other types of blocking might occur, such as by a shared-left turn lane and opposing traffic receiving a green light simultaneously, see e.g.~\cite{chai2014traffic,levinson1989capacity,liu2011arterial,liu2008lane,ma2017two,wu2011modelling,yang2018analytical,yao2013optimal}. As such, the bFCTL queue (either with multiple lanes or not) is a relevant addition to the literature because it enables a more suitable modelling of traffic lights at intersections with crossing pedestrians and bikes, which leads to traffic-light control strategies for more realistic situations. In order to model a situation where two opposing streams of vehicles potentially block one another as in e.g.~\cite{yang2018analytical}, the bFCTL queue would have to be extended. For more references on the topics discussed in this paragraph see also the review paper by~\cite{cheng2016review}. Another related study is~\cite{oblakova2019exact} who introduce a model with ``distracted'' drivers, which can be considered as an FCTL queue with independent blockages, but this blocking mechanism is a special case of the one discussed in the present paper.
As mentioned before, we call the model that we consider in this paper the bFCTL queue with multiple lanes. On the one hand we thus allow for the modelling of vehicle streams that are spread over multiple lanes and on the other hand we allow for vehicles to be (temporarily) blocked during the green phase. The key observation to constructing the mathematical model is that we can model multiple parallel (say $m$) lanes as \emph{one} single queue where batches of (up to) $m$ delayed vehicles can depart in one time slot, for more details see Section~\ref{sec:materials}. The resulting queueing model is one-dimensional just like the standard FCTL queue, which allows us to obtain the probability generating function (PGF) of the steady-state queue-length distribution of the bFCTL queue with multiple lanes and to provide an exact characterization of the capacity.
In summary, our main contributions are as follows:
\begin{itemize}
\item[(i)] We extend the general applicability of the Fixed-Cycle Traffic-Light (FCTL) queue. We allow for traffic streams with multiple lanes and for vehicles to be blocked during the green phase. We refer to this model variation as the blocked Fixed-Cycle Traffic-Light (bFCTL) queue with multiple lanes.
\item[(ii)] We provide an exact capacity analysis for the bFCTL queue relieving the need for simulation studies.
\item[(iii)] We provide a way to compute the PGF of the steady-state queue-length distribution of the bFCTL queue and show that it can be used to obtain several performance measures of interest.
\item[(iv)] We provide a queueing-theoretic framework for the study of shared lanes with potential blockages by pedestrians. This e.g. allows for the study of several performance measures and allows us to model the impact of randomness on the performance measures.
\end{itemize}
\subsection*{Paper outline}
The remainder of this paper is organized as follows. In Section~\ref{sec:materials}, we give a detailed model description. This is followed by a capacity analysis, a derivation of the PGF of the steady-state queue-length distribution, and a derivation of some of the main performance measures in Section~\ref{sec:queue_length_derivation}. In Section~\ref{sec:results}, we provide an overview of relevant performance measures for some numerical examples and point out various interesting results. We wrap up with a conclusion and some suggestions for future research in Section~\ref{sec:discussion}.
\section{Detailed model description}\label{sec:materials}
In this section we provide a detailed model description of the bFCTL queue with multiple lanes.
\begin{figure}[h!]
\centering
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\scriptsize (a)
&
\hspace{0.2cm} \scriptsize (b)
\end{tabular}
\caption{Visualization of (a) the bFCTL model in terms of an intersection with a traffic stream spread over $m$ lanes and (b) the corresponding queueing model, where the server takes batches of $m$ vehicles into service simultaneously unless there are less than $m$ vehicles present; in that case all vehicles are taken into service.}
\label{fig:vis_queue}
\end{figure}
We assume that there are multiple lanes for a traffic stream, that is a group of vehicles coming from the same road and heading into one (or several) direction(s), governed by a \emph{single} traffic light. A visualization can be found in Figure~\ref{fig:vis_queue}(a). As can be seen in Figure~\ref{fig:vis_queue}(a), we assume that there are $m$ lanes and that vehicles spread themselves among the available lanes in such a way that $m$ vehicles can depart if there are at least $m$ vehicles. In practice, this assumption makes sense as drivers gladly minimize their delay by choosing free lanes. The traffic-light model is then turned into a queueing model with a \emph{single} queue with batch services of vehicles, see Figure~\ref{fig:vis_queue}(b). The batches generally consist of $m$ delayed vehicles (we consider delayed vehicles as is done in the study of the FCTL queue, see e.g.~\cite{boon2019pollaczek}), except if less than $m$ delayed vehicles are present at the moment that a batch is taken into service: then all vehicles are taken into service. We further assume that the time axis is divided into time intervals of constant length, where each interval corresponds to the time it takes for a batch of delayed vehicles to depart from the queue. We will refer to these intervals as slots.
We now turn to discuss two concrete, motivational examples that fit the framework of the bFCTL queue with multiple lanes. After that, we describe the assumptions of the bFCTL queue more formally.
\begin{example}[Shared right-turn lane]
In this example we consider the scenario as in Figure~\ref{fig:vis}(b). We have batches of vehicles of size $1$, i.e. batches are individual vehicles.
We distinguish between vehicles that are going straight ahead and vehicles that turn right. We do so because only right-turning vehicles can be blocked by crossing pedestrians. The probability that an arbitrary vehicle at the head of the queue is a turning vehicle is $p$. Such a turning vehicle is blocked by a pedestrian in slot $i$ with probability $q_i$, i.e.\,a pedestrian is present on the crossing with probability $q_i$. If a turning vehicle is blocked, all vehicles behind it are also blocked. Then, we proceed to the next slot, $i+1$, and check whether there are any pedestrians crossing (with probability $q_{i+1}$): if there are pedestrians crossing, all vehicles in the queue keep being blocked and otherwise, the turning vehicle at the head of the queue may depart and the blockage of all other vehicles is removed.
Moreover, if the queue becomes empty during the green period, it will in general not start building again (cf. the FCTL assumption for the regular FCTL queue, see e.g.~\cite{van2006delay}), \emph{except} if there arrives a turning vehicle and there is a crossing pedestrian. The turning vehicle is then blocked and any vehicles arriving in the same slot behind this vehicle are also blocked.
\end{example}
\begin{example}[Two turning lanes]\label{ex:two}
In this example we consider the scenario as in Figure~\ref{fig:vis}(c). We have batches of vehicles of size $2$.
In this example, there is no need to make a distinction between vehicles: each vehicle is a turning vehicle with probability $1$, i.e. $p=1$. During each slot $i$, there are pedestrians on the crossing with probability $q_i$ and if there is a pedestrian, all vehicles in the batch are blocked, as are all other vehicles in the queue: there are no vehicles that can complete the right turn. All vehicles in the queue keep being blocked until there are no pedestrians crossing anymore.
Also in this example, the queue of vehicles might dissolve entirely during the green period. If that happens, it only starts building again if there are vehicles arriving \emph{and} if there are pedestrians crossing. In such cases, all arriving vehicles get blocked and remain blocked until there are no pedestrians anymore.
\end{example}
We are now set to formalize the assumptions for the bFCTL queue with multiple lanes. We number them for clarity and provide additional remarks if necessary. We start with a standard assumption for FCTL queues and a standard assumption on the independence of arriving vehicles, see, e.g.~\cite{van2006delay}.
\begin{assumption} \label{ass:disctime} [Discrete-time assumption]
We divide time into discrete slots.
The red and green times, $r$ and $g$ respectively, are fixed multiples of those discrete slots and the total cycle length, $c=g+r$, thus consists of an integer number of slots. Each slot corresponds to the duration of the departure of a batch of maximally $m$ delayed vehicles, where $m$ is the maximum number of vehicles that can cross the intersection simultaneously. Any arriving vehicle that finds at least $m$ other vehicles waiting in front of the traffic light is delayed and joins the queue.
\end{assumption}
\begin{assumption}[Independence of arrivals]
All arrivals are assumed to be independent. In particular, the arrivals during slot $i$ do not affect the arrivals in slot $j$ when $i\neq j$.
\end{assumption}
The next three assumptions, Assumptions~\ref{as:division}, \ref{ass:removal}, and \ref{ass:adapted}, relate to the blockages of vehicles and that allow us to explicitly model such blockages.
\begin{assumption}[Green period division]\label{as:division}
For the green period we distinguish between two parts, $g_1$ and $g_2$, with $g=g_1+g_2$. During the first part of the green period, blockages might occur (see also Assumption~\ref{ass:removal} below). During the second part of the green period there are no blockages at all. We further assume that $g_2>0$ for technical reasons.
\end{assumption}
We make a division of the green period into two parts as is done in e.g.~\cite{shaoluen2020random}. Moreover, such a division is often present in reality and it slightly eases the computations later on. This e.g. means that during the second part of the green period there is a ``no walk'' sign flashing, during which pedestrians are not allowed to cross the intersection. We note that if $g_1=0$ (and $m=1$), we obtain the standard FCTL queue.
Further, we assume that the second part of the green period is strictly positive, mainly for technical reasons. This basically implies that at least one batch of vehicles can depart from the queue during each cycle and that there is \emph{no} batch of vehicles in the queue at the end of the cycle that has caused a blockage before. If $g_2$ would be zero and if a batch of vehicles is blocked at the end of slot $g_1$, this would allow for a blockage to carry over to the next cycle, leading to a slightly more complex model. Moreover, the red and green times could be taken random in the regular FCTL queue when the times are independent of one another, see e.g.~\cite{boon2021optimal}. At the expense of additional complexity, our framework for the bFCTL queue could be adjusted to account for such sources of randomness. This would allow one to model (to some extent) randomness in, for example, crossing times of pedestrians.
Next, we make an assumption about the blocking of batches of vehicles during the first part of the green period. We take into account that (i) not all batches of vehicles at the head of the queue are potentially blocked (e.g. because only turning batches of vehicles can be blocked); that (ii) if a batch of vehicles is blocked, all vehicles behind it are blocked as well; that (iii) once a blockage occurs, it carries over to the next slot; and that (iv) blockages occur only in the combined event of having a right-turning batch of vehicles at the head of the queue \emph{and} pedestrians crossing the road.
\begin{assumption}[Potential blocking of batches]\label{ass:removal}
A batch of vehicles, arriving at the head of the queue in time slot $i$, turns right with probability $p_i$. Independently, in time slot $j$, pedestrians cross the road with probability $q_j$, blocking right-turning traffic. As a consequence, whenever a new batch arrives at the head of the queue, this batch will be served in that particular time slot if (i) the batch goes straight ahead, \emph{or} (ii) the batch turns right but there are no crossing pedestrians. Once a batch (of right-turning vehicles) is blocked, it will remain blocked until the next time slot when no pedestrians cross the road. Note that this will be time slot $g_1+1$ at the latest. If the batch at the head of the queue is blocked, it will also block all the other batches in the queue, including those that would go straight. Both $p_i$ and $q_i$ are allowed to depend on the slot $i$.
\end{assumption}
\begin{remark
\label{rem:pi}
We make a couple of remarks on the values of the $p_i$.
First, we note that $p_i$ is not representing the probability that the batch at the head of the queue is a turning batch, but rather the probability that a \emph{newly arriving} batch that gets to the head of the queue in slot $i$, is a turning batch. In practice, this will usually \emph{not} depend on the slot in which the batch gets to the head of the queue. This would imply that $p_i=p$ (see, e.g., Example~\ref{ex:two}) and that we could drop the subscript $i$. However, we are able to let $p_i$ depend on the slot in the derivation of the formulas and opt to provide the general case where $p_i$ is allowed to depend on $i$.
Moreover, in the case that $m>1$, we will often assume that either $p_i=0$, as is the case in Figure~\ref{fig:vis}(a), or $p_i=1$, as is the case in Figure~\ref{fig:vis}(c). This is mainly due to the fact that \emph{all} vehicles in a batch have to be treated similarly: the framework of the bFCTL queue does not allow for batches consisting of one right-turning vehicle that is blocked and one straight-going vehicle that is allowed to depart because it is not blocked. I.e. a case with mixed traffic and \emph{multiple} lanes, such as the shared right-turn lane example in Figure~\ref{fig:vis}(b) but with $m>1$, is not modeled by the bFCTL queue. We do not consider this to be a severe restriction as it will often be the case in practice that $p_i=0$ or $p_i=1$ if $m>1$. We stress that the case with $m=1$ as depicted by the blue rectangle in Figure~\ref{fig:vis}(b) can be studied by the bFCTL queue.
\end{remark}
\begin{remark
\label{rem:blockages}
We would like to stress that the blockage of a batch of vehicles carries over to the next slot. E.g. if a vehicle is a right-turning vehicle in Figure~\ref{fig:vis}(b) and is blocked, it is still at the head of the queue in the next slot. So, as soon as a blockage actually takes place, we are essentially in a different state of the system than in the case where there is no blockage: if there is a blockage in time slot $i$ then we are sure that there is a right-turning batch at the head of the queue in time slot $i+1$. This is why we have two mechanisms for the blocking: on the one hand we have the $p_i$ to check whether \emph{new} batches that get to the head of the queue are right turning and on the other hand we have the pedestrians crossing in slot $i$ accounted for by the $q_i$.
\end{remark}
We need one final assumption which is a slightly adapted version of the standard FCTL assumption. We require a slight change because of the potential blocking of vehicles during the first part of the green phase and because of the possibility that there is more than one delayed vehicle departing in a single slot during the green period because of the batch-service structure.
\begin{assumption}[bFCTL assumption]\label{ass:adapted}
We assume that any vehicle arriving during a slot where $m-1$ or less vehicles are in the queue, may depart from the queue immediately together with the $m-1$ or less delayed vehicles. There are two exceptions: (i) if this batch of $m-1$ or less vehicles is blocked or (ii) if the queue was empty and there is an arriving vehicle that gets blocked, in which case that vehicle gets blocked together with any arriving vehicles after that vehicle. In the former case, all arriving vehicles together with the delayed vehicles remain at the queue. In the latter case, the first blocked vehicle is delayed and any arriving vehicles behind it (if any) are also delayed and blocked where we restrict ourselves to the situation where the queue is empty. If the queue was not empty, then we assume that either all arriving vehicles in that slot are blocked and delayed (because the batch at the head of the queue is blocked) or that all arriving vehicles are allowed to depart along with the batch of delayed vehicles (because the batch at the head of the queue is not blocked). Summarizing, if the queue length at the start of the slot is at least $1$ but at most $m-1$, we either have no departures (in case of a blockage) or \emph{all} vehicles are allowed to cross the intersection (including arriving vehicles). If the queue length is $0$, we only have a non-zero queue at the end of the slot if a vehicle gets blocked: then the blocked vehicle and any vehicles arriving behind it are queued.
\end{assumption}
\begin{remark
\label{rem:bFCTL}
The bFCTL assumption allows one to model a situation where arriving vehicles get blocked if the queue was already empty before the start of the slot. Although, in principle, one can use any distribution for the number of arriving vehicles that are blocked, there are only few logical choices in practice. For example, in the case of Figure~\ref{fig:vis}(b), the number of (potentially) blocked vehicles that arrive at the queue during slot $i$ would correspond to the number of vehicles counting from the first right-turning vehicle among all vehicles arriving in slot $i$: these vehicles will be blocked if there is a crossing pedestrian in slot $i$. In Figure~\ref{fig:vis}(c), any arriving vehicle is a turning vehicle. So, if there is a crossing pedestrian, all arriving vehicles in slot $i$ are blocked.
\end{remark}
The combination of all the above assumptions enables us to view the process as a discrete-time Markov chain, which in turn allows us to obtain the capacity and the PGF of the steady-state queue-length distribution of the bFCTL queue with multiple lanes. We do so in the next section.
\section{Capacity analysis, PGFs, and performance measures for the bFCTL queue}\label{sec:queue_length_derivation}
In this section we provide an exact analysis for the bFCTL queue. We start with an exact characterization of the capacity in Subsection~\ref{subsec:capacity}. In Subsection~\ref{subsec:pgf_derivation}, we obtain the steady-state queue-length distribution in terms of PGFs where we thus focus on the \emph{transforms} of the queue-length distribution, because we cannot directly obtain closed-form expressions for the probabilities. We can use the methods devised in e.g.~\cite{abate1992numerical} and~\cite{choudhury1996numerical} to obtain numerical values from the PGFs for the queue-length probabilities and moments respectively. Without giving details, we stress that our recursive approach in Subsection~\ref{subsec:pgf_derivation} also allows us to provide a transient analysis in which case we can also take time-varying parameters into account. In Subsection~\ref{sec:performance_measures}, we study several important performance measures of the bFCTL queue.
\subsection{Capacity analysis for the bFCTL queue}\label{subsec:capacity}
In this subsection we develop a computational algorithm to determine the capacity for the bFCTL queue. The capacity is defined as the maximum number of vehicles that can cross the intersection in the given lane group, per time unit. In the standard FCTL queue, the capacity can simply be determined by multiplying the saturation flow with the ratio of the green time and the cycle length. In the bFCTL model, however, there are subtle dependencies which carry over from one cycle to the next cycle. We will capture these dependencies by means of a Markov reward model. The Markov chain with the associated transition probabilities that we use is depicted in Figure~\ref{fig:MC}. We are interested in the number of departures of delayed vehicles in each time slot. For this reason, the Markov chain that we consider here only has states $(i,s)$ for $i=1,\dots,g_1$ representing the slots during the first part of the green period and $s=u,b$, representing the case where vehicles are not blocked ($s=u$) and the case where vehicles are blocked ($s=b$). We also have states $i$ for $i=g_1+1,\dots,g_1+g_2+r$ representing the slots during the second part of the green period and the red period. Finally, we create an artificial state $0$ to gather the rewards from states $(1,b)$ and $(1,u)$. The long-term mean number of departures of delayed vehicles can now be determined by means of a Markov reward analysis.
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[->,auto,node distance=2.2cm]
\tikzstyle{every state}=[fill=white,draw=black,text=black,minimum size=1.1cm]
\node[draw,circle,minimum height = 1cm] (A) {$0$};
\node[draw,circle,minimum height = 1cm] (B) [above right of=A] {$(1,u)$};
\node[draw,circle,minimum height = 1cm] (C) [below right of=A] {$(1,b)$};
\node[draw,circle,minimum height = 1cm] (D) [right = 1.5 cm of B] {$(2,u)$};
\node[draw,circle,minimum height = 1cm] (E) [right = 1.5 cm of C] {$(2,b)$};
\node[minimum size=1cm] (H) [right=1.5 cm of D] {$\dots$};
\node[minimum size=1cm] (I) [right=1.5 cm of E] {$\dots$};
\node[draw,circle,minimum height = 1cm] (J) [right=1.5 cm of H] {$(g_1,u)$};
\node[draw,circle,minimum height = 1cm] (K) [right=1.5 cm of I] {$(g_1,b)$};
\node[draw,circle,minimum height = 1cm] (L) [below right of=J] {$g_1+1$};
\node[draw,circle,minimum height = 1.75cm] (LL) [below=2.5 cm of L] {$g_1+2$};
\node[minimum size=1cm] (M) [left=0.5 cm of LL] {$\dots$};
\node[draw,circle,minimum height = 1.75cm] (N) [left=0.5 cm of M] {\begin{tabular}{c}$g_1+$\\$g_2$\end{tabular}};
\node[draw,circle,minimum height = 2cm] (O) [left=0.5 cm of N] {\begin{tabular}{c}$g_1+$\\$g_2+1$\end{tabular}};
\node[minimum size=1cm] (P) [left=0.5 cm of O] {$\dots$};
\node[draw,circle,minimum height = 1.75cm] (Q) [left=0.5 cm of P] {\begin{tabular}{c}$g_1+$\\$g_2+r$\end{tabular}};
\path (A) edge node {\footnotesize{$1-p_1q_1$}} (B);
\path (A) edge node {\footnotesize{$p_1q_1$}} (C);
\path (B) edge node {\footnotesize{$1-p_2q_2$}} (D);
\path (B) edge node[pos=0.78] {\footnotesize{$p_2q_2$}} (E);
\path (C) edge node[pos=0.6,xshift=1.5cm] {\footnotesize{$1-q_2$}} (D);
\path (C) edge node {\footnotesize{$q_2$}} (E);
\path (D) edge node {\footnotesize{$1-p_3q_3$}} (H);
\path (D) edge node[pos=0.78] {\footnotesize{$p_3q_3$}} (I);
\path (E) edge node[pos=0.6,xshift=1.5cm] {\footnotesize{$1-q_3$}} (H);
\path (E) edge node {\footnotesize{$q_3$}} (I);
\path (H) edge node {\footnotesize{$1-p_{g_1}q_{g_1}$}} (J);
\path (H) edge node[pos=0.78] {\footnotesize{$p_{g_1}q_{g_1}$}} (K);
\path (I) edge node[pos=0.6,xshift=1.7cm] {\footnotesize{$1-q_{g_1}$}} (J);
\path (I) edge node {\footnotesize{$q_{g_1}$}} (K);
\path (K) edge node[pos=0.1,xshift=0.8cm,yshift=-0.5cm] {\footnotesize{$1$}} (L);
\path (J) edge node {\footnotesize{$1$}} (L);
\path (L) edge node {\footnotesize{$1$}} (LL);
\path (LL) edge node {\footnotesize{$1$}} (M);
\path (M) edge node {\footnotesize{$1$}} (N);
\path (N) edge node {\footnotesize{$1$}} (O);
\path (O) edge node {\footnotesize{$1$}} (P);
\path (P) edge node {\footnotesize{$1$}} (Q);
\end{tikzpicture}
\caption{Markov chain used to study the capacity of the bFCTL queue.}\label{fig:MC}
\end{figure}
We use Markov reward theory to obtain the capacity of the bFCTL queue. In order to use Markov reward theory, we work backwards from state $g_1+g_2+r$ to obtain the reward in state $0$. Indeed, we get the mean number of vehicles that is able to depart from the queue in an arbitrary cycle when we compute the reward in state $0$. The rewards that we assign to each transition are as follows: if we make a transition to a state $(i,u)$ for $i=1,\dots,g_1$, we receive a reward $m$ reflecting the maximum of $m$ delayed vehicles departing from the queue. We also get a reward $m$ if we make a transition from state $g_1+i$ to state $g_1+i+1$ for $i=1,\dots,g_2-1$. For all other transitions, we receive no reward as there are no vehicles departing. We denote the received reward up to state $(i,s)$ with $r_{i,s}$ with $i=1,\dots,g_1$ and $s=u,b$ and the received reward up to state $i$ with $r_i$ for $i=0$ and $i=g_1+1,\dots,g_1+g_2+r$. Then we get the following relations between the rewards in the various states. We start with defining the total reward in state $g_1+g_2+r$ to be $0$ (there are no vehicle departures while being in state $g_1+g_2+r$), i.e.
\begin{equation} \label{eq:rc}
r_{g_1+g_2+r} = 0.
\end{equation}
For states $i=g_1+g_2,\dots,g_1+g_2+r-1$, we obtain
\begin{equation}
r_{i} = r_{i+1},
\end{equation}
as there are no departures during the red period. However, for states $i=g_1+1,\dots,g_1+g_2-1$, we have
\begin{equation}\label{eqnm1}
r_{i} = m + r_{i+1}
\end{equation}
as there are (potentially) $m$ delayed vehicles departing. For state $(g_1,b)$ we have that
\begin{equation}
r_{g_1,b} = r_{g_1+1},
\end{equation}
as there are no departures when the vehicles are blocked. For state $(g_1,u)$ we obtain
\begin{equation}
r_{g_1,u} = m + r_{g_1+1}\label{eqnm2}
\end{equation}
as there are, at most, $m$ delayed vehicles departing from the queue when we transition from state $(g_1,u)$ to $g_1+1$. Similarly, for states $(i,b)$ with $i=1,\dots,g_1-1$, we get
\begin{equation}
r_{i,b} = q_{i+1}r_{i+1,b}+(1-q_{i+1})r_{i+1,u}
\end{equation}
and for states $(i,u)$ with $i=1,\dots,g_1-1$, we get
\begin{equation}\label{eqnm3}
r_{i,u} = m+p_{i+1}q_{i+1}r_{i+1,b}+(1-p_{i+1}q_{i+1})r_{i+1,u}.
\end{equation}
Finally, for state $0$, we get
\begin{equation}\label{eq:r0}
r_{0} = p_1q_1r_{1,b}+(1-p_1q_1)r_{1,u}.
\end{equation}
Then we have that $r_0$ is the average reward received when traversing the Markov chain as depicted in Figure~\ref{fig:MC}. This average reward translates to the mean number of delayed vehicles that are able to depart from the queue during a cycle, which is exactly the capacity of this lane group. We can thus compute the capacity of the bFCTL queue for each set of input parameters. Along with the mean number arrivals per cycle, we can also check whether the bFCTL queue renders a stable queueing model. If we denote the mean number of arrivals in slot $i$ by $\mathbb{E}[Y_i]$, the mean number of arrivals per cycle is $\sum_{i=1}^c\mathbb{E}[Y_i]$ and the bFCTL queue is stable if $r_0>\sum_{i=1}^c\mathbb{E}[Y_i]$. The procedure to check for stability is summarized in Algorithm~\ref{alg:stability}.
\begin{algorithm}[H]
\caption{Algorithm to check for stability of the bFCTL queue.} \label{alg:stability}
\begin{algorithmic}[1]
\State Input: $\mathbb{E}[Y_i]$ for $i=1,\dots,c$, $g_1$, $g_2$, $c$, $p_{i}$ for $i=1,\dots,g_1$, and $q_{i}$ for $i=1,\dots,g_1$.
\State Use Equations~\eqref{eq:rc} up to \eqref{eq:r0} to determine $r_0$.
\If {$\sum_{i=1}^c\mathbb{E}[Y_i]< r_0$}
\State The bFCTL queue is stable.
\Else
\State The bFCTL queue is not stable.
\EndIf
\end{algorithmic}
\end{algorithm}
\begin{remark}\label{rem:capacity}
One of our model restrictions (Assumption~\ref{ass:disctime}) is that vehicles depart at the end of each time slot, meaning that we do not correct for the fact that turning vehicles might need more time to accelerate. A simple method to account for this effect, which reduces the capacity in practice, is to modify the reward structure of the Markov chain. One can modify the value of $m$ in Equations~\eqref{eqnm1}, \eqref{eqnm2}, and \eqref{eqnm3} to account for the lower departure rate of turning vehicles. For example, one can use
\begin{equation}\label{eqn:capacityCorrection}
m^* = p_i m_\textit{turn} + (1-p_i)m_\textit{through},
\end{equation}
where $m_\textit{through}$ and $m_\textit{turn}$ represent the average number of through-vehicles and turning vehicles, respectively, crossing the intersection per time unit. For this capacity calculation, these numbers do not need to be integers. See Section~\ref{subsec:capacity2} for a numerical example and a comparison to the HCM capacity formula.
\end{remark}
\subsection{Derivation of the PGFs for the bFCTL queue}\label{subsec:pgf_derivation}
First, we need to introduce some further concepts and notation before we continue our quest to obtain the relevant PGFs of the queue-length distribution. We introduce two states, one corresponding to a situation where the queue is blocked and one where this is not the case, cf. Assumption~\ref{ass:removal} and Remark~\ref{rem:blockages} and as is done in Subsection~\ref{subsec:capacity}. We denote the random variable of being in either of the two states with $S$ and $S$ takes the values $b$ (blocked) and $u$ (unblocked). By definition, blocked states only occur during the first part of the green period and if there are vehicles in the queue. We define $S$ to be equal to $u$ if the queue is empty. We denote the joint steady-state queue length (measured in number of vehicles) and the state $S$ at the end of slot $i=1,\dots,g_1$ with the tuple $(X_i,S)$ and we denote its PGF with $X_{i,j}(z)$ where $i=1,\dots,g_1$ and $j=u,b$. We note that $X_{i,b}(z)$ and $X_{i,u}(z)$ are partial generating functions: we e.g. have $X_{i,b}(z) = \mathbb{E}[z^{X_{i}}\mathds{1}\{S=b\}]$, where $\mathds{1}\{S=b\}=1$ if $S=b$ and $0$ otherwise. For the slots $i=1,\dots,c$ we denote the steady-state queue length with $X_i$ and its PGF with $X_i(z)$, so for $i=1,\dots,g_1$ we have that $X_i(z)=X_{i,u}(z)+X_{i,b}(z)$.
We note that, as we are looking at the steady-state distribution of the number of vehicles in the queue, we need to require stability of the queueing model. We can check whether or not the stability condition is satisfied by means of Algorithm~\ref{alg:stability} devised in Subsection~\ref{subsec:capacity}.
We further denote with $Y_i$ the number of arrivals during slot $i$ and with $Y_{i,b}$ we denote the total number of arrivals of potentially blocked vehicles during slot $i$, see also Assumption~\ref{ass:adapted}.
We denote their PGFs respectively with $Y_i(z)$ and $Y_{i,b}(z)$. Later in this subsection, we provide $Y_{i,b}(z)$ for several concrete examples.
In the next part of this subsection we provide the recursion between the $X_{i,j}(z)$, $i=1,\dots,g_1$ and $j=u,b$, and the $X_i(z)$, $i=g_1+1,\dots,c$. Afterwards, we wrap up with some technicalities that need to be overcome to obtain a full characterization of all the PGFs.
\subsubsection{Recursion for the $X_{i,j}(z)$}
We start with the relation between $X_{1,b}(z)$ and $X_c(z)$. We distinguish several cases while making a transition from slot $c$ to a blocked state in slot $1$. We get
\begin{equation}\label{eq:X1b}
\begin{aligned}
X_{1,b}(z) = & p_1q_1 \mathbb{E}[z^{X_c+Y_1}\mathds{1}\{X_c>0\}] + q_1 \mathbb{E}[z^{Y_{1,b}}\mathds{1}\{X_c=0\}\mathds{1}\{Y_{1,b}>0\}] +\\& 0\cdot\mathbb{E}[ \mathds{1}\{X_c=0\}\mathds{1}\{Y_{1,b}=0\}]\\
= & p_1q_1 X_{c}(z) Y_1(z) + q_1\mathbb{P}(X_{c}=0)\left(Y_{1,b}(z)-Y_{1,b}(0)-p_1Y_1(z)\right).
\end{aligned}
\end{equation}
We explain this relation as follows: if the queue is nonempty at the end of slot $c$, we need both a right-turning batch of vehicles and a crossing pedestrian in slot $1$ to get a blockage, which happens with probability $p_1q_1$. The queue length at the end of slot $1$ is then $X_c+Y_1$. The second term can be understood as follows: if $X_c=0$, the queue at the end of slot $c$ is empty and then we get to a blocked state if there is a pedestrian crossing (which happens with probability $q_1$) and if $Y_{1,b}>0$, in which case the queue length is $Y_{1,b}$. Note that we further have that the case $X_{1,b}=0$ cannot occur (by definition) as indicated by the term on the second line of Equation~\eqref{eq:X1b}.
Similarly, we derive $X_{1,u}(z)$:
\begin{align}\label{eq:X1u}
X_{1,u}(z) = &\nonumber (1-p_1q_1) \mathbb{E}[z^{X_c+Y_1-m}\mathds{1}\{X_c\geq m\}] + (1-p_1q_1)\mathbb{E}[z^0\mathds{1}\{1\leq X_c\leq m-1\}] + \\&(1-q_1)\mathbb{E}[z^0 \mathds{1}\{X_c=0\}]+q_1\mathbb{E}[z^0 \mathds{1}\{X_c=0\}\mathds{1}\{Y_{1,b}=0\}] \\
= &\nonumber(1-p_1q_1) X_{c}(z) \frac{Y_1(z)}{z^m} + (1-p_1q_1)\sum_{l=1}^{m-1}\mathbb{P}(X_{c}=l)\left(1-\frac{Y_{1}(z)}{z^{m-l}}\right)+\\&\mathbb{P}(X_c=0)\left(1-q_1+q_1Y_{1,b}(0)-(1-p_1q_1)\frac{Y_1(z)}{z^m}\right).\nonumber
\end{align}
This relation can be understood in the following way: first, if there are at least $m$ vehicles at the end of slot $c$ and if there is no blockage (which occurs with probability $1-p_1q_1$, i.e. the complement of a blockage occurring), then the queue length at the end of slot $1$ is $X_c+Y_1-m$. Secondly, if there is at least $1$ but at most $m-1$ vehicles at the end of slot $c$, we have an empty queue at the end of slot $1$ if there is no blockage (which is the case with probability $1-p_1q_1$). Thirdly, if the queue is empty at the end of slot $c$, then the queue remains empty if there are no pedestrians crossing (occurring with probability $1-q_1$) or if there is a pedestrian crossing (occurring with probability $q_1$) while $Y_{1,b}=0$. This fully explains Equation~\eqref{eq:X1u}.
In a similar way, we obtain the following relations for slots $i=2,\dots,g_1$:
\begin{equation}\label{eq:Xib}
\begin{aligned}
X_{i,b}(z) = & p_iq_i \mathbb{E}[z^{X_{i-1}+Y_i}\mathds{1}\{S=u\}] + q_i \mathbb{E}[z^{X_{i-1}+Y_i}\mathds{1}\{S=b\}] + \\& q_i\mathbb{E}[z^{Y_{i,b}}\mathds{1}\{X_{i-1}=0\}\mathds{1}\{S=u\}\mathds{1}\{Y_{i,b}>0\}]\\
= &p_iq_i X_{i-1,u}(z) Y_i(z) + q_iX_{i-1,b}(z)Y_i(z) +\\&q_i\mathbb{P}(X_{i-1}=0,S=u)\left(Y_{i,b}(z)-Y_{i,b}(0)-p_iY_{i}(z)\right),
\end{aligned}
\end{equation}
where we have to take both transitions from slot $i-1$ while being blocked (the case $S=b$) and not being blocked (the case $S=u$) into account, and
\begin{align}
X_{i,u}(z) = & (1-p_iq_i) \mathbb{E}[z^{X_{i-1}+Y_i-m}\mathds{1}\{X_{i-1}\geq m\}\mathds{1}\{S=u\}] +\nonumber\\& (1-q_i) \mathbb{E}[z^{X_{i-1}+Y_i-m}\mathds{1}\{X_{i-1}\geq m\}\mathds{1}\{S=b\}]+ \nonumber\\& (1-p_iq_i)\mathbb{E}[z^0\mathds{1}\{1\leq X_{i-1}\leq m-1\}\mathds{1}\{S=u\}] + \nonumber\\& (1-q_i)\mathbb{E}[z^0\mathds{1}\{1\leq X_{i-1}\leq m-1\}\mathds{1}\{S=b\}] +\nonumber\\& (1-q_i)\mathbb{E}[z^0\mathds{1}\{X_{i-1}=0\}\mathds{1}\{S=u\}]+\nonumber\\&q_i\mathbb{E}[z^0\mathds{1}\{X_{i-1}=0\}\mathds{1}\{S=u\}\mathds{1}\{Y_{i-1,b}=0\}] \\
= &(1-p_iq_i) X_{i-1,u}(z) \frac{Y_i(z)}{z^m} + (1-q_i)X_{i-1,b}(z)\frac{Y_i(z)}{z^m}+ \nonumber\\&(1-p_iq_i)\sum_{l=1}^{m-1}\mathbb{P}(X_{i-1}=l,S=u)\left(1-\frac{Y_{i}(z)}{z^{m-l}}\right)+\nonumber\\& (1-q_i)\sum_{l=1}^{m-1}\mathbb{P}(X_{i-1}=l,S=b)\left(1-\frac{Y_{i}(z)}{z^{m-l}}\right)+\nonumber\\&\mathbb{P}(X_{i-1}=0,S=u)\left(1-q_i+q_iY_{i,b}(0)-(1-p_iq_i)\frac{Y_i(z)}{z^m}\right).\nonumber
\end{align}
In order to derive $X_{g_1+1}(z)$, we note that we need to take the cases into account where the queue was blocked or not during slot $g_1$. We then get
\begin{align}
\begin{aligned}
X_{g_1+1}(z) = & \mathbb{E}[z^{X_{g_1}+Y_{g_1+1}-m}\mathds{1}\{X_{g_1}\geq m\}\mathds{1}\{S=u\}]+\\&\mathbb{E}[z^{X_{g_1}+Y_{g_1+1}-m}\mathds{1}\{X_{g_1}\geq m\}\mathds{1}\{S=b\}]+\\&\mathbb{E}[z^0\mathds{1}\{X_{g_1}\leq m-1\}\mathds{1}\{S=u\}]+\mathbb{E}[z^0\mathds{1}\{X_{g_1}\leq m-1\}\mathds{1}\{S=b\}] \\
= & \left(X_{g_1,u}(z) +X_{g_1,b}(z) \right)\frac{Y_{g_1+1}(z)}{z^m}+\\&\sum_{l=0}^{m-1}\left(\mathbb{P}(X_{g_1}=l,S=u\right)\left(1-\frac{Y_{g_1+1}(z)}{z^{m-l}}\right)+\\&\sum_{l=1}^{m-1}\left(\mathbb{P}(X_{g_1}=l,S=b\right)\left(1-\frac{Y_{g_1+1}(z)}{z^{m-l}}\right).
\end{aligned}
\end{align}
For $i=g_1+2,\dots,g_1+g_2$, we obtain the following
\begin{equation}
\begin{aligned}
X_{i}(z) = & \mathbb{E}[z^{X_{i-1}+Y_{i}-m}\mathds{1}\{X_{i-1}\geq m\}]+\mathbb{E}[z^0\mathds{1}\{X_{i-1}\leq m-1\}] \\
= & X_{i-1}(z) \frac{Y_{i}(z)}{z^m}+\sum_{l=0}^{m-1}\left(\mathbb{P}(X_{i-1}=l\right)\left(1-\frac{Y_{i}(z)}{z^{m-l}}\right),
\end{aligned}
\end{equation}
while for slots $i=g_1+g_2+1,\dots,c$ we get
\begin{equation}\label{eq:Xg1+g2+i}
\begin{aligned}
X_{i}(z) = \mathbb{E}[z^{X_{i-1}+Y_{i}}]= X_{i-1}(z)Y_i(z).
\end{aligned}
\end{equation}
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The combination of all equations above, provides us with a recursion with which we can express $X_{g_1+g_2}(z)$ in terms of $Y_i(z)$, $Y_{i,b}(z)$, $\mathbb{P}(X_{i}=l,S=u)$ and $\mathbb{P}(X_{i}=l,S=b)$ for $i=1,\dots,g_1$ and $l=0,\dots,m-1$, and $\mathbb{P}(X_{i} = l)$ for $i=g_1+1,\dots,g_1+g_2-1$, $i=c$, and $l=0,\dots,m-1$, with the following general form:
\begin{equation}\label{eq:generalbFCTL}
X_{g_1+g_2}(z) = \frac{X_n(z)}{X_d(z)},
\end{equation}
with known $X_n(z)$ and $X_d(z)$. We refrain from giving $X_n(z)$ and $X_d(z)$ in the general case because of their complexity and only provide them under simplifying assumptions later in this subsection. The $Y_{i}(z)$ are known, but we still need to obtain the $Y_{i,b}(z)$, the $\mathbb{P}(X_{i}=l,S=u)$ and $\mathbb{P}(X_{i}=l,S=b)$ for $i=1,\dots,g_1$ and $l=0,\dots,m-1$, and the $\mathbb{P}(X_{i} = l)$ for $i=g_1+1,\dots,g_1+g_2-1$, $i=c$, and $l=0,\dots,m-1$. We start with the $Y_{i,b}(z)$ and then come back to the unknown probabilities.
The occurrence of the PGF $Y_{i,b}(z)$ directly relates to Assumption~\ref{ass:adapted}. As mentioned before in Remark~\ref{rem:bFCTL}, one could, a priori, use any positively distributed, discrete random variable. However, when we have a specific example in mind, there is usually one logical definition, see also Remark~\ref{rem:Yib} below.
\begin{remark}\label{rem:Yib
In general, we define $Y_{i,b}$ to be the random variable of the total number of arrivals of potentially blocked vehicles during slot $i$, cf. Assumption~\ref{ass:adapted}. In case $m=1$, such as in Figure~\ref{fig:vis}(b), the interpretation of the $Y_{i,b}(z)$ is straightforward. We simply count the number of arriving vehicles starting from the first vehicle that is a turning vehicle. We get the following expression for $Y_{i,b}(z)$:
\begin{align*}
Y_{i,b}(z) & = \sum_{k=0}^\infty \mathbb{P}(Y_{i,b} = k)z^k\\
& = \sum_{j=0}^\infty \mathbb{P}(Y_i=j)(1-p_i)^j + \sum_{k=1}^\infty \sum_{j=k}^\infty \mathbb{P}(Y_i = j) (1-p_i)^{j-k} p_i z^k \\
& = Y_i(1-p_i) + \sum_{j=1}^\infty p_i \mathbb{P}(Y_i=j)(1-p_i)^j\sum_{k=1}^j \left(\frac{z}{1-p_i}\right)^k \\
& = Y_i(1-p_i) + \sum_{j=1}^\infty p_i \mathbb{P}(Y_i=j)(1-p_i)^j z\frac{1-\left(\frac{z}{1-p_i}\right)^j}{1-p_i-z}\\
& = Y_i(1-p_i) + \frac{p_i z}{1-p_i-z}\sum_{j=1}^\infty \mathbb{P}(Y_i=j)\left((1-p_i)^j-z^j\right)\\
& = Y_i(1-p_i) + \frac{p_iz}{1-p_i-z}\left(Y_i(1-p_i)-Y_i(z)\right),
\end{align*}
where in the second step we condition on the total number of arrivals and take into account how we can get to $k$ blocked vehicles; in the third step we interchange the order of the summation; and in the fourth step we compute a geometric series.
If $m>1$, the interpretation as above for the case $m=1$ is not necessarily meaningful. It is more difficult to compute the $Y_{i,b}$ in a logical and consistent way. This has to do with the fact that if $m>1$ we consider batches of vehicles that are either all blocked or not, whereas the $Y_{i,b}$'s are about individual vehicles. As mentioned before in Remark~\ref{rem:pi}, if $m>1$ we often have that either $p_i=0$ or $p_i=1$. If $p_i=0$, the general expression for $Y_{i,b}(z)$ reduces to:
\[
Y_{i,b}(z) = Y_{i}(1) + 0\cdot(Y_i(1)-Y_i(z)) = Y_{i}(1)=1,
\]
which makes sense as there are no turning vehicles in case $p_i=0$. If $p_i=1$, we have that:
\[
Y_{i,b}(z) = Y_{i}(0) - (Y_{i}(0)-Y_{i}(z))=Y_{i}(z),
\]
which is also logical: every arriving vehicle is a turning vehicle if $p_i=1$, so we have that $Y_{i,b}(z)=Y_{i}(z)$.
\end{remark}
Except for the constants $\mathbb{P}(X_{i}=l,S=u)$ and $\mathbb{P}(X_{i}=l,S=b)$ for $i=1,\dots,g_1$ and $l=0,\dots,m-1$, and $\mathbb{P}(X_{i} = l)$ for $i=g_1+1,\dots,g_1+g_2-1$, $i=c$, and $l=0,\dots,m-1$, we are now done. We explain how to find the (so far) unknown constants in the next part of this subsection.
%
%
%
\subsubsection{Finding the unknowns in $X_{g_1+g_2}(z)$}\label{subsubsec:completion}
As mentioned before, we still need to find several unknowns before the expression for $X_{g_1+g_2}(z)$ is complete.
The standard framework for the FCTL queue as described in e.g.~\cite{van2006delay} is also applicable to the bFCTL queue with multiple lanes with some minor differences. Although we are dealing with more complex formulas, the key ideas are identical. We have $m(g_1+g_2)+(m-1)g_1$ unknowns in the numerator $X_n(z)$ of $X_{g_1+g_2}(z)$ in Equation~\eqref{eq:generalbFCTL} and we have $m(g_1+g_2)$ roots with $|z|\leq 1$ for the denominator $X_d(z)$ of $X_{g_1+g_2}(z)$, assuming stability of the queueing model. An application of Rouch\'{e}'s theorem, see e.g.~\cite{adan2006application}, shows that $X_d(z)$ indeed has $m(g_1+g_2)$ roots on or within the unit circle assuming stability. One root is $z=1$, which leads to a trivial equation and as a substitute for this root, we put in the additional requirement that $X_{g_1+g_2}(1)=1$. The remaining $(m-1)g_1$ equations are implicitly given in Equations~\eqref{eq:X1b} and \eqref{eq:Xib}. We give them here separately for completeness. We have for $k=1,...,m-1$
\begin{equation*}\label{eq:bFCTL_prob_blocked_first}
\mathbb{P}(X_{1} = k,S=b) =
p_1q_1 \sum_{l=1}^k \mathbb{P}(X_{c} = l)\mathbb{P}(Y_1 = k-l) + q_1\mathbb{P}(X_{c}=0)\mathbb{P}(Y_{1,b} = k),
\end{equation*} and for $i=2,\dots,g_1$ and $k=1,\dots,m-1$
\begin{equation*}\label{eq:bFCTL_prob_blocked}
\begin{aligned}
\mathbb{P}(X_{i} = k, S=b) = &
\sum_{l=1}^{k} \left\{p_iq_i\mathbb{P}(X_{i-1}=l,S=u)+q_i\mathbb{P}(X_{i-1}=l,S=b)\right\}\mathbb{P}(Y_i = k-l)\\&+ q_i\mathbb{P}(X_{i-1} = 0,S=u)\mathbb{P}(Y_{i,b} = k),
\end{aligned}
\end{equation*}
which provides us with the $(m-1)g_1$ additional equations. In total, we obtain a set of $m(g_1+g_2)+(m-1)g_1$ linear equations with $m(g_1+g_2)+(m-1)g_1$ unknowns, which we can solve to find the unknown $\mathbb{P}(X_{i}=l,S=u)$, for $i=1,\dots,g_1$ and $l=0,\dots,m-1$, the unknown $\mathbb{P}(X_{i}=l,S=b)$, for $i=1,\dots,g_1$ and $l=1,\dots,m-1$, and the unknown $\mathbb{P}(X_{i}=l)$, for $i=g_1+1,\dots,g_1+g_2-1$, $i=c$, and $l=0,\dots,m-1$. Due to the complicated structure of our formulas, we do not obtain a similar, easy-to-compute Vandermonde system as for the standard FCTL queue (see~\cite{van2006delay}), but a linear solver is in general able to find the unknowns (we did not encounter any numerical issues/problems in the examples that we studied).
There are several ways to obtain the roots of $X_d(z)$ in Equation~\eqref{eq:generalbFCTL}. Because those roots are subsequently used in solving a system of linear equations, we need to find the required roots with a sufficiently high precision, certainly if $m(g_1+g_2)+(m-1)g_1$ is large. In some cases, the roots can be found analytically, e.g. in case the number of arrivals per slot has a Poisson or geometric distribution. In other cases, the roots have to obtained numerically. There are several ways to do so. An algorithm to find roots is given in~\cite{boon2019pollaczek}, Algorithm~1, while two other methods, one based on a Fourier series representation and one based on a fixed point iteration, are described in~\cite{janssen2005analytic}.
\subsection{Performance measures}\label{sec:performance_measures}
Now that we have a complete characterization of $X_{g_1+g_2}(z)$, we can find the PGFs of the queue-length distribution at the end of the other slots by employing Equations~\eqref{eq:X1b} up to \eqref{eq:Xg1+g2+i}. This basically implies that we can find any type of performance measure related to the queue-length distribution. As an example we find the PGF of the queue-length distribution at the end of an arbitrary slot. We denote this PGF with $X(z)$ and obtain the following expression:
\begin{equation*}
X(z) = \frac{1}{c}\sum_{i=1}^c X_{i}(z) .
\end{equation*}
Another important performance measure is the delay distribution. The mean of the delay distribution, $\mathbb{E}[D]$, can easily be derived from the mean queue length at the end of an arbitrary slot by means of Little's law with a time-varying arrival rate (for a proof of Little's law in this setting see e.g.~\cite{stidham1972lambda}):
\begin{equation*}
\mathbb{E}[D] = \frac{X^\prime(1)}{\frac{1}{c}\sum_{i=1}^cY_{i}^\prime(1)}.
\end{equation*}
The PGF of the delay distribution can be derived (as is done for the FCTL queue in~\cite{van2006delay}), but such a derivation is more difficult. In the regular FCTL queue, the number of slots an arriving car has to wait is deterministic when conditioned on the number of vehicles in the queue and the time slot in which the car arrives. This is not the case for the bFCTL queue as the occurrence of blockages is random. By proper conditioning on the various blocked slots and queue lengths, one can obtain the delay distribution from the distribution of the queue length. We do not pursue this here.
If we want to obtain probabilities and moments from a PGF, we need to differentiate the PGF and respectively put $z=0$ or $z=1$. In our experience, this has not proven to be a problem. However, differentiation might become prohibitive in various settings, e.g. when $m(g_1+g_2)+(m-1)g_1$ becomes large or if we want to obtain tail probabilities. There are ways to circumvent such problems. If we are pursuing probabilities and do not want to rely on differentiation, we might use the algorithm developed by Abate and Whitt in~\cite{abate1992numerical} to numerically obtain probabilities from a PGF. For obtaining moments of random variables from a PGF, an algorithm was developed in~\cite{choudhury1996numerical} which finds the first $N$ moments of a PGF numerically. Essentially, this shows that, from the PGF, we can obtain any type of quantity related to the steady-state distribution of the queue length, in the form of a numerical approximation.
All formulas computed in this section have been verified by comparing the numerical results with a simulation which mimics our discrete-time queueing model. More information about this simulation is given in Appendix~\ref{a:simulation}.
\section{Examples}\label{sec:results}
We start in Subsection~\ref{subsec:special} with several special cases of the bFCTL queue for which we provide explicit expressions for the PGF of the overflow queue and relate those special cases to the existing literature. Subsequently, we make a comparison between the capacity obtained in the HCM~\cite{HCM} and the capacity in our model in Subsection~\ref{subsec:capacity2}. After that, we investigate the influence of several parameters on the performance measures in numerical examples. We consider performance measures like the mean and variance of the steady-state queue-length distribution, both at specific moments and at the end of an arbitrary slot, the mean delay, and several interesting queue-length probabilities. We study the influence of the $p_i$ and $q_i$ in Subsection~\ref{subsec:parameter}. In Subsection~\ref{subsec:layout}, we compare the case of turning and straight-going traffic on a single lane, as present in the bFCTL queue where blockages of all vehicles might occur, and cases where we have dedicated lanes for the right-turning and straight-going traffic where only turning vehicles are blocked. Note that we will consider each lane \emph{separately} in those examples, so there is no conflict with e.g. Remark~\ref{rem:pi}.
\subsection{Special cases of the bFCTL queue}\label{subsec:special}
We study several special cases of the bFCTL queue, e.g. cases where the bFCTL queue reduces to the FCTL queue.
If $q_i=1$, an explicit expression for the PGF of the distribution of the overflow queue, $X_{g_1+g_2}(z)$, can be written down relatively easily. When it is further assumed, for the ease of exposition, that all $p_i=p$, $Y_i\overset{d}= Y$, $Y_{i,b}\overset{d} = Y_b$ and $m=1$, the following expression for $X_{g_1+g_2}(z)$ is obtained:
\begin{equation}\label{eq:explicit_Xg_q=0}
X_{g_1+g_2}(z) = \frac{X_n(z)}{X_d(z)},
\end{equation} with
\begin{align}
\label{eq:Xn_q=0}
X_n&(z) =z^{g_1+g_2}\sum_{i=0}^{g_2-1}\left(\frac{Y(z)}{z}\right)^{g_2-i-1}\left(1-\frac{Y(z)}{z}\right)\mathbb{P}(X_{g_1+i}=0)+ \nonumber\\
& z^{g_1}Y(z)^{g_2}\sum_{i=0}^{g_1-1}\nonumber \Bigg\{ \mathbb{P}(X_{i}=0,S=u)\Bigg[\left(Y_b(0) - (1-p)\frac{Y(z)}{z}\right)\left((1-p)\frac{Y(z)}{z}\right)^{g_1-i-1}+\\&\left(Y_b(z)-Y_b(0)-p Y(z)\right)Y(z)^{g_1-i-1}\Bigg]+\nonumber\\
&p Y(z)^{g_1-i}\sum_{j=0}^{i-1} \mathbb{P}(X_{j}=0,S=u)\left(Y_b(0) - (1-p)\frac{Y(z)}{z}\right)\left((1-p)\frac{Y(z)}{z}\right)^{i-j-1}\Bigg\},
\end{align}
where $\mathbb{P}(X_{0}=0,S=u)$ is to be interpreted as $\mathbb{P}(X_{c}=0)$, and
\begin{equation}\label{eq:Xd_q=0}
X_d(z) = z^{g_1+g_2} - \left(\left(1-p\right)^{g_1}+p z^{g_1}\sum_{i=0}^{g_1-1}\left(\frac{1-p}{z}\right)^i\right) Y(z)^c.
\end{equation}
The reason that we provide an explicit formula for this particular case is that this formula is significantly easier than the formula in the case where $q_i< 1$ for one or more $i=1,\dots,g_1$. The stability condition (cf. Algorithm~\ref{alg:stability} in Subsection~\ref{subsec:capacity}) for this example is relatively easy to derive and reads as follows:
\begin{equation*}
\begin{cases}
\mu c < g_1+g_2, & \textrm{if } p = 0,\\
\mu c < g_2, & \textrm{if } p = 1,\\
\mu c < g_2+\left(1-(1-p)^{g_1}\right)\frac{1-p}{p}, & \textrm{otherwise},
\end{cases}
\end{equation*}
where $\mu$ is the mean arrival rate per slot, i.e. $\mu=\mathbb{E}[Y]$.
This can be understood as follows: if $p=0$ there are no turning vehicles and we obtain the regular FCTL queue with green period $g_1+g_2$. If $p=1$ all vehicles are turning vehicles and there are no departures during the first part of the green period because $q_i=1$, so we obtain the FCTL queue with green period $g_2$. The other case can be understood as follows: on the left-hand side we have the average number of arrivals per cycle whereas on the right-hand side we have the average number of slots available for delayed vehicles to depart. Indeed, on the right-hand side we have $g_2$, the number of green slots during the second part of the green period which are all available for vehicles to depart, and the number of green slots available for departures during the first green period:
\[
\sum_{i=1}^{g_1}(1-p)^i = \left(1-(1-p)^{g_1}\right)\frac{1-p}{p}.
\]
If $p_i=0$ for all $i$, i.e. there are no blockages occurring at all (regardless of the $q_i$), the FCTL queue with multiple lanes (with green period $g=g_1+g_2$) is obtained. Note that we do not have to include the state $S$, because there are no blockages of batches of vehicles. If $m=1$, we obtain the regular FCTL queue as studied in e.g.~\cite{van2006delay}. This can e.g. be observed when putting $p_i=0$ and $m=1$ in Equations~\eqref{eq:explicit_Xg_q=0}, \eqref{eq:Xn_q=0}, and \eqref{eq:Xd_q=0}. The expression for $X_{g_1+g_2}(z)$ or, alternatively, $X_g(z)$ is (after rewriting):
\begin{align}\label{eq:FCTL}
X_g(z) & = \frac{(z-Y(z))z^{g-1}\sum_{i=0}^{g-1}\mathbb{P}(X_i=0)\left(\frac{Y(z)}{z}\right)^{g-i-1}}{z^g-Y(z)^c},
\end{align}
where $\mathbb{P}(X_{0}=0)$ is to be interpreted as $\mathbb{P}(X_{c}=0)$. For general $m$, we have the following formula:
\begin{equation}
X_g(z) = \frac{z^{mg}\sum_{i=0}^{g-1}\sum_{l=0}^{m-1} \mathbb{P}(X_{i}=l)\left(1-\frac{Y(z)}{z^{m-l}}\right)\left(\frac{Y(z)}{z}\right)^{g-i-1}}{z^{mg}-Y(z)^c},
\end{equation}
where the $\mathbb{P}(X_{0}=l)$, $l=0,\dots,m-1$, are to be interpreted as $\mathbb{P}(X_{c}=l)$. The stability condition for this case can be verified to be
\[
\mu c < m g
\]
which is in accordance with Algorithm~\ref{alg:stability}.
It can also be verified that the bFCTL queue reduces to the regular FCTL queue with green time $g = g_2$ and red time $r+g_1$, if $p_i = 1$ and $q_i = 1$.
We note that for the FCTL queue with a single lane and no blockages (i.e. $p_i=0$ or $p_i=1$ and $q_i=1$) there is an alternative characterization of the PGF in terms of a complex contour integral, see~\cite{boon2019pollaczek}. It remains an open question whether such a contour-integral representation exists for the bFCTL with multiple lanes, as the polynomial structure in terms of $Y(z)/z$ as present in Equation~\eqref{eq:FCTL} is not present in the general bFCTL queue. This feature of the FCTL queue seems essential to obtain a contour-integral expression as is done in~\cite{boon2019pollaczek}.
In \cite{boon2019pollaczek}, a decomposition result is presented in Theorem~2. It shows that several related queueing processes can in fact be decomposed in the independent sum of the FCTL queue and some other queueing process. It is likely that the bFCTL queue with multiple lanes allows for some of those generalizations as well. We mention randomness in the green and red time distributions as a relevant potential extension.
\subsection{Capacity} \label{subsec:capacity2}
In order to compare our model and the existing literature (focusing on the HCM~\cite{HCM}), we provide several examples in this subsection.
The formula for the capacity of a permitted right-turn lane in a shared lane in the HCM is
\begin{equation*}
s_{sr} = \frac{s_{th}}{1+P_r\left(\frac{E_R}{f_{Rpb}}-1\right)},
\end{equation*}
cf.~\cite{HCM} equation (31-105). Here, $s_{sr}$ is the saturation flow of the shared lane, $s_{th}$ the saturation flow of an exclusive through lane, $P_r$ the right-turning portion of vehicles, $E_R$ the equivalent number of through vehicles for a protected right-turn vehicle and $f_{Rpb}$ is the bicycle-pedestrian adjustment factor for right-turn groups. The latter is defined as the average amount of time during the green period during which right-turning vehicles are not blocked, i.e., in our model, there are no pedestrians crossing. There is a procedure provided in the HCM to compute this factor, but in our model this simply corresponds to the $q_i$ and we will determine the $f_{Rpb}$ factor on the $q_i$. Further, in order to make a comparison with our model, we turn the saturation flow of the shared lane into a number of vehicles per cycle.
More concretely, we choose the green period to be $30$ seconds, split into the two phases as follows: $g_1=20$ and $g_2=10$. We pick the cycle length to be $90$ seconds, the time slots to have length $2$ seconds and we focus on a single shared lane, so we have at most $1$ vehicle departing per time slot. Further, we choose the right-turning portion vehicles to be $1$ or $0.9$ in our examples. Lastly, for the HCM formula, we assume that vehicles heading straight have a crossing time of $1$ second. To account for this effect in our bFCTL model, we use the correction discussed in Remark~\ref{rem:capacity}. In this example we have:
\[
m^* = p_i m_\textit{turn} + (1-p_i)m_\textit{through} = p_i\times 1 + (1-p_i)\times 2.
\]
This enables us to compute the capacity in our model and in the HCM up to the $q_i$.
We first focus on the cases with $p_i=1$ and we display the capacity according to the HCM in Figure~\ref{f:cap1}(a). Note that $f_{Rpb}$ is at least $1/3$ because $g_1=20$ and $g_2=10$, implying that during at least a part $1/3$ of the cycle, turning vehicles are not blocked. In Figure~\ref{f:cap1}(a) we also depict two capacities according to the bFCTL queue. In case $(1)$ we assume that all the $q_i$ are the same and are chosen in such a way that the $f_{Rpb}$ in the HCM formula is matched. E.g. in case $f_{Rpb}=1/3$, we choose $q_i=0$ as there are no pedestrians and in case $f_{Rpb}=2/3$, we choose $q_i=1/2$. In case $(2)$, we consider a step function for the $q_i$ such that
\begin{equation*}
q_i = \begin{cases}
1 & \textrm{if } i < k\\
0 & \textrm{if } i > k\\
k^* & \textrm{otherwise,}
\end{cases}
\end{equation*}
for some values of $k$ and $k^*$ such that the $q_i$ match with the value for $f_{Rpb}$ that is used in the formula for the HCM.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\textwidth]{capacity_p_1rev.pdf}
&
\includegraphics[width=0.45\textwidth]{capacity_p_2rev.pdf}\\
(a) & (b) \\[1ex]
\end{tabular}
\caption{Capacity in vehicles per cycle for the example according to the HCM and to the bFCTL queue with two different choices for the $q_i$ (case $(1)$ and case $(2)$) as detailed in the text. We have $p_i=1$ in (a) and $p_i=0.2$ in (b).}
\label{f:cap1}
\end{figure}
Figure~\ref{f:cap1}(a) makes sense: if $f_{Rpb}$ is for example equal to $1$, there are no pedestrians crossing (i.e. $q_i=0$), and then the number of vehicles departing per cycle is $(g_1+g_2)/2=15$. The capacity according to the bFCTL queue when $p_i=1$ is equal to (after simplification)
\begin{equation}\label{eq:cap}
\frac{g_1+g_2}{2} - \sum_{i=1}^{g_1} q_i.
\end{equation}
\textbf{This shows that when $\sum_{i=1}^{g_1} q_i$ is translated into the factor $f_{Rpb}$ in the HCM, we have an identical capacity.} E.g. if the $q_i=0$, then also in the bFCTL queue, the capacity is equal to $15$ vehicles per cycle. Equation~\eqref{eq:cap} also indicates that it does not matter in which slots the pedestrians are crossing if $p_i=1$ (when looking at the capacity). In this case, the $q_i$ only influence the capacity through their sum, however in general the individual $p_i$ and $q_i$ have an impact on the capacity (and the queue-length process). Similar observations hold if $p_i=0$, i.e. there are no turning vehicles.
\textbf{If the $p_i$ are not equal to $1$, there are differences between the capacity in the HCM and the bFCTL queue.} We study an example where $p_i=0.2$. The results are depicted in Figure~\ref{f:cap1}(b). The values for the capacity obtained with the function in the HCM are slightly lower than the values that we obtain in both cases of the bFCTL queue.
In contrast with the previous example, there are differences between all three choices which relate to various causes.
The main reason for the occurring difference between cases $(1)$ and $(2)$ in the bFCTL queue, is that the individual $q_i$ are determining the capacity rather than the total value of the $q_i$'s alone as was the case when $p_i=1$. \textbf{Here we thus see that our detailed description of the queueing model in terms of slots is necessary to fully understand the capacity (and, more generally, the queueing process).}
In this subsection we have been working under several assumptions. If one would, e.g., also incorporate start-up delays as is done in~\cite{shaoluen2020random}, we would see that the capacity in the HCM results in an overestimation of the capacity as is more generally observed~\cite{shaoluen2020random}. We also expect that the distribution of the $q_i$ over the different slots has a bigger impact on the capacity and queueing process if start-up delays are incorporated. Implementing such effects into our model is possible (probably in a similar way as including a departure variable as discussed above), but is beyond the scope of the present paper.
\subsection{The bFCTL queue with turning vehicles and pedestrians}\label{subsec:parameter}
In this subsection, we study the bFCTL queue with a single lane, so $m=1$. The setting in this subsection is as depicted in Figure~\ref{fig:vis}(b). We mainly focus on the distribution of $X_{g_1+g_2}$, to which we refer as the overflow queue, as this is the distribution from which some interesting performance measures can be derived.
This distribution reflects the probability distribution of the queue size at the moment that the green light switches to a red light. We also briefly consider some other performance measures.
\subsubsection{Influence of the number of turning vehicles}\label{sec:influenceofturning}
First, we vary the fraction of right-turning vehicles $p_i$ and study its influence on $X_{g_1+g_2}$. We choose the $p_i$ to be the same for each $i$, so we have $p_i=p$, and we vary $p$. We choose the value of the $q_i=q$ to be $1$, so there are always pedestrians on the pedestrian crossing during the first part of the green period with length $g_1$. In this way, we can effectuate the influence of the fraction of turning vehicles on the performance measures. Further, we choose $g_1$ to be either $2$ or $10$ and we choose $g_2=r=2g_1$. The arrival process is taken to be Poisson with mean $0.39$. Note that the lane is close to its point of saturation, because the capacity can be shown to be equal to $0.4$.
We display results for $\mathbb{P}(X_{g_1+g_2}\leq j)$ for $j=0,\dots,10$ in Figure~\ref{fig:ex3}.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[height=4.7cm]{linechartXg1g2p1_039.pdf}
&
\includegraphics[height=4.7cm]{linechartXg1g2p2_039.pdf}\\
(a) & (b) \\[1ex]
\end{tabular}
\caption{Cumulative Distribution Function (CDF) of the overflow queue for various values of $p_i=p$, $q_i=q=1$, and Poisson arrivals with mean $0.39$. In (a) we have $g_2=r=2g_1=4$ and in (b) we have $g_2=r=2g_1=20$.}
\label{fig:ex3}
\end{figure}
As can be observed from Figure~\ref{fig:ex3}, \textbf{the fraction of turning vehicles may dramatically influence the number of queueing vehicles}. There is virtually no queue at the end of the green period when there are no turning vehicles ($p=0$), whereas in \textbf{more than 50\%} of the cases there is a queue of at least $10$ vehicles at the end of the green period when all vehicles are turning vehicles ($p=1$). The blockages of the turning vehicles in the latter case effectively reduce the green period by a factor $1/3$ in our examples (as $q=1$), which causes the huge difference in performance. We note that the distribution of $X_{g_1+g_2}$ coincides with the overflow queue distribution in the FCTL queue when $p=0$ (when we take $g_1+g_2$ as the green period and $r$ as the red period in the FCTL queue) and when $p=1$ and $q=1$ (with $g_2$ the green period and $r+g_1$ the red period).
When comparing Figures~\ref{fig:ex3}(a) and ~\ref{fig:ex3}(b), we see that \textbf{the influence of $p$ is not uniform across the two examples}. In case $p=0$ or $p=1$, the probability of a large overflow queue is larger for the case where $g_1=2$. This might be clarified by noting that a larger cycle reduces the amount of within-cycle variance which reduces the probabilities of a large queue length. If $0<p<1$ this does not seem to be the case. This might be due to the fact that a relatively big part of the first green period is eaten away by turning vehicles that are blocked when $g_1=10$. For example, when $p>0$ and the first vehicle is a turning vehicle, immediately the entire period $g_1$ is wasted because $q=1$. This is of course also the case when $g_1=2$, but the blockage is resolved sooner and during the second part of the green period the blocked vehicle may depart relatively soon in comparison with the case where $g_1=10$.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.47\textwidth]{barChartXI0_039.pdf}
&
\includegraphics[width=0.47\textwidth]{barChartMeanXi_039.pdf}\\
(a) & (b) \\[1ex]
\end{tabular}
\caption{In~(a) $\mathbb{P}(X_{i}=0)$ for slot number $i=1,\dots,10$ is displayed for two different values of $p_i$, where orange corresponds to $p_i=p=0$ and blue to $p_i=p=0.6$, with $2g_1=g_2=r = 4$, $q_i=q=1$, and with Poisson arrivals with mean $0.39$. In~(b) the same two examples are studied, but the mean queue length $\mathbb{E}[X_{i}]$ at the end of slot $i$ is shown. }
\label{fig:ex3xk}
\end{figure}
In Figure~\ref{fig:ex3xk}(a), we see the probability of an empty queue after slot $i$, where $i=1,2,\dots,c$, for two different values of $p$. For the case $p=0$ (in orange) we have a monotone increasing sequence of probabilities during the green period as one would expect: this setup corresponds to a regular FCTL queue and once the queue empties during the green period, it stays empty. We see that for the case $p=0.6$ (in blue) the probabilities of an empty queue after slot $i$ are much lower (as there are more turning vehicles which might be blocked and hence cause the queue to be non-empty). In fact, the probability of an empty queue even decreases when going from slot $2$ to slot $3$. This can be clarified by the fact that the queue might start building again even when the queue is (almost) empty: e.g. if the queue is empty during the first green period and there is an arrival of a turning vehicle, that vehicle will be blocked as $q=1$ in which case the queue is no longer empty.
The same type of behaviour is reflected in the mean queue length at the end of a slot, as can be observed in Figure~\ref{fig:ex3xk}(b). Even though the green period already started, the queue in the example with $p=0.6$ still grows (in expected value) during the first part of the green period, see the first two blue bars. This is caused by the fact that vehicles might be blocked, \textbf{which demonstrates the possibly severe impact of blocked vehicles on the performance of the system}.
\subsubsection{Influence of the pedestrians}
Secondly, we investigate the influence of the presence of pedestrians by studying various values for the $q_i$. A high value of the $q_i$ corresponds to a high density of pedestrians as $q_i$ corresponds to the probability that a turning vehicle is not allowed to depart during the first green period. Conversely, a low value of the $q_i$ corresponds to a low density of pedestrians and a relatively high probability of a turning vehicle departing during the first green period. We choose $p_i=p=0.5$ and take $g_1=g_2=r=10$. We take Poisson arrivals with mean $0.36$. We study one set of examples where the $q_i$ are constant over the various slots, see Figure~\ref{fig:ex4xk}(a). We also study the influence of the dependence of the $q_i$ on $i$ by investigating two cases with all parameters as before in Figure~\ref{fig:ex4xk}(b). In one case we take $q_i=0.5$ for all $i$, but in the other case we take $q_i=1-(i-1)/g_1$. The latter case reflects a decreasing number of pedestrians blocking the turning flow of vehicles during the first part of the green period.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[height=4.5cm]{linechartXg1g2q1.pdf}
&
\includegraphics[height=4.5cm]{barChartMeanXIqi.pdf}\\
(a) & (b) \\[1ex]
\end{tabular}
\caption{In (a) the CDF of the overflow queue is displayed for various values of the $q_i$ with all $q_i=q$ the same, $p_i=p=0.5$, Poisson arrivals with mean $0.36$, and $g_1=g_2=r=10$. In (b) the $\mathbb{E}[X_{i}]$ are compared for slot number $i=1,\dots,30$ with in orange $q_i=0.5$ and in blue $q_i=1-(i-1)/g_1$ for $i=1,\dots,g_1$. Further, it is assumed that $p_i=p=0.5$, that the number of arrivals in each slot follows a Poisson distribution with mean $0.36$, and that $g_1=g_2=r=10$.}
\label{fig:ex4xk}
\end{figure}
We note that it is \textbf{important to estimate the correct blocking probabilities $q_i$ from data}, when applying our analysis to a real-life situation \textbf{as the $q_i$ have an impact on the performance measures}. In Figure~\ref{fig:ex4xk}(a), we clearly see that the more pedestrians, the longer the queue length at the end of the green period is. Indeed, if there are more pedestrians, there are relatively many blockages of vehicles which causes the queue to be relatively large.
Moreover, \textbf{it is important to capture the dependence of the $q_i$ on the slot $i$ in the right way}, see Figure~\ref{fig:ex4xk}(b). Even though, on average over all slots, the mean number of pedestrians present is similar in the two cases, we see a clear difference between the two examples. In the case with decreasing $q_i$ (in blue), we see an initial increase of the mean queue length during the first green slots of the cycle, caused by a relatively large fraction of turning vehicles ($p=0.5$) \emph{and} a high value of $q_i$. This is not the case in the other example where $q_i=0.5$ for all $i$. After some slots of the first green period, the decrease in the mean queue length is quicker for the example where the $q_i$ decrease when $i$ increases, which can (at least partly) be explained by the decreasing $q_i$. During the remaining part of the cycle, the queue in front of the traffic light behaves more or less the same in both examples and even the mean overflow queue, $\mathbb{E}[X_{g_1+g_2}]$, is not that much different for the two examples. This implies, as can also be observed in Figure~\ref{fig:ex4xk}(b), that the mean queue length during the red period is comparable as well for our setting. This does not hold for the mean queue length at the end of an arbitrary slot and the mean delay, because of the differences in the queue length during the first part of the green period.
\subsection{Shared right-turn lanes and dedicated lanes}\label{subsec:layout}
We continue with a study of several numerical examples that focus on the differences between shared right-turn lanes and dedicated lanes for turning traffic. We do so in order to provide relevant insights in the benefit of splitting the vehicles in different streams. Firstly, we study the difference between a single shared right-turn lane (as visualized in Figure~\ref{fig:vis2}(a)) and a case where the straight-going and turning vehicles are split into two different lanes. In the latter case, we thus have two lanes, one for the straight-going traffic and one for the turning traffic (as visualized in Figure~\ref{fig:vis2}(b)) which we can analyze as two separate bFCTL queues.
\begin{figure}[h!]
\centering
\begin{tabular}{ccc}
\begin{tikzpicture}[scale=0.1,rotate=90]
\draw[black,fill=lightgray](30,20) rectangle (40,50);
\draw[black,fill=lightgray](0,30) rectangle (50,40);
\draw[thick,white,dash pattern=on 7 off 4](0,35) to (50,35);
\draw[thick,white,dash pattern=on 7 off 4](35,0) to (35,60);
\draw[lightgray,fill=lightgray](30,30) rectangle (40,40);
\draw[white,->,thick](24,32.5) to [out = 0, in = 120] (28,30.5);
\draw[white,->,thick](24,32.5) to [out = 0, in = 180] (28.5,32.5);
\draw[white,fill=white](31,25.5) rectangle (32,28.5);
\draw[white,fill=white](33,25.5) rectangle (34,28.5);
\draw[white,fill=white](36,25.5) rectangle (37,28.5);
\draw[white,fill=white](38,25.5) rectangle (39,28.5);
\end{tikzpicture}
&\hspace{-1cm} \begin{tikzpicture}[scale=0.1,rotate=90]
\draw[black,fill=lightgray](30,20) rectangle (40,60);
\draw[black,fill=lightgray](0,30) rectangle (60,50);
\draw[thick,white,dash pattern=on 7 off 4](0,35) to (60,35);
\draw[thick,white,dash pattern=on 7 off 4](0,45) to (60,45);
\draw[thick,white](0,40) to (60,40);
\draw[thick,white,dash pattern=on 7 off 4](35,0) to (35,60);
\draw[lightgray,fill=lightgray](30,30) rectangle (40,50);
\draw[white,->,thick](24,32.5) to [out = 0, in = 120] (28,30.5);
\draw[white,->,thick](24,37.5) to [out = 0, in = 180] (28.5,37.5);
\draw[white,fill=white](31,25.5) rectangle (32,28.5);
\draw[white,fill=white](33,25.5) rectangle (34,28.5);
\draw[white,fill=white](36,25.5) rectangle (37,28.5);
\draw[white,fill=white](38,25.5) rectangle (39,28.5);
\end{tikzpicture}
&\hspace{-1cm} \begin{tikzpicture}[scale=0.1,rotate=90]
\draw[black,fill=lightgray](30,20) rectangle (40,60);
\draw[black,fill=lightgray](0,30) rectangle (60,50);
\draw[thick,white,dash pattern=on 7 off 4](0,35) to (60,35);
\draw[thick,white,dash pattern=on 7 off 4](0,45) to (60,45);
\draw[thick,white](0,40) to (60,40);
\draw[thick,white,dash pattern=on 7 off 4](35,0) to (35,60);
\draw[lightgray,fill=lightgray](30,30) rectangle (40,50);
\draw[white,->,thick](24,32.5) to [out = 0, in = 120] (28,30.5);
\draw[white,->,thick](24,32.5) to [out = 0, in = 180] (28.5,32.5);
\draw[white,->,thick](24,37.5) to [out = 0, in = 180] (28.5,37.5);
\draw[white,fill=white](31,25.5) rectangle (32,28.5);
\draw[white,fill=white](33,25.5) rectangle (34,28.5);
\draw[white,fill=white](36,25.5) rectangle (37,28.5);
\draw[white,fill=white](38,25.5) rectangle (39,28.5);
\end{tikzpicture}
\\
\hspace{-0.9cm}\scriptsize (a)
&
\hspace{-2.85cm}\scriptsize (b)
&
\hspace{-2.85cm}\scriptsize (c)
\end{tabular}
\caption{The various lane configurations considered in Subsection~\ref{subsec:layout}. In~(a) we have a single lane with a shared right-turn lane. In~(b) we have two dedicated lanes: one for straight-going vehicles and one for right-turning traffic, whereas in~(c) we have a two-lane setup with one lane for straight-going vehicles only and a shared right turn.}
\label{fig:vis2}
\end{figure}
Secondly, we compare two two-lane settings. The first is visualized in Figure~\ref{fig:vis2}(b), while the other is a two-lane scenario where one lane is a dedicated lane for straight-going traffic and the other is a shared right-turn lane as depicted in Figure~\ref{fig:vis2}(c). We thus allow for straight-going traffic to mix with some of the right-turning vehicles in the latter case. We do so in order to make sure that the shared right-turn lane together with the lane for vehicles heading straight has the same capacity as the two lanes where the two streams of vehicles are split (as opposed to the first example in this subsection). In both two-lane scenarios we, again, analyze the two lanes as two separate bFCTL queues.
\subsubsection{One lane for the shared right-turn}
We start with comparing the traffic performance of a single shared right-turn lane as in Figure~\ref{fig:vis2}(a), case ($1$), and a two-lane scenario where the turning vehicles and the straight-going vehicles are split as in Figure~\ref{fig:vis2}(b), case ($2$). We refer in the latter case to the lane which has right-turning vehicles as lane $1$ and to the other lane we refer as lane $2$. We assume that the arrival process is Poisson and that the arrival rate of turning vehicles, $\mu_1$, and straight-going vehicles, $\mu_2$, are the same in both cases. The total arrival rate of vehicles is $\mu=\mu_1+\mu_2$ in case~($1$). We choose $p_i=0.3$ for the shared right-turn lane, whereas in the two-lane case we have $p_i=1$ for lane $1$ and $p_i=0$ for lane $2$ and arrival rates $\mu_1=0.3\mu$ at lane $1$ and $\mu_2=0.7\mu$ at lane $2$. Further, we choose $q_i=1$, $g_1=8$, $g_2=20$, and $r=20$. We compute the mean queue length at the end of an arbitrary time slot for both lanes in case ($2$), denoted with $\mathbb{E}[X^{(i)}]$ for lane $i$, and the total mean queue length at the end of an arbitrary time slot, denoted with $\mathbb{E}[X^{t}]$, and which equals $\mathbb{E}[X^{(1)}]+\mathbb{E}[X^{(2)}]$. For case ($1$) we denote the mean queue length at the end of an arbitrary time slot with $\mathbb{E}[X^t]$. The delay of an arbitrary car is denoted with $\mathbb{E}[D]$ for both cases ($1$) and ($2$). We study an example with various values of $\mu$ in Figure~\ref{f:single}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.7\textwidth]{table_1_fig_cars.pdf}\\
(a)\\[1ex]
\includegraphics[width=0.7\textwidth]{table_1_fig_delaywebster.pdf}\\
(b) \\[1ex]
\caption{The total Poisson arrival rate, $\mu$, on the horizontal axis and in (a) the mean queue length at the end of an arbitrary time slot for the various cases and lanes where $\mathbb{E}[X^{t}]=\mathbb{E}[X^{(1)}]+\mathbb{E}[X^{(2)}]$ for case (2), and in (b) the mean delay for the various cases.}
\label{f:single}
\end{figure}
In Figure~\ref{f:single}, we can clearly see that the total mean queue length at the two lanes in case ($2$) is lower than the mean queue length at the single lane in case ($1$). This makes sense from various points of view: in case ($2$), we have twice as many lanes as in case ($1$), so we would expect a smaller total mean queue length in case ($2$). Moreover, in case ($1$), it might happen that straight-going vehicles are blocked. Such blockages cannot occur in case ($2$), as all turning traffic is on lane $1$ and all vehicles that go straight are on lane $2$. These two reasons are the main drivers for the performance difference in cases ($1$) and ($2$). From the point of view of the traffic performance, \textbf{it thus makes sense to split the traffic on a shared right-turn lane into two separate streams of vehicles on two lanes while assuming one lane available for departures in case ($1$) and two lanes in case ($2$)}. We observe similar results when looking at the mean delay and comparing cases ($1$) and ($2$).
\begin{remark}
We emphasized before that the blocking mechanism makes it impossible to use existing methods to analyze the queue lengths and delays. However, in this particular example we have chosen the parameter settings in such a way that case (2) \emph{can} be analyzed using existing methods. The reason is that we have two separate lanes, each with its own ``extreme'' blocking mechanism: lane 1 contains \emph{only} turning vehicles and \emph{all} of them are blocked during $g_1$. Essentially, this turns this lane into a regular FCTL queue with an extra long red period ($r + g_1$) and a shorter green period ($g_2$). Lane 2 contains only vehicles going straight, none of which are blocked. This means that this lane is essentially a regular FCTL queue as well. As a consequence, these two lanes can be analyzed separately using standard FCTL methods. When applying the method described in \cite{van2006delay}, the mean delay would be exactly the same as computed in Figure~\ref{f:single}(b). Moreover, this means that we can also use Webster's well-known approximation for the mean delay for case (2). This has also been visualized in Figure~\ref{f:single}(b) and, indeed, the approximation is remarkably accurate. Still, we stress that this is only possible because we have chosen an extreme blocking mechanism ($q_i=1$) in combination with Poisson arrivals (Webster's approximation only works for Poisson arrival processes).
\end{remark}
\subsubsection{Two lanes for the shared right-turn}
Now we turn to an example where we still have two dedicated lanes as in case ($2$) of the previous example, one for turning traffic and one for straight-going traffic, see Figure~\ref{fig:vis2}(b), but we compare it with a two-lane example where the vehicles mix, see Figure~\ref{fig:vis2}(c). All turning vehicles will be on lane $1$, but we allow some straight-going traffic to be present on lane $1$ too. Lane $1$ is thus a shared right-turn lane. On lane $2$, we only have vehicles that are heading straight. This could, e.g., model a scenario in which some straight-going vehicles desire to take a specific lane, strategically anticipating on an upcoming exit. Anticipation in lane changing behaviour is more generally investigated in e.g.~\cite{choudhury2013modelling} in urban scenarios. We could adapt the value of $p$ depending on this number of strategic vehicles. In order to make a comparison between the various cases that we study and that is as fair as possible, we assume the following: the total arrival rate and the fraction of turning vehicles are the same.
We assume that the probability that an arbitrary vehicle is a turning vehicle is $0.3$ and we vary the total Poisson arrival rate $\mu$ to study the influence of the strict splitting of the turning vehicles. In case ($1$), we thus have an arrival rate at the right-turning lane that satisfies $\mu_1=0.3\mu$, whereas on the other lane we have an arrival rate $\mu_2 = 0.7\mu$. At lane $1$ we have $p_i=1$ and at lane $2$ we have $p_i=0$. In case ($2$) we distinguish between two subcases. In subcase ($2$a) we assume that the total arrival rate at both lanes is the same and thus $\mu_1=\mu_2=0.5\mu$. In subcase ($2$b), we assume that the arrival rate is split in the ratio $2:3$, so $\mu_1= 0.4\mu$ and $\mu_2=0.6\mu$. This implies that in subcase ($2$a) we choose $p_i=0.6$ (the fraction of turning vehicles is then $p\mu_1=0.6\cdot 0.5\mu=0.3\mu$) and in subcase ($2$b) we choose $p_i=0.75$ (the fraction of turning vehicles is then $p\mu_1=0.75\cdot0.4\mu=0.3\mu$), to make sure that we match the number of turning vehicles in case ($1$). Further, we choose $q_i=1$, $g_1=8$, $g_2=16$ and $r=16$. Then, we study the mean queue length at the end of an arbitrary time slot of both lanes, $\mathbb{E}[X^{(1)}]$ and $\mathbb{E}[X^{(2)}]$, and the total average mean queue length at the end of an arbitrary time slot, denoted with $\mathbb{E}[X^t]$. We obtain Figure~\ref{f:split}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.65\textwidth]{table_2_fig_cars_1.pdf}\\
(a)\\[1ex]
\includegraphics[width=0.65\textwidth]{table_2_fig_cars_2.pdf}\\
(b)\\[1ex]
\includegraphics[width=0.65\textwidth]{table_2_fig_cars_t.pdf}\\
(c) \\[1ex]
\caption{The total Poisson arrival rate, $\mu$ and the mean queue length at the end of an arbitrary time slot for the various cases, split among lane $1$ (a), lane $2$ (b) and the total among the two lanes (c) where $\mathbb{E}[X^{t}]=\mathbb{E}[X^{(1)}]+\mathbb{E}[X^{(2)}]$.}
\label{f:split}
\end{figure}
In Figure~\ref{f:split}, we see only small differences in the total mean queue lengths at the end of an arbitrary time slot for low arrival rates. At both lanes, there are few vehicles in the queue. This is different for the examples in Figure~\ref{f:split} with a higher arrival rate. In all examples for case ($1$) we see that the mean queue length at lane $2$, the straight-going traffic lane, is higher than for lane $1$. This is due to the relatively high fraction of vehicles that \emph{have} to use lane $2$ due to the strict splitting between turning and straight-going vehicles. In some sense, lane $1$, which only has turning vehicles, has overcapacity that cannot be used for the busier lane $2$ with only straight-going traffic. This is different for the other two cases, where the traffic is split more evenly across the two lanes. As one would expect, the longest queue in subcase ($2$a) is present at lane $1$, as the arrival rate at both lanes is the same and because vehicles are only blocked at lane $1$, the shared right-turn lane. This points towards another potential improvement and this is found in subcase ($2$b) where we balance the arrival rate differently. \emph{The right balance leads to a more economic use of both lanes and, hence, also the best performance} in this example when looking at $\mathbb{E}[X^t]$.
The results in Figures~\ref{f:single} and \ref{f:split} might seem conflicting at a first glance, but they are not. In the case of a single, shared right-turn lane as in Figure~\ref{f:single}, we see a higher mean queue length than for the two dedicated lanes case in Figure~\ref{f:single}. This is the other way around in Figure~\ref{f:split} (considering case ($2$b)). This is mainly explained by the fact that in case ($2$b) in Figure~\ref{f:split}, we have two lanes and thus twice as many potential departures as in case ($1$) in Figure~\ref{f:single}. This is one of the main factors in the explanation of the differences in the mean performance between the examples studied in Figures~\ref{f:single} and \ref{f:split}.
The two examples in this subsection tell us that \textbf{a separate or dedicated lane for turning traffic does not necessarily improve the traffic flow}. The intuition behind this is that a dedicated lane might have overcapacity which is not employed (e.g. in the case of an asymmetric load on both lanes). This issue is less present when the two dedicated lanes are turned into two lanes, one exclusively for straight-going traffic and one shared lane. This is confirmed by our simulations. As such, \textbf{an in-depth study is needed to obtain the best layout of the intersection and the best traffic-light control}. As a side-remark, we surpass the possibility here that in Figure~\ref{f:split}, case ($1$), we might control the two lanes in a different way, e.g. by prolonging the green period for one of the lanes. This is not possible in cases ($2$a) and ($2$b).
\begin{comment}
\subsection{FCTL queue with multiple lanes}\label{subsec:multiFCTL}
The regular FCTL queue has only a single lane from which vehicles might depart, yet at bigger intersections, this is not realistic. There might be several lanes for, e.g., straight-going traffic which all receive green simultaneously. For a visualization, see Figure~\ref{fig:vis}(a).
\begin{table}[htb!]
\begin{center}
\caption{The bFCTL queue with $m$ lanes, $g=5$, $r=5$, Poisson arrivals, and no blockages. The load $\rho$, the number of lanes $m$, the mean arrival rate $\mu$, and several performance measures are displayed.}
\label{t:multiFCTLpois}
\[\begin{array}{|c|cc|ccHccHc|}\hline
\rho & m & \mu & \mathbb{E}[X_g] & \textrm{Var}[X_g] &
& \mathbb{P}(X_g\geq10) & \mathbb{E}[X^t]
&
& \mathbb{E}[D] \\\hline
0.2 & 1 & 0.1 & 0.000583 & 0.000788 & 1. & <0.00001 & 0.170 & 0.205 & 1.701 \\
& 2 & 0.2 & <0.00001 & 0.000010 & 1. & <0.00001 & 0.317 & 0.449 & 1.587 \\
& 5 & 0.5 & <0.00001 & <0.00001 & 1. & <0.00001 & 0.762 & 1.57 & 1.523 \\
& 10 & 1.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 1.505 & 4.75 & 1.505 \\
& 15 & 1.5 & <0.00001 & <0.00001 & 1. & <0.00001 & 2.252 & 9.56 & 1.502 \\
& 20 & 2.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 3.001 & 16.0 & 1.500 \\\hline
0.4 & 1 & 0.2 & 0.0217 & 0.0384 & 0.985 & <0.00001 & 0.404 & 0.565 & 2.021 \\
& 2 & 0.4 & 0.00324 & 0.00663 & 0.998 & <0.00001 & 0.711 & 1.25 & 1.778 \\
& 5 & 1.0 & 0.000013 & 0.000033 & 1. & <0.00001 & 1.661 & 4.86 & 1.661 \\
& 10 & 2.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 3.240 & 15.9 & 1.620 \\
& 15 & 3.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 4.816 & 33.2 & 1.605 \\
& 20 & 4.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 6.390 & 56.7 & 1.598 \\\hline
0.6 & 1 & 0.3 & 0.180 & 0.429 & 0.903 & 0.000029 & 0.817 & 1.4 & 2.724 \\
& 2 & 0.6 & 0.0770 & 0.215 & 0.963 & 0.000019 & 1.279 & 2.63 & 2.131 \\
& 5 & 1.5 & 0.00788 & 0.0298 & 0.997 & <0.00001 & 2.834 & 9.97 & 1.890 \\
& 10 & 3.0 & 0.00019 & 0.00101 & 1. & <0.00001 & 5.505 & 33.4 & 1.835 \\
& 15 & 4.5 & <0.00001 & 0.000030 & 1. & <0.00001 & 8.181 & 70.5 & 1.818 \\
& 20 & 6.0 & <0.00001 & <0.00001 & 1. & <0.00001 & 10.85 & 121. & 1.809 \\\hline
0.8 & 1 & 0.4 & 1.097 & 4.181 & 0.645 & 0.00842 & 2.025 & 5.73 & 5.063 \\
& 2 & 0.8 & 0.795 & 3.465 & 0.763 & 0.00662 & 2.598 & 7.45 & 3.247 \\
& 5 & 2.0 & 0.359 & 2.038 & 0.917 & 0.00417 & 4.707 & 18.6 & 2.354 \\
& 10 & 4.0 & 0.109 & 0.836 & 0.982 & 0.00242 & 8.621 & 56.4 & 2.155 \\
& 15 & 6.0 & 0.0343 & 0.332 & 0.996 & 0.00130 & 12.68 & 118. & 2.113 \\
& 20 & 8.0 & 0.0109 & 0.127 & 0.999 & 0.00057 & 16.79 & 203. & 2.099 \\\hline
0.98 & 1 & 0.49 & 23.22 & 614.8 & 0.0918 & 0.638 & 24.44 & 617 & 49.88 \\
& 2 & 0.98 & 22.59 & 613.1 & 0.126 & 0.621 & 25.02 & 619 & 25.53 \\
& 5 & 2.45 & 21.02 & 606.9 & 0.219 & 0.580 & 27.06 & 631 & 11.04 \\
& 10 & 4.90 & 18.47 & 589.0 & 0.363 & 0.517 & 30.51 & 666 & 6.227 \\
& 15 & 7.35 & 15.90 & 558.9 & 0.489 & 0.451 & 33.93 & 718.3 & 4.616 \\
& 20 & 9.80 & 13.45 & 517.4 & 0.596 & 0.381 & 37.44 & 788.7 & 3.820 \\\hline
\end{array}\]
\end{center}
\end{table}
Our framework for the bFCTL queue with multiple lanes allows us to model such examples, which we demonstrate in this subsection. We study both the case of a Poisson distributed number of arrivals studied in~\cite{van2006delay}. We thus study a case where $g=g_1+g_2=5$, $p_i=0$ for all $i$, $r=5$, and with Poisson distributed arrivals in each slot with mean $\mu$. We study various cases of $\mu$ and analyze the overflow queue, denoted with $X_g$, the mean queue length at the end of an arbitrary time slot $\mathbb{E}[X^t]$, and the mean delay $\mathbb{E}[D]$. We also vary $m$ to study the influence of having multiple lanes in the FCTL queue. In order to make a comparison between the various cases with different $m$, we scale the arrival rate proportionally with $m$ so that the load or vehicle-to-capacity ratio, $\rho=(c\mu)/(mg)$, is fixed for different values of $m$. Then, we obtain Table~\ref{t:multiFCTLpois}.
We note that there is a difference between analyzing $m$ FCTL queues separately and the joint analysis of the $m$ lanes as presented here. \textbf{It is thus important to perform an analysis that accounts for the number of lanes that vehicles from a single stream can use.} This can most prominently be observed by fixing $\rho$ and considering various values of $m$: the mean and variance of the overflow queue (measured in number of vehicles) then decrease
if we have Poisson arrivals in each slot.
\textbf{This indicates that having more lanes at a single intersection while $\rho$ is fixed, is not necessarily beneficial when looking at the total number of vehicles in the queue}: a high variability in the number of arrivals per slot might result in an increase of the number of vehicles in the queue when the number of lanes is increased. However, in all cases the mean delay decreases if $\rho$ is fixed and $m$ increases.
\end{comment}
\section{Conclusion and discussion}\label{sec:discussion}
In this paper, we have established a recursion for the PGFs of the queue-length distribution at the end of each slot which can be used to provide a full queue-length analysis of the bFCTL queue with multiple lanes. This is an extension of the regular FCTL queue so that we can account for temporal blockages of vehicles receiving a green light, for example because of a crossing pedestrian at the turning lane or because of a (separate) bike lane, and to account for a vehicle stream that is spread over multiple lanes. These features might impact
the traffic-light performance as we have shown by means of various numerical examples. The blocking of turning vehicles and the number of lanes corresponding to a vehicle stream therefore has to be taken into account when choosing the settings for a traffic light.
We briefly touched upon how one should design the layout of an intersection. Interestingly, it might be suboptimal to have a dedicated lane for turning traffic. It seems that mixing turning and straight-going traffic has benefits over a strict separation of those two traffic streams when there are two lanes for this turning and straight-going traffic. We advocate a further investigation into the influence of separating or mixing different streams of vehicles in front of traffic lights. It might be possible to find the optimal division of straight-going and turning vehicles over the various lanes, e.g. by enumerating several possibilities. A more structured optimization seems difficult because of the intricate expressions involved, but would definitely be worthwhile to investigate. Some research on the splitting of different traffic streams has already been done in e.g.~\cite{kikuchi2007lengths,tian2006probabilistic,wu1999capacity} and \cite{zhang2008modeling} and the present study can be seen as an alternative way of modelling the situation at hand.
A possible extension of the results on the bFCTL queue is a study of (the PGF of) the delay distribution. We have refrained from deriving the delay distribution because of its (notational) complexity. Using proper conditioning, one can obtain (the PGF of) the delay distribution for the bFCTL queue.
The work in~\cite{shaoluen2020random}, in which a simulation study of a similar model is performed, has been a source of inspiration for the study in this paper.
There are some extensions possible when comparing our work with~\cite{shaoluen2020random}. We e.g. did not study the influence of start-up delays as is done in~\cite{shaoluen2020random}. Investigating such start-up delays at the beginning of the green period is easily done in our framework: we simply need to adjust the $Y_i$ for the first few slots. Another approach to deal with start-up delays is presented in~\cite{maesnetworks}. Start-up delays which depend on the blocking of vehicles and different slot lengths for different combinations of turning/straight going vehicles, are harder to tackle. One could e.g. introduce additional states (besides states $u$ and $b$) to deal with this. Although the developed recursion does not directly allow for such a generalization, it seems possible to account for this at the expense of a more complex recursion. For the ease of exposition, we have refrained from doing so and we leave a full study on this topic for future research.
A further possible extension of the bFCTL queue would be to consider different blocking behaviours: instead of e.g. a fixed probability $q_i$ for each slot $i$, a more general blocking process might be considered. For example, if there are no pedestrians during slot $i$ for the model depicted in Figure~\ref{fig:vis}(b), then the probability that there are also no pedestrians in slot $i+1$, might be relatively high. In other words, there might be \emph{dependence} between the various slots when considering the presence of pedestrians. We gave an example where there is dependence between the current and the next slot, but it is also possible to consider such dependencies among more than two slots. It is worthwhile to investigate generalizations of the blocking process in order to further increase the general applicability of the bFCTL queue with multiple lanes.
Another generalization for the blocking mechanism, is to block only a \emph{part} of the $m$ vehicles that are at the head of the queue. Indeed, we restrict ourselves to the cases where either all vehicles in a batch of size $m$ are blocked (or not). In various real-life examples, it might be the case that only part of the $m$ vehicles are blocked. It would be interesting to investigate whether such a model can be analyzed. Further, a situation with ``a right turn is always permitted'' scenario might be investigated. In such a case, right-turning vehicles are always free to turn, but might be blocked by straight-going vehicles in front them, which have to wait for a red traffic light, or are blocked by pedestrians. Straight-going vehicles might be blocked by turning traffic waiting for pedestrians. It seems that such a case, at the expense of additional complexity, can be tackled by a similar type of recursion as the one that is developed in this paper by extending and generalizing the blocking mechanism (and, thus, the recursion) to the red period.
\paragraph{Discussion.} We end this paper with a discussion on its practical applicability. Although we have extended the standard model for traffic signals with fixed settings, there are still quite some possible improvements, as discussed in the above paragraphs. Still, to the best of our knowledge, this paper is the first to present analytical results for traffic intersections with blocking mechanisms, based on a queueing theoretic approach. Note that standard formulas like Webster's approximation for the mean delay \cite{webster1958traffic} cannot be used in these situations. From a practical point of view, the most relevant extension to the current analysis would be to deal with start-up delays that depend on the blocking of vehicles. One way to do this, is by considering different slot lengths for different combinations of turning/straight going vehicles, inspired by an analysis in ~\cite{maesnetworks}. This would make it possible to compute a saturation flow adjustment factor due to the right-turning movements at shared lane conditions (see also Biswas et al. \cite{biswas2018}).
Finally, we also advocate an investigation whether the bFCTL queue with a vehicle-actuated mechanism (rather than the fixed green and red times that we consider) results in a tractable model.
\paragraph*{Acknowledgements} We would like to thank Onno Boxma for several interesting discussions that a.o. improved the readability of this manuscript. We are also thankful to Joris Walraevens who suggested the current exposition of the PGF recursion and to the reviewers who suggested several improvements of the paper.
\paragraph*{Funding} The work in this paper is supported by the Netherlands Organization for Scientific Research (NWO) under grant number 438-13-206.
\paragraph*{Disclosure statement} The authors report there are no competing interests to declare.
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{"url":"https:\/\/codegolf.stackexchange.com\/questions\/210162\/is-it-almost-prime","text":"# Is it almost-prime?\n\nSandbox\n\nDefinition: A positive integer n is almost-prime, if it can be written in the form n=p^k where p is a prime and k is also a positive integers. In other words, the prime factorization of n contains only the same number.\n\nInput: A positive integer 2<=n<=2^31-1\n\nOutput: a truthy value, if n is almost-prime, and a falsy value, if not.\n\nTruthy Test Cases:\n\n2\n3\n4\n8\n9\n16\n25\n27\n32\n49\n64\n81\n1331\n2401\n4913\n6859\n279841\n531441\n1173481\n7890481\n40353607\n7528289\n\n\nFalsy Test Cases\n\n6\n12\n36\n54\n1938\n5814\n175560\n9999999\n17294403\n\n\nPlease do not use standard loopholes. This is so the shortest answer in bytes wins!\n\n\u2022 To clarify: the truthy and falsy values need not be consistent, right? \u2013\u00a0Luis Mendo Aug 26 '20 at 0:42\n\u2022 This is A000961 in the OEIS. \u2013\u00a0Giuseppe Aug 26 '20 at 13:21\n\u2022 The usual name for this kind of number is \"prime power\". \u2013\u00a0Andreas Rejbrand Aug 26 '20 at 17:39\n\u2022 It feels odd to me that you include prime numbers as being \"almost prime,\" but this is still a good challenge! :) \u2013\u00a0Captain Man Aug 26 '20 at 18:26\n\u2022 This should use the terminology \"prime power\". en.wikipedia.org\/wiki\/Almost_prime already has a definition. \u2013\u00a0qwr Aug 28 '20 at 5:27\n\n# Sagemath, 2 bytes\n\nGF\n\n\nOutputs via exception.\n\nTry it online!\n\nThe Sagemath builtin $$\\\\text{GF}\\$$ creates a Galois Field of order $$\\n\\$$. However, remember that $$\\\\mathbb{F}_n\\$$ is only a field if $$\\n = p^k\\$$ where $$\\p\\$$ is a prime and $$\\k\\$$ a positive integer. Thus the function throws an exception if and only if its input is not a prime power.\n\n\u2022 Galois fields was the first thing I thought of, but I had no idea that Sagemath had a builtin for it. \u2013\u00a0Don Thousand Aug 26 '20 at 14:30\n\u2022 @DonThousand you should see what you can do with SM, it's support for Elliptic curve math\/crypto is fantastic. \u2013\u00a0Woodstock Aug 26 '20 at 17:37\n\n# Python 2, 42 bytes\n\nf=lambda n,p=2:n%p and f(n,p+1)or p**n%n<1\n\n\nTry it online!\n\nSince Python doesn't have any built-ins for primes, we make do with checking divisibility.\n\nWe find the smallest prime p that's a factor of n by counting up p=2,3,4,... until n is divisible by p, that is n%p is zero. There, we check that this p is the only prime factor by checking that a high power of p is divisible by n. For this, p**n suffices.\n\nAs a program:\n\n43 bytes\n\nn=input()\np=2\nwhile n%p:p+=1\nprint p**n%n<1\n\n\nTry it online!\n\nThis could be shorter with exit codes if those are allowed.\n\n46 bytes\n\nlambda n:all(n%p for p in range(2,n)if p**n%n)\n\n\nTry it online!\n\n# Shakespeare Programming Language, 329 bytes\n\n,.Ajax,.Page,.Act I:.Scene I:.[Enter Ajax and Page]\nAjax:Listen tothy.\nPage:You cat.\nScene V:.\nPage:You is the sum ofYou a cat.\nIs the remainder of the quotient betweenI you nicer zero?If soLet usScene V.\nScene X:.\nPage:You is the cube ofYou.Is you worse I?If soLet usScene X.\nYou is the remainder of the quotient betweenYou I.Open heart\n\n\nTry it online!\n\nOutputs 0 if the input is almost prime, and a positive integer otherwise. I am not sure this is an acceptable output; changing it would cost a few bytes.\n\nExplanation:\n\n\u2022 Scene I: Page takes in input (call this n). Initialize Ajax = 1.\n\u2022 Scene V: Increment Ajax until Ajax is a divisor of Page; call the final value p This gives the smallest divisor of Page, which is guaranteed to be a prime.\n\u2022 Scene X: Cube Ajax until you end up with a power of p, say p^k which is greater than n. Then n is almost-prime iff n divides p^k.\n\u2022 You can save a byte by removing the space after \"the remainder of\", right? \u2013\u00a0Hello Goodbye Aug 29 '20 at 1:56\n\u2022 @HelloGoodbye No, that space is needed, as the name of the function is the_remainder_of_the_quotient_between. If you remove the space in the middle of the name, the function is not applied. \u2013\u00a0Robin Ryder Aug 29 '20 at 5:04\n\u2022 Right, okay. I'd forgotten about that function \u2013\u00a0Hello Goodbye Aug 30 '20 at 17:44\n\n# MATL, 4 bytes\n\nYf&=\n\n\u2022 For almost-primes the output is a matrix containing only 1s, which is truthy.\n\u2022 Otherwise the output is a matrix containing several 1s and at least one 0, which is falsy.\n\nTry it online! Or verify all test cases, including truthiness\/falsihood test.\n\n### How it works\n\n % Implicit input\nYf % Prime factors. Gives a vector with the possibly repeated prime factors\n&= % Matrix of all pair-wise equality comparisons\n% Implicit output\n\n\n# R, 3632 29 bytes\n\n-3 bytes by outputting a vector of booleans without extracting the first element\n\n!(a=2:(n=scan()))[!n%%a]^n%%n\n\n\nTry it online!\n\nOutputs a vector of booleans. In R, a vector of booleans is truthy iff the first element is TRUE.\n\nFirst, find the smallest divisor p of n. We can do this by checking all integers (not only primes), as the smallest divisor of an integer (apart from 1) is always a prime number. Here, let a be all the integers between 2 and n, then p=a[!n%%a][1] is the first element of a which divides n.\n\nThen n is almost prime iff n divides p^n.\n\nThis fails for any moderately large input, so here is the previous version which works for most larger inputs:\n\n# R, 36 33 bytes\n\n!log(n<-scan(),(a=2:n)[!n%%a])%%1\n\n\nTry it online!\n\nCompute the logarithm of n in base p: this is an integer iff n is almost prime.\n\nThis will fail due to floating point inaccuracy for certain (but far from all) large-ish inputs, in particular for one test case: $$\\4913=17^3\\$$.\n\n\u2022 Brilliant and 25 bytes less than my own best attempt without peeking... The log trick is super! \u2013\u00a0Dominic van Essen Aug 26 '20 at 8:05\n\u2022 Although a nice approach, I'm afraid it fails for test case 4913 due to floating point inaccuracies (2.9999999999999996 is not an integer). I've just looked in the meta, and apparently you have to work around this if your language supports an accurate decimal type. I don't know R, so I don't know if this applies to it, but I was about to port your approach to Java to golf my about to post answer, but that apparently wouldn't be allowed unless I use java.math.BigDecimal instead of regular doubles.. :\/ \u2013\u00a0Kevin Cruijssen Aug 26 '20 at 8:48\n\u2022 @KevinCruijssen I assumed this was OK, as it would work with a theoretical infinite-precision computer. I am probably not aware of the meta consensus you referred to, could you link to it? \u2013\u00a0Robin Ryder Aug 26 '20 at 9:06\n\u2022 @RobinRyder Ah, I thought I added a link. Here it is. \u2013\u00a0Kevin Cruijssen Aug 26 '20 at 9:09\n\u2022 @DominicvanEssen While you were commenting, I made a similar change: I don't need \u2026^(3*n) but simply \u2026^n, which gains 4 bytes (but fails for any moderately large input). \u2013\u00a0Robin Ryder Aug 26 '20 at 9:58\n\n# C (gcc), 43 bytes\n\nf(n,i){for(i=1;n%++i;);n=i<n&&f(n\/i)^i?:i;}\n\n\nTry it online!\n\nReturns p if n is almost-prime, and 1 otherwise.\n\nf(n,i){\nfor(i=1;n%++i;); \/\/ identify i = the least prime factor of n\nn=i<n&&f(n\/i)^i \/\/ if n is neither prime nor almost-prime\n? \/\/ return 1\n:i; \/\/ return i\n}\n\n\n# Wolfram Language (Mathematica), 11 bytes\n\nPrimePowerQ\n\n\nTry it online!\n\n@Sisyphus saved 1 byte\n\n# 05AB1E, 2 bytes\n\n\u00d2\u00cb\n\n\nTry it online!\n\n### Commented:\n\n\u00d2 -- Are all the primes in the prime decomposition\n\u00cb -- Equal?\n\n\u2022 A pure ASCII alternative: fg - since only 1 is truthy in 05AB1E. \u2013\u00a0user96495 Aug 26 '20 at 2:02\n\n# J, 9 8 bytes\n\n1=#@=@q:\n\n\nTry it online!\n\n-1 byte thanks to xash\n\nTests if the self-classify = of the prime factors q: has length # equal to one 1=\n\n# APL (Dyalog Classic), 3331 26 bytes\n\n{\u2375\u220a\u220a(((\u22a2~\u2218.\u00d7\u2368)1\u2193\u2373)\u2375)\u2218*\u00a8\u2373\u2375}\n\n\n-5 bytes from Kevin Cruijssen's suggestion.\n\nWarning: Very, very slow for larger numbers.\n\n## Explanation\n\n{\u2375\u220a\u220a(((\u22a2~\u2218.\u00d7\u2368)1\u2193\u2373)\u2375)\u2218*\u00a8\u2373\u2375} \u2375=n in all the following steps\n\u2373\u2375 range from 1 to n\n\u2218*\u00a8 distribute power operator across left and right args\n(((\u22a2~\u2218.\u00d7\u2368)1\u2193\u2373)\u2375) list of primes till n\n\u220a flatten the right arg(monadic \u220a)\n\u2375\u220a is n present in the primes^(1..n)?\n\n\nTry it online!\n\n\u2022 I'm not entirely sure, but I think you can drop the *0.5 and just use a range or 1 to n? \u2013\u00a0Kevin Cruijssen Aug 26 '20 at 7:46\n\u2022 It is way too slow with that but yeah, sure. \u2013\u00a0Razetime Aug 26 '20 at 7:52\n\u2022 Well, codegolf is all about saving bytes, so compilation warnings, code standards, and performance are all irrelevant in code-golfing. Even if the performance would go from $O(1)$ to $O(n^n)$, if we can save even a single byte it's worth it, haha. ;) But if TIO would be to slow to run, you could leave the *0.5 in the TIO and mention it's only used to speed up the online code. (PS: The 0.5 could have been golfed to .5 if it was indeed required.) \u2013\u00a0Kevin Cruijssen Aug 26 '20 at 7:55\n\n# Pyth, 5 bytes\n\n!t{PQ\n\n\nTry it online!\n\nExplanation:\n\nQ - Takes integer input\nP - List of prime factors\n{ - Remove duplicate elements\nt - Removes first element\n! - Would return True if remaining list is empty, otherwise False\n\n\u2022 I have absolutely no idea what I'm doing, but does this work? \u2013\u00a0Unrelated String Aug 26 '20 at 3:48\n\u2022 @UnrelatedString that does, but I am not sure if im allowed to do that since it returns 0 (falsy value) for right cases and a truthy value for the others (which isn't fixed either). \u2013\u00a0Manish Kundu Aug 26 '20 at 3:51\n\u2022 Well, in that case, !s.+PQ is still a byte shorter. \u2013\u00a0Unrelated String Aug 26 '20 at 3:53\n\n## Setanta, 61 59 bytes\n\ngniomh(n){p:=2nuair-a n%p p+=1nuair-a n>1 n\/=p toradh n==1}\n\n\nTry it here\n\nNotes:\n\n\u2022 The proper keyword is gn\u00edomh, but Setanta allows spelling it without the accents so I did so to shave off a byte.\n\u2022 What an interesting language! Makes me want to go try to learn Irish again, and Setanta, for that matter! \u2013\u00a0Giuseppe Aug 26 '20 at 16:51\n\nf n=mod(until((<1).mod n)(+1)2^n)n<1\n\n\nTry it online!\n\n36 bytes\n\nf n=and[mod(gcd d n^n)n<2|d<-[1..n]]\n\n\nTry it online!\n\n39 bytes\n\nf n=all((elem[1,n]).gcd n.(^n))[2..n]\n\n\nTry it online!\n\n39 bytes\n\nf n=mod n(n-sum[1|1<-gcd n<$>[1..n]])<1 Try it online! 40 bytes f n=and[mod(p^n)n<1|p<-[2..n],mod n p<1] Try it online! # JavaScript (ES6), 43 bytes ## Without BigInts Returns a Boolean value. f=(n,k=1)=>n%1?!~~n:f(n<0?n\/k:n%++k?n:-n,k) Try it online! A recursive function that first looks for the smallest divisor $$\\k>1\\$$ of $$\\n\\$$ and then divides $$\\-n\\$$ by $$\\k\\$$ until it's not an integer anymore. (The only reason why we invert the sign of $$\\n\\$$ when $$\\k\\$$ is found is to distinguish between the two steps of the algorithm.) If $$\\n\\$$ is almost-prime, the final result is $$\\-\\dfrac{1}{k}>-1\\$$. So we end up with $$\\\\lceil n\\rceil=0\\$$. If $$\\n\\$$ is not almost-prime, there exists some $$\\q>k\\$$ coprime with $$\\k\\$$ such that $$\\n=q\\times k^{m}\\$$. In that case, the final result is $$\\-\\dfrac{q}{k}<-1\\$$. So we end up with $$\\\\lceil n\\rceil<0\\$$. # JavaScript (ES11), 33 bytes ## With BigInts With BigInts, using @xnor's approach is probably the shortest way to go. Returns a Boolean value. f=(n,k=1n)=>n%++k?f(n,k):k**n%n<1 Try it online! # Retina 0.8.2, 50 bytes .+$*\n^(?=(11+?)\\1*$)((?=\\1+$)(?=(1+)(\\3+)$)\\4)+1$\n\n\nTry it online! Link includes faster test cases. Based on @Deadcode's answer to Match strings whose length is a fourth power. Explanation:\n\n.+\n$* Convert the input to unary. ^(?=(11+?)\\1*$)\n\n\nStart by matching the smallest factor $$\\ p \\$$ of $$\\ n \\$$. ($$\\ p \\$$ is necessarily prime, of course.)\n\n(?=\\1+$)(?=(1+)(\\3+)$)\n\n\nWhile $$\\ p | \\frac n { p^i } \\$$, find $$\\ \\frac n { p^i } \\$$'s largest proper factor, which is necessarily $$\\ \\frac n { p^{i+1} } \\$$.\n\n\\4\n\n\nThe factorisation also captures $$\\ (p - 1) \\frac n { p^{i+1} } \\$$, which is subtracted from $$\\ \\frac n { p^i } \\$$, leaving $$\\ \\frac n { p^{i+1} } \\$$ for the next pass through the loop.\n\n(...)+1$ Repeat the division by $$\\ p \\$$ as many times as possible, then check that $$\\ \\frac n { p^k } = 1 \\$$. # Io, 48 bytes Port of @RobinRyder's R answer. method(i,c :=2;while(i%c>0,c=c+1);i log(c)%1==0) Try it online! ## Explanation method(i, \/\/ Take an input c := 2 \/\/ Set counter to 2 while(i%c>0, \/\/ While the input doesn't divide counter: c=c+1 \/\/ Increment counter ) i log(c)%1==0 \/\/ Is the decimal part of input log counter equal to 0? ) # Assembly (MIPS, SPIM), 238 bytes, 6 * 23 = 138 assembled bytes main:li$v0,5\nsyscall\nmove$t3,$v0\nli$a0,0 li$t2,2\nw:bgt$t2,$t3,d\ndiv$t3,$t2\nmfhi$t0 bnez$t0,e\nadd$a0,$a0,1\ns:div$t3,$t2\nmfhi$t0 bnez$t0,e\ndiv$t3,$t3,$t2 b s e:add$t2,$t2,1 b w d:move$t0,$a0 li$a0,0\nbne$t0,1,p add$a0,$a0,1 p:li$v0,1\nsyscall\n\n\nTry it online!\n\n\u2022 You can save bytes if you use $2 through$9, rather than named registers. \u2013\u00a0insou Sep 20 '20 at 8:27\n\n# Brachylog, 2 bytes\n\nAre all prime factors equal?\n\n\u1e0b=\n\n\nTry it online!\n\n# GAP 4.7, 31 bytes\n\nn->Length(Set(FactorsInt(n)))<2\n\nThis is a lambda. For example, the statement\n\nFiltered([2..81], n->Length(Set(FactorsInt(n)))<2 );\n\nyields the list [ 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81 ].\n\nTry it online!\n\n\u2022 Please link your answers to Try It Online or any other interpreter so they can be verified properly. \u2013\u00a0Razetime Aug 26 '20 at 5:49\n\n# MathGolf, 10 bytes\n\n\u2552g\u00b6m\u00c9k\u2552#\u2500\u2567\n\n\nPort of @Razetime's APL (Dyalog Classic) answer, so make sure to upvote him as well!\n\nTry it online.\n\nExplanation:\n\n\u2552 # Push a list in the range [1, (implicit) input-integer)\ng # Filter it by:\n\u00b6 # Check if it's a prime\nm # Map each prime to,\n\u00c9 # using the following three operations:\nk\u2552 # Push a list in the range [1, input-integer) again\n# # Take the current prime to the power of each value in this list\n\u2500 # After the map, flatten the list of lists\n\u2567 # And check if this list contains the (implicit) input-integer\n# (after which the entire stack joined together is output implicitly)\n\n\n# Factor, 35 bytes\n\n: f ( n -- ? ) factors all-equal? ;\n\n\nTry it online!\n\n\u2022 @petStorm Here it is :) Thanks! \u2013\u00a0Galen Ivanov Aug 26 '20 at 11:13\n\n# Japt, 6 bytes\n\nI feel like this should be 1 or 2 bytes shorter ...\n\nk \u00e4\u00b6 \u00d7\n\n\nTry it - includes all test cases\n\n# Java, 69 (or 64?) bytes\n\nn->{int c=0,t=1;for(;t++<n;)if(n%t<1)for(c++;n%t<1;)n\/=t;return c<2;}\n\n\nTry it online.\n\nExplanation:\n\nn->{ \/\/ Method with integer parameter and boolean return-type\nint c=0, \/\/ Counter-integer, starting at 0\nt=1;for(;t++<n;) \/\/ Loop t in the range (1,n]:\nif(n%t<1) \/\/ If the input is divisible by t:\nfor(c++; \/\/ Increase the counter by 1\nn%t<1;) \/\/ Loop as long as the input is still divisible by t\nn\/=t; \/\/ And divide n by t every iteration\nreturn c<2;} \/\/ Return whether the counter is 1\n\n\nIf we would be allowed to ignore floating point inaccuracies, a port of @RobinRyder's R answer would be 64 bytes instead:\n\nn->{int m=1;for(;n%++m>0;);return Math.log(n)\/Math.log(m)%1==0;}\n\n\nTry it online.\n\nExplanation:\n\nn->{ \/\/ Method with integer parameter and boolean return-type\nint m=1; \/\/ Minimum divisor integer m, starting at 1\nfor(;n%++m>0;); \/\/ Increase m by 1 before every iteration with ++m\n\/\/ And continue looping until the input is divisible by m\nreturn Math.log(n)\/Math.log(m)\n\/\/ Calculate log_m(n)\n%1==0;} \/\/ And return whether it has no decimal values after the comma\n\n\nBut unfortunately this approach fails for test case 4913 which would become 2.9999999999999996 instead of 3.0 due to floating point inaccuracies (it succeeds for all other test cases).\nA potential fix would be 71 bytes:\n\nn->{int m=1;for(;n%++m>0;);return(Math.log(n)\/Math.log(m)+1e9)%1<1e-8;}\n\n\nTry it online.\n\n# Jelly, 3 bytes\n\n\u00c6fE\n\n\nTry it online!\n\n# Burlesque, 6 bytes\n\nrifCsm\n\n\nTry it online!\n\nExplanation:\n\nri # Read integer from input\nfC # Find its prime factorisation\nsm # Are all values the same?","date":"2021-01-26 18:49:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"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\": 33, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.35212597250938416, \"perplexity\": 3275.920317915851}, \"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-2021-04\/segments\/1610704803308.89\/warc\/CC-MAIN-20210126170854-20210126200854-00336.warc.gz\"}"}
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\section{Acknowledgement}
This work is supported by
\section{Background}\label{sec:background}
\subsection{Object Detection}
Object detection has received significant attention and achieved striking improvements in recent years, as demonstrated in popular object detection competitions such as PASCAL
VOC detection challenge~\cite{everingham2010pascal, everingham2015pascal}, ILSVRC large scale detection challenge~\cite{russakovsky2015imagenet} and MS COCO large scale detection challenge~\cite{lin2014microsoft}. Object detection aims at outputting instances of semantic objects with a certain class label such as humans, cars. It has wide applications in many computer vision tasks including face detection, face recognition, pedestrian detection, video object co-segmentation, image retrieval, object tracking and video surveillance. Different from image classification, object detection is not to classify the whole image.
Position and category information of the objects are both needed which means we have to segment instances of objects from backgrounds and label them with position and class. The inputs are images or video frames while the outputs are lists where each item represents position and category information of candidate objects. In general, object detection seeks to extract discriminative features to help in distinguishing the classes.
Methods for object detection generally fall into 3 categories: 1) traditional machine learning based approaches; 2) region proposal based deep learning approaches; 3) end-to-end deep learning approaches. For traditional machine learning based approaches, one of the important steps is to design features. Many methods have been proposed to first design features~\cite{viola2001rapid,viola2004robust,lowe1999object,dalal2005histograms} and apply techniques such as support vector machine (SVM)~\cite{hearst1998support} to do the classification. The main steps of traditional machine learning based approaches are:
\begin{itemize}
\item Region Selection: using sliding windows at different sizes to select candidate regions from whole images or video frames;
\item Feature Extraction: extract visual features from candidate regions using techniques such as Harr feature for face detection, HOG feature for pedestrian detection or general object detection;
\item Classifier: train and test classifier using techniques such as SVM.
\end{itemize} The tradition machine learning based approaches have their limitations. The scheme using sliding windows to select RoIs (Regions of Interests) increases computation time with a lot of window redundancies. On the other hand, these hand-crafted features are not robust due to the diversity of objects, deformation, lighting condition, background and etc., while the feature selection has a huge effect on classification performance of candidate regions.
Recent advances in deep learning, especially in computer vision have shown that Convolutional Neural Networks (CNNs) have a strong capability of representing objects and help to boost the performance of numerous vision tasks, comparing to traditional heuristic features \cite{dalal2005histograms}. For deep learning based approaches, there are convolutional neural networks (CNN) to extract features of region proposals or end-to-end object detection without specifically defining features of a certain class. The well-performed deep learning based approaches of object detection includes Region Proposals (R-CNN)~\cite{girshick2014rich}, Fast R-CNN~\cite{girshick2015fast}, Faster R-CNN~\cite{ren2015faster}), Single Shot MultiBox Detector (SSD)~\cite{liu2016ssd}, and You Only Look Once (YOLO)~\cite{redmon2016you}.
Usually, we adopt region proposal methods (Category 2) for producing multiple object proposals, and then apply a robust classifier to further refine the generated proposals, which are also referred as two-stage method. The first work of the region proposal based deep learning approaches is R-CNN~\cite{girshick2014rich} proposed to solve the problem of selecting a huge number of regions. The main pipeline of R-CNN~\cite{girshick2014rich} is: 1) gathering input images; 2) generating a number of region proposals (e.g. 2000); 3) extracting CNN features; 4) classifying regions using SVM. It usually adopts Selective Search (SS), one of the state-of-art object proposals method \cite{Uijlings13} applied in numerous detection task on several fascinating systems\cite{girshick2014rich,girshick2015fast,ren2015faster}, to extract these regions from the image and names them region proposals. Instead of trying to classify all the possible proposals, R-CNN select a fixed set of proposals (e.g. 2000) to work with. The selective search algorithm used to generate these region proposals includes: (1) Generate initial sub-segmentation, generate many candidate regions; (2) Use greedy algorithm to recursively combine similar regions into larger ones; (3) Use the generated regions to produce the final candidate region proposals.
These candidate region proposals are warped into a square and fed into a convolutional neural network (CNN) which acts as the feature extractor. The output dense layer consists of the extracted features to be fed into an SVM~\cite{hearst1998support} to classify the presence of the object within that candidate region proposal. The main problem of R-CNN~\cite{girshick2014rich} is that it is limited by the inference speed, due to a huge amount of time spent on extracting features of each individual region proposal. And it cannot be applied in applications requiring a real-time performance (such as online video analysis). Later, Fast R-CNN~\cite{girshick2015fast} is proposed to improve the speed by avoiding feeding raw region proposals every time. Instead, the convolution operation is done only once per image and RoIs over the feature map are generated. Faster R-CNN \cite{ren2015faster} further exploits the shared convolutional features to extract region proposals used by the detector. Sharing convolutional features leads to substantially faster speed for object detection system.
The third type is end-to-end deep learning approaches which do not need region proposals (also referred as one-stage method). The pioneer works are SSD~\cite{liu2016ssd} and YOLO~\cite{redmon2016you} . An SSD detector \cite{liu2016ssd} works by adding a sequence of feature maps of progressively decreasing the spatial resolution
to replace the two stage's second classification stage, allowing a fast computation and multi-scale detection on one single input. YOLO detecor is an object detection algorithm much different from the region based algorithms. In YOLO~\cite{redmon2016you}, it regards object detection as an end-to-end regression problem and uses a single convolutional network to predict the bounding boxes and the corresponding class probabilities. It first takes the image and splits it into an $S \times S$ grid, within each of the grid we take $m$ bounding boxes. For each of the bounding box with multi scales, the convolutional neural network outputs a class probability and offset values for the bounding box. Then it selects bounding boxes which have the class probability above a threshold value and uses them to locate the object within the image. YOLO~\cite{redmon2016you} is orders of magnitude faster (45 frames per second) than other object detection approaches but the limitation is that it struggles with small objects within the image.
\subsection{Multiple Object Tracking}
Video object tracking is to locate objects over video frames and it has various important applications in robotics, video surveillance and video scene understanding. Based on the number of moving objects that we wish to track, there are Single Object Tracking (SOT) problem and Multiple Object Tracking (MOT) problem. In addition to detecting objects in video frame, the MOT solution requires to robustly associate multiple detected objects between frames to get a consistent tracking and this data association part remains very challenging. In MOT tasks, for each frame in a video, we aim at localizing and identifying all objects of interests, so that the identities are consistent throughout the video. Typically, the main challenge lies on speed, data association, appearance change, occlusions, disappear / re-enter objects and etc. In practice, it is desired that the tracking could be performed in real-time so as to run as fast as the frame-rate of the video. Also, it is challenging to provide a consistent labeling of the detected objects in complex scenarios such as objects change appearance, disappear, or involve severe occlusions.
In general, Multiple Object Tracking (MOT) can be regarded as a multi-variable estimation problem~\cite{luo2014multiple}. The objective of multiple object tracking can be modeled by performing MAP (maximal a posteriori) estimation in order to find the \textit{optimal} sequential states of all the objects, from the conditional distribution of the sequential states given all the observations:
\begin{equation}
\label{eq:map}
\widehat{\mathbf{S}}_{1:t} = \underset{\mathbf{S}_{1:t}}\argmax \ P\left(\mathbf{S}_{1:t}|\mathbf{O}_{1:t}\right).
\end{equation} where $\mathbf{s}_t^i$ denotes the state of the $i$-th object in the $t$-th frame. $\mathbf{S}_t = (\mathbf{s}_t^1, \mathbf{s}_t^2, ..., \mathbf{s}_t^{M_t})$ denotes states of all the $M_t$ objects in the $t$-th frame. $\mathbf{S}_{1:t} = \{\mathbf{S}_1, \mathbf{S}_2, ..., \mathbf{S}_t\}$ denotes all the sequential states of all the objects from the first frame to the $t$-th frame. In tracking-by-detection, $\mathbf{o}_t^i$ denotes the collected observations for the $i$-th object in the $t$-th frame. $\mathbf{O}_t = (\mathbf{o}_t^1, \mathbf{o}_t^2, ..., \mathbf{o}_t^{M_t})$ denotes the collected observations for all the $M_t$ objects in the $t$-th frame. $\mathbf{O}_{1:t} = \{\mathbf{O}_1, \mathbf{O}_2, ..., \mathbf{O}_t\}$ denotes all the collected sequential observations of all the objects from the first frame to the $t$-th frame. Different Multiple Object Tracking (MOT) algorithms can be thought as designing different approaches to solving the above MAP problem, either from a \emph{probabilistic inference} perspective, e.g. Kalman filter or a \emph{deterministic optimization} perspective, e.g. Bipartite graph matching, and machine learning approaches.
Multiple Object Tracking (MOT) approaches can be categorized by different types of models. A distinction based on \textit{Initialization Method} is that of Detection Based Tracking (DBT) versus Detection Free Tracking (DFT). DBT refers that before tracking, object detection is performed on video frames. DBT methods involve two distinct jobs between the detection and tracking of objects. In this paper, we focus on DBT, also refers as tracking-by-detection for MOT. The reason is that DBT methods are widely used due to excellent performance with deep learning based object detectors, while DFT methods require manually annotations of the targets and bad results could arise when a new unseen object appears. Another important distinction based on \textit{Processing Mode} is that of Online versus Offline models. An Online model receives video input on a frame-by-frame basis, and gives output per frame. This means only information from past frames and the current frame can be used. Offline models have access to the entire video, which means that information from both past and future frames can be used.
Tracking-by-detection methods are usually utilized in online tracking models. A simple and classic pipeline is as (1) Detect objects of interest; (2) Predict new locations of objects from previous frames; (3) Associate objects between frames by similarity of detected and predicted locations. Well-performed CNN architectures can be used for object detection such as Faster R-CNN~\cite{ren2015faster}, YOLO~\cite{redmon2016you} and SSD~\cite{liu2016ssd}. For prediction of new locations of tracked objects, approaches model the velocity of objects, and predict the position in future frames using optical flow, or recurrent neural networks, or Kalman filters. The association task is to determine which detection corresponds to which object, or a detection represents a new object.
One popular dataset for Multiple Object Tracking (MOT) is MOTChallenge~\cite{leal2015motchallenge}. In MOTChallenge~\cite{leal2015motchallenge}, detections for each frame are provided in the dataset, and the tracking capability is measured as opposed to the detection quality. Video sequences are labeled with bounding boxes for each pedestrian collected from multiple sources. This motivates the use of tracking-by-detection paradigm. MDPs~\cite{xiang2015learning} is a tracking-by-detection method and achieved the state-of-the-art performance on MOTChallenge~\cite{leal2015motchallenge} Benchmark when it was proposed. Major contributions can be solving MOT by learning a MDP policy in a reinforcement learning fashion which benefits from both advantages of offline-learning and online-learning for data association. It also can handle the birth / death and appearance / disappearance of targets by simply treating them as state transitions in the MDP while leveraging existing online single object tracking methods. SORT~\cite{bewley2016simple} is a simple and real-time Multiple Object Tracking (MOT) method where state-of-the-art tracking quality can be achieved with only classical tracking methods. It is the most widely used real-time online Multiple Object Tracking (MOT) method and is very efficient for real-time applications in practice. Due to the simplicity of SORT~\cite{bewley2016simple}, the tracker updates at a rate of 260 Hz which is over 20x faster than other state-of-the-art trackers. On the MOTChallenge~\cite{leal2015motchallenge}, SORT~\cite{bewley2016simple} with a state-of-the-art people detector ranks on average higher than MHT~\cite{kim2015multiple} on standard detections. DeepSort~\cite{wojke2017simple} is an extension of SORT~\cite{bewley2016simple} which integrates appearance information to improve the performance of SORT~\cite{bewley2016simple} which can track through longer periods of occlusion, making SORT~\cite{bewley2016simple} a strong competitor to state-of-the-art online tracking algorithms.
\subsection{Near Accident Detection}
In addition to vehicle detection and vehicle tracking, analysis of the interactions or behavior of tracked vehicles has emerged as an active and challenging research area in recent years~\cite{sivaraman2011learning,hermes2009long,wiest2012probabilistic}. Near Accident Detection is one of the highest levels of semantic interpretation in characterizing the interactions of vehicles on the road. The basic task of near accident detection is to locate near accident regions and report them over video frames. In order to detect near accident on traffic scenes, robust vehicle detection and vehicle tracking are the prerequisite tasks.
Most of the near accident detection approaches are based on motion cues and trajectories. The most typical motion cues are optical flow and trajectory. Optical flow is widely utilized in video processing tasks such as video segmentation~\cite{huang2018supervoxel}. A trajectory is defined as a data sequence containing several concatenated state vectors from tracking, an indexed sequence of positions and velocities over a given time window. In recent years, researches have tried to make long-term classification and prediction of vehicle motion. Based on vehicle tracking algorithms such as Kalman filtering, optimal estimation of the vehicle state can be computed one frame ahead of time. Trajectory modeling approaches try to predict vehicle motion more frames ahead of time, based on models of typical vehicle trajectories~\cite{sivaraman2011learning,hermes2009long,wiest2012probabilistic}. In~\cite{sivaraman2011learning}, it used clustering to model the typical trajectories in highway driving and hidden Markov modeling for classification. In~\cite{hermes2009long}, trajectories are classified using a rotation-invariant version of the longest common subsequence as the similarity metric between trajectories. In~\cite{wiest2012probabilistic}, it uses variational Gaussian mixture modeling to classify and predict the long-term trajectories of vehicles.
Over the past two decades, for automatic traffic accident detection, a great deal of literature emerged in various ways. Several approaches have been developed based on decision trees, Kalman filters, or time series analysis, with varying degrees of success in their performance~\cite{srinivasan2003traffic,srinivasan2001hybrid,xu1998real,shuming2002traffic,jiansheng2014vision,bhonsle2000database}. Ohe et al.~\cite{ohe1995method} use neural networks to detect traffic incidents immediately by utilizing one minute average traffic data as input, and determine whether an incident has occurred or not. In~\cite{ikeda1999abnormal}, the authors propose a system to distinguish between different types of incidents for automatic incident detection. In~\cite{kimachi1994incident}, it investigates the abnormal behavior of vehicle related to accident based on the concepts of fuzzy theory where accident occurrence relies on the behavioral abnormality of multiple continual images. Zeng et al.~\cite{zeng2008data} propose an automatic accident detection approach using D-S evidence theory data fusion based on the probabilistic output of multi-class SVMs. In~\cite{sadeky2010real}, it presents a real-time automatic traffic accidents detection method using Histogram of Flow Gradient (HFG) and the trajectory of vehicles by which the accident was occasioned is determined in case of occurrence of an accident. In~\cite{kamijo2000traffic}, it develops an extendable robust event recognition system for Traffic Monitoring and Accident Detection based on the hidden Markov model (HMM). ~\cite{chen2010automatic} proposed a method using SVM based on traffic flow measurement. A similar approach using BP-ANN for accident detection has been proposed in~\cite{srinivasan2004evaluation,ghosh2003wavelet}. In~\cite{saunier2010large}, it presented a refined probabilistic framework for the analysis of road-user interactions using the identification of potential collision points for estimating collision probabilities. Other methods for Traffic Accident Detection has also been presented using Matrix Approximation~\cite{xia2015vision}, optical flow and Scale Invariant Feature Transform (SIFT)~\cite{chen2016vision}, Smoothed Particles Hydrodynamics (SPH)~\cite{ullah2015traffic}, and adaptive traffic motion flow modeling~\cite{maaloul2017adaptive}. With advances in object detection with deep neural networks, several convolutional neural networks (CNNs) based automatic traffic accident detection methods~\cite{singh2018deep,sultani2018real} and recurrent neural networks (RNNs) based traffic accident anticipation methods~\cite{chan2016anticipating,suzuki2018anticipating} have been proposed along with some traffic accident dataset~\cite{sultani2018real,suzuki2018anticipating,kataoka2018drive,shah2018accident} of surveillance videos or dashcam videos~\cite{chan2016anticipating}. However, either most of these methods do not have real-time performances for online accident detection without using future frames, or most of these methods mentioned above give unsatisfactory results. Besides that, no proposed dataset contains videos with top-down views such as drone/Unmanned Aerial Vehicles (UAVs) videos, or omnidirectional camera videos for traffic analysis.
\section{Conclusion}\label{sec:conclusion}
We have proposed a two-stream Convolutional Network architecture that performs real-time detection, tracking, and near accident detection of road users in traffic video data. The two-stream Convolutional Networks consist of a spatial stream network to detect individual vehicles and likely near accident regions at the single frame level, by capturing appearance features with a state-of-the-art object detection method. The temporal stream network leverages motion features of detected candidates to perform multiple object Tracking and generate individual trajectories of each tracking target. We detect near accident by incorporating appearance features and motion features to compute probabilities of near accident candidate regions. We have present a challenging Traffic Near Accident dataset (TNAD), which contains different types of traffic interaction videos that can be used for several vision-based traffic analysis tasks. On the TNAD dataset, experiments have demonstrated the advantage of our framework with an overall competitive qualitative and quantitative performance at high frame rates. The future direction of the work is the image stitching mehtods for our proposed multi-camera fisheye videos.
\begin{acks}
The authors would like to thank City of Gainesville for providing real traffic fisheye video data.
\end{acks}
\section{Experiments}\label{sec:experiments}
In this section, we first introduce our novel traffic near accident dataset (TNAD) and describe the preprocessing, implementation detail and experiments settings. Finally, we present qualitative and quantitative evaluation in terms of the performance of object detection, multiple object tracking, and near accident detection, and comparison between other methods and our framework.
\subsection{Traffic Near Accident Dataset (TNAD)}
As we mentioned in Section~\ref{sec:background}, there is no such a comprehensive traffic near accident dataset containing top-down views videos such as drone/Unmanned Aerial Vehicles (UAVs) videos, or omnidirectional camera videos for traffic analysis. Therefore, we have built our own dataset, traffic near accident dataset (TAND) which is depicted in Figure~\ref{fig:dataset}. Intersections tend to experience more and severe near accident due to factors such as angles and turning collisions. Traffic Near Accident Dataset (TNAD) containes 3 types of video data of traffic intersections that could be utilized for not only near accident detection but also other traffic surveillance tasks including turn movement counting.
\begin{figure}
\includegraphics[width=1\linewidth]{samples/dataset.png}
\caption{Samples of Traffic Near Accident Dataset (TNAD). Our dataset consists of a large number of diverse intersection surveillance videos and different near accident (cars and motorcycles). Yellow rectangles and lines represent the same object in multi-camera video. White circles represent the near accident regions.}
\label{fig:dataset}
\end{figure}
The first type is drone video that monitoring an intersection with top-down view. The second type of intersection videos is real traffic videos acquired by omnidirectional fisheye cameras that monitoring small or large intersections. It is widely used in transportation surveillance. These video data can be directly used as input for our vision-intelligent framework, and also pre-processing of fisheye correction can be applied to them for better surveillance performance. The third type of video is video data simulated by game engine for the purpose to train and test with more near accident samples. The traffic near accident dataset (TAND) consists of 106 videos with total duration over 75 minutes with frame rates between 20 fps to 50 fps. The drone video and fisheye surveillance videos are recorded in Gainesville, Florida at several different intersections. Our videos are challenging than videos in other datasets due to the following reasons:
\begin{itemize}
\item Diverse intersection scene and camera perspectives: The intersections in drone video, fisheye surveillance video, and simulation video are much different. Additionally, the fisheye surveillance video has distortion and fusion technique is needed for multi-camera fisheye videos.
\item Crowded intersection and small object: The number of moving cars and motorbikes per frame are large and these objects are relatively smaller than normal traffic video.
\item Diverse accidents: Accidents involving cars and motorbikes are all included in our dataset.
\item Diverse Lighting condition: Different lighting conditions such as daylight and sunset are included in our dataset.
\end{itemize}
We manually annotate the spatial location and temporal locations of near accidents and the still/moving objects with different vehicle class in each video. 32 videos with sparse sampling frames (only 20\% frames of these 32 videos are used for supervision) are used only for training the object detector. The remaining 74 videos are used for testing.
\subsection{Fisheye and multi-camera video}
The fisheye surveillance videos are recorded from real traffic data in Gainesville. We have collected 29 single-camera fisheye surveillance videos and 19 multi-camera fisheye surveillance videos monitoring a large intersection. We conduct two experiments, one directly using these raw videos as input for our system and another is first to do preprocessing for correcting fisheye distortion on video level and feed them into our system. As the original survellance video has many visual distortions especially near the circular boundaries of cameras, our system performs better on these after preprocessing videos. Therefore we keep the distortion correction preprocessing in the experiments for fisheye videos.
For large intersection, two fisheye cameras placed at opposite directions are used for surveillance and each of them mostly shows half of roads and real traffic for the large intersection. In this paper, we do not investigate the real stitching problem (we'll leave it for further work). First, we do fisheye distortion correction and combine the two video with similar points. Then we apply a simple object level stitching methods by assigning the object identity for the same objects across the left and right video using similar features and appearing/vanishing positions.
\subsection{Model Training}
The layer configuration of our spatial and temporal convolutional neural networks (based on Darknet-19~\cite{wojke2017simple}) is schematically shown in Table~\ref{net}. We adopt the Darknet-19~\cite{wojke2017simple} for classification and detection with deepSORT using data association metric combining deep appearance feature. We implement our framework on Tensorflow and do multi-scale training and testing with a single GPU (Nvidia Titan X Pascal). Training a single spatial convolutional network takes 1 day on our system with 1 Nvidia Titan X Pascal card. For classification and detection training, we use the same training strategy as YOLO9000~\cite{wojke2017simple}. We train the network on our TNAD dataset with 4 class of vehicle (motorcycle, bus, car, and truck) for 160 epochs using stochastic gradient descent with a starting learning rate of 0.1 for classification, and $10^{-3}$ for detection (dividing it by 10 at 60 and 90 epochs.), weight decay of 0.0005 and momentum of 0.9 using the Darknet neural network framework~\cite{wojke2017simple}.
\begin{figure}
\includegraphics[width=1\linewidth]{samples/objectdetection.png}
\caption{Sample results of object detection on TNAD dataset. \textbf{Left and middle left:} results of directly using YOLOv2~\cite{redmon2017yolo9000} detector pretrained on generic objects (VOC dataset)~\cite{Everingham15}. \textbf{Middle right and right:} results of our spatial network with multi-scale training based on YOLOv2~\cite{redmon2017yolo9000}.}
\label{fig:objectdetection}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth, height=8cm]{samples/tracking.png}
\caption{Tracking and trajectory comparison with Urban Tracker~\cite{jodoin2014urban} and TrafficIntelligence~\cite{jackson2013flexible} on drone videos of TNAD dataset.\textbf{Left:} results of Urban Tracker~\cite{jodoin2014urban}(BSG with Multilayer and Lobster Model). \textbf{Middle:} results of TrafficIntelligence~\cite{jackson2013flexible}. \textbf{Right:} results of our spatial network.}
\label{fig:tracking}
\end{figure}
\begin{figure}
\includegraphics[width=1\linewidth]{samples/nearaccident.png}
\caption{Sample results of tracking, trajectory and near accident detection of our two-stream Convolutional Networks on simulation videos of TNAD dataset.\textbf{Left:} tracking results based on DeepSORT~\cite{wojke2017simple}. \textbf{Middle:} trajectory results of our spatial network. ~\cite{jackson2013flexible}.\textbf{Right:} near accident detection results of our two-stream Convolutional Networks.}
\label{fig:nearaccident}
\end{figure}
\begin{table}[htbp]
\caption{Frame level near accident detection results }\label{tab:quantitative}
\resizebox{0.95\columnwidth}{!}{
\begin{tabular}{|c|c|c|c|c|}
\hline
Video ID & near Accident (pos/neg) & \vtop{\hbox{\strut \# of frame for positive Near accident}\hbox{\strut (groundtruth)/total frame}} & \vtop{\hbox{\strut \# of frame for correct localization }\hbox{\strut (IoU >= 0.6)}}
& \vtop{\hbox{\strut \# of frame for incorent localization}\hbox{\strut (IoU<0.6)}}\tabularnewline
\hline
\hline
1 & pos & 12/245 & 12 & 0 \tabularnewline
\hline
2 & neg & 0/259 & 0 &0 \tabularnewline
\hline
3 & neg & 0/266 & 0 & 0 \tabularnewline
\hline
4& pos & 16/267 & 13 & 0 \tabularnewline
\hline
5 & pos & 6/246 & 4 & 0\tabularnewline
\hline
6 & pos & 4/243 & 4 & 0\tabularnewline
\hline
7 & neg & 0/286 & 0 & 0\tabularnewline
\hline
8 & pos & 2/298 & 0 & 0\tabularnewline
\hline
9 & pos & 27/351 & 23 & 6\tabularnewline
\hline
10 & neg & 0/301 & 0 & 0\tabularnewline
\hline
11 & neg & 0/294 & 0 & 0\tabularnewline
\hline
12 & pos & 6/350 & 6 & 6 \tabularnewline
\hline
13 & neg & 0/263 & 0 & 0 \tabularnewline
\hline
14 & pos & 5/260 & 5 & 0 \tabularnewline
\hline
15 & pos & 4/326 & 4 & 0 \tabularnewline
\hline
16 & neg & 0/350 & 0 & 0 \tabularnewline
\hline
17 & neg & 0/318 & 0 & 1\tabularnewline
\hline
18 & pos & 10/340 & 8 & 0\tabularnewline
\hline
19 & pos & 6/276 & 0 & 0\tabularnewline
\hline
20 & pos & 8/428 & 4 & 0\tabularnewline
\hline
21 & neg & 0/259 & 0 & 0 \tabularnewline
\hline
22 & pos & 10/631 & 8 & 0 \tabularnewline
\hline
23 & pos & 35/587 & 30 & 2 \tabularnewline
\hline
24 & neg & 0/780 & 0 & 0 \tabularnewline
\hline
25 & neg & 0/813 & 0 & 0\tabularnewline
\hline
26 & neg & 0/765 & 0 & 0 \tabularnewline
\hline
27 & pos & 8/616 & 8 & 0 \tabularnewline
\hline
28 & pos & 10/243 & 10 & 1\tabularnewline
\hline
29 & pos & 6/259 & 6 & 0 \tabularnewline
\hline
30 & pos & 17/272 & 15 & 3 \tabularnewline
\hline
\end{tabular}
}
\end{table}
\subsection{Qualitative results}
We present some example experimental results of object detection, multiple object tracking and near accident detection on our traffic near accident dataset (TNAD) for drone videos, fisheye videos, and simulation videos. For object detection (Figure~\ref{fig:objectdetection}), we present some detection results of directly using YOLO detector~\cite{redmon2017yolo9000} trained on generic objects (VOC dataset)~\cite{Everingham15} and results of our spatial network with multi-scale training based on YOLOv2~\cite{redmon2017yolo9000}. For multiple object tracking (Figure~\ref{fig:tracking}), we present comparison of our temporal network based on DeepSORT~\cite{wojke2017simple} with Urban Tracker~\cite{jodoin2014urban} and TrafficIntelligence~\cite{jackson2013flexible}. For near accident detection (Figure~\ref{fig:nearaccident}), we present near accident detection results along with tracking and trajectories using our two-stream Convolutional Networks method. The object detection results shows that, with multi-scale training on TNAD dataset, the performance of the detector significant improves. It can perform well vehicle detection on top-down view surveillance videos even for small objects. In addition, we can achieve fast detection rate at 20 to 30 frame per second. Overall, this demonstrates the effectiveness of our spatial neural network. For the tracking part, since we use a tracking-by-detection paradigm, our methods can handle still objects and measure their state where Urban Tracker~\cite{jodoin2014urban} and TrafficIntelligence~\cite{jackson2013flexible} can only handle tracking for moving objects. On the other hand, Urban Tracker~\cite{jodoin2014urban} and TrafficIntelligence~\cite{jackson2013flexible} can compute dense trajectories of moving objects with good accuracy but they have slower tracking speed around 1 frame per second. For accident detection, our two-stream Convolutional Networks are able to do spatial localization and temporal localization for diverse accidents regions involving cars and motorcycles. The three sub-tasks (object detection, multiple object tracking and near accident detection) can always achieve real-time performance at high frame rate, 20 to 30 frame per second according to the frame resolution (e.g. 28 fps for 960$\times$480 image frame). Overall, the qualitative results demonstrate the effectiveness of our spatial neural network and temporal network respectively.
\begin{table}[]
\caption{Quantitative evaluation.}\label{tab:prerecall}
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{Benchmark Result}} & \multicolumn{2}{c|}{Predicted} \\ \cline{3-4}
\multicolumn{2}{|c|}{} & Negative & Positive \\ \hline
\multirow{2}{*}{Actual} & Negative & 11081 & 19 \\ \cline{2-4}
& Positive & 32 & 160 \\ \hline
\end{tabular}
\end{table}
\subsection{Quantitative results}
Since our frame has three tasks and our dataset are much different than other object detection dataset, tracking dataset and near accident dataset such as dashcam accident dataset~\cite{chan2016anticipating}, it is difficult to compare individual quantitative performance for all three tasks with other methods. One of our motivation is to propose a vision-based solution for Intelligent Transportation System, we focus more on near accident detection and present quantitative analysis of our two-stream Convolutional Networks. The simulation videos are for the purpose to train and test with more near accident samples and we have 57 simulation videos with a total over 51,123 video frames. We sparsely sample only 1087 frames from them for whole training processing. We present the analysis of near accident detection for 30 testing videos (18 has positive near accident, 12 has negative near accident). Table~\ref{tab:quantitative} shows frame level near accident detection performance on 30 testing simulation videos. The performance of precision, recall and F-measure are presented in Table~\ref{tab:prerecall}. If a frame contains a near accident scenario and we can successfully localize it with Intersection of union (IoU) is large or equal than 0.6, this is a True Positive (TP). If we cannot localize it or localize it with Intersection of Union (IoU) is less than 0.6, this is a False Negative (FN). If a frame has no near accident scenario but we detect a near accident region, this is a False Positive (FP). Otherwise, this is a True Negative (TN). We compute the $\text{precision} = \frac{\text{TP}}{\text{TP + FP}}$, $\text{recall} = \frac{\text{TP}}{\text{TP + FN}}$, and F-measure $\text{F-measure} = \frac{2\times \text{precision} \times \text{recall}}{\text{precision + recall}}= \frac{\text{2TP}}{\text{2TP + FP + FN}}$. Our precision is about 0.894, recall is about 0.8333 and F1 score is about 0.863. The three sub-tasks (object detection, multiple object tracking and near accident detection) can always achieve real-time performance at high frame rate, 20~30 frame per second according to the frame resolution (e.g. 28 fps for 960$\times$480 image frame). In conclusion, we have demonstrated that our two-stream Convolutional Networks have an overall competitive performance for near accident detection on our TNAD dataset.
\section{Introduction}\label{sec:intro}
The technologies of Artificial Intelligence (AI) and Internet of Things (IoTs) are ushering in a new promising era of ''Smart Cities'', where billions of people around the world can improve the quality of their life in aspects of transportation, security, information and communications and etc. One example of the data-centric AI solutions is computer vision technologies that enables vision-based intelligence at the edge devices across multiple architectures. Sensor data from smart devices or video cameras can be analyzed immediately to provide real-time analysis for the Intelligent Transportation System (ITS). At traffic intersections, it has more volume of road users (pedestrians, vehicles), traffic movement, dynamic traffic event, near accidents and etc. It is a critically important application to enable global monitoring of traffic flow, local analysis of road users, automatic near accident detection.
As a new technology, vision-based intelligence has a wide range of applications in traffic surveillance and traffic management~\cite{coifman1998real,valera2005intelligent,buch2011review,kamijo2000traffic,veeraraghavan2003computer, he2017single}. Among them, many research works have focused on traffic data acquirement with aerial videos~\cite{angel2002methods,salvo2017traffic}, where the aerial view provides better perspectives to cover a large area and focus resources for surveillance tasks. Unmanned Aerial Vehicles (UAVs) and omnidirectional cameras can acquire useful aerial videos for traffic surveillance especially at intersections with a broader perspective of the traffic scene, with the advantage of being both mobile, and able to be present in both time and space. UAVs has been exploited in a wide range of transportation operations and planning applications including emergency vehicle guidance, track vehicle movements. A recent trend of vision-based intelligence is to apply computer vision technologies to these acquired intersection aerial videos ~\cite{scotti2005dual,wang2006intelligent} and process them at the edge across multiple ITS architecture.
From global monitoring of traffic flow for solving traffic congestion to quest for better traffic information, an increasing reliance of ITS has resulted in a need for better object detection (such as wide-area detectors for pedestrian, vehicles), and multiple vehicle tracking that yields traffic parameters such as flow, velocity and vehicle trajectories. Tracks and trajectories are measures over a length of path rather than at a single point. It is possible to tackle related surveillance tasks including traffic movement measurements (e.g. turn movement counting) and routing information. The additional information from vehicle trajectories could be utilized to improve near accident detection, by either detecting stopped vehicles with their collision status or identifying acceleration / deceleration patterns or conflicting trajectories that are indicative of near accidents. Based on the trajectories, it is also possible to learn and forecast vehicle trajectory to enable near accident anticipation.
Generally, a vision-based surveillance tool for intelligent transportation system should meet several requirements:
\begin{enumerate}
\item Segment vehicles from the background and from other vehicles so that all vehicles (stopped or moving) are detected;
\item Classify detected vehicles into categories: cars, buses, trucks, motorcycles and etc;
\item Extract spatial and temporal features (motion, velocity, trajectory) to enable more specific tasks including vehicle tracking, trajectory analysis, near accident detection, anomaly detection and etc;
\item Function under a wide range of traffic conditions (light traffic, congestion, varying speeds in different lanes) and a wide variety of lighting conditions (sunny, overcast, twilight, night, rainy, etc.);
\item Operate in real-time.
\end{enumerate} Over the decades, although an increasing number of research on vision-based system for traffic surveillance have been proposed, many of these criteria still cannot be met. Early solutions~\cite{hoose1992impacts} do not identify individual vehicles as unique targets and progressively track their movements. Methods have been proposed to address individual vehicle detection and vehicles tracking problems~\cite{koller1993model,mclauchlan1997real,coifman1998real} with tracking strategies including model based tracking, region based tracking, active contour based tracking, feature based tracking and optical flow employment. Compared to traditional hand-crafted features, deep learning methods~\cite{ren2015faster,girshick2016region,redmon2016you, tian2016detecting} in object detection have illustrated the robustness with specialization of the generic detector to a specific scene. Leuck~\cite{leuck1999automatic} and Gardner~\cite{gardner1996interactive} use three-dimensional (3-D) models of vehicle shapes to estimate vehicle images projected onto a two-dimensional (2-D) image plane. Recently, automatic traffic accident detection has become an important topic. One typical approach uses object detection or tracking before detecting accident events~\cite{sadeky2010real,kamijo2000traffic,jiansheng2014vision, jiang2007abnormal,hommes2011detection}, with Histogram of Flow Gradient
(HFG), Hidden Markov Model (HMM) or, Gaussian Mixture Model (GMM). Other approaches~\cite{liu2010anomaly,ihaddadene2008real,wang2010anomaly,wang2012real,tang2005traffic,karim2002incident,xia2015vision,chen2010automatic,chen2016vision} use low-level features (e.g.\ motion features) to demonstrate better robustness. Neural networks have also been employed to automatic accident detection~\cite{ohe1995method,yu2008back,srinivasan2004evaluation,ghosh2003wavelet}.
In this paper, we first propose a Traffic Near Accident Dataset (TNAD). Intersections tend to experience more and severe near accident, due to factors such as angles and turning collisions. Observing this, the TNAD dataset is collected to contain three types of video data of traffic intersections that could be utilized for not only near accident detection but also other traffic surveillance tasks including turn movement counting. The first type is drone video that monitoring an intersections with top-down view. The second type of intersection videos is real traffic videos acquired by omnidirectional fisheye cameras that monitoring small or large intersections. It is widely used in transportation surveillance. These video data can be directly used as inputs for any vision-intelligent framework. The pre-processing of fisheye correction can be applied to them for better surveillance performance.
As there exist only a few samples of near accident in the reality per hour. The third type of video is proposed by simulating with game engine for the purpose to train and test with more near accident samples.
We propose a uniformed vision-based framework with the two-stream Convolutional Network architecture that performs real-time detection, tracking, and near accident detection of traffic road users. The two-stream Convolutional Networks consist of a spatial stream network to detect individual vehicles and likely near accident regions at the single frame level, by capturing appearance features with a state-of-the-art object detection method~\cite{redmon2016you}. The temporal stream network leverages motion features extracted from detected candidates to perform multiple object Tracking and generate corresponding trajectories of each tracking target. We detect near accident by incorporating appearance features and motion features to compute probabilities of near accident candidate regions. Experiments demonstrate the advantage of our framework with an overall competitive performance at high frame rates. The contributions of this work can be summarized as:
\begin{itemize}
\item A uniformed framework that performs real-time object detection, tracking and near accident detection.
\item The first work of an end-to-end trainable two-stream deep models to detect near accident with good accuracy.
\item A Traffic and Near Accident Detection Dataset (TNAD) containing different types of intersection videos that would be used for several vision-based traffic analysis tasks.
\end{itemize}
The organization of the paper is as follows. Section~\ref{sec:background} describes background on Object Detection, Multiple Object Tracking and Near Accident Detection. Section~\ref{sec:method} describes the overall architecture, methodologies, and implementation of our vision-based intelligent framework. This is followed in Section~\ref{sec:experiments} by an introduction of our Traffic Near Accident Detection Dataset (TNAD) and video preprocessing techniques. Section~\ref{sec:experiments} presents a comprehensive evaluation of our approach and other state-of-the-art near accident detection methods both qualitatively and quantitatively. Section~\ref{sec:conclusion} concludes by summarizing our contributions and also discusses the scope for future work.
\section{Two-Stream architecture for Near Accident Detection}\label{sec:method}
We present our vision-based two-stream architecture for real-time near accident detection based on real-time object detection and multiple object tracking. The goal of near accident detection is to detect likely collision scenarios across video frames and report these near accident records. As videos can be decomposed into spatial and temporal components. We divide our framework into a two-stream architecture as shown in Fig~\ref{fig:two}. The spatial part consists of individual frame appearance information about scenes and objects in the video. The temporal part contains motion information for moving objects. For spatial stream convolutional neural network, we utilized a standard convolutional network of a state-of-the-art object detection method~\cite{redmon2016you} to detect individual vehicles and likely near accident regions at the single frame level. The temporal stream network is leveraging object candidates from object detection CNNs and integrates their appearance information with a fast multiple object tracking method to extract motion features and compute trajectories. When two trajectories of individual objects start intersecting or become closer than a certain threshold, we'll label the region covering two objects as high probability near accident regions. Finally, we take average near accident probability of spatial stream network and temporal stream network and report the near accident record.
\subsection{Preliminaries}
\textbf{Convolutional Neural Networks:} Convolutional Neural Networks (CNNs) have a strong capability of representing objects and helps to boost the performance of numerous vision tasks, comparing to traditional heuristic features \cite{dalal2005histograms}. A Convolutional Neural Networks (CNN) is a a class of deep neural networks which is widely applied for visual imagery analysis in computer vision. A standard CNN usually consists of an input and an output layer, as well as multiple hidden layers (convolutional layers, pooling layers, fully connected layers and normalization layers) as shown in Figure~\ref{fig:cnn}. The input to a convolutional layer is an original image $\boldsymbol{X}$. We denote the feature map of $i$-th convolutional layer as $\boldsymbol{H}_i$ and $\boldsymbol{H}_0=\boldsymbol{X}$. Then $\boldsymbol{H}_i$ can be described as
\begin{equation}
{\boldsymbol{H}_i} = f\left( {{\boldsymbol{H}_{i - 1}} \otimes {\boldsymbol{W}_i} + {\boldsymbol{b}_i}} \right)
\end{equation}
where $\boldsymbol{W}_i$ is the weight for $i$-th convolutional kernel, and $\otimes$ is the convolution operation of the kernel and $i-1$-th image or feature map. Output of convolution operation are summed with a bias $\boldsymbol{b}_i$. Then the feature map for $i$-th layer can be computed by applying a nonlinear activation function to it. Take an example of using a $32\times32$ RGB image with a simple ConvNet for CIFAR-10 classification~\cite{krizhevsky2009learning}.
\begin{itemize}
\item Input layer: the original image with raw pixel values as width is 32, height is 32, and color channels (R,G,B) is 3.
\item Convolutional layer: compute output of neurons which are connected to local regions in the image through activation functions. If we use 12 filters, we have result in volume such as $[32\times32\times12]$.
\item Pooling layer: perform a downsampling operation, resulting in volume such as $[16\times16\times12]$.
\item Fully connected layer: compute the class scores, resulting in volume of size $[1\times1\times10]$, where these 10 numbers are corresponding to 10 class score.
\end{itemize}
In this way, CNNs transform the original image into multiple high-level feature representations layer by layer and compute the final final class scores.
\begin{figure}
\includegraphics[width=\textwidth]{samples/cnn.png}
\caption[]{Architecture of Convolutional Neural Networks for image classification.\footnotemark}
\label{fig:cnn}
\end{figure}
\footnotetext{The credit should be given to Adit Deshpande and his blog.}
\subsection{Spatial stream network}
In our framework, each stream is implemented using a deep convolutional neural network. Near accident scores are combined by the averaging score. Since our spatial stream ConvNet is essentially an object detection architecture, we build it upon the recent advances in object detection with YOLO detector~\cite{redmon2016you}, and pre-train the network from scratch on our dataset containing multi-scale drone, fisheye and simulation videos. As most of our videos contain traffic scenes with vehicles and traffic movement in top-down view, we specify different vehicle classes such as motorcycle, car, bus, and truck as object classes for training the detector. Additionally, near accident or collision can be detected from single still frame either from the beginning of a video or stopped vehicles associated in an accident after collision. Therefore, we train our detector to localize these likely near accident scenarios. Since the static appearance is a useful cue, the spatial stream network effectively performs object detection by operating on individual video frames.
\subsubsection{YOLO object detection}
You Only Look Once (YOLO)~\cite{redmon2016you} is a state-of-the-art, real-time object detection system. This end-to-end deep learning approach does not need region proposals and is much different from the region based algorithms. The pipeline of YOLO~\cite{redmon2016you} is pretty straightforward: YOLO~\cite{redmon2016you} passes the whole image through the neural network only once where the title comes from (You Only Look Once) and returns bounding boxes and class probabilities for predictions. Figure~\ref{fig:2} demonstrates the detection model and system of YOLO~\cite{redmon2016you}. In YOLO~\cite{redmon2016you}, it regards object detection as an end-to-end regression problem and uses a single convolutional network predicts the bounding boxes and the class probabilities for these boxes. It first takes the image and split it into an $S \times S$ grid, within each of the grid we take $m$ bounding boxes. For each grid cell,
\begin{itemize}
\item it predicts B boundary boxes and each box has one box confidence score
\item it detects one object only regardless of the number of boxes B
\item it predicts C conditional class probabilities (one per class for the likeliness of the object class)
\end{itemize}
\begin{figure}
\includegraphics[width=0.9\linewidth]{samples/cnn2.png}
\caption{The YOLO Detection System~\cite{redmon2016you}. It (1) resizes the input image to $448 \times 448$, (2) runs a single convolutional network on the image, and (3) thresholds the resulting detections by the model's confidence.}
\label{fig:2}
\end{figure}
For each of the bounding box, the convolutional neural network (CNN) outputs a class probability and offset values for the bounding box. Then it selects bounding boxes which have the class probability above a threshold value and uses them to locate the object within the image. In detail, each boundary box contains 5 elements: $(x, y, w, h)$ and a box confidence. The $(x, y)$ are coordinates which represent the center
of the box relative to the bounds of the grid cell. The $(w,h)$ are width and height. These elements are normalized as $x$, $y$, $w$ and $h$ are all between 0 and 1. The confidence prediction represents the intersection over union (IoU) between the predicted box and any ground truth box which reflects how likely the box contains an object (objectness) and how accurate is the boundary box. The mathematical definitions of those scoring and probability terms are:
\begin{center}
box confidence score $\equiv P_{r}(object)\cdot IoU$\\
conditional class probability $\equiv P_{r}(class_{i}|object)$\\
class confidence score $\equiv P_{r}(class_{i})\cdot IoU$\\
class confidence score $=$ box confidence score $\times$ conditional class probability
\end{center}
where $\equiv P_{r}(object)$ is the probability the box contains an object. $IoU$ is the IoU between the predicted box and the ground truth. $\equiv P_{r}(class_{i})$ is the probability the object belongs to $class_{i}$. $\equiv P_{r}(class_{i}|object)$ is the probability the object belongs to $class_{i}$ given an object is presence. The network architecture of YOLO~\cite{redmon2016you} simply contains 24 convolutional layers followed by two fully connected layers, reminiscent of AlexNet and even earlier convolutional architectures. Some convolution layers use $1 \times 1$ reduction layers alternatively to reduce the depth of the features maps. For the last convolution layer, it outputs a tensor with shape $(7, 7, 1024)$ which is flattened. YOLO~\cite{redmon2016you} performs a linear regression using two fully connected layers to make boundary box predictions and to make a final prediction using threshold of box confidence scores. The final loss adds localization, confidence and classification losses together.
\begin{table*}[h]
\begin{center}
\begin{tabular}{c|c|c|c}
Type & Filters & Size/Stride & Output\\
\hline
Convolutional & 32 & $3 \times 3$ & $224 \times 224 $ \\
Maxpool & &$2 \times 2 / 2$ & $112 \times 112 $ \\
Convolutional & 64 & $3 \times 3$ & $112 \times 112 $ \\
Maxpool & & $2 \times 2 / 2$ & $56 \times 56 $ \\
Convolutional & 128 &$3 \times 3$ & $56 \times 56 $ \\
Convolutional & 64 &$1 \times 1$ & $56 \times 56 $ \\
Convolutional & 128 &$3 \times 3$ & $56 \times 56 $ \\
Maxpool & & $2 \times 2 / 2$ & $28 \times 28 $ \\
Convolutional & 256 & $3 \times 3$ & $28 \times 28 $ \\
Convolutional & 128 & $1 \times 1$ & $28 \times 28 $ \\
Convolutional & 256& $3 \times 3$ & $28 \times 28 $ \\
Maxpool & & $2 \times 2 / 2$ & $14 \times 14 $ \\
Convolutional & 512 & $3 \times 3$ & $14 \times 14 $ \\
Convolutional & 256& $1 \times 1$ & $14 \times 14 $ \\
Convolutional & 512 & $3 \times 3$ & $14 \times 14$ \\
Convolutional & 256& $1 \times 1$ & $14 \times 14$ \\
Convolutional & 512 & $3 \times 3$ & $14 \times 14 $ \\
Maxpool & & $2 \times 2 / 2$ & $7 \times 7 $ \\
Convolutional & 1024 & $3 \times 3$ & $7 \times 7 $ \\
Convolutional & 512 & $1 \times 1$ & $7 \times 7 $ \\
Convolutional & 1024 & $3 \times 3$ & $7 \times 7$ \\
Convolutional & 512 & $1 \times 1$ & $7 \times 7$ \\
Convolutional & 1024 & $3 \times 3$ & $7 \times 7$ \\
\hline
\hline
Convolutional & 1000 & $1 \times 1$ & $7 \times 7$ \\
Avgpool & & Global & $1000$ \\
Softmax & & &\\
\end{tabular}
\end{center}
\caption{\small \textbf{Darknet-19~\cite{redmon2017yolo9000}.}}
\label{net}
\end{table*}
\scriptsize
\begin{multline}
\lambda_\textbf{coord}
\sum_{i = 0}^{S^2}
\sum_{j = 0}^{B}
\mathlarger{\mathbbm{1}}_{ij}^{\text{obj}}
\left[
\left(
x_i - \hat{x}_i
\right)^2 +
\left(
y_i - \hat{y}_i
\right)^2
\right]
+ \lambda_\textbf{coord}
\sum_{i = 0}^{S^2}
\sum_{j = 0}^{B}
\mathlarger{\mathbbm{1}}_{ij}^{\text{obj}}
\left[
\left(
\sqrt{w_i} - \sqrt{\hat{w}_i}
\right)^2 +
\left(
\sqrt{h_i} - \sqrt{\hat{h}_i}
\right)^2
\right]
\\
+ \sum_{i = 0}^{S^2}
\sum_{j = 0}^{B}
\mathlarger{\mathbbm{1}}_{ij}^{\text{obj}}
\left(
C_i - \hat{C}_i
\right)^2
+ \lambda_\textrm{noobj}
\sum_{i = 0}^{S^2}
\sum_{j = 0}^{B}
\mathlarger{\mathbbm{1}}_{ij}^{\text{noobj}}
\left(
C_i - \hat{C}_i
\right)^2
+ \sum_{i = 0}^{S^2}
\mathlarger{\mathbbm{1}}_i^{\text{obj}}
\sum_{c \in \textrm{classes}}
\left(
p_i(c) - \hat{p}_i(c)
\right)^2
\end{multline}
\normalsize
where $\mathbbm{1}_i^{\text{obj}}$ denotes if object appears in cell $i$ and $\mathbbm{1}_{ij}^{\text{obj}}$ denotes that the $j$th bounding box predictor in cell $i$ is ``responsible'' for that prediction. YOLO~\cite{redmon2016you} is orders of magnitude faster (45 frames per second) than other object detection approaches which means it can process streaming video in realtime and achieves more than twice the mean average precision of other real-time systems. For the implementation, we leverage the extension of YOLO~\cite{redmon2016you}, Darknet-19, a classification model that used as the base of YOLOv2~\cite{redmon2017yolo9000}. The full network description of it is shown in Table~\ref{net}. Darknet-19~\cite{redmon2017yolo9000} has 19 convolutional layers and 5 maxpooling layers and it uses batch normalization to stabilize training, speed up convergence, and regularize the model~\cite{ioffe2015batch}.
\subsection{Temporal stream network}
The spatial stream network is not able to extract motion features and compute trajectories due to single-frame inputs. To leverage these useful information, we present our temporal stream network, a ConvNet model which performs a tracking-by-detection multiple object tracking algorithm~\cite{bewley2016simple,wojke2017simple} with data association metric combining deep appearance features. The inputs are identical to the spatial stream network using the original video. Detected object candidates (only vehicle classes) are used to for tracking handling, state estimation, and frame-by-frame data association using SORT~\cite{bewley2016simple} and DeepSORT~\cite{wojke2017simple}, the real-time multiple object tracking methods. The multiple object tracking models each state of objects and describes the motion of objects across video frames. With tracking, we obtain results by stacking trajectories of moving objects between several consecutive frames which are useful cues for near accident detection.
\subsubsection{SORT}
Simple Online Realtime Tracking (SORT)~\cite{bewley2016simple} is a simple, popular and fast Multiple Object Tracking (MOT) algorithm. The core idea is to perform a Kalman filtering~\cite{kalman1960new} in image space and do frame-by-frame data association using the Hungarian methods~\cite{kuhn1955hungarian} with an association metric that measures bounding box overlap. Despite only using a rudimentary combination of the Kalman Filter~\cite{kalman1960new} and Hungarian algorithm~\cite{kuhn1955hungarian} for the tracking components, SORT~\cite{bewley2016simple} achieves an accuracy comparable to state-of-the-art online trackers. Moreover, due to the simplicity of it, SORT~\cite{bewley2016simple} can updates at a rate of 260 Hz on single machine which is over 20x faster than other state-of-the-art trackers.
\textbf{Estimation Model.} The state of each target is modelled as:
\begin{equation}
\mathbf{x} = [u,v,s,r,\dot{u},\dot{v},\dot{s}]^T,
\end{equation}
where $u$ and $v$ represent the horizontal and vertical pixel location of the centre of the target, while the scale $s$ and $r$ represent the scale (area) and the aspect ratio (usually considered to be constant) of the target's bounding box respectively. When a detection is associated to a target, it updates the target state using the detected bounding box where the velocity components are solved optimally via a Kalman filter framework~\cite{kalman1960new}. If no detection is associated to the target, its state is simply predicted without correction using the linear velocity model.
\textbf{Data Association.} In order to assign detections to existing targets, each target's bounding box geometry is estimated by predicting its new location in the current frame. The assignment cost matrix is defined as the IoU distance between each detection and all predicted bounding boxes from the existing targets. Then the assignment problem is solved optimally using the Hungarian algorithm~\cite{kuhn1955hungarian}. Additionally, a minimum IoU is imposed to reject assignments where the detection to target overlap is less than $IoU_{min}$. The IoU distances of the bounding boxes are found so as to handle short term occlusion caused by passing targets.
\textbf{Creation and Deletion of Track Identities.} When new objects enter or old objects vanish in video frames, unique identities for objects need to be created or destroyed accordingly. For creating trackers, we consider any detection with an overlap less than $IoU_{min}$ to signify the existence of an untracked object. Then the new tracker undergoes a probationary period where the target needs to be associated with detections to accumulate enough evidence in order to prevent tracking of false positives. Tracks could be terminated if they are not detected for $T_{Lost}$ frames to prevent an unbounded growth in the number of trackers and localization errors caused by predictions over long durations without corrections from the detector.
\subsubsection{DeepSORT}
DeepSORT~\cite{wojke2017simple} is an extension of SORT~\cite{bewley2016simple} which integrates appearance information through a pre-trained association metric to improve the performance of SORT~\cite{bewley2016simple}. It adopts a conventional single hypothesis tracking methodology with recursive Kalman filtering~\cite{kalman1960new} and frame-by-frame data association. DeepSORT~\cite{wojke2017simple} helps to solve a large number of identities switching problem in SORT~\cite{bewley2016simple} and it can track objects through longer periods of occlusions. During online application, it establishs measurement-to-track associations using nearest neighbor queries in visual appearance space.
\textbf{Track Handling and State Estimation}. The track handling and state estimation using Kalman filtering~\cite{kalman1960new} is mostly identical to the SORT~\cite{bewley2016simple}. The tracking scenario is defined using eight dimensional state space~$(u, v, \gamma, h, \dot{x}, \dot{y}, \dot{\gamma}, \dot{h})$ that contains the bounding box center position $(u, v)$, aspect ratio $\gamma$, height $h$, and their respective velocities in image coordinates. It uses a standard Kalman filter~\cite{kalman1960new} with a constant velocity motion and linear observation model, where it takes the bounding coordinates~$(u, v, \gamma, h)$ as direct observations of the object state.
\textbf{Data Association}. To solve the frame-by-frame association problem between the predicted Kalman states and the newly arrived measurements, it uses the Hungarian algorithm~\cite{kuhn1955hungarian}. In formulation, it integrates both motion and appearance information through combination of two appropriate metrics. For motion information, the (squared) Mahalanobis distance between predicted Kalman states and newly arrived measurements is utilized:
\begin{equation}
d^{(1)}(i,j)
=
(\bm{d}_j - \bm{y}_i)^{\bm{T}}(\bm{S}_i)^{-1}(\bm{d}_j - \bm{y}_i)
\end{equation}
where the projection of the $i$-th track distribution into
measurement space is ~$(\bm{y}_i, \bm{S}_i)$ and the $j$-th bounding box detection is $\bm{d}_j$. The second metric measures the smallest cosine distance between the~$i$-th track and~$j$-th detection in appearance space:
\begin{equation}
d^{(2)}(i, j)
= \min \left \{ 1 - {\bm{r}_j}^{T} \bm{r}^{(i)}_k | \bm{r}^{(i)}_k\in \mathcal{R}_i \right \}
\end{equation}
Then this association problem is built with combination of both metrics using a weighted sum where the influence of each metric on combined association cost can be controlled through hyperparameter $\lambda$.
\begin{equation}
c_{i,j}
=
\lambda \, d^{(1)}(i, j) + (1 - \lambda) d^{(2)}(i, j)
\end{equation}
\textbf{Matching Cascade}. Rather than solving measurement-to-track associations in a global way, it adopts a matching cascade introduced in~\cite{wojke2017simple} to solve a series of subproblems. In some situation, when occlusion happens to a object for a longer period of time, the subsequent Kalman filter~\cite{kalman1960new} predictions would increase the uncertainty associated with the object location. In consequent, probability mass spreads out in state space and the observation likelihood becomes less peaked. Intuitively, the association metric should account for this spread of probability mass by increasing the measurement-to-track distance. Therefore, the matching cascade strategy gives priority to more frequently seen objects to encode the notion of probability spread in the association likelihood.
\subsection{Near Accident Detection}
When utilizing the multiple object tracking algorithm, we compute the center of each object in several consecutive frames to form stacking trajectories as our motion representation. These stacking trajectories can provide accumulated information through image frames, including the number of objects, their motion history and timing of their interactions such as near accident. We stack the trajectories of all object by every $L$ consecutive frames as illustrated in Figure~\ref{fig:traj} where $\mathbf{p}_t^i$ denotes the center position of the $i$-th object in the $t$-th frame. $\mathbf{P}_t = (\mathbf{p}_t^1, \mathbf{p}_t^2, ..., \mathbf{p}_t^{M_t})$ denotes trajectories of all the $M_t$ objects in the $t$-th frame. $\mathbf{P}_{1:t} = \{\mathbf{P}_1, \mathbf{P}_2, ..., \mathbf{P}_t\}$ denotes all the sequential trajectories of all the objects from the first frame to the $t$-th frame. As we only exam every $L$ consecutive frames, the stacking trajectories are sequentially as
\begin{equation}
\mathbf{P}_{1:L} = \{\mathbf{P}_1, \mathbf{P}_2, ..., \mathbf{P}_L\}, \mathbf{P}_{L+1:2L} = \{\mathbf{P}_{L+1}, \mathbf{P}_{L+2}, ..., \mathbf{P}_{2L}\},\cdots
\end{equation}
$\mathbf{O}_t = (\mathbf{o}_t^1, \mathbf{o}_t^2, ..., \mathbf{o}_t^{M_t})$ denotes the collected observations for all the $M_t$ objects in the $t$-th frame. $\mathbf{O}_{1:t} = \{\mathbf{O}_1, \mathbf{O}_2, ..., \mathbf{O}_t\}$ denotes all the collected sequential observations of all the objects from the first frame to the $t$-th frame. We use a simple detection algorithm which finds collisions between simplified forms of the objects, using the center of bounding boxes.
Our algorithm is depicted in Algorithm~\ref{alg:detect}. Once the collision is detected, we set the region covering collision associated objects to be a new bounding box with class probability of near accident to be 1. By averaging the near accident probability of output from spatial stream network and temporal stream network, we are able to compute the final outputs of near accident detection.
\begin{figure}
\includegraphics[width=0.9\linewidth]{samples/trajectory.png}
\caption{The stacking trajectories extracted from tracking. Consecutive frames and the corresponding displacement vectors are shown with the same colour.}
\label{fig:traj}
\end{figure}
\begin{algorithm}
\KwIn{current frame $t_{current}$, collision state list $Collision$}
\KwOut{collision state list $Collision$}
\For{$t_{L} \leftarrow t_{previous}$ to $t_{current}$ in steps of $L$ frames}{
\For{each pair of object trajectory ($\mathbf{p}_{:t_{L}}^1$, $\mathbf{p}_{:t_{L}}^2)$}{
\If{($\mathbf{p}_{:t_{L}}^1$ intersects $\mathbf{p}_{:t_{L}}^2$ as of $t_{L}$)}{add $\mathbf{o}_1$, $\mathbf{o}_2$ to $Collision$}
}
\If{($Collisions$)}{$t_{previous} \leftarrow t_{L}$;
return TRUE}
}
$t_{previous} \leftarrow t_{d}$; return FALSE
\caption{Collision Detection}\label{alg:detect}
\end{algorithm}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,344
|
package org.onosproject.isis.controller.topology;
import org.onlab.packet.Ip4Address;
/**
* Representation of an ISIS link information.
*/
public interface LinkInformation {
/**
* Gets link id.
*
* @return link id
*/
String linkId();
/**
* Sets link id.
*
* @param linkId link id
*/
void setLinkId(String linkId);
/**
* Gets whether link information is already created or not.
*
* @return true if link information is already created else false
*/
boolean isAlreadyCreated();
/**
* Sets link information is already created or not.
*
* @param alreadyCreated true if link information is already created else false
*/
void setAlreadyCreated(boolean alreadyCreated);
/**
* Returns link destination ID.
*
* @return link destination ID
*/
String linkDestinationId();
/**
* Sets link destination id.
*
* @param linkDestinationId link destination id
*/
void setLinkDestinationId(String linkDestinationId);
/**
* Gets link source id.
*
* @return link source id
*/
String linkSourceId();
/**
* Sets link source id.
*
* @param linkSourceId link source id
*/
void setLinkSourceId(String linkSourceId);
/**
* Gets interface ip address.
*
* @return interface ip address
*/
Ip4Address interfaceIp();
/**
* Sets interface ip address.
*
* @param interfaceIp interface ip address
*/
void setInterfaceIp(Ip4Address interfaceIp);
/**
* Gets neighbor ip address.
*
* @return neighbor ip address
*/
Ip4Address neighborIp();
/**
* Sets neighbor ip address.
*
* @param neighborIp neighbor ip address
*/
void setNeighborIp(Ip4Address neighborIp);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,286
|
Exited investments
Current investments
Home / Portfolio companies / Current investments
Image Analysis Ltd.
Image Analysis Group (UK) is a leader in the rapidly growing market for Magnetic Resonance Imaging (MRI) data quantification and analysis, helping partners such as hospitals, CROs and biotech companies design and manage their clinical studies for the most effective use of resources while providing earlier, more convincing evidence of treatment effect with smaller patient cohorts. The company's solution improves the accuracy of diagnostic assessment and research decisions for companies developing treatments for inflammatory musculoskeletal conditions, neuro-inflammation, oncology, cardiac perfusion and rare diseases.
Immusoft
Immusoft (Seattle, USA) develops a first-in-class hybrid cell/gene therapy, which uses a clinically validated, non-viral vector for safe, reliable insertion of functional genes into B-cells. The ISP™ platform enables safe insertion of genes encoding the correct human homolog of a missing or defective protein into a patient's immune cells using the Sleeping Beauty (SB) Transposon system – a non-viral vector. If successful, the therapy could become a breakthrough in multiple indications with lysosomal storage and protein production disorders.
AGCT
AGCT (St. Petersburg, Russia) develops a gene therapy against HIV and HIV-linked cancers. The setup is a spin-out from one of the major oncology clinics in Russia and is supported by IP on genome editing from a top EU university. The therapy has the potential to bring cure to previously incurable patients with HIV and lymphomas and could also be used in a number of less severe indications.
Pipeline Therapeutics
Pipeline Therapeutics a US-based creating a portfolio of first-in-class small molecules in neuro-regeneration: Synaptogenesis (with a lead program in Hearing Loss), Remyelination, and Axonal Repair. PIPE-505 is the first small molecule developed specifically for the treatment of sensorineural hearing loss (SNHL) associated with cochlear synaptopathy. The therapeutic focus, regeneration of the cochlear synapse, should augment signal-to-noise processing and manifest as improved speech-in-noise comprehension, a chief auditory complaint and unmet need of patients with SNHL.
Botkin.AI
Botkin.AI is a leading developer of medical solutions based on a proprietary platform using artificial intelligence technologies. The products developed by the company facilitate the identification and analysis of pathologies in computer tomography, X-ray and mammography images. The use of Botkin.AI allow to improve early diagnosis of various diseases and increase the efficiency of physicians' work.
PanDx
PanDx (Cambridge, UK) develops a catheter for the treatment of acute pancreatitis and other severe GI conditions. By combining a number of novel and established treatment approaches, the catheter is positioned to significantly decrease mortality and reduce the treatment costs, as well as discover new ways of diagnosing diverse hepato-biliary diseases.
Immusoft and Takeda Collaborate to Discover and Develop Cell Therapies for Rare Neurometabolic Disorders
Pipeline Therapeutics Appoints Scott Oross as General Counsel
Immusoft Announces Formation Of Scientific Advisory Board
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,993
|
Q: Bing Map: Issue with directions manager drag and drop - I am currently working on bing map's directions manager. I am creating a trip/route by adding few waypoints. After getting a result from the directionsManager, if i drag and drop a waypoint(marker on the map), I am facing an inconsistency issue with bing map's direction manager. I do not get the address of the waypoint back from the directionsManager.getAllWaypoints().
Issue: i do not get the address after drag and drop always (I do get it most of the times but there are instances wheen i dont get it as well); however i do get the lat long of that wypoint.
I am checking for the updated address in the event handler 'directionsUpdated', using directionsManager.getAllwaypoints(). i have tried event 'dragDropCompleted' that also doesn't help.
Observation:
* 'directionsUpdated' occurs after 'dragDropCompleted'.
* I do see the address after the complete map is loaded, but i need it before that.
I am using ajax api.
Thanks in advance.
A: If you just wanted to drag a pushpin on a map and get the address it was dropped on you can do this fairly easily. Here is an example of how to make a pushpin draggable and fire an event when it is dropped: https://www.bingmapsportal.com/ISDK/AjaxV7#Pushpins13
Once you have this you can either use the Search Module in the Bing Maps control or the Bing Maps REST services to reverse geocode the coordinate the pushpin is dropped on. Here are some examples of how to do the reverse geocoding:
https://www.bingmapsportal.com/ISDK/AjaxV7#SearchModule3
https://www.bingmapsportal.com/ISDK/AjaxV7#RESTServices1
For the second example set the query value to "[latitude],[longitude]", make sure to replace the square brackets with the actual numbers.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,613
|
{"url":"https:\/\/math.stackexchange.com\/questions\/3201195\/how-can-i-calculate-frac-partial-log-sigma-partial-rho-where-sigm","text":"# How can I calculate $\\frac{\\partial \\log |\\Sigma|}{\\partial \\rho }$ where $\\Sigma=(1-\\rho)I+\\rho\\mathbf{1}\\mathbf{1}^\\top$?\n\nI need to calculate the $$\\dfrac{\\partial \\log |\\Sigma|}{\\partial \\rho }$$ when $$\\Sigma = (1-\\rho) I + \\rho \\mathbf{1} \\mathbf{1}^\\top$$ and $$\\Sigma$$ has dimension $$p \\times p$$.\n\nI try to use the formula presented here and here but the result is not right.\n\n\u2022 Without trying anything fancy, you can simply calculate $|\\Sigma|$ and proceed as usual. \u2013\u00a0StubbornAtom Apr 25 at 15:38\n\nThe derivative wrt $$\\rho$$ is $$\\operatorname{trace}((-I+11^T){\\Sigma}^{-1})$$.","date":"2019-05-22 16:40:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 6, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9945608973503113, \"perplexity\": 133.74581038424776}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232256887.36\/warc\/CC-MAIN-20190522163302-20190522185302-00142.warc.gz\"}"}
| null | null |
from __future__ import absolute_import
from .base import Filter
from ua_parser.user_agent_parser import Parse
MIN_VERSIONS = {
'Chrome': 0,
'IE': 10,
'Firefox': 0,
'Safari': 6,
'Edge': 0,
'Opera': 15,
}
class LegacyBrowsersFilter(Filter):
id = 'legacy-browsers'
name = 'Filter out known errors from legacy browsers'
description = 'Older browsers often give less accurate information, and while they may report valid issues, the context to understand them is incorrect or missing.'
default = False
def get_user_agent(self, data):
try:
for key, value in data['sentry.interfaces.Http']['headers']:
if key.lower() == 'user-agent':
return value
except LookupError:
return ''
def test(self, data):
if data.get('platform') != 'javascript':
return False
value = self.get_user_agent(data)
if not value:
return False
ua = Parse(value)
if not ua:
return False
browser = ua['user_agent']
if not browser['family']:
return False
try:
minimum_version = MIN_VERSIONS[browser['family']]
except KeyError:
return False
try:
major_browser_version = int(browser['major'])
except (TypeError, ValueError):
return False
if minimum_version > major_browser_version:
return True
return False
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,455
|
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1, minimum-scale=1, maximum-scale=1, user-scalable=no">
<title>SITNA - Ejemplo de LayerOptions.filter</title>
<link rel="stylesheet" href="examples.css" />
</head>
<body>
<div class="instructions">
<p>Ejemplo de uso de <a href="../doc/global.html#LayerOptions">LayerOptions</a>.filter.</p>
<p>Filtrado de capas WMS mediante filtros GML o CQL.</p>
</div>
<script src="../"></script>
<script type="text/javascript" src="examples.js"></script>
<div id="mapa"></div>
<script>
// Establecemos un layout simplificado apto para hacer demostraciones de controles.
SITNA.Cfg.layout = "layout/ctl-container";
// Añadimos el control de tabla de contenidos en la primera posición.
SITNA.Cfg.controls.TOC = {
div: "slot1"
};
// Añadimos la capa de IDENA de "Estaciones de aforo del Gobierno de Navarra" cuyo titular es "Gobierno de Navarra"
// Y añadimos la capa "Estaciones meteorológicas" de IDENA mostrando solo aquellas que están por encima de 1000 m.
SITNA.Cfg.workLayers = [
{
id: "layer1",
title: "Estaciones de aforo del Gobierno de Navarra",
type: SITNA.Consts.layerType.WMS,
url: "//idena.navarra.es/ogc/wms",
layerNames: "IDENA:HIDROG_Sym_EstacAforo",
filter: '<ogc:Filter xmlns:ogc="http://www.opengis.net/ogc"><ogc:PropertyIsEqualTo><ogc:PropertyName>TITULAR</ogc:PropertyName><ogc:Literal><![CDATA[Gobierno de Navarra]]></ogc:Literal></ogc:PropertyIsEqualTo></ogc:Filter>'
},
{
id: "layer2",
title: "Estaciones meteorológicas por encima de 1000m",
type: SITNA.Consts.layerType.WMS,
url: "//idena.navarra.es/ogc/ows",
layerNames: "IDENA:estacMeteor",
filter: 'ALTITUD>1000'
}
];
var map = new SITNA.Map("mapa");
</script>
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,045
|
Education
Elias Koutsoupias is a Greek computer scientist working in algorithmic game theory.
Koutsoupias received his bachelor's degree in electrical engineering from the National Technical University of Athens and his doctorate in computer science in 1994 from the University of California, San Diego under the supervision of Christos Papadimitriou. He subsequently taught at the University of California, Los Angeles, the University of Athens, and is now a professor at the University of Oxford.
Career
In 2012, he was one of the recipients of the Gödel Prize for his contributions to algorithmic game theory, specifically the introduction of the price of anarchy concept with Papadimitriou in the paper 'Worst-case equilibria'. His work has also spanned complexity theory, design and analysis of algorithms, online algorithms, networks, uncertainty decisions and mathematical economics. In 2019, he gave a lecture on game theory at CERN.
In 2016, Koutsoupias worked with Aggelos Kiayias and Maria Kyropoulou on the paper "Blockchain Mining Games". He contributed aspects of game theory for stake pools in the Ouroboros consensus protocol. This was used in the Cardano blockchain, and Koutsoupias became a senior research fellow at IOHK, the blockchain engineering company developing Cardano.
Selected publications
References
External links
Homepage
Greek computer scientists
Year of birth missing (living people)
Living people
National and Kapodistrian University of Athens alumni
Greek expatriates in the United States
Academics of the University of Oxford
University of California, Los Angeles faculty
National Technical University of Athens alumni
University of California, San Diego alumni
Game theorists
Gödel Prize laureates
People associated with Cardano
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,580
|
\section{Introduction}
\label{introduction}
The Standard model of the electroweak and strong interactions (extended by the inclusion of the massive
neutrinos) fits well all the existing experimental data. It assumes around 25 parameters and constraints,
the origin of which is not yet understood. Questions like: Why has Nature chosen
$SU(3)\times SU(2)\times U(1)$ to describe the charges of spinors and $SO(1,3)$ to describe
the spin of spinors?,
Why are the left handed spinors weak charged, while the right handed spinors are weak chargeless?,
Where do the Yukawa couplings (together with the weak scale and the families of quarks and leptons)
come from?, and many others, remain unanswered.
The advantage of the approach, unifying spins and
charges\cite{norma92,norma93,normasuper94,norma95,norma97,%
pikanormaproceedings1,holgernorma00,norma01,pikanormaproceedings2,Portoroz03}, is that
it might offer possible
answers to the open questions of the Standard electroweak model. We demonstrated in
references\cite{pikanormaproceedings1,%
norma01,pikanormaproceedings2,Portoroz03} that a left handed $SO(1,13)$
Weyl spinor multiplet includes, if the representation is interpreted
in terms of the subgroups $SO(1,3)$, $SU(2)$, $SU(3)$ and the sum of the two $U(1)$'s,
all the spinors of
the Standard model - that is the left handed $SU(2)$ doublets and the right handed $SU(2)$
singlets of (with the group $SU(3)$ charged) quarks and (chargeless) leptons.
Right handed neutrinos - weak and hyper chargeless - are also included.
In the gauge theory of gravity (in our case in $d=(1+13)$-dimensional space),
the Poincar\' e group is gauged,
leading to spin connections and vielbeins, which determine the gravitational
field\cite{mil,norma93,norma01}.
There are vielbein fields and spin connection fields, which might manifest - after
the appropriate compactification
(or some other kind of making the rest of d-4 space unobservable at
low energies) - in the four dimensional
space-time as all the gauge fields of the known charges, as well as the Yukawa couplings within
each family. No additional Higgs field is needed to generate masses of families and
to ''dress'' the right handed spinors with the weak charge. It is a part of the
starting Lagrangean in $d\ge1+13$, which manifests in $d=1+3$ as the Yukawa coupling and does
what the Higgs does in the Standard model.
If assuming a second kind of the Clifford algebra objects,
the corresponding gauge fields manifest as the Yukawa couplings among families
(contributing also to
Yukawa couplings within each family).
In the refs.\cite{pikanormaproceedings2,Portoroz03
holgernorma02,technique03,astridragannorma,bmnBled04} it was shown,
that the approach unifying spins and charges might explain the Yukawa couplings if
an appropriate break of both symmetries, connected with the two kinds of the
Clifford algebra objects, appears.
An even number of families is predicted, in particular,
the fourth family of quarks and leptons might appear under certain conditions at low energies
in agreement with\cite{okun}.
The approach seems to have, like all the Kaluza-Klein-like theories, a very
serious disadvantage, namely that there might not exist any massless,
mass protected spinors, which are, after the break of symmetries,
chirally coupled to the desired (Kaluza-Klein) gauge fields\cite{witten}.
This would mean that there are no observable spinors at low energies.
Since the idea that it is only one internal degree of freedom -
the spin - and the Kaluza-Klein idea that the gravity is the only gauge field,
are beautiful and attractive,
we have tried hard to find any exampl
, which would give hope to Kaluza-Klein-like
theories by demonstrating that a kind of a break of symmetries leads to
massless, mass protected spinors,
chirally coupled to the Kaluza-Klein gauge fields, observable at low energies.
We discuss in ref.\cite{holgernorma05} such a case - a toy model of a spinor,
living in $d(=1+5)$-dimensional space, which breaks into a finite disk with
the boundary, which allows spinors of only one handedness.
Although not yet realistic, the toy model looks promising.
In the present paper we analyze how do families of quarks and leptons,
and accordingly also the Yukawa couplings, appear within the
approach unifying spins and charges.
We comment on the type of
contributions to the Yukawa couplings and discuss some general properties
of the mass matrices, which follow from the assumptions of the approach,
trying to find out whether the approach could show a possible answer to the questions:
What is the origin of the families of quarks and leptons?, What does determine the Yukawa couplings?,
Why only the left handed quarks and leptons carry the weak charge?.
Since we do not know, which way of breaking the starting symmetries of the approach is the appropriate
one and since results of the investigation drastically depend on the way of breaking symmetries
and might as well depend on non adiabatic processes following the break of symmetries,
this paper (and also the paper which follows this one and represent some numerical
investigations) can only be understood as an attempt to see whether the
approach unifying spins and
charges has a chance to explain the origin of families of quarks and leptons and their
properties and to which extend might it help to understand the appearance of families and the
Yukawa couplings.
We are not (yet) performing the calculations of breaking the symmetry
$SO(1,13)$ to $SO(1,7) \times U(1) \times SU(3)$ within our approach. (Some very rough
estimates can be found in ref.\cite{hnrunBled02}.) The break of symmetries
influences both kinds of gauge fields, although we can not yet tell indeed in which way.
Therefore, we can not tell the strength of
the fields which appear in the Yukawa couplings as "the vacuum expectation values"
and which lead further to
$SO(1,3) \times U(1) \times SU(3)$. We only can evaluate (after making some assumptions) several
relations among the spin connection fields. Using then these very preliminary relations and
the known experimental data,
we can make a prediction for the number of families at "physical energies" and discuss
properties of quarks and leptons within this approach: their masses and mixing matrices.
Accordingly the results can be taken only as a first step in analyzing properties of
families of quarks and leptons within {\em the approach unifying spins and charges,
which might offers a mechanism for generating families and
correspondingly the Yukawa couplings.}
We shall present some numerical results in the paper, following
this one.
In Sect.\ref{lagrangesec} of this paper we present the action for a Weyl spinor in
$(1+13)$-dimensional space
within our approach and suggest a break of the symmetry $SO(1,13)$
to $SO(1,7) \times SO(6)$ and further. We assume that the break of $SO(1,13)$ to
$SO(1,7) \times U(1) \times SU(3)$
does lead to massless Weyl spinors with the $U(1)\times SU(3)$ charges.
The main point of this paper is to demonstrate that while
one Weyl spinor representation of $SO(1,13)$, if analyzed with respect to subgroups
$SO(1,3)\times SU(2)\times
U(1)\times U(1)\times SU(3)$, contains all the spinors needed in the Standard model
(the right handed weak chargeless quarks and leptons
and left handed weak charged quarks and leptons), {\it the starting action for a Weyl spinor,
which carries
only (two kinds of) the spin} {\it and no charges and interacts with only the gravitational field,
includes the Yukawa couplings}, which transform
the right handed weak chargeless spinors into left handed weak charged spinors and contribute to the
mass terms just as it is suggested by the Standard model, without assuming the existence
of a Higgs weak charged doublet. There are, namely, the generators of the Lorentz transformations
within the group $SO(1,7)$ in our model ($S^{0s},\;s=7,8$, for example), which take care of what
in the Standard model the Higgs doublet, together with $\gamma^0$, does.
In Subsect.\ref{break} we comment on a possible break of the starting symmetry $SO(1,13)$ the
internal symmetry of which is connected by $S^{ab}$, while in
Subsect.\ref{so1,13} of Sect.\ref{lagrangesec} we discuss properties of the group $SO(1,13)$ in
terms of subgroups, which appear in the Standard electroweak model.
In the same section, Subsect.\ref{technique}, we present briefly the
technique\cite{holgernorma02,technique03},
which turns out to be very helpful when discussing spinor representations, since it allows to
generate as well as present spinor representations and families of spinor representations
in a very transparent way. In particular, the technique
helps to point out very clearly how do the Yukawa couplings appear in our approach.
In Subsect.\ref{techniquefamilies} we comment on the
appearance of families within our technique\cite{norma93,technique03}.
In Sect.\ref{Yukawawithin} we discuss in details within our approach the appearance of
the Yukawa couplings within one family,
while in Sect.\ref{Yukawafamilies} we discuss the number of families as well as the Yukawa couplings
among the families.
In Sect.\ref{example} we present, after making several assumptions and simplifications,
a possible explicit expression for the mass matrices for four families of quarks and leptons
in terms of the spin connection fields.
\section{ Weyl spinors in $d= (1+13)$ manifesting families of quarks and leptons in $d= (1+3)$}
\label{lagrangesec}
We start with a left handed Weyl spinor in $(1+13)$-dimensional space. A spinor carries no charges,
only two kinds of spins and interacts accordingly with only gauge gravitational
fields
- with spin connections and vielbeins. We assume two kinds of the Clifford
algebra objects defining two kinds of the generators of the Lorentz algebra and allow accordingly
two kinds of gauge fields\cite{norma92,norma93,normasuper94,norma95,norma97,%
pikanormaproceedings1,holgernorma00,norma01,pikanormaproceedings2,Portoroz03}. One kind is the ordinary
gauge field (gauging the Poincar\' e symmetry in $d=1+13$). The corresponding spin connection field
appears for spinors as a gauge field of $S^{ab}= \frac{1}{4} (\gamma^a \gamma^b - \gamma^b \gamma^a)$,
where $\gamma^a$ are the ordinary Dirac operators.
The contribution of these fields to the mass matrices manifests in only the diagonal terms
(connecting right handed weak chargeless quarks or leptons with left handed weak charged partners within
one family of spinors).
The second kind of gauge fields is in our approach responsible for the appearance of families and
consequently for the Yukawa couplings
among families of spinors (contributing also to diagonal matrix elements) and will be used
in this paper to explain the origin of the families of quarks and leptons.
The corresponding spin connection fields appear for spinors as a gauge field
of $\tilde{S}^{ab}$ ($\tilde{S}^{ab} = \frac{1}{2} (\tilde{\gamma}^a \tilde{\gamma}^b-
\tilde{\gamma}^b \tilde{\gamma}^a)$) with $\tilde{\gamma}^a$, which are
the Clifford algebra objects\cite{norma93,technique03}, like $\gamma^a$, but anticommute with $\gamma^a$.
Accordingly we write the action for a Weyl (massless) spinor
in $d(=1+13)$ - dimensional space as follows\footnote{Latin indices
$a,b,..,m,n,..,s,t,..$ denote a tangent space (a flat index),
while Greek indices $\alpha, \beta,..,\mu, \nu,.. \sigma,\tau ..$ denote an Einstein
index (a curved index). Letters from the beginning of both the alphabets
indicate a general index ($a,b,c,..$ and $\alpha, \beta, \gamma,.. $ ),
from the middle of both the alphabets
the observed dimensions $0,1,2,3$ ($m,n,..$ and $\mu,\nu,..$), indices from the bottom of the alphabets
indicate the compactified dimensions ($s,t,..$ and $\sigma,\tau,..$). We assume the signature $\eta^{ab} =
diag\{1,-1,-1,\cdots,-1\}$.
}
\begin{eqnarray}
S &=& \int \; d^dx \; {\mathcal L
\nonumber\\
{\mathcal L} &=& \frac{1}{2} (E\bar{\psi}\gamma^a p_{0a} \psi) + h.c. = \frac{1}{2}
(E\bar{\psi} \gamma^a f^{\alpha}{}_a p_{0\alpha}\psi) + h.c. ,
\nonumber\\
p_{0\alpha} &=& p_{\alpha} - \frac{1}{2}S^{ab} \omega_{ab\alpha} -
\frac{1}{2}\tilde{S}^{ab} \tilde{\omega}_{ab\alpha}.
\label{lagrange}
\end{eqnarray}
Here $f^{\alpha}{}_a$ are vielbeins (inverse to the gauge field of the generators of translations
$e^{a}{}_{\alpha}$, $e^{a}{}_{\alpha} f^{\alpha}{}_{b} = \delta^{a}_{b}$,
$e^{a}{}_{\alpha} f^{\beta}{}_{a} = \delta_{\alpha}{}^{\beta}$),
with $E = det(e^{a}{}_{\alpha})$, while
$\omega_{ab\alpha}$ and $\tilde{\omega}_{ab\alpha} $ are the two kinds of the spin
connection fields, the gauge
fields of $S^{ab}$ and $\tilde{S}^{ab}$, respectively, corresponding to the two kinds of
the Clifford algebra
objects\cite{holgernorma02,Portoroz03}, namely $\gamma^a$ and $\tilde{\gamma}^{a}$, with the property
$\{\gamma^a,\tilde{\gamma}^b\}_+ =0$, which leads to
$\{ S^{ab}, \tilde{S}^{cd}\}_-=0$. We shall discuss the properties
of these two kinds of $\gamma^a$'s in Subsects.\ref{technique} and \ref{techniquefamilies}.
To see that one Weyl spinor in $d=(1+13)$ with the spin as the only internal degree of freedom, can
manifest in
four-dimensional ''physical'' space as the ordinary ($SO(1,3)$) spinor with all the known charges
of one family of quarks and leptons of the Standard model, one has to analyze one
Weyl spinor (we make a choice of the left handed one) representation in terms of the subgroups
$SO(1,3) \times
U(1) \times SU(2) \times SU(3)$. We shall do this in Subsect.\ref{so1,13} of this section.
(The reader can see this analyses in several references, like the one in\cite{Portoroz03}.)
To see that the Yukawa couplings are the part of the starting Lagrangean of Eq.(\ref{lagrange}), we rewrite
the Lagrangean in Eq.(\ref{lagrange}) as follows\cite{Portoroz03}
\begin{eqnarray}
{\mathcal L} &=& \bar{\psi}\gamma^{m} (p_{m}- \sum_{A,i}\; g^{A}\tau^{Ai} A^{Ai}_{m}) \psi
+ \nonumber\\
& & \sum_{s=7,8}\;
\bar{\psi} \gamma^{s} p_{0s} \; \psi + {\rm the \;rest}.
\label{yukawa}
\end{eqnarray}
Index $A$ determines the charge groups ($SU(3), SU(2)$ and the two $U(1)$'s), index $i$ determines
the generators within one charge group. $\tau^{Ai}$ denote the generators of the charge groups
(expressible\cite{norma01} in terms of $S^{st},\; s,t \in 5,6,..,14$), while $A^{Ai}_{m}, m=0,1,2,3,$
denote the corresponding
gauge fields (expressible in terms of $\omega_{st m}$).
The second term can be rewitten in terms of the kinetic part ($\psi^{\dagger} \gamma^0\gamma^s p_s \psi$)
and the part $ - \psi^{\dagger} \gamma^0\gamma^{s} S^{ t t'} f^{\sigma}_s \omega_{t t' \sigma} \psi -
\psi^{\dagger} \gamma^0 \gamma^{s} \tilde{S}^{t t'} f^{\sigma}_s \tilde{\omega}_{t t' \sigma} \psi $, which
looks like a mass term (also the kinetic term, if nonzero, contributes to the mass term), since
$f^{\sigma}_s \omega_{t t' \sigma}$ and $f^{\sigma}_s \tilde{\omega}_{t t' \sigma}, $
$ s,t \in 5,6,7,8,\; \sigma \in (5),(6),(7),(8))$, behave in $d(=1+3)-$
dimensional space like scalar fields, while the operator $\gamma^0\gamma^{s}, s=7,8$, for example,
transforms a right handed
weak chargeless spinor (for example $e_R$) into a left handed weak charged spinor (in this case to $e_L$),
without changing the spin in $d=1+3$ (Subsect.\ref{technique}, Eq.(\ref{graphgammaaction})
and the third and the fifth
row of Table II or the fourth and the sixth row of the same table) - just
what the Yukawa couplings with the Higgs doublet included do in the Standard model formulation.
The reader will
find the detail explanation in Subsects.\ref{technique},\ref{techniquefamilies}.
It should be pointed out that no Higgs weak charge doublet is needed here,
as $S^{0s}, s=7,8$ does its job.
One can always rewrite the Lagrangean from Eq.(\ref{lagrange}) in the way of Eq.(\ref{yukawa}).
The question is, of course, what are the terms, which are in Eq.(\ref{yukawa}) written
under ''the rest'' and whether they can be assumed as negligible at ''low energy world''.
We have no proof that any break of symmetry, presented in Subsect.\ref{break},
leads to such an effective Lagrangean, which would after the first break (or several successive breaks) of
the starting symmetry of $SO(1,13)$ manifest any massless spinors, which would then, after further breaks,
manifest in the "physical space" the masses corresponding to
the Yukawa couplings of Eq.(\ref{yukawa}), while all the rest terms are negligibly small.
We just assume instead, that we start with the Lagrangean of
Eq.(\ref{yukawa}) and then study properties of the system, described by such a Lagrange density
at ''physical'' energies.
We also would like to point out that the fact that the generators of families of spinors
$\tilde{S}^{ab}$ and the generators of the Lorentz transformations of spinors $S^{ab}$ commute
($\{\tilde{S}^{ab},S^{ab}\}_- =0 $), suggests that most of properties of
quarks and leptons must be the same within this approach. There are namely only the generators
of families, which define off diagonal elements of the Yukawa couplings. But they do not at all
distinguish among quarks and leptons. Since also in the diagonal matrix elements differ quarks
and leptons in only one parameter times the identity, the question arises:
What is then the reason for so different mixing matrices of
quarks and leptons as observed? Might it be that there are the nonperturbative effects
(like in the hadron case when quarks ''dress nonadiabatically''
into the clouds of quarks and antiquarks and the gluon field before forming a hadron) which are
responsible
for so different properties of quarks and leptons? Could instead be that very peculiar breaks
of symmetries cause the difference in off diagonal matrix elements for quarks and leptons? Or
one must take the appearance of the Majorana fermions into account?
The approach by itself gives different off diagonal
elements of mass matrices for $u$-quarks and $d$-quarks, and for $\nu$ and electrons (although
it still relates them).
We shall discuss this point later in this paper, as well as in the paper following this one.
\subsection{Break of symmetries}
\label{break}
There are several ways of breaking the group $SO(1,13)$ down to subgroups of the Standard model.
(One of) the most probable breaks, suggested by the approach unifying spins and charges, is the
following one
\[
\begin{array}{c}
\begin{array}{c}
\underbrace{%
\begin{array}{rrcll}
& & \mathrm{SO}(1,13) \\
& & \downarrow \\
& & \mathrm{SO}(1,7) \otimes \mathrm{SU}(3) \otimes U(1) \\
& & & \\
\end{array}} \\
\end{array}\\
\downarrow \\
\mathrm{SO}(1,3)\otimes\mathrm{U}(1)\otimes\mathrm{SU}(3)\\
\end{array}
\]
%
We start from a massless left handed Weyl spinor in $d=1+13$. We assume that the first break of symmetries
leads again to massless spinors in $d=1+7$, chirally coupled with the $SU(3)$ and $U(1)$ charge
to the corresponding fields, which follow from the spin connection and vielbein fields in $d=1+13$.
(The reader can find more about this kind of breaking the
starting symmetry in ref.\cite{hnrunBled02}.) We have no justification for such an assumption
(except that we have shown on one toy model\cite{holgernorma05} that in that very special case such an
assumption is justified). And we have no calculation, which would help to guess the strength of
the ''vacuum expectation values'' of the fields.
The Yukawa like terms themselves then break further the symmetry, ending up with the
''physical'' degrees of freedom. An additional non yet solved problem is, how does the break of
symmetries influences
the part $\tilde{S}^{st}\tilde{\omega}_{sts^,}$.
\subsection{Spin and charges of one left handed Weyl representation of SO(1,13)}
\label{so1,13}
We discuss in this subsection the properties of one Weyl spinor representation when
analysing the representation in terms of subgroups of the group $SO(1,13).$
The group $SO(1,13)$ of the rank $7$ has as possible subgroups the groups
$SO(1,3)$ (the ''complexified''
$SU(2) \times SU(2)$), $SU(2), SU(3)$ and the two $U(1)$'s, with the sum of the ranks of all
these subgroups equal to $7$. These subgroups
are candidates for describing the spin, the weak charge, the colour charge and the two hyper charges,
respectively (only one is needed in the Standard model).
The generators of these groups can be written in terms of the generators $S^{ab}$ as follows
\begin{eqnarray}
\tau^{Ai} = \sum_{a,b} \;c^{Ai}{ }_{ab} \; S^{ab},
\nonumber\\
\{\tau^{Ai}, \tau^{Bj}\}_- = i \delta^{AB} f^{Aijk} \tau^{Ak}.
\label{tau}
\end{eqnarray}
We could count the two $SU(2)$ subgroups of the group $SO(1,3)$ in the same way as the rest of subgroups.
Instead we shall
use $A=1,2,3,4,$ to represent only the subgroups describing charges and $f^{Aijk}$ to describe
the corresponding structure
constants. Coefficients $c^{Ai}{ }_{ab}$, with $a,b \in \{5,6,...,14\}$, have to be determined so that the
commutation relations of Eq.(\ref{tau})
hold\cite{norma97}.
The weak charge ($SU(2)$ with the generators $\tau^{1i}$) and one $ U(1)$ charge (with the generator
$\tau^{21}$) content of the compact group $SO(4)$
(a subgroup of $SO(1,13)$) can be demonstrated when expressing
\begin{eqnarray}
\tau^{11}: = \frac{1}{2} ( {\mathcal S}^{58} - {\mathcal S}^{67} ),\quad
\tau^{12}: = \frac{1}{2} ( {\mathcal S}^{57} + {\mathcal S}^{68} ),\quad
\tau^{13}: = \frac{1}{2} ( {\mathcal S}^{56} - {\mathcal S}^{78} ),
\nonumber\\
\tau^{21}: = \frac{1}{2} ( {\mathcal S}^{56} + {\mathcal S}^{78} ).
\label{su12w}
\end{eqnarray}
To see the colour charge and one additional $U(1)$ content in the group $SO(1,13)$ we write $\tau^{3i}$ and
$\tau^{41}$, respectively, in terms of the generators ${\mathcal S}^{ab}$
\begin{eqnarray}
\tau^{31}: &=& \frac{1}{2} ( {\mathcal S}^{9\;12} - {\mathcal S}^{10\;11} ),\quad
\tau^{32}: = \frac{1}{2} ( {\mathcal S}^{9\;11} + {\mathcal S}^{10\;12} ),\quad
\tau^{33}: = \frac{1}{2} ( {\mathcal S}^{9\;10} - {\mathcal S}^{11\;12} ),\quad
\nonumber
\\
\tau^{34}:&=& \frac{1}{2} ( {\mathcal S}^{9\;14} - {\mathcal S}^{10\;13} ),\quad
\tau^{35}: = \frac{1}{2} ( {\mathcal S}^{9\;13} + {\mathcal S}^{10\;14} ),\quad
\tau^{36}: = \frac{1}{2} ( {\mathcal S}^{11\;14} - {\mathcal S}^{12\;13}),\quad
\nonumber\\
\tau^{37}: &=& \frac{1}{2} ( {\mathcal S}^{11\;13} + {\mathcal S}^{12\;14} ),\quad
\tau^{38}: = \frac{1}{2\sqrt{3}} ( {\mathcal S}^{9\;10} + {\mathcal
S}^{11\;12} - 2{\mathcal S}^{13\;14}),
\nonumber\\
\tau^{41}: &=& -\frac{1}{3}( {\mathcal S}^{9\;10} + {\mathcal S}^{11\;12}
+ {\mathcal S}^{13\;14} ).
\label{su3u1so6}
\end{eqnarray}
To reproduce the Standard model groups one must introduce the two superpositions of the two $U(1)$'s
generators as follows
\begin{eqnarray}
Y = \tau^{41} + \tau^{21}, \quad Y' = \tau^{41} - \tau^{21}.
\label{yyprime}
\end{eqnarray}
The above choice of subgroups of the group $SO(1,13)$ manifests the Standard model charge structure
of one Weyl spinor of the
group $SO(1,13)$, with one additional hyper charge.
We may very similarly proceed also with the generators $\tilde{S}^{ab}$ by assuming that
a kind of a break makes the starting $SO(1,13)$ group to manifest in terms of some
$\tilde{\tau}^{\tilde{A}i}$ like
\begin{eqnarray}
\tilde{\tau}^{\tilde{A}i} = \sum_{a,b} \;\tilde{c}^{\tilde{A}i}{ }_{ab} \; \tilde{S}^{ab},
\nonumber\\
\{\tilde{\tau}^{\tilde{A}i}, \tilde{\tau}^{\tilde{B}j}\}_- = i \delta^{\tilde{A}\tilde{B}}
\tilde{f}^{\tilde{A}ijk} \tilde{\tau}^{\tilde{A}k}.
\label{tildetau}
\end{eqnarray}
We shall try to guess the way of breaking through the comparison of the results with the
experimental data in the paper following this one.
\subsection{Spinor representation in terms of Clifford algebra objects}
\label{technique}
In this subsection we briefly present our technique\cite{holgernorma02} for generating spinor representations
in any dimensional space. The advantage of this technique is simplicity in using it and transparency
in understanding detailed properties of spinor representations. We also show how families of spinors enter
into our approach\cite{norma93,technique03}.
We start by defining two kinds of the Clifford algebra objects, $\gamma^a$ and $\tilde{\gamma^a}$,
with the properties
\begin{eqnarray}
\{\gamma^a,\gamma^b\}_{+} = 2\eta^{ab} = \{\tilde{\gamma}^a,\tilde{\gamma}^b\}_{+},
\quad \{\gamma^a,\tilde{\gamma}^b\}_{+} = 0.
\label{clifford}
\end{eqnarray}
The operators $\tilde{\gamma}^a$ are introduced formally as operating on any Clifford algebra object $B$
from the left hand side, but they also can be expressed in terms of the ordinary $\gamma^a$ as
operating from the right hand side as follows
\begin{eqnarray}
\tilde{\gamma}^a B : = i(-)^{n_B} B \gamma^a,
\label{tildegclifford}
\end{eqnarray}
with $(-)^{n_B} = +1$ or $-1$, when the object $B$ has a Clifford even or odd character, respectively.
Accordingly two kinds of generators of the Lorentz transformations follow, namely
$S^{ab}: = (i/4) (\gamma^a \gamma^b - \gamma^b \gamma^a)$ and
$\tilde{S}^{ab}: = (i/4) (\tilde{\gamma}^a \tilde{\gamma}^b - \tilde{\gamma}^b \tilde{\gamma}^a)$, with the property
$\{ S^{ab},\tilde{S}^{cd}\}_{-}=0$.
We define a basis of spinor representations as
eigen states of the chosen Cartan subalgebra of the Lorentz algebra $SO(1,13)$, with the operators
$S^{ab}$ and $\tilde{S}^{ab}$ in the two Cartan subalgebra sets, with the same indices in both cases.
By introducing the notation
\begin{eqnarray}
\stackrel{ab}{(\pm i)}: &=& \frac{1}{2}(\gamma^a \mp \gamma^b), \quad
\stackrel{ab}{[\pm i]}: = \frac{1}{2}(1 \pm \gamma^a \gamma^b), \;{\rm for} \; \eta^{aa} \eta^{bb} =-1, \nonumber\\
\stackrel{ab}{(\pm )}: &= &\frac{1}{2}(\gamma^a \pm i \gamma^b), \quad
\stackrel{ab}{[\pm ]}: = \frac{1}{2}(1 \pm i\gamma^a \gamma^b), \;{\rm for} \; \eta^{aa} \eta^{bb} =1,
\label{eigensab}
\end{eqnarray}
it can be shown that
\begin{eqnarray}
S^{ab} \stackrel{ab}{(k)} &=& \frac{k}{2} \stackrel{ab}{(k)}, \quad
S^{ab} \stackrel{ab}{[k]} = \frac{k}{2} \stackrel{ab}{[k]}, \nonumber\\
\tilde{S}^{ab} \stackrel{ab}{(k)} &= & \frac{k}{2} \stackrel{ab}{(k)}, \quad
\tilde{S}^{ab} \stackrel{ab}{[k]} = - \frac{k}{2} \stackrel{ab}{[k]}.
\label{eigensabev}
\end{eqnarray}
The above binomials are all ''eigen vectors'' of the generators $S^{ab}$, as well as of $\tilde{S}^{ab}$.
We further find
\begin{eqnarray}
\gamma^a \stackrel{ab}{(k)}&=&\eta^{aa}\stackrel{ab}{[-k]},\quad
\gamma^b \stackrel{ab}{(k)}= -ik \stackrel{ab}{[-k]}, \nonumber\\
\gamma^a \stackrel{ab}{[k]}&=& \stackrel{ab}{(-k)},\quad \quad \quad
\gamma^b \stackrel{ab}{[k]}= -ik \eta^{aa} \stackrel{ab}{(-k)}
\label{graphgammaaction}
\end{eqnarray}
and
\begin{eqnarray}
\tilde{\gamma^a} \stackrel{ab}{(k)} &=& - i\eta^{aa}\stackrel{ab}{[k]},\quad
\tilde{\gamma^b} \stackrel{ab}{(k)} = - k \stackrel{ab}{[k]}, \nonumber\\
\tilde{\gamma^a} \stackrel{ab}{[k]} &=& \;\;i\stackrel{ab}{(k)},\quad \quad \quad
\tilde{\gamma^b} \stackrel{ab}{[k]} = -k \eta^{aa} \stackrel{ab}{(k)}.
\label{gammatilde}
\end{eqnarray}
Using the following useful relations
\begin{eqnarray}
\stackrel{ab}{(k)}^{\dagger}=\eta^{aa}\stackrel{ab}{(-k)},\quad
\stackrel{ab}{[k]}^{\dagger}= \stackrel{ab}{[k]},
\label{graphher}
\end{eqnarray}
we may define
\begin{eqnarray}
<\; \stackrel{ab}{(k)}^{\dagger} | \stackrel{ab}{(k)}\;>=1= <\;\stackrel{ab}{[k]}^{\dagger}
| \stackrel{ab}{[k]}\;>.
\label{scalar}
\end{eqnarray}
We shall later make use of the relations
\begin{eqnarray}
\stackrel{ab}{(k)}\stackrel{ab}{(k)}& =& 0, \quad \quad \stackrel{ab}{(k)}\stackrel{ab}{(-k)}
= \eta^{aa} \stackrel{ab}{[k]}, \quad
\stackrel{ab}{[k]}\stackrel{ab}{[k]} = \stackrel{ab}{[k]}, \quad \quad
\stackrel{ab}{[k]}\stackrel{ab}{[-k]}= 0,
\nonumber\\
\stackrel{ab}{(k)}\stackrel{ab}{[k]}& =& 0,\quad \quad \quad \stackrel{ab}{[k]}\stackrel{ab}{(k)}
= \stackrel{ab}{(k)}, \quad \quad
\stackrel{ab}{(k)}\stackrel{ab}{[-k]} = \stackrel{ab}{(k)},
\quad \quad \stackrel{ab}{[k]}\stackrel{ab}{(-k)} =0, \quad \quad
\label{graphbinoms}
\end{eqnarray}
as well as the relations, following from Eqs.(\ref{gammatilde},\ref{graphbinoms}),
\begin{eqnarray}
\stackrel{ab}{\tilde{(k)}} \stackrel{ab}{(k)}& =& 0,
\quad \quad \stackrel{ab}{\tilde{(-k)}} \stackrel{ab}{(k)}
= -i \eta^{aa} \stackrel{ab}{[k]},
\quad \stackrel{ab}{\tilde{(-k)}}\stackrel{ab}{[-k]}= i \stackrel{ab}{(-k)},\quad
\stackrel{ab}{\tilde{(k)}} \stackrel{ab}{[-k]} = 0, \nonumber\\
\stackrel{ab}{\tilde{(k)}} \stackrel{ab}{[k]}& =& i \stackrel{ab}{(k)}, \;\;
\stackrel{ab}{\tilde{(-k)}}\stackrel{ab}{[+k]}= 0, \;\;\quad \quad \quad \stackrel{ab}{\tilde{(-k)}}\stackrel{ab}{(-k)}=0,
\;\;\stackrel{ab}{\tilde{(k)}}\stackrel{ab}{(-k)} = -i \eta^{aa} \stackrel{ab}{[-k]}.
\label{graphbinomsfamilies}
\end{eqnarray}
Here
\begin{eqnarray}
\stackrel{ab}{\tilde{(\pm i)}} = \frac{1}{2} (\tilde{\gamma}^a \mp \tilde{\gamma}^b), \quad
\stackrel{ab}{\tilde{(\pm 1)}} = \frac{1}{2} (\tilde{\gamma}^a \pm i\tilde{\gamma}^b), \nonumber\\
\stackrel{ab}{\tilde{[\pm i]}} = \frac{1}{2} (1 \pm \tilde{\gamma}^a \tilde{\gamma}^b), \quad
\stackrel{ab}{\tilde{[\pm 1]}} = \frac{1}{2} (1 \pm i \tilde{\gamma}^a \tilde{\gamma}^b).
\label{deftildefun}
\end{eqnarray}
The reader should notice that $\gamma^a$'s transform the binomial
$\stackrel{ab}{(k)}$ into the binomial $\stackrel{ab}{[-k]}$,
whose ''eigen value'' with respect to $S^{ab}$ changes sign, while
$\tilde{\gamma}^a$'s transform the binomial $\stackrel{ab}{(k)}$
into $\stackrel{ab}{[k]}$ with unchanged ''eigen value''
with respect to $S^{ab}$.
We define the operators of handedness of the group $SO(1,13)$ and of the subgroups $SO(1,3), SO(1,7),
SO(6)$ and $SO(4)$ as follows
\begin{eqnarray}
\Gamma^{(1,13)} &=& i 2^{7} \; S^{03} S^{12} S^{56} \cdots S^{13 \; 14}, \quad
\Gamma^{(1,3)}\;= - i 2^2 S^{03} S^{12}, \nonumber \\
\Gamma^{(1,7)}\;&=& - i2^{4} S^{03} S^{12} S^{56} S^{78},\quad \quad \quad
\Gamma^{(1,9)}\; = i2^{5} S^{03} S^{12} S^{9\;10} S^{11\;12} S^{13 \; 14},\nonumber\\
\Gamma^{(6)}\;\;&=& - 2^3 S^{9 \;10} S^{11\;12} S^{13 \; 14},\quad\quad\;\;\;
\Gamma^{(4)}\;\;= 2^2 S^{56} S^{78}.
\label{handedness}
\end{eqnarray}
We shall represent one Weyl left handed spinor as products of binomials $\stackrel{ab}{(k)}$ or
$\stackrel{ab}{[k]}$, which are ''eigen vectors'' of the members of the Cartan subalgebra set.
We make the following choice of the Cartan subalgebra set of the algebra $S^{ab}$
\begin{eqnarray}
S^{03}, S^{12}, S^{56}, S^{78}, S^{9 \;10}, S^{11\;12}, S^{13\; 14}.
\label{cartan}
\end{eqnarray}
We are now prepared to make a choice of a starting basic vector of one Weyl representation of the group
$SO(1,13)$, which is the
eigen state of all the members of the Cartan subalgebra (Eq.(\ref{cartan})) and is left handed
($\Gamma^{(1,13)} =-1$)
\begin{eqnarray}
&&\stackrel{03}{(+i)}\stackrel{12}{(+)}|\stackrel{56}{(+)}\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{(-)}\stackrel{13\;14}{(-)} |\psi \rangle =
\nonumber\\
&&(\gamma^0 -\gamma^3)(\gamma^1 +i \gamma^2)| (\gamma^5 + i\gamma^6)(\gamma^7 +i \gamma^8)||
(\gamma^9 +i\gamma^{10})(\gamma^{11} -i \gamma^{12})(\gamma^{13}-i\gamma^{14})|\psi \rangle.
\nonumber\\
\label{start}
\end{eqnarray}
The signs "$|$" and "$||$" are to point out the $SO(1,3)$ (up to $|$), $SO(1,7)$ (up to $||$)
and $SO(6)$ (after $||$) substructure of the starting basic vector of the left handed multiplet of
$SO(1,13)$, which has $2^{14/2-1}= 64 $ vectors. Here $|\psi\rangle$ is any vector, which is not
transformed
to zero and therefore we shall not write down $|\psi \rangle$ any longer.
One easily finds that the eigen values of the chosen
Cartan subalgebra elements of $S^{ab}$ and $\tilde{S}^{ab}$ (Eq.(\ref{cartan}))
are $(+i/2, 1/2, 1/2,1/2,1/2,-1/2,-1/2)$
and $(+i/2, 1/2, 1/2,1/2,1/2,-1/2,-1/2)$,
respectively. This state has with respect to the operators $S^{ab}$ the following properties:
With respect to the group $SO(1,3)$ is a right handed spinor ($\Gamma^{(1,3)} =1$)
with spin up ($S^{12} =1/2$), it is weak chargeless (it is an $SU(2)$
singlet - $\tau^{13} = 0$) and it carries a colour charge (it is the member
of the $SU(3)$ triplet with ($\tau^{33} =1/2, \tau^{38} = 1/(2 \sqrt{3})$),
it has $\tau^{21} = 1/2$ and $\tau^{41}= 1/6$ and correspondingly the two hyper charges
equal to $Y=2/3$ and $Y'= -1/3$, respectively. We further find
according to Eq.(\ref{handedness}) that $\Gamma^{(4)} =1$ (the handedness of the group $SO(4)$, whose
subgroups are $SU(2)$ and $U(1)$),
$\Gamma^{(1,7)}= 1$ and $ \Gamma^{(6)} = -1$. The starting vector (Eq.(\ref{start})) can be recognized
in terms of the Standard model subgroups as the right handed weak chargeless $u$-quark carrying
one of the three colours.
To obtain all the basic vectors of one Weyl spinor, one only has to apply on the starting basic vector of Eq.(\ref{start})
the generators $S^{ab}$. All the quarks and the leptons of one family of the Standard model appear in this multiplet
(together with the corresponding anti quarks and anti leptons).
We present in Table I all the quarks of one particular colour (the right handed weak
chargeless $u_R,d_R$ and left handed weak charged $u_L, d_L$, with the colour $(1/2,1/(2\sqrt{3}))$ in
the Standard model notation). They all are members of one $SO(1,7)$ multiplet.
\begin{center}
\begin{tabular}{|r|c||c||c|c||c|c|c||c|c|c||r|r|}
\hline
i&$$&$|^a\psi_i>$&$\Gamma^{(1,3)}$&$ S^{12}$&$\Gamma^{(4)}$&
$\tau^{13}$&$\tau^{21}$&$\tau^{33}$&$\tau^{38}$&$\tau^{41}$&$Y$&$Y'$\\
\hline\hline
&& ${\rm Octet},\;\Gamma^{(1,7)} =1,\;\Gamma^{(6)} = -1,$&&&&&&&&&& \\
&& ${\rm of \; quarks}$&&&&&&&&&&\\
\hline\hline
1&$u_{R}^{c1}$&$\stackrel{03}{(+i)}\stackrel{12}{(+)}|\stackrel{56}{(+)}\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{(-)}\stackrel{13\;14}{(-)}$
&1&1/2&1&0&1/2&1/2&$1/(2\sqrt{3})$&1/6&2/3&-1/3\\
\hline
2&$u_{R}^{c1}$&$\stackrel{03}{[-i]}\stackrel{12}{[-]}|\stackrel{56}{(+)}\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{(-)}\stackrel{13\;14}{(-)}$
&1&-1/2&1&0&1/2&1/2&$1/(2\sqrt{3})$&1/6&2/3&-1/3\\
\hline
3&$d_{R}^{c1}$&$\stackrel{03}{(+i)}\stackrel{12}{(+)}|\stackrel{56}{[-]}\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{(-)}\stackrel{13\;14}{(-)}$
&1&1/2&1&0&-1/2&1/2&$1/(2\sqrt{3})$&1/6&-1/3&2/3\\
\hline
4&$d_{R}^{c1}$&$\stackrel{03}{[-i]}\stackrel{12}{[-]}|\stackrel{56}{[-]}\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{(-)}\stackrel{13\;14}{(-)}$
&1&-1/2&1&0&-1/2&1/2&$1/(2\sqrt{3})$&1/6&-1/3&2/3\\
\hline
5&$d_{L}^{c1}$&$\stackrel{03}{[-i]}\stackrel{12}{(+)}|\stackrel{56}{[-]}\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{(-)}\stackrel{13\;14}{(-)}$
&-1&1/2&-1&-1/2&0&1/2&$1/(2\sqrt{3})$&1/6&1/6&1/6\\
\hline
6&$d_{L}^{c1}$&$\stackrel{03}{(+i)}\stackrel{12}{[-]}|\stackrel{56}{[-]}\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{(-)}\stackrel{13\;14}{(-)}$
&-1&-1/2&-1&-1/2&0&1/2&$1/(2\sqrt{3})$&1/6&1/6&1/6\\
\hline
7&$u_{L}^{c1}$&$\stackrel{03}{[-i]}\stackrel{12}{(+)}|\stackrel{56}{(+)}\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{(-)}\stackrel{13\;14}{(-)}$
&-1&1/2&-1&1/2&0&1/2&$1/(2\sqrt{3})$&1/6&1/6&1/6\\
\hline
8&$u_{L}^{c1}$&$\stackrel{03}{(+i)}\stackrel{12}{[-]}|\stackrel{56}{(+)}\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{(-)}\stackrel{13\;14}{(-)}$
&-1&-1/2&-1&1/2&0&1/2&$1/(2\sqrt{3})$&1/6&1/6&1/6\\
\hline\hline
\end{tabular}
\end{center}
Table I. The 8-plet of quarks - the members of $SO(1,7)$ subgroup, belonging to one Weyl left
handed ($\Gamma^{(1,13)} = -1 = \Gamma^{(1,7)} \times \Gamma^{(6)}$) spinor representation of
$SO(1,13)$.
It contains the left handed weak charged quarks and the right handed weak chargeless quarks of a particular
colour ($(1/2,1/(2\sqrt{3}))$). Here $\Gamma^{(1,3)}$ defines the handedness in $(1+3)$ space,
$ S^{12}$ defines the ordinary spin (which can also be read directly from the basic vector),
$\tau^{13}$ defines the weak charge, $\tau^{21}$ defines the $U(1)$ charge, $\tau^{33}$ and
$\tau^{38}$ define the colour charge and $\tau^{41}$ another $U(1)$ charge, which together with the
first one defines $Y$ and $Y'$.
The reader can find the whole Weyl representation in the ref.\cite{Portoroz03}.
In Table II we present the leptons of one family of the Standard model. All the leptons belong
to the same multiplet with respect to the group $SO(1,7)$.
They are colour chargeless and differ accordingly from the quarks in Table I in the
second $U(1)$ charge and in the colour charge. The quarks and the leptons are
equivalent with respect to the group $SO(1,7)$.
\begin{center}
\begin{tabular}{|r|c||c||c|c||c|c|c||c|c|c||r|r|}
\hline
i&$$&$|^a\psi_i>$&$\Gamma^{(1,3)}$&$ S^{12}$&$\Gamma^{(4)}$&
$\tau^{13}$&$\tau^{21}$&$\tau^{33}$&$\tau^{38}$&$\tau^{41}$&$Y$&$Y'$\\
\hline\hline
&& ${\rm Octet},\;\Gamma^{(1,7)} =1,\;\Gamma^{(6)} = -1,$&&&&&&&&&& \\
&& ${\rm of \; leptons}$&&&&&&&&&&\\
\hline\hline
1&$\nu_{R}$&$\stackrel{03}{(+i)}\stackrel{12}{(+)}|\stackrel{56}{(+)}\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{[+]}\stackrel{13\;14}{[+]}$
&1&1/2&1&0&1/2&0&$0$&-1/2&0&-1\\
\hline
2&$\nu_{R}$&$\stackrel{03}{[-i]}\stackrel{12}{[-]}|\stackrel{56}{(+)}\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{[+]}\stackrel{13\;14}{[+]}$
&1&-1/2&1&0&1/2&0&$0$&-1/2&0&-1\\
\hline
3&$e_{R}$&$\stackrel{03}{(+i)}\stackrel{12}{(+)}|\stackrel{56}{[-]}\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{[+]}\stackrel{13\;14}{[+]}$
&1&1/2&1&0&-1/2&0&$0$&-1/2&-1&0\\
\hline
4&$e_{R}$&$\stackrel{03}{[-i]}\stackrel{12}{[-]}|\stackrel{56}{[-]}\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{[+]}\stackrel{13\;14}{[+]}$
&1&-1/2&1&0&-1/2&0&$0$&-1/2&-1&0\\
\hline
5&$e_{L}$&$\stackrel{03}{[-i]}\stackrel{12}{(+)}|\stackrel{56}{[-]}\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{[+]}\stackrel{13\;14}{[+]}$
&-1&1/2&-1&-1/2&0&0&$0$&-1/2&-1/2&-1/2\\
\hline
6&$e_{L}$&$\stackrel{03}{(+i)}\stackrel{12}{[-]}|\stackrel{56}{[-]}\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{[+]}\stackrel{13\;14}{[+]}$
&-1&-1/2&-1&-1/2&0&0&$0$&-1/2&-1/2&-1/2\\
\hline
7&$\nu_{L}$&$\stackrel{03}{[-i]}\stackrel{12}{(+)}|\stackrel{56}{(+)}\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{[+]}\stackrel{13\;14}{[+]}$
&-1&1/2&-1&1/2&0&0&$0$&-1/2&-1/2&-1/2\\
\hline
8&$\nu_{L}$&$\stackrel{03}{(+i)}\stackrel{12}{[-]}|\stackrel{56}{(+)}\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{[+]}\stackrel{13\;14}{[+]}$
&-1&-1/2&-1&1/2&0&0&$0$&-1/2&-1/2&-1/2\\
\hline\hline
\end{tabular}
\end{center}
Table II. The 8-plet of leptons - the members of $SO(1,7)$ subgroup, belonging to one Weyl left
handed ($\Gamma^{(1,13)} = -1 = \Gamma^{(1,7)} \times \Gamma^{(6)}$) spinor representation of
$SO(1,13)$, is presented.
It contains the left handed weak charged leptons and the right handed weak chargeless leptons, all
colour chargeless. The two $8$-plets in Table I and II are equivalent with respect to the groups
$SO(1,7)$. They only differ in properties with respect to the group $SU(3)$ and $U(1)$ and consequently
in $Y$ and $Y'$.
\subsection{Appearance of families}
\label{techniquefamilies}
While the generators of the Lorentz group $S^{ab}$, with a pair of $(ab)$,
which does not belong
to the Cartan subalgebra (Eq.(\ref{cartan})), transform one vector of one Weyl
representation into another vector of the same Weyl representation,
transform the generators $\tilde{S}^{ab}$ (again if the pair $(ab)$ does not belong to the Cartan set)
a member of one family into the same member of another family, leaving all the other quantum numbers
(determined by $S^{ab}$)
unchanged\cite{norma92,norma93,norma95,norma01,holgernorma00,Portoroz03}.
This is happening since the application of $\gamma^a$ (from the left) changes the operator
$\stackrel{ab}{(+)}$ (or the operator $\stackrel{ab}{(+i)}$)
into the operator $\stackrel{ab}{[-]}$ (or the operator $\stackrel{ab}{[-i]}$, respectively),
while the operator
$\tilde{\gamma}^a$ (which is understood, up to a factor $\pm i$, as the application of
$\gamma^a$ from the right hand side)
changes $\stackrel{ab}{(+)}$ (or $\stackrel{ab}{(+i)}$) into $\stackrel{ab}{[+]}$
(or into $\stackrel{ab}{[+i]}$, respectively), without changing the ''eigen values'' of the Cartan
subalgebra set of the operators $S^{ab}$. Bellow, as an example, the
application of $\tilde{S}^{01}$ on the
state of Eq.({\ref{start}}) (up to a constant) is presented:
\begin{eqnarray}
\stackrel{03}{(+i)} \stackrel{12}{(+)}| \stackrel{56}{(+)} \stackrel{78}{(+)}||
\stackrel{9 10}{(+)} \stackrel{11 12}{(-)} \stackrel{13 14}{(-)} \nonumber\\
\stackrel{03}{[+i]} \stackrel{12}{[+]}| \stackrel{56}{(+)} \stackrel{78}{(+)}||
\stackrel{9 10}{(+)} \stackrel{11 12}{(-)} \stackrel{13 14}{(-)}.
\label{twofamilies}
\end{eqnarray}
One can easily see that both vectors of (\ref{twofamilies}) describe a right handed $u$-quark of
the same colour. They are equivalent with respect to the operators $S^{ab}$. They only differ
in properties, determined by the operators $\tilde{S}^{ab}$ and accordingly also with respect to the
Cartan subalgebra set ($\tilde{S}^{03}, \tilde{S}^{12},\cdots,\tilde{S}^{13\;14}$). The first row vector
has the following ''eigen values'' of this Cartan subalgebra set
$(i/2,1/2,1/2,1/2,1/2,-1/2,-1/2)$, while the corresponding ''eigen values'' of the
vector in the second row
are $(-i/2,-1/2,1/2,1/2,1/2,-1/2,-1/2)$. Therefore, the operators $\tilde{S}^{ab}$
can be used to generate
families of quarks and leptons of the Standard model.
We present here some useful relations, concerning families
\begin{eqnarray}
\tilde{S}^{ac}\stackrel{ab}{(k)}\stackrel{cd}{(l)}& = &\frac{i}{2} \eta^{aa} \eta^{cc}
\stackrel{ab}{[k]}\stackrel{cd}{[l]}, \quad\;\;
\tilde{S}^{ad}\stackrel{ab}{(k)}\stackrel{cd}{(l)} = \frac{1}{2} l \eta^{aa}
\stackrel{ab}{[k]}\stackrel{cd}{[l]}, \nonumber\\
\tilde{S}^{bc}\stackrel{ab}{(k)}\stackrel{cd}{(l)} &=& \frac{1}{2} k \eta^{cc}
\stackrel{ab}{[k]}\stackrel{cd}{[l]},\quad \quad\;\;
\tilde{S}^{bd}\stackrel{ab}{(k)}\stackrel{cd}{(l)} = -\frac{i}{2} k l
\stackrel{ab}{[k]}\stackrel{cd}{[l]},
\nonumber\\
\tilde{S}^{ac}\stackrel{ab}{[k]}\stackrel{cd}{[l]}& = & -\frac{i}{2}
\stackrel{ab}{(k)}\stackrel{cd}{(l)}, \quad \quad \;\;\;\;\;
\tilde{S}^{ad}\stackrel{ab}{[k]}\stackrel{cd}{[l]} = \frac{1}{2} l \eta^{cc}
\stackrel{ab}{(k)}\stackrel{cd}{(l)}, \nonumber\\
\tilde{S}^{bc}\stackrel{ab}{[k]}\stackrel{cd}{[l]}& = & \frac{1}{2} k \eta^{aa}
\stackrel{ab}{(k)}\stackrel{cd}{(l)}, \quad \quad\;\;
\tilde{S}^{bd}\stackrel{ab}{[k]}\stackrel{cd}{[l]} = \frac{i}{2} kl \eta^{aa} \eta^{cc}
\stackrel{ab}{(k)}\stackrel{cd}{(l)},
\nonumber\\
\tilde{S}^{ac}\stackrel{ab}{(k)}\stackrel{cd}{[l]}& = & -\frac{i}{2} \eta^{aa}
\stackrel{ab}{[k]}\stackrel{cd}{(l)}, \quad \;\;\;\;
\tilde{S}^{ad}\stackrel{ab}{(k)}\stackrel{cd}{[l]} = \frac{1}{2} l \eta^{aa} \eta^{cc}
\stackrel{ab}{[k]}\stackrel{cd}{(l)}, \nonumber\\
\tilde{S}^{bc}\stackrel{ab}{(k)}\stackrel{cd}{[l]}& = & -\frac{1}{2} k
\stackrel{ab}{[k]}\stackrel{cd}{(l)}, \quad\quad \;\;\;\;
\tilde{S}^{bd}\stackrel{ab}{(k)}\stackrel{cd}{[l]} = -\frac{i}{2} k l \eta^{cc}
\stackrel{ab}{[k]}\stackrel{cd}{(l)},
\nonumber\\
\tilde{S}^{ac}\stackrel{ab}{[k]}\stackrel{cd}{(l)}& = &\frac{i}{2} \eta^{cc}
\stackrel{ab}{(k)}\stackrel{cd}{[k]}, \quad \quad \;\;\;\;
\tilde{S}^{ad}\stackrel{ab}{[k]}\stackrel{cd}{(l)} = \frac{1}{2} l
\stackrel{ab}{(k)}\stackrel{cd}{[k]},\nonumber\\
\tilde{S}^{bc}\stackrel{ab}{[k]}\stackrel{cd}{(l)}& = &- \frac{1}{2} k \eta^{aa} \eta^{cc}
\stackrel{ab}{(k)}\stackrel{cd}{[k]}, \;\;
\tilde{S}^{bd}\stackrel{ab}{[k]}\stackrel{cd}{(l)} = \frac{i}{2} k l \eta^{aa}
\stackrel{ab}{(k)}\stackrel{cd}{[k]}.
\label{tildesac}
\end{eqnarray}
\section{Mass matrices in the Approach unifying spins and charges - terms within each family}
\label{Yukawawithin}
We are now prepared to look at the terms, which manifest as the Yukawa couplings in our approach
unifying spins and charges. Let us at first neglect the terms $\tilde{S}^{ab}\tilde{\omega}_{abc}$
(Eqs.(\ref{lagrange},\ref{yukawa})) to which $\tilde{S}^{ab}$ contribute and see how does the ordinary
Poincar\' e gauging gravity contribute to
the Yukawa couplings. It contributes to the matrix elements within families only.
Let us look at the Lagrange density (Eq.(\ref{yukawa})) for one Weyl spinor of a
particular handedness - say the left handed one ($\Gamma^{(1+7)}=-1$) - and of all possible
$SU(3)\times U(1)$ charges.
The Lagrange density (Eq.(\ref{yukawa})) manifests couplings of a spinor with the colour,
the weak and the hyper charges
fields and it also manifests the mass term
\begin{eqnarray}
{\mathcal L} &=& \bar{\psi} \gamma^{m} (p_{m}- \sum_{A,i} \; g^{A}
\tau^{Ai} A^{Ai}_{m}) \; \psi + \nonumber\\
& & + \psi^+ \gamma^0 \gamma^s \;(p_s - \sum_{A,i} \; g^{A}
\tau^{Ai} A^{Ai}_{s} ) \psi
+ {\rm terms \; with}\; \tilde{S}^{ab}\tilde{\omega}_{abc} + {\rm the \; rest}.
\label{yukawa1}
\end{eqnarray}
We use the notation $f^{\alpha}{}_{a} p_{\alpha} = p_a$. Since for simplicity we assume that in the
''physical space'' there is no gravity, it follows: $f^{\mu}{}_{m} =
\delta^{\mu}{}_{m}$. We also (just) assume that the term of the type $\tilde{\tau}^{Ai} \tilde{A}^{Ai}_{m}) $
is negligible at "physical energies".
We recognize that the terms with $\gamma^0 \gamma^7$ or $\gamma^0 \gamma^8$ transform a right
handed weak chargeless spinors with the
spin $1/2$ (like it is the $u_R$ quark from the first row in Table I or the $e_R$ electron from the
third row in Table II) into a left handed weak charged spinors with the spin $1/2$ (in our example into
the $u_L$ quark from the seventh's row in Table I or the $e_L$ electron from the fifth's row in Table II).
We can rearrange the first term in the Lagrangean of Eq.(\ref{yukawa1}), with $m \in \{0,1,2,3\},$
to manifest the Standard model structure
\begin{eqnarray}
{\mathcal L}_{f} = \bar{\psi} \gamma^{m} \{p_{m} &-& \frac{g}{2} (\tau^{+} W^{+}_{m} +
\tau^{-} W^{-}_{m}) + \frac{g^2}{\sqrt{g^2 +g'^2}} Q' Z_{m} + \nonumber\\
&+& \frac{g g'}{\sqrt{g^2 +g'^2}} Q A_{m} +\nonumber\\
&+& \sum_{i} g^3 \tau^{3i} A^{3i}_{m} + \; g^{Y'} A^{Y'}_{m}\}\; \psi,
\label{lagrangesmf}
\end{eqnarray}
with
\begin{eqnarray}
Q &=& \tau^{13} + Y = S^{56} + \tau^{41}, \nonumber\\
Q' &=& \tau^{13} - (\frac{g'}{g})^2 Y = \frac{1}{2}
(1- (\frac{g'}{g})^2) S^{56} - \frac{1}{2}(1+ (\frac{g'}{g})^2) S^{78} - (\frac{g'}{g})^2 \tau^{41}.
\label{Q}
\end{eqnarray}
We assume that (due to an appropriate break of symmetries) the term $A^{Y'}_{m} $ is non observable at
''physical'' energies (not yet).
We rearrange the mass term of the Lagrange density of Eq.(\ref{yukawa1})
in a similar way as the ''dynamical'' part of the Lagrange density in the ''physical space''
(the first term of this equation),
so that the fields $A^{Ai}_{s}, s\in \{5,6,7,8\}$,
instead of $\omega_{t t' s} $, appear.
The charge $Q$ is conserved, as seen in Eq.(\ref{Q}), if we assume that no terms with
either $\gamma^5$ or $\gamma^6$ or $\tau^{3i}$
contribute to the mass term.
Since all the operators in Eq.(\ref{yukawa1}) are to be applied on right handed spinors,
which are weak chargeless objects (as seen from Table I and II), the part with $\sum_{i} \;
\tau^{1i} A^{1i}_{s} $ contributes zero to mass matrices and we shall leave it out.
We also expect, that at the ''observable'' energies the contribution of
the components $p_s$ of momenta, with $s=5,6,\cdots, $
are negligible. Accordingly we neglect also this term.
What then stays in the Lagrange density for the mass terms of spinors if the terms with
$\tilde{S}^{ab}$ are not yet taken into account, is as follows\footnote{We shall from now on simplify
the notation
from $g^{A} A^{Ai}_{a}$ to $A^{Ai}_{a}$.}
\begin{eqnarray}
- {\mathcal L}_{Y} &=& - \psi^+ \gamma^0 \sum_{s=7,8} \gamma^s \; (
Y A^{Y}_{s} +
Y' A^{Y'}_{s} ) \psi \nonumber\\
&+& {\rm terms \; with}\; \tilde{S}^{ab}\tilde{\omega}_{abc}.
\label{yukawa3}
\end{eqnarray}
These terms distinguish among the spinors: they are different for quarks than for leptons,
as well as different
for the
$u$ quarks than for the $d$ quarks and different for the electrons than for the neutrinos,
since according to Table I and II,
different spinors carry different values of the two hyper charges.
The first two terms in Eq.(\ref{yukawa3})
contribute to mass matrices within one family only.
The expression for the mass terms (the Yukawa couplings in the Standard model language)
within one family (Eq.(\ref{yukawa3})) can be further rewritten,
if introducing the following superposition of operators
\begin{eqnarray}
(\gamma^7 \pm i \gamma^8 ) = 2 \stackrel{78}{(\pm)}.
\label{pm}
\end{eqnarray}
It then follows
\begin{eqnarray}
- {\mathcal L}_{Y} &=& - \psi^+ \gamma^0 \; \sum_{y=Y,Y'}\;
\{ \stackrel{78}{(+)} \; y A^{y}_{+}\; + \; \stackrel{78}{(-)} \; y A^{y}_{-} \}\psi \nonumber\\
&+& {\rm terms \; with}\; \tilde{S}^{ab}\tilde{\omega}_{abc},
\label{yukawa4}
\end{eqnarray}
with $A^{y}_{\pm} = - (A^{y}_{7} \mp i A^{y}_{8})$ and $y=Y,Y'$.
According to Eq.(\ref{graphbinoms}),
saying that
\begin{eqnarray}
\stackrel{78}{(+)} \stackrel{78}{(+)} &=& 0, \quad \quad \quad \stackrel{78}{(-)} \stackrel{78}{(+)} \;
= \; - \stackrel{78}{[-]}, \nonumber\\
\stackrel{78}{(+)} \stackrel{78}{[-]} &=& \stackrel{78}{(+)}, \quad \quad \;\stackrel{78}{(-)}
\stackrel{78}{[-]} \; = \; 0,
\label{graphbinomssel}
\end{eqnarray}
we conclude, after reading also Table I and Table II, that the term with the
fields $A^{y}_{+}$, $y=Y, Y'$,
contributes only to the masses of the $d-$quarks and the electrons,
while $A^{y}_{-}$, $y=Y, Y'$, contributes
only to the masses of the $u-$quarks and the neutrinos.
\section{Mass matrices in the Approach unifying spins and charges - terms within and among families}
\label{Yukawafamilies}
We have seen in Subsect.\ref{techniquefamilies} that while the operators $S^{ab}$ transform the
members of one Weyl representation among themselves, the operators $\tilde{S}^{ab}$ transform one
member of a family into the same member of another family, changing nothing but the family index.
Each spinor basic vector has accordingly two indices: one index tells to
which family a spinor belongs, another index tells which member of a particular family a spinor
represents.
There are two types of terms in the Lagrange density of Eq.(\ref{lagrange}), which contribute to the
mass matrices. We have studied in the previous Sect.\ref{Yukawawithin} only the terms, determined by
the generators of the Lorentz group ($S^{ab}$) and the corresponding gauge fields. After making a few
assumptions we ended up with quite a
simple expression for the contribution to {\em the masses of quarks and leptons within one family}.
The assumption, that there are two kinds of gauge fields connected with two kinds of the generators
of the Lorentz transformations, is new\cite{norma93,holgernorma00,norma01,pikanormaproceedings2} and
requires accordingly additional cautions, when using it. On the other hand, the idea of the
existence of {\em two kinds of Clifford algebra objects
leads to the concept of families} and is therefore too exciting
not to be used to try to describe families of quarks and leptons
and to see what can the approach say about families of quarks and leptons.
As already said, the two kinds of generators are in our approach accompanied by the
two kinds of gauge fields,
gauging $S^{ab}$ and $\tilde{S}^{ab}$, respectively.
We shall assume that breaks of the symmetries of
the Poincar\' e group in $d=1+13$ influence this additional spin connection field as well.
Since we do not know, how these breaks could occur for any of these two kinds of degrees of freedom,
the calculations which will follow can be understood only as an attempt to study these degrees
of freedom, that is families of quarks and leptons, within the proposed approach and
and not yet as a severe prediction of properties of families of this approach.
On the other side, many a property of families, presented in this paper, not connected
with the break of symmetries, can follow from any concept of
construction of families, if operators for generating families
commute with the generators of spins and charges and if the generators are accompanied by
gauge fields. What our proposal might offer
in addition to the general concept of families is the prediction of the number of families and
possible relations among matrix elements of mass matrices due to connected effects followed by
breaks of symmetries.
Our approach suggests an even number of families. Namely: The number of all the orthogonal
basic states is in our approach for a particular $d$ equal to $2^d$. Since we start
with one Weyl spinor and neither $S^{ab}$ nor $\tilde{S}^{ab}$ change
the Clifford oddness or evenness of basic states, we stay with $2^{d-2}$ basic states. Each Weyl
representation has $2^{d/2-1}$ members. For any $d$ there are accordingly $2^{d/2-1}$ families.
After the assumption that some spontaneous breaks of symmetries lead to massless spinors in $d=(1+7)$-
dimensional space, in which spinors carry the $SU(3)$ and $U(1)$ charges (the two $U(1)$'s, one from
$SO(6)$ and another from $SO(1,7)$, being connected), there are accordingly only
$2^{8/2-1} = 8$ families at most, seen at low energies. The Yukawa-like terms further break symmetry,
leading to $Q= S^{56} + \tau^{41}$ and $SO(1,3), SU(3)$ as conserved quantities and to massive spinors
in the ''physical'' space with $d=(1+3)$.
Since the application of $\tilde{S}^{ab}$ generates the equivalent representations with respect to the
Lorentz group ($\{S^{ab},\tilde{S}^{cd}\}_{-}
= 0$ and accordingly $ \tilde{S}^{ab} \stackrel{ab}{(k)} = \tilde{S}^{ab} \stackrel{ab}{[-k]}$), it
follows that if we know properties with respect to $\tilde{S}^{ab}$ for one of basic states of
a Weyl spinor,
we know them for the whole Weyl spinor - {\em up to the influence of the break of symmetries} and
up to the fact that the contribution to the Yukawa couplings of the two kinds of generators are
in our approach related.
To demonstrate properties of families we shall make use of the first
state in Table I and Table II. The two tables differ only in the $SU(3)$ and $ U(1)$
charges (both kinds originate in $SO(6)$) and these two
charges do not concern the $SO(1,7)$ part.
Accordingly we shall tell only families, connected with $SO(1,7)$.
\begin{eqnarray}
I.\;\;& & \stackrel{03}{(+i)} \stackrel{12}{(+)} |\stackrel{56}{(+)} \stackrel{78}{(+)}||...\quad
V.\; \stackrel{03}{[+i]} \stackrel{12}{(+)} |\stackrel{56}{(+)} \stackrel{78}{[+]}||... \nonumber\\
II.\;& &\stackrel{03}{[+i]} \stackrel{12}{[+]} |\stackrel{56}{(+)} \stackrel{78}{(+)}||... \quad
VI. \stackrel{03}{(+i)} \stackrel{12}{[+]} |\stackrel{56}{(+)} \stackrel{78}{[+]}||... \nonumber\\
III.& &\stackrel{03}{(+i)} \stackrel{12}{(+)} |\stackrel{56}{[+]} \stackrel{78}{[+]}||...\quad
VII. \stackrel{03}{(+i)} \stackrel{12}{[+]} |\stackrel{56}{[+]} \stackrel{78}{(+)}||... \nonumber\\
IV.& &\stackrel{03}{[+i]} \stackrel{12}{[+]} |\stackrel{56}{[+]} \stackrel{78}{[+]}||...\quad
VIII.\stackrel{03}{[+i]} \stackrel{12}{(+)} |\stackrel{56}{[+]} \stackrel{78}{(+)}||.... .
\label{eightfamilies}
\end{eqnarray}
The diagonal terms, to which $S^{st}\omega_{sts'}$ contribute, depend on a state of the
Weyl representation,
as we have commented in Sect.\ref{Yukawawithin}, and distinguish between quarks and
leptons, as well as
between the $u$ and the $d$ quarks and between the electrons and the neutrinos.
The diagonal and the
off diagonal elements, to which $\tilde{S}^{ab} \tilde{\omega}_{abs}$ contribute,
distinguish between the $u$ and the $d$ quarks and between the electrons and the neutrinos, due to
the factor $\stackrel{78}{(\pm)}$ (determined by the ordinary $\gamma^a$ operators), but do not
distinguish between the quarks and the leptons. However, the way of breaking
symmetries might influence the commutation relations among
both kinds of generators.
(We could, in some rough estimation, take these effects into account for instance just by putting an
additional index to gauge
fields $\tilde{\omega}_{abc}$. We shall discuss such possibilities in the paper which will
present the numerical results.)
The diagonalization of the $u$ ($\nu$) mass matrix
leads accordingly to a different transformation
matrix than the diagonalization of $d$ ($e$) mass matrix. The mixing matrix
for quarks is correspondingly
not the unit matrix (as expected, if it should agree with the experimental data) but might not differ
from the mixing matrix for leptons.
Let us now write down the whole expression for the Yukawa couplings, with
$\tilde{S}^{ab} \tilde{\omega}_{ab \alpha}$
included
\begin{eqnarray}
- {\mathcal L}_{Y} &=& \psi^{\dagger} \gamma^0 \gamma^s p_{0 s} \psi\; = \psi^{\dagger} \; \gamma^0 \{
\stackrel{78}{(+)} \; p_{0+}\; + \; \stackrel{78}{(-)} \;p_{0-} \}\psi,
\label{yukawatildeyukawa}
\end{eqnarray}
with
\begin{eqnarray}
p_{0\pm} = (p_{7} \mp i \; p_{8}) - \frac{1}{2} S^{ab} \omega_{ab\pm} -
\frac{1}{2} \tilde{S}^{ab} \tilde{\omega}_{ab\pm}, \nonumber\\
\omega_{ab\pm} = \omega_{ab7} \mp i \; \omega_{ab8}, \quad
\tilde{\omega}_{ab\pm} = \tilde{\omega}_{ab 7} \mp i \; \tilde{\omega}_{ab8}.
\label{yukawatildeyukawadet}
\end{eqnarray}
We shall rewrite diagonal matrix elements, to which $\tilde{S}^{ab}$ contribute, in a similar way as
we did in the previous sections for the contribution of $S^{ab}$.
We therefore introduce the appropriate superposition of the operators
$\tilde{S}^{ab}$
\begin{eqnarray}
\tilde{N}^{\pm}_{3}: &=& \frac{1}{2} ( \tilde{{\mathcal S}}^{12} \pm i
\tilde{{\mathcal S}}^{03} ), \nonumber\\
\tilde{\tau}^{13} : &=& \frac{1}{2} ( \tilde{{\mathcal S}}^{56} - \tilde{{\mathcal S}}^{78} ), \nonumber\\
\tilde{Y} &=& \tilde{\tau}^{41} + \tilde \tau^{21}, \quad
\tilde{Y'} = \tilde{\tau}^{41} - \tilde{\tau}^{21}, \nonumber\\
\tilde{\tau}^{21}: &=& \frac{1}{2} ( \tilde{{\mathcal S}}^{56} + \tilde{{\mathcal S}}^{78} ), \quad
\tilde{\tau}^{41}: = -\frac{1}{3} ( \tilde{{\mathcal S}}^{9 \;10} + \tilde{{\mathcal S}}^{11\; 12}
+ \tilde{{\mathcal S}}^{13\; 14} ).
\label{tildetau}
\end{eqnarray}
We allow also terms with $\tilde{S}^{mn}$,$ m,n=0,1,2,3,$ which in diagonal matrix
elements of a mass matrix appear as $\tilde{N}^{\pm}_{3}$.
Taking into account that
\begin{eqnarray}
-\frac{1}{2} S^{st} \omega_{st\pm} &=& Y A^{Y}_{\pm} + Y' A^{Y'}_{\pm} + \tau^{13} A^{13}_{\pm}, \nonumber\\
-\frac{1}{2} \tilde{S}^{st} \tilde{\omega}_{st\pm} &=& \tilde{Y} \tilde{A}^{\tilde{Y}}_{\pm} +
\tilde{Y'} \tilde{A}^{\tilde{Y'}}_{\pm} + \tilde{\tau}^{13} \tilde{A}^{13}_{\pm}, \nonumber\\
-\frac{1}{2} \tilde{S}^{mn}\tilde{\omega}_{mn\pm} &=& \tilde{N}^{+3} \tilde{A}^{+3}_{\pm} +
\tilde{N}^{-3} \tilde{A}^{-3}_{\pm},
\label{tildetaufields}
\end{eqnarray}
with the pairs $(m,n) =(0,3),(1,2)$; $(s,t) = (5,6),(7,8),$ belonging to the Cartan sub algebra and
$\Omega_{\pm} = \Omega_7 \mp i \Omega_8$, where $\Omega_7, \Omega_8$
stay for any of the above fields, we find
\begin{eqnarray}
A^{13}_{\pm} &=& - (\omega_{56\pm} - \omega_{78 \pm}), \nonumber\\
A^{Y}_{\pm} &=& -\frac{1}{2} (A^{41}_{\pm} + (\omega_{56\pm} + \omega_{78\pm})),\nonumber\\
A^{Y'}_{\pm} &=& -\frac{1}{2} (A^{41}_{\pm} - (\omega_{56\pm} + \omega_{78\pm})), \nonumber\\
\tilde{A}^{13}_{\pm} &=& - (\tilde{\omega}_{56\pm} - \tilde{\omega}_{78 \pm}), \nonumber\\
\tilde{A}^{\tilde{Y}}_{\pm} &=& -\frac{1}{2} (\tilde{A}^{41}_{\pm} + (\tilde{\omega}_{56\pm}
+ \tilde{\omega}_{78\pm})),\nonumber\\
\tilde{A}^{\tilde{Y'}}_{\pm} &=& -\frac{1}{2} (\tilde{A}^{41}_{\pm} - (\tilde{\omega}_{56\pm} +
\tilde{\omega}_{78\pm})), \nonumber\\
\tilde{A}^{\tilde{N}^{+}_{3}}_{\pm} &=& - (\tilde{\omega}_{12 \pm} - i \; \tilde{\omega}_{03\pm}),\nonumber\\
\tilde{A}^{\tilde{N}^{-}_{3}}_{\pm} &=& - (\tilde{\omega}_{12 \pm} + i \; \tilde{\omega}_{03\pm}),
\label{tildetaufields1}
\end{eqnarray}
where the fields $A^{y}_{\pm},\; y= 13,41,Y,Y',$ and $\tilde{A}^{\tilde{y}}_{\pm},\; \tilde{y}
= \tilde{N}^{+}_{3},
\tilde{N}^{-}_{3}, 13, 41,
\tilde{Y},\tilde{Y'},$ are uniquely expressible with the
corresponding spin connection fields. Let us repeat that $\omega_{ab c} =
f^{\alpha}{}_{c} \;\omega_{ab \alpha}$ and $\tilde{\omega}_{ab c} = f^{\alpha}{}_{c} \;
\tilde{\omega}_{ab \alpha}$.
The operators, which contribute to non diagonal terms in mass matrices, are superpositions of
$\tilde{S}^{ab}$ and can be written in terms of nilpotents
\begin{eqnarray}
\stackrel{ab}{\tilde{(k)}}\stackrel{cd}{\tilde{(l)}},
\label{lowertilde}
\end{eqnarray}
with indices $(ab)$ and $(cd)$ which belong to the Cartan sub algebra indices (Eq.(\ref{cartan})).
We may write accordingly
\begin{eqnarray}
\sum_{(a,b) } -\frac{1}{2} \stackrel{78}{(\pm)}\tilde{S}^{ab} \tilde{\omega}_{ab\pm} =
- \sum_{(ac),(bd), \; k,l}\stackrel{78}{(\pm)}\stackrel{ac}{\tilde{(k)}}\stackrel{bd}{\tilde{(l)}}
\; \tilde{A}^{kl}_{\pm} ((ac),(bd)),
\label{lowertildeL}
\end{eqnarray}
where the pair $(a,b)$ in the first sum runs over all the indices, which do not characterise
the Cartan sub algebra, with $ a,b = 0,\dots, 8$, while the two pairs $(ac)$ and $(bd)$
denote only the Cartan sub algebra pairs
(for $SO(1,7)$ we only have the pairs $(03), (12)$; $(03), (56)$ ;$(03), (78)$;
$(12),(56)$; $(12), (78)$; $(56),(78)$ ); $k$ and $l$ run over four
possible values so that $k=\pm i$, if $(ac) = (03)$
and $k=\pm 1$ in all other cases, while $l=\pm 1$.
The fields $\tilde{A}^{kl}_{\pm} ((ac),(bd))$ can then
be expressed by $\tilde{\omega}_{ab \pm}$ as follows
\begin{eqnarray}
\tilde{A}^{++}_{\pm} ((ab),(cd)) &=& -\frac{i}{2} (\tilde{\omega}_{ac\pm} -\frac{i}{r} \tilde{\omega}_{bc\pm}
-i \tilde{\omega}_{ad\pm} -\frac{1}{r} \tilde{\omega}_{bd\pm} ), \nonumber\\
\tilde{A}^{--}_{\pm} ((ab),(cd)) &=& -\frac{i}{2} (\tilde{\omega}_{ac\pm} +\frac{i}{r} \tilde{\omega}_{bc\pm}
+i \tilde{\omega}_{ad\pm} -\frac{1}{r} \tilde{\omega}_{bd\pm} ),\nonumber\\
\tilde{A}^{-+}_{\pm} ((ab),(cd)) &=& -\frac{i}{2} (\tilde{\omega}_{ac\pm} + \frac{i}{r} \tilde{\omega}_{bc\pm}
-i \tilde{\omega}_{ad\pm} +\frac{1}{r} \tilde{\omega}_{bd\pm} ), \nonumber\\
\tilde{A}^{+-}_{\pm} ((ab),(cd)) &=& -\frac{i}{2} (\tilde{\omega}_{ac\pm} - \frac{i}{r} \tilde{\omega}_{bc\pm}
+i \tilde{\omega}_{ad\pm} +\frac{1}{r} \tilde{\omega}_{bd\pm} ),
\label{Awithomega}
\end{eqnarray}
with $r=i$, if $(ab) = (03)$ and $r=1$ otherwise.
We simplify the index $kl$ in the exponent
of fields $\tilde{A}^{kl}{}_{\pm} ((ac),(bd))$ to $\pm $, omitting $i$.
Any break of symmetries in the $\tilde{S}^{ab}$ sector would cause relations among the
corresponding $\tilde{\omega}_ {ab\pm}$. Namely, if $-\frac{1}{2}\tilde{S}^{ab}
\tilde{\omega}_{ ab\pm} = \tilde{\tau}^{Ai} \tilde{A}^{Ai}_{\pm}$ is not just
a unitary transformation of basic states, but means due to a break of symmetries that,
let us say, a particular $\tilde{A}^{A'i}=0$, then this can only happen, if
$\tilde{\omega}_{ab \pm}$ are related.
The Lagrange density, representing the mass matrices of fermions
(the Yukawa couplings in the Standard model)
(Eq.(\ref{yukawatildeyukawa})), can be rewritten as follows
\begin{eqnarray}
{\mathcal L}_{Y} = \psi^+ \gamma^0 \;
\{ & &\stackrel{78}{(+)} ( \sum_{y=Y,Y'}\; y A^{y}_{+} +
\sum_{\tilde{y}=\tilde{N}^{+}_{3},\tilde{N}^{-}_{3},\tilde{\tau}^{13},\tilde{Y},\tilde{Y'}}
\tilde{y} \tilde{A}^{\tilde{y}}_{+}\;)\; + \nonumber\\
& & \stackrel{78}{(-)} ( \sum_{y=Y,Y'}\;y A^{y}_{-} +
\sum_{\tilde{y}= \tilde{N}^{+}_{3},\tilde{N}^{-}_{3},\tilde{\tau}^{13},\tilde{Y},\tilde{Y'}}
\tilde{y} \tilde{A}^{\tilde{y}}_{-}\;) + \nonumber\\
& & \stackrel{78}{(+)} \sum_{\{(ac)(bd) \},k,l} \; \stackrel{ac}{\tilde{(k)}} \stackrel{bd}{\tilde{(l)}}
\tilde{{A}}^{kl}_{+}((ac),(bd)) \;\;+ \nonumber\\
& & \stackrel{78}{(-)} \sum_{\{(ac)(bd) \},k,l} \; \stackrel{ac}{\tilde{(k)}}\stackrel{bd}{\tilde{(l)}}
\tilde{{A}}^{kl}_{-}((ac),(bd))\}\psi,
\label{yukawa4tilde}
\end{eqnarray}
with pairs $((ac),(bd))$, which run over all the members of the Cartan sub algebra, while $k=\pm i,$ if
$ (ac) =(03)$, otherwise $k=\pm 1$ and $l= \pm 1$.
The terms $\tilde{{A}}^{kl}_{\pm}((ac),(bd))$ are expressible in terms of $\tilde{\omega}_{ab \pm}$
as presented in Eq.(\ref{Awithomega}), while any break of symmetries relates $\tilde{\omega}_{ab \pm}$
in a very particular way.
We omitted the term with $\tau^{13}$, as well as the terms
$p_{\pm}$, since, as already explained in the previous section, the first one when being applied
on the right handed spinors contributes zero, while for the second ones we assume that at
low energies their contribution is negligible.
The mass matrix, which follows from these Lagrange density and depends strongly on all possible breaks
of symmetries, is in general not Hermitean.
Let us now repeat the assumptions we have made up to now. They are either the starting assumptions
of our approach unifying spins and charges, or we made them to be able to connect the starting
Lagrange density at low energies with the observable phenomena.
a.i. We use the approach, unifying spins and charges, which assumes, that in $d=1+13$ massless
spinors carry two types of
spins: the ordinary (in $d=1 + 13$) one, which we describe by $S^{ab} =
\frac{1}{4}(\gamma^a \gamma^b - \gamma^b \gamma^a)$ and the additional one,
described by $\tilde{S}^{ab} =
\frac{1}{4}(\tilde{\gamma}^a \tilde{\gamma}^b - \tilde{\gamma}^b \tilde{\gamma}^a)$.
The two types of the
Clifford algebra objects anti commute ($\{\gamma^a, \tilde{\gamma}^b \}_+ =0$).
Spinors carry no charges in $d=1+13$.
The operators $S^{ab}$ determine (after an appropriate break of symmetries) at low energies the ordinary
spin in $d= 1+3$
and all the known charges, while $\tilde{S}^{ab}$ generate families of spinors. Accordingly spinors
interact with only the gravitational fields, the gauge fields of the Poincar\' e group ($p_{\alpha}$,
$S^{ab}$), and the gauge fields of the operators $\tilde{S}^{ab}$
($p_{0a} = f^{\alpha}{}_{a} - \frac{1}{2} (S^{cd} \omega_{cda} + \tilde{S}^{cd} \tilde{\omega}_{cda})).$
a.ii. The break of symmetries of $SO(1,13)$ into $SO(1,7)\times SU(3)\times U(1)$
occurs in a way that only massless spinors
in $d=1+7$ with the charge $ SU(3)$ and $ U(1)$ survive, with the one $U(1)$ from $SO(1,7)$
and the next $U(1)$
from $SO(6)$ aligned, while $S^{56}$ does not contribute to the Yukawa-like terms, so that
$Q= \tau^{41} + S^{56}$ is conserved in $d=1+3$.
a.iii. The break of symmetries influences both: the Poincar\' e symmetry and the symmetry
described by $\tilde{S}^{ab}$, it might be that to some extend in a similar way.
The study of both kinds of breaking
symmetries stays as an open problem.
a.iv. The terms which include $p_{s}, s = 5,..,14,$ do not
contribute at low energies.
\section{An example of mass matrices for four families}
\label{example}
Let us make, for simplicity, two further assumptions besides the four (a.i-a.iv.) ones,
presented at the end of Sect.\ref{Yukawafamilies}:
b.i. There are no terms, which would in Eq.(\ref{eightfamilies}) transform
$\stackrel{\tilde{56}}{(+)}$ into $\stackrel{\tilde{56}}{[+]}$.
This assumption (which could also be understood as a break of symmetry, which
requires that terms of the type $\tilde{S}^{5a}\tilde{\omega}_{5ab}$ and
$\tilde{S}^{6a}\tilde{\omega}_{6ab}$ are negligible and might be a part of
requirement a.iii.)
leaves us with only four families
of quarks and leptons. (This assumption might be justified with a break of symmetry
in the $\tilde{S}^{ab}$ sector from $SO(1,7)$ to $SO(1,5) \times U(1)$, with all
the contributions of the terms $\tilde{S}^{5a}\tilde{\omega}_{5ab}$ and
$\tilde{S}^{6a}\tilde{\omega}_{6ab}$ equal to zero.)
b.ii. The rough estimation will be done on ''a tree level''.
Since we do not know either how does the break of symmetries occur or how
does the break influence the
strength of the fields $\omega_{abc}$ and $\tilde{\omega}_{abc}$,
we can not really say, to which extend are the above assumptions justified. For
none of them we have a justification. Also the nonperturbative effects could be very
strong and the tree level might not mean a lot.
But yet a simplified version can help
us to understand to what conclusions might the proposed approach lead with respect to
families of quarks and leptons and their properties.
Our approach (which predicts an even number of families) suggests that under the assumptions a. and b.
there are the following four families of quarks and leptons
\begin{eqnarray}
I.\;& & \stackrel{03}{(+i)} \stackrel{12}{(+)} |\stackrel{56}{(+)} \stackrel{78}{(+)}||...\nonumber\\
II.\;& &\stackrel{03}{[+i]} \stackrel{12}{[+]} |\stackrel{56}{(+)} \stackrel{78}{(+)}||... \nonumber\\
III.& & \stackrel{03}{[+i]} \stackrel{12}{(+)} |\stackrel{56}{(+)} \stackrel{78}{[+]}||... \nonumber\\
IV. & & \stackrel{03}{(+i)} \stackrel{12}{[+]} |\stackrel{56}{(+)} \stackrel{78}{[+]}||... .
\label{fourfamilies}
\end{eqnarray}
We see from Table I (and II) that due to the properties of the nilpotents $\stackrel{78}{(\pm)}$
(Eq.\ref{graphbinoms}),
to the $u$ quark (and to the $\nu$ lepton) mass matrix only the operator $\stackrel{78}{(-)}$
(accompanied by the fields $A_-, \tilde{A}_-$)
contributes, while to the $d$ quark (and to the $e$ lepton) mass matrix only $\stackrel{78}{(+)}$
(accompanied by the fields $A_+, \tilde{A}_+$) contributes.
This means that {\em the off diagonal matrix elements of the Yukawa couplings are different
for $u$-quarks ($\nu$) and for $d$-quarks (e)}, although still related, while the quarks have
the same off diagonal matrix elements as the corresponding leptons (unless some breaks
of symmetries do not destroys this symmetry).
Assuming that after the appropriate breaks of symmetries the fields
contributing to the Yukawa couplings obtain some nonzero expectation values (which are in general
related in a very particular way) and
integrating the Lagrange density $L_Y$ over the coordinates and the internal (spin) degrees of freedom,
we end up
with the mass matrices for four families of quarks and leptons (Eq.(\ref{fourfamilies})), whose
structure is presented in Table III.
\begin{center}
\begin{tabular}{|r||c|c|c|c|}
\hline
$$&$ I_{R} $&$ II_{R} $&$ III_{R} $&$ IV_{R}$\\
\hline\hline
&&&& \\
$I_{L}$ & $ A^I_{\mp} $ & $ \tilde{A}^{++}_{\mp} ((03),(12)) $ & $ \pm \tilde{A}^{++}_{\mp} ((03),(78))$ &
$ \mp \tilde{A}^{++}_{\mp} ((12),(78))$ \\
&&&& \\
\hline
&&&& \\
$II_{L}$ & $ \tilde{A}^{--}_{\mp} ((03),(12)) $ & $ A^{II}_{\mp} $ & $ \pm \tilde{A}^{-+}_{\mp} ((12),(78))$ &
$ \mp \tilde{A}^{-+}_{\mp} ((03),(78))$ \\
&&&& \\
\hline
&&&& \\
$III_{L}$ & $ \pm \tilde{A}^{--}_{\mp} ((03),(78)) $ & $\mp \tilde{A}^{+-}_{\mp} ((12),(78)) $ & $ A^{III}_{\mp}$ &
$ \tilde{A}^{-+}_{\mp} ((03),(12))$ \\
&&&& \\
\hline
&&&& \\
$IV_{L}$ & $\pm \tilde{A}^{--}_{\mp} ((12),(78)) $ & $\mp \tilde{A}^{+-}_{\mp} ((03),(78)) $ &
$ \tilde{A}^{+-}_{\mp} ((03),(12))$ & $ A^{IV}_{\mp} $ \\
&&&& \\
\hline\hline
\end{tabular}
\end{center}
Table III. The mass matrices for four families of quarks and leptons in the approach unifying spins
and charges, obtained under the assumptions a.i.- a.iv. and b.i.- b.ii..
The values $ A^{I'}_{-}, I'= I,II,III,IV,$ and $ \tilde{A}^{lm}_{-} ((ac),(bd)); l,m = \pm,$ determine
matrix elements for the $u$ quarks and the neutrinos, the values $ A^{I'}_{+}, I'= I,II,III,IV,$ and $
\tilde{A}^{lm}_{+} ((ac),(bd)); l,m =\pm,$ determine the matrix elements for the $d$ quarks and
the electrons.
Diagonal matrix elements are different for quarks than for leptons and distinguish also between the $u$
and the $d$ quarks and between the $\nu$ and the $e$ leptons (Eqs.\ref{yukawa4tilde},
\ref{tildetaufields1}).
They also differ from family to family. Non diagonal matrix elements distinguish among families and
among $(u, \nu)$ and $(d,e)$.
The presented matrix should be understood as a very preliminary estimate of the mass matrices
of quarks and leptons.
The explicit forms of the diagonal matrix elements for the above choice of assumptions in terms of
$\omega_{abc}$,
$\tilde{\omega}_{abc}$ and $\tilde{A}^{41}_{\pm}$ is as follows
\begin{eqnarray}
A^{I}_{u} &=& \frac{2}{3} A^{Y}_{-} - \frac{1}{3} A^{Y'}_{-} + \tilde{\omega}^{I}_{-},
\nonumber\\
A^{I}_{\nu} &=& - A^{Y'}_{-} + \tilde{\omega}^{I}_{-},
\nonumber\\
A^{I}_{d} &=& -\frac{1}{3} A^{Y}_{+} + \frac{2}{3} A^{Y'}_{+} + \tilde{\omega}^{I}_{+},
\nonumber\\
A^{I}_{e} &=& - A^{Y}_{+} + \tilde{\omega}^{I}_{+},
\nonumber\\
A^{II}_{u} &= & A^{I}_{u} + (i \tilde{\omega}_{03-} + \tilde{\omega}_{12 -}), \quad
A^{II}_{\nu} = A^{I}_{\nu} + (i \tilde{\omega}_{03-} + \tilde{\omega}_{12 -}), \nonumber\\
A^{II}_{d} &= & A^{I}_{d} + (i \tilde{\omega}_{03+} + \tilde{\omega}_{12 +}), \quad
A^{II}_{e} = A^{I}_{e} + (i \tilde{\omega}_{03+} + \tilde{\omega}_{12 +}), \nonumber\\
A^{III}_{u} &= & A^{I}_{u} + (i \tilde{\omega}_{03-} + \tilde{\omega}_{78 -}), \quad
A^{III}_{\nu} = A^{I}_{\nu} + (i \tilde{\omega}_{03-} + \tilde{\omega}_{78 -}), \nonumber\\
A^{III}_{d} &= & A^{I}_{d} + (i \tilde{\omega}_{03+} + \tilde{\omega}_{78 +}), \quad
A^{III}_{e} = A^{I}_{e} + (i \tilde{\omega}_{03+} + \tilde{\omega}_{78 +}), \nonumber\\
A^{IV}_{u} &= & A^{I}_{u} + ( \tilde{\omega}_{12-} + \tilde{\omega}_{78 -}), \quad
A^{IV}_{\nu} = A^{I}_{\nu} + ( \tilde{\omega}_{12-} + \tilde{\omega}_{78 -}), \nonumber\\
A^{IV}_{d} &= & A^{I}_{d} + ( \tilde{\omega}_{12+} + \tilde{\omega}_{78 +}), \quad
A^{IV}_{e} = A^{I}_{e} + ( \tilde{\omega}_{12+} + \tilde{\omega}_{78 +}),
\label{diagmfour}
\end{eqnarray}
with $-
\tilde{\omega}^{I}{}_{\pm} = \frac{1}{2} (i \tilde{\omega}_{03\pm} +
\tilde{\omega}_{12\pm} +\tilde{\omega}_{56\pm} + \tilde{\omega}_{78\pm}
+ \frac{1}{3} \tilde{A}^{41}_{\pm})$.
The explicit forms of non diagonal matrix elements are written in Eq.(\ref{Awithomega}).
As allready stated, the break of symmetries, which is not taken into account in Table III,
would strongly relate ''vacuum expactation values'' of $\tilde{\omega}_{ab \pm}$.
To evaluate briefly the structure of mass matrices we make one further assumption:
b.iii. Let the mass matrices be real and symmetric (while all the $\omega_{abc}$ and
$\tilde{\omega}_{abc}$ are assumed to be real).
We then obtain for the $u$ quarks (and neutrinos) the mass matrices as presented in Table IV.
\begin{center}
\begin{tabular}{|r||c|c|c|c|}
\hline
$u$&$ I_{R} $&$ II_{R} $&$ III_{R} $&$ IV_{R}$\\
\hline\hline
&&&& \\
$I_{L}$ & $ A^I_{u} $ & $ \tilde{A}^{++}_{u} ((03),(12))= $ & $ \tilde{A}^{++}_{u} ((03),(78)) =$ &
$ - \tilde{A}^{++}_{u} ((12),(78)) = $ \\
$$&$$ &$ \frac{1}{2}(\tilde{\omega}_{327} +\tilde{\omega}_{018})$&$ \frac{1}{2}(\tilde{\omega}_{387} +\tilde{\omega}_{078}) $&$
\frac{1}{2}(\tilde{\omega}_{277} +\tilde{\omega}_{187})$ \\
&&&& \\
\hline
&&&&\\
$II_{L}$ & $ \tilde{A}^{--}_{u} ((03),(12))= $ & $ A^{II}_{u}= $ & $ \tilde{A}^{-+}_{u} ((12),(78)) = $ &
$ - \tilde{A}^{-+}_{u} ((03),(78)) = $ \\
$$&$ \frac{1}{2}(\tilde{\omega}_{327} +\tilde{\omega}_{018})$&$A^{I}_{u} + (\tilde{\omega}_{127} - \tilde{\omega}_{038})$
&$ -\frac{1}{2}(\tilde{\omega}_{277} -\tilde{\omega}_{187}) $&$ \frac{1}{2}(\tilde{\omega}_{387} - \tilde{\omega}_{078})$ \\
\hline
&&&& \\
$III_{L}$ & $ \tilde{A}^{--}_{u} ((03),(78)) =$ & $- \tilde{A}^{+-}_{u} ((12),(78))= $ & $ A^{III}_{u}=$ &
$ \tilde{A}^{-+}_{u} ((03),(12)) =$ \\
$$&$ \frac{1}{2}(\tilde{\omega}_{387} +\tilde{\omega}_{078})$&$ -\frac{1}{2}(\tilde{\omega}_{277} -\tilde{\omega}_{187})$
&$A^{I}_{u} + (\tilde{\omega}_{787} - \tilde{\omega}_{038})$&$ -\frac{1}{2}(\tilde{\omega}_{327} -\tilde{\omega}_{018}) $\\
&&&& \\
\hline
&&&& \\
$IV_{L}$ & $ \tilde{A}^{--}_{u} ((12),(78)) =$ & $- \tilde{A}^{+-}_{u} ((03),(78)) = $ &
$ \tilde{A}^{+-}_{u} ((03),(12))$ & $ A^{IV}_{u} =$ \\
$$&$ \frac{1}{2}(\tilde{\omega}_{277} +\tilde{\omega}_{187})$&$ \frac{1}{2}(\tilde{\omega}_{387} -\tilde{\omega}_{078})$
&$ -\frac{1}{2}(\tilde{\omega}_{327} -\tilde{\omega}_{018}) $&$A^{I}_{u} + (\tilde{\omega}_{127} + \tilde{\omega}_{787})$\\
&&&& \\
\hline\hline
\end{tabular}
\end{center}
Table IV. The mass matrix of four families of the $u$-quarks (and neutrinos) obtained within the approach
unifying spins and charges and under the assumptions a.i.-a.iv., b.i.-b.iii..
Neutrinos and $u$-quarks distinguish in $ A^I_{u} \ne A^I_{\nu} $.
The break of symmetries, not yet taken into account, would relate $\tilde{\omega}_{ab 7,8}$ and would
accordingly reduce the number of free parameters.
The corresponding mass matrix for $d$-quarks (and electrons) is presented in Table V.
\begin{center}
\begin{tabular}{|r||c|c|c|c|}
\hline
$d-$&$ I_{R} $&$ II_{R} $&$ III_{R} $&$ IV_{R}$\\
\hline\hline
&&&& \\
$I_{L}$ & $ A^I_{d} $ & $ \tilde{A}^{++}_{d} ((03),(12))= $ & $ - \tilde{A}^{++}_{d} ((03),(78)) =$ &
$ \tilde{A}^{++}_{d} ((12),(78)) = $ \\
$$&$$ &$ \frac{1}{2}(\tilde{\omega}_{327} - \tilde{\omega}_{018})$&$ -\frac{1}{2}(\tilde{\omega}_{387} -
\tilde{\omega}_{078}) $&$ -\frac{1}{2}(\tilde{\omega}_{277} +\tilde{\omega}_{187})$ \\
&&&& \\
\hline
&&&&\\
$II_{L}$ & $ \tilde{A}^{--}_{d} ((03),(12))= $ & $ A^{II}_{d}= $ & $ -\tilde{A}^{-+}_{d} ((12),(78)) = $ &
$ \tilde{A}^{-+}_{d} ((03),(78)) = $ \\
$$&$ \frac{1}{2}(\tilde{\omega}_{327} -\tilde{\omega}_{018})$&$A^{I}_{d} + (\tilde{\omega}_{127} + \tilde{\omega}_{038})$
&$ \frac{1}{2}(\tilde{\omega}_{277} -\tilde{\omega}_{187}) $&$ -\frac{1}{2}(\tilde{\omega}_{387} +\tilde{\omega}_{078})$ \\
\hline
&&&& \\
$III_{L}$ & $ - \tilde{A}^{--}_{d} ((03),(78)) =$ & $ \tilde{A}^{+-}_{d} ((12),(78))= $ & $ A^{III}_{d}=$ &
$ \tilde{A}^{-+}_{d} ((03),(12)) =$ \\
$$&$ -\frac{1}{2}(\tilde{\omega}_{387} -\tilde{\omega}_{078})$&$ \frac{1}{2}(\tilde{\omega}_{277} -\tilde{\omega}_{187})$
&$A^{I}_{d} + (\tilde{\omega}_{787} + \tilde{\omega}_{038})$&$ -\frac{1}{2}(\tilde{\omega}_{018} +\tilde{\omega}_{327}) $\\
&&&& \\
\hline
&&&& \\
$IV_{L}$ & $ -\tilde{A}^{--}_{d} ((12),(78)) =$ & $ \tilde{A}^{+-}_{d} ((03),(78)) = $ &
$ \tilde{A}^{+-}_{d} ((03),(12))$ & $ A^{IV}_{d} $ \\
$$&$ -\frac{1}{2}(\tilde{\omega}_{277} +\tilde{\omega}_{187})$&$ -\frac{1}{2}(\tilde{\omega}_{387} +\tilde{\omega}_{078})$
&$ -\frac{1}{2}(\tilde{\omega}_{018} +\tilde{\omega}_{327}) $&$A^{I}_{d} + (\tilde{\omega}_{127} + \tilde{\omega}_{787})$\\
&&&& \\
\hline\hline
\end{tabular}
\end{center}
Table V. The mass matrix of four families of the $d$-quarks and electrons. The
quarks and the leptons distinguish in this approximation in $ A^I_{d} \ne A^I_{e}$.
Other comments are the same as in Table IV.
The relation between the mass matrix of $u$-quarks and the mass matrix of neutrinos is,
under the assumptions and simplifications made during deriving both tables, as follows:
All the off-diagonal elements are for the neutrinos the same as for the $u$-quarks, while
the diagonal matrix elements depend on
the eigen values of $Y$ and $Y'$. Accordingly,
both $A^{I}_{\alpha},
\alpha =u,\nu,$ can be understood as independent parameters, expressible in terms of
$A^{Y}_{-}, A^{Y'}_{-}, $ and $
\tilde{\omega}^{I}{}_{-}$.
Similarly,
the relation between the mass matrix of $d$-quarks and the mass matrix for electrons is,
under the same assumptions and simplification as used for finding the expressions for the mass
matrices for the $u-$quarks and the neutrinos, as follows:
All the off-diagonal elements are the same for both - the $d$-quarks and the electrons, while
the diagonal matrix elements distinguish in
the eigen values of $Y$ and $Y'$.
Again, both $A^{I}_{\beta},
\beta =d,e,$ can be understood as independent parameters, expressible in terms of
$A^{Y}_{+}, A^{Y'}_{+}, $ and $
\tilde{\omega}^{I}{}_{+}$.
The requirement about reality and symmetry of
mass matrices, relates $A^{Y}_{+} = A^{Y}_{-}, A^{Y'}_{+} = A^{Y'}_{-},
\tilde{\omega}^{I}{}_{+} = \frac{1}{2} \tilde{\omega}_{03 8} + \tilde{\omega},
\tilde{\omega}^{I}{}_{-} = -\frac{1}{2} \tilde{\omega}_{03 8} + \tilde{\omega}$,
where $\tilde{\omega} = \tilde{\omega}_{12 7} +
\tilde{\omega}_{56 7} + \tilde{\omega}_{78 7}
+ \frac{1}{3} \tilde{A}^{41}$ and $\tilde{A}^{41}$ is the real part of either $\tilde{A}^{41}_{+}
$ or $\tilde{A}^{41}_{-} $. The same assumption relates also off diagonal
elements for $u$-quarks and $d$-quarks (or neutrinos and electrons),
as seen from both tables, so that there are $13$ free parameters, which determine $4\times 4(4+1)/2$ mass
matrix elements, and from these mass matrices $4 \times 4$ masses of quarks and leptons and
$2 (4(4+1)/2-1) $
elements of the two mixing matrices should follow.
Further break of symmetries would further
relate the $\tilde{\omega}_{ab \pm}$ fields, reducing strongly the number of free parameters on
Table IV and Table V.
A very peculiar boundary conditions could - when breaking symmetries - even cause differences
in off diagonal matrix elements of
quarks and leptons. Also could the nonperturbative effects beyond the ''tree level'' be responsible
for the differences observed in the measured properties of quarks and leptons or for
what in many references are trying
to achieve with additional Higgs fields. We did not take into account any Majorana fermions.
Most of the above assumptions were proposed to be able to make a rough
estimation of properties of the mass matrices,
predicted by the approach unifying spins and charges.
We shall present the calculations with the parameters presented in Tables IV and V in the paper,
which will follow this one.
\section{Concluding discussions}
\label{conclusions}
In this paper we discuss about a possible origin of the families of quarks and leptons and
of their Yukawa couplings as proposed by the approach unifying spins and
charges\cite{norma92,norma93,normasuper94,%
norma95,norma97,pikanormaproceedings1,holgernorma00,norma01,%
pikanormaproceedings2,Portoroz03}.
The approach assumes that a Weyl spinor of a chosen handedness carries in
$d (=1+13)-$ dimensional space nothing but two kinds of spin degrees of freedom. One kind belongs to
the Poincar\' e group in $d=1+13$,
another one generates families.
The idea of generating families with the second kind of the Clifford algebra objects
(which commute with the generators of the Lorentz transformations for spinors) is new, as it is
new also the idea that there are the generators of the Lorentz transformations (accompanied by the
spin connection fields in $d > 4$) which are (together with $ \gamma^0$)
responsible for the Yukawa couplings within a family,
transforming a right handed weak chargeless quark or lepton into a left handed weak charged
one.
Spinors interact with only the gravitational fields,
manifested by vielbeins and spin connections, the gauge fields of the momentum
$p_{\alpha}$ and the two kinds of the generators of the Lorentz group $S^{ab}$ and $\tilde{S}^{ab}$,
respectively.
To derive the mass matrices from the starting Lagrangean - that is to calculate the Yukawa couplings
of the Standard model - no additional (Higgs) field is needed. In order to make a simple and
transparent evaluation of properties of the mass matrices and consequently
some estimations and rough predictions for the
masses and mixing matrices for quarks
and leptons, observed at ''physical'' energies,
we made several assumptions, approximations and simplifications,
not necessary all of them are ''physical'' (and some of them
should soon be relaxed in further studies):
i. The break of symmetries of the group $SO(1,13)$ into $SO(1,7)\times SU(3)\times U(1)$ occurs
in a way that only
massless spinors in $d=1+7$ with the charge $ SU(3)\times U(1)$ survive. And yet
the two $U(1)$ charges,
following from $SO(6)$ and $SO(1,7)$, respectively, are related.
(Our work on the compactification
of a massless spinor in $d=1+5$ into $d=1+3$ and a finite disk gives us
some hope that this assumption
might be fulfilled\cite{holgernorma05}.)
The requirement that the terms
with $S^{5a}, S^{6a}$ do
not contribute to the
mass term at low energies, assures that the charge
$Q= \tau^{41} + S^{56}$ is conserved.
ii. The break of symmetries influences the Poincar\' e symmetry and the symmetries described by
$\tilde{S}^{ab}$. But it is assumed that the gauge symmetries connected with $\tilde{S}^{ab}$
do not manifest as gauge fields (additional to the known charge gauge fields) in $d=1+3$.
It is also assumed that there are no terms, which would in Eq.(\ref{eightfamilies}) transform
$\stackrel{\tilde{56}}{(+)}$ into $\stackrel{\tilde{56}}{[+]}$ (which can be explained by the
related break of symmetries in $S^{ab}$ and $\tilde{S}^{ab}$ sektor). This assumption reduces the
number of families by a factor $2$. It really means that the group $SO(1,7)$, whose generators are
$\tilde{S}^{ab}$, is broken into $SO(1,5)\times U(1)$ in a way that terms
$\tilde{S}^{5a}\tilde{\omega}_{5ab}$ and $\tilde{S}^{6a}\tilde{\omega}_{6ab}$ bring no
contribution to the mass matrices. Otherwise no additional break of symmetry was taken into account.
We also assume that terms which
include the components $p_s, s=5,..,14$ of the momentum $p^a$ do not
contribute at low energies to the mass matrices. We leave for further studies to find out
how do different ways of breaking symmetries of the Poincar\' e group in $d=1+13$ and the
$SO(1,13)$ group of $\tilde{S}^{ab}$ influence the mass matrices.
iii. We make estimations on a ''tree level''.
iv. We assume the mass matrices to be real and symmetric.
Our starting Weyl spinor representation of a chosen handedness in $d(=1+13)-$dimensional
space manifests, if analyzed in terms of the
subgroups $SO(1,3), SU(3), SU(2)$ and two $U(1)'$s (the sum of the ranks of the subgroups is the
same as the rank of the starting group) of the group $SO(1,13)$, the spin and all the charges
of one family of quarks and leptons. It includes left handed weak charged quarks and leptons
and right handed weak chargeless quarks and leptons in the same representation
and does accordingly {\em answer one of
the open questions of the Standard model: Why only the left handed fermions carry the
weak charge while the right handed ones are weak chargeless}, how can it at all happen
that handedness, which concerns only the spin (in $d=1+3$), is so strongly related to
a (weak) charge?
We use our technique\cite{holgernorma02,technique03} to present spinor representations
in a transparent way so that one easily sees how does a part of the covariant
derivative of a spinor in $d=1+13$ manifest in $d=1+3$ as Yukawa couplings.
We use the same technique to represent also families of spinors. Since the starting action in
$d={1+13}$ manifests in $d=1+3$ the (even number) of families and the Yukawa couplings,
it {\em offers a possible
answer to the questions, why families of quarks and leptons and
the corresponding Yukawa couplings manifest in nature}.
We found the off diagonal mass matrix elements of the quarks and the
leptons strongly related.
We expect that these relations might very probably turn out to be too strong
(since there are in our case the same off diagonal and diagonal matrix elements,
which determine the orthogonal rotations of the matrices for the $u-$quarks and neutrinos
and the $d-$quarks and electron into the diagonal forms and
consequently also the corresponding mixing matrices for quarks are the same as for leptons,
while the experimental data
shows quite a difference in the mixing matrices of quarks and leptons). We suspect that some
particular breaks of symmetries (with the help of very peculiar
boundary conditions added) might be responsible in our approach
for the differences between quarks and leptons
in off diagonal
and also those diagonal matrix elements, which are generated by the family generators.
Or might the reason for the difference be in the Majorana like neutrinos (as suspected in
many references), which are not treated in these
studies.
If the symmetry of mass matrices as presented in this paper for four families
breaks further, the relations among the parameters determining
the mass matrices follow, reducing the presented number of independent parameters
$\tilde{\omega}_{abcd}$.
In particular, an exact break of the symmetry of $SO(1,5)$ in the $\tilde{S}^{ab}$
sector into $SU(3)\times U(1)$
would manifest in decoupling of the fourth family from the first three (by
relating $\tilde{\omega}_{abc}$ so that the corresponding matrix elements would be zero),
while the break of
$SO(1,5)$ into $SU(2)\times SU(2)\times U(1)$ would manifest in decoupling of the first
two families from the second two families.
We treat the quarks and the leptons in an equivalent way, with no Majorana neutrinos included.
Not all the above assumptions and simplifications are needed in order
to be able to estimate
mass matrices of quarks and leptons with not too much effort. And in addition, it might happen
that this too simplified estimates lead to unrealistic conclusions. (In particular,
the $CP$ violation can under
the assumption v. hardly be possible.) One also can not expect, that ''a tree level'' estimate is
good enough to evaluate properties of quarks and leptons. Nonperturbative effects might
strongly influence the results and they might even be a very strong reason for the difference
in properties
of the families of the $u$-quarks, the $d$-quarks and both kinds of leptons.
Corrections bellow the tree level might bring also the contributions, which
several references try to simulate by more than one Higgs
field.
For the four predicted families of quarks and leptons we present the explicit expressions
for the mass matrices in the above mentioned approximations.
Since in our approach the generators of the Poincar\' e group and of those,
generating families, commute,
many a property of mass matrices, presented in this paper, would be true also for, let us say,
models, in which
the generators of the Poincar\' e group and those of generating families, commute.
However, in our case breaks of symmetries in the two sectors are related
and these relations might be very important for the properties of quarks and leptons.
On the other hand we should find the explanation why the additional gauge fields, connected
with the $\tilde{S}^{ab}$ sector does not manifest in $d=1+3$.
We shall present numerical estimates for the Yukawa couplings after
relating our results with the known experimental data in the paper\cite{matjazdragannorma},
following this one, together with further
discussions of the properties of families of quarks and leptons as following from
the approach unifying spins and charges, in order to be able to see
whether this approach shows the right way
beyond the Standard model of electroweak and colour interactions.
\section*{Acknowledgments}
It is a pleasure to thank all the participants of the workshops entitled
"What comes beyond the Standard model",
taking place at Bled annually in July, starting at 1998, for many very fruitful discussions,
in particular to H.B. Nielsen.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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namespace cpp_nbt {
class nbt_base {
public:
virtual ~nbt_base() = default;
virtual void swap(nbt_base & with) = 0;
nbt_base & operator=(const nbt_base & from);
virtual bool operator==(const nbt_base & to);
virtual void read(std::istream & from) = 0;
virtual void write(std::ostream & to) const = 0;
virtual unsigned char id() const = 0;
virtual std::unique_ptr<nbt_base> clone() const = 0;
};
}
namespace std {
void swap(cpp_nbt::nbt_base & lhs, cpp_nbt::nbt_base & rhs);
}
#endif // NBT_BASE_HPP
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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\section{Introduction}
Capability to observe the Sunyaev-Zel'dovich (SZ) effect has improved immensely in recent years. Dedicated instruments now produce high resolution images of single objects \citep[e.g.][]{kit04,halversonetal09,nor09} and moderately large samples of high--quality SZ measurements of previously--known clusters \citep[e.g.,][]{mro09,pla09}. In addition, large-scale surveys for clusters using the SZ effect are underway, both from space with the Planck mission \citep[]{2007NewAR..51..287V,2003NewAR..47.1017L} and from the ground with several dedicated telescopes, such as the South Pole Telescope \citep{car09} leading to the first discoveries of clusters solely through their SZ signal \citep{stanisetal09}. These results open the way for a better understanding of the $SZ-Mass$ relation and, ultimately, for cosmological studies with large SZ cluster catalogues.
The SZ effect probes the hot gas in the intracluster medium (ICM). Inverse Compton scattering of cosmic microwave background (CMB) photons by free electrons in the ICM creates a unique spectral distortion \citep{sun70,sun72} seen as a frequency--dependent change in the CMB surface brightness in the direction of galaxy clusters that can be written as $\Delta i_\nu(\hat{n}) = y(\hat{n}) j_\nu(x)$, where $j_\nu$ is a universal function of the dimensionless frequency $x=h\nu/kT_{\rm cmb}$. The Compton $y$--parameter is given by the integral of the electron pressure along the line--of--sight in the direction $\hat{n}$,
\begin{equation}
y = \int_{\hat{n}}\; \frac{kT_e}{m_e c^2} n_e\sigma_T dl,
\end{equation}
where $\sigma_T$ is the Thomson cross section.
Most notably, the integrated SZ flux from a cluster directly measures the total thermal energy of the gas. Expressing this flux in terms of the integrated Compton $y$--parameter $Y_{\rm SZ}$ -- defined by $\int\, d\Omega\,\Delta\,i_\nu(\hat{n})\,=\,Y_{\rm SZ} j_\nu(x)$ -- we see that $Y_{\rm SZ}\,\propto\,\int\,d\Omega\,dl n_eT_e\propto \int\,n_eT_e dV$. For this reason, we expect $Y_{\rm SZ}$ to closely correlate with total cluster mass, $M$, and to provide a low--scatter mass proxy.
This expectation, borne out by both numerical simulations \citep[e.g.,][]{das04,mot05,kra06} and indirectly from X--ray observations using $Y_{\rm X}$, the product of the gas mass and mean temperature \citep[][]{2007ApJ...668....1N,2007A&A...474L..37A,vik09}, strongly motivates the use of SZ cluster surveys as cosmological probes.
Theory predicts the cluster abundance and its evolution -- the mass function -- in terms of $M$ and the cosmological parameters. With a good mass proxy, we can measure the mass function and its evolution and hence constrain the cosmological model, including the properties of dark energy. In this context the relationship between the integrated SZ flux and total mass, $Y_{\rm SZ}-M$, is fundamental as the required link between theory and observation. Unfortunately, despite its importance, we are only beginning to observationally constrain the relation \citep{bonamente08,marrone09}.
Several authors have extracted the cluster SZ signal from WMAP data \citep{2003ApJS..148....1B, 2007ApJS..170..288H, 2009ApJS..180..225H}. However, the latter are not ideal for SZ observations: the instrument having been designed to measure primary CMB anisotropies on scales larger than galaxy clusters, the spatial resolution and sensitivity of the sky maps render cluster detection difficult. Nevertheless, these authors extracted the cluster SZ signal by either cross--correlating with the general galaxy distribution \citep{2003ApJ...597L..89F, 2004MNRAS.347L..67M, 2004ApJ...613L..89H, 2006A&A...449...41H} or `stacking' existing cluster catalogues in the optical or X-ray \citep{2006ApJ...648..176L, 2007MNRAS.378..293A, 2008ApJ...675L..57A, bie07, 2009MNRAS.tmp.1927D}. These analyses indicate that
an isothermal $\beta$-model is not a good description of the SZ profile, and some suggest that the SZ signal strength is lower than expected from the X-ray properties of the clusters \citep{2006ApJ...648..176L, bie07}.
Recent in--depth X-ray studies of the ICM pressure profile demonstrate regularity in shape and simple scaling with cluster mass. Combining these observations with numerical simulations leads to a universal pressure profile \citep{2007ApJ...668....1N,arnaud09} that is best
fit by a modified NFW profile. The isothermal $\beta$--model, on the other hand, does not provide an adequate fit.
From this newly determined X--ray pressure profile, we can infer the expected SZ profile, $y(r)$, and the $Y_{\rm SZ}-M$ relation at low
redshift \citep{arnaud09}.
It is in light of this recent progress from X--ray observations that we present a new analysis of the SZ effect in WMAP with the aim of
constraining the SZ scaling laws.
We build a multifrequency matched filter \citep{2002MNRAS.336.1057H, 2006A&A...459..341M} based on the known spectral shape of the thermal SZ effect and the shape of the universal pressure profile of \citet{arnaud09}. This profile was derived from {\gwpfont REXCESS}\ \citep{boe07}, a sample expressly designed to measure the structural and scaling properties of the local X--ray cluster population by means of an unbiased, representative sampling in luminosity. Using the multifrequency matched filter, we search for the SZ effect in WMAP from a catalogue of 893 clusters detected by ROSAT, maximising the signal--to--noise by adapting the filter scale to the expected characteristic size of each cluster. The size is estimated through the luminosity--mass relation derived from the {\gwpfont REXCESS}\ sample by \citet{2009A&A...498..361P}.
We then use our SZ measurements to directly determine the $Y_{\rm SZ}-L_X$ and $Y_{\rm SZ}-M$ relations and compare to expectations based on the universal X--ray pressure profile. As compared to the previous analyses of \citet{bonamente08} and \citet{marrone09}, the large number of systems in our WMAP/ROSAT sample allows us to constrain both the normalisation and slope of the $Y_{\rm SZ}-L_X$ and $Y_{\rm SZ}-M$ relations over a wider mass range and in the larger aperture of $r_{500}$. In addition, the analysis is based on a more realistic pressure profile than in these analyses, which were based on an isothermal $\beta$--model. Besides providing a direct constraint on these relations, the good agreement with X--ray predictions implies that there is in fact no deficit in SZ signal strength relative to expectations from the X-ray properties of these clusters.
The discussion proceeds as follows. We first present the WMAP 5--year data and the ROSAT cluster sample used, a combination
of the REFLEX and NORAS catalogues. We then present the SZ model
based on the X--ray--measured pressure profile (Sec.~\ref{cluster_model}). In Sec.~\ref{sz_flux}, we discuss our SZ measurements, after first describing how we extract the signal
using the matched filter. Section~\ref{err_bud} details the error budget. We compare our measured scaling laws to the X--ray predictions in Sect.~\ref{yl_relation} and~\ref{ym_relation} and then conclude in Sec.~\ref{discussion}. Finally, we collect useful SZ definitions
and unit conversions in the Appendices.
Throughout this paper, we use the WMAP5--only cosmological parameters set as our `fiducial cosmology', i.e. $h=0.719$, $\Omega_M=0.26$, $\Omega_\Lambda=0.74$, where $h$ is the Hubble parameter at redshift $z\,=\,0$ in units of $100\,$\,km/s/Mpc. We note $h_{70} = h/0.7$ and $E(z)$ is the Hubble parameter at redshift $z$ normalised to its present value. $M_{\rm 500}$ is defined as the mass within the radius $r_{\rm 500}$ at which the mean mass density is 500 times the critical density, $ \rho_{crit}(z)$, of the universe at the cluster redshift: $M_{500} = {4 \over 3} \pi \, \rho_{crit}(z) \, 500 \, r_{500}^3$.
\section{The WMAP-5yr data and the NORAS/REFLEX cluster sample}
\label{data}
\subsection{The WMAP-5yr data}
We work with the WMAP full resolution coadded five year sky temperature maps at each frequency channel (downloaded from the LAMBDA archive\footnote{http://lambda.gsfc.nasa.gov/}). There are five full sky maps corresponding to frequencies 23, 33, 41, 61, 94 GHz (bands K, Ka, Q, V, W respectively). The corresponding beam full widths at half maximum are approximately 52.8, 39.6, 30.6 21.0 and 13.2 arcmins.
The maps are originally at HEALPix\footnote{http://healpix.jpl.nasa.gov} resolution nside=512 (pixel= 6.87 arcmin). Although this is reasonably adequate to sample WMAP data, it is not adapted to the multifrequency matched filter algorithm we use to extract the cluster fluxes. We oversample the original data, to obtain nside=2048 maps, by zero-padding in harmonic space. In detail, this is performed by computing the harmonic transform of the original maps, and then performing the back transform towards a map with nside=2048, with a maximum value of $\ell$ of $\ell_{\rm max}$ = 750, 850, 1100, 1500, 2000 for the K, Ka, Q, V, W bands respectively. The upgraded maps are smooth and do not show pixel edges as we would have obtained using the HEALPix upgrading software, based on the tree structure of the HEALPix pixelisation scheme. This smooth upgrading scheme is important as the high spatial frequency content induced by pixel edges would have been amplified through the multifrequency matched filters implemented in harmonic space.
In practice, the multifrequency matched filters are implemented locally on small, flat patches (gnomonic projection on tangential maps), which permits adaptation of the filter to the local conditions of noise and foreground contamination. We divide the sphere into 504 square tangential overlapping patches (100 ${\rm deg}^2$ each, pixel=1.72 arcmin). All of the following analysis is done on these sky patches.
The implementation of the matched filter requires knowledge of the WMAP beams. In this work, we assume symmetric beams, for which the transfer function $b_\ell$ is computed, in each frequency channel, from the noise-weighted average of the transfer functions of individual differential assemblies (a similar approach was used in \citealt{2009A&A...493..835D}).
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{13999fg1.eps}
\caption{Inferred masses for the 893 NORAS/REFLEX clusters as a function of redshift. The cluster sample is flux limited. The right vertical axis gives the corresponding X-ray luminosities scaled by $E(z)^{-7/3}$. The dashed blue lines delineate the mass range over which the $L_{500}$-$M_{500}$ relation from~\citet{2009A&A...498..361P} was derived.}
\label{fig:m_z_relation}
\end{figure}
\subsection{The NORAS/REFLEX cluster sample and derived X--ray properties}
\label{rass}
We construct our cluster sample from the largest published X-ray catalogues: NORAS \citep{2000ApJS..129..435B} and REFLEX \citep{2004A&A...425..367B}, both constructed from the ROSAT All-Sky Survey. We merge the cluster lists given in Tables 1, 6 and 8 from ~\citet{2000ApJS..129..435B} and Table 6 from~\citet{2004A&A...425..367B} and since the luminosities of the NORAS clusters are given in a standard cold dark matter (SCDM) cosmology ($h=0.5$, $\Omega_M=1$), we converted them to the WMAP5 cosmology. We also convert the luminosities of REFLEX clusters from the basic $\Lambda$CDM cosmology ($h=0.7$, $\Omega_M=0.3$, $\Omega_\Lambda=0.7$) to the more precise WMAP5 cosmology. Removing clusters appearing in both catalogues leaves 921 objects, of which 893 have measured redshifts. We use these 893 clusters in the analysis detailed in the next Section.
The NORAS/REFLEX luminosities $L_{\rm X}$, measured in the soft $[0.1$--$2.4]\,{\rm keV}$ energy band, are given within various apertures depending on the cluster. We convert the luminosities $L_{\rm X}$ to $L_{500}$, the luminosities within $r_{\rm 500}$, using an iterative scheme. This scheme is based on the mean electron density profile of the {\gwpfont REXCESS}\ cluster sample \citep{cro08}, which allows conversion of the luminosity between various apertures, and the {\gwpfont REXCESS}\ $L_{500}$--$M_{500}$ relation \citep{2009A&A...498..361P}, which implicitly relates $r_{\rm 500}$ and $L_{500}$. The procedure thus simultaneously yields an estimate of the cluster mass, $M_{500}$, and the corresponding angular extent $ \theta_{500} = r_{\rm 500}/D_{\rm ang}(z)$, where $D_{ang}(z)$ is the angular distance at redshift $z$. In the following we consider values derived from relations both corrected and uncorrected for Malmquist bias. The relations are described by the following power law models\footnote{Since we consider a standard self-similar model, we used the power law relations given in Appendix B of \citet{arnaud09}. They are derived as in \citet{2009A&A...498..361P} with the same luminosity data but for masses derived from a standard slope $M_{500}$--$Y_{\rm X}$ relation.}:
\begin{equation}
\label{lx_m500}
E(z)^{-7/3} \, L_{500} = C_M \, \left ( M_{500} \over 3 \times 10^{14} h_{70}^{-1} M_\odot \right )^{\alpha_M}
\end{equation}
where the normalisation $C_M$, the exponent $\alpha_M$ and the dispersion (nearly constant with mass) are given in Table~\ref{tab:lx_m_param}.
The $L_{500}$--$M_{500}$ relation was derived in the mass range $[10^{14}$--$10^{15}]\,M_\odot$. These limits are shown in Fig.~\ref{fig:m_z_relation}. Note that we assume the relation is valid for lower masses.
\begin{table}
\caption[]{Values for the parameters of the $L_X-M$ relation derived from {\gwpfont REXCESS}\ data \citep{2009A&A...498..361P,arnaud09}}
\label{tab:lx_m_param}
\begin{center}
\begin{tabular}{cccc}
\hline
\hline
Corrected for MB & $\log \left ( C_M \over 10^{44} h_{70}^{-2} {\rm [erg s^{-1}]} \right ) $ & $\alpha_M$ & $\sigma_{\log L - \log M}$ \\
\hline
no & 0.295 & 1.50 & 0.183 \\
yes & 0.215 & 1.61 & 0.199 \\
\hline
\end{tabular}
\end{center}
\end{table}
The final catalogue of 893 objects contains the position of the clusters (longitude and latitude), the measured redshift $z$, the derived X-ray luminosity $L_{500}$, the mass $M_{500}$ and the angular extent $ \theta_{500}$. The clusters uniformly cover the celestial sphere at Galactic latitudes above $|b|>20 \, {\rm deg}$. Their luminosities $L_{500}$ range from 0.002 to $35.0 \, 10^{44} {\rm erg/s}$, and their redshifts from 0.003 to 0.460. Figure~\ref{fig:m_z_relation} shows the masses $M_{500}$ as a function of redshift $z$ for the cluster sample (red crosses). The corresponding corrected luminosities $L_{500}$ can be read on the right axis. The typical luminosity correction from measured $L_X$ to $L_{500}$ is about $10\%$. The progressive mass cut-off with redshift (only the most massive clusters are present at high $z$) reflects the flux limited nature of the sample.
\section{The cluster SZ model}
\label{cluster_model}
In this Section we describe the cluster SZ model, based on X-ray observations of the {\gwpfont REXCESS}\ sample combined with numerical simulations, as presented in \citet{arnaud09}. We use the standard self-similar model presented in their Appendix B. Given a cluster mass $M_{500}$ and redshift $z$, the model predicts the electronic pressure profile. This gives both the SZ profile shape and $Y_{500}$,
the SZ flux integrated in a sphere of radius $r_{500}$.
\subsection{Cluster shape}
The dimensionless universal pressure profile is taken from Eq.~ B1 and Eq.~B2 of \citet{arnaud09}:
\begin{equation}
\label{cluster_profile}
{P(r) \over P_{500}} = {P_0 \over x^\gamma (1+x^\alpha)^{(\beta-\gamma)/\alpha}}
\end{equation}
where $x=r/r_s$ with $r_s=r_{500}/c_{500}$ and $c_{500}=1.156$, $\alpha=1.0620$, $\beta=5.4807$, $\gamma=0.3292$ and with $P_{500}$ defined in Eq.~\ref{p500} below.
This profile shape is used to optimise the SZ signal detection. As described below in Sect.~\ref{sz_flux}, we extract the $Y_{SZ}$ flux from WMAP data for each ROSAT system fixing $c_{500}$, $\alpha$, $\beta$, $\gamma$ to the above values, but leaving the normalisation free.
\subsection{Normalisation}
\begin{figure*}[t]
\centering
\includegraphics[scale=0.5]{13999fg2a.eps}
\includegraphics[scale=0.5]{13999fg2b.eps}
\caption{{\it Left:} Estimated SZ flux from a cylinder of aperture radius $5 \times r_{\rm 500}$ ($Y^{cyl}_{5r500}$) as a function of the X-ray luminosity in an aperture of $r_{500}$ ($L_{500}$), for the 893 NORAS/REFLEX clusters. The individual clusters are barely detected. The bars give the total 1 $\sigma$ error. {\it Right:} Red diamonds are the weighted average signal in 4 logarithmically--spaced luminosity bins. The two high luminosity bins exhibit significant SZ cluster flux. Note that we have divided the vertical scale by 30 between Fig. left and right. The thick and thin bars give the 1 $\sigma$ statistical and total errors, respectively. Green triangles (shifted up by 20\% with respect to diamonds for clarity) show the result of the same analysis when the fluxes of the clusters are estimated at random positions instead of true cluster positions. }
\label{fig:cy_lx_raw_binned}
\end{figure*}
The model allows us to compute the physical pressure profile as a function of mass and $z$, thus the $Y_{SZ}$-$M_{500}$ relation by integration of $P(r)$ to $r_{500}$. For the shape parameters given above, the normalisation parameter $P_0=8.130 \, h_{70}^{3/2} = 7.810$ and the self-similar definition of
$P_{500}$ \citep[][Eq.~5 and Eq.~B2]{arnaud09},
\begin{equation}
\label{p500}
P_{500} = 1.65 \times 10^{-3} E(z)^{8/3} \left ( {M_{500} \over 3\times10^{14}\, h_{70}^{-1} M_\odot} \right )^{2/3} h_{70}^2 \; {\rm keV \, cm^{-3}},
\end{equation}
one obtains:
\begin{equation}
\label{ym_ss_rel}
Y_{500} \, [{\rm arcmin}^2]= Y^*_{500} \, \left ({M_{500} \over 3\times10^{14}\,h^{-1}\,M_\odot} \right )^{5/3} \, E(z)^{2/3} \, \left ({D_{ang}(z) \over 500 \, {\rm Mpc}} \right )^{-2},
\end{equation}
\noindent where $Y^*_{500}=1.54\times10^{-3} \, \left ({h \over 0.719} \right )^{-5/2} \, {\rm arcmin}^2$. Equivalently, one can write:
\begin{equation}
Y_{500} \, \, [{\rm Mpc}^2] = Y^*_{500} \; \left ({M_{500} \over 3\times10^{14}\,h^{-1}\,M_\odot} \right )^{5/3} \; E(z)^{2/3}
\end{equation}
\noindent where $Y^*_{500}=3.27\times10^{-5} \, \left ({h \over 0.719} \right )^{-5/2} \, {\rm Mpc}^2$. Details of unit conversions are given in Appendix~\ref{sz_conv}.
The mass dependence ($M_{500}^{5/3}$) and the redshift dependence ($E(z)^{2/3}$) of the relation are self-similar by construction. This model is used to predict the $Y_{500}$ value for each cluster. These predictions are compared to the WMAP-measured values in Figs.~\ref{fig:cy_lx_relation}, \ref{fig:cy_m_relation}, \ref{fig:evolution} and \ref{fig:cy_m_loglog}.
\section{Extraction of the SZ flux}
\label{sz_flux}
\subsection{Multifrequency Matched Filters}
\label{mmf_sec}
We use multifrequency matched filters to estimate cluster fluxes from the WMAP frequency maps. By incorporating prior
knowledge of the cluster signal, i.e., its spatial and spectral
characteristics, the method maximally enhances the signal--to--noise of a SZ cluster source by optimally filtering the data. The universal profile shape described in Sec.~\ref{cluster_model} is assumed, and we evaluate the effects of uncertainty in this profile as outlined in Sec.~\ref{err_bud} where we discuss our overall error budget. We fix the position and the characteristic radius $\theta_{\rm s}$ of each cluster and estimate only its flux. The position is taken from the NORAS/REFLEX catalogue and $\theta_{\rm s}=\theta_{500}/c_{500}$ with $\theta_{500}$ derived from X-ray data as described in Sec.~\ref{rass}.
Below, we recall the main features of the multifrequency matched filters. More details can be found in~\citet{2002MNRAS.336.1057H} or~\citet{2006A&A...459..341M}.
Consider a cluster with known radius $\theta_{\rm s}$ and unknown central $y$--value $\yo$ positioned at a known point $\vec{x}_{\rm o}$ on the sky. The region is covered by the five WMAP maps $M_i(\vec{x})$ at frequencies $\nu_i$=23, 33, 41, 61, 94 GHz ($i=1,...,5$). We arrange the survey maps into a column vector $\vec{M}(\vec{x})$ whose $i^{th}$ component is the map at frequency $\nu_i$. The maps contain the cluster SZ signal plus noise:
\begin{equation}
\vec{M}(\vec{x}) = \yo\vec{\jnu}T_{\thetas}(\vec{x}-\vec{x}_{\rm o}) + \vec{N}(\vec{x})
\end{equation}
where $\vec{N}$ is the noise vector (whose components are noise maps at the different observation frequencies) and $\vec{\jnu}$ is a vector with components given by the SZ spectral function $\jnu$ evaluated at each frequency. Noise in this context refers to both instrumental noise as well as all signals other than the cluster thermal SZ effect; it thus also comprises astrophysical foregrounds, for example, the primary CMB anisotropy, diffuse Galactic emission and extragalactic point sources. $T_{\thetas}(\vec{x}-\vec{x}_{\rm o})$ is the SZ template, taking into account the WMAP beam, at projected distance $(\vec{x}-\vec{x}_{\rm o})$ from the cluster centre, normalised to a central value of unity before convolution. It is computed by integrating along the line--of--sight and normalising the universal pressure profile (Eq.~\ref{cluster_profile}). The profile is truncated at $5 \times r_{500}$ (i.e. beyond the virial radius) so that what is actually measured is the flux within a cylinder of aperture radius $5 \times r_{500}$.
X-ray observations are typically well-constrained out to $r_{500}$. Our decision to integrate out to $5 \times r_{500}$ is motivated by the fact that for the majority of clusters the radius $r_{500}$ is of order the Healpix pixel size (nside=512, pixel=6.87 arcmin). Integrating only out to $r_{500}$ would have required taking into account that only a fraction of the flux of some pixels contributes to the true SZ flux in a cylinder of aperture radius $r_{500}$. We thus obtain the total SZ flux of each cluster by integrating out to $5 \times r_{500}$, and then convert this to the value in a sphere of radius $r_{500}$ for direct comparison with the X-ray prediction.
The multifrequency matched filters $\vec{\Psi_{\thetas}}(\vec{x})$ return a minimum variance unbiased estimate, $\hat{\yo}$, of
$\yo$ when centered on the cluster:
\begin{equation}
\label{eq:filter}
\hat{\yo} = \int d^2x\; \vec{\Psi_{\thetas}}^t(\vec{x}-\vec{x}_{\rm o})
\cdot \vec{M}(\vec{x})
\end{equation}
where superscript $t$ indicates a transpose (with complex conjugation when necessary). This is just a linear combination of the maps, each
convolved with its frequency--specific filter $(\Psi_{\thetas})_i$.
The result expressed in Fourier space is:
\begin{equation}
\vec{\Psi_{\thetas}}(\vec{k}) = \sigma_{\thetas}^2 \vec{P}^{-1}(\vec{k})\cdot \vec{F_{\thetas}}(\vec{k})
\end{equation}
where
\begin{eqnarray}
\vec{F_{\thetas}}(\vec{k}) & \equiv & \vec{\jnu} T_{\thetas}(\vec{k})\\
\label{eq:sigt}
\sigma_{\thetas} & \equiv & \left[\int d^2k\;
\vec{F_{\thetas}}^t(\vec{k})\cdot \vec{P}^{-1} \cdot
\vec{F_{\thetas}}(\vec{k})\right]^{-1/2}
\end{eqnarray}
with $\vec{P}(\vec{k})$ being the noise power spectrum, a matrix in
frequency space with components $P_{ij}$ defined by $\langle
N_i(\vec{k})N_j^*(\vec{k}')\rangle_N=P_{ij}(\vec{k})
\delta(\vec{k}-\vec{k}')$.
The quantity $\sigma_{\thetas}$ gives the total noise variance through the filter, corresponding to the statistical errors quoted in this paper. The other uncertainties are estimated separately as described in Sec.~\ref{disp_err}. The noise power spectrum $\vec{P}(\vec{k})$ is directly estimated from the maps: since the SZ signal is subdominant at each frequency, we assume $\vec{N}(\vec{x}) \approx \vec{M}(\vec{x})$ to do this calculation. We undertake the Fourier transform of the maps and average their cross-spectra in annuli with width $\Delta l=180$.
\subsection{Measurements of the SZ flux}
The derived total WMAP flux from a cylinder of aperture radius $5 \times r_{\rm 500}$ ($Y^{cyl}_{5r500}$) for the 893 individual NORAS/REFLEX clusters is shown as a function of the measured X--ray luminosity $L_{\rm 500}$ in the left-hand panel of Fig.~\ref{fig:cy_lx_raw_binned}. The clusters are barely detected individually. The average signal--to--noise ratio (S/N) of the total population is 0.26 and only 29 clusters are detected at $S/N>2$, the highest detection being at 4.2. However, one can distinguish the deviation towards positive flux at the very high luminosity end.
In the right-hand panel of Fig.~\ref{fig:cy_lx_raw_binned}, we average the 893 measurements in four logarithmically--spaced luminosity bins (red diamonds plotted at bin center). The number of clusters are 7, 150, 657, 79 from the lowest to the highest luminosity bin. Here and in the following, the bin average is defined as the weighted mean of the SZ flux in the bin (weight of $1/\sigma_{\thetas}^2$). The thick error bars correspond to the statistical uncertainties on the WMAP data only, while the thin bar gives the total errors as discussed in Sec.~\ref{disp_err}.
The SZ signal is clearly detected in the two highest luminosity bins (at 6.0 and 5.4 $\sigma$, respectively). As a demonstrative check, we have undertaken the analysis a second time using random cluster positions. The result is shown by the green triangles in Fig.~\ref{fig:cy_lx_raw_binned} and is consistent with no SZ signal, as expected.
In the following Sections, we study both the relation between the SZ signal and the X-ray luminosity and that with the mass $M_{500}$. We consider $Y_{500}$, the SZ flux from a sphere of radius $r_{500}$, converting the measured $Y^{cyl}_{5r500}$ into $Y_{500}$ as described in Appendix~\ref{sz_ref}. This allows a more direct comparison with the model derived from X--ray observations (Sec.~\ref{cluster_model}). Before presenting the results, we first discuss the overall error budget.
\section{Overall error budget}
\label{err_bud}
\subsection{Error due to dispersion in X-ray properties}
\label{disp_err}
The error $\sigma_{\thetas}$ on $Y_{\rm 500}$ given by the multifrequency matched filter only includes the statistical SZ measurement error, due to the instrument (beam, noise) and to the astrophysical contaminants (primary CMB, Galaxy, point sources). However, we must also take into account: 1) uncertainties on the cluster mass estimation from the X-ray luminosities via the $L_{500}-M_{500}$ relation, 2) uncertainties on the cluster profile parameters. These are sources of error on individual $Y_{\rm 500}$ estimates (actual parameters for each individual cluster may deviate somewhat from the average cluster model). These deviations from the mean, however, induce additional {\it random} uncertainties on statistical quantities derived from $Y_{\rm 500}$, i.e. bin averaged $Y_{500}$ values and the $\Yv$--$\LX$\ scaling relation parameters. Their impact on the $\Yv$--$\Mv$\ relation, which depends directly on the $M_{\rm 500}$ estimates, is also an additional random uncertainty.
The uncertainty on $M_{\rm 500}$ is dominated by the intrinsic dispersion in the $\LX$--$\Mv$\ relation. Its effect is estimated by a Monte Carlo (MC) analysis of 100 realisations. We use the dispersion at $z=0$ as estimated by \citet{2009A&A...498..361P}, given in Table~\ref{tab:lx_m_param}. For each realisation, we draw a random mass $\log M_{500}$ for each cluster from a Gaussian distribution with mean given by the $\LX$--$\Mv$\ relation and standard deviation $\sigma_{\log L - \log M}/\alpha_M$. We then redo the full analysis (up to the fitting of the $Y_{SZ}$ scaling relations) with the new values of $M_{500}$ (thus $\theta_{\rm s}$).
The second uncertainty is due to the observed dispersion in the cluster profile shape, which depends on radius as shown in~\citet[][$\sigma_{\log P} \sim 0.10$ beyond the core]{arnaud09}. Using new 100 MC realisations, we estimate this error by drawing a cluster profile in the log--log plane from a Gaussian distribution with mean given by Eq.~\ref{cluster_profile} and standard deviation depending on the cluster radius as shown in the lower panel of Fig.~2 in~\citet{arnaud09}.
The total error on $Y_{500}$ and on the scaling law parameters is calculated from the quadratic sum of the standard deviation of both the above MC realisations and the error due to the SZ measurement uncertainty.
\subsection{The Malmquist bias}
\label{mal_bias}
The NORAS/REFLEX sample is flux limited and is thus subject to the Malmquist bias (MB). This is a source of systematic error. Ideally we should use a $\LX$--$\Mv$\ relation which takes into account the specific bias of the sample, i.e. computed from the true $\LX$--$\Mv$\ relation, with dispersion and bias according to each survey selection function. We have an estimate of the true, ie MB corrected, $\LX$--$\Mv$\ relation, from the published analysis of {\gwpfont REXCESS}\ data (Table 1). However, while the REFLEX selection function is known and available, this is not the case for the NORAS sample. This means that we cannot perform a fully consistent analysis.
In order to estimate the impact of the Malmquist Bias we thus present, in the following, results for two cases.
In the first case, we use the published $\LX$--$\Mv$\ relation derived directly from the {\gwpfont REXCESS}\ data, i.e. not corrected for the {\gwpfont REXCESS}\ MB (hereafter the {\gwpfont REXCESS}\ $\LX$--$\Mv$\ relation). Note that the {\gwpfont REXCESS}\ is a sub-sample of REFLEX. Using this relation should result in correct masses if the Malmquist bias for the NORAS/REFLEX sample is the same as that for the {\gwpfont REXCESS}. The $\Yv$--$\Mv$\ relation derived in this case would also be correct and could be consistently compared with the X--ray predicted relation. We recall that this relation was derived from pressure and mass measurements that are not sensitive to the Malmquist bias. However $L_{\rm 500}$ would remain uncorrected so that the $\Yv$--$\LX$\ relation derived in this case should be viewed as a relation uncorrected for the Malmquist bias.
In the second case, we use the MB corrected $\LX$--$\Mv$\ relation (hereafter the intrinsic $\LX$--$\Mv$\ relation). This reduces to assuming that the Malmquist bias is negligible for the NORAS/REFLEX sample.
The comparison of the two analyses provides an estimate of the direction and amplitude of the effect of the Malmquist bias on our results. The {\gwpfont REXCESS}\ $\LX$--$\Mv$\ relation is expected to be closer to the $\LX$--$\Mv$\ relation for the NORAS/REFLEX sample than the intrinsic relation. The discussions and figures correspond to the results obtained when using the former, unless explicitly specified.
\begin{figure*}[tp]
\centering
\includegraphics[scale=0.5]{13999fg3a.eps}
\includegraphics[scale=0.5]{13999fg3b.eps}
\caption{{\it Left:} Bin averaged SZ flux from a sphere of radius $r_{500}$ ($Y_{500}$) as a function of X-ray luminosity in a aperture of $r_{500}$ ($L_{500}$). The WMAP data (red diamonds and crosses), the SZ cluster signal expected from the X--ray based model (blue stars) and the combination of the $\Yv$--$\Mv$\ and $\LX$--$\Mv$\ relations (dash and dotted dashed lines) are given for two analyses, using respectively the intrinsic $\LX$--$\Mv$\ and the {\gwpfont REXCESS}\ $\LX$--$\Mv$\ relations. As expected, the data points do not change significantly from one case to the other showing that the $Y_{500}$--$L_{500}$ relation is rather insensitive to the finer details of the underlying $\LX$--$\Mv$\ relation.
{\it Right:} Ratio of data points to model for the two analysis. The points for the analysis undertaken with the intrinsic $\LX$--$\Mv$\ are shifted to lower luminosities by 20\% for clarity.}
\label{fig:cy_lx_relation}
\end{figure*}
\begin{table*}
\caption[]{Fitted parameters for the observed $Y_{SZ}$-$L_{500}$ relation. The X-ray based model gives $Y^{*L}_{500}=0.89|1.07 \times 10^{-3} \, \left ({h/0.719} \right )^{-5/2} {\rm arcmin}^2$, \\ $\alpha^L_Y=1.11|1.04$ and $\beta^L_Y=2/3$ for the {\gwpfont REXCESS}\ and intrinsic $\LX$--$\Mv$\ relation, respectively.}
\label{fit_param_un_l}
\begin{center}
\begin{tabular}{cccc}
\hline
\hline
$L_{500}-M_{500}$ &$Y^{*L}_{500}$ [$10^{-3} \, \left ({h/0.719} \right )^{-2} {\rm arcmin}^2 $] & $\alpha^L_Y$ & $\beta^L_Y$ \\
\hline
{\gwpfont REXCESS}\ & $ 0.92 \pm 0.08 \, {\rm stat} \; [\pm 0.10 \, {\rm tot} ] $ & 1.11 (fixed) & 2/3 (fixed) \\
& $ 0.88 \pm 0.10 \, {\rm stat} \; [ \pm 0.12 \, {\rm tot} ] $ & $1.19 \pm 0.10 \, {\rm stat} \; [ \pm 0.10 \, {\rm tot}]$ & 2/3 (fixed) \\
& $ 0.90 \pm 0.13 \, {\rm stat} \; [ \pm 0.16 \, {\rm tot} ] $ & 1.11 (fixed) & $1.05 \pm 2.18 \, {\rm stat} \; [ \pm 2.25 \, {\rm tot} ]$\\
\hline
Intrinsic & $ 0.95 \pm 0.09 \, {\rm stat} \; [ \pm 0.11 \, {\rm tot} ] $ & 1.04 (fixed) & 2/3 (fixed) \\
& $ 0.89 \pm 0.10 \, {\rm stat} \; [ \pm 0.12 \, {\rm tot} ] $ & $1.19 \pm 0.10 \, {\rm stat} \; [ \pm 0.10 \, {\rm tot}]$ & 2/3 (fixed) \\
& $ 0.89 \pm 0.13 \, {\rm stat} \; [ \pm 0.16 \, {\rm tot} ] $ & 1.04 (fixed) & $2.06 \pm 2.14 \, {\rm stat} \; [ \pm 2.21 \, {\rm tot}]$\\
\hline
\end{tabular}
\end{center}
\end{table*}
The choice of the $\LX$--$\Mv$\ relation has an effect both on the estimated $L_{\rm 500}$, $M_{\rm 500}$ and $Y_{\rm 500}$ values and on the expectation for the SZ signal from the NORAS/REFLEX clusters. However, for a cluster of given luminosity measured a given aperture, $L_{\rm 500}$ depends weakly on the exact value of $r_{500}$ due to the steep drop of X--ray emission with radius. As a result, and although $L_{\rm 500}$ and $M_{\rm 500}$ (or equivalently $r_{\rm 500}$) are determined jointly in the iterative procedure described in Sec.~\ref{rass}, changing the underlying $\LX$--$\Mv$\ relation mostly impacts the $M_{\rm 500}$ estimate: $L_{\rm 500}$ is essentially unchanged (median difference of $\sim 0.8\%$) and the difference in $M_{\rm 500}$ simply reflects the difference between the relations at fixed luminosity. This has an impact on the measured $Y_{500}$ via the value of $r_{500}$ (the profile shape being fixed) but the effect is also small ($<1\%$). This is due to the rapidly converging nature of the $Y_{SZ}$ flux (see Fig.~11 of \citealt{arnaud09}). On the other hand, all results that depend directly on $M_{\rm 500}$, namely the derived $\Yv$--$\Mv$\ relation or the model value for each cluster, that varies as $M_{\rm 500}^{5/3}$ (Eq.~\ref{ym_ss_rel}), depend sensitively on the $\LX$--$\Mv$\ relation. $M_{\rm 500}$ derived from the intrinsic relation is higher, an effect increasing with decreasing cluster luminosity (see Fig. B2 of \citealt{2009A&A...498..361P}).
\subsection{Other possible sources of uncertainty}
The analysis presented in this paper has been performed on the entire NORAS/REFLEX cluster sample without removal of clusters hosting radio point sources. To investigate the impact of the point sources on our result, we have cross-correlated the NVSS~\citep{condon98} and SUMMS~\citep{mauch03} catalogues with our cluster catalogue. We conservatively removed from the analysis all the clusters hosting a total radio flux greater than 1~Jy within $5 \times r_{500}$. This leaves 328 clusters in the catalogue, removing the measurements with large uncertainties visible in Figure~\ref{fig:cy_lx_raw_binned} left. We then performed the full analysis on these 328 objects up to the fitting of the scaling laws, finding that the impact on the fitted values is marginal. For example, for the {\gwpfont REXCESS}\ case, the normalisation of the $\Yv$--$\Mv$\ relation decreases from 1.60 to 1.37 (1.6 statistical $\sigma$) and the slope changes from 1.79 to 1.64 (1 statistical $\sigma$). The statistical errors on these parameters decrease respectively from 0.14 to 0.30 and from 0.15 to 0.40 due to the smaller number of remaining clusters in the sample.
The detection method does not take into account superposition effects along the line of sight, a drawback that is inherent to any SZ observation. Thus we cannot fully rule out that our flux estimates are not partially contaminated by low mass systems surrounding the clusters of our sample. Numerical simulations give a possible estimate of the contamination: \cite{hallman2007} suggest that low-mass systems and unbound gas may contribute up to $16.3\%^{+7\%}_{-6.4\%}$ of the SZ signal. This would lower our estimated cluster fluxes by $\sim 1.5 \sigma$.
\section{The $Y_{\rm SZ}$-$L_{500}$ relation}
\label{yl_relation}
\subsection{WMAP SZ measurements vs. X--ray model}
\label{Szmes_Xray}
We first consider bin averaged data, focusing on the luminosity range $L_{\rm 500} \gtrsim10^{43}$\,ergs/s where the SZ signal is significantly detected (Fig.~\ref{fig:cy_lx_raw_binned} right). The left panel of Fig.~\ref{fig:cy_lx_relation} shows $Y_{500}$ from a sphere of radius $r_{\rm 500}$ as a function of $L_{500}$, averaging the data in six equally--spaced logarithmic bins in X--ray luminosity. Both quantities are scaled according to their expected redshift dependence. The results are presented for the analyses based on the {\gwpfont REXCESS}\ (red diamonds) and intrinsic (red crosses) $\LX$--$\Mv$\ relations. For the reasons discussed in Sec.~\ref{mal_bias}, the derived data points do not differ significantly between the two analyses (Fig.~\ref{fig:cy_lx_relation} left), confirming that the measured $\Yv$--$\LX$\ relation is insensitive to the finer details of the underlying $\LX$--$\Mv$\ relation.
\begin{figure}
\centering
\includegraphics[scale=0.5]{13999fg4.eps}
\caption{Estimated SZ flux $Y_{500}$ (in a sphere of radius $r_{500}$) as a function of the mass $M_{500}$ averaged in 4 mass bins. Red diamonds are the WMAP data. Blue stars correspond to the X-ray based model predictions and are shifted to higher masses by 20\% for clarity. The model is in very good agreement with the data.}
\label{fig:cy_m_relation}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[scale=0.5]{13999fg5a.eps}
\includegraphics[scale=0.5]{13999fg5b.eps}
\caption{Evolution of the $Y_{500}$-$M_{500}$ relation. {\it Left:} The WMAP data from Fig.~\ref{fig:cy_m_relation} are divided into three redshift bins: z$<$0.08 (blue diamonds), 0.08$<$z$<$0.18 (green crosses), z$>$0.18 (red triangles). We observe the expected trend: at fixed mass, $Y_{500}$ decreases with redshift. This redshift dependence is mainly due to the angular distance ($Y_{500} \propto D_{ang}(z)^{-2}$). The stars give the prediction of the model. {\it Right:} We divide $Y_{500}$ by $M_{500}^{5/3} D_{ang}(z)^{-2}$ and plot it as a function of $z$ to search for evidence of evolution in the $Y_{500}$-$M_{500}$ relation. The thick bars give the 1 $\sigma$ statistical errors from WMAP data. The thin bars give the total 1 sigma errors.}
\label{fig:evolution}
\end{figure*}
We also apply the same averaging procedure to the model $Y_{500}$ values derived for each cluster in Sec.~\ref{cluster_model}. The expected values for the same luminosity bins are plotted as stars in the left-hand hand panel of Fig.~\ref{fig:cy_lx_relation}. The $\Yv$--$\LX$\ relation expected from the combination of the $\Yv$--$\Mv$\ (Eq.~\ref{ym_ss_rel}) and $\LX$--$\Mv$\ (Eq.~\ref{lx_m500}) relations is superimposed to guide the eye.
The right-hand panel of Fig.~\ref{fig:cy_lx_relation} shows the ratio between the measured data points and those expected from the model. As discussed in Sec.~\ref{mal_bias}, the model values depend on the assumed $\LX$--$\Mv$\ relation. The difference is maximum in the lowest luminosity bin where the intrinsic relation yields $\sim 40\%$ higher value than the {\gwpfont REXCESS}\ relation (Fig.~\ref{fig:cy_lx_relation} left panel). The SZ model prediction and the data are in good agreement, but the agreement is better when the {\gwpfont REXCESS}\ $\LX$--$\Mv$\ is used in the analysis (Fig.~\ref{fig:cy_lx_relation} right panel). This is expected if indeed the agreement is real and the effective Malmquist bias for the NORAS/REFLEX sample is not negligible and is similar to that of the {\gwpfont REXCESS}.
\begin{figure*}[t]
\centering
\includegraphics[scale=0.5]{13999fg6a.eps}
\includegraphics[scale=0.5]{13999fg6b.eps}
\caption{{\it Left:} Zoom on the $> 5 \times 10^{13} M_\odot$ mass range of the $Y_{500}-M_{500}$ relation shown in Fig.~\ref{fig:cy_m_relation}. The data points and model stars are now scaled with the expected redshift dependence and are placed at the mean mass of the clusters in each bin. {\it Right:} Ratio between data and model.}
\label{fig:cy_m_loglog}
\end{figure*}
\subsection{$Y_{500}$-$L_{500}$ relation fit}
\label{yl_fit_sec}
Working now with the individual flux measurements, $Y_{500}$, and $L_{500}$ values, we fit an $\Yv$--$\LX$\ relation of the form:
\begin{equation}
Y_{500} = Y^{*L}_{500} \; \left ({E(z)^{-7/3} L_{500} \over 10^{44} h^{-2} {\rm erg/s}} \right )^{\alpha^L_Y} \; E(z)^{\beta^L_Y} \; \left ({D_{ang}(z) \over 500 \, {\rm Mpc}} \right )^{-2}
\end{equation}
using the statistical error on $Y_{500}$ given by the multifrequency matched filter. The total error is estimated by Monte Carlo (see Sec.~\ref{disp_err}) but is dominated by the statistical error. The results are presented in Table~\ref{fit_param_un_l}. As already stated in Sec.~\ref{Szmes_Xray}, the fitted values depend only weakly on the choice of $\LX$--$\Mv$\ relation.
\section{The $Y_{\rm SZ}$--$M_{500}$ relation and its evolution}
\label{ym_relation}
In this Section, we study the mass and redshift dependence of the SZ signal and check it against the X--ray based model. Furthermore, we fit the $\Yv$--$\Mv$\ relation and compare it with the X--ray predictions.
\subsection{Mass dependence and redshift evolution}
Figure~\ref{fig:cy_m_relation} shows the bin averaged SZ flux measurement as a function of mass compared to the X--ray based model prediction. As expected, the SZ cluster flux increases as a function of mass and is compatible with the model. In order to study the behaviour of the SZ flux with redshift, we subdivide each of the four mass bins into three redshift bins corresponding to the following ranges: $z<0.08$, $0.08<z<0.18$, $z>0.18$. The result is shown in the left panel of Fig.~\ref{fig:evolution}.
In a given mass bin the SZ flux decreases with redshift, tracing the $D_{ang}(z)^{-2}$ dependence of the flux. In particular, in the highest mass bin ($10^{15} M_\odot$), the SZ flux decreases from 0.007 to $0.001 \, {\rm arcmin^2}$ while the redshift varies from $z<0.08$ to $z>0.18$. The mass and the redshift dependence are in good agreement with the model (stars) described in Sec.~\ref{cluster_model}.
Since the $D_{ang}(z)^{-2}$ dependence is the dominant effect in the redshift evolution, we multiply $Y_{500}$ by $D_{ang}(z)^{2}$ and divide it by the self-similar mass dependence $M_{500}^{5/3}$. The expected self-similar behaviour of the new quantity $Y_{500}D_{ang}(z)^{2}/M_{500}^{5/3}$ as a function of redshift is $E(z)^{2/3}$ (see Eq.~\ref{ym_ss_rel}). The right panel of Fig.~\ref{fig:evolution} shows $Y_{500}D_{ang}(z)^{2}/M_{500}^{5/3}$ as a function of redshift for the three redshift bins $z<0.08$, $0.08<z<0.18$, $z>0.18$. The points have been centered at the average value of the cluster redshifts in each bin. The model is displayed as blue stars. Since the model has a self-similar redshift dependence and $E(z)^{2/3}$ increases only by a factor of 5\% over the studied redshift range, the model stays nearly constant. The blue dotted line is plotted through the model and varies as $E(z)^{2/3}$. The data points are in good agreement with the model, but clearly, the redshift leverage of the sample is insufficient to put strong constraints on the evolution of the scaling laws.
\begin{table*}[t]
\caption[]{Fitted parameters for the observed $Y_{SZ}$-$M_{500}$ relation. The X-ray based model gives $Y^*_{500}=1.54 \times 10^{-3} \, \left ({h/0.719} \right )^{-5/2} {\rm arcmin}^2$, $\alpha_Y=5/3$ and $\beta_Y=2/3$.}
\label{fit_param_un}
\begin{center}
\begin{tabular}{cccc}
\hline
\hline
$\LX$--$\Mv$\ relation & $Y^*_{500}$ [$10^{-3} \, \left ({h/0.719} \right )^{-2} \, {\rm arcmin}^2$] & $\alpha_Y$ & $\beta_Y$ \\
\hline
{\gwpfont REXCESS}\ & $1.60 \pm 0.14 \, {\rm stat} \; [ \pm 0.19 \, {\rm tot} ] $ & 5/3 (fixed) & 2/3 (fixed) \\
& $ 1.60 \pm 0.15 \, {\rm stat} \; [ \pm 0.19 \, {\rm tot} ] $ & $1.79 \pm 0.15 \, {\rm stat} \; [ \pm 0.17 \, {\rm tot}]$ & 2/3 (fixed) \\
& $ 1.57 \pm 0.23 \, {\rm stat} \; [ \pm 0.29 \, {\rm tot} ] $ & 5/3 (fixed) & $1.05 \pm 2.18 \, {\rm stat} \; [ \pm 2.52 \, {\rm tot}]$\\
\hline
intrinsic & $ 1.37 \pm 0.12 \, {\rm stat} \; [ \pm 0.17 \, {\rm tot} ] $ & 5/3 (fixed) & 2/3 (fixed) \\
& $ 1.36 \pm 0.13 \, {\rm stat} \; [ \pm 0.17 \, {\rm tot} ] $ & $1.91 \pm 0.16 \, {\rm stat} \; [ \pm 0.18 \, {\rm tot}]$ & 2/3 (fixed) \\
& $ 1.28 \pm 0.19 \, {\rm stat} \; [ \pm 0.24 \, {\rm tot} ] $ & 5/3 (fixed) & $2.06 \pm 2.14 \, {\rm stat} \; [ \pm 2.48 \, {\rm tot}]$\\
\hline
\end{tabular}
\end{center}
\end{table*}
We now focus on the mass dependence of the relation. We scale the SZ flux with the expected redshift dependence and plot it as a function of mass. The result is shown in Fig.~\ref{fig:cy_m_loglog} for the high mass end. The figure shows a very good agreement between the data points and the model, which is confirmed by fitting the relation to the individual SZ flux measurements (see next Section).
\subsection{$Y_{500}$--$M_{500}$ relation fit}
\label{ym_fit_sec}
Using the individual $Y_{500}$ measurements and $M_{500}$ estimated from the X--ray luminosity, we fit a relation of the form:
\begin{equation}
Y_{500} = Y^*_{500} \; \left ({M_{500} \over 3 \times 10^{14} h^{-1} M_\odot} \right )^{\alpha_Y} \; E(z)^{\beta_Y} \; \left ({D_{ang}(z) \over 500 \, {\rm Mpc}} \right )^{-2}
\end{equation}
The results are presented in Table~\ref{fit_param_un} for the analysis undertaken using the {\gwpfont REXCESS}\ and that using the intrinsic $\LX$--$\Mv$\ relation. The pivot mass $3 \times 10^{14} h^{-1} M_\odot$, close to that used by \citet{arnaud09}, is slightly larger than the average mass of the sample ($2.8|2.5 \, \times 10^{14} M_\odot$ for the {\gwpfont REXCESS}|intrinsic $\LX$--$\Mv$\ relation, respectively). We use a non-linear least-squares fit built on a gradient-expansion algorithm (IDL curvefit function). In the fitting procedure, only the statistical errors given by the matched multifilter ($\sigma_{Y_{500}}$) are taken into account. The total errors on the final fitted parameters, taking into account uncertainties in X--ray properties, are estimated by Monte Carlo as described in Sec.~\ref{err_bud}.
We first discuss the results obtained using the {\gwpfont REXCESS}\ $\LX$--$\Mv$\ relation, which is expected to be close to the optimal case (see discussion in Sec.~\ref{mal_bias}).
First, we keep the mass and redshift dependence fixed to the self-similar expectation ($\alpha_{\rm Y}=5/3$, $\beta_{\rm Y}=2/3$) and we fit only the normalisation. We obtain $Y^*_{500} = 1.60 \times 10^{-3} \, \left ({h/0.719} \right )^{-2} \, {\rm arcmin}^2$, in agreement with the X-ray prediction $Y^*_{500}=1.54 \times 10^{-3} \, \left ({h/0.719} \right )^{-5/2} \, {\rm arcmin}^2$ (at $ 0.4 \sigma$). Then, we relax the constraint on $\alpha_{\rm Y}$ and fit the normalisation and mass dependence at the same time. We obtain a value for $\alpha_{\rm Y}=1.79$, slightly greater than the self-similar expectation (5/3) by $0.8\,\sigma$. To study the redshift dependence of the effect, we fix the mass dependence to $\alpha_{\rm Y}=5/3$ and fit $Y^*_{500}$ and $\beta_{\rm Y}$ at the same time. We obtain a somewhat stronger evolution $\beta_{\rm Y}=1.05$ than the self-similar expectation (2/3). The difference, however, is not significant ($0.2 \sigma$). As already mentioned above (see also Fig.~\ref{fig:evolution} right), the redshift leverage is too small to get interesting constraints on $\beta_{\rm Y}$.
As cluster mass estimates depend on the assumption of the underlying $\LX$--$\Mv$\ relation, so does the derived $\Yv$--$\Mv$\ relation as well. However, the effect is small. The normalisation is shifted from $ \left (1.60 \pm 0.14 \, {\rm stat} \; [ \pm 0.19 \, {\rm tot} ] \right ) \, 10^{-3} \, {\rm arcmin}^2$ to $ \left (1.37 \pm 0.12 \, {\rm stat} \; [ \pm 0.17 \, {\rm tot} ] \right ) \, 10^{-3} \, {\rm arcmin}^2$ when using the intrinsic $\LX$--$\Mv$\ relation. The difference is less than two statistical sigmas, and for the mass exponent, it is less than one.
\section{Discussion and conclusions}
\label{discussion}
In this paper we have investigated the SZ effect and its scaling with mass and X-ray luminosity using WMAP 5-year data of the largest published X-ray-selected cluster catalogue to date, derived from the combined NORAS and REFLEX samples. Cluster SZ flux estimates were made using an optimised multifrequency matched filter. Filter optimisation was achieved through priors on the pressure distribution (i.e., cluster shape) and the integration aperture (i.e., cluster size). The pressure distribution is assumed to follow the universal pressure profile of \citet{arnaud09}, derived from X-ray observations of the representative local {\gwpfont REXCESS}\ sample. This profile is the most realistic available for the general population at this time, and has been shown to be in good agreement with recent high-quality SZ observations from SPT \citep{pla09}. Furthermore, our analysis takes into account the dispersion in the pressure distribution. The integration aperture is estimated from the $L_{500}-M_{500}$ relation of the same {\gwpfont REXCESS}\ sample. We emphasise that these two priors determine only the input spatial distribution of the SZ flux for use by the multifrequency matched filters; the priors give no information on the amplitude of the measurement. As the analysis uses minimal X-ray data input, the measured and X-ray predicted SZ fluxes are essentially independent.
We studied the $Y_{SZ}-L_X$ relation using both bin averaged analyses and individual flux measurements. The fits using individual flux measurements give quantitative results for calibrating the scaling laws. The bin averaged analyses allow a direct quantitative check of SZ flux measurements versus X-ray model predictions based on the universal pressure profile derived by \citet{arnaud09} from {\gwpfont REXCESS}. An excellent agreement is found.
Using WMAP 3-year data, both \citet{2006ApJ...648..176L} and \citet{bie07} found that the SZ signal strength is lower than predicted given expectations from the X-ray properties of their clusters, concluding that that there is some missing hot gas in the intra-cluster medium. The excellent agreement between the SZ and X-ray properties of the clusters in our sample shows that there is in fact no deficit in SZ signal strength relative to expectations from X-ray observations. Due to the large size and homogeneous nature of our sample, and the internal consistency of our baseline cluster model, we believe our results to be robust in this respect. We note that there is some confusion in the literature regarding the phrase `missing baryons'. The `missing baryons' mentioned by \citet{2007MNRAS.378..293A} in the WMAP 3-year data are missing with respect to the universal baryon fraction, but not with respect to the expectations from X-ray measurements. \citet{2007MNRAS.378..293A} actually found good agreement between the strength of the SZ signal and the X-ray properties of their cluster sample, a conclusion that agrees with our results. This good convergence between SZ direct measurements and X--ray data is an encouraging step forward for the prediction and interpretation of SZ surveys.
Using $L_{500}$ as a mass proxy, we also calibrated the $\Yv$--$\Mv$\ relation, finding a normalisation in excellent agreement with X-ray predictions based on the universal pressure profile, and a slope consistent with self-similar expectations. However, there is some indication that the slope may be steeper, as also indicated from the {\gwpfont REXCESS}\ analysis when using the best fitting empirical $M_{\rm 500}$--$Y_{\rm X}$ relation \citep{arnaud09}. $M_{500}$ depends on the assumed $L_{500}-M_{500}$ relation, making the derived $\Yv$--$\Mv$\ relation sensitive to Malmquist bias which we cannot fully account for in our analysis. However, we have shown that the effect of Malmquist bias on the present results is less than $2 \sigma$ (statistical).
Regarding evolution, we have shown observationally that the SZ flux is indeed sensitive to the angular size of the cluster through the diameter distance effect. For a given mass, a low redshift cluster has a bigger integrated SZ flux than a similar system at high redshift, and the redshift dependence of the integrated SZ flux is dominated by the angular diameter distance ($\propto D_{ang}^2(z)$ ). However, the redshift leverage of the present cluster sample is too small to put strong contraints on the evolution of the $\Yv$--$\LX$\ and $\Yv$--$\Mv$\ relations. We have nevertheless checked that the observed evolution is indeed compatible with the self-similar prediction.
In this analysis, we have compensated for the poor sensitivity and resolution of the WMAP experiment (regarding SZ science) with the large number of known ROSAT clusters, leading to self-consistent and robust results. We expect further progress using upcoming Planck all-sky data. While Planck will offer the possibility of detecting the clusters used in this analysis to higher precision, thus significantly reducing the uncertainty on individual measurements, the question of evolution will not be answered with the present RASS sample due to its limited redshift range. A complementary approach will thus be to obtain new high sensitivity SZ observations of a smaller sample. The sample must be representative, cover a wide mass range, and extend to higher z (e.g., XMM-Newton follow-up of samples drawn from Planck and ground based SZ surveys). This would deliver efficient constraints not only on the normalisation and slope of the $Y_{SZ}-L_X$ and $Y_{SZ}-M$ relations, but also their evolution, opening the way for the use of SZ surveys for precision cosmology.
\begin{acknowledgements}
The authors wish to thank the anonymous referee for useful comments. J.-B. Melin thanks R. Battye for suggesting introduction of the $h$ dependance into the presentation of the results. The authors also acknowledge the use of the HEALPix package~(\cite{2005ApJ...622..759G}) and of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. We also acknowledge use of the Planck Sky Model, developed by the Component Separation Working Group (WG2) of the Planck Collaboration, for the estimation of the radio source flux in the clusters and for the development of the matched multifilter, although the model was not directly used in the present work.
\\
\end{acknowledgements}
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{
"redpajama_set_name": "RedPajamaArXiv"
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Dr. Robert R. Redfield's salary is being reduced following reports that he was being paid more than the Health and Human Services secretary, the head of the Food and Drug Administration and the director of the National Institutes of Health.
The government will lower the $375,000 salary of the new director of the Centers for Disease Control and Prevention, Dr. Robert R. Redfield, after reports that he was being paid considerably more than previous directors, the Department of Health and Human Services confirmed on Monday, though it declined to say what his new pay will be.
Dr. Redfield, who became the C.D.C. director in March, had been given the higher salary under a provision called Title 42. It was created by Congress to allow federal agencies to offer compensation that is competitive with the private sector in order to attract top-notch scientists with expertise that the departments would not otherwise have. News reports of his earnings sparked complaints from Senate Democrats and watchdog groups.
Title 42 was not used for Dr. Redfield's predecessor, Dr. Brenda Fitzgerald, an obstetrician-gynecologist, who was paid $197,300 a year until she resigned in January, or for her predecessor, Dr. Thomas R. Frieden, an infectious disease specialist and the former health commissioner of New York City, whose salary was $219,700.
Dr. Redfield, an H.I.V./AIDS researcher at the University of Maryland School of Medicine and co-founder of its Institute of Human Virology, was also being paid more than his boss, Alex M. Azar II, the H.H.S. secretary; Dr. Scott Gottlieb, head of the Food and Drug Administration; and Dr. Francis Collins, director of the National Institutes of Health.
Each of those political appointees is paid less than $200,000 a year, and Dr. Gottlieb and Mr. Azar took much larger pay cuts in their government jobs than did Dr. Redfield, who reported $757,100 in salary and bonuses from the University of Maryland Department of Medicine for 2017 through mid-March of 2018.
domains: Who Is Neri Oxman?
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{
"redpajama_set_name": "RedPajamaC4"
}
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Q: Text processing Aptly output file I have a text file made from the output of the repository management tool aptly, which lists my published repositories, from which I need to extract information.
The file format is as follows:
Published repositories:
* test_repo_one/xenial [i386,amd64] publishes {main: [xenial-main_20190311]: Snapshot from mirror [xenial-main]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}, {multiverse: [xenial-multiverse_20190311]: Snapshot from mirror [xenial-multiverse]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}, {restricted: [xenial-restricted_20190311]: Snapshot from mirror [xenial-restricted]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}, {universe: [xenial-universe_20190311]: Snapshot from mirror [xenial-universe]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}
* test_repo_one/xenial-security [i386,amd64] publishes {main: [xenial-security-main_20190311]: Snapshot from mirror [xenial-security-main]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}, {multiverse: [xenial-security-multiverse_20190311]: Snapshot from mirror [xenial-security-multiverse]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}, {restricted: [xenial-security-restricted_20190311]: Snapshot from mirror [xenial-security-restricted]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}, {universe: [xenial-security-universe_20190311]: Snapshot from mirror [xenial-security-universe]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}
* test_repo_two/trusty [i386,amd64] publishes {main: [trusty-main_20190312]: Snapshot from mirror [trusty-main]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}, {multiverse: [trusty-multiverse_20190312]: Snapshot from mirror [trusty-multiverse]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}, {restricted: [trusty-restricted_20190312]: Snapshot from mirror [trusty-restricted]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}, {universe: [trusty-universe_20190312]: Snapshot from mirror [trusty-universe]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}
...
The last line of the output ends in a new line.
The "Published repositories:" line is not required.
For each of the lines starting ' *' I need to remove extraneous information, leaving only snapshot names. There is no way to do this in aptly. The desired output for the first of these lines is.
test_repo_one/xenial [xenial-main_20190311] [xenial-multiverse_20190311] [xenial-restricted_20190311] [xenial-universe_20190311]
The square brackets are not essential either so a solution that retains or removes these is fine. I'd prefer a sed or awk solution but anything that works would be highly appreciated.
A: Two answers in one
I've posted two answers here:
*
*A bash script which is hopefully easier to understand
*A one-liner using common Linux utilities grep, sed and cut
How the Bash script looks in operation
I've turned off gnome-terminal line wrap to make input and output files easier to read.
───────────────────────────────────────────────────────────────────────────────────────────
rick@alien:~/askubuntu$ tput rmam # Turn off line wrap
───────────────────────────────────────────────────────────────────────────────────────────
rick@alien:~/askubuntu$ cat aptfilein
Published repositories:
* test_repo_one/xenial [i386,amd64] publishes {main: [xenial-main_20190311]: Snapshot from mirr}
* test_repo_one/xenial-security [i386,amd64] publishes {main: [xenial-security-main_20190311]: }
* test_repo_two/trusty [i386,amd64] publishes {main: [trusty-main_20190312]: Snapshot from mirr}
...
───────────────────────────────────────────────────────────────────────────────────────────
rick@alien:~/askubuntu$ time aptfileparse.sh
5 lines read from aptfilein
3 lines written to aptfileout
real 0m0.025s
user 0m0.016s
sys 0m0.004s
───────────────────────────────────────────────────────────────────────────────────────────
rick@alien:~/askubuntu$ cat aptfileout
test_repo_one/xenial [xenial-main_20190311] [xenial-multiverse_20190311] [xenial-restricted_201]
test_repo_one/xenial-security [xenial-security-main_20190311] [xenial-security-multiverse_20190]
test_repo_two/trusty [trusty-main_20190312] [trusty-multiverse_20190312] [trusty-restricted_201]
───────────────────────────────────────────────────────────────────────────────────────────
rick@alien:~/askubuntu$
The actual Bash script
Remember to make the script executable with chmod a+x script.sh
#!/bin/bash
# NAME: aptfileparse.sh
# PATH: ~/askubuntu
# DESC: Parse Apt File giving new lines.
# DATE: July 1, 2019.
# NOTE: For: https://askubuntu.com/questions/1127821/text-processing-aptly-output-file
# Program would be ~10 lines shorter (but harder to read) with arrays.
: <<'END'
/* -----------------------------------------------------------------------------
INPUT FILE LAYOUT
=================
Published repositories:
* test_repo_one/xenial [i386,amd64] publishes {main: [xenial-main_20190311]: Snapshot from mirror [xenial-main]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}, {multiverse: [xenial-multiverse_20190311]: Snapshot from mirror [xenial-multiverse]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}, {restricted: [xenial-restricted_20190311]: Snapshot from mirror [xenial-restricted]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}, {universe: [xenial-universe_20190311]: Snapshot from mirror [xenial-universe]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}
* test_repo_one/xenial-security [i386,amd64] publishes {main: [xenial-security-main_20190311]: Snapshot from mirror [xenial-security-main]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}, {multiverse: [xenial-security-multiverse_20190311]: Snapshot from mirror [xenial-security-multiverse]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}, {restricted: [xenial-security-restricted_20190311]: Snapshot from mirror [xenial-security-restricted]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}, {universe: [xenial-security-universe_20190311]: Snapshot from mirror [xenial-security-universe]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}
* test_repo_two/trusty [i386,amd64] publishes {main: [trusty-main_20190312]: Snapshot from mirror [trusty-main]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}, {multiverse: [trusty-multiverse_20190312]: Snapshot from mirror [trusty-multiverse]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}, {restricted: [trusty-restricted_20190312]: Snapshot from mirror [trusty-restricted]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}, {universe: [trusty-universe_20190312]: Snapshot from mirror [trusty-universe]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}
...
OUTPUT FILE LAYOUT
==================
test_repo_one/xenial [xenial-main_20190311] [xenial-multiverse_20190311] [xenial-restricted_20190311] [xenial-universe_20190311]
Five fields to extract: name, main, multiverse, restricted, universe
----------------------------------------------------------------------------- */
END
INPUT="aptfilein"
OUTPUT="aptfileout"
> "$OUTPUT" # Erase previous output file
# Read all input lines
while IFS= read -r line ; do
let CountIn++
! [[ "$line" =~ " *" ]] && continue # skip lines not starting " *"
# Get name
line="${line#" * "}" # remove leading " * "
lout="${line%%" "*}" # name is up to next " "
line="${line#" "*}" # remove name from line
# Get main
line="${line#*"{main: "}" # remove leading "{main: "
lout="$lout ${line%%":"*}" # main is up to next ":"
line="${line#":"*}" # remove name from line
# Get multiverse
line="${line#*"{multiverse: "}" # remove leading "{multiverse: "
lout="$lout ${line%%":"*}" # maultiverse is up to next ":"
line="${line#":"*}" # remove multiverse from line
# Get restricted
line="${line#*"{restricted: "}" # remove leading "{restricted: "
lout="$lout ${line%%":"*}" # restricted is up to next ":"
line="${line#":"*}" # remove restricted from line
# Get universe
line="${line#*"{universe: "}" # remove leading "{universe: "
lout="$lout ${line%%":"*}" # universe is up to next ":"
line="${line#":"*}" # remove universe from line
# Append line to output file with leading space
echo " $lout" >> "$OUTPUT"
let CountOut++
done < "$INPUT"
echo "$CountIn lines read from $INPUT"
echo "$CountOut lines written to $OUTPUT"
One-liner with common utilities
One-liners are popular in the Linux community and there are some excellent awk and perl answers posted in this Q&A. Here is an example using common utilities most experienced command line users are familiar with:
$ time grep ^" \*" aptfilein | sed 's/ \* //;s/ /: /;s/^/ /' | cut -d':' -f1,3,6,9,12 --output-delimiter=''
test_repo_one/xenial [xenial-main_20190311] [xenial-multiverse_20190311] [xenial-restricted_20190311] [xenial-universe_20190311]
test_repo_one/xenial-security [xenial-security-main_20190311] [xenial-security-multiverse_20190311] [xenial-security-restricted_20190311] [xenial-security-universe_20190311]
test_repo_two/trusty [trusty-main_20190312] [trusty-multiverse_20190312] [trusty-restricted_20190312] [trusty-universe_20190312]
real 0m0.011s
user 0m0.003s
sys 0m0.008s
*
*grep ^" \*" aptfilein - the grep command selects lines containing a search string. The carrot (^) denotes the string must start at the beginning of the line. The backslash (\) denotes the asterisk/splat (*) is to be taken literally and not act as a wildcard character that selects everything. In summary this grep command selects all lines beginning with * in file aptfilein.
*sed is a "stream editor" that edits lines coming in and changes them and passes them out. There are three sed changes here 's/ \* //;s/ /: /;s/^/ /'. The changes are between quotes (') and delineated (separated) by a semi-colon (;) deliminator. They are broken down in next three points.
*s/ \* // - search first occurrence of * and change it to null. This will erase the * that begins at each line.
*s/ /: / - searches for the first space and changes it into a colon (:) followed by a space. This is necessary to change our first field into a key. For example test_repo_one/xenial becomes test_repo_one/xenial: .
*s/^/ / - tells sed to insert a space at the beginning of each line.
*cut -d':' -f1,3,6,9,12 --output-delimiter='' - Uses the cut command to select key fields # 1, 3, 6, 9 and 12. The key fields are delimited by a colon as argument -d':' stipulates. Normally output fields are delimited the same but this is overridden to null using --output-delimiter=''` parameter.
Note: The one-liner is faster than bash which is slower at string processing.
A: A Perl approach:
$ perl -lne 'next unless /^\s*\*\s*(\S+)/; $n=$1; @k=(/\{.+?:\s*\[(.+?)\]/g); print "$n @k"' file
test_repo_one/xenial xenial-main_20190311 xenial-multiverse_20190311 xenial-restricted_20190311 xenial-universe_20190311
test_repo_one/xenial-security xenial-security-main_20190311 xenial-security-multiverse_20190311 xenial-security-restricted_20190311 xenial-security-universe_20190311
test_repo_two/trusty trusty-main_20190312 trusty-multiverse_20190312 trusty-restricted_20190312 trusty-universe_20190312
Explanation
*
*perl -lne: read the input file line by line (-n), remove trailing newlines (-l) and run the script given by -e on each line. The -l also adds a newline to each print call.
*next unless /^\s*\*\s*(\S+)/; : find the name of the repo, so the first stretch of non-whitespace characters (\S+) on a line that starts with 0 or more whitespace characters (^\s*), then a * (\*), and 0 or more whitespace characters again. The longest stretch of non-whitespace after that is what we want. If this line doesn't match this regex, the next will move us onto the next line.
*$n=$1 : save what was captured by the match above (the (\S+) in parentheses, $1) as $n.
*@k=(/\{.+?:\s*\[(.+?)\]/g): find all cases where we have a {, any other characters and then a :, followed by whitespace and a [ and capture anything between the [ and the ]. Save all matching strings in the array @k.
*print "$n @k" : finally, print the name of the repo, the $n, and the array @k from above.
If you prefer to have the square brackets included, you can use:
$ perl -lne 'next unless /^\s*\*\s*(\S+)/; $n=$1; @k=(/\{.+?:\s*(\[.+?\])/g); print "$n @k"' file
test_repo_one/xenial [xenial-main_20190311] [xenial-multiverse_20190311] [xenial-restricted_20190311] [xenial-universe_20190311]
test_repo_one/xenial-security [xenial-security-main_20190311] [xenial-security-multiverse_20190311] [xenial-security-restricted_20190311] [xenial-security-universe_20190311]
test_repo_two/trusty [trusty-main_20190312] [trusty-multiverse_20190312] [trusty-restricted_20190312] [trusty-universe_20190312]
A: My awk approach:
$ cat 1.txt
Published repositories:
* test_repo_one/xenial [i386,amd64] publishes {main: [xenial-main_20190311]: Snapshot from mirror [xenial-main]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}, {multiverse: [xenial-multiverse_20190311]: Snapshot from mirror [xenial-multiverse]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}, {restricted: [xenial-restricted_20190311]: Snapshot from mirror [xenial-restricted]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}, {universe: [xenial-universe_20190311]: Snapshot from mirror [xenial-universe]: http//gb.archive.ubuntu.com/ubuntu/ xenial [src]}
* test_repo_one/xenial-security [i386,amd64] publishes {main: [xenial-security-main_20190311]: Snapshot from mirror [xenial-security-main]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}, {multiverse: [xenial-security-multiverse_20190311]: Snapshot from mirror [xenial-security-multiverse]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}, {restricted: [xenial-security-restricted_20190311]: Snapshot from mirror [xenial-security-restricted]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}, {universe: [xenial-security-universe_20190311]: Snapshot from mirror [xenial-security-universe]: http//gb.archive.ubuntu.com/ubuntu/ xenial-security[src]}
* test_repo_two/trusty [i386,amd64] publishes {main: [trusty-main_20190312]: Snapshot from mirror [trusty-main]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}, {multiverse: [trusty-multiverse_20190312]: Snapshot from mirror [trusty-multiverse]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}, {restricted: [trusty-restricted_20190312]: Snapshot from mirror [trusty-restricted]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}, {universe: [trusty-universe_20190312]: Snapshot from mirror [trusty-universe]: http//gb.archive.ubuntu.com/ubuntu/ trusty[src]}
$ awk '$1=="*"{split ($0, a, /:/); print $2 a[2] a[5] a[8] a[11]}' 1.txt
test_repo_one/xenial [xenial-main_20190311] [xenial-multiverse_20190311] [xenial-restricted_20190311] [xenial-universe_20190311]
test_repo_one/xenial-security [xenial-security-main_20190311] [xenial-security-multiverse_20190311] [xenial-security-restricted_20190311] [xenial-security-universe_20190311]
test_repo_two/trusty [trusty-main_20190312] [trusty-multiverse_20190312] [trusty-restricted_20190312] [trusty-universe_20190312]
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Q: Can I allow my database to be accessed from anywhere if I have "good" credentials? I'm deploying a Node/React app to DigitalOcean Apps(similar to Heroku) and a Database to MongoDB Atlas. DigitalOcean do not provide external fixed IP addresses.
I want to secure the connection between the Server and database. Obviously I can't use the whitelist feature, since my server doesn't have a fixed IP.
My questions are:
*
*Can I allow the database to be accessed from anywhere if I put my app to production?
*What can I do to fix this problem? -how can I secure the connection in my server-database setup?
|
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"redpajama_set_name": "RedPajamaStackExchange"
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@stinaface Giiirrrlll @instagram just shared this!!!
@tamtam1936 Congratulations on your feature this is so pretty.
@laderalady I love that place! but how in the world did you find a time when there were no people???
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{
"redpajama_set_name": "RedPajamaC4"
}
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Пегі Люїндюла (, 25 травня 1979, Кіншаса) — французький футболіст, що грав на позиції нападника.
Насамперед відомий виступами за «Олімпік» (Ліон), «Парі Сен-Жермен» та національну збірну Франції.
Дворазовий володар Кубка Франції. Триразовий чемпіон Франції. Дворазовий володар Суперкубка Франції.
Клубна кар'єра
У дорослому футболі дебютував 1997 року виступами за команду клубу «Ніор», в якій провів один сезон, взявши участь у 27 матчах чемпіонату.
Протягом 1998—2001 років захищав кольори команди клубу «Страсбур». За цей час виборов титул володаря Кубка Франції.
Своєю грою за останню команду привернув увагу представників тренерського штабу клубу «Олімпік» (Ліон), до складу якого приєднався 2001 року. Відіграв за команду з Ліона наступні три сезони своєї ігрової кар'єри. Більшість часу, проведеного у складі ліонського «Олімпіка», був основним гравцем атакувальної ланки команди. У складі ліонського «Олімпіка» був одним з головних бомбардирів команди, маючи середню результативність на рівні 0,36 голу за гру першості. За цей час додав до переліку своїх трофеїв три титули чемпіона Франції, ставав володарем Суперкубка Франції.
Згодом з 2004 по 2007 рік грав у складі команд клубів «Олімпік» (Марсель), «Осер» та «Леванте».
До складу клубу «Парі Сен-Жермен» приєднався 2007 року. Наразі встиг відіграти за паризьку команду 130 матчів в національному чемпіонаті.
Виступи за збірні
Протягом 1999–2002 років залучався до складу молодіжної збірної Франції. На молодіжному рівні зіграв у 26 офіційних матчах, забив 14 голів.
У 2004 році дебютував в офіційних матчах у складі національної збірної Франції. Провів у формі головної команди країни 6 матчів, забивши 1 гол.
Титули і досягнення
Володар Кубка Франції (2):
«Страсбур»: 2000–01
«Парі Сен-Жермен»: 2009–10
Чемпіон Франції (3):
«Олімпік» (Ліон): 2001-02, 2002-03, 2003-04
Володар Кубку французької ліги (1):
«Парі Сен-Жермен»: 2007-08
Володар Суперкубка Франції (2):
«Олімпік» (Ліон): 2002, 2003
Посилання
Статистика виступів на footballdatabase.eu
Французькі футболісти
Гравці молодіжної збірної Франції з футболу
Гравці збірної Франції з футболу
Футболісти «Ніора»
Футболісти «Страсбура»
Футболісти «Олімпіка» (Ліон)
Футболісти «Олімпіка» (Марсель)
Футболісти «Осера»
Футболісти «Леванте»
Футболісти «Парі Сен-Жермен»
Футболісти «Нью-Йорк Ред Буллз»
Французькі футбольні легіонери
Футбольні легіонери в Іспанії
Футбольні легіонери у США
Уродженці Кіншаси
Конголезькі емігранти до Франції
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\section{Introduction}
Rare B decays induced by the flavor-changing neutral current~(FCNC) occur at loop level in the Standard Model~(SM) and thus proceed at a low rate.
They can provide useful information on the parameters of the SM and test its predictions. Meanwhile, they offer a valuable possibility of an indirect search of
new physics~(NP) for their sensitivity to the gauge structure and new contributions. Experimentally, the fruitful running of BABAR, Belle and Tevatron in the past decade provides a very fertile ground for testing SM and probing possible NP effects. As particle physics is entering the era of LHC, $B_s$ physics has attracted much more attention.
Recently, CDF collaboration has reported the first observation of the rare semileptonic $\bar{B}_s \to \phi\mu^+\mu^-$ decay and measured its branching fraction to be~\cite{CDFPhill}
\begin{equation} \label{eq:Heff}
{\cal B}(\bar{B}_s \to \phi\mu^+\mu^-)=[1.44\pm0.33({\rm stat.})\pm0.46({\rm syst.})]\times10^{-6}\,\quad {\rm CDF\,.}
\end{equation}
Theoretically, many evaluations for $\bar{B}_s \to \phi\mu^+\mu^-$ decay have been done within both SM and various NP scenarios (for example, Refs.~\cite{Erkol,Bsphill}). The SM prediction for ${\cal B}^{SM}(\bar{B}_s \to \phi\mu^+\mu^-)$~($\sim1.65\times10^{-6}$(QCDSR)~\cite{Erkol}, for example) agrees well with CDF measurement$~(1.44\pm0.57)\times10^{-6}$ for large experimental error. If more exact measurement on $\bar{B}_s \to \phi\mu^+\mu^-$ is gotten by the running LHC-b and future super-B, the possible NP space will be strongly constrained or excluded. So, it is worth evaluating the effects of the possible NP, such as a family non-universal $Z^{\prime}$ boson, on $\bar{B}_s \to \phi\mu^+\mu^-$ decay.
A new family non-universal $Z^\prime$ boson could be naturally derived in certain
string constructions~\cite{string}, $E_6$ models~\cite{E6} and so
on. Searching for such an extra $Z^{\prime}$ boson is an important
mission in the experimental programs of Tevatron~\cite{Tevatron} and
LHC~\cite{LHC}. The general framework for non-universal $Z^{\prime}$ model has been developed in Ref.~\cite{Langacker}. Within such model, FCNC in $b\to s$ and $d$ transitions could be induced by family non-universal $U(1)^\prime$ gauge symmetries at tree level. Its effects on $b\to s$ transition have attracted much more attention and been widely studied. Interestingly, the behavior of a family non-universal $Z^\prime$ boson is helpful to resolve many puzzles in $B_{(u,d,s)}$ decays, such as ``$\pi K$ puzzle''~\cite{Barger1,Chang1}, anomalous $\bar{B}_s-B_s$ mixing phase~\cite{Liu,Chang2} and mismatch in $A_{FB}(B\to K^{\ast}\mu^{+}\mu^{-})$ spectrum at low $q^2$ region~\cite{Chang3,CDLv}.
Within a family non-universal $Z^\prime$ model, $\bar{B}_s \to \phi\mu^+\mu^-$ decay involves $b-s-Z^{\prime}$ and $\mu-\mu-Z^{\prime}$ couplings, which have been strictly bounded by the constraints from $\bar{B}_s-B_s$ mixing, $B\to\pi K^{(\ast)}$, $\rho K$, $\bar{B}_d\to X_s\mu\mu$, $K^{(\ast)}\mu\mu$ decays and so on~\cite{Chang1,Chang2,Chang3}.
So, it is worth evaluating the effects of a non-universal $Z^\prime$ boson on $\bar{B_s}\to\phi\mu^+\mu^-$ decay and checking whether such settled values of $Z^\prime$ couplings are permitted by CDF measurement on ${\cal B}(\bar{B}_s \to \phi\mu^+\mu^-)$.
Our paper is organized as follows. In Section~2,
we briefly review the theoretical framework for $b\to s l^+ l^-$ decay within both SM and a family
non-universal $Z^{\prime}$ model. In Section~3, the effects of
a non-universal $Z^{\prime}$ boson on $\bar{B}_s \to \phi\mu^+\mu^-$ decay are investigated in detail. Our conclusions are summarized in
Section~4. Appendix~A and B include all of the theoretical input
parameters.
\section{The theoretical framework for $b\to s l^+ l^-$ decays}\label{theo}
In the SM, neglecting the doubly Cabibbo-suppressed contributions, the effective Hamiltonian governing semileptonic $b\to s \ell^+\ell^-$ transition is given by~\cite{Altmannshofer:2008dz,Chetyrkin:1996vx}
\begin{equation} \label{eq:Heff}
{\cal H}_{{\text{eff}}} = - \frac{4\,G_F}{\sqrt{2}}V_{tb}V_{ts}^{\ast}
\sum_{i=1}^{10} C_i(\mu) O_i(\mu) \,.
\end{equation}
Here we choose the operator basis given by Ref.~\cite{Altmannshofer:2008dz}, in which
\begin{eqnarray}\label{O910}
O_9=\frac{e^2}{g_s^2}(\bar{d}\gamma_\mu P_Lb)(\bar{l}\gamma^\mu l)\,,\quad O_{10}=\frac{e^2}{g_s^2}(\bar{d}\gamma_\mu P_Lb)(\bar{l}\gamma^\mu\gamma_5 l)\,.
\end{eqnarray}
Wilson coefficients $C_i$ can be calculated perturbatively~\cite{Beneke:2001at,bobeth,bobeth02,Huber:2005ig}, with the numerical results listed in Table~\ref{wc}.
The effective coefficients $C_{7,9}^{eff}$, which are particular combinations of $C_{7,9}$ with the other $C_i$, are defined as~\cite{Altmannshofer:2008dz}
\begin{eqnarray}\label{eq:effWC}
&& C_7^{\rm eff} = \frac{4\pi}{\alpha_s}\, C_7 -\frac{1}{3}\, C_3 -
\frac{4}{9}\, C_4 - \frac{20}{3}\, C_5\, -\frac{80}{9}\,C_6\,,
\nonumber\\
&& C_9^{\rm eff} = \frac{4\pi}{\alpha_s}\,C_9 + Y(q^2)\,, \qquad
C_{10}^{\rm eff} = \frac{4\pi}{\alpha_s}\,C_{10}\,,
\end{eqnarray}
in which $Y(q^2)$ denotes the matrix element of four-quark operators and given by
\begin{eqnarray}
Y(q^2)&=&h(q^2,m_c)\big(\frac{4}{3}C_1+C_2+6C_3+60C_5\big)-\frac{1}{2}h(q^2,m_b)\big(7C_3+\frac{4}{3}C_4+76C_5+\frac{64}{3}C_6\big)\,\nonumber\\
&&-\frac{1}{2}h(q^2,0)\big(C_3+\frac{4}{3}C_4+16C_5+\frac{64}{3}C_6\big)+\frac{4}{3}C_3+\frac{64}{9}C_5+\frac{64}{27}C_6\,.
\end{eqnarray}
We have neglected the long-distance contribution mainly due to $J/\Psi$ and $\Psi^{\prime}$ in the decay chain $\bar{B}_s \to \phi \Psi^{(\prime)}\to\phi l^+l^-$,
which could be vetoed experimentally~\cite{CDFPhill}. For the recent detailed discussion of such resonance effects, we refer to Ref.~\cite{resEff}.
\begin{table}[htbp]
\begin{center}
\caption{The SM Wilson coefficients at the scale $\mu=m_b$.}
\label{wc}
\vspace{0.3cm}
\doublerulesep 0.5pt \tabcolsep 0.07in
\begin{tabular}{lccccccccccc}
\hline \hline
$C_1(m_b)$& $C_2(m_b)$& $C_3(m_b)$& $C_4(m_b)$& $C_5(m_b)$& $C_6(m_b)$& $C_7^{\rm eff}(m_b)$& $C_9^{\rm eff}(m_b)-Y(q^2)$& $C_{10}^{\rm eff}(m_b)$\\\hline
$-0.284$ & $1.007$ & $-0.004$ & $-0.078$ & $0.000$ & $0.001$ & $-0.303$ & $4.095$ & $-4.153$\\
\hline \hline
\end{tabular}
\end{center}
\end{table}
Although there are quite a lot of interesting observables in semileptonic $b\to s \ell^+ \ell^-$ decay, we shall focus only on the dilepton invariant mass spectrum and the forward-backward asymmetry in this paper. Adopting the same convention and notation as \cite{Ali:1999mm}, the dilepton invariant mass spectrum and forward-backward asymmetry for $\bar{B}_s\to \phi\ell^+\ell^-$ decay is given as
\begin{eqnarray}
\frac{{\rm d} \Gamma^{\phi}}{{\rm d}\hat{s}} & = &
\frac{G_F^2 \, \alpha^2 \, m_{B_s}^5}{2^{10} \pi^5}
\left| V_{ts}^\ast V_{tb} \right|^2 \, {\hat{u}}(\hat{s}) \,
\Bigg\{
\frac{|A|^2}{3} \hat{s} {\lambda} (1+2 \frac{\mh_\l^2}{\hat{s}}) +|E|^2 \hat{s}
\frac{{\hat{u}}(\hat{s})^2}{3}
\Bigg.
\nonumber \\
& & + \Bigg. \frac{1}{4 \mh_{\phi}^2} \left[
|B|^2 ({\lambda}-\frac{{\hat{u}}(\hat{s})^2}{3} + 8 \mh_{\phi}^2 (\hat{s}+ 2 \mh_\l^2) )
+ |F|^2 ({\lambda} -\frac{ {\hat{u}}(\hat{s})^2}{3} + 8 \mh_{\phi}^2 (\hat{s}- 4 \mh_\l^2))
\right]
\Bigg.
\nonumber \\
& & +\Bigg.
\frac{{\lambda} }{4 \mh_{\phi}^2} \left[ |C|^2 ({\lambda} - \frac{{\hat{u}}(\hat{s})^2}{3})
+ |G|^2 \left({\lambda} -\frac{{\hat{u}}(\hat{s})^2}{3}+4 \mh_\l^2(2+2 \mh_{\phi}^2-\hat{s}) \right)
\right]
\Bigg.
\nonumber \\
& & - \Bigg.
\frac{1}{2 \mh_{\phi}^2}
\left[ {\rm Re}(BC^\ast) ({\lambda} -\frac{ {\hat{u}}(\hat{s})^2}{3})(1 - \mh_{\phi}^2 -
\hat{s})
\nonumber \right. \Bigg.\\
& & + \left. \Bigg.
{\rm Re}(FG^\ast) (({\lambda} -\frac{ {\hat{u}}(\hat{s})^2}{3})(1 - \mh_{\phi}^2 - \hat{s}) +
4 \mh_\l^2 {\lambda}) \right]
\Bigg.
\nonumber \\
& & - \Bigg.
2 \frac{\mh_\l^2}{\mh_{\phi}^2} {\lambda} \left[ {\rm Re}(FH^\ast)-
{\rm Re}(GH^\ast) (1-\mh_{\phi}^2) \right] +\frac{\mh_\l^2}{\mh_{\phi}^2} \hat{s} {\lambda} |H|^2
\Bigg\} \,;
\label{eq:dwbvll}
\end{eqnarray}
\begin{eqnarray}\label{EqAFB}
\frac{{\rm d} {\cal A}_{\rm FB}^{\phi}}{{\rm d} \hat{s}}& =&
-\frac{G_F^2 \, \alpha^2 \, m_{B_s}^5}{2^{8} \pi^5}
\left| V_{ts}^\ast V_{tb} \right|^2 \, \hat{s} \, {\hat{u}}(\hat{s})^2 \, \nonumber \\
& & \hspace{-0.5cm} \times \left[ {\rm Re}(\cn^{\rm eff}{\c_{10}^{\rm eff}}^{\ast}) V A_1+ \frac{\mh_b}{\hat{s}}
{\rm Re}(\cs^{\rm eff}{\c_{10}^{\rm eff}}^{\ast}) {\Big (}V T_2 (1-\mh_{\phi})+ A_1 T_1
(1+\mh_{\phi}){\Big)} \right]\,,
\end{eqnarray}
with $s=q^2$ and $\hat{s}=s/m_{B_s}^2$. Here the auxiliary functions $A\,,B\,,C\,,E\,,F$ and $G$, with the explicit expressions given in Ref.~\cite{Ali:1999mm}, are combinations of the effective Wilson coefficients in Eq.~(\ref{eq:effWC}) and the $B_s\to\phi$ transition form factors, which are calculated with light-cone QCD sum rule approach in Ref.~\cite{Ball:2004rg} and given in Appendix B.
From the experimental point of view, the normalized forward-backward asymmetry is more useful, which is defined
as~\cite{Ali:1999mm}
\begin{equation} \label{eq:AFB2}
\frac{{\rm d} \bar{{\cal A}}_{\rm FB}}{{\rm d} \hat{s}} =
\frac{{\rm d} {\cal A}_{\rm FB}}{{\rm d} \hat{s}}/\frac{{\rm d} \Gamma}{{\rm d}\hat{s}}\,.
\end{equation}
A new family non-universal $Z^\prime$ boson could be naturally derived in many extension of SM. One of the possible way to get such non-universal $Z^\prime$ boson is to include an addition $U^{\prime}(1)$ gauge symmetry, which has been formulated in detail by Langacker and Pl\"{u}macher~\cite{Langacker}. Under the assumption that the couplings of right-handed quark flavors with $Z^{\prime}$ boson are diagonal, the $Z^{\prime}$ part of the effective Hamiltonian for $b\to s l^+ l^-$ transition can be written as~\cite{Liu}
\begin{equation}\label{ZPHbsll}
{\cal H}_{eff}^{Z^{\prime}}(b\to sl^+l^-)=-\frac{2G_F}{\sqrt{2}}
V_{tb}V^{\ast}_{ts}\Big[-\frac{B_{sb}^{L}B_{ll}^{L}}{V_{tb}V^{\ast}_{ts}}
(\bar{s}b)_{V-A}(\bar{l}l)_{V-A}-\frac{B_{sb}^{L}B_{ll}^{R}}{V_{tb}V^{\ast}_{ts}}
(\bar{s}b)_{V-A}(\bar{l}l)_{V+A}\Big]+{\rm h.c.}\,.
\end{equation}
With the assumption that no significant RG running effect between $M_{Z^{\prime}}$ and $M_W$ scales, $Z^{\prime}$ contributions could be treated as modification to wilson coefficients, i.e. $C_{9,10}^{\prime}(M_W)=C_{9,10}^{SM}(M_W)+\triangle C_9^{\prime}(M_W)$. As a result, Eq.~(\ref{ZPHbsll}) could also be reformulated as
\begin{equation}\label{ZPHbsllMo}
{\cal H}_{eff}^{Z^{\prime}}(b\to sl^+l^-)=-\frac{4G_F}{\sqrt{2}}
V_{tb}V^{\ast}_{ts}\left[\triangle C_9^{\prime} O_9+\triangle C_{10}^{\prime} O_{10}\right]+{\rm h.c.}\,,
\end{equation}
with
\begin{eqnarray}\label{C910Zp}
\triangle C_9^{\prime}(M_W)&=&-\frac{g_s^2}{e^2}\frac{B_{sb}^L
}{V_{ts}^{\ast}V_{tb}} S_{ll}^{LR}\,,\quad S_{ll}^{LR}=(B_{ll}^{L}+B_{ll}^{R})\,,\nonumber \\
\triangle C_{10}^{\prime}(M_W)&=&\frac{g_s^2}{e^2}\frac{B_{sb}^L
}{V_{ts}^{\ast}V_{tb}} D_{ll}^{LR}\,,\quad\,\, D_{ll}^{LR}=(B_{ll}^{L}-B_{ll}^{R})\,.
\end{eqnarray}
$B_{sb}^L$ and $B_{ll}^{L,R}$ denote the effective chiral $Z^{\prime}$ couplings to quarks and leptons, in which the off-diagonal element $B_{sb}^L$ can contain a new weak phase and could be written as $|B_{sb}^L|e^{i\phi_s^{L}}$.
To include $Z^{\prime}$ contributions, one just needs to make the replacements
\begin{eqnarray}\label{C910ERp}
C_{9}^{\rm eff}&\to&\bar{C}_9^{\rm eff}=\frac{4\pi}{\alpha_s}C_9^{\prime}+Y(q^2)\;,\nonumber\\
C_{10}^{\rm eff}&\to&\bar{C}_{10}^{\rm eff}=\frac{4\pi}{\alpha_s}C_{10}^{\prime}\;,
\end{eqnarray}
in the formalisms relevant to $\bar{B}_s\to \phi \ell^{+}\ell^{-}$.
\section{Numerical analyses and discussions}
\begin{table}[t]
\begin{center}
\caption{Predictions for ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)[\times 10^{-6}]$ and $A_{FB}(\bar{B}_s\to \phi \mu^+\mu^-)[\times 10^{-2}]$ within the SM and the non-universal $Z^{\prime}$ model.}
\label{tab_pred}
\vspace{0.5cm}
\small
\doublerulesep 0.7pt \tabcolsep 0.1in
\begin{tabular}{lccccccccccc} \hline \hline
&Exp.~\cite{HFAG} & SM & S1 &S2 &Scen.~I &Scen.~II & Scen.~III
\\\hline
${\cal B}$ &$1.44\pm0.57$ &$1.46\pm0.10$ &$2.47\pm1.18$ &$1.40\pm0.27$ &$2.86$ &$1.26$ &$1.92$ \\
${\cal B}^L$&--- &$0.34\pm0.04$ &$0.56\pm0.27$ &$2.61\pm0.19$ &$0.64$ &$0.28$ &$0.44$\\
${\cal B}^H$&--- &$0.29\pm0.02$ &$0.51\pm0.25$ &$1.26\pm0.08$ &$0.59$ &$0.26$ &$0.39$\\
\hline
$A_{FB}$ &--- &$25.6\pm1.2$ &$19.4\pm10.9$ &$24\pm0.03$ &$29.9$ &$26.6$ &$8.9$\\
$A_{FB}^L$ &--- &$5.7\pm0.6$ &$6.0\pm7.4$ &$0.09\pm0.02$ &$13.3$ &$6.9$ &$1.4$\\
$A_{FB}^H$ &--- &$34.1\pm0.2$ &$22.5\pm12.9$ &$-0.07\pm0.01$&$35.0$ &$34.8$ &$13.1$\\
\hline \hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[ht]
\begin{center}
\epsfxsize=15cm \centerline{\epsffile{spectr.eps}}
\centerline{\parbox{16cm}{\caption{\label{spectr}\small Dimuon invariant mass distribution and normalized forward-backward asymmetry of the $\bar{B}_s\to \phi \mu^+\mu^-$ decay within SM and three limiting scenarios.}}}
\end{center}
\end{figure}
\begin{table}[t]
\begin{center}
\caption{The inputs parameters for the $Z^{\prime}$ couplings~\cite{Chang2,Chang3}. }
\label{NPPara_value}
\vspace{0.5cm}
\doublerulesep 0.7pt \tabcolsep 0.1in
\begin{tabular}{lccccccccccc} \hline \hline
& $|B_{sb}^L|(\times10^{-3})$ & $\phi_{s}^L[^{\circ}]$ &$S^{LR}_{\mu\mu}(\times10^{-2})$ & $D^{LR}_{\mu\mu}(\times10^{-2})$\\\hline
S1 & $1.09\pm0.22$ & $-72\pm7$ &$-2.8\pm3.9$ & $-6.7\pm2.6$ \\
S2 & $2.20\pm0.15$ & $-82\pm4$ &$-1.2\pm1.4$ & $-2.5\pm0.9$ \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
With the relevant theoretical formulas collected in
Section~\ref{theo} and the input parameters summarized in the
Appendix, we now proceed to present our numerical analyses and
discussions.
In Table~\ref{tab_pred}, we present our theoretical predictions for integrated branching fraction and
forward-backward asymmetry of $\bar{B}_s\to \phi \mu^+\mu^-$ decay. Within the SM, we again find our prediction ${\cal B}^{SM}(\bar{B}_s\to \phi \mu^+\mu^-)=1.46\times 10^{-6}$ is perfectly consistent with CDF measurement $(1.44\pm0.57)\times 10^{-6}$. The forward-backward asymmetry for $\bar{B}_s\to \phi \mu^+\mu^-$ decay is evaluated at $\sim25\%$, which hasn't be measured by the experiment. In addition, in Table~\ref{tab_pred}, we also calculate their results ${\cal B}^{L,H}$ and $A_{FB}^{L,H}$ at both low~($1GeV^2<s<6GeV^2$) and high~($14.4GeV^2<s<25GeV^2$) integration regions, which are sufficiently below and above the threshold for charmonium resonances $J/\psi,\psi^{\prime}$ respectively. The dimuon invariant mass distribution and
forward-backward asymmetry spectrum are shown in Fig.~\ref{spectr}. As Fig.~\ref{spectr}(b) shows, similar to the situation in $\bar{B}^0\to K^{\ast}\mu^+\mu^-$ decay, the zero crossing exist in $A_{FB}$ spectrum at $s_0\sim3~GeV^2$, whose position is well-determined and free from hadronic uncertainties at the leading order in $\alpha_s$~\cite{Beneke:2001at,Ali:1999mm,Burdman:1998mk}. In $\bar{B}^0\to K^{\ast}\mu^+\mu^-$ decay, the $A_{FB}$ spectrum measured by Belle collaboration~\cite{:2009zv} indicates that there might be no zero crossing, which presents a challenge to the SM in low $s$ region.
If the future measurement on $A_{FB}(\bar{B}_s\to \phi\mu^+\mu^-)$ spectrum presents a similar result as the one in $\bar{B}^0\to K^{\ast}\mu^+\mu^-$ decay, it will be a significant NP signal.
Within a family non-universal $Z^{\prime}$ model, the $Z^{\prime}$ contributions to $\bar{B}_s\to \phi \mu^+\mu^-$ decay involve
four new $Z^{\prime}$ parameters $|B_{sb}^L|$, $\phi_{s}^L$, $S^{LR}_{\mu\mu}$ and $D^{LR}_{\mu\mu}$. Combining the constraints from $\bar{B}_s-B_s$ mixing, $B\to\pi K^{(\ast)}$ and $\rho K$ decays, $|B_{sb}^L|$ and $\phi_{s}^L$ have been strictly constrained~\cite{Chang1,Chang2}. After having included the constraints from $\bar{B}_d\to X_s\mu\mu$, $K\mu\mu$ and $K^{\ast}\mu\mu$, as well as $B_s\to\mu\mu$ decays, we have also gotten the allowed ranges for $S^{LR}_{\mu\mu}$ and $D^{LR}_{\mu\mu}$ in Ref.~\cite{Chang3}.
For convenience, we recollect their numerical results in Table~\ref{NPPara_value}, in which S1 and S2 correspond to UTfit collaboration's two fitting results for $\bar{B}_s-B_s$ mixing~\cite{UTfit}. Our following evaluations and discussions are based on these given ranges for $Z^{\prime}$ couplings. With the values of $Z^{\prime}$ parameters listed in Table~\ref{NPPara_value} as inputs, we present our predictions for the observables in the third and fourth columns of Table~\ref{tab_pred}.
\begin{figure}[t]
\begin{center}
\epsfxsize=15cm \centerline{\epsffile{Br_ZpCoupling.eps}}
\centerline{\parbox{16cm}{\caption{\label{Br_ZpCoupling}\small The dependence of ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)$ on $S_{\mu\mu}^{LR}$ and $D_{\mu\mu}^{LR}$ within their allowed ranges in S1 and S2 with different $\phi_s^L$ values. The black dashed line corresponds to the SM result.}}}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\epsfxsize=15cm \centerline{\epsffile{AFBLHZP.eps}}
\centerline{\parbox{16cm}{\caption{\label{AFBLHZP}\small The dependence of $A_{FB}(\bar{B}_s\to \phi\mu^+\mu^-)$ on $S_{ud}^{L,R}$ and $D_{ud}^{L,R}$ at $s=1.5\,{\rm GeV^2}$~(a) and $s=15\,{\rm GeV^2}$~(b) with $|B_{db}^L|=1.09(\times10^{-3})$, $\phi_s^L=-72^{\circ}$~(S1) and the central values of the other theoretical input parameters. The blue planes correspond to SM results.}}}
\end{center}
\end{figure}
As illustrated in Fig.~\ref{Br_ZpCoupling}, integrated branching fraction for $\bar{B}_s\to \phi \mu^+\mu^-$ is sensitive to the $Z^{\prime}$ contributions. Obviously, $\bar{B}_s\to \phi \mu^+\mu^-$ is enhanced by the $Z^{\prime}$ contributions with large negative $S_{\mu\mu}^{LR}$, $D_{\mu\mu}^{LR}$ and $\phi_s^L$. Moreover, compared Fig.~\ref{Br_ZpCoupling}~(a,b) with (c,d), we find the effects of solution S1 is more significant than the one of S2. So, for simplicity, we just pay our attention to the solution S1 in the following. As Fig.~\ref{Br_ZpCoupling} shows, the $Z^{\prime}$ contributions with a small negative weak phase $\phi_s^L$ are helpful to reduce ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)$. However, because the range $\phi_s^L>-65^{\circ}$ is excluded by the constraints from $\bar{B}_s-B_s$ mixing and $B\to \pi K$ decays~\cite{Chang1,Chang2}, ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)$ is hardly to be reduced so much by $Z^{\prime}$ contributions.
In order to see the $Z^{\prime}$ effect on $A_{FB}(\bar{B}_s\to \phi \mu^+\mu^-)$ explicitly, with $Y(q^2)$ being excluded, we can
rewrite ${\rm Re}(\bar{C}_9^{\rm eff}\bar{C}_{10}^{{\rm eff}\ast})$ and ${\rm Re}(\bar{C}_7^{\rm eff}\bar{C}_{10}^{{\rm eff}\ast})$ in Eq.~(\ref{EqAFB}), which dominates $A_{FB}(\bar{B}_s\to \phi \mu^+\mu^-)$ in high and low $s$ regions respectively, as
\begin{eqnarray}\label{Ldomi}
{\rm Re}(\bar{C}_9^{\rm eff}\bar{C}_{10}^{{\rm eff}\ast})&\simeq&{\rm Re}(\bar{C}_9^{\rm eff})\,{\rm Re}(\bar{C}_{10}^{{\rm eff}\ast})+\left(\frac{4\pi}{\alpha_s}\right)^2\,{\rm Im}(\triangle C_9^{\prime})\,{\rm Im}(\triangle C_{10}^{\prime})\,,\\\label{Hdomi}
{\rm Re}(\bar{C}_7^{\rm eff}\bar{C}_{10}^{{\rm eff}\ast})&\simeq&{\rm Re}(\bar{C}_7^{\rm eff}){\rm Re}(\bar{C}_{10}^{{\rm eff}\ast})\,.
\end{eqnarray}
Combining Eq.~(\ref{C910Zp}) and Eq.~(\ref{Hdomi}), due to the tiny $Z^{\prime}$ contribution to $C_7^{\rm eff}$, the only solution to enhance $A_{FB}$ in low $s$ region is a larger negative $D^{LR}_{\mu\mu}$, which also can be found in Fig.~\ref{AFBLHZP}(a). In high $s$ region, as Fig.~\ref{AFBLHZP}~(b) shows, $A_{FB}$ could be reduced significantly and enhanced a bit by $Z^{\prime}$ contributions.
Based on the analyses above, in order to evaluate the exact strength of $Z^{\prime}$ effects, our following analyses can be divided into three limiting scenarios:
\subsubsection*{\small Scenario~I}
In order to get the maximum ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)$, within the allowed ranges for $Z^{\prime}$ couplings listed in Table~\ref{NPPara_value}, we choose a set of extreme values
\begin{eqnarray}\label{ScenarioI}
|B_{sb}^L|=1.31\times10^{-3}\,, \phi_{s}^L=-79^{\circ}\,, S^{LR}_{\mu\mu}= -6.7\times10^{-2}\,, D^{LR}_{\mu\mu}=-9.3\times10^{-2} \quad {\rm Scen.~I}\,,
\end{eqnarray}
named Scenario~I. With the central values of the other theoretical input parameters, we get ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)=2.86\times10^{-6}$, which is $2.5\sigma$ larger than CDF result $(1.44\pm0.57)\times10^{-6}$. Compared with the SM prediction $1.46\times10^{-6}$, we find ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)$ could be enhanced by about $96\%$ at most by $Z^{\prime}$ contributions.
This scenario is the most helpful solution to moderate the discrepancy for $A_{FB}(\bar{B}_s\to K^{\ast} \mu^+\mu^-)$ between SM prediction and experimental data in low $s$ region~\cite{Chang3,CDLv}. As Fig.~\ref{AFBLHZP}~(a) shows, we find Scenario~I also provides the most helpful solution to enhance $A_{FB}(\bar{B}_s\to \phi \mu^+\mu^-)$ in low $s$ region. Compared with the SM results, we find $A_{FB}^{(L)}(\bar{B}_s\to \phi \mu^+\mu^-)$ could be enhanced by about $17\%(133.3\%)$ at most. However,
in the high $s$ region, the effect of Scenario~I on $A_{FB}(\bar{B}_s\to \phi \mu^+\mu^-)$, as Fig.~\ref{AFBLHZP}~(b) shows, is not significant.
In addition, due to the strong constraints on $D^{LR}_{\mu\mu}$ from $\bar{B}_d\to X_s\mu\mu$ decay, the much larger value $|D^{LR}_{\mu\mu}|>9.3\times10^{-2}$ is forbidden~\cite{Chang2}, which means the sign of ${\rm Re}(\bar{C}_7^{\rm eff}\bar{C}_{10}^{{\rm eff}\ast})$ can hardly be flipped by $Z^{\prime}$ contributions~\cite{Chang3}. So, as Fig.~\ref{spectr}~(b) shows, the zero crossing in $A_{FB}$ spectrum also exists and moves to $s_0\sim1GeV^2$ point in this scenario.
\subsubsection*{\small Scenario~II}
From Fig.~\ref{Br_ZpCoupling}, one may find that ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)$ can hardly be reduced by $Z^{\prime}$ contributions so much within the allowed $Z^{\prime}$ parameters' ranges. The most minimal value of ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)$ appears at
\begin{eqnarray}\label{ScenarioII}
|B_{sb}^L|=1.31\times10^{-3}\,, \phi_{s}^L=-65^{\circ}\,, S^{LR}_{\mu\mu}= -2\times10^{-2}\,, D^{LR}_{\mu\mu}=-4\times10^{-2} \quad {\rm Scen.~II}\,,
\end{eqnarray}
named Scenario~II. In this scenario, compared with SM prediction, we find ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)$ could be reduced just by about $14\%$ at most by $Z^{\prime}$ contributions. Due to the small $Z^{\prime}$ contributions, its effect on $A_{FB}(\bar{B}_s\to \phi \mu^+\mu^-)$ is also tiny.
\subsubsection*{\small Scenario~III}
As Fig.~\ref{AFBLHZP}~(b) shows, $A_{FB}(\bar{B}_s\to \phi \mu^+\mu^-)$ would be reduced rapidly in high $s$ region when $S^{LR}_{\mu\mu}$ is enlarged. So, we present a limiting scenario for the minimal $A_{FB}^H(\bar{B}_s\to \phi \mu^+\mu^-)$,
\begin{eqnarray}\label{ScenarioIII}
|B_{sb}^L|=1.31\times10^{-3}\,, \phi_{s}^L=-65^{\circ}\,, S^{LR}_{\mu\mu}= 1.1\times10^{-2}\,, D^{LR}_{\mu\mu}=-9.3\times10^{-2} \quad {\rm Scen.~III}\,,
\end{eqnarray}
named Scenario~III. Compared with SM prediction, $A_{FB}^{(H)}(\bar{B}_s\to \phi \mu^+\mu^-)$ is reduced by about $62\%$~($62\%$). However, as Fig.~\ref{AFBLHZP}~(b) shows, in the low $s$ region, $A_{FB}$ is just enhanced a bit. So, this scenario also leads to the minimal $A_{FB}(\bar{B}_s\to \phi \mu^+\mu^-)\sim8.9\%$, which is $65\%$ smaller than SM prediction. While, in this scenario, our prediction ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)=1.92\times10^{-6}$ also agrees with CDF measurement within $1\sigma$. So, although Scenario~III presents a strange effects on $A_{FB}$ spectrum, it is not excluded by current measurement either.
Moreover, different from Scenario~I, zero crossing in $A_{FB}$ spectrum moves to positive side in this scenario.
\section{Conclusion}
In conclusion, motivated by recent measurement on ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)$ by CDF Collaboration, after revisiting $\bar{B}_s\to \phi \mu^+\mu^-$ decay within SM, we have investigated the effects of a family non-universal $Z^{\prime}$ boson with the given $Z^{\prime}$ couplings. Our conclusions can be summarized as:
\begin{itemize}
\item Branching fraction and forward-backward asymmetry for $\bar{B}_s\to \phi \mu^+\mu^-$ decay are sensitive to $Z^{\prime}$ contributions. All of the $Z^{\prime}$ couplings listed in Table~\ref{NPPara_value} survive under the constraint from ${\cal B}(\bar{B}_s\to \phi \mu^+\mu^-)$ measured by CDF within errors.
\item We present three limiting scenarios: ${\cal B}(\bar{B}_s\to \pi^- K^+)$ and $A_{FB}^{(L)}(\bar{B}_s\to \phi \mu^+\mu^-)$ could be enhanced by about $96\%$ and $17\%\,(133\%)$ at most by $Z^{\prime}$ contributions (Scenario~I); However, ${\cal B}(\bar{B}_s\to \pi^- K^+)$ is hardly to be reduced ( reduced by $14\%$ at most in Scenario~II) by $Z^{\prime}$ contributions; Moreover, in Scenario~III, $A_{FB}^{(H)}(\bar{B}_s\to \phi \mu^+\mu^-)$ reaches its minimal value, which is $65\%(62\%)$ lower than SM prediction.
\item The zero crossing in $A_{FB}(\bar{B}_s\to \phi \mu^+\mu^-)$ spectrum always exists in the three scenarios.
\end{itemize}
The refined measurements for the $B_{s}$ leptonic
decay $\bar{B}_s\to \phi \mu^+\mu^-$ in the upcoming LHC-b and proposed super-B
will provide a powerful testing ground for the SM and possible NP scenarios. Our analyses
of the $Z^{\prime}$ effects on the observables for $\bar{B}_s\to \phi \mu^+\mu^-$ decay are useful for
probing or refuting the effects of a family non-universal $Z^{\prime}$ boson.
\section*{Acknowledgments}
The work is supported by the National Science Foundation
under contract Nos.11075059, 10735080 and 11005032.
\begin{appendix}
\section*{Appendix A: Theoretical input parameters}
For the CKM matrix elements, we adopt the UTfit collaboration's
fitting results~\cite{UTfitCKM}
\begin{eqnarray}
\overline{\rho}&=&0.132\pm0.02\,(0.135\pm0.04), \quad
\overline{\eta}=0.367\pm0.013\,(0.374\pm0.026),\nonumber\\
A&=&0.8095\pm0.0095\,(0.804\pm0.01),\quad
\lambda=0.22545\pm0.00065\,(0.22535\pm0.00065).
\end{eqnarray}
As for the quark masses, we take~\cite{PDG10,PMass}
\begin{eqnarray}
&&m_u=m_d=m_s=0, \quad m_c=1.61^{+0.08}_{-0.12}\,{\rm GeV},\nonumber\\
&&m_b=4.79^{+0.19}_{-0.08}\,{\rm GeV}, \quad m_t=172.4\pm1.22\,{\rm GeV}.
\end{eqnarray}
\section*{Appendix B: Transition form factors from light-cone QCD sum rule}
In order to calculate the $\bar{B}_s\to\phi \ell^+ \ell^-$ decay amplitude, we have to evaluate the $\bar{B}_s\to\phi$ matrix elements of quark bilinear currents.
They can be expressed in terms of ten form factors, which depend on the momentum transfer $q^2$ between the $B_s$ and the $\phi$ mesons~($q=p - k$)~\cite{Ball:2004rg}:
\begin{eqnarray}\label{eq:SLFF}
\langle \phi(k) | \bar d\gamma_\mu(1-\gamma_5) b | \bar
B_s(p)\rangle &=& -i \epsilon^*_\mu (m_{B_s}+m_{\phi}) A_1(q^2) + i
(2p-q)_\mu (\epsilon^* \cdot q)\,
\frac{A_2(q^2)}{m_{B_s}+m_{\phi}}\, \nonumber \\
& & + i q_\mu (\epsilon^* \cdot q) \, \frac{2m_{\phi}}{q^2}\,
\Big[A_3(q^2)-A_0(q^2)\Big] \, \nonumber \\
& & + \epsilon_{\mu\nu\rho\sigma}\epsilon^{*\nu} p^\rho k^\sigma\,
\frac{2V(q^2)}{m_{B_s}+m_{\phi}}\,,
\end{eqnarray}
with $A_3(q^2) = \frac{m_{B_s}+m_{\phi}}{2m_{\phi}}\, A_1(q^2) -
\frac{m_{B_s}-m_{\phi}}{2m_{\phi}}\, A_2(q^2)$ and $A_0(0) = A_3(0)$,
\begin{eqnarray}\label{eq:pengFF}
\langle \phi(k) | \bar s \sigma_{\mu\nu} q^\nu (1+\gamma_5) b |
\bar{B}_s(p)\rangle &=& i\epsilon_{\mu\nu\rho\sigma} \epsilon^{*\nu}
p^\rho k^\sigma \, 2 T_1(q^2)\, \nonumber\\
& & + T_2(q^2) \Big[\epsilon^*_\mu (m_{B_s}^2-m_{\phi}^2)
- (\epsilon^* \cdot q) \,(2p-q)_\mu \Big] \, \nonumber\\
& & + T_3(q^2) (\epsilon^* \cdot q) \left[q_\mu -
\frac{q^2}{m_{B_s}^2-m_{\phi}^2}\, (2p-q)_\mu \right]\,,
\end{eqnarray}
with $T_1(0) = T_2(0)$. $\epsilon_\mu$ is the polarization vector of the $\phi$ meson. The physical range in $s=q^2$ extends from $s_{\rm min} = 0$ to $s_{\rm max} =(m_{B_s}-m_{\phi})^2$.
\begin{table}[t]
\begin{center}
\caption{\label{FFfit} Fit parameters for $B_s\to\phi$ transition form
factors~\cite{Ball:2004rg}.}
\vspace{0.3cm}
\begin{tabular}{crrrrrl}\hline\hline
&$F(0)$ &$r_1$ &$m_R^2$ &$r_2$ &$m^2_{\rm fit}$& \\ \hline
$V^{B_s\to\phi}$ &$0.434$ &$1.484$ &$5.32^2$ &$-1.049$ &$39.52$ &Eq.~(\ref{r12mRfit})\\\hline
$A_0^{B_s\to\phi}$ &$0.474$ &$3.310$ &$5.28^2$ &$-2.835$ &$31.57$ &Eq.~(\ref{r12mRfit})\\\hline
$A_1^{B_s\to\phi}$ &$0.311$ &--- &--- &$0.308$ &$36.54$ &Eq.~(\ref{r2mfit})\\\hline
$A_2^{B_s\to\phi}$ &$0.234$ &$-0.054$&--- &$0.288$ &$48.94$ &Eq.~(\ref{r12mfit})\\\hline
$T_1^{B_s\to\phi}$ &$0.349$ &$1.303$ &$5.32^2$ &$-0.954$ &$38.28$ &Eq.~(\ref{r12mRfit})\\\hline
$T_2^{B_s\to\phi}$ &$0.349$ &--- &--- &$0.349$ &$37.21$ &Eq.~(\ref{r2mfit})\\\hline
$\tilde{T}_3^{B\to \phi}$&$0.349$ &$0.027$ &--- &$0.321$ &$45.56$ &Eq.~(\ref{r12mfit})\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
These transition form factors have been updated recently within the light-cone QCD sum rule approach~\cite{Ball:2004rg}. For the $q^2$ dependence of the form factors, they can be parameterized in terms of simple formulae with two or three parameters. The form factors $V$, $A_0$ and $T_1$ are parameterized by
\begin{eqnarray}
F(s)=\frac{r_1}{1-s/m^2_{R}}+\frac{r_2}{1-s/m^2_{\rm fit}}.
\label{r12mRfit}
\end{eqnarray}
For the form factors $A_2$ and $\tilde{T}_3$, it is more appropriate to expand to the second order around the pole, yielding
\begin{eqnarray}
F(s)=\frac{r_1}{1-s/m^2}+\frac{r_2}{(1-s/m)^2}\,, \label{r12mfit}
\end{eqnarray}
where $m=m_{\rm fit}$ for $A_2$ and $\tilde{T}_3$. The fit formula for $A_1$ and $T_2$ is
\begin{eqnarray}
F(s)=\frac{r_2}{1-s/m^2_{\rm fit}}.\label{r2mfit}
\end{eqnarray}
The form factor $T_3$ can be obtained through the relation $T_3(s)=\frac{m_{B_s}^2-m_{\phi}^2}{s}\big[\tilde{T}_3(s)-T_2(s)\big]$. All the relevant fitting parameters for these form factors are taken from Ref.~\cite{Ball:2004rg} and are recollected in Table~\ref{FFfit}.
\end{appendix}
|
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{"url":"https:\/\/mathsci.kaist.ac.kr\/~sangil\/seminar\/20140707\/","text":"Ilkyoo Choi, Choosability of Toroidal Graphs with Forbidden Structures\n\nChoosability of Toroidal Graphs with Forbidden Structures\n2014\/07\/07 Monday 4PM-5PM\nRoom 1409\nThe choosability $$\\chi_\\ell(G)$$ of a graph G is the minimum k such that having k colors available at each vertex guarantees a proper coloring. Given a toroidal graph G, it is known that $$\\chi_\\ell(G)\\leq 7$$, and $$\\chi_\\ell(G)=7$$ if and only if G contains $$K_7$$. Cai, Wang, and Zhu proved that a toroidal graph G without 7-cycles is 6-choosable, and $$\\chi_\\ell(G)=6$$ if and only if G contains $$K_6$$. They also prove that a toroidal graph G without 6-cycles is 5-choosable, and conjecture that $$\\chi_\\ell(G)=5$$ if and only if G contains $$K_5$$. We disprove this conjecture by constructing an infinite family of non-4-colorable toroidal graphs with neither $$K_5$$ nor cycles of length at least 6; moreover, this family of graphs is embeddable on every surface except the plane and the projective plane. Instead, we prove the following slightly weaker statement suggested by Zhu: toroidal graphs containing neither $$K^-_5$$ (a $$K_5$$ missing one edge) nor 6-cycles are 4-choosable. This is sharp in the sense that forbidding only one of the two structures does not ensure that the graph is 4-choosable.\n\nTags:","date":"2019-10-22 04:36:28","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\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7108402252197266, \"perplexity\": 401.2910665074465}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987798619.84\/warc\/CC-MAIN-20191022030805-20191022054305-00232.warc.gz\"}"}
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Malta Limited Liability Company
European Companies, News, Registration, Tax Optimisation
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Malta Limited Liability Company.
Registration of Companies in Malta
The following are just a number of the advantages that companies registered in Malta offer (Malta Limited Liability Company).
Malta is the only EU state with the full imputation system
Effective tax rate of 5% on company's trading profits ‐ following refunds (10% in the case of passive interest and royalties)
Participation exemption on dividends or gains from qualifying holdings (no tax paid in Malta)
Certainty –Malta reached an agreement with the EU in 2006 regarding changes implemented to its corporate tax regime o Low annual company maintenance costs
No thin capitalisation or anti‐controlled foreign company regimes
No duty payable on transfers of shares in companies having the majority of their business interests situated outside Malta
Company law based on the UK company law and in line with EU Directives
Malta is an EU Member since 2004 and Maltese Companies enjoy the benefit of relevant Directives
Possibility for companies to denominate their share capital in major foreign currencies – tax is paid and refunded in the same currency thereby minimising exchange rate risks
Political and economic stability – bipartisan system and both parties have a political track record of achieving consensus on issues related to international business
Malta is a member of the Eurozone having adopted the euro in January 2008
English is an official language in Malta and all legislation and official documentation must be both in Maltese and in English
50 Double Tax Treaties signed and ratified and more are under negotiation (if double tax relief is resorted to, a refund of 2/3 of the Malta Tax Paid is available)
Private Limited Liability Company in Malta
Business corporate entities in Malta are incorporated and regulated by the Companies Act 1995 with the most common business entity being the limited liability company. A company has separate legal personality from the shareholders and it is unlimitedly liable for its obligations. A limited liability company's capital is divided into shares which are in turn subscribed to by the company's shareholders.
The company provides its shareholders with the benefit of limiting their liability to the amount if any, still unpaid on the shares that they hold. A private company is a company in which the transfer of its shares is restricted and the number of its shareholders is limited to fifty. A company's name must have the suffix Limited or Ltd., while the authorised and issued share capital of the company is normally not lower than €1,200 which must be at least 20% paid up. As a general rule each company must have at least two shareholders, however there is an exception to this rule and subject to certain conditions, companies may also be registered as single member companies – such a company is usually only feasible for very small operations.
A company's Memorandum and Articles of Association must contain the following information:
The name of the Company
Whether it is a public or private limited liability company
The Registered Office (which must be situated in Malta)
The Objects of the Company
Details of the issued and authorised share capital of the Company
The name and residence of the shareholders of the Company
The number of the directors together with the name and residence of the first directors of the Company (if the directors are corporate entities similar information would be required)
The name and residence of the first company secretary of the Company
The company's Memorandum and Articles of Association of a company may either be signed by the company's subscriber or by their authorised agent. A private limited liability company is established on the submission of the Memorandum & Articles of Association to the Registrar of Companies together with the registration fee which increases with the authorised share capital of the company. The incorporation of a company is normally completed within twenty‐four to forty‐eight hours from submission of the relevant documents.
The management and administration of a company is vested in its Directors who enjoy all powers except those that have been specifically reserved to the shareholders of the company, exercised in general meetings. A private company must have at least one director and unless the company is an exempt company the sole director may not be the company secretary of the company (who must be an individual).
The directors are bound to act honestly and in good faith in the best interests of the company. The company's directors also have a number of duties that must be fulfilled including the keeping of proper accounts, having the company's accounts audited annually, presenting the audited accounts at the annual general meeting, the preparation and submission of the official forms indicating changes within the company, the filing of income tax and VAT returns.
The company's share capital may be denominated in all major currencies. As a result the company's accounts will be prepared in the currency of its share capital while tax and refunds are all paid in the same currency as the company's share capital. This effectively reduces exchange rate risks.
Taxation of Companies in Malta – Malta Limited Liability Company
Malta's fiscal regime applicable to companies has been one of Malta's primary magnets for international business over the last decade. Starting with the introduction of an onshore company tax regime in the mid‐1990's providing an attractive effective tax rate for international businesses and further developments in line with an agreement reached with the European Union in 2006, the taxation of Maltese companies has gone through dramatic changes implemented with effect from 2007.
In terms of Maltese law, a company registered in Malta is domiciled and resident in Malta and therefore taxable on all of its profits. However should the board of directors meet regularly outside of Malta, a foreign jurisdiction may attempt to tax the profits of the Maltese company on the basis that its management and control is exercised outside of Malta. For this reason it is recommended that approximately four board meetings are held in Malta and/or a Maltese resident Director is appointed.
Profits of Maltese companies are taxed according to the nature of their source. In order to differentiate the different sources of income for tax purposes, Maltese law provides for the distribution of a company's profits into five different tax accounts. These accounts are the Immovable Property
Account, the Final Tax Account, the Foreign Income Account, the Maltese Taxed Account, and the Untaxed Account. The standard corporate tax rate in Malta is 35% however due to the full imputation system the tax paid by the company is attributed to that due by the shareholder on the company's profits therefore the shareholder will pay no further tax in Malta on receipt of dividends. Malta is the only EU state that has maintained the full imputation system.
Furthermore, Malta's corporate tax system provides a number of refunds of the tax paid by the company on the distribution of dividends.
These refunds vary according to which tax account the dividends had been distributed from. The amount of refund is set at 6/7 of the tax paid by the company on income distributed from the Foreign Income Account and the Maltese Taxed Account – thereby the effective rate of tax in Malta paid on the company's profits would be reduced to 5%. The refund is reduced to 5/7 if the income from which the dividend has been distributed was derived from passive interest and royalties (with an effective rate of 10%). In the event that the company had claimed double taxation relief, a refund of 2/3 of the tax paid by the company would be available.
In order to fully benefit from this refund structure we recommend that a two tier company structure is established whereby a holding company will receive dividends from an operating company and also tax refunds from the relevant authorities.
Further to the 2007 amendments, Maltese companies also benefit from a participating exemption regime in which dividends or gains derived from a qualifying shareholding in a foreign company (whether located in the EU or not – subject to certain conditions) will either be taxed at 35% and on distribution of a dividend the shareholder may apply for a full refund of the Malta tax paid or alternatively, said dividends or gains may not be taxed in the hands of the company – thereby leaving the income with a nil liability of Maltese tax.
Malta Limited Liability Company – Additional Info. Why use a Maltese Company?
Maltese companies are versatile and through the innovate fiscal regime applicable to companies registered in Malta, they may be used in a wide number of corporate structures and for various business operations.
Maltese companies may be used for trade in general and benefit from a low effective rate of taxation in a number of group structures – including group financing companies, service companies, sales & promotion companies, support companies and holding companies just to name a few.
Is the 5% tax rate in compliance with EU law? Prior to negotiations with the European Union concluded in 2006, Malta provided a tax regime available to onshore companies that was aimed at international business through what was known as the International Trading Company (ITC). Through a series of refunds the profits of the ITC were taxed at an effective rate of 4.17%.
The ITC was necessarily limited to trading outside of Malta in its objects clause and the refunds were only available to non‐residents. Both of these discriminatory features were deemed to be unacceptable by the EU and the new system has removed these discriminatory provisions. Under the new system, refunds are available to both residents and non‐residents and on income derived from activities carried out both within and outside of Malta subject to certain conditions and restrictions.
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A Maltese company may be designated with any name but must be followed by private limited company or Limited or its abbreviation Ltd.
Before the actual registration of a company, the shareholders may reserve their desired company name for a period of three months. The name of the company may not be already taken or similar to that of another company already registered in Malta, nor may it have a name that is deemed to be offensive or otherwise undesirable by the Registrar of Companies. Furthermore, in the interests of customer protection, the name may not include words denoting a particular activity that is regulated in terms of Maltese law – such as fiduciary, trustee or other similar names.
Is there an audit requirement for Malta Limited Liability Company?
Presently, all Maltese companies must ensure that their financial statements are audited by a qualified and registered auditor in Malta in accordance with International Standards on Auditing. The auditor must also submit a report on his/her findings to the shareholders of the company at general meeting. A company must submit its audited financial statements to the Registrar of Companies annually, within ten months from the end of the company's financial year, provided that for the first accounting period, the company may submit accounts covering a period of not more than eighteen months. Furthermore, the company may change the financial year end of the company of the current and all subsequent years by submitting the relative Form to the Registrar of Companies. Depending on the size and status of the company, certain information may be omitted from the submitted financial statements in the form of abridged financial statements.
Are there any annual requirements for Malta Limited Liability Company?
There are a number of annual requirements that Maltese companies must adhere to, all of which may assist you with. Malta Limited Liability Company – requirements:
Maltese companies must hold a general meeting of the shareholders every year, at a date no later than 15 months from the previous general During the general meeting the directors are to present the company's audited financial statement together with the auditor's and director's report.
Maltese companies must also submit an annual return to the Registrar of Companies providing information related to the share capital, directors and similar information together with its annual registration fee which varies according to the authorised share capital of the company. This must either be signed and submitted by one of the company's officers or submitted electronically by an individual duly authorised either in the company's memorandum, by a resolution of the board of directors or an extra‐ordinary resolution.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,880
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Anomalurus é um gênero de roedor da família Anomaluridae.
Espécies
Anomalurus beecrofti Fraser, 1853
Anomalurus derbianus (Gray, 1842)
Anomalurus pelii (Schlegel e Müller, 1845)
Anomalurus pusillus Thomas, 1887
Referências
DIETERLEN, F. Family Anomaluridae. In: WILSON, D. E.; REEDER, D. M. (Eds.). Mammal Species of the World: A Taxonomic and Geographic Reference. 3. ed. Baltimore: Johns Hopkins University Press, 2005. v. 2, p. 1532-1534.
Anomaluridae
Gêneros de mamíferos
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,264
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\section{Introduction}
Sobolev-type metrics on the space of plane immersed curves were independently introduced in
\cite{Charpiat2007,Michor2006c, Mennucci2007}. They are used in computer vision, shape
classification and tracking, mainly in the form of their induced metric on shape space, which is
the orbit space under the action of the reparameterization group.
See \cite{Kurtek2012,Sundaramoorthi2011} for applications of
Sobolev-type metrics and \cite{Bauer2013_preprint,Michor2007} for an overview of their mathematical
properties. Sobolev-type metrics were also generalized to immersions of higher dimensional
manifolds in \cite{Bauer2011b,Bauer2012d}.
It was shown in \cite{Michor2007} that the geodesic equation of a Sobolev-type metric of order
$n\geq 1$ is locally well-posed and this result was extended in \cite{Bauer2011b} to a larger class
of metrics and immersions of arbitrary dimension. The main result of this paper is to show global
well-posedness of the geodesic equation for Sobolev-type metrics of order $n\geq 2$ with constant
coefficients. In particular we prove the following theorem:
\begin{theorem}\label{mainTheorem}
Let $n\geq 2$ and the metric $G$ on $\on{Imm}(S^1,{\mathbb R}^2)$ be given by
\[
G_c(h,k) = \int_{S^1} \sum_{j=0}^n a_j \langle D_s^j h, D_s^j k \rangle \,\mathrm{d} s\,,
\]
with $a_j\geq 0$ and $a_0,a_n \neq 0$. Given initial conditions $(c_0, u_0) \in T\on{Imm}(S^1,{\mathbb R}^2)$ the solution of the geodesic equation
\begin{equation*}
\begin{split}
\p_t \left(\sum_{j=0}^n (-1)^j \,|c'|\, D_s^{2j} c_t\right) &=
-\frac{a_0}2 \,|c'|\, D_s\left( \langle c_t, c_t \rangle v \right) \\
&\qquad{} + \sum_{k=1}^n \sum_{j=1}^{2k-1} (-1)^{k+j} \frac{a_k}{2}\, |c'|\, D_s
\left(\langle D_s^{2k-j} c_t, D_s^j c_t \rangle v \right)\,.
\end{split}
\end{equation*}
for the metric $G$ with initial values $(c_0,u_0)$ exists for all time.
\end{theorem}
Here $\operatorname{Imm}(S^1,\mathbb R^2)$ denotes the
space of all smooth, closed, plane curves with nowhere zero tangent vectors; this space is open in
$C^\infty(S^1,\mathbb R^2)$. We assume that $c\in \operatorname{Imm}(S^1,\mathbb R^2)$ and $h$, $k$ are vector
fields along $c$, $\,\mathrm{d} s=|c'|\,\mathrm{d}\th$ is the arc-length measure, $D_s=\frac 1{|c'|}\p_\th$ is the
derivative with respect to arc-length, $v = c'/|c'|$ is the unit length tangent vector to $c$
and $\langle \;,\; \rangle$ is the Euclidean inner product on $\mathbb R^2$.
Thus if $G$ is a Sobolev-type metric of order at least 2, then the Riemannian manifold
$(\on{Imm}(S^1,{\mathbb R}^2), G)$ is geodesically complete.
If the Sobolev-type metric is invariant under the reparameterization group $\on{Diff}(S^1)$,
also the induced metric on shape space $\operatorname{Imm}(S^1,\mathbb R^2)/\on{Diff}(S^1)$ is geodesically complete.
The latter space is an infinite dimensional orbifold; see
\cite[2.5 and 2.10]{Michor2006c}.
Theorem \ref{mainTheorem} seems to be the first result about geodesic completeness on manifolds
of mappings outside the realm of diffeomorphism groups and manifolds of metrics.
In the first paragraph of \cite[p. 140]{Ebin1970} a proof is sketched that a right
invariant $H^s$-metric on the group of volume preserving diffeomorphisms on a compact manifold $M$ is
geodesically complete, if $s\ge \dim(M)/2+1$. In \cite{TrouveYounes05} there is an implicit
result that a topological group of diffeomorphisms constructed from a reproducing kernel Hilbert
space of vector
fields whose reproducing kernel is at least $C^1$, is geodesically complete. For a certain metric
on a group of diffeomorphisms on $\mathbb R^n$ with $C^1$ kernel geodesic completeness is shown in
\cite[Thm.~2]{Michor2013}. Metric completeness and existence of minimizing geodesics have also been studied on the diffeomorphism group in \cite{Bruveris2014}. The manifold of all Riemannian metrics with fixed volume form is
geodesically complete for the $L^2$-metric (also called the Ebin metric).
Sobolev-type metrics of order 1 are not geodesically complete, since it is possible to shrink a
circle to a point along a geodesic in finite time, see \cite[Sect.~6.1]{Michor2007}. Similarly a
Sobolev metric of order 2 or higher with both $a_0, a_1=0$ is a geodesically incomplete metric on
the space $\on{Imm}(S^1,{\mathbb R}^2)/\on{Tra}$ of plane curves modulo translations. In this case it is
possible to blow up a circle along a geodesic to infinity in finite time; see Rem.\
\ref{rem:incomplete}.
In order to prove long-time existence of geodesics, we need to study properties of the geodesic distance.
In particular we show the following theorem regarding continuity of curvature $\ka$ and its derivatives.
\begin{theorem}
\label{thm:l2_bound_intro}
Let $G$ be a Sobolev-type metric of order $n\geq 2$ with constant coefficients and $\on{dist}^G$
the induced geodesic distance. If $0 \leq k \leq n-2$, then the functions
\begin{align*}
D_s^k (\ka) \sqrt{|c'|} &: (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to L^2(S^1,{\mathbb R}) \\
D_s^{k+1} (\log|c'|) \sqrt{|c'|} &: (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to L^2(S^1,{\mathbb R})
\end{align*}
are continuous and Lipschitz continuous on every metric ball.
\end{theorem}
A similar statement can be derived for the $L^\infty$-continuity of curvature and its derivatives;
see Rem.\ \ref{rem:Linfty_continuity}.
The full proof of Thm.~\ref{mainTheorem} is surprisingly complicated. One reason is that we have
to work on the Sobolev completion (always with respect to the original parameter $\th$ in $S^1$)
of the space of immersions in order to apply results on ODEs on
Banach spaces. Here the operators (and their inverses and adjoints) acquire non-smooth coefficients.
Since we we want the Sobolev order as low as possible, the geodesic equation involves $H^{-n}$; see Sect.~\ref{lem:Ds_smooth_sobolev}.
Eventually we use that the metric operator has constant coefficients.
We have to use estimates with precise constants which are uniformly bounded on metric balls.
In \cite{Bauer2011b} the authors studied Sobolev metrics on immersions of higher dimensional
manifolds. One might hope that similar methods to those used in this article can be applied to show
the geodesic completeness of the spaces $\on{Imm}(M,N)$ with $M$ compact and $(N, \bar{g})$ a
suitable Riemannian manifold. A crucial ingredient in the proof for plane curves are the Sobolev
inequalities Lem. \ref{lem:poincare} and Lem. \ref{lem:poincare2} with explicit constants, which
only depend on the curve through the length. The lack of such inequalities for general $M$ will one
of the factors complicating life in higher dimensions.
\section{Background Material and Notation}
\subsection{The Space of Curves}
The space
\[
\on{Imm}(S^1, {\mathbb R}^2) = \left\{ c \in C^\infty(S^1, {\mathbb R}^2) \,:\, c'(\th) \neq 0 \right\}
\]
of immersions is an open set in the Fr\'echet space $C^\infty(S^1, {\mathbb R}^2)$ with respect to the $C^\infty$-topology and thus itself a smooth Fr\'echet manifold. The tangent space of $\on{Imm}(S^1, {\mathbb R}^2)$ at the point $c$ consists of all vector fields along the curve $c$. It can be described as the space of sections of the pullback bundle $c^\ast T{\mathbb R}^2$,
\begin{equation*}
T_c \on{Imm}(S^1,{\mathbb R}^2) = \Ga(c^\ast T{\mathbb R}^2) =
\left\{h: \quad \begin{aligned}\xymatrix{
& T{\mathbb R}^2 \ar[d]^{\pi} \\
S^1 \ar[r]^c \ar[ur]^h & {\mathbb R}^2
} \end{aligned} \right\}\,.
\end{equation*}
In our case, since the tangent bundle $T{\mathbb R}^2$ is trivial, it can also be identified with the space of ${\mathbb R}^2$-valued functions on $S^1$,
\[
T_c \on{Imm}(S^1, {\mathbb R}^2) \cong C^\infty(S^1, {\mathbb R}^2)\,.
\]
For a curve $c \in \on{Imm}(S^1, {\mathbb R}^2)$ we denote the parameter by $\th \in S^1$ and differentiation $\p_\th$ by $'$, i.e., $c'=\p_\th c$. Since $c$ is an immersion, the unit-length tangent vector $v = c' / |c'|$ is well-defined. Rotating $v$ by $\tfrac \pi 2$ we obtain the unit-length normal vector $n = Jv$, where $J$ is rotation by $\tfrac \pi 2$. We will denote by $D_s = \p_\th / |c_\th|$ the derivative with respect to arc-length and by $\,\mathrm{d} s = |c_\th| \,\mathrm{d} \th$ the integration with respect to arc-length. To summarize we have
\begin{align*}
v &= D_s c\,, & n &= Jv\,, & D_s &= \frac 1{|c_\th|} \p_\th\,, & \,\mathrm{d} s &= |c_\th| \,\mathrm{d} \th\,.
\end{align*}
The curvature can be defined as
\[
\ka = \langle D_s v, n \rangle
\]
and we have the Frenet-equations
\begin{align*}
D_s v &= \ka n \\
D_s n &= -\ka v\,.
\end{align*}
The length of a curve will be denoted by $\ell_c = \int_{S^1} 1 \,\mathrm{d} s$.
We define the turning angle $\al : S^1 \to {\mathbb R}/2\pi\mathbb Z$ of a curve $c$ by $v(\th) = (\cos \al(\th), \sin \al(\th))$. Then curvature is given by $\ka = D_s \al$.
\subsection{Variational Formulae}\label{variational-formulae}
We will need formulas that express, how the quantities $v$, $n$ and $\ka$ change, if we vary the
underlying curve $c$. For a smooth map $F$ from $\on{Imm}(S^1,{\mathbb R}^2)$ to any convenient vector space
(see \cite{KM97})
we denote by
\[
D_{c,h} F = \left.\frac{\,\mathrm{d}}{\,\mathrm{d} t}\right|_{t=0} F(c+th)
\]
the variation in the direction $h$.
The proof of the following formulas can be found for example in \cite{Michor2007}.
\begin{align*}
D_{c,h} v &= \langle D_s h, n \rangle n\quad\implies\quad D_{c,h}\al= \langle D_s h, n \rangle\\
D_{c,h} n &= -\langle D_s h, n \rangle v \\
D_{c,h} \ka &= \langle D_s^2 h, n \rangle - 2\ka \langle D_s h, v \rangle \\
D_{c,h}\left( |c'|^k \right) &= k\,\langle D_s h, v \rangle \,|c'|^k\,.
\end{align*}
With these basic building blocks, one can use the following lemma to compute the variations of higher derivatives.
\begin{lemma}
\label{lem:Ds_rmap}
If $F$ is a smooth map $F : \on{Imm}(S^1,{\mathbb R}^2) \to C^\infty(S^1, {\mathbb R}^d)$, then the variation of the composition $D_s \o F$ is given by
\[
D_{c,h} \left(D_s \o F\right) = D_s \left(D_{c,h} F\right) - \langle D_s h, v \rangle D_s F(c)\,.
\]
\end{lemma}
\begin{proof}
The operator $\p_\th$ is linear and thus commutes with the derivative with respect to $c$. Thus we have
\begin{align*}
D_{c,h} \left(D_s \o F\right) &= D_{c,h} \left( |c'|\i \p_\th F(c) \right)\\
&= |c'|\i \p_\th\left(D_{c,h}F\right) + \left(D_{c,h} \,|c'|\i\right) \p_\th F(c) \\
&= D_s \left( D_{c,h} F \right) - \langle D_s h, v \rangle\, |c'|\i \p_\th F(c) \\
&= D_s \left( D_{c,h} F \right) - \langle D_s h, v \rangle\, D_s F(c) \,. \qedhere
\end{align*}
\end{proof}
\subsection{Sobolev Norms}
In this paper we will only consider Sobolev spaces of integer order. For $n\geq 1$ the $H^n(d\th)$-norm on $C^\infty(S^1,{\mathbb R}^d)$ is given by
\begin{equation}
\label{eq:Hn_dtheta}
\| u \|_{H^n(d\th)}^2 = \int_{S^1} |u|^2 + |\p_\th^n u|^2 \,\mathrm{d} \th\,.
\end{equation}
Given $c \in \on{Imm}(S^1,{\mathbb R}^2)$, we define the $H^n(ds)$-norm on $C^\infty(S^1,{\mathbb R}^d)$ by
\begin{equation}
\label{eq:Hn_ds}
\| u \|_{H^n(ds)}^2 = \int_{S^1} |u(s)|^2 + |D_s^{n}u(s)|^2 \,\mathrm{d} s\,.
\end{equation}
Note that in \eqref{eq:Hn_ds} integration and differentiation are performed with respect to the arc-length of $c$, while in \eqref{eq:Hn_dtheta} the parameter $\th$ is used. In particular the $H^n(ds)$-norm depends on the curve $c$. The norms $H^n(d\th)$ and $H^n(ds)$ are equivalent, but the constants do depend on $c$. We prove in Lem.\ \ref{lem:Hk_local_equivalence}, that if $c$ doesn't vary too much, the constants can be chosen independently of $c$.
The $L^2(d\th)$- and $L^2(ds)$-norms are defined similarly,
\begin{align*}
\| u \|^2_{L^2(d\th)} &= \int_{S^1} |u|^2 \,\mathrm{d} \th\,, &
\| u \|^2_{L^2(ds)} = \int_{S^1} |u|^2 \,\mathrm{d} s\,,
\end{align*}
and they are related via $\left\| u \sqrt{|c'|} \right\|_{L^2(d\th)} = \| u \|_{L^2(ds)}$. Whenever we write $H^n(S^1,{\mathbb R}^d)$ or $L^2(S^1,{\mathbb R}^d)$, we always endow them with the $H^n(d\th)$- and $L^2(d\th)$-norms.
For $n \geq 2$ we shall denote by
\[
\on{Imm}^n(S^1,{\mathbb R}^2) = \{ c \,:\, c \in H^n(S^1,{\mathbb R}^2), c'(\th) \neq 0 \}
\]
the space of Sobolev immersions of order $n$. Because of the Sobolev embedding theorem, see
\cite{Adams2003}, we have $H^2(S^1,{\mathbb R}^2) \hookrightarrow C^1(S^1,{\mathbb R}^2)$ and thus
$\on{Imm}^n(S^1,{\mathbb R}^2)$ is well-defined. We will see in Sect.\ \ref{sec:Sobolev_immersions} that the
$H^n(ds)$-norm remains well-defined if $c \in \on{Imm}^n(S^1,{\mathbb R}^2)$.
The following result on point-wise multiplication will be used repeatedly. It can be found, among
other places in \cite[Lem.\ 2.3]{Inci2013}. We will in particular use that $k$ can be negative.
\begin{lemma}
\label{lem:sob_mult}
Let $n \geq 1$ and $k \in \mathbb Z$ with $|k|\leq n$ Then multiplication is a bounded bilinear map
\[
\cdot: H^{n}(S^1,{\mathbb R}^d) \times H^k(S^1,{\mathbb R}^d) \to H^k(S^1,{\mathbb R})\,,\quad
(f, g) \mapsto \langle f, g \rangle
\]
\end{lemma}
The last tool, that we will need is composition of Sobolev diffeomorphisms. For $n \geq 1$, define
\[
\mc D^n(S^1) = \{ \ph \,:\, \ph \text{ is $C^1$-diffeomorphism of $S^1$ and } \ph \in H^n(S^1,S^1)\}
\]
the group of Sobolev diffeomorphisms. The following lemma can be found in \cite[Thm.\ 1.2]{Inci2013}.
\begin{lemma}\label{Sobolev-composition}
Let $n \geq 2$ and $0 \leq k \leq n$. Then the composition map
\[
H^k(S^1,{\mathbb R}^d) \times \mc D^n(S^1) \to H^k(S^1,{\mathbb R}^d)\,,\quad
(f, \ph) \mapsto f \o \ph
\]
is continuous.
\end{lemma}
Let $n\geq 2$ and fix $\ph \in \mc D^n(S^1)$. Denote by $R_\ph(h) = h \o \ph$ the composition with $\ph$. From Lem.\ \ref{Sobolev-composition} we see that $R_\ph$ is a bounded linear map $R_\ph : H^n \to H^n$. The following lemma tells us that the transpose of this map respects Sobolev orders.
\begin{lemma}\label{Sobolev-transpose}
Let $n \geq 2$, $\ph \in \mc D^n(S^1)$ and $-n \leq k \leq n-1$. Then the restrictions of $R_\ph^\ast$ are bounded linear maps
\[
R_\ph^\ast \upharpoonright H^k(S^1,{\mathbb R}^d) : H^k(S^1,{\mathbb R}^d) \to H^k(S^1,{\mathbb R}^d)\,.
\]
On $L^2(S^1,{\mathbb R}^d)$ we have the identity $R_{\ph\i}^\ast(f) = R_{\ph}(f)\, \ph'$.
\end{lemma}
\begin{proof}
For $-n \leq k \leq 0$, we obtain from Lem.\ \ref{Sobolev-composition} that $R_\ph$ is a map $R_\ph : H^{-k} \to H^{-k}$ and by $L^2$-duality we obtain that $R_\ph^\ast : H^k \to H^k$ as required.
Now let $0 \leq k \leq n-1$, $f \in H^k$ and $g \in H^n$. We replace $\ph$ by $\ph\i$ to simplify the formulas. By definition of the transpose
\begin{multline*}
\left\langle R_{\ph\i}^\ast f, g \right\rangle_{H^{-n}\times H^n}
= \left\langle f, R_{\ph\i}\, g \right\rangle_{H^{-n}\times H^n} = \\
= \int_{S^1} \left\langle f(\th), g(\ph\i(\th)) \right\rangle\,\mathrm{d} \th
= \int_{S^1} \left\langle f(\ph(\th)), g(\th) \right\rangle \ph'(\th) \,\mathrm{d} \th = \\
= \left\langle \left(R_\ph f\right) \ph', g\right\rangle_{H^{-p}\times H^p}\,.
\end{multline*}
Thus we obtain $R_{\ph\i}^\ast(f) = R_{\ph}(f)\, \ph'$ and using Lem.\ \ref{lem:sob_mult} we see that for $f \in H^k$ we also have $R_{\ph\i}^\ast(f) \in H^k$.
\end{proof}
\subsection{Notation}\label{notation:lesssim}
We will write
\[
f \lesssim_{A} g
\]
if there exists a constant $C > 0$, possibly depending on $A$, such that the inequality $f \leq C g$ holds.
\subsection{Gronwall Inequalities}
The following version of Gronwall's inequality can be found in \cite[Thm.\ 1.3.2]{Pachpatte1998} and \cite{Jones1964}.
\begin{theorem}
\label{thm:gronwall}
Let $A$, $\Ph$, $\Ps$ be real continuous functions defined on $[a,b]$ and $\Ph \geq 0$. We suppose that on $[a,b]$ we have the following inequality
\[
A(t) \leq \Ps(t) + \int_a^t A(s)\Ph(s) \,\mathrm{d} s\,.
\]
Then
\[
A(t) \leq \Ps(t) + \int_a^t \Ps(s)\Ph(s) \operatorname{exp}\left(\int_s^t \Ph(u) \,\mathrm{d} u \right) \,\mathrm{d} s
\]
holds on $[a,b]$.
\end{theorem}
We will repeatedly use the following corollary.
\begin{corollary}
\label{cor:gronwall_applied}
Let $A$, $G$ be real, continuous functions on $[0,T]$ with $G\geq 0$ and $\al, \be$ non-negative constants. We suppose that on $[0,T]$ we have the inequality
\[
A(t) \leq A(0) + \int_0^t (\al + \be A(s)) G(s) \,\mathrm{d} s\,.
\]
Then
\[
A(t) \leq A(0) + \left(\al + (A(0) + \al N) \be e^{\be N}\right) \int_0^t G(s) \,\mathrm{d} s
\]
holds in $[0,T]$ with $N = \int_0^T G(t) \,\mathrm{d} t$.
\end{corollary}
\begin{proof}
Apply the Gronwall inequality with $[a,b]=[0,T]$, $\Ps(t) = A(0) + \al \int_0^t G(s) \,\mathrm{d} s$ and $\Ph(s) = \be G(s)$, and note that $G(s) \geq 0$ implies $\int_s^t G(u) \,\mathrm{d} u \leq N$.
\end{proof}
\subsection{Poincar\'e Inequalities}
In the later sections it will be necessary to estimate the $H^k(ds)$-norm of a function by the $H^{n}(ds)$-norm with $k<n$, as well as the $L^\infty$-norm by the $H^k(ds)$-norm. In particular, we will need to know, how the curve $c$ enters into the estimates. The basic result is the following lemma, which is adapted from \cite[Lem.\ 18]{Mennucci2008}.
\begin{lemma}
\label{lem:poincare_basic}
Let $c \in \on{Imm}^2(S^1,{\mathbb R}^2)$ and $h:S^1\to {\mathbb R}^d$ be absolutely continuous. Then
\[
\sup_{\th \in S^1} \left| h(\th) - \frac 1{\ell_c} \int_{S^1} h \,\mathrm{d} s \right| \leq \frac 12 \int_{S^1} |D_sh| \,\mathrm{d} s\,.
\]
\end{lemma}
\begin{proof}
Since $h(0) = h(2\pi)$, the following equality holds,
\[
h(\th) - h(0) = \frac 12 \left( \int_0^\th h'(\si) \,\mathrm{d} \si - \int_\th^{2\pi} h'(\si) \,\mathrm{d} \si \right) \,,
\]
and hence after integration
\[
\frac 1{\ell_c} \int_{S^1} h \,\mathrm{d} s - h(0) = \frac{1}{2\ell_c} \int_{S^1} \left( \int_0^\th h'(\si) \,\mathrm{d} \si - \int_\th^{2\pi} h'(\si) \,\mathrm{d} \si \right) \,\mathrm{d} s\,.
\]
Next we take the absolute value
\begin{align*}
\left| \frac 1{\ell_c} \int_{S^1} h \,\mathrm{d} s - h(0) \right| &\leq \frac{1}{2\ell_c} \int_{S^1} \left( \int_0^\th |h'(\si)| \,\mathrm{d} \si + \int_\th^{2\pi} |h'(\si)| \,\mathrm{d} \si \right) \,\mathrm{d} s \\
&\leq \frac{1}{2\ell_c} \int_{S^1} |h'(\si)|\,\mathrm{d} \si \int_{S^1} 1 \,\mathrm{d} s = \frac 12 \int_{S^1} |D_sh| \,\mathrm{d} s
\end{align*}
Now we replace 0 by an arbitrary $\th \in S^1$ and repeat the above steps.
\end{proof}
This lemma permits us to prove the inequalities that we will use throughout the remainder of the paper.
\begin{lemma}
\label{lem:poincare}
Let $c \in \on{Imm}^2(S^1,{\mathbb R}^2)$ and $h \in H^2(S^1,{\mathbb R}^d)$. Then
\begin{itemize}
\item
$\| h\|_{L^\infty}^2 \leq \displaystyle\frac 2{\ell_c} \| h \|_{L^2(ds)}^2 + \displaystyle \frac {\ell_c} 2 \| D_s h \|_{L^2(ds)}^2\,,$
\item
$\| D_s h \|_{L^\infty}^2 \leq \displaystyle \frac {\ell_c}4 \| D_s^2 h \|_{L^2(ds)}^2\,,$
\item
$\| D_s h \|_{L^2(ds)}^2 \leq \displaystyle \frac {\ell_c^2}4 \| D_s^2 h \|_{L^2(ds)}^2\,.$
\end{itemize}
\end{lemma}
\begin{proof}
From Lem.\ \ref{lem:poincare_basic} we obtain the inequality
\[
\| h \|_{L^\infty} \leq \frac 1{\ell_c} \int_{S^1} |h| \,\mathrm{d} s + \frac 12 \int_{S^1} |D_s h| \,\mathrm{d} s\,.
\]
Next we use $(a+b)^2 \leq 2a^2 + 2b^2$ and Cauchy-Schwarz in
\begin{align*}
\| h \|^2_{L^\infty} &\leq \frac 2{\ell_c^2} \left(\int_{S^1} |h| \,\mathrm{d} s\right)^2
+ \frac 12 \left( \int_{S^1} |D_s h| \,\mathrm{d} s\right)^2 \\
&\leq \frac 2{\ell_c} \left( \int_{S^1} |h|^2 \,\mathrm{d} s \right)
+ \frac{\ell_c}{2} \left( \int_{S^1} |D_s h|^2 \,\mathrm{d} s \right)\,,
\end{align*}
thus proving the first statement. To prove the second statement we note that $\int_{S^1} D_s h \,\mathrm{d} s = 0$ and thus by Lem.\ \ref{lem:poincare_basic}
\[
\| D_s h \|_{L^\infty} \leq \frac 12 \int_{S^1} |D_s^2 h| \,\mathrm{d} s\,.
\]
Hence
\[
\| D_s h \|^2_{L^\infty} \leq \frac 14 \left(\int_{S^1} |D_s^2 h| \,\mathrm{d} s\right)^2
\leq \frac{\ell_c}{4} \| D_s^2 h\|^2_{L^2(ds)}\,.
\]
To prove the third statement we estimate
\[
\| D_s h\|^2_{L^2(ds)} \leq \| D_s h \|^2_{L^\infty} \int_{S^1} 1 \,\mathrm{d} s
\leq \frac{\ell_c^2}4 \| D_s^2 h\|^2_{L^2(ds)}\,.
\]
This completes the proof.
\end{proof}
The next lemma allows us to estimate the $H^k(ds)$-norm using a combination of the $L^2(ds)$- and the $H^n(ds)$-norms, without introducing constants that depend on the curve.
\begin{lemma}
\label{lem:poincare2}
Let $n\geq 2$, $c \in \on{Imm}^n(S^1,{\mathbb R}^2)$ and $h \in H^n(S^1,{\mathbb R}^d)$. Then for $0 \leq k \leq n$,
\[
\| D_s^k h \|^2_{L^2(ds)} \leq \| h\|^2_{L^2(ds)} + \| D_s^n h \|^2_{L^2(ds)}\,.
\]
\end{lemma}
\begin{proof}
Let us write $D_c$ and $L^2(c)$ for $D_s$ and $L^2(ds)$ respectively to emphasize the dependence on the curve $c$. Since $\left\|D_c^k h\right\|_{L^2(c)} = \left\|D_{c\o\ph}^k (h\o\ph)\right\|_{L^2(c\o\ph)}$, we can assume that $c$ has a constant speed parametrization, i.e. $|c'| = \ell_c/2\pi$. The inequality we have to show is
\[
\int_{0}^{2\pi} \left( \frac{2\pi}{\ell_c}\right)^{2k-1} \big|h^{(k)}(\th)\big|^{2} \,\mathrm{d} \th \leq
\int_{0}^{2\pi} \frac{\ell_c}{2\pi} \left|h(\th)\right|^2 + \left( \frac{2\pi}{\ell_c}\right)^{2n-1} \big|h^{(n)}(\th)\big|^{2} \,\mathrm{d} \th\,.
\]
Let $\ph(x) = \frac{2\pi}{\ell_c} x$. After a change of variables this becomes
\begin{equation}
\label{eq:ineq1}
\int_{0}^{\ell_c} \big|(h\o\ph)^{(k)}(x)\big|^{2} \,\mathrm{d} x \leq
\int_{0}^{\ell_c} \left|h\o\ph(x)\right|^2 + \big|(h\o\ph)^{(n)}(x)\big|^{2} \,\mathrm{d} x\,.
\end{equation}
Let $f = h\o\ph$ and assume w.l.o.g. that $f$ is ${\mathbb R}$-valued.
Define $f_k(x) = \ell_c^{-1/2} \operatorname{exp}\left(i\frac{2\pi k}{\ell_c}x\right)$,
which is an orthonormal basis of $L^2([0,\ell_c], {\mathbb R})$.
Then $f = \sum_{k \in \mathbb Z} \wh f(k) f_k$ and \eqref{eq:ineq1} becomes
\[
\sum_{k \in \mathbb Z} \left(\tfrac {2\pi k}{\ell_c}\right)^{2k} \big| \wh f(k) \big|^2
\leq \sum_{k \in \mathbb Z} \left[ 1 + \left(\tfrac {2\pi k}{\ell_c}\right)^{2n}\right] \big| \wh f(k) \big|^2\,.
\]
Since for $a \geq 0$ we have the inequality $a^k \leq 1+a^n$, the last inequality is satisfied, thus concluding the proof.
\end{proof}
An alternative way to estimate the $H^k(ds)$-norm is given by the following lemma, which is the
periodic version of the Gagliardo-Nirenberg inequalities (see \cite{Nirenberg1959}).
\begin{lemma}
\label{lem:poincare3}
Let $n\geq 2$, $c \in \on{Imm}^n(S^1,{\mathbb R}^2)$ and $h \in H^n(S^1,{\mathbb R}^d)$. Then for $0 \leq k \leq n$,
\[
\| D_s^k h \|_{L^2(ds)} \leq \| h\|^{1-k/n}_{L^2(ds)} \, \| D_s^n h \|^{k/n}_{L^2(ds)}\,.
\]
If $c \in \on{Imm}^2(S^1,{\mathbb R}^2)$, the inequality also holds for $n=0,1$.
\end{lemma}
\subsection{The Geodesic Equation on Weak Riemannian Manifolds}
Let $V$ be a convenient vector space, $M \subseteq V$ an open subset and $G$ a possibly weak Riemannian metric on $M$. Denote by $\bar{L} : TM \to (TM)'$ the canonical map defined by
\[
G_c(h,k) = \langle \bar{L}_c h, k \rangle_{TM}\,,
\]
with $c \in M$, $h,k \in T_c M$ and with $\langle \cdot,\cdot \rangle_{TM}$ denoting the canonical pairing between $(TM)'$ and $TM$. We also define $H_c(h,h) \in (T_cM)'$ via
\[
D_{c,m} G_c(h,h) = \langle H_c(h,h), m \rangle_{TM}\,,
\]
with $D_{c,m}$ denoting the directional derivative at $c$ in direction $m$.
In fact $H$ is a smooth map
\[
H : TM \to (TM)'\,,\quad (c,h) \mapsto (c, H_c(h,h))\,.
\]
With these definitions we can state how to calculate the geodesic equation.
\begin{lemma}
\label{lem:convenient_geod_eq}
The geodesic equation -- or equivalently the Levi-Civita covariant derivative -- on $(M,G)$ exists if and only if $\tfrac 12 H_c(h,h) - \left(D_{c,h} \bar{L}_c\right)\!(h)$ is in the image of $\bar{L}_c$ for all $(c,h) \in TM$ and the map
\[
TM \to TM\,,\quad (c,h) \mapsto \bar{L}_c\i \left(\tfrac 12 H_c(h,h) - \left(D_{c,h} \bar{L}_c\right)\!(h)\right)
\]
is smooth. In this case the geodesic equation can be written as
\[
\begin{aligned}
c_t &= \bar{L}_c\i p \\
p_t &= \frac12 H_c(c_t,c_t)
\end{aligned}
\qquad\text{ or }\qquad
c_{tt} = \frac12 \bar L_c\i\left(H_c(c_t,c_t) - \left(\p_t\bar L_c\right)\!(c_t)\right)\,.
\]
\end{lemma}
This lemma is an adaptation of the result given in \cite[2.4.1]{Bauer2013d_preprint} and the same proof can be repeated; see also \cite[Sect.\ 2.4]{Micheli2013}.
\section{Sobolev Metrics with Constant Coefficients}
In this paper we will consider \emph{Sobolev-type metrics} with constant coefficients. These are metrics of the form
\[
G_c(h,k) = \int_{S^1} \sum_{j=0}^n a_j \langle D_s^j h, D_s^j k \rangle \,\mathrm{d} s\,,
\]
with $a_j\geq 0$ and $a_0, a_n \neq 0$. We call $n$ the \emph{order} of the metric. The metric can be defined either on the space $\on{Imm}(S^1,{\mathbb R}^2)$ of ($C^\infty$-)smooth immersions or for $p \geq n$ on the spaces $\on{Imm}^p(S^1,{\mathbb R}^2)$ of Sobolev $H^p$-immersions.
\subsection{The Space of Smooth Immersions}
Let us first consider $G$ on the space of smooth immersions. The metric can be represented via the associated family of operators, $L$, which are defined by
\[
G_c(h,k) = \int_{S^1} \langle L_c h, k \rangle \,\mathrm{d} s = \int_{S^1} \langle h, L_c k \rangle \,\mathrm{d} s\,,
\]
The operator $L_c : T_c \on{Imm}(S^1,{\mathbb R}^2) \to T_c \on{Imm}(S^1,{\mathbb R}^2)$ for a Sobolev metric with constant coefficients can be calculated via integration by parts and is given by
\[
L_c h = \sum_{j=0}^n (-1)^j a_j D_s^{2j} h\,.
\]
The operator $L_c$ is self-adjoint, positive and hence injective. Since $L_c$ is elliptic, it is
Fredholm $H^k\to H^{k-2n}$ with vanishing index and thus surjective. Furthermore its inverse is smooth as well.
We want to distinguish between the operator $L_c$ and the canonical embedding from $T_c\on{Imm}$ into $(T_c \on{Imm})'$,
which we denote by $\bar{L}_c$. They are related via
\[
\bar{L}_c h = L_c h \otimes \,\mathrm{d} s = L_c h \otimes |c'| \,\mathrm{d} \th\,.
\]
Later we will simply write $\bar{L}_c h = L_c h\,|c'|$, especially when the order of multiplication and differentiation becomes important in Sobolev spaces.
\subsection{The Space of Sobolev Immersions}
\label{sec:Sobolev_immersions}
Assume $n \geq 2$ and let $G$ be a Sobolev metric of order $n$. We want to extend $G$ from the
space $\on{Imm}(S^1,{\mathbb R}^2)$ to a smooth metric on the Sobolev-completion $\on{Imm}^n(S^1,{\mathbb R}^2)$.
First we have to look at the action of the arc-length derivative and its transpose (with respect to
$H^0(d\th)$) on Sobolev
spaces. Remember that we always use the $H^n(d\th)$-norm on Sobolev completions.
We can write $D_s$ as the composition $D_s = \tfrac{1}{|c'|} \o \p_\th$, where $\tfrac{1}{|c'|}$ is
interpreted as the multiplication operator $f \mapsto \tfrac{1}{|c'|} f$. Its transpose is
$D_s^\ast = \p_\th^\ast \o \left(\tfrac{1}{|c'|}\right)^\ast = -\p_\th \o \tfrac{1}{|c'|}$. These
operators are smooth in the following sense.
\begin{lemma}
\label{lem:Ds_smooth_sobolev}
Let $n\geq 2$ and $k \in \mathbb Z$ with $|k| \leq n-1$. Then the maps
\begin{align*}
D_s &: \on{Imm}^n(S^1,{\mathbb R}^2) \times H^{k+1}(S^1,{\mathbb R}^d) \to H^k(S^1,{\mathbb R}^d)\,,\quad
(c,h) \mapsto D_s h = \tfrac{1}{|c'|} h' \\
D_s^\ast &: \on{Imm}^n(S^1,{\mathbb R}^2) \times H^{k}(S^1,{\mathbb R}^d) \to H^{k-1}(S^1,{\mathbb R}^d)\,,\quad
(c, h) \mapsto D_s^\ast h = -\left(\tfrac 1{|c'|} h\right)'
\end{align*}
are smooth.
\end{lemma}
\begin{proof}
For $n\geq 2$, the map $c \mapsto \tfrac{1}{|c'|}$ is the composition of the following smooth maps,
\[
\begin{array}{ccccc}
\on{Imm}^n(S^1,{\mathbb R}^2) & \to &
\{ f : f > 0 \} \subset H^{n-1}(S^1,{\mathbb R}) & \to &
H^{n-1}(S^1,{\mathbb R}) \\
c & \mapsto & |c'| & \mapsto & \tfrac{1}{|c'|}
\end{array}\,.
\]
Since $\tfrac{1}{|c'|} \in H^{n-1}(S^1,{\mathbb R}^2)$, Lem.\ \ref{lem:sob_mult}. concludes the proof.
\end{proof}
Using Lem.\ \ref{lem:Ds_smooth_sobolev} we see that
\[
G_c(h,h) = \int_{S^1} \sum_{k=0}^n a_k \langle D_s^k h, D_s^k h \rangle \,\mathrm{d} s
\]
is well-defined for $(c,h) \in T\on{Imm}^n(S^1,{\mathbb R}^2)$. As the tangent bundle is isomorphic to $T\on{Imm}^n(S^1,{\mathbb R}^2) \cong \on{Imm}^n(S^1,{\mathbb R}^2) \times H^n(S^1,{\mathbb R}^2)$, we can also write the metric as
\[
G_c(h,h) = \left\langle \sum_{k=0}^n a_k\, (D_s^k)^\ast\, |c'|\, D_s^k h, h \right\rangle_{H^{-n}\times H^n}\,.
\]
Again we note that $|c'|$ has to be interpreted as the multiplication operator $f \mapsto |c'| \,f$ on the spaces $H^k$ with $|k| \leq n-1$. Thus the operator $\bar{L}_c:H^n \to H^{-n}$ is given by
\begin{equation*}
\bar{L}_c = \sum_{k=0}^n a_k\, (D_s^k)^\ast \o |c'| \o D_s^k\,.
\end{equation*}
While it is tempting to ``simplify'' the expression for $\bar{L}_c$ using the identity
\[
D_s^\ast \o |c'| = -|c'| \o D_s\,,
\]
one has to be careful, since the identity is only valid, when interpreted as an operator $H^k \to H^{k-1}$ with $-n+2 \leq k \leq n-1$. The left hand side however makes sense also for $k=-n+1$. Thus we have the operator
\[
(D_s^n)^\ast \o |c'| : L^2 \to H^{-n}\,,
\]
but the domain has to be at least $H^1$ for the operator
\[
(-1)^n\, |c'| \o D_s^n : H^1 \to H^{-n+1}\,.
\]
So the expression
\[
\bar{L}_c h = \sum_{k=0}^n (-1)^k a_k\, |c'|\,D_s^{2k} h \,,
\]
is only valid, when we restrict $\bar{L}_c$ to $H^{n+1}$, i.e., $\bar{L}_c : H^{n+1} \to H^{-n+1}$.
\subsection{The Geodesic Equation}
By Lem.\ \ref{lem:convenient_geod_eq}, we need to calculate $H_c(h,h)$. This is achieved in the following lemma.
\begin{lemma}
Let $n \geq 2$ and let $G$ be a Sobolev metric of order $n$. On $\on{Imm}^n(S^1,{\mathbb R}^2)$ we have
\begin{equation}
\label{eq:H_gradient_Sobolev}
H_c(h,h) = -a_0\, |c'|\, D_s\left( \langle h, h \rangle v \right)
- \sum_{k=1}^n \sum_{j=1}^{2k-1} (-1)^{k+j} a_k\, D_s^\ast \o
\left( |c'| \langle D_s^{2k-j} h, D_s^j h \rangle v \right)\,.
\end{equation}
On $\on{Imm}^p(S^1,{\mathbb R}^2)$ with $p \geq n+1$ as well as $\on{Imm}(S^1,{\mathbb R}^2)$ we have the equivalent expression,
\begin{align*}
H_c(h,h) &=\Bigg( -2 \langle L_c h, D_s h \rangle v - a_0 \langle h, h \rangle \ka n+{}\\
&\qquad{}+ \sum_{k=1}^n \sum_{j=1}^{2k-1} (-1)^{k+j} a_k \langle D_s^{2k-j} h, D_s^j h \rangle \ka n \Bigg) \otimes \,\mathrm{d} s\,.
\end{align*}
\end{lemma}
\begin{proof}
For $k\geq 1$ the variation of the $k$-th arc-length derivative is
\[
D_{c,m} D_s^k h = - \sum_{j=1}^k D_s^{k-j} \left( \langle D_s m, v \rangle D_s^j h \right)\,,
\]
and the formula is valid for $(c,m) \in T\on{Imm}^n(S^1,{\mathbb R}^2)$ and $h \in H^{-n+k}(S^1,{\mathbb R}^d)$. So
\begin{align*}
D_{c,m} G_c(h,h)
&=\!\! \int_{S^1} \sum_{k=0}^n a_k\, \langle D_s^k h, D_s^k h \rangle
\langle D_s m, v \rangle \, |c'|
+ 2 \sum_{k=1}^n a_k \left\langle D_s^k h, D_{c,m} D_s^k h \right\rangle |c'| \,\mathrm{d} \th \\
&= \left\langle \sum_{k=0}^n a_k \,|c'| \langle D_s^k h, D_s^k \rangle v,
D_s m \right\rangle_{H^{-n+1}\times H^{n-1}} \\
&\qquad\qquad -2 \sum_{k=1}^n \sum_{j=1}^k a_k\, \left\langle
|c'| D_s^k h, D_s^{k-j} \langle D_s m, v \rangle D_s^j h \right\rangle_{H^{-n+k}\times H^{n-k}}\,.
\end{align*}
Each term in the second sum is equal to
\begin{align*}
\Big\langle |c'| D_s^k h,
D_s^{k-j} \langle D_s m, v \rangle &D_s^j h \Big\rangle_{H^{-n+k}\times H^{n-k}} = \\
&= \left\langle \left(D_s^{k-j}\right)^\ast |c'|\, D_s^k h,
\langle D_s m, v \rangle D_s^j h \right\rangle_{H^{-n+j}\times H^{n-j}} \\
&= (-1)^{k-j} \left\langle |c'|\, D_s^{2k-j} h,
\langle D_s m, v \rangle D_s^j h \right\rangle_{H^{-n+j}\times H^{n-j}} \\
&= (-1)^{k-j} \left\langle |c'|\, \langle D_s^{2k-j} h,
D_s^j h \rangle v, D_s m \right\rangle_{H^{-n+1}\times H^{n-1}}\,.
\end{align*}
So
\begin{align*}
H_c&(h,h) =\\
&=
\sum_{k=0}^n a_k D_s^\ast \o \left( |c'| \langle D_s^k h, D_s^k h \rangle v \right)
- 2\sum_{k=1}^n \sum_{j=1}^k (-1)^{k-j} a_k D_s^\ast \o \left(
|c'|\langle D_s^{2k-j} h, D_s^j h \rangle v \right) \\
&= -a_0\, |c'| D_s\left( \langle h, h \rangle v \right)
- \sum_{k=1}^n \sum_{j=1}^{2k-1} (-1)^{k+j} a_k\, D_s^\ast \o
\left( |c'| \langle D_s^{2k-j} h, D_s^j h \rangle v \right)\,.
\end{align*}
This proves the first formula.
If $(c,h) \in T\on{Imm}^p(S^1,{\mathbb R}^2)$ with $p \geq 1$, we can commute $D_s^\ast \o |c'| = -|c'| \o D_s$ to obtain
\[
H_c(h,h)
= -a_0\, |c'| D_s\left( \langle h, h \rangle v \right)
+ \sum_{k=1}^n \sum_{j=1}^{2k-1} (-1)^{k+j} a_k\, |c'|\, D_s \left( \langle D_s^{2k-j} h, D_s^j h \rangle v \right)\,.
\]
Parts of the expression simplify as follows
\begin{align*}
&\sum_{k=1}^n \sum_{j=1}^{2k-1} (-1)^{k+j} a_k
D_s \left( \langle D_s^{2k-j} h, D_s^j h \rangle \right)
- a_0 D_s \Big( \langle h, h \rangle \Big) \\
&= \sum_{k=1}^n \sum_{j=1}^{2k-1} (-1)^{k+j} a_k
\left( \langle D_s^{2k-j+1} h, D_s^j h \rangle
+ \langle D_s^{2k-j} h, D_s^{j+1} h \rangle \right)
- 2a_0 \langle h, D_s h \rangle \\
&= \sum_{k=1}^n a_k \!\left( \sum_{j=0}^{2k-2} (-1)^{k+j+1}
\langle D_s^{2k-j} h, D_s^j h \rangle
+ \sum_{j=1}^{2k-1} (-1)^{k+j}
\langle D_s^{2k-j} h, D_s^{j+1} h \rangle \right)
\!\!-\! 2a_0 \langle h, D_s h\rangle \\
&= \sum_{k=1}^n (-1)^{k+1} 2a_k \langle D_s^{2k} h, D_s h \rangle
- 2a_0 \langle h, D_s h\rangle \\
&= -2 \langle L_c h, D_s h \rangle\,,
\end{align*}
And by collecting the remaining terms we arrive at the desired result.
\end{proof}
Now that we have computed $H_c(h,h)$, we can write the geodesic equation of the metric $G$. It is
\begin{equation}
\label{eq:geod_eq}
\begin{split}
\p_t \left(\bar{L}_c c_t\right) &=
-\frac{a_0}2 \,|c'|\, D_s\left( \langle c_t, c_t \rangle v \right) \\
&\qquad{} - \sum_{k=1}^n \sum_{j=1}^{2k-1} (-1)^{k+j} \frac{a_k}{2}\, D_s^\ast \o
\left( |c'| \langle D_s^{2k-j} c_t, D_s^j c_t \rangle v \right)\,.
\end{split}
\end{equation}
\subsection{Local Well-Posedness}
It has been shown in \cite[Thm.\ 4.3]{Michor2007} that the geodesic equation of a Sobolev metric is
well-posed on $\on{Imm}^p(S^1,{\mathbb R}^2)$ for $p \geq 2n+1$. For a metric of order $n\geq 2$ we extend
the result to $p \geq n$. This will later simplify the proof of geodesic completeness.
\begin{theorem}
\label{thm:geod_ex}
Let $n\geq 2$, $p\geq n$ and let $G$ be a Sobolev metric of order $n$ with constant
coefficients. Then the geodesic equation \eqref{eq:geod_eq} has unique local solutions in the space
$\on{Imm}^{p}(S^1,{\mathbb R}^2)$ of Sobolev $H^p$-immersions. The solutions depend $C^\infty$ on $t$ and
the initial conditions. The domain of existence (in $t$) is uniform in $p$ and thus the geodesic
equation also has local solutions in $\on{Imm}(S^1,{\mathbb R}^2)$, the space of smooth immersions.
\end{theorem}
\begin{proof}
Fix $p \geq n$. For the geodesic equation to exist, we need to verify the assumptions in Lem.\ \ref{lem:convenient_geod_eq}. We first note that $\bar{L}_c$ is a map $\bar L_c : H^p \to H^{p-2n}$. By inspecting \eqref{eq:H_gradient_Sobolev} we see that $H_c(h,h) \in H^{p-2n}$ as well. Thus it remains to show that $\bar L_c$ maps $H^p$ onto $H^{p-2n}$ and that the inverse is smooth. This is shown in \ref{lem:Lc_inv_smooth}.
Regarding local existence, we rewrite the geodesic equation as a differential equation on $T\on{Imm}^n(S^1,{\mathbb R}^2)$,
\begin{align*}
c_t &= u \\
u_t &= \frac12 \bar L_c\i\left(H_c(u,u) - \left(D_{c,u} \bar L_c\right)\!(u)\right)\,.
\end{align*}
This is a smooth ODE on a Hilbert space and therefore by Picard-Lindel\"of it has local solutions,
that depend smoothly on $t$ and the initial conditions. That the intervals of existence are uniform
in the Sobolev order $p$, can be found in \cite[App. A]{Bauer2013d_preprint}. The result goes back
to \cite[Thm.\ 12.1]{Ebin1970} and a different proof can be found in \cite{Michor2007}.
\end{proof}
The following lemma shows that the operator $\bar L_c$ has a smooth inverse on appropriate Sobolev spaces. For $p=n$, we can use Lem. \ref{lem:Hk_local_equivalence} and the lemma of Lax-Milgram to show that $\bar L_c : H^n \to H^{-n}$ is invertible. For $p>n$ more work is necessary. Although $\bar L_c$ is an elliptic, positive differential operator, it has non-smooth coefficients. In fact, since $|c'| \in H^{n-1}$, some of the coefficients are only distributions. To overcome this, we will exploit the reparametrization invariance of the metric to transform $\bar L_c$ into a differential operator with constant coefficients.
\begin{lemma}
\label{lem:Lc_inv_smooth}
Let $n\geq 2$ and $G$ be a Sobolev metric of order $n$. For $p \geq n$ and $c \in \on{Imm}^p(S^1,{\mathbb R}^2)$, the associated operators
\[
\bar L_c : H^p(S^1,{\mathbb R}^d) \to H^{p-2n}(S^1,{\mathbb R}^d)\,,
\]
are isomorphisms and the map
\[
\bar L\i : \on{Imm}^p(S^1,{\mathbb R}^2) \times H^{p-2n}(S^1,{\mathbb R}^d) \to H^{p}(S^1,{\mathbb R}^d)\,,\quad
(c,h) \mapsto \bar L_c\i h
\]
is smooth.
\end{lemma}
\begin{proof}
Given a curve $c \in \on{Imm}^p(S^1,{\mathbb R}^2)$, we can write it as $c = d \o \ps$, where $d$ has constant speed, $|d'| = \ell_c / 2\pi$, and $\ps$ is a diffeomorphism of $S^1$. The pair $(d, \ps)$ is determined only up to rotations; we can remove the ambiguity by requiring that $c(0) = d(0)$. Then $\ps$ is given by
\[
\ps(\th) = \frac{2\pi}{\ell_c} \int_0^\th |c'(\si)| \,\mathrm{d} \si\,.
\]
Concerning regularity, we have $\ps$ and $\ps^{-1} \in H^p(S^1,S^1)$ thus $\ps\in\mc D^p(S^1)$, and $d \in \on{Imm}^p(S^1,{\mathbb R}^2)$.
The reparametrization invariance of the metric $G$ implies
\[
\langle \bar{L}_c h, m \rangle_{H^{-p}\times H^p} =
\left\langle \bar{L}_{c \o \ps\i}(h \o \ps\i), m \o\ps\i \right\rangle_{H^{-p}\times H^p}\,.
\]
Introduce the notation $R_\ph(h) = h \o \ph$. If $\ph \in \mc D^p(S^1)$ is a diffeomorphism,
the map $R_\ph$ is an invertible linear map $R_\ph : H^p \to H^p$, by Lem.\ \ref{Sobolev-composition}. Furthermore by Lem.\ \ref{Sobolev-transpose} the transpose $R_\ph^\ast$ is an invertible map $R_\ph^\ast : H^{p-2n} \to H^{p-2n}$.
Thus we get
\[
\bar{L}_c h = R_{\ps\i}^\ast \o \bar{L}_d \o R_{\ps\i}(h)\,.
\]
Because $|d'|=\ell_c/2\pi$, the operator $\bar{L}_d$ is equal to
\[
\bar L_d = \sum_{k=0}^n (-1)^k a_k \left( \frac{2\pi}{\ell_c}\right)^{2k-1} \p_\th^{2k}\,.
\]
This is a positive, elliptic differential operator with constant coefficients and thus $\bar{L}_d : H^p \to H^{p-2n}$ is invertible.
Thus the composition $\bar{L}_c : H^p \to H^{p-2n}$ is invertible.
Smoothness of $(c,h) \mapsto \bar{L}_c\i h$ follows from the smoothness of $(c,h) \mapsto \bar{L}_c h$
and the implicit function theorem on Banach spaces.
\end{proof}
The remainder of the paper will be concerned with the analysis of the geodesic distance function induced by Sobolev metrics. These results will be used to show that geodesics for metrics of order $2$ and higher exist for all times.
\section{Lower Bounds on the Geodesic Distance}
To prepare the proof of geodesic completeness we first need to use geodesic distance to estimate quantities, that are derived from the curve and that appear in the geodesic equation. These include the length $\ell_c$, curvature $\ka$, its derivatives $D_s^k \ka$ as well as the length element $|c'|$ and its derivatives $D_s^k \log |c'|$. We want to show that they are bounded on metric balls of a Sobolev metric of sufficiently high order.
We start with the length $\ell_c$. The argument given in \cite[Sect.\ 4.7]{Michor2007} can be used to show the following slightly stronger statement.
\begin{lemma}
\label{lem:ell_lip}
Let the metric $G$ on $\on{Imm}(S^1,{\mathbb R}^2)$ satisfy
\[
\int_{S^1} \langle D_s h, v \rangle^2 \,\mathrm{d} s \leq A\, G_c(h,h)\,,
\]
for some $A>0$. Then we have the estimate
\[
\left\| \sqrt{|c_1'|} - \sqrt{|c_2'|} \right\|_{L^2(d\th)} \leq \frac {\sqrt{A}}2 \on{dist}^G(c_1, c_2)\,,
\]
in particular the function
$\sqrt{|c'|} : (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to L^2(S^1,{\mathbb R})$
is Lipschitz.
\end{lemma}
\begin{proof}
Take two curves $c_1, c_2 \in \on{Imm}(S^1,{\mathbb R}^2)$ and let $c(t,\th)$ be a smooth path between them. Then the following relation holds pointwise for each $\th \in S^1$,
\[
\sqrt{|c_2'|}(\th) - \sqrt{|c_1'|}(\th) = \int_0^1 \p_t \left(\sqrt{|c'|} \right)(t,\th) \,\mathrm{d} t\,.
\]
The derivative $\p_t \sqrt{|c'|}$ is given by
\[
\p_t \sqrt{|c'|} = \frac 12 \langle D_s c_t, v\rangle \sqrt{|c'|}\,,
\]
and so
\begin{align*}
\left\| \sqrt{|c_1'|} - \sqrt{|c_2'|} \right\|_{L^2(d\th)}
&\leq \frac 12 \int_0^1 \left\| \langle D_s c_t, v \rangle
\sqrt{|c'|} \right\|_{L^2(d\th)} \,\mathrm{d} t \\
&\leq \frac 12 \int_0^1 \big\| \langle D_sc_t,v \rangle \big\|_{L^2(ds)} \,\mathrm{d} t \\
&\leq \frac {\sqrt{A}}2 \int_0^1 \sqrt{ G_c(c_t,c_t)} \,\mathrm{d} t \\
&\leq \frac {\sqrt{A}}2 \on{Len}^G(c)\,.
\end{align*}
Since this estimate holds for every smooth path $c$, by taking the infimum we obtain
\[
\left\| \sqrt{|c_1'|} - \sqrt{|c_2'|} \right\|_{L^2} \leq \frac {\sqrt{A}}2 \inf_{c} \on{Len}^G(c)
= \frac {\sqrt{A}}2 \on{dist}^G(c_1,c_2)\,. \qedhere
\]
\end{proof}
We recover the statement of \cite[Sect.\ 4.7]{Michor2007} by applying the reverse triangle
inequality. The following corollary is a disguised version of the fact, that on a normed space the
norm function is Lipschitz.
\begin{corollary}
\label{cor:ell_lip}
If the metric $G$ on $\on{Imm}(S^1,{\mathbb R}^2)$ satisfies
\[
\int_{S^1} \langle D_s h, v \rangle^2 \,\mathrm{d} s \leq A\, G_c(h,h)\,,
\]
for some $A>0$, then the function $\sqrt{\ell_c} : (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to {\mathbb R}_{>0}$ is Lipschitz.
\end{corollary}
\begin{proof}
The statement follows from
\[
\ell_c = \int_{S^1} |c'(\th)| \,\mathrm{d} \th = \left\| \sqrt{|c'|} \right\|_{L^2(d\th)}^2\,,
\]
and the inequality
\begin{align}
\label{eq:ell_lip_const}
\left|\sqrt{\ell_{c_1}} - \sqrt{\ell_{c_2}} \right|
&= \left| \left\| \sqrt{|c'_1|} \right\|_{L^2(d\th)} - \left\| \sqrt{|c'_2|} \right\|_{L^2(d\th)} \right|
\\\notag
&\leq \left\| \sqrt{|c_1'|} - \sqrt{|c_2'|} \right\|_{L^2(d\th)} \leq \frac {\sqrt{A}}2
\on{dist}^G(c_1, c_2)\,. \qedhere
\end{align}
\end{proof}
\begin{remark}
Lemma \ref{lem:ell_lip} and Cor.\ \ref{cor:ell_lip} apply in particular to Sobolev metrics of order $n\geq 1$.
For $n=1$ this is clear from $\langle D_s h, v \rangle^2 \leq |D_s h|^2$.
For $n\geq 2$ we use Lem.\ \ref{lem:poincare2} to estimate
\[
\int_{S^1} \langle D_s h, v \rangle^2 \,\mathrm{d} s \leq \| D_s h \|^2_{L^2(ds)}
\leq \| h\|^2_{L^2(ds)} + \| D_s^n h \|^2_{L^2(ds)}
\leq \max\left(a_0^{-1}, a_n^{-1}\right) G_c(h,h)\,.
\]
We could have also used Lem.\ \ref{lem:poincare3},
\[
\int_{S^1} \langle D_s h, v \rangle^2 \,\mathrm{d} s \leq \| D_s h \|^2_{L^2(ds)}
\leq \| h \|^{2-2/n}_{L^2(ds)}\, \| D_s^n h \|^{2/n}_{L^2(ds)}
\leq a_0^{(1-n)/n} a_n^{-1/n} G_c(h,h)\,,
\]
to reach the same conclusion.
\end{remark}
The following lemma shows a similar statement for $\ell_c^{-1/2}$. We do not get global Lipschitz continuity, instead the function $\ell_c^{-1/2}$ is Lipschitz on every metric ball. This implies that $\ell_c\i$ is bounded on every metric ball. We will show later in Cor.\ \ref{cor:lenpt_inv_bound} that the pointwise quantities $|c'(\th)|$ and $|c'(\th)|\i$ are also bounded on metric balls.
\begin{lemma}
\label{lem:ellc_inv_lipschitz}
Let the metric $G$ on $\on{Imm}(S^1,{\mathbb R}^2)$ satisfy
\[
\int_{S^1} \left| h \right|^2 + \left|D_s^n h \right|^2 \,\mathrm{d} s \leq A\, G_c(h,h)
\]
for some $n\geq 2$ and some $A>0$. Given $c_0 \in \on{Imm}(S^1,{\mathbb R}^2)$ and $N>0$ there exists a
constant $C = C(c_0, N)$ such that for all $c_1, c_2 \in \on{Imm}(S^1,{\mathbb R}^2)$ with $\on{dist}^G(c_0, c_i) < N$,
$i=1,2$, we have
\[
\left| \ell_{c_1}^{-1/2} - \ell_{c_2}^{-1/2} \right| < C(c_0,N)\, \on{dist}^G(c_1,c_2)\,.
\]
In particular the function $\ell_c^{-1/2} : (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to {\mathbb R}_{>0}$
is Lipschitz on every metric ball.
\end{lemma}
\begin{proof}
Fix $c_1, c_2$ with $\on{dist}^G(c_0, c_i) < N$ and let $c(t,\th)$ be a path between them,
such that $\on{dist}^G(c_0, c(t)) < 2N$. Then
\begin{align*}
\p_t \left( \ell_c^{-1/2} \right) &= - \tfrac 12 \ell_c^{-3/2} \int_{S^1} \langle D_s c_t, v \rangle\, |c'| \,\mathrm{d} \th\,,
\end{align*}
and by taking absolute values
\begin{align*}
\left| \p_t \left( \ell_c^{-1/2} \right) \right|&\leq \tfrac 12 \ell_c^{-3/2} \int_{S^1} \left| \langle D_s c_t, v \rangle\right| \, |c'| \,\mathrm{d} \th \\
&\leq \tfrac 12 \ell_c^{-3/2} \sqrt{\int_{S^1} |c'| \,\mathrm{d} \th} \sqrt{\int_{S^1} \langle D_s c_t, v \rangle^2\, |c'| \,\mathrm{d} \th} \\
&\leq \tfrac 12 \ell_c^{-1} \| D_s c_t \|_{L^2(ds)}
\leq \tfrac 12 \ell_c^{-1} \left(
\frac{\ell_c}{2} \right)^{n-1} \| D_s^n c_t \|_{L^2(ds)}
\quad\text{ by \ref{lem:poincare},} \\
&\leq 2^{-n}\, \ell_c^{n-2}\, \sqrt{A}\, \sqrt{G_c(c_t, c_t)} \,.
\end{align*}
By Cor.\ \ref{cor:ell_lip} the length $\ell_c$ is bounded along the path $c(t,\th)$ and and since $n\geq 2$ so is $\ell_c^{n-2}$. Thus
\begin{align*}
\left| \ell_{c_1}^{-1/2} - \ell_{c_2}^{-1/2} \right|
&\leq \int_{0}^1 \left| \p_t \left( \ell_c^{-1/2} \right) \right| \,\mathrm{d} t \\
&\leq 2^{-n}\sqrt{A} \int_0^1 \ell_c^{n-2} \sqrt{G_c(c_t,c_t)} \,\mathrm{d} t \\
&\lesssim_{c_0,N} \on{Len}^G(c)\,;\quad\text{ see \ref{notation:lesssim} for notation.}
\end{align*}
After taking the infimum over all paths connecting $c_1$ and $c_2$ we obtain
\[
\left| \ell_{c_1}^{-1/2} - \ell_{c_2}^{-1/2} \right| \lesssim_{c_0,N} \on{dist}^G(c_1, c_2)\,.
\qedhere
\]
\end{proof}
\begin{remark*}
We can compute the constant $C=C(c_0,N)$ in Lem.\ \ref{lem:ellc_inv_lipschitz} explicitly. Indeed from
\[
\left| \ell_{c_1}^{-1/2} - \ell_{c_2}^{-1/2} \right|
\leq 2^{-n}\sqrt{A} \int_0^1 \ell_c^{n-2} \sqrt{G_c(c_t,c_t)} \,\mathrm{d} t\,,
\]
we obtain, following the proof,
\[
\left| \ell_{c_1}^{-1/2} - \ell_{c_2}^{-1/2} \right|
\leq 2^{-n}\sqrt{A} \left( \sup_{\on{dist}^G(c,c_0) < N} \ell_c^{n-2} \right)
\on{dist}^G(c_1,c_2)\,.
\]
Now, using \eqref{eq:ell_lip_const}, we can estimate $\ell_c$ via
\[
\sqrt{\ell_c} \leq
\sqrt{\ell_{c_0}} + \left| \sqrt{\ell_c} - \sqrt{\ell_{c_0}} \right| \leq
\sqrt{\ell_{c_0}} + \frac 12 \sqrt{A} \on{dist}^G(c,c_0) \leq
\sqrt{\ell_{c_0}} + \frac 12 \sqrt{A}N\,.
\]
Thus we can use
\[
C(c_0, N) = 2^{-n} \sqrt{A} \left(\sqrt{\ell_{c_0}} + \tfrac 12 \sqrt{A}N\right)^{2n-4}
\]
for the constant.
\end{remark*}
\begin{corollary}
\label{cor:ellc_inv_bounded}
Let $G$ satisfy the assumptions of Lem.\ \ref{lem:ellc_inv_lipschitz}. Then $\ell_c^{-1}$ is bounded on every metric ball of $(\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G)$.
\end{corollary}
\begin{proof}
Fix $c_0 \in \on{Imm}(S^1,{\mathbb R}^2)$ and $N>0$ and let $c \in \on{Imm}(S^1,{\mathbb R}^2)$ with $\on{dist}^G(c_0, c) < N$. Then
\[
\ell_c^{-1/2} \leq \ell_{c_0}^{-1/2} + \left| \ell_{c_0}^{-1/2} - \ell_{c}^{-1/2} \right| \lesssim_{c_0,N} \ell_{c_0} + \on{dist}^G(c_0, c) \lesssim_{c_0, N} 1\,,
\]
and thus $\ell_c^{-1/2}$ is bounded on metric balls, which implies that $\ell_c\i$ is bounded as well.
\end{proof}
The variations of the turning angle $\al$ and of $\log |c'|$ are given by
\begin{align*}
D_{c,h} \left(\log |c'| \right) &= \langle D_s h, v \rangle \\
D_{c,h} \al &= \langle D_s h, n \rangle\,.
\end{align*}
As a preparation for the proof of Thm.\ \ref{thm:ka_bound} we compute explicit expressions for the variations of their derivatives.
\begin{lemma}
\label{lem:ka_log_var}
Let $c \in \on{Imm}(S^1, {\mathbb R}^2)$, $h \in T_c \on{Imm}(S^1, {\mathbb R}^2)$ and $k \geq 0$. Then
\begin{align}
\label{eq:Dk_logcp_var}
D_{c,h} \left(D_s^k \log |c'|\right) &= D_s^k \langle D_s h, v \rangle
- \sum_{j=0}^{k-1} \binom{k}{j+1} \left( D_s^{k-j} \log |c'| \right) D_s^j \langle D_s h, v\rangle \\
\label{eq:Dk_al_var}
D_{c,h} \left(D_s^k \al\right) &= D_s^k \langle D_s h, n \rangle
- \sum_{j=0}^{k-1} \binom{k}{j+1} \left( D_s^{k-j} \al \right) D_s^j \langle D_s h, v\rangle\,.
\end{align}
\end{lemma}
\begin{proof} Recall Lem.\ \ref{lem:Ds_rmap}:
if $F: \on{Imm}(S^1, {\mathbb R}^2) \to C^\infty(S^1, {\mathbb R}^d)$ is smooth then
\[
D_{c,h} \left(D_s\o F\right) = D_s\left(D_{c,h} F \right) - \langle D_s h, v \rangle D_s F(c)\,.
\]
For $k=0$, by Sect. \ref{variational-formulae} we have
\begin{align*}
D_{c,h} \left(\log |c'| \right) &= \langle D_s h, v \rangle\,,
\quad
D_{c,h} \al = \langle D_s h, n \rangle\,,
\quad
D_{c,h}D_s = -\langle D_sh,v \rangle D_s\,,
\\
D_{c,h}(D_s^k) &= - \sum_{j=0}^{k-1} D_s^j \o \langle D_sh,v \rangle \o D_s^{k-j}\,.
\end{align*}
Thus we get
\[
D_{c,h} \left(D_s^k \log |c'|\right) = D_s^k \langle D_s h, v \rangle - \sum_{j=0}^{k-1} D_s^j \Big( \langle D_s h, v\rangle \big( D_s^{k-j} \log |c'| \big) \Big)\,.
\]
Next we use the identity \cite[(26.3.7)]{NIST2010},
\[
\sum_{j=i}^{k-1} \binom{j}{i} = \binom{k}{i+1} \,,
\]
and the product rule for differentiation to obtain
\begin{align*}
D_{c,h} \left(D_s^k \log |c'|\right) &= D_s^k \langle D_s h, v \rangle - \sum_{j=0}^{k-1} \sum_{i=0}^j \binom{j}{i} \left( D_s^{k-j+j-i} \log |c'| \right) D_s^i \langle D_s h, v\rangle \\
&= D_s^k \langle D_s h, v \rangle - \sum_{i=0}^{k-1} \sum_{j=i}^{k-1} \binom{j}{i} \left( D_s^{k-i} \log |c'| \right) D_s^i \langle D_s h, v\rangle \\
&= D_s^k \langle D_s h, v \rangle - \sum_{i=0}^{k-1} \binom{k}{i+1} \left( D_s^{k-i} \log |c'| \right) D_s^i \langle D_s h, v\rangle\,,
\end{align*}
which completes the first part of the proof.
Along the same lines we also get the variation of $D_s^k \al$.
\end{proof}
\begin{theorem}
\label{thm:ka_bound}
Assume that the metric $G$ on $\on{Imm}(S^1,{\mathbb R}^2)$ satisfies
\begin{equation}
\label{eq:G_stronger_Dk2}
\int_{S^1} |h|^2 + |D_s^{n} h|^2 \,\mathrm{d} s \leq A\, G_c(h,h)\,.
\end{equation}
for some $n\geq 2$ and some $A>0$.
For each $c_0 \in \on{Imm}(S^1,{\mathbb R}^2)$ and $N>0$ there exists a constant $C = C(c_0,N)$ such that for all $c_1, c_2 \in \on{Imm}(S^1,{\mathbb R}^2)$ with $\on{dist}^G(c_0,c_i) < N$ and all $0\leq k \leq n-2$ we have
\begin{gather*}
\left\| (D_{c_1}^k\ka_1) \sqrt{|c_1'|} - (D_{c_2}^k\ka_2) \sqrt{|c_2'|} \right\|_{L^2(d\th)} \leq C \on{dist}^G(c_1, c_2) \\
\left\| (D_{c_1}^{k+1}\log |c_1'|) \sqrt{|c_1'|} - (D_{c_2}^{k+1}\log |c_2'|) \sqrt{|c_2'|} \right\|_{L^2(d\th)} \leq C \on{dist}^G(c_1, c_2)\,.
\end{gather*}
In particular the functions
\begin{align*}
(D_s^k \ka) \sqrt{|c'|} &: (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to L^2(S^1,{\mathbb R}) \\
(D_s^{k+1} \log|c'|) \sqrt{|c'|} &: (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to L^2(S^1,{\mathbb R})
\end{align*}
are continuous and Lipschitz continuous on every metric ball.
\end{theorem}
\begin{proof}
We have $\on{dist}^G(c_1,c_2) < 2N$ by the triangle inequality.
Let $c(t,\th)$ be a path between $c_1$ and $c_2$ with $\on{Len}^G(c) \leq 3N$. Then
\begin{align*}
\on{dist}^G(c_0, c(t)) &\leq \on{dist}^G(c_0, c_1) + \on{dist}^G(c_1, c(t)) \\
&\leq N + \on{Len}^G(c|_{[0,t]}) \\
&\leq N + 3N \leq 4N\,;
\end{align*}
thus any path of this kind remains within a ball of radius $4N$ around $c_0$.
We will prove the theorem for each $n$ by induction over $k$. The proof of the continuity of
$(D_s^k \ka)\sqrt{|c'|}$ does not depend on the continuity of $(D_s^{k+1}\log|c'|) \sqrt{|c'|}$.
Thus, even if we prove both statements in parallel,
we will assume that we have established the
continuity and local Lipschitz continuity of $ (D_s^k \ka) \sqrt{|c'|}$
when estimating
$\|\p_t \big((D_s^{k+1} \log |c'|) \sqrt{|c'|}\big)\|_{L^2(d\th)}$ below; in particular we will need
that
\begin{equation}\label{eq:kabounded}
\big\| D_s^k \ka\big\|_{L^2(ds)}\quad\text{ remains bounded along the path.}
\end{equation}
The proof consists of two steps. First we show that the following estimates hold along $c(t,\th)$:
\begin{align}
\label{eq:pt_ka}
\left \|\p_t \big((D_s^k \ka) \sqrt{|c'|}\big)\right\|_{L^2(d\th)}
&\lesssim_{c_0, N} \left(1 + \| D_s^k \ka \|_{L^2(ds)}\right) \sqrt{G_c(c_t, c_t)} \\
\label{eq:pt_logcp}
\left \|\p_t \big((D_s^{k+1} \log |c'|) \sqrt{|c'|}\big)\right\|_{L^2(d\th)}
&\lesssim_{c_0, N} \left(1 + \| D_s^{k+1} \log |c'|\|_{L^2(ds)}\right) \sqrt{G_c(c_t, c_t)}\,.
\end{align}
Then we apply Gronwall's inequality to prove the theorem.
{\bf Step 1.} For $k=0$ we have
\begin{align*}
\p_t \big( \ka \sqrt{|c'|}\big) &=
\langle D_s^2 c_t, n \rangle \sqrt{|c'|} - \tfrac 32 \ka \langle D_s c_t, v \rangle \sqrt{|c'|} \\
\p_t \big( (D_s \log |c'|) \sqrt{|c'|}\big) &=
\langle D_s^2 c_t, v \rangle \sqrt{|c'|} + \ka \langle D_s c_t, n \rangle \sqrt{|c'|}\, -
\\&\qquad\qquad
- \tfrac 12 (D_s \log|c'|)\langle D_s c_t, v \rangle \sqrt{|c'|}\,,
\end{align*}
and therefore
\begin{align*}
\left \|\p_t \big(\ka \sqrt{|c'|}\big)\right\|_{L^2(d\th)} &\leq
\| D_s^2 c_t \|_{L^2(ds)} + \tfrac 32 \| \ka \|_{L^2(ds)} \| D_s c_t \|_{L^\infty} \\
\left \|\p_t \big((D_s \log |c'|) \sqrt{|c'|}\big)\right\|_{L^2(d\th)} &\leq
\| D_s^2 c_t \|_{L^2(ds)} + \| \ka \|_{L^2(ds)} \| D_s c_t \|_{L^\infty} + {}\\
&\qquad {}+\tfrac12 \| D_s \log |c'| \|_{L^2(ds)} \| D_s c_t \|_{L^\infty}\,.
\end{align*}
Note that the length $\ell_c$ is bounded along $c(t,\th)$ by Cor.\ \ref{cor:ell_lip}. Using the
Poincar\'e inequalities from Lem.\ \ref{lem:poincare} and assumption \eqref{eq:G_stronger_Dk2} we obtain
\begin{align*}
\left\|\p_t \big(\ka \sqrt{|c'|}\big)\right\|_{L^2(d\th)} &\lesssim_{c_0, N}
\left( 1 + \| \ka \|_{L^2(ds)} \right) \sqrt{G_c(c_t, c_t)} \\
\left\|\p_t \big((D_s\log |c'|) \sqrt{|c'|}\big)\right\|_{L^2(d\th)} &\lesssim_{c_0, N}
\left( 1 + \| D_s \log |c'| \|_{L^2(ds)}\right) \sqrt{G_c(c_t, c_t)}\,.
\end{align*}
For the second estimate we used the boundedness of $\|\ka \|_{L^2(ds)}$ from \eqref{eq:kabounded}.
This concludes the proof of step 1 for $k=0$.
Now consider $k>0$ and assume that the theorem has been shown for $k-1$. Along $c(t,\th)$ the following objects are bounded
\begin{itemize}
\item
$\ell_c$ by Cor.\ \ref{cor:ell_lip}, allowing us to use Poincar\'e inequalities,
\item
$\| D_s^{k-1} \ka \|_{L^2(ds)}$ and $\| D_s^k \log |c'| \|_{L^2(ds)}$ by induction, and
\item
$\| D_s^j \ka \|_{L^\infty}$ and $\| D_s^{j+1} \log |c'| \|_{L^\infty}$ for $0 \leq j \leq k-2$ via Poincar\'e inequalities.
\end{itemize}
We also have the following bounds, which are valid for both $v$ and $n$:
\begin{itemize}
\item
$\| D_s^j \langle D_s c_t, v \rangle \|_{L^2(ds)} \lesssim_{c_0, N} \sqrt{G_c(c_t,c_t)}$ for $0\leq j \leq k\,.$
This is clear for $j\leq k-1$, since the highest derivative of $\ka$ that appears due to the Frenet
equations is $D_s^{k-2} \ka$ and thus all terms involving $\ka$ can be bounded by the
$L^\infty$-norm. For $j=k$ we have
\begin{align*}
D_s^k \langle D_s c_t, v \rangle = \langle D_sc_t, D_s^k v \rangle + \sum_{j=1}^k \binom{k}{j} \langle D_s^{j+1} c_t, D_s^{k-j} v \rangle
\end{align*}
and
\[
D_s^k v = (D_s^{k-1} \ka) n + \text{lower order derivatives in $\ka$}\,.
\]
Thus
\[
\| D_s^k v \|_{L^2(ds)} \leq \| D_s^{k-1}\ka\|_{L^2(ds)} + \dots \lesssim_{c_0, N} 1\,.
\]
Hence we get
\begin{align*}
\| D_s^k \langle D_s c_t, v \rangle \|_{L^2(ds)} &\leq
\| D_s c_t \|_{L^\infty} \| D_s^k v \|_{L^2(ds)} + \sum_{j=1}^k \binom{k}{j}
\| D^{j+1}_s c_t \|_{L^2(ds)} \| D_s^{k-j} v \|_{L^\infty} \\
&\lesssim_{c_0, N} \sqrt{G_c(c_t, c_t)}\,.
\end{align*}
\item
$\| D_s^{k+1} \langle D_s c_t, v \rangle \|_{L^2(ds)} \lesssim_{c_0, N} (1 + \| D_s^k \ka \|_{L^2(ds)} )\sqrt{G_c(c_t,c_t)}\,.$
We obtain this bound from
\begin{align*}
D_s^{k+1} \langle D_s c_t, v \rangle &=
\langle D_s^{k+2} c_t, v\rangle + \langle D_sc_t, D_s^{k+1} v \rangle +
\sum_{j=1}^k \binom{k+1}j \langle D_s^{k+2-j} c_t, D_s^{j} v\rangle\,.
\end{align*}
Taking the $L^2(ds)$-norm we get
\begin{multline*}
\| D_s^{k+1} \langle D_s c_t, v \rangle \|_{L^2(ds)} \leq
\| D_s^{k+2} c_t \|_{L^2(ds)} + \| D_s c_t \|_{L^\infty} \| D_s^{k+1} v \|_{L^2(ds)} +
\\
+
\sum_{j=1}^k \binom{k+1}j \| D_s^{k+2-j} c_t \|_{L^2(ds)} \| D_s^{j} v \|_{L^\infty}
\\
\lesssim_{c_0,N}\! \sqrt{G_c(c_t, c_t)} +
\left(1 + \| D_s^k \ka \|_{L^2(ds)}\right) \sqrt{G_c(c_t, c_t)} +
\sqrt{G_c(c_t, c_t)}\,,
\end{multline*}
thus showing the claim.
\end{itemize}
Equation \eqref{eq:Dk_al_var} from Lem.\ \ref{lem:ka_log_var}, rewritten for $\ka$, is
\[
D_{c,h} \left(D_s^k \ka \right) = D_s^{k+1} \langle D_s h, n \rangle
- \sum_{j=0}^{k} \binom{k+1}{j+1} \left( D_s^{k-j} \ka \right) D_s^j \langle D_s h, v\rangle\,.
\]
Thus we get
\begin{multline*}
\p_t \left((D_s^k \ka) \sqrt{|c'|}\right) =
(D_s^{k+1} \langle D_s c_t, n \rangle) \sqrt{|c'|} - (k+\tfrac 12) (D_s^k \ka) \,\langle D_s c_t, v \rangle \sqrt{|c'|} -{} \\
{}- \binom{k+1}{2} \left(D_s^{k-1} \ka\right)\left(D_s \langle D_s c_t, v \rangle\right) \sqrt{|c'|}
-\sum_{j=2}^k \binom{k+1}{j+1} \left(D_s^{k-j} \ka\right)\, D_s^j \langle D_s c_t, v \rangle \sqrt{|c'|}\,,
\end{multline*}
and hence, by taking norms,
\begin{align*}
\left \|\p_t ((D_s^k \ka) \sqrt{|c'|})\right\|_{L^2(d\th)} &\leq
\left\| D_s^{k+1} \langle D_s c_t, n \rangle \right\|_{L^2(ds)}
+ (k+\tfrac 12) \left\| D_s^k \ka \right\|_{L^2(ds)}
\Big\| \langle D_s c_t, v \rangle \Big\|_{L^\infty} \\
&\qquad{}+ \binom{k+1}{2} \left\| D_s^{k-1} \ka\right\|_{L^2(ds)}
\Big\| D_s \langle D_s c_t, v \rangle \Big\|_{L^\infty} \\
&\qquad{}+\sum_{j=2}^k \binom{k+1}{j+1} \left\| D_s^{k-j} \ka\right\|_{L^\infty}
\Big\| D_s^j \langle D_s c_t, v \rangle\Big\|_{L^2(ds)}\\
&\lesssim_{c_0, N} \left(1 + \| D_s^k \ka \|_{L^2(ds)}\right) \sqrt{G_c(c_t, c_t)}\,.
\end{align*}
For $(D_s^{k+1} \log |c'|) \sqrt{|c'|}$ we proceed similarly. The time derivative is
\begin{multline*}
\p_t ((D_s^{k+1} \log |c'|) \sqrt{|c'|}) =
D_s^{k+1} \langle D_s c_t, v \rangle \sqrt{|c'|} -
\\
- (k+\tfrac 12) (D_s^{k+1} \log |c'|)\, \langle D_s c_t, v \rangle \sqrt{|c'|}
\\
- \binom{k+1}{2} \left(D_s^{k} \log |c'|\right)\, D_s \langle D_s c_t, v \rangle \sqrt{|c'|}
\\
-\sum_{j=2}^k \binom{k+1}{j+1} \left(D_s^{k+1-j} \log |c'|\right) D_s^j \langle D_s c_t, v \rangle \sqrt{|c'|}\,,
\end{multline*}
which can be estimated by
\begin{align*}
\Big\| \p_t \big((D_s^{k+1} &\log |c'|) \sqrt{|c'|}\big)\Big\|_{L^2(d\th)} \\
&{}\leq
\left\| D_s^{k+1} \langle D_s c_t, v \rangle \right\|_{L^2(ds)}
+ (k+\tfrac 12) \left\| D_s^{k+1} \log |c'|\right\|_{L^2(ds)}
\Big\| D_s c_t \Big\|_{L^\infty} \\
&\qquad {}+ \binom{k+1}{2} \left\| D_s^{k} \log |c'|\right\|_{L^2(ds)}
\Big\| D_s \langle D_s c_t, v \rangle \Big\|_{L^\infty}\\
&\qquad {}+ \sum_{j=2}^k \binom{k+1}{j+1}
\left\| D_s^{k+1-j} \log |c'| \right\|_{L^\infty}
\Big \| D_s^j \langle D_s c_t, v \rangle \Big\|_{L^2(ds)} \\
&\lesssim_{c_0, N} \left(1 + \| D_s^{k+1} \log |c'| \|_{L^2(ds)} \right)
\sqrt{G_c(c_t,c_t)}\,.
\end{align*}
{\bf Step 2.} The proof of this step depends only on the estimates \eqref{eq:pt_ka} and \eqref{eq:pt_logcp}.
We have a path $c(t,\th)$ between $c_1$ and $c_2$.
We write again $D_{c_1}$ and $D_{c(t)}$ for $D_{s_{c_1}}$ and $D_{s_{c(t)}}$, respectively.
Define the functions
\begin{align}
\label{eq:def_A_ka}
A(t) &= \left\| (D_{c_1}^k\ka_1) \sqrt{|c_1'|} - (D_{c(t)}^k\ka(t)) \sqrt{|c(t)'|} \right\|_{L^2(d\th)}\\
\label{eq:def_B_logcp}
B(t) &= \left\| (D_{c_1}^{k+1}\log |c_1'|) \sqrt{|c_1'|} - (D_{c(t)}^{k+1}\log |c(t)'|) \sqrt{|c(t)'|} \right\|_{L^2(d\th)}\,.
\end{align}
From
\[
(D_{c}^k\ka) \sqrt{|c'|}(t,\th) - (D_{c_1}^k\ka_1) \sqrt{|c_1'|}(\th) =
\int_0^t \p_t (D_s^k \ka) \sqrt{|c'|}) (\ta, \th) \,\mathrm{d} \ta
\]
we get, by taking norms,
\begin{align*}
A(t) &\leq \int_0^t \left \|\p_t (D_s^k \ka \sqrt{|c'|})\right\|_{L^2(d\th)} \,\mathrm{d} \ta \\
& \lesssim_{c_0, N} \int_0^t \left(1 + \| D_s^k \ka \|_{L^2(ds)}\right) \sqrt{G_c(c_t, c_t)} \,\mathrm{d} \ta \\
& \lesssim_{c_0, N} \int_0^t \left(1 + \| D_s^k \ka_1 \|_{L^2(ds)} + A(\ta) \right) \sqrt{G_c(c_t, c_t)} \,\mathrm{d} \ta\,.
\end{align*}
Now we use Gronwall's inequality, Cor.\ \ref{cor:gronwall_applied}, to obtain
\[
A(t) \lesssim_{c_0, N} \left(1 + \| D_s^k \ka_1 \|_{L^2(ds)}\right) \int_0^t \sqrt{G_c(c_t, c_t)} \,\mathrm{d} \ta\,.
\]
Taking the infimum over all paths and evaluating at $t=1$ then yields almost the desired inequality
\begin{equation}
\label{eq:almost_ineq}
\left\| (D_{c_1}^k\ka_1) \sqrt{|c_1'|} - (D_{c_2}^k\ka_2) \sqrt{|c_2'|} \right\|_{L^2(d\th)}
\lesssim_{c_0, N} \left(1 + \| D_s^k \ka_1 \|_{L^2(ds)}\right) \on{dist}^G(c_1, c_2)\,.
\end{equation}
To bound $\| D_s^k \ka_1 \|_{L^2(ds)}$, which appears on the right hand side, we apply \eqref{eq:almost_ineq} with $c_2 = c_0$.
\begin{align*}
\| D_s^k \ka_1 \|_{L^2(ds)} &\leq
\left\| D_{c_1}^k(\ka_1) \sqrt{|c_1'|} - D_{c_0}^k(\ka_0) \sqrt{|c_0'|} \right\|_{L^2(d\th)}
+ \| D_s^k \ka_0 \|_{L^2(ds)} \\
&\lesssim_{c_0, N} \left(1 + \| D_s^k \ka_0 \|_{L^2(ds)}\right) \on{dist}^G(c_0, c_1) +
\| D_s^k \ka_0 \|_{L^2(ds)}
\lesssim_{c_0, N} 1\,.
\end{align*}
This concludes the proof for $(D_s^k \ka) \sqrt{|c'|}$.
For $(D_s^{k+1} \log |c'|) \sqrt{|c'|}$ proceed in the same way with $B(t)$ in place of $A(t)$
using the estimate \eqref{eq:pt_logcp}.
\end{proof}
\begin{remark}
Theorem \ref{thm:ka_bound} makes no statement about the continuity or local Lipschitz continuity of
the function $\log |c'|\, \sqrt{|c'|}$, when $G$ is a Sobolev metric of order 1. In fact it appears
that one needs a metric of order $n\geq 2$. In that case one can use the variational formula
\[
D_{c,h} \left( \log |c'|\, \sqrt{|c'|}\right) =
\left( 1 + \tfrac 12 \log |c'| \right) \langle D_s h, v \rangle \sqrt{|c'|}\,,
\]
and the same method of proof -- with $n \geq 2$ one can estimate $\langle D_s h, v \rangle$ using the $L^\infty$-norm -- to show that,
\[
\left(\log |c'|\right) \sqrt{|c'|} : (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to L^2(S^1,{\mathbb R}^2)
\]
is continuous and Lipschitz continuous on every metric ball.
\end{remark}
\begin{remark}
\label{rem:Linfty_continuity}
In a similar way we can also obtain continuity in $L^\infty$ instead of $L^2$. Assume the metric
satisfies \eqref{eq:G_stronger_Dk2} with $n\geq 3$. Then for all $1 \leq k \leq n-2$ the functions
\begin{align*}
D_s^{k-1} \ka &: (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to L^\infty(S^1,{\mathbb R}) \\
D_s^{k} \log|c'| &: (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to L^\infty(S^1,{\mathbb R})
\end{align*}
are continuous and Lipschitz continuous on every metric ball. To prove this we follow the proof of Thm.\ \ref{thm:ka_bound} and replace the estimates \eqref{eq:pt_ka}, \eqref{eq:pt_logcp} with
\begin{align*}
\left \|\p_t \big(D_s^{k-1} \ka \big)\right\|_{L^\infty} &\lesssim_{c_0, N} \left(1 + \| D_s^{k-1} \ka \|_{L^\infty}\right) \sqrt{G_c(c_t, c_t)} \\
\left \|\p_t \big(D_s^{k} \log |c'| \big)\right\|_{L^\infty} &\lesssim_{c_0, N} \left(1 + \| D_s^{k} \log |c'|\|_{L^\infty}\right) \sqrt{G_c(c_t, c_t)}\,,
\end{align*}
which can be established in the same way.
We also have $L^\infty$-continuity of $\log |c'|$, when $n=2$.
Since we will use it in the proof of geodesic completeness, we shall provide an explicit proof in Lem.\ \ref{lem:lenpt_inv_bound}.
\end{remark}
\begin{lemma}
\label{lem:lenpt_inv_bound}
Let the metric $G$ on $\on{Imm}(S^1,{\mathbb R}^2)$ satisfy
\[
\int_{S^1} |h|^2 + |D_s^n h|^2 \,\mathrm{d} s \leq A\, G_c(h, h)\,,
\]
for some $n\geq 2$ and some $A > 0$. Given $c_0 \in \on{Imm}(S^1,{\mathbb R}^2)$ and $N > 0$, there exists a constant $C = C(c_0, N)$ such that for all $c_1, c_2 \in \on{Imm}(S^1,{\mathbb R}^2)$ with $\on{dist}^G(c_0, c_i) < N$ we have
\[
\left\| \log |c_1'| - \log |c_2'| \right\|_{L^\infty} \leq C \on{dist}^G(c_1,c_2)\,.
\]
In particular the function
\[
\log |c'| : (\on{Imm}(S^1,{\mathbb R}^2), \on{dist}^G) \to L^\infty(S^1,{\mathbb R})
\]
is continuous and Lipschitz continuous on every metric ball.
\end{lemma}
\begin{proof}
Fix $\th \in S^1$ and $c_1 \in \on{Imm}(S^1,{\mathbb R}^2)$ satisfying $\on{dist}^G(c_0, c_1) < N$ and let $c(t,\th)$ be a path between $c_0$ and $c_1$ with $\on{Len}^G(c) \leq 2N$. Then
\[
\p_t \left(\log |c'(\th)|\right) =
\langle D_s c_t(\th), v(\th) \rangle\,.
\]
After integrating and taking norms we get
\begin{align*}
\left| \log |c_1'(\th)| - \log |c_0'(\th)| \right|
&\leq \int_0^1 |D_s c_t(t, \th)| \,\mathrm{d} t\,.
\end{align*}
Using Poincar\'e inequalities and Cor.\ \ref{cor:ell_lip} we can estimate
\begin{multline}
\label{eq:Ds_ct_explicit_bound}
|D_s c_t(\th)| \leq \frac{\sqrt{\ell_c}}2 \| D_s^2 c_t \|_{L^2(ds)} \leq
\\
\leq \frac{\sqrt{\ell_c}}2 \sqrt{ \| c_t \|_{L^2(ds)}^2 + \| D_s^n c_t \|_{L^2(ds)}^2} \leq
\tfrac 12 \sqrt{\ell_c A} \sqrt{G_c(c_t, c_t)}\,.
\end{multline}
Thus by taking the infimum over all paths between $c_0$ and $c_1$ we get
\[
\left\| \log |c_1'| - \log |c_0'| \right\|_{L^\infty} \lesssim_{c_0, N}
\on{dist}^G(c_0, c_1) \,. \qedhere
\]
\end{proof}
\begin{remark*}
An explicit value for the constant is given by
\[
C(c_0, N) = \tfrac 12 \sqrt{A} \left(\sqrt{\ell_{c_0}} + \tfrac 12 \sqrt{A}N\right)\,.
\]
This can be found by combining the estimates \eqref{eq:Ds_ct_explicit_bound} and \eqref{eq:ell_lip_const}.
\end{remark*}
This corollary gives us upper and lower bounds on $|c'(\th)|$ in terms of the geodesic distance.
Therefore, a geodesic $c(t,\th)$ for a Sobolev metric with order at
least 2 cannot leave $\on{Imm}(S^1,{\mathbb R}^2)$ by having $c'(t,\th)=0$ for some $(t,\th)$.
\begin{corollary}
\label{cor:lenpt_inv_bound}
Under the assumptions of Lem.\ \ref{lem:lenpt_inv_bound}, given $c_0 \in \on{Imm}(S^1,{\mathbb R}^2)$ and $N>0$, there exists a constant $C=C(c_0,N)$, such that
\[
\| c' \|_{L^\infty} \leq C\,\qquad\text{ and }\qquad \left\| \frac1{|c'|} \right\|_{L^\infty} \leq C\,
\]
hold for all $c \in \on{Imm}(S^1,{\mathbb R}^2)$ with $\on{dist}^G(c_0,c)<N$.
\end{corollary}
\begin{proof}
By Lem.\ \ref{lem:lenpt_inv_bound} we have
\[
\| \log |c'(\th)| \leq
\| \log |c_0'(\th)| + \left\| \log |c'| - \log |c_0'| \right\|_{L^\infty}
\lesssim_{c_0, N} 1\,.
\]
Now apply $\on{exp}$ and take the supremum over $\th$ to obtain $\| c'\|_{L^\infty} \lesssim_{c_0,N} 1$. Similarly by starting from
\[
-\| \log |c'(\th)| \leq
-\| \log |c_0'(\th)| + \left\| \log |c'| - \log |c_0'| \right\|_{L^\infty}
\lesssim_{c_0, N} 1\,.
\]
we obtain the bound $\left\| |c'|\i \right\| \lesssim_{c_0,N}$.
\end{proof}
\begin{remark*}
Using the explicit constant for Lem.\ \ref{lem:lenpt_inv_bound}, we can obtain the following more explicit inequalities,
\begin{align*}
|c'(\th)| &\leq |c_0'(\th)| \operatorname{exp}\left( \tfrac 12 \sqrt{A}N \left( \sqrt{\ell_{c_0}} + \tfrac 12 \sqrt{A}N\right)\right) \\
|c'(\th)|\i &\leq |c_0'(\th)|\i \operatorname{exp}\left( \tfrac 12 \sqrt{A}N \left( \sqrt{\ell_{c_0}} + \tfrac 12 \sqrt{A}N\right)\right)\,.
\end{align*}
for Cor.\ \ref{cor:lenpt_inv_bound}.
\end{remark*}
\begin{remark}
To simplify the exposition, the results in this section were formulated on the space
$\on{Imm}(S^1,{\mathbb R}^2)$ of smooth immersions.
If $G$ is a Sobolev metric of order $n$ with $n\geq 2$,
we can replace $\on{Imm}(S^1,{\mathbb R}^2)$ by $\on{Imm}^n(S^1,{\mathbb R}^2)$ in all statements of this section with the same proofs.
\end{remark}
\section{Geodesic Completeness for Sobolev Metrics}
On the space $H^n(S^1,{\mathbb R}^d)$ we have two norms: the $H^n(d\th)$-norm as well as the $H^n(ds)$-norm, which depends on the choice of a curve $c \in \on{Imm}(S^1,{\mathbb R}^2)$. Although the norms are equivalent, the constant in the inequality
\[
C\i \| h \|_{H^k(d\th)} \leq \| h \|_{H^k(ds)} \leq C \| h \|_{H^k(d\th)}\,,
\]
depends in general on the curve and its derivatives. The next lemma shows, that if $c$ remains in a metric ball with respect to the geodesic distance, then the constant depends only on the center and the radius of the ball.
\begin{lemma}
\label{lem:Hk_local_equivalence}
Let the metric $G$ on $\on{Imm}(S^1,{\mathbb R}^2)$ satisfy
\[
\int_{S^1} |h|^2 + |D_s^n h|^2 \,\mathrm{d} s \leq A\, G_c(h,h)
\]
for some $n \geq 2$ and some $A>0$. Given $c_0 \in \on{Imm}(S^1,{\mathbb R}^2)$ and $N > 0$, there exists a constant $C = C(c_0, N)$ such that for $0 \leq k \leq n$,
\[
C\i \| h \|_{H^k(d\th)} \leq \| h \|_{H^k(ds)} \leq C \| h \|_{H^k(d\th)}\,,
\]
holds for all $c \in \on{Imm}(S^1,{\mathbb R}^2)$ with $\on{dist}^G(c_0,c) < N$ and all $h \in H^k(S^1,{\mathbb R}^d)$.
\end{lemma}
\begin{proof}
By definition,
\begin{align*}
\| u \|_{H^k(d\th)}^2 &= \| h \|^2_{L^2(d\th)} + \| \p_\th^k h \|^2_{L^2(d\th)} \\
\| u \|_{H^k(ds)}^2 &= \| h \|^2_{L^2(ds)} + \| D_s^k h \|^2_{L^2(ds)}\,.
\end{align*}
The estimates
\[
\left( \min_{\th \in S^1} |c'(\th)| \right) \| h \|^2_{L^2(d\th)}
\leq \| h \|^2_{L^2(ds)} \leq
\| c'\|_{L^\infty} \| h \|^2_{L^2(d\th)}\,,
\]
together with Cor.\ \ref{cor:lenpt_inv_bound} take care of the $L^2$-terms. Thus it remains to compare the derivatives $\| \p_\th^k h \|^2_{L^2(d\th)}$ and $\| D_s^k h \|^2_{L^2(ds)}$. From the identities
\begin{align*}
h' &= |c'| D_s h \\
h'' &= |c'|^2 D_s^2 h + (\p_\th|c'|) D_sh \\
h''' &= |c'|^3 D_s^3 h + 3\,|c'|(\p_\th|c'|) D_s^2 h + (\p_\th^2 |c'|) D_sh \\
h''''\! &= |c'|^4 D_s^4 h + 6\, |c'|^2 (\p_\th |c'|) D_s^3 h +
\left( 3\left(\p_\th |c'|\right)^2 \!+ 4\, |c'|(\p_\th^2 |c'|)\right)D_s^2 h + (\p_\th^3 |c'|) D_s h\,,
\end{align*}
we generalize to
\begin{equation}
\label{eq:hk_to_Dshk}
\p_\th^k h = \sum_{j=1}^k \sum_{\al \in A_j} c_{j,\al} \prod_{i=0}^{k-1} \left(\p_\th^i |c'|\right)^{\al_i} D_s^j h\,,
\end{equation}
where $c_{j,\al}$ are some constants and $\al=(\al_0,\ldots,\al_{k-1})$ are multi-indices that are summed over the index sets
\[
A_j = \left\{ \al\,:\, \sum_{i=0}^{k-1} i\al_i = k-j,\, \sum_{i=0}^{k-1} \al_i = j \right\}\,.
\]
Equation \eqref{eq:hk_to_Dshk} is related to Fa\`a di Bruno's formula \cite{Bruno1855} and can be proven by induction.
The length $\ell_c$ is bounded on the metric ball by Cor.\ \ref{cor:ell_lip}.
Then Lem.\ \ref{thm:ka_bound} together with Poincar\'e inequalities shows that
\begin{itemize}
\item
$\| D_s^{n-1} \log|c'| \|_{L^2(ds)}$ and
\item
$\| D_s^{k} \log|c'| \|_{L^\infty}$ for $1 \leq k \leq n-2$
\end{itemize}
are bounded as well. Repeated application of the chain rule for differentiation yields
\begin{equation*}
D_s^k|c'| = D_s^k\left( \operatorname{exp} \log |c'|\right) = |c'|\, D_s^k \log |c'| + \text{lower $D_s$-derivatives of $\log |c'|$}\,.
\end{equation*}
Thus also $\| D_s^{n-1} |c'| \|_{L^2(ds)}$ and $\| D_s^{k} |c'| \|_{L^\infty}$ for $1 \leq k \leq n-2$ are bounded on metric balls. Next we apply formula \eqref{eq:hk_to_Dshk} to $h = |c'|$ obtaining
\begin{equation}
\label{eq:cp_dth}
\p_\th^k|c'| = |c'|^k D_s^k |c'| + \text{lower $D_s$-derivatives of $|c'|$}\,.
\end{equation}
Together with Lem.\ \ref{lem:lenpt_inv_bound} this implies that
\begin{itemize}
\item
$\| \p_\th^{n-1} |c'| \|_{L^2(d\th)}$ and
\item
$\| \p_\th^k |c'| \|_{L^\infty}$ for $0 \leq k \leq n-2$
\end{itemize}
are bounded on metric balls.
We proceed by induction over $k$. The case $k=0$ has been dealt with at the beginning of the proof. Assume $k \leq n-1$ and the equivalence of the norms has been shown for $k-1$. Then the highest derivative of $|c'|$ is $\p_\th^{k-1} |c'|$ and so in \eqref{eq:hk_to_Dshk} we can estimate every term involving $|c'|$ using the $L^\infty$-norm. Thus using Poincaré inequalities and the equivalence of $L^2(d\th)$ and $L^2(ds)$-norms we get
\[
\| \p_\th^k h \|^2_{L^2(d\th)} \lesssim_{c_0, N} \| D_s^k h \|^2_{L^2(ds)}\,.
\]
For the other inequality write
\[
D_s^k h = |c'|^{-k}\, \p_\th^k h - |c'|^{-k} \sum_{j=1}^{k-1} \sum_{\al \in A_j} c_{j,\al} \prod_{i=0}^{k-1} \left(\p_\th^i |c'|\right)^{\al_i} D_s^j h\,,
\]
and use the induction assumption $\| D_s^j h \|^2_{L^2(ds)} \lesssim_{c_0, N} \| \p_\th^j h \|^2_{L^2(d\th)}$ for $0 \leq j < k$.
The only remaining case is $k=n$. There we have to be a bit more careful, since then $\p_\th^{n-1}
|c'|$ appears in \eqref{eq:hk_to_Dshk}, which cannot be bound using the $L^\infty$-norm. However
$\p_\th^{n-1} |c'|$ appears only in the summand $\left( \p_\th^{n-1} |c'| \right) D_s h$, i.e. if
$\al_{n-1} \neq 0$, then $\al_{n-1} = 1$, $\al_i = 0$ for $i \neq n-1$ and $\al \in A_1$. This term
we can estimate via
\[
\left\| \left( \p_\th^{n-1} |c'| \right) D_s h \right\|_{L^2(d\th)} \leq
\left\| \p_\th^{n-1} |c'| \right\|_{L^2(d\th)} \| D_s h\|_{L^\infty}\,,
\]
and then depending on which direction we want to estimate, we can use either of
\begin{align*}
\| D_s h \|_{L^\infty} &\leq 2\i \sqrt{\ell_c}\, \left\| D_s^2 h \right\|_{L^2(ds)} \\
\| D_s h \|_{L^\infty} &\leq \left\| |c'|\i \right\|_{L^\infty} \left\| \p_\th h \right\|_{L^\infty} \leq C \left\| |c'|\i \right\|_{L^\infty}
\left\| \p_\th^2 h \right\|_{L^2(d\th)}\,.
\end{align*}
From here we proceed as for $k< n$.
\end{proof}
We saw in Lem.\ \ref{lem:sob_mult} that multiplication is a bounded bilinear map on the spaces $H^k(S^1,{\mathbb R}^d)$ with the $H^k(d\th)$-norm. Since the $H^k(d\th)$-norm and the $H^k(ds)$-norm are equivalent, this holds also for the $H^k(ds)$-norm. A consequence of Lem.\ \ref{lem:Hk_local_equivalence} is that the constant in the inequality
\[
\left\| \langle f, g \rangle \right\|_{H^k(ds)} \leq C \| f \|_{H^k(ds)} \| g \|_{H^k(ds)}\,,
\]
again depends only on the center and radius of the geodesic ball.
\begin{corollary}
\label{cor:Hk_multiplication}
Under the assumptions of Lem.\ \ref{lem:Hk_local_equivalence} there exists a constant $C=C(c_0, N)$ such that for $c \in \on{Imm}(S^1,{\mathbb R}^2)$ with $\on{dist}^G(c_0, c) < N$ and $1 \leq k \leq n$,
\[
\left\| \langle f, g \rangle \right\|_{H^k(ds)} \leq C \| f \|_{H^k(ds)} \| g \|_{H^k(ds)}\,,
\]
holds for all $f, g \in H^k(S^1,{\mathbb R}^d)$.
\end{corollary}
\begin{proof}
We use Lem.\ \ref{lem:Hk_local_equivalence} and the boundedness of multiplication on $H^k(d\th)$,
\begin{align*}
\left\| \langle f, g \rangle \right\|_{H^k(ds)}
&\lesssim_{c_0,N} \left\| \langle f, g \rangle \right\|_{H^k(d\th)} \\
&\lesssim_{c_0,N} \| f \|_{H^k(d\th)} \| g \|_{H^k(d\th)}
\lesssim_{c_0, N} \| f \|_{H^k(ds)} \| g \|_{H^k(ds)}\,. \qedhere
\end{align*}
\end{proof}
This last lemma shows that the identity
\[
\on{Id}: \left((\on{Imm}^n(S^1,{\mathbb R}^2), \on{dist}^G\right) \to \left(H^n(S^1,{\mathbb R}^2), H^n(d\th)\right)
\]
maps bounded sets to bounded sets and that the same holds for the function
\[
\left((\on{Imm}^n(S^1,{\mathbb R}^2), \on{dist}^G\right) \to {\mathbb R}\,,\quad
c \mapsto \| c\|_{H^n(ds)}\,,
\]
when $G$ is stronger than a Sobolev metric of order $n$.
\begin{lemma}
\label{lem:c_bounded}
Let the metric $G$ on $\on{Imm}(S^1,{\mathbb R}^2)$ satisfy
\[
\int_{S^1} |h|^2 + |D_s^n h|^2 \,\mathrm{d} s \leq A\, G_c(h,h)
\]
for some $n \geq 2$ and some $A>0$. Given $c_0 \in \on{Imm}(S^1,{\mathbb R}^2)$ and $N > 0$, there exists a constant $C = C(c_0, N)$, such that
\[
\| c \|_{H^n(d\th)} \leq C\,,\qquad
\| c \|_{H^n(ds)} \leq C\,,
\]
hold for all $c \in \on{Imm}(S^1,{\mathbb R}^2)$ with $\on{dist}^G(c_0, c) < N$.
\end{lemma}
\begin{proof}
It is only necessary to prove the boundedness in one of the norms, since Lem.\ \ref{lem:Hk_local_equivalence} will imply the other one.
We have
\[
\| c \|^2_{H^n(ds)} = \| c \|^2_{L^2(ds)} + \| D_s^n c \|^2_{L^2(ds)}
= \| c \|^2_{L^2(ds)} + \| D_s^{n-2} \ka \|^2_{L^2(ds)}\,.
\]
The boundedness of $\| D_s^{n-2} \ka \|^2_{L^2(ds)}$ on metric balls has been shown in Thm.\ \ref{thm:ka_bound}. For $\| c \|_{L^2(ds)}$ we choose a path $c(t)$ from $c_0$ to $c = c(1)$ with $\on{Len}^G(c(t)) < 2N$. Then
\begin{align*}
\| c \|_{L^2(ds)} &\lesssim_{c_0,N} \| c \|_{L^2(d\th)}
\leq \| c -c_0 \|_{L^2(d\th)} + \| c_0 \|_{L^2(d\th)} \\
&\lesssim_{c_0, N} \left\| \int_0^1 \p_t c(t) \,\mathrm{d} t \right\|_{L^2(d\th)}
\leq \int_0^1 \left\| \p_t c(t) \right\|_{L^2(d\th)} \,\mathrm{d} t \\
&\lesssim_{c_0, N} \int_0^1 \left\| \p_t c(t) \right\|_{L^2(ds)} \,\mathrm{d} t
\leq \on{Len}^G(c(t)) \lesssim_{c_0, N} 1. \qedhere
\end{align*}
\end{proof}
\begin{remark}
The proof of Lem.\ \ref{lem:Hk_local_equivalence} shows that under the assumptions of Lem.\ \ref{lem:c_bounded} we can choose $C=C(c_0,N)$ such that the additional inequality
\[
\left\| |c'| \right\|_{H^{n-1}(d\th)} \leq C\,,
\]
holds as well.
\end{remark}
Now we are ready to prove the main theorem.
\begin{theorem}
\label{thm:long_time}
Let $n\geq 2$ and let $G$ be a Sobolev metric with constant coefficients $a_i\ge 0$ of order $n$
and $a_0,a_n>0$.
Given $(c_0, u_0) \in T\on{Imm}^n(S^1,{\mathbb R}^2)$ the solution of the geodesic equation for the metric $G$ with initial values $(c_0, u_0)$ exists for all time.
\end{theorem}
\begin{corollary}
Let the metric $G$ be as in Thm.\ \ref{thm:long_time}. Then the Riemannian manifolds $(\on{Imm}^n(S^1,{\mathbb R}^2),G)$ and $\on{Imm}(S^1,{\mathbb R}^2), G)$ are geodesically complete.
\end{corollary}
\begin{proof}
The geodesic completeness of $\on{Imm}(S^1,{\mathbb R}^2)$ follows from Thm.\ \ref{thm:geod_ex}, since given smooth initial conditions the intervals of existence are uniform in the Sobolev order.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:long_time}]
The geodesic equation is equivalent to the following ODE on $(T\on{Imm}^n)' \cong \on{Imm}^n \times H^{-n}$,
\begin{align*}
c_{t} &= \bar{L}_c\i p \\
p_t &= \tfrac 12 H_c\left(\bar{L}_c\i p, \bar{L}_c\i p\right)\,,
\end{align*}
with $p(t) = \bar{L}_{c(t)} u(t)$. Fix initial conditions $(c(0), p(0))$. In order to show that the geodesic with these initial conditions exists for all time, we need to show that on any finite interval $[0, T)$, on which the geodesic $(c(t), p(t))$ exists, we have that
\begin{enumerate}[(A)]
\item
\label{completeness_cond1}
the closure of ${c([0,T))}$ in $H^n(S^1,{\mathbb R}^2)$ is contained in $\on{Imm}^n(S^1,{\mathbb R}^2)$ and,
\item
\label{completeness_cond2}
$\left\| \bar{L}_c\i p \right\|_{H^n(d\th)}$,
$\tfrac 12 \left\| H_c(\bar{L}_c\i p, \bar{L}_c\i p) \right\|_{H^{-n}(d\th)}$ are bounded on $[0,T)$.
\end{enumerate}
Then we can apply \cite[Thm.\ 10.5.5]{Dieudonne1969} to conclude that $[0,T)$ is not the maximal interval of existence. Since this holds for every $T$, the geodesic must exist on $[0,\infty)$.
Assume now that $T>0$ is fixed. We will pass freely between the momentum and the velocity via $u(t) = \bar{L}\i_{c(t)} p(t)$. Since $c(t)$ is a geodesic, we have
\[
\on{dist}^G(c_0, c(t)) \leq \sqrt{G_{c(0)}(u(0), u(0))}\, T
\quad\text{and}\quad
G_{c(t)}(u(t), u(t)) = G_{c(0)}(u(0), u(0))\,.
\]
In particular the geodesic remains in a metric ball around $c_0$. It follows from Cor.\ \ref{cor:lenpt_inv_bound} that there exists a $C>0$ with $|c'(t,\th)|\geq C$ for $(t,\th) \in [0,T) \times S^1$. Since the set $\{ c : |c'(\th)| \geq C \}$ is $H^2$-closed -- and hence also $H^n$-closed -- in $\on{Imm}^n(S^1,{\mathbb R}^2)$, we can conclude that condition (\ref{completeness_cond1}) is satisfied.
The first part of condition (\ref{completeness_cond2}) follows easily from
\begin{multline*}
\left\| \bar{L}_c\i p \right\|^2_{H^n(d\th)} = \| u \|^2_{H^n(d\th)}
\lesssim_{c_0,T} \| u \|^2_{H^n(ds)} \leq{} \\
\leq \max(a_0\i, a_n\i) G_c(u,u)
= \max(a_0\i, a_n\i) G_{c(0)}(u(0),u(0))\,,
\end{multline*}
using Lem.\ \ref{lem:Hk_local_equivalence} and that the velocity is constant along a geodesic.
It remains to show that $\| H_c(u, u) \|_{H^{-n}(d\th)}$ remains bounded along $c(t)$. To estimate
this norm, pick $m \in H^n(d\th)$ and consider the pairing
\begin{multline*}
\langle H_c(u,u), m \rangle_{H^{-n}\times H^n} = D_{c,m} G_c(u,u)
= \int_{S^1} \sum_{k=0}^n a_k \langle D_s^k u, D_s^k u \rangle
\langle D_s m, v \rangle \,\mathrm{d} s -
\\
- 2 \sum_{k=1}^n \sum_{j=1}^k a_k \langle D_s^k u,
D_s^{k-j}\left( \langle D_s m, v \rangle D_s^j u \right) \rangle \,\mathrm{d} s\,.
\end{multline*}
Using Poincar\'e inequalities, Lem.\ \ref{lem:Hk_local_equivalence}, and that $\ell_c$ is bounded
along $c(t)$, we can estimate the first term,
\begin{align*}
\left| \int_{S^1} \sum_{k=0}^n a_k \langle D_s^k u, D_s^k u \rangle
\langle D_s m, v \rangle \,\mathrm{d} s \right|
&\leq \| D_s m \|_{L^\infty}\, G_c(u,u) \\
&\lesssim_{c_0,T} \| m \|_{H^n(ds)}
\lesssim_{c_0,T} \| m \|_{H^n(d\th)}\,.
\end{align*}
For the second term we additionally need Cor.\ \ref{cor:Hk_multiplication}. For each $1\leq k \leq n$ and $1 \leq j \leq k$ we have,
\begin{align*}
\bigg| \int_{S^1} \Big\langle D_s^k u,
D_s^{k-j}\big( \langle D_s m, v \rangle &D_s^j u \big) \Big\rangle \,\mathrm{d} s \bigg|
\leq \left\| D_s^k u \right\|_{L^2(ds)} \left\| D_s^{k-j}\left(
\langle D_s m, v \rangle D_s^j u \right) \right\|_{L^2(ds)} \\
&\leq \| u \|_{H^k(ds)} \left\| \langle D_s m, v \rangle
D_s^j u \right\|_{H^{k-j}(ds)} \\
&\lesssim_{c_0,T} \| u \|_{H^k(ds)} \| D_s m \|_{H^{k-j}(ds)} \| v \|_{H^{k-j}(ds)}
\| D_s^j u \|_{H^{k-j}(ds)} \\
&\lesssim_{c_0,T} \| u \|^2_{H^n(ds)} \| c \|_{H^{n}(ds)} \| m \|_{H^{n}(ds)}\,.
\end{align*}
We know that $\| u \|^2_{H^n(ds)}$ is bounded along $c(t)$ and using Lem.\ \ref{lem:c_bounded} we see that $\| c \|_{H^{n}(ds)}$ is bounded as well. Hence we obtain
\[
\left| \langle H_c(u,u), m \rangle_{H^{-n}\times H^n} \right|
\lesssim_{c_0,T} \| m \|_{H^n(d\th)}\,,
\]
which implies
$$
\| H_c(u,u) \|_{H^{-n}(d\th)} \lesssim_{c_0,T} 1\,,
$$
i.e., $\| H_c(u,u) \|_{H^{-n}(d\th)}$ is bounded along the geodesic.
\end{proof}
\begin{remark}
\label{rem:incomplete}
If $G$ is a Sobolev-type metric of order $n\geq 2$ with $a_0=0$, $a_1=0$, then $G$ is a Riemannian metric on the space $\on{Imm}(S^1,{\mathbb R}^2)/\on{Tra}$ of plane curves modulo translations. We will show that for these metrics it is possible to blow up circles to infinity along geodesics in finite time, making them geodesically incomplete. Thus a non-vanishing zero or first order term is necessary for geodesic completeness.
The 1-dimensional submanifold consisting of all concentric circles, which are parametrized by
constant speed, is a geodesic with respect to the metric,
because Sobolev-type metrics are invariant under the motion group.
Let $c(t,\th) = r(t)\, (\cos \th,
\sin\th)$. Then $c_t(t,\th) = r_t(t)\, (\cos \th, \sin \th)$ and $|c'(t,\th)| = r(t)$. Thus
\[
G_c(c_t,c_t) = 2\pi \sum_{j=2}^n a_j r(t)^{1-2j} r_t(t)^2\,,
\]
and the length of the curve is
\[
\on{Len}^G(c) = \int_0^1 \sqrt{2\pi \sum_{j=2}^n a_j r(t)^{1-2j} r_t(t)^2} \,\mathrm{d} t
= \sqrt{2\pi} \int_{r(0)}^{r(1)} \sqrt{\sum_{j=2}^n a_j \si^{1-2j}} \,\mathrm{d} \si\,.
\]
Since the integral converges for $r(1) \to \infty$, it follows that the path consisting of growing circles can reach infinity with finite length.
\end{remark}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,147
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Accueil › Sade "Lovers Rock" CD (2000)
Sade "Lovers Rock" CD (2000)
Details: "Lovers Rock is the fifth studio album by Sade, released in 2000 on Epic Records. The album was titled after a style of reggae music known as lovers rock, noted for its romantic sound and content, which front woman Sade Adu listened to in her youth. Fantastic condition CD.
Description: "… Britishness had long shaped Sade, but here it's literalized. The album's title and the sounds throughout point to the specific style of romantic reggae that shaped much of London youth culture in the 1970s. Lovers Rock was also, for Adu, who was then approaching middle age, a full circle meaning: She was spiritually indebted to the genre, as her career in music had accidentally been kickstarted by a chance run-in at a lovers rock concert. And the relationship that underscored much of it had led her to spend years in Jamaica during that sabbatical. I've always been compelled by the absence of a possessive in a noun (Lovers) that often requires it.
The album stood out all the more in 2000, as pop became a shiny, slick vision of some imagined, tech-mitigated Jetsons-style future. Unlike other '80s acts desperately seeking reinvention in the trends of the day, Sade avoided obvious dialogue with the charts. There are nods to hip-hop in the drums on "Flow" and digital experimenting in the synths on "By Your Side," but for the most part, the band opted out. And yet Lovers Rock proved to be predictive. Echos of its style, the kind of mellow, vibey pop that could very well be described as Sade-core, have been palpable throughout the past decade, from the rhythmic R&B of Jessie Ware and Rhye to an entire generation of rappers. Among the most obviously indebted to Sade is Drake, whose grotesque registry of commemorative tattoos includes two of Adu's face. The "dark, sexy" sound he introduced early in his career are direct reflections of the band's influence. "I'll call them 'Sade moments,' where you hear it, it hits you, and you feel something," he told MTV in 2010."—Pitchfork
Grade: NM (CD/Cover)
2. Flow
3. King Of Sorrow
4. Somebody Already Broke My Heart
5. All About Love
6. Slave Song
7. The Sweetest Gift
8. Every Word
9. Immigrant
10. Lovers Rock
11. It's Only Love That Gets You Through
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,689
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/**
* This file is part of the CernVM File System.
*/
#ifndef CVMFS_INTERRUPT_H_
#define CVMFS_INTERRUPT_H_
/**
* Allows to query for interrupts of active file system requests. Used
* to hande canceled fuse requests with the inherited class FuseInterruptCue.
*/
class InterruptCue {
public:
InterruptCue() { }
virtual ~InterruptCue() { }
virtual bool IsCanceled() { return false; }
};
#endif // CVMFS_INTERRUPT_H_
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,480
|
Le gouvernement finlandais () est l'institution disposant du pouvoir exécutif en Finlande. Le pays fonctionnant sur la base d'un régime parlementaire, il ne peut gouverner sans disposer de la confiance de la Diète nationale (en ).
L'actuel gouvernement finlandais est le gouvernement Marin.
Ministères
Le gouvernement finlandais compte douze ministères mais les fonctions d'un ministère peuvent être exercées par plusieurs ministres.
Conseil d'État
Administration locale
.
Références
Voir aussi
Article connexe
Politique en Finlande
Vice-Premier ministre de Finlande
Liens externes
Site du gouvernement
Liste des gouvernements de Finlande
Politique étrangère de la Finlande
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,299
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Tinkering With Trait Deeds = Removing A Way To Earn LoTRO Points ???
Thread: Tinkering With Trait Deeds = Removing A Way To Earn LoTRO Points ???
One of the things we have been looking into is an update to the Virtue system to make it more relevant. Rather than the current system where Deeds reward Virtue points directly, we will shift to a system where completing Deeds (and possibly other content) will award Virtue experience points. Players will be able to choose which Virtue advances, whether that Virtue is slotted or not. Characters with existing Virtue ranks will be converted to the new system with their Virtue level intact. Additionally, all Virtues will provide small passive benefits to your character, even if they are not slotted. These changes will allow players more choice when earning Virtue benefits.
Looking back, many things changed and initially advertised or announced as being done for our/the players sake, has often turned out to be a sour fruit to bite.
The removing of earnable Lootbox keys, that was not what players asked for but we asked for the removal of lootboxes all together. Not what we got, quite the opposite, making it even more pay for stuff. As well as old style lootboxes LvL 10-100, gone and items from them like defense tomes nowhere to be found but the store and stat tomes only very rarely and randomly from instances. Things that used to be able to get via keys earned in game and open the old lootboxes LvL 10-100.
The so called balance patches, that in many cases have become NERF patches, totally ruining some builds, very often the blueline on DPS classes (like burglar and hunter for example), leaving only redline left to use (and yellow for burglar in some instances), even only those lines required if wanting to get a spot in groups. Leaving less choices up to the player how they prefer to play. Another example, forcing minstrels to become only healers and no one inviting a dps minstrel anymore (from what I understand). The recent RK NERF doing the reverse of that, forcing RK's to be dp classes and NERFing their healing capability etc. the list can go on. The only class that seem to have gotten pure improveents would be Beorning.
Not to mention the whole new instance lock system with favoured loot that sounded so promising upon peoples request to remove daily locks, but instead lead to now being able to run instances in ered Mithrin even less than once/day with any loot worth mentioning and still having daily locks on solo versions. Nothing of all that has turned out to what players acctually requested. As well as a bugged system when loot that is supposed to be able to be shared witing the group or account bound from the Ered Mithrin instances ends up character bound if lose connection or log out while the 1 hr timer is ticking down. You can not mail it to an alt during that timer either. Evn trading to others often lead to crashing and getting logged out and then stuck with those items on that character and no way to transfer to an alt afterwards.
All this, and more, leads me to be very sceptical to requests I see from people about revamping the LI system for example "Be Careful What You Wish For" and as I now read that SSG will revamp the trait system I'm starting to wonder what will now be removed/NERFed??? Since earning traits by doing deeds is the main source for many to earn LoTRO Points (lotro announced as free to play remember), I wonder if anyone have thought about that in such a revamp, the LoTRO points earned could be removed or NERFed in the process from doing such deeds in the future ??? Could that be the real reason behind such a revamp ??? A stealth NERF of ways to earn LoTRO Points from deeds, making it harder, take longer or earning less, possibly none etc ???
We have already seen signs of such with deeds in newer regions already rewarding less than they used to in older areas. No more 5 LP for basic deed, then 10 LP for advanced deed. Now it's just 5 LP and 5 LP. Wasn't there even one of the "newer" regions where we only earned 5 LP and nothing more ???
I'm sorry to sound so sceptical and negative, but I tend to try and learn from history and looking back this is one such conclusion I draw that I find could unfortionetly possibly be the case. Hopefully my doubts are in vain. No one will be happier than me if my worries are unfounded.
Last edited by Lord.Funk; Feb 22 2019 at 11:46 PM. Reason: The usual typos and such.
Exactly right. They offered completely dubious reasoning for this change. It's why the letter was so long coming, how the ^&$& do we spin this?
T3 T4 or T8 no real extra work just a setting to set.
Masterful spin in that virtue paragraph, with curiously vague wording. I am inclined to think in the same direction as OP.
Who knows what they were planning before this thread, but I can bet I know what they WILL do now sadly.
I wonder if it will win the same industry accolades as they got for the server merge spin, err datacentre upgrade?
If this announcement to revamp the trait system ends up being just another covert way to nerf our ability to earn LP...? Then what is happening to their "free to play" model? Are they changing it to a "REALLY HARD to play for free" format now?
This is just my opinion, but changing game systems to force people to pay to play your game I think will eventually backfire. The more you tighten your grip on us players the more will slip away.
How many have you lost alrdy over the whole lootbox debacle? I have no family or friends left that continue to play this game and I am the only active person left in our kin. Is this your intention? Sure seems like it is!
I know you need to make money and I will support that with my wallet if you would just focus on improving the systems alrdy in place without taking more stuff/privileges away.
No, you're not imagining it.
The heaviest class nerfs have been directed squarely at low level (1-25), mobile DPS-traited classes proficient at farming LP, in my opinion. I pointed this out in a couple of posts over the past year but it never really gained much traction. If SSG diminishes LP generation with this planned revamp of deeding I'm sure there will be a sizable contingent of angry LOTRO'ers. I sincerely hope this doesn't happen as I believe it will hurt the game overall. I will be watching this matter closely this year, and I suspect many others will too.
I'm going to totally reserve judgement on this new virtue system until we have all the details. I'm worried about it - for sure, but that comes from how previous changes to systems for the "better" almost always turned out, to be the exact opposite of bettering anything.
They've made no mention of stopping LP from deeds so far . . . . but that of course doesn't mean that they won't go there. I think it may be quite difficult to completely stop it, after all, players have bought regions that they may not have otherwise bought, for the purpose of grinding out the deeds within them on all of their characters, even future characters. I think removing that option would be a bit shady.
True, with Mordor they reduced the number of LP from deeds, which of course, they can do without any problem as it's not taking something away from people after they have paid for it. Of course, nobody actually knew about this change until after they had made the purchase, so perhaps stating it beforehand would have been a nicer approach, but hey . . .
Whether they restrict or reduce LP from deeds in the next update remains to be seen, but either way, I'm hoping the next expansion doesn't come with the same kind of price tag as Mordor did (base pack price was fine, I'm talking about the other packs). It was drastically over-priced IMO. I'd like to buy the next one, but just like with Mordor, I'll sit it out if they over-price it and maybe buy it later if it goes on sale (without actually counting on that however, I'd be fine with never buying it).
I'm think this line of thinking is paranoid. There's nothing there that suggests LP points from deeds will be affected. LP points and virtues are separate rewards from deeds, there's no reason to think a change in one will reflect in the other. It's easy to imagine that the change will as simple as possible: when you complete a deed, instead of immediately gaining a specific virtue level, you get points you use to advance the virtue you want. It doesn't seem likely 1the whole deed system will change, just the virtue rewards.
You know, it's generally a good advice to expect the best in people. If you approach any problem with the mentality that people are greedy and dishonest and out to get you, the only result is that you become distrustful and hostile towards anyone you meet. Chill out, people.
Turbine then SSG changed my outlook on such matters.
Im not super worried yet. Virtues have been nearly unchanged in 11 years. It worth looking at them for an upgrade. The legendary server reminded me how annoying it can be to grind virtues on all characters. And hopefully virtue experience is account shared. So while you can complete to deed on multiple characters, you only "Have to" complete it on one.
Wow. That may be the most untrue thing I have ever read on the internet.
People may not be greedy (in a negative way) and dishonest, but they are going to act in their own self interest which may or may not be compatible with yours (and they are generally going to be indifferent to yours in the absence of some established relationship and even then human nature will often win out). This is why in business we have contracts and the second reason (after mutual defense) that we tolerate governments (the enforcement of those contracts).
Given the frankly hostile changes to minis that happened recently without a mention in the flowery patch notes, it's entirely reasonable for people to speculate about what they are not telling us. particularly if it fits into an established pattern of behavior from the company. And given the frequency of threads in these forums that amount to "Let's get rid of F2P players and then somehow PROFIT," it's entirely possible that someone at SSG might be tempted to think that's an acceptable plan going forward. These forums exist, among other things, to give SSG feedback and OP is doing just that.
Of course the easiest way for this worry to be dealt with is for someone blue to reassure us that only the virtue award is being modified and LP for all deeds will remain at current levels.
I'm glad to have set a record. If it's the case, you must be the happiest person in the planet. Here's the thing: I believe firmly that kind of absurd hyperbole contributes in nothing to a productive discussion; on the contrary, it makes it impossible to engage in ay meaning conversation about anything whatsoever. For example, I hardly think the changes in minstrel class could be classified as some kind of war crime or high-level betrayal of public trust.
I explicitly meant it as advice, which cannot be "true" or "untrue", and I believe it to be good advice. It has nothing to do with what people objectively will do or not do, but only with what is our general attitude towards other people. I'm pointing to the fact that if you expect people to do the worst, it will affect your outlook negatively. If I personally thought that SSG was composed of nothing but greedy self-interested people, and that every single aspect of this game could be reduced to a crude money transaction, I'd not be here. If that's all you expect, is it worth it? If you see everything through the lenses of commercial exchange between selfish individuals only out to protect themselves, I don't understand how any of this could be fun, or engaging, or worthwhile in any aspect. Wouldn't it be better to go play in the stock exchange, or watch someone play with a spreadsheet?
Other than that, what you present is a very superficial hobbesian perspective of the basis of the modern state, and I'm personally much prefer Locke's view.
Can anyone explain to me, where the connection between Virtue tiers and Trait points lies, that you are criticizing both at the same time, in this thread? Yes, you can get both by deeding. But only the virtue type is gained by nothing else than deeds. Unfortunately, you cannot chose; it is always a certain virtue, that is linked to a given deed. Which is something my lore master doesn't like, because eg Discipline is completely worthless for him. But why do you think, that a change means that the deeds will be removed? Let us consider this: Until now, we selected five of about twenty virtues, with fixed values. There isn't much versatility in that, nor do they actually help with the gameplay.
I don't know what they are planning; but I can imagine that virtue trees will come, and that these work in a similar way as the trait trees already do; and you will have a blue tree (resistances), a red tree (physical mastery), and a yellow tree (tactical mastery). This would mean: in instances you will have to adapt according to the tiers, from aggressive red/yellow on tier 1 or 2, to defensive blue on tier 4, to finish said instance. Is that so bad?
Last edited by Polymachos; Feb 23 2019 at 12:59 PM.
The OP didn't mention trait points, at all.
"as I now read that SSG will revamp the trait system I'm starting to wonder . . . "
The trait system was raised, not trait points. Hover over the wizard icon in game - to open the "traits" panel. Virtues are very much within the "trait system". They are called Virtue "traits", and sit below Class traits and Race traits.
I hope the OP is wrong in his surmise, but it wouldn't surprise me if he isn't. The stinginess about LP in upper levels is one of those small things that irks me very much, because it seems petty in the extreme. Truth to tell, I've been a completionist until the upper levels, but I just don't feel like killing tons of things for no reward whatsoever. It's not like it becomes easier to do slayers at higher levels, given the increase in the amounts required. And I can't see those few LP SSG are withholding are going to make or break the game. IMHO, it should go to 5 for basic slayers, 10 for advanced all the way up to level cap.
I'd be bummed if LP were no longer awarded for deeds, but I don't think the producer's letter said one way or the other. It said that instead of a specific deed leveling a specific virtue, deeds will now earn virtue points so that the player can decide which virtues to focus on. It didn't say whether we'd continue to get faction rep and other rewards for deeds, but I expect we will.
That is quite the jump the OP made there. I wouldn't put anymore thought into this until stated otherwise. I feel like you just gave them an idea they didn't previously have.
Actually it isn't. Cut off the free faucet to have you drink bottled water bought at your local store. From a marketing stand it makes perfect sense. Why not cut off the free flow by changing the implement by which it is being served? Also, I doubt this thread gave anyone in their marketing an idea they haven't thought of already. If players cant see the means for accruing free LP is and has been being diminished then they aren't looking very hard.
I'm think this line of thinking is paranoid.
Remember things can only be paranoid if not becoming true and the fear/skepticism comes out of pure fantasy and have no basis in past experiences or an established pattern. To take an extreme example, that very few could even believe or imagine was true, the whole Echelon monitoring and surveillance issue. https://en.wikipedia.org/wiki/ECHELON People who suspected it to be true were deemed as paranoid. Some even tried to warn us as far back as in 1972. It was all deemed as paranoia until it was uncovered that it indeed was true. Many other such examples in real life and history. It is not being paranoid if a fear/skeptical questions are based on logic and an established pattern. I already named at least 3 examples in recent history, all related to LoTRO, that have turned out sour and not at all how they were requested by players nor how they were intitially presented as something good and beneficial to all. When implemented they took away and made things worse for the players in many cases. Someone else also brought up a 4th example with the server move/merges. The whole "European server" thing going down the drain and instead server mergers and all still based in US.
That is why I also said that I am skeptical about people requesting an LI revamp. Who can say if it will be a system that will be better for us, the players ??? The current system is grindy as *piiip* but at least once You have Your LI it acctually works and gives nice boost that are so necessary today that most (probably all) classes would be really crippled without them. Imagine what we could get ??? Something even more store oriented, a big push towards the store. Or not even a system that reduces cooldowns on skills. Not being as flexible at all. Ending up to be the biggest NERF to most classes and replaced by something worse. It was acctually that alot of people were unhappy that their so called Legendary Weapon did not grow with them, as originally announced it would (not making it feel legendary at all) having to get a new one all the time and start over, and requested a system that would grow with them, that lead to the imbuement system in the first place. We simpy do not know what it might lead to now requesting it to be revamped. Sure it could be a better system as well. However again, looking at an established pattern, it would be safer to simply ask for a reduction in number of emp scrolls and star-lits needed, rather than a revamp. It would be alot let risky than a revamp.
I think it is better to bring up any question and notice a potential problem before it is too late. It's too late when a new system stares us in the face. Be it a revamped virtue system or something that replaces LI's. Bring up our concerns and hopefully affect the outcome in a way that we would like to see it done, or at least mitigate the damage a bit.
I was optimistic about the "favoured loot system", along with many others. It sounded so good on paper. A solution to the daily locks. However what we got was something that gives us even fewer runs (with any loot worth mentioning) per week. Not at all how it was announced or presented. I and many others felt really bit and dissepointed by that. Not even a solution to the issue that people had. Not at all making it possible to run insances more times, but instead fewer. Same with the removal of lootboxes, instead we got the removal of earnable lootbox keys. Quite the opposite of what most requested. It has even lead to old things You could get from old boxes, like defence tomes, now only being availbale from store, stat tomes being almost impossible to get for Your class and the right tier without buying them in store etc. Not at all what the players requested. I shouldn't even have to reapeat it, as i already explained all that and it's historical facts as far as this game goes. Very recent history.
So my skepticism, fear, worries or questions etc, are purely based on an already established pattern. I have never said it's a fact that it will happen with LP. I raised the question, wanting to make people aware of it. Make people think about it, hopefully by simply raisig it that it maybe might even mitigate it. Maybe making SSG/Daybreak (whover calls the shots we simply do not know) think twice before doing something like that. Hoping by seeing players reaction they would refrain from even having any such plans. Letting SSG/Daybreak know that some of us are watching what they do (hopefully wake up more people as well) and not simply fall for weird unclear wordings once more and swallow or accept it at face value, but that we think further than that. I also said that no one will be happier than me if my worries are unfounded in this case.
Even if they turn out to be unfounded, it will not stop me from having similar thoughts or worries in the future. I already have it as far as SSG starting to tinker with the LI system as well (as I have explained my reasons for). At this point it would be naive, to the point of being blind, if one is not skeptical. Again due to an established pattern by Turbine/SSG/Daybreak (whoever called the shots in each specific case) in the recent past. I have come to think of that saying more and more each time I see requests in here or I think of coming with a request myself "Careful What You Wish For". Is it a shame to have to think like that ??? Certainly, but it didn't come out of nowhere. If You get bit by some type of creature a couple of times, I assure You that You would start being more cautious and not assume they are all benevolent. It's simply called not being naive and learn from history and experience.
Yes, I agree, it's a wonderful advice, if the world was a Utopia. However if someone on internet asks You to send them say US$ 1.000 with a promise to pay You back, will You then just assume the best in them and send them the money ??? I do hope You would not, or at least base it on their previous behaviour if You have any experince with them, before You make a decision. You see not being naive and use caution can sometimes save You alot of dissepointments and heartache. Yes, there is of course the risk of unhealthy skepticism and even paranoia, but with pure optimism and naivety also comes huge risks of being used, abused, scammed etc and very dissepointed after the fact. In fact, it might be that having once been that naive that has led some (me included) to now not be so naive anymore and more skeptical. They/we/I have learned their lesson from reality so to speak. Not a bad think if You ask me.
I will not make the mistake again of accepting something at face value. Especially not when not indepth explained.
to accept someone or something without considering whether they really are what they claim to be"
You see skepticism is nothing new. It exists for a reason and it can be very healthy indeed and not always simply be brushed away as paranoia or lead to a negative outcome. In fact being naive can be much more damaging. It all depends on the situation and past experience.
I see it as healthy though that some question my skepticism, as that is what being skeptical is all about. Not accept anything at face value. So by quetioning someone else being skeptical, You Yourself are being skeptical too. It's a good thing.
Also remember that I am still open to the possibilty that these are unfounded fears/worries. It is still justified to raise the question though. Rather safe than sorry, as they say.
Last edited by Lord.Funk; Feb 24 2019 at 03:23 AM.
Calling people's input paranoia derails the conversation, in my opinion.
"I'm sorry to sound so sceptical...No one will be happier than me if my worries are unfounded."
He is also correct that a healthy skepticism has its benefits. Lets not forget that one of the first deeds Middle Earthers can earn is called "The Wary", which can only be awarded if one is ware of their surroundings.
I'll be reserving judgment, like Arnenna, but I'll still be watching the matter closely as stated before.
What concerns me is that they might botch the trait system in the same way that they botched the skill tree system (having to do deeds using skills we don't want to earn points to spend on the skills we do want). If we were to earn 'trait points' by doing deeds, does that mean that for each deed we complete we would earn 1/3 of a trait point? Because currently if I want to increase my Agility stat, I do the slug slayer deed in the Shire for the Determination Virtue and that gives me +9 to Agility, Wolf slayer in Ered Luin does the same. If they start messing with this system does that mean that I will have to do 3 deeds to earn 1/3 of a traitpoint on each just to get the same result as doing 1 deed does now?
I wasn't being serious about this thread giving them any ideas. I am being serious that I do not think at all that they are taking away the ability to gain LOTRO points via completing deeds. They said they were adjusting deeds in how they gave out virtues. Instead of now giving out a specific virtue level for a specific deed you will be getting experience points where you can level up the virtues of your choice.
It wouldn't make sense considering all the backlash they have been getting lately to not share another thing that would get backlash now and wait until down the road to share it where it would look far worse and like they were hiding it. On your comment about LP diminishing, you can farm about a hundred an hour at ultra low levels. If you are worried LP are going away you might go make use of this now.
The problem with long eloquent prose is that the deniers can close their eyes and move on without taking any of it on board.
But someone with a critical brain isn't supposed to be playing this game any more. It's only for people who get impressed with the surface veneer, and not even that much given that Mordor Trailer. But if you are impressed by gameplay on the live streams you are exactly who SSG wants playing. I can understand those who buy into the collectable game box but flounder when it's extended to a digital download. After the official stream this Friday I stayed watching and we got to see a dozen or so Mordor Doors in the vault, a brilliant reminder that the owner hadn't the first clue how to make it through that content on level yet always eager to profess a skill and understanding of the game that just isn't there. Her toadying to "devs" rewarded by more influence.
When Skynet becomes self aware it does all go to pot (time travel does screw up tense). Does trump have it in him to become self aware? When these players become self aware their world comes crashing down, so let them stay safe and cosy. Are you Neo or Mr Anderson, stay plugged in or risk it ending? Or maybe a better future?
I'm thinking that at the high level areas, players that get there are expected to place down some real world money. A $20 purchase every 4 months is a little over 4,500 points. That's a lot of grinding deeds.
I feel that if you can afford internet and a comp that can play LOTRO, you can afford $20 every 3 or 4 months. I don't see having reduced LP in the new areas as being an issue.
Not only that, you can make a ton of points with low level alts in a fraction of the time.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,872
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I was so nervous about ordering a prom dress online due to many reasons such as not fitting or being the wrong color but this dress is exactly what is shown in the picture and so much better. It's so pretty and fits amazing on me. I did a custom size and added a few inches to my measurements to make sure that it wouldn't be too tight on me and it fits so nice with enough room for me to breathe and move around in. It is such a pretty dress!!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,447
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package org.pentaho.di.trans.steps.memgroupby;
import static org.mockito.Matchers.any;
import static org.mockito.Mockito.when;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collection;
import java.util.List;
import junit.framework.Assert;
import org.junit.Before;
import org.junit.BeforeClass;
import org.junit.Test;
import org.mockito.Mockito;
import org.pentaho.di.core.exception.KettleException;
import org.pentaho.di.core.logging.LoggingObjectInterface;
import org.pentaho.di.core.row.RowMetaInterface;
import org.pentaho.di.core.row.ValueMetaInterface;
import org.pentaho.di.core.row.value.ValueMetaInteger;
import org.pentaho.di.trans.steps.mock.StepMockHelper;
public class MemoryGroupByNewAggregateTest {
static StepMockHelper<MemoryGroupByMeta, MemoryGroupByData> mockHelper;
static List<Integer> strings;
static List<Integer> statistics;
MemoryGroupBy step;
MemoryGroupByData data;
@BeforeClass
public static void setUpBeforeClass() throws Exception {
mockHelper =
new StepMockHelper<MemoryGroupByMeta, MemoryGroupByData>( "Memory Group By", MemoryGroupByMeta.class,
MemoryGroupByData.class );
when( mockHelper.logChannelInterfaceFactory.create( any(), any( LoggingObjectInterface.class ) ) ).thenReturn(
mockHelper.logChannelInterface );
when( mockHelper.trans.isRunning() ).thenReturn( true );
// In this step we will distinct String aggregations from numeric ones
strings = new ArrayList<Integer>();
strings.add( MemoryGroupByMeta.TYPE_GROUP_CONCAT_COMMA );
strings.add( MemoryGroupByMeta.TYPE_GROUP_CONCAT_STRING );
// Statistics will be initialized with collections...
statistics = new ArrayList<Integer>();
statistics.add( MemoryGroupByMeta.TYPE_GROUP_MEDIAN );
statistics.add( MemoryGroupByMeta.TYPE_GROUP_PERCENTILE );
}
@Before
public void setUp() throws Exception {
data = new MemoryGroupByData();
data.subjectnrs = new int[16];
int[] arr = new int[16];
String[] arrF = new String[16];
for ( int i = 0; i < arr.length; i++ ) {
// set aggregation types (hardcoded integer values from 1 to 18)
arr[i] = i + 1;
data.subjectnrs[i] = i;
}
Arrays.fill( arrF, "x" );
MemoryGroupByMeta meta = new MemoryGroupByMeta();
meta.setAggregateType( arr );
meta.setAggregateField( arrF );
ValueMetaInterface vmi = new ValueMetaInteger();
when( mockHelper.stepMeta.getStepMetaInterface() ).thenReturn( meta );
RowMetaInterface rmi = Mockito.mock( RowMetaInterface.class );
data.inputRowMeta = rmi;
when( rmi.getValueMeta( Mockito.anyInt() ) ).thenReturn( vmi );
data.aggMeta = rmi;
step = new MemoryGroupBy( mockHelper.stepMeta, data, 0, mockHelper.transMeta, mockHelper.trans );
}
@Test
public void testNewAggregate() throws KettleException {
Object[] r = new Object[16];
Arrays.fill( r, null );
Aggregate agg = new Aggregate();
step.newAggregate( r, agg );
Assert.assertEquals( "All possible aggregation cases considered", 16, agg.agg.length );
// all aggregations types is int values, filled in ascending order in perconditions
for ( int i = 0; i < agg.agg.length; i++ ) {
int type = i + 1;
if ( strings.contains( type ) ) {
Assert.assertTrue( "This is appendable type, type=" + type, agg.agg[i] instanceof Appendable );
} else if ( statistics.contains( type ) ) {
Assert.assertTrue( "This is collection, type=" + type, agg.agg[i] instanceof Collection );
} else {
Assert.assertNull( "Aggregation initialized with null, type=" + type, agg.agg[i] );
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,978
|
{"url":"http:\/\/www.phy.ntnu.edu.tw\/ntnujava\/msg.php?id=7633","text":"Why the angle $\\theta$ has to be smaller than 5 degree to be able to satisfy the small angle approximation?\nWhy 5.5 can not be a small angle? Does 5 a magic number???\n\n[quote]\ni don't understand the question on \"damping factor are the same for all pendulums\"\n[\/quote]\nWhat I mean is \"The period and damping rate are the same for all pendulums.\"","date":"2020-08-13 12:33:02","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\": 1, \"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\": 0.8931554555892944, \"perplexity\": 591.9655767836952}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439738982.70\/warc\/CC-MAIN-20200813103121-20200813133121-00485.warc.gz\"}"}
| null | null |
Andreas Josenhans (* 9. September 1950 in Pforzheim) ist ein kanadischer Segler deutscher Herkunft.
Werdegang
Er wuchs als Sohn einer Hochschulprofessorin und eines Arztes in der kanadischen Provinz Nova Scotia auf. Andreas Josenhans nahm 1976 an den Olympischen Sommerspielen in Montreal teil und wurde im Soling Achter. In derselben Bootsklasse gewann er 1977 und 1980 mit Glen Dexter und Sandy McMillan den Weltmeistertitel, 1978 wurde man WM-Zweiter. Zusammen mit Buddy Melges wurde Josenhans des Weiteren 1978 und 1979 Weltmeister im Starboot.
Josenhans ließ sich im US-Bundesstaat Connecticut nieder. Beruflich wurde er als Segelmacher tätig. Er gewann 1992 als Besatzungsmitglied der America 3 von Bootsbesitzer Bill Koch den America's Cup. Später lebte er in der Nähe von Lunenburg in Nova Scotia. Im Oktober 2021 wurde Josenhans in die Ruhmeshalle des kanadischen Segelverbands aufgenommen.
Einzelnachweise
Regattasegler (Kanada)
Weltmeister (Segeln)
Teilnehmer der Olympischen Sommerspiele 1976
Olympiateilnehmer (Kanada)
Kanadier
Geboren 1950
Mann
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 321
|
Cory's Insights – Thu 3 Dec, 2015
Here's why investors booed European Central Bank's stimulus plans
If you care to know why investors were disappointing with the announcement from the ECB this article sums it up nicely. Essentially Draghi and the ECB disappointed on all fronts of the announcement.
Click here to visit the original posting page.
Now here's the article….
"Always leave them wanting more" may be good advice for show business professionals, but that approach ensured a round of boos for Mario Draghi after the European Central Bank chief offered up a round of additional stimulus measures that fell short of investor expectations for more aggressive action.
The euro EURUSD, +2.8449% soared to a four-week high, punishing bears who had loaded up on bets a more aggressive ECB and a presumably soon-to-tighten Federal Reserve would help pound the shared currency toward parity with the dollar. European government bonds also jumped and stocks weakened. See the recap of our live blog of Draghi's news conference.
Adding to the turmoil, the Financial Times published a pre-written story in error minutes before the ECB's rate announcement. The story incorrectly said the ECB had made no change to its deposit rate, which triggered an initial round of euro short covering.
The ECB did expand its bond buying program, but not to the extent that many investors had been looking for. While the central bank extended its 60 billion euro a month bond buying program through March 2017 form its initially planned end date of September 2016, it didn't increase the size of those monthly purchases.
The decision to expand the scope of items it can purchase to include regional government bonds was a disappointment to sovereign bond traders absent an increase in the overall size of purchases. And earlier, the ECB lowered its already negative deposit rate to minus 0.3% from minus 0.2%—a move that was also at the low end of expectations, while leaving the bank's main lending rate unchanged at 0.05%. The ECB also said it would reinvest principal payments from securities it bought and extended its program of fixed-rate refinancing loans through the end of next year.
Economists were quick to pin the blame on Draghi. After all, in October the ECB chief, whose ability to jawbone the markets is deservedly renown, inspired a rally in bonds and stocks and helped sink the euro when he dropped strong hints that some very aggressive actions were in the pipeline. Subsequent comments by Draghi and other ECB officials only stoked those expectations.
This reaction from Jonathan Loynes, chief European economist at Capital Economics, captured the mood: "In short, the ECB has comprehensively failed to live up to its own hype and markets and forecasters will take future communications from Mr. Draghi and colleagues with a corresponding bucket of salt."
It's possible that Draghi—and investors—underestimated the roadblocks thrown in the ECB's way by the Governing Council's hawks, said Carsten Brzeski, economist at ING Bank in Brussels, in a note.
Draghi, of course, signaled no such resistance. While the Governing Council didn't unanimously back Thursday's actions, the moves were supported by a "large majority," Draghi said.
He defended the ECB's actions, arguing that the stimulus measures had already worked to disconnect consumers' inflation expectations from falling oil prices. Without those measures, inflation next year would be a half percentage point lower than otherwise, he argued. Taking into consideration what's already working, the ECB took what actions it deemed remained necessary and left the door open to doing even more if needed, he argued.
Draghi has a point on the economy, analysts said. Purchasing managers index readings have been gathering strength, with manufacturing activity rising to a 19-month high in November and services hitting a 4 1/2 year high, noted Matthew Weller, senior market strategist at Forex.com.
"Draghi decided that discretion was the better part of valor and opted to keep his powder dry in case it's more desperately needed in the future," Weller said, in a note.
For now, it's clear investor faith in the ECB has been shaken. In fact, the market's sharp reaction Thursday could make the ECB's own staff projections, which had trimmed the outlook for annual inflation in 2016 to 1% from 1.1%, already look obsolete, wrote Lena Komileva of G-plus Economics.
This could create expectations the ECB will have to correct its error and announce an even stronger easing package, possibly as early as January, she said. This is part of a recipe for increased market volatility as the Federal Reserve prepares markets for a possible December rate increase and China moves toward more easing. See live blog of Janet Yellen testimony before Congress.
"The cross currents of global central banks' communication, with the ECB's own path now a source of risk for investors, and focus shifting to the Fed and Asian central banks, is a breeding ground for more financial volatility into the year-end," Komileva said.
On December 3, 2015 at 9:39 am,
http://www.telegraph.co.uk/finance/economics/12031008/european-central-bank-qe-cut-rates-stimulus-ecb-live.html
The Big Zero just announced he was mystified as to reasons for the Californian shooting.
Here's my suggestion to him:
Read somenewspapers you a..-hole.
http://www.dailymail.co.uk/news/article-3344350/Devout-Muslim-citizen-Saudi-wife-living-American-Dream-identified-heavily-armed-duo-burst-office-holiday-party-slaughtered-14-leaving-baby-mother.html
This article is entirely wrong on almost every front. It is a rationalization after the fact that seeks to explain market moves based on news events. What has happened today was in fact blazingly obvious on the various charts I have been discussing since Tuesday morning.
Don't say I didn't warn you guys.
Here is a partial recap of those recent remarks as a refresher where I wrote:
"To be explicit; I think the US Dollar rolls over by Monday morning, crude oil and gold take off abruptly and Treasuries reverse the move up we saw today……..(Here is) the Euro on a monthly chart. Notice that it has reached the bottom of its declining channel and is forming a clear double bottom as it prepares to move higher. This is one of the easiest charts we have to read right now. Few here will believe it. All I can say to them is let the charts do the talking for us. We are about to get a dollar / euro reversal whether the doubters believe it or not".
And indeed, that big bond move has now been reversed, WTI oil is up over a dollar already and the Euro took the elevator ride that shocked the pants off the bears. And all of that was PREDICTABLE.
How do I know that? Well I predicted it that's how.
OK, so I did write that my target band was for Monday morning and on that point I was not quite perfect as the sharp expected move arrived today (Thursday). Not that it mattered much.
In fact we were on top of the trade here and I was showing my wife the action this morning as it evolved (trying to impress here as usual). I told her the Euro would hit 1.05 and then take off.
Thirty minutes later that is exactly what we got. A rocket ride of over 400 points. KA-CHING!!!
Thank you very much Mr Market.
On December 3, 2015 at 2:27 pm,
irishtony says:
Bird..Hello…Now that you showed your good lady what a clever little bird you are & she loves you for it….It means you have got to take her out for a great meal to the restaurant of HER choice.
Cute, but no. She's going shopping.
On December 3, 2015 at 10:44 am,
Frank from moscow CCF says:
GOLD ON A ROLL……………..UP $11……..
Jerry !!!…Calm down , Its only up $11, When it's up $111 , then you can do the Hooky- Dooky.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,035
|
{"url":"http:\/\/crypto.stackexchange.com\/questions?page=4&sort=unanswered&pagesize=15","text":"# All Questions\n\n78 views\n\n### Implementation Attacks on Hashes\n\nSo I am familiar with attacking implementations of block ciphers via side channel attacks, cache-timing attacks, etc. What implementation attacks are there against hashes or hashed based functions ...\n237 views\n\n### Challenge-response based on public-key decryption, why send public key\n\nQuoting the handbook of applied cryptography, chapter 10.3.3 (i): Identification based on PK decryption and witness. Consider the following protocol: $A \\leftarrow B: h(r), B, P_A(r,B)$ ...\n70 views\n\n### Where is the OID and ASN.1 specified for AES_CMAC?\n\nI have a requirement to implement AES_CMAC as the authentication algorithm in a CMS library. I just can't seem to find the OID and the ASN.1 definition for it. I expected to see the OID under ...\n95 views\n\n### Protocol: Coin Flipping over phone\n\nProtocol: Coin Flipping over telephone Hypothesis: Alice and Bob have agreed: a one to one function $f$ which is easy to compute $f(x)$ from $x$ and while given any value $f(x)$ it is nearly ...\n43 views\n\n### Malicious KGC attack in the identity-based authenticated key exchange protocol\n\nThere are two users $A$ and $B$ who are in different domain. Let's assume that $A\\in KGC_1$ and $B\\in KGC_2$. If they want to establish secure session, then they must run an identity-based ...\n71 views\n\n98 views\n\n### How does use of HMAC affect hash function combiners?\n\nIn the comments of this question, Ricky Demer proposes a function $HMAX$ to combine two $HMAC$s based on different hash functions: \\$HMAX_{H0,H1}(\u27e8k_0,k_1\u27e9,m) = HMAC_{H0}(k_0,m) \\oplus ...\n70 views\n\n### Any functioning system for interactive proof?\n\nMy problem is to outsource an array of data and ask the prover to sort the data. I am wonder if there is any working system out there that support interactive proof for the above computation? Or if ...\n63 views\n\n### Protocol\/algorithms based on a variable-length input PRF\n\nAre the proofs based on a PRF assumption still valid when using a variable-length input PRF ? The answer might be obvious, but I have a doubt.\n107 views\n\n### What role plays Quantum Fourier Transformation in Shor's integer factorization algorithm?\n\nI cannot seem to understand the role or goal of Quantum Fourier Transformation in Shor's integer factorization algorithm. Is it used to collapse all quantum states into one, in which it has a factor ...\n441 views\n\n### Question about modifying the MD5 plain text to cause a collision\n\nI was trying to understand MD5 and got stuck with this question, its from Michael sipser's book on Information security Principles and Practice, 2nd edition The MD5 collision in Problem 25 is said ...\n30 views\n\n### Detail about reactive simulatability framework\n\nI'm trying to understand the framework of reactive simulatability using *\"PfWa2_00AsyncModel\"* from this publication which is a tech report behind the paper A model for Asynchronous Reactive Systems ...\n269 views\n\n### How to perform benchmark of block\/stream ciphers?\n\nI would like to perform some benchmarking of different block and stream ciphers for general data (lossless and lossy data) encryption and decryption (with focus on power consumption). To get general ...\n160 views\n\n### Precise meaning of various terms related to universal hash functions\n\nI've been reading about universal hashing, but I'm confused by all these different terms and notations. Could someone help me understand the precise meaning or relation between the following terms: ...\n171 views\n\n### Integer factorization based password authentication\n\nAfter looking at this security issue at DjangoProject, I started to think in a password-based authentication that places the burden of PBKDF2 (or whatever is the hashing function) on the client. So I ...\n84 views\n\n### partial-domain permutation and strong assumption\n\nQuoting \"Verified Security of Redundancy-Free Encryption from Rabin and RSA\", Chapter 2 (on page 4): OAEP was proved IND-CCA-secure by Fujisaki et al. [28] under the assumption that the underlying ...","date":"2015-10-10 19:54:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8495285511016846, \"perplexity\": 2707.525136947967}, \"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-2015-40\/segments\/1443737962531.78\/warc\/CC-MAIN-20151001221922-00087-ip-10-137-6-227.ec2.internal.warc.gz\"}"}
| null | null |
Caucus Memberships
Cosponsored Legislation
Visiting Washington, D.C.
Request a U.S. Flag
Mobile Office Hours
The 59th Presidential Inauguration
Congressional Youth Cabinet
Skype with the Senator
Federal Spending
Immigration Reform & Border Security
Dr. Boozman's Check-up
Senator John Boozman
Facebook Twitter Email Print
Arkansas Delegation Announces Disaster Declaration for Farmers, Ranchers Affected by Flooding
WASHINGTON – U.S. Senators Mark Pryor and John Boozman, along with Congressmen Rick Crawford (AR-1), Tim Griffin (AR-2), Steve Womack (AR-3), and Tom Cotton (AR-4) today announced that 23 Arkansas counties have been designated as disaster areas, allowing farmers and ranchers to receive assistance to recover from losses caused by severe weather in the state. In June, east Arkansas was ravaged by flash flooding, which devastated thousands of acres of crops and pastures.
"Last month's flash floods not only damaged Arkansans' homes and businesses, but destroyed crops and washed away land used for livestock," said Pryor. "Storms like these remind us of the importance of supporting disaster relief and helping our neighbors in need. I'm pleased the USDA will provide this assistance to help our farmers and ranchers recover."
"The severe weather conditions in Arkansas last month brought flash flooding, hail, and other destructive forces of nature that were devastating to agriculture producers in the path of these storms. As a result of this declaration, farmers will now be eligible to apply for emergency loans through the FSA to help them recover their losses and continue operations. Considering that agriculture makes up a large portion of our state's economy, this relief is essential to many Arkansans who faced significant losses as a result of these storms," Boozman said.
"Arkansas' First District has some of the most productive agricultural growers in the country, but this year's widespread crop damage illustrates the uncertainty they face every day. Heavy rains and high water have hurt a wide range of crops grown in the First District, creating significant producer losses with few options to salvage the fall harvest," said Crawford. "While a Secretarial Disaster Declaration does not solve the ongoing flooding dilemma facing our producers, it does provide emergency loan assistance to lessen uncertainty and get our growers back in their fields next year. I'm grateful Secretary Vilsack recognized that need."
"Agriculture is an integral part of Arkansas's economy, and today's declaration will provide essential assistance to help Arkansas's farmers and ranchers recover from challenges caused by the recent severe weather," said Griffin.
"Arkansas's farmers and ranchers are vital to our state's economy. As they recover from this latest wave of severe weather, I'm grateful they will have much-needed assistance from USDA," said Womack.
"I appreciate Secretary Vilsack's quick approval of Governor Beebe's disaster declaration request for the 23 impacted counties," said Cotton. "I have heard from many farmers about the impact of the recent flooding, and I look forward to working with our friends in Arkansas to make sure farmers are able to access the emergency funds they need."
Under this designation, agricultural producers in Cross, Jackson, Independence, Lee, Lonoke, Monroe, Prairie, St. Francis, White, and Woodruff Counties will be eligible to apply for assistance from the Farm Service Agency, including emergency loans. Arkansas, Cleburne, Craighead, Crittenden, Faulkner, Izard, Jefferson, Lawrence, Phillips, Poinsett, Pulaski, Sharp, and Stone Counties have been named contiguous disaster counties. Arkansans can reach out to their local FSA offices for more information.
Arkansas Delegation Requests Disaster Assistance for State's Farmers, Ranchers
Press Releases Agriculture
Permalink: https://www.boozman.senate.gov/public/index.cfm/2014/7/arkansas-delegation-announces-disaster-declaration-for-farmers-ranchers-affected-by-flooding
What They're Saying: Boozman Delivers Investments for Arkansas Water Infrastructure
Paycheck Protection Program Re-Launches with Improvements
Boozman, Cotton Urge SBA to Halt PPP Funds to Planned Parenthood
ICYMI: Boozman-Backed Provision Expands PPP Eligibility to Local News Outlets
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,374
|
'use strict'
const {ClayResource} = require('clay-resource')
const clayDriverMemory = require('clay-driver-memory')
// Extends ClayResource class
class UserResource extends ClayResource {
/* ... */
}
void async function () {
const driver = clayDriverMemory()
const userResource = UserResource.fromDriver(driver, 'User')
const user = await userResource.create({name: 'Taka Okunishi'})
/* ... */
}()
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,265
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In this Research Paper, Dr. Christophe Paulussen explores whether the current international legal framework is sufficiently equipped to effectively deal with the threat of terrorism and counter-terrorism practices or whether it is in need of change. The paper specifically looks at whether the current jus ad bellum (the law regulating when inter-state force may be used) and jus in bello (the law of war, the law regulating the conduct of warfare) are still suitable in the current climate. This Paper clarifies a few concepts that are often heard, and sometimes misunderstood, in the counter-terrorism discussion: counter-insurgency (and then in particular its correlation with counter-terrorism) and asymmetrical warfare. The final section offers some concluding remarks.
How to cite: Paulussen, C. "Testing the Adequacy of the International Legal Framework in Countering Terrorism: The War Paradigm", The International Centre for Counter-Terrorism 3, no. 9 (2012).
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,614
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Diglossa gloriosa е вид птица от семейство Тангарови (Thraupidae).
Разпространение
Видът е разпространен във Венецуела.
Източници
Цветарници
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,523
|
\section{Introduction}\label{S:introduction}
Wormholes are tunnel-like structures in
spacetime that link widely separated
regions of our Universe or different universes
altogether \cite{MT88}. This spacetime
geometry can be described by the metric
\begin{equation}\label{E:line}
ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{dr^2}{1-b(r)/r}
+r^{2}(d\theta^{2}+\text{sin}^{2}\theta\,
d\phi^{2}),
\end{equation}
using units in which $c=G=1$. In this line
element, $b=b(r)$ is called the \emph{shape
function} and $\Phi=\Phi(r)$ is called the
\emph{redshift function}; the latter must be
finite everywhere to avoid the presence of
an event horizon. For the shape function we
must have $b(r_0)=r_0$, where $r=r_0$ is the
radius of the \emph{throat} of the wormhole.
Another important requirement is the
\emph{flare-out condition} at the throat:
$b'(r_0)<1$; also, $b(r)<r$ near the throat.
In classical general relativity, the flare-out
condition can only be satisfied by violating
the null energy condition (NEC):
\begin{equation}
T_{\mu\nu}k^{\mu}k^{\nu}\ge 0
\end{equation}
for all null vectors $k^{\mu}$, where
$T_{\mu\nu}$ is the stress-energy tensor.
In particular, for the outgoing null vector
$(1,1,0,0)$, the violation becomes
\begin{equation}
T_{\mu\nu}k^{\mu}k^{\nu}=\rho +p_r<0.
\end{equation}
Here $T^t_{\phantom{tt}t}=-\rho$ is the energy
density, $T^r_{\phantom{rr}r}= p_r$ is the
radial pressure, and
$T^\theta_{\phantom{\theta\theta}\theta}=
T^\phi_{\phantom{\phi\phi}\phi}=p_t$ is
the lateral pressure.
According to Lobo \cite{HLMS}, these ideas can
be extended to$f(R)$ modified gravity by
referring back to the Raychaudhury equation.
In this theory, the Ricci scalar $R$ in the
Einstein-Hilbert action
\begin{equation*}
S_{\text{EH}}=\int\sqrt{-g}\,R\,d^4x
\end{equation*}
is replaced by a nonlinear function $f(R)$:
\begin{equation*}
S_{f(R)}=\int\sqrt{-g}\,f(R)\,d^4x.
\end{equation*}
To extend these ideas, the stress-energy tensor
$T_{\mu\nu}$ has to be replaced by
$T^{\text{eff}}_{\mu\nu}$, the
\emph{effective} stress-energy tensor
arising from the modified theory,
leading to the Einstein field equations
$G_{\mu\nu}=\kappa^2T^{\text{eff}}_{\mu\nu}$.
The NEC now becomes
\begin{equation}
T^{\text{eff}}_{\mu\nu}k^{\mu}k^{\nu}\ge 0.
\end{equation}
As a result, the violation of the (generalized)
NEC becomes
$T^{\text{eff}}_{\mu\nu}k^{\mu}k^{\nu}<0$,
which reduces to $T_{\mu\nu}k^{\mu}k^{\nu}<0$
in classical general relativity. According
to Ref. \cite{HLMS}, it now becomes possible
in principle to allow the matter threading the
wormhole to satisfy the NEC while retaining
the violation of the generalized NEC, i.e.,
$T^{\text{eff}}_{\mu\nu}k^{\mu}k^{\nu}<0$.
So the necessary condition for maintaining
a traversable wormhole has been met.
According to Ref. \cite{HLMS}, the
higher-order curvature terms leading to
the violation may be interpreted as a
gravitational fluid that supports the
wormhole.
The purpose of this paper is to show that
noncommutative geometry, an offshoot of
string theory, is not only an example of
such a modified gravitational theory, it
provides a motivation for the choice of
the function $f(R)$.
\section{Noncommutative geometry}
An important outcome of string theory is
the realization that coordinates may become
noncommutative operators on a $D$-brane
\cite{eW96, SW99}. Noncommutativity replaces
point-like objects by smeared objects
\cite{SS03, NSS06, NS10} with the aim of
eliminating the divergences that normally
occur in general relativity. Moreover,
noncommutative geometry results in a
fundamental discretization of spacetime
due to the commutator
$[\textbf{x}^{\mu},\textbf{x}^{\nu}]
=i\theta^{\mu\nu}$, where $\theta^{\mu\nu}$ is
an antisymmetric matrix.
An effective way to model the smearing is
to assume that the energy density of a
static, spherically symmetric, and
particle-like gravitational source has
the form \cite{NM08, LL12}
\begin{equation}\label{E:rho}
\rho(r)=\frac{\mu\sqrt{\beta}}
{\pi^2(r^2+\beta)^2}.
\end{equation}
Here the mass $\mu$ is diffused throughout the
region of linear dimension $\sqrt{\beta}$ due
to the uncertainty. Noncommutative
geometry is an intrinsic property of spacetime
and does not depend on any particular features
such as curvature. Eq. (\ref{E:rho})
immediately yields the mass distribution
\begin{equation}\label{E:mass}
m_{\beta}(r)=\int^r_04\pi (r')^2\rho(r')
dr'=\frac{2M}{\pi}\left(\text{tan}^{-1}
\frac{r}{\sqrt{\beta}}-
\frac{r\sqrt{\beta}}{r^2+\beta}\right),
\end{equation}
where $M$ is now the total mass of the source.
\section{Wormholes in modified gravity}
In this section we adopt the point of view that
noncommutative geometry is a modified gravity
theory, but first we make the important
observation that the Einstein field equations
$G_{\mu\nu}=\kappa^2T^{\text{eff}}_{\mu\nu}$
mentioned in Sec. \ref{S:introduction} show
that the noncommutative effects can be
implemented by modifying only the
stress-energy tensor, while leaving the
Einstein tensor unchanged. As a result,
the length scales can be macroscopic.
The next step is to show that our
noncommutative-geometry background is a
special case of $f(R)$ modified gravity.
To that end, we need to list the gravitational
field equations in the form used by Lobo and
Oliveira \cite{LO09}. Here we assume that
$\Phi'(r)\equiv 0$; otherwise, according to
Ref. \cite{LO09}, the analysis becomes
intractable. (It is also assumed that for
notational convenience, $\kappa =1$ in
the field equations.)
\begin{equation}\label{E:Lobo1}
\rho(r)=F(r)\frac{b'(r)}{r^2},
\end{equation}
\begin{equation}\label{E:Lobo2}
p_r(r)=-F(r)\frac{b(r)}{r^3}
+F'(r)\frac{rb'(r)-b(r)}{2r^2}
-F''(r)\left(1-\frac{b(r)}{r}\right),
\end{equation}
and
\begin{equation}\label{E:Lobo3}
p_t(r)=-\frac{F'(r)}{r}\left(1-\frac{b(r)}{r}
\right)+\frac{F(r)}{2r^3}[b(r)-rb'(r)],
\end{equation}
where $F=\frac{df}{dR}$. If $F(r)\equiv 1$,
then Eqs. (\ref{E:Lobo1}) - (\ref{E:Lobo3})
reduce to the usual field equations with
$\kappa =1$ and $\Phi'(r)\equiv 0$.
So from $\rho(r)=b'(r)/r^2$ and Eq.
(\ref{E:rho}), we obtain the shape function
\begin{equation}\label{E:shape}
b(r)=\frac{M\sqrt{\beta}}{2\pi^2}
\left(\frac{1}{\sqrt{\beta}}\text{tan}^{-1}
\frac{r}{\sqrt{\beta}}-\frac{r}{r^2+\beta}-
\frac{1}{\sqrt{\beta}}\text{tan}^{-1}
\frac{r_0}{\sqrt{\beta}}+\frac{r_0}{r_0^2
+\beta}\right)+r_0,
\end{equation}
where $M$ is now the mass of the wormhole,
and from
\begin{equation}\label{E:bprime}
b'(r)=\frac{M\sqrt{\beta}}{\pi^2}
\frac{r^2}{(r^2+\beta)^2},
\end{equation}
we see that $b'(r_0)<1$; the flare-out
condition is thereby met.
According to Ref. \cite{LO09}, the Ricci
scalar is
\begin{equation}\label{E:R1}
R(r)=\frac{2b'(r)}{r^2}.
\end{equation}
\section{Avoiding exotic matter}
Since we wish the matter threading the
wormhole to obey the null energy
condition, we require that $\rho + p_r
\ge 0$ and $\rho +p_t\ge 0$, as well
as $\rho\ge 0$. From Eqs. (\ref{E:Lobo1})
and (\ref{E:Lobo2}), we therefore need to
satisfy the following conditions:
\begin{equation}\label{E:Con1}
\rho=\frac{Fb'}{r^2}\ge 0
\end{equation}
and
\begin{equation}\label{E:Con2}
\rho +p_r=
\frac{(2F+rF')(b'r-b)}{2r^3}
-F''\left(1-\frac{b}{r}\right)
\ge 0.
\end{equation}
Using Eqs. (\ref{E:Lobo1}) and
(\ref{E:rho}), we get
\begin{equation}\label{E:F(r)}
F(r)=\frac{r^2}{b'(r)}\rho(r)=
\frac{1}{b'(r)/r^2}
\frac{\mu\sqrt{\beta}}{\pi^2}
\frac{1}{(r^2+\beta)^2}.
\end{equation}
Eq. (\ref{E:R1}) now implies that
\begin{equation}\label{E:r}
r(R)=\sqrt{\frac{2b'}{R}}.
\end{equation}
Substituting in Eq. (\ref{E:F(r)}) yields
\begin{equation}\label{E:F(R)}
F(R)=\frac{2\mu\sqrt{\beta}}{\pi^2}
\frac{1}{R\left(\frac{2b'}{R}+\beta
\right)^2}
\end{equation}
and
\begin{equation}\label{E:derivative}
F'(R)=-\frac{4\mu\sqrt{\beta}}{\pi^2}
\frac{2b'+2\beta R}{(2b'R+\beta R^2)^3}.
\end{equation}
Inequality (\ref{E:Con1}) is evidently
satisfied. Substituting in Inequality
(\ref{E:Con2}), we obtain (at or near the
throat)
\begin{equation}\label{E:Omega1}
\rho+p_r=\frac{(2F+rF')(b'r-b)}{2r^3}
=\frac{1}{r^3}\frac{2\mu\sqrt{\beta}}{\pi^2}
\frac{1}{(2b'R+\beta R^2)^2}\left(1-
\frac{r}{R}\frac{2b'+2\beta R}{2b'+\beta R}
\right)(b'r-b).
\end{equation}
To show that $\rho+p_r|_{r=r_0} > 0$,
recall from Eq. (\ref{E:bprime}) that
$b'(r_0)\ll 1$ since $\beta$ is
extremely small. So
\begin{equation}
\left.\frac{r}{R}\right |_{r=r_0}=
\frac{r_0^3}{2b'(r_0)}>1,
\end{equation}
as long as the throat size is not
microscopic, i.e., $r_0>[2b'(r_0)]^{1/3}$.
Since we also have
$(2b'+2\beta R)(2b'+\beta R)>1$, it now follows
that
\begin{equation}\label{E:Omega2}
\rho+p_r|_{r=r_0} > 0.
\end{equation}
Next, we find that
\begin{equation}
\rho+p_t|_{r=r_0}=F(r_0)\frac{b'(r_0)+1}
{2r_0^2}>0.
\end{equation}
The NEC is therefore satisfied at the throat
for the null vectors $(1,1,0,0)$,
$(1,0,1,0)$, and $(1,0,0,1)$. It is
shown in Ref. \cite{pK18} that the result
can be extended to any null vector
\[
(1,a,b,c),\quad 0\le a,b,c\le 1,\quad
a^2+b^2+c^2=1.
\]
Since $F=\frac{df}{dR}$, Eq. (\ref{E:F(R)})
also yields
\begin{equation}\label{E:f(R)}
f(R)=\int^R_0\frac{2\mu\sqrt{\beta}}{\pi^2}
\frac{1}{R'\left(\frac{2b'}{R'}+\beta
\right)^2}dR'=
\frac{2\mu\sqrt{\beta}}{\pi^2}
\frac{(\beta R+2b')\text{ln}\,
(\beta R+2b')-\beta R}
{\beta^2(\beta R+2b')}+C.
\end{equation}
To check the violation of the generalized
NEC, i.e.,
$T^{\text{eff}}_{\mu\nu}k^{\mu}k^{\nu}<0$,
we follow Lobo and Oliveira \cite {LO09}:
\begin{multline}
\left. \rho^{\text{eff}}+p_r^{\text{eff}}
|_{r=r_0}
=\frac{1}{F}\frac{rb'-b}{r^3}+
\frac{1}{F}\left(1-\frac{b}{r}\right)
\left(F''-F'\frac{b'r-b}{2r^2(1-b/r}
\right)\right|_{r-r_0}\\
=\frac{1}{F}\frac{b'(r_0)-1}
{r_0^2}+\frac{1-b'(r_0)}{2r_0}
\frac{F'}{F}.
\end{multline}
Since $F'(R)<0$, it follows that
\begin{equation}
\rho^{\text{eff}}+p_r^{\text{eff}}
|_{r=r_0}<0.
\end{equation}
So the generalized NEC is violated
thanks to the stress-energy tensor
$T^{\text{eff}}_{\mu\nu}$.
\section{The high radial tension}\label{S:tension}
Although not part of this study, noncommutative
geometry plays another important role in
wormhole physics. According to Ref. \cite{MT88},
for a moderately-sized wormhole, the radial
tension at the throat has the same magnitude
as the pressure at the center of a massive
neutron star. Attributing this outcome to
exotic matter is rather problematical since
exotic matter was introduced primarily to
ensure the violation of the null energy
condition.
It is shown in Ref. \cite{pK20} that such an
outcome can be accounted for by the
noncommutative geometry background.
Recalling that noncommutative geometry is
an offshoot of string theory, this approach
can be viewed as a foray into quantum
gravity.
\section{Conclusion}
A fundamental geometric property of traversable
wormholes is the flare-out condition $b'(r_0)<1$.
In classical general relativity, the flare-out
condition can only be met by violating the NEC,
$T_{\mu\nu}k^{\mu}k^{\nu}<0$, for all null
vectors $k^{\mu}$. In $f(R)$ modified gravity,
the stress-energy tensor $T_{\mu\nu}$ is
replaced by the effective stress-energy tensor
$T^{\text{eff}}_{\mu\nu}$ arising from the
modified theory. So it is possible in principle
to have a violation of the generalized NEC,
$ T^{\text{eff}}_{\mu\nu}k^{\mu}k^{\nu}<0$,
while maintaining the NEC,
$T_{\mu\nu}k^{\mu}k^{\nu}\ge 0$, for the
material threading the wormhole. According
to Ref. \cite{HLMS}, the higher-order
curvature terms leading to the violation
may be interpreted as a gravitational fluid
that supports the wormhole.
The purpose of this paper is to show that
noncommutative geometry, an offshoot of
string theory, can be viewed as a special
case of $f(R)$ modified gravity, where
$f(R)$ is given by Eq. (\ref{E:f(R)}).
The result is a zero-tidal force
traversable wormhole without exotic matter.
The noncommutative-geometry background
also provides a motivation for the choice
of $f(R)$.
Sec. \ref{S:tension} reiterates another
aspect of noncommutative geometry, the
ability to account for the enormous
radial tension in a Morris-Thorne
wormhole, as shown in Ref. \cite{pK20}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,478
|
\section{Introduction}
Considering the important role of the Higgs boson in particle physics,
hunting for it has been one of the major tasks of the running Large Hadron Collider (LHC).
Recently, both the ATLAS and CMS collaborations have reported some evidence for a light Higgs boson
near 125 GeV \cite{ATLAS,CMS} with a di-photon signal rate slightly above the SM prediction \cite{Carmi}.
As is well known, in new physics beyond the SM model several Higgs bosons are predicted,
among which the SM-like one may be near 125 GeV \cite{125Higgs,125other,cao125,Cao:2011sn}.
Recently, in our studies \cite{cao125,Cao:2011sn}
we examined the mass of the SM-like Higgs boson in several supersymmetric (SUSY) models including
the minimal supersymmetric standard model (MSSM)\cite{Haber,Djouadi}, the next-to-minimal supersymmetric
standard model (NMSSM)\cite{NMSSM1,NMSSM2} and the constrained MSSM\cite{mSUGRA,nuhm2}.
At tree-level, these SUSY models are hard to predict a Higgs boson near 125 GeV,
and sizable radiative corrections, which mainly come from the top and top-squark loops,
are necessary to enhance the Higgs boson mass\cite{Carena:1995bx}. Due to the different
properties of these SUSY models, the loop contributions to the Higgs boson mass are different
for giving a 125 GeV Higgs boson. Therefore, different models have different lower bounds on
the top-squark mass which is associated with the fine-tuning problem \cite{tuning}.
On the other hand, since the di-photon Higgs signal is the most promising discovery
channel for a light Higgs boson at the LHC\cite{diphoton1}, in our recent study \cite{di-photon} we
performed a comparative study for the di-photon Higgs signal in different SUSY models,
namely the MSSM, NMSSM and the nearly mininal supersymmertric
standard model (nMSSM) \cite{xnMSSM,cao-xnmssm}.
In this note we briefly review these stuides on
a 125 GeV Higgs boson and its di-photon signal rate
in different SUSY models.
This note is organized as follows. In the next section we briefly describe the Higgs sector
and the di-photon Higgs signal in these SUSY models. Then we present the numerical results
and discussions in Sec. III. Finally, the conclusions are given in Sec. IV.
\section{The Higgs sector and di-photon signal rate in SUSY models}
\subsection{The Higgs sector in SUSY models}
Different from the SM, the Higgs sector in the supersymmetric models
is usually extended by adding Higgs doublets and/or singlets.
The most economical realization is the MSSM, which consists of
two Higgs doublet $H_u$ and $H_d$. In order to solve the $\mu-$problem
and the little hierarchy problem in the MSSM, the singlet extension of MSSM,
such as the NMSSM\cite{NMSSM1} and nMSSM\cite{xnMSSM,cao-xnmssm}
has been intensively studied\cite{Barger}.
The differences between these models come from their
superpotentials and the corresponding soft-breaking terms, which are given by:
\begin{eqnarray}
W_{\rm MSSM}&=& W_F + \mu \hat{H_u}\cdot \hat{H_d}, \label{MSSM-pot}\\
W_{\rm NMSSM}&=&W_F + \lambda\hat{H_u} \cdot \hat{H_d} \hat{S}
+ \frac{1}{3}\kappa \hat{S^3},\\
W_{\rm nMSSM}&=&W_F + \lambda\hat{H_u} \cdot \hat{H_d} \hat{S}
+\xi_F M_n^2\hat S,\\
V_{\rm soft}^{\rm MSSM}&=&\tilde m_u^2|H_u|^2 + \tilde m_d^2|H_d|^2
+ (B\mu H_u\cdot H_d + h.c.),\\
V_{\rm soft}^{\rm NMSSM}&=&\tilde m_u^2|H_u|^2 + \tilde m_d^2|H_d|^2
+ \tilde m_S^2|S|^2 +(A_\lambda \lambda SH_u\cdot H_d
+\frac{A_\kappa}{3}\kappa S^3 + h.c.),\\
V_{\rm soft}^{\rm nMSSM}&=&\tilde m_u^2|H_u|^2 + \tilde m_d^2|H_d|^2
+ \tilde m_S^2|S|^2 +(A_\lambda \lambda SH_u\cdot H_d
+\xi_S M_n^3 S + h.c.),
\end{eqnarray}
where $W_F$ is the MSSM superpotential without the $\mu$ term,
$\lambda$, $\kappa$ and $\xi_F$ are the dimensionless parameters and
$\tilde{m}_{u}$, $\tilde{m}_{d}$, $\tilde{m}_{S}$, $B$,
$A_\lambda$, $A_\kappa$ and $\xi_S M_n^3$ are soft-breaking parameters.
Note that in the NMSSM and nMSSM the $\mu$-term is replaced by the
$\mu_{\rm eff}=\lambda s$ when the singlet Higgs field $\hat S$ develops
a VEV $s$. The differences between the NMSSM and nMSSM reflect the last term
in the superpotential, where the cubic singlet term $\kappa \hat{S}^3$ in the NMSSM
is replaced by a tadpole term $\xi_F M_n^2 \hat{S}$ in the nMSSM.
This replacement in the superpotential makes the nMSSM has no discrete symmetry and
thus free of the domain wall problem that the NMSSM suffers from.
Actually, due to the tadpole term $\xi_F M_n^2$ does not
induce any interaction, the nMSSM is identical to the NMSSM with $\kappa=0$,
except for the minimization conditions of the Higgs potential and the tree-level
Higgs mass matrices.
With the superpotentials and the soft-breaking terms giving above, one
can get the Higgs potentials of these SUSY models, and then
can derive the Higgs mass matrics and eigenstates.
At the minimum of the potential, the Higgs fields $H_u$, $H_d$ and $S$ are expanded as
\begin{eqnarray}
H_u = \left ( \begin{array}{c} H_u^+ \\
v_u +\frac{ \phi_u + i \varphi_u}{\sqrt{2}}
\end{array} \right),~~
H_d & =& \left ( \begin{array}{c}
v_d + \frac{\phi_d + i \varphi_d}{\sqrt{2}}\\
H_d^- \end{array} \right),~~
S = s + \frac{1}{\sqrt{2}} \left(\sigma + i \xi \right),
\end{eqnarray}
with $v=\sqrt{v_u^2+v_d^2}=$ 174 GeV.
By unitary rotation the mass eigenstates can be given by
\begin{eqnarray} \left( \begin{array}{c} h_1 \\
h_2 \\ h_3 \end{array} \right) = S \left( \begin{array}{c} \phi_u
\\ \phi_d\\ \sigma\end{array} \right),~ \left(\begin{array}{c} a\\
A\\ G^0 \end{array} \right) = P \left(\begin{array}{c} \varphi_u
\\ \varphi_d \\ \xi \end{array} \right),~ \left(\begin{array}{c} H^+
\\G^+ \end{array} \right) =U \left(\begin{array}{c}H_u^+\\ H_d^+
\end{array} \right). \label{rotation}
\end{eqnarray}
where $h_1,h_2,h_3$ are physical CP-even Higgs bosons ($m_{h_1}<m_{h_2}<m_{h_3}$), $a,A$ are CP-odd
Higgs bosons, $H^+$ is the charged Higgs boson, and $G^0$, $G^+$
are Goldstone bosons eaten by $Z$ and $W^+$. Due to the absence of
the singlet field $S$, the MSSM only has two CP-even Higgs bosons
and one CP-odd Higgs bosons, as well as one pair of charged Higgs bosons.
At the tree-level, the Higgs masses in the MSSM are conventionally
parameterized in terms of the mass of the CP-odd Higgs boson ($m_A$) and $\tan\beta\equiv v_u/v_d$
and the loop corrections typically come from top and
stop loops due to their large Yukawa coupling. For small splitting between the
stop masses, an approximate formula of the lightest Higgs boson mass is given by\cite{Carena:2011aa},
\begin{equation}\label{mh}
m^2_{h} \simeq M^2_Z\cos^2 2\beta +
\frac{3m^4_t}{4\pi^2v^2} \ln\frac{M^2_S}{m^2_t} +
\frac{3m^4_t}{4\pi^2v^2}\frac{X^2_t}{M_S^2} \left( 1 - \frac{X^2_t}{12M^2_S}\right),
\end{equation}
where $M_S = \sqrt{m_{\tilde{t}_1}m_{\tilde{t}_2}}$ and
$X_t \equiv A_t - \mu \cot \beta$.
The formula manifests that larger $M_S$ or $\tan\beta$
is necessary to push up the Higgs boson mass. And the Higgs boson mass can reach a maximum
when $X_t/M_S=\sqrt{6}$ for given $M_S$ (i.e. the so-called $m_h^{max}$ scenario).
Note that the lightest Higgs boson is the SM-like Higgs boson $h$ (with the largest
coupling to vector bosons) in most of the MSSM parameter space.
Different from the case in the MSSM, the Higgs sector in the NMSSM depends on the following
six parameters,
\begin{eqnarray}
\lambda, \quad \kappa, \quad M_A^2= \frac{2 \mu (A_\lambda + \kappa s)}{\sin 2 \beta},
\quad A_\kappa, \quad \tan \beta=\frac{v_u}{v_d}, \quad \mu = \lambda s.
\end{eqnarray}
and in the nMSSM the input parameters in the Higgs sector are
\begin{eqnarray}
\lambda, \quad \tan\beta, \quad \mu \quad A_\lambda, \quad \tilde m_S,
\quad M_A^2=\frac{2(\mu A_\lambda + \lambda \xi_F M_n^2)}{\sin 2 \beta}.
\end{eqnarray}
Because the coupling $\lambda\hat{H_u} \cdot \hat{H_d} \hat{S}$ in the superpotential,
the tree-level Higgs boson mass has an additional contribution in the NMSSM and nMSSM,
\begin{eqnarray}
\Delta m_h^2= \lambda^2 v^2 \sin^2 2\beta
\end{eqnarray}
In order to push up the tree-level Higgs boson mass,
$\lambda$ has to be as large as possible and $\tan\beta$
has to be small. The requirement of the absence of a landau singularity
below the GUT scale implies that $\lambda\lesssim$ 0.7 at the weak scale,
and the upper bound on $\lambda$ at the weak scale depends strongly on
$\tan\beta$ and grows with increasing $\tan\beta$\cite{king}.
However, this can still lead to
a larger tree-level Higgs boson mass than in the MSSM.
Therefore, the radiative corrections to $m_h^2$ may be reduced in the NMSSM and nMSSM,
which may induce light top-squark and ameliorate the fine-tuning problem\cite{tuning2}.
In the NMSSM and nMSSM, due to the mixing between the doublet Higgs fields and the
singlet Higgs field, the SM-like Higgs boson $h$ may either be the lightest CP-even
Higgs boson or the next-to-lightest CP-even Higgs boson, which corresponds to the so-called
pull-down case or the push-up case\cite{cao125}, respectively.
Although the mass of the SM-like Higgs boson in the nMSSM is quite similar to that
in the NMSSM, the Higgs signal is quite different.
This is because the peculiarity of the neutralino sector in the nMSSM, where
the lightest neutralino $\tilde{\chi}^0_1$ as the lightest supersymmetric particle(LSP)
acts as the dark matter candidate, and its mass takes the form\cite{rarez}
\begin{eqnarray}
m_{\tilde{\chi}^0_1} \simeq \frac{2\mu \lambda^2 v^2}{\mu^2+\lambda^2 v^2}
\frac{\tan \beta}{\tan^2 \beta+1}
\end{eqnarray}
This expression implies that $\tilde{\chi}_1^0$ must be lighter than
about $60$ GeV for $\lambda < 0.7$ (perturbativity bound) and
$\mu > 100 {\rm GeV}$ (from lower bound on chargnio mass).
And $\tilde{\chi}^0_1$ must annihilate by exchanging a resonant light
CP-odd Higgs boson to get the correct relic density.
For such a light neutralino, the SM-like Higgs boson around 125GeV tends
to decay predominantly into light neutralinos or other light Higgs bosons\cite{cao-xnmssm}.
\subsection{The di-photon Higgs signal}
Considering the di-photon signal is of prime importance to searching for Higgs boson near 125 GeV,
it is necessary to estimate its signal rate, and we define the normalized
production rate as
\begin{eqnarray}
R_{\gamma\gamma} &\equiv & \sigma_{SUSY} ( p p \to h \to
\gamma \gamma)/\sigma_{SM} ( p p \to h \to \gamma \gamma ) \nonumber \\
&\simeq& [\Gamma(h\to gg) Br(h\to \gamma\gamma)] /[\Gamma(h_{SM}\to
gg) Br(h_{SM}\to \gamma\gamma)] \nonumber \\
&=& [\Gamma(h\to gg) \Gamma(h\to \gamma\gamma)] /[\Gamma(h_{SM}\to
gg) \Gamma(h_{SM}\to \gamma\gamma)] \times
\Gamma_{tot}(h_{SM})/\Gamma_{tot}(h) \nonumber\\
&=& C_{hgg}^2 C_{h\gamma\gamma}^2\times
\Gamma_{tot}(h_{SM})/\Gamma_{tot}(h) \label{definition}
\end{eqnarray}
where $C_{hgg}$ and $C_{h\gamma\gamma}$ are the couplings of Higgs
to gluons and photons in SUSY with respect to their SM values, respectively.
In SUSY, the $hgg$ coupling arises mainly from the loops mediated by the third
generation quarks and squarks, while the $h\gamma\gamma$ coupling has additional
contributions from loops mediated by W-boson, charged Higgs boson, charginos and
the third generation leptons and sleptons. Their decay widths are given by\cite{Djouadi}
\begin{eqnarray}
\Gamma(h\to gg)&=&\frac{G_F \alpha_s^2 m_h^3}{36 \sqrt{2}\pi^3}
\left| \frac{3}{4}\sum_q g_{hqq}\, A_{1/2}^h(\tau_q)
+\frac{3}{4}{\cal A}^{gg}
\right|^2 \label{hgg}\\
\Gamma(h\to \gamma\gamma)&=&\frac{G_{F}\alpha^{2}m_{h}^{3}}{128\sqrt{2}\pi}
\left| \sum_f N_{c}\, Q_{f}^{2}\, g_{hff}\, A_{1/2}^{h}(\tau_{f})+
g_{hWW}\, A_{1}^{h}(\tau_{W}) + {\cal A}^{\gamma\gamma}\right|^2, \label{hgaga}
\end{eqnarray}
with $\tau_i = m_h^2/(4m_i^2)$, and
\begin{eqnarray}
{\cal A}^{gg} &=& \sum_{i}
\frac{g_{h\tilde{q}_i\tilde{q}_i}}{m_{\tilde{q}_i}^2}A_{0}^h(\tau_{\tilde{q}_i}),\nonumber\\
{\cal A}^{\gamma\gamma} &=&
g_{hH^{+}H^{-}}\frac{m_{W}^{2}}{m^{2}_{H^{\pm}}}A_{0}^{h}(\tau_{H^{\pm}})
+\sum_{f}\frac{g_{h\tilde{f}\tilde{f}}}{m_{\tilde{f}}^2}A_{0}^h(\tau_{\tilde{f}})
+\sum_i g_{h\chi_i^+\chi_i^-}\frac{m_W}{m_{\chi_i}} A_{1/2}^h(\tau_{\chi_i}) \label{agg},
\end{eqnarray}
where $m_{\tilde{f}}$ and $m_{\chi_i}$ are the mass of sfermion and chargino, respectively.
In the limit $\tau_i \ll 1$, the asymptotic behaviors of $A_i^h$ are given by
\begin{equation}\label{asymp}
A_0^h \to - \frac13\ , \quad A_{1/2}^h \to -\frac43\ ,\quad
A_{1}^h \to +7 \ ,
\end{equation}
One can easily learn that the W-boson contribution to $h\gamma\gamma$ is
by far dominant, however, for light stau or squarks with large mixing,
the $h\gamma\gamma$ coupling can be enhanced. While light squarks with
large mixing can suppress the $hgg$ coupling. Therefore, light stau with
large mixing may enhance the di-photon signal rate\cite{Carena:2011aa}, while light squarks
with large mixing have little effect on the di-photon signal rate.
\section{Numerical Results and discussions}
In our work the Higgs boson mass are calculated by the package
NMSSMTools\cite{NMSSMTools}, which include the dominant one-loop
and leading logarithmic two-loop corrections.
Considering the Higgs hints at the LHC, we focus on the Higgs boson
mass between 123 GeV and 127 GeV, furthermore, we consider the
following constraints:
\begin{itemize}
\item [(1)]The constraints from LHC experiment for the non-standard Higgs boson.
\item [(2)]The constraints from LEP and Tevatron on the masses
of the Higgs boson and sparticles, as well as on the neutralino pair productions.
\item [(3)]The indirect constraints from B-physics (such as the latest experimental result
of $B_s\to \mu^+\mu^-$) and from the electroweak
precision observables such as $M_W$, $\sin^2 \theta_{eff}^{\ell}$ and $\rho_{\ell}$,
or their combinations $\epsilon_i (i=1,2,3)$ \cite{Altarelli}.
\item [(4)]The constraints from the muon $g-2$:
$a_\mu^{exp}-a_\mu^{SM} = (25.5\pm 8.2)\times 10^{-10}$ \cite{g-2}. We require
SUSY to explain the discrepancy at $2\sigma$ level.
\item [(5)]The dark matter constraints from WMAP relic density
(0.1053 $< \Omega h^2 <$ 0.1193) \cite{WMAP} and the direct detection exclusion limits
on the scattering cross section from XENON100 experiment (at $90\%$ C.L.) \cite{XENON}.
\end{itemize}
Note that most of the above constraints have been encoded in the package NMSSMTools.
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{fig1a.ps}\hspace{0.2cm}
\includegraphics[width=7cm]{fig1b.ps}
\vspace*{-0.5cm}
\caption{The scatter plots of the samples in the MSSM and NMSSM (with
$\lambda>$ 0.53) satisfying various constraints listed in the text
(including $ 123{\rm GeV} \leq m_h \leq 127 {\rm GeV} $), showing
the correlation between the mass of the lighter top-squark and $X_t/M_{S}$
with $M_S \equiv \sqrt{m_{\tilde t_1} m_{\tilde t_2}}$ and $X_t \equiv A_t - \mu \cot \beta$.
In the right panel the circles (green) denote the pull-down case (the lightest Higgs boson
being the SM-like Higgs), and the times (red) denote the push-up case
(the next-to-lightest Higgs boson being the SM-like Higgs).}
\label{fig1}
\end{figure}
Natural supersymmetry are usually characterized by a small superpotential parameter $\mu$,
and the third generation squarks with mass $\lesssim 0.5-1.5$ TeV\cite{naturalsusy}. Therefore,
we only consider the case with
\begin{eqnarray}
100{\rm ~GeV}\leq (M_{Q_3},M_{U_3})\leq 1 {\rm ~TeV} ,~~|A_{t}|\leq 3 {\rm ~TeV}.
\label{narrow}
\end{eqnarray}
For the case with $\lambda<0.2$ in the NMSSM, the property of the NMSSM is similar to
the case in the MSSM\cite{cao125}. In order to distinguish the features
between MSSM and NMSSM, we only consider the case with $\lambda> m_Z/v \simeq 0.53$ in
the NMSSM. We scan over the parameter space of the MSSM and NMSSM under
the above experimental constraints and study the property of the
Higgs boson for the samples surviving the constraints.
\begin{figure}[htbp]
\centering
\includegraphics[width=6.5cm]{fig2.ps}
\vspace*{-0.5cm}
\caption{Same as Fig.\ref{fig1}, but only for the NMSSM, showing the correlation between
$m_{\tilde t_1}$ and the ratio $m_{\tilde t_2}/m_{\tilde t_1}$.}
\label{fig2}
\end{figure}
In Fig.\ref{fig1} we display the surviving samples in the MSSM and NMSSM (with $\lambda>$ 0.53),
showing the correlation between the lighter top-squark mass and the ratio $X_t/M_{S}$ with
$M_S \equiv \sqrt{m_{\tilde t_1} m_{\tilde t_2}}$. From the figure we see that
for a moderate light $\tilde t_1$, large $X_t$ is necessary to satisfy $m_h\sim$ 125 GeV,
and for large $m_{\tilde t_1}$, the ratio $X_t/M_{S}$ decreases.
In the MSSM, $|X_t/M_{S}|>$ 1.6 for $m_{\tilde t_1} <$ 1 TeV, i.e. no-mixing scenario($X_t=0$)
cannot survive, and the top-squark mass is usually larger than 300 GeV.
This implies that a large top-squark mass or a near-maximal stop mixing
is necessary to satisfy the Higgs mass near 125 GeV.
However, the case is very different in the NMSSM, $X_t\approx 0$ may also survive,
and the lighter top-squark mass can be as light as about 100 GeV, which may alleviate the fine-tuning
problem and make the NMSSM seems more natural. In the case of light $m_{\tilde t_1}$,
$|X_t/M_{S}|$ is usually larger than $\sqrt{6}$, which corresponds to a large splitting between
$m_{\tilde t_1}$ and $m_{\tilde t_2}$, as the Fig.\ref{fig2} shown.
\begin{figure}[t]
\includegraphics[width=6.5cm]{fig3a.ps}\hspace{0.2cm}
\includegraphics[width=6.5cm]{fig3b.ps}
\includegraphics[width=6.5cm]{fig3c.ps}\hspace{0.2cm}
\includegraphics[width=6.5cm]{fig3d.ps}
\vspace*{-0.5cm}
\caption{Same as Fig.1, projected in the planes of $m_{\tilde t_1}$ versus the reduced
couplings $C_{h\gamma\gamma}$ and $C_{hgg}$, respectively. }
\label{fig3}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{fig4a.ps}\hspace{0.2cm}
\includegraphics[width=7cm]{fig4b.ps}
\vspace*{-0.5cm}
\caption{Same as Fig.\ref{fig1}, but only for the MSSM, showing the correlation between
$m_{\tilde \tau_1}$ and the reduced coupling $C_{h\gamma\gamma}$, $\mu$ and $\tan\beta$,
respectively. The purple points correspond to $R_{\gamma\gamma}>1$.}
\label{fig4}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{fig5a.ps}\hspace{0.2cm}
\includegraphics[width=7cm]{fig5b.ps}
\vspace*{-0.5cm}
\caption{Same as Fig.\ref{fig1}, but showing the dependence of the
di-photon signal rate $R_{\gamma\gamma}$
on the effective $h b\bar{b}$ coupling
$C_{h b\bar{b}}\equiv C^{\rm SUSY}_{h b\bar{b}}/C^{\rm SM}_{h b\bar{b}}$.
(taken for Ref.\cite{cao125})}
\label{fig5}
\end{figure}
Due to the clean background, the di-photon signal is crucial for searching for
the Higgs boson near 125 GeV. As discussed in the Sec.II, the signal rate is
relevant with the coupling $C_{h\gamma\gamma}$ and $C_{hgg}$ and the total width
of the SM-like Higgs boson. Both the coupling $C_{h\gamma\gamma}$ and $C_{hgg}$ are
affected by the contributions from the squark loops, especially the light top-squark loop,
so in the Fig.\ref{fig3} we give the relationship between the lighter top-squark mass
and the coupling $C_{h\gamma\gamma}$ and $C_{hgg}$, respectively. The figure shows that
the light $m_{\tilde t_1}$ may suppress the coupling $C_{hgg}$ significantly, especially
in the NMSSM. While the light top-squark has little effect on the coupling $C_{h\gamma\gamma}$
because there are additional contributions, as the Eq.(\ref{hgg})and Eq.(\ref{hgaga}) shown.
As analyzed in the Sec.II, light stau may enhance the coupling $C_{h\gamma\gamma}$, so in
Fig.\ref{fig4} we give the correlation between $m_{\tilde \tau_1}$ and the coupling
$C_{h\gamma\gamma}$ in the MSSM. The figure clearly shows that the coupling
$C_{h\gamma\gamma}$ can enhance to 1.25 for $m_{\tilde \tau_1}\sim$ 100 GeV.
Fig.4 also manifests that the enhancement of the coupling
$C_{h\gamma\gamma}$ corresponds to large $\mu\tan\beta$, which leads to large mixing.
These results exactly verifies the discussions in the Sec.II.
Since $h\to b\bar b$ is the main decay mode of the light Higgs boson, the total
width of the SM-like Higgs boson may be affected by the
effective $hb\bar b$ coupling $C_{hb\bar{b}}$, as discussed in \cite{di-photon}.
Under the effect of the combination $C_{hgg} C_{h\gamma \gamma}/C_{hb\bar{b}}$, the di-photon Higgs
signal rate may be either enhanced or suppressed, as shown in Fig.\ref{fig5}, which
also manifest that for the signal rate larger than 1, the effective $hb\bar b$ coupling
is enhanced a little in the MSSM, while it is suppressed significantly in the NMSSM.
Therefore, we can conclude that the reason for the enhancement in the signal rate is very different
between the MSSM and NMSSM. In the MSSM the enhancement of the signal is mainly due to the
enhancement of the coupling $C_{h\gamma\gamma}$, while in the NMSSM it is mainly due to the suppression
of the $hb\bar b$ coupling.
\begin{figure}[htb]
\centering
\includegraphics[width=7cm]{fig6.ps}
\vspace*{-0.5cm}
\caption{Same as Fig.\ref{fig2}, showing the signal rate $R_{\gamma\gamma}$ versus
$S_d^2$ with $S_d=C_{hb\bar b}\cos\beta$.}
\label{fig6}
\end{figure}
Due to the presence of the singlet field in the NMSSM, the doublet component in the
SM-like Higgs boson $h$ may be different from the case in the MSSM, which will affect
the coupling $hb\bar b$, and accordingly affect the total width of $h$. At the tree-level,
$C_{h b\bar{b}}=S_d/\cos\beta$. In Fig.\ref{fig6} we show the dependence of the
signal rate $R_{\gamma\gamma}$ on $S_d^2$. Obviously, for the signal rate larger than 1,
$S_d^2$ is usually very small, which leads to large suppression on the reduced coupling
$hb\bar b$. The figure also shows that the push-up case is more effective to enhance
the signal rate than the pull-down case. This is because the push-up case is easier
to realize the large mixing between the singlet field and the doublet field\cite{cao125}.
As the case in the NMSSM, nMSSM can also accommodate a 125 GeV SM-like Higgs\cite{di-photon}. However,
due to the peculiar property of the lightest neutralino $\tilde\chi_1^0$ in the nMSSM\cite{rarez},
the decay mode of the SM-like Higgs is very different from the case in the MSSM and NMSSM.
As discussed in the Sec.II, $h\to \tilde\chi_1^0\tilde\chi_1^0$ may be dominant
over $h\to b\bar b$ in a major part of parameter space in the nMSSM \cite{di-photon,zhu}, which
induce a severe suppression on the di-photon Higgs signal.
Although the Higgs mass can be easily reach to 125 GeV, the di-photon signal
is not consistent with the LHC experiment. Therefore, the nMSSM may be excluded
by LHC experiment.
\begin{figure}[htbp]
\centering
\includegraphics[width=12cm]{fig7.ps}
\vspace*{-0.5cm}
\caption{The scatter plots of the surviving sample in the CMSSM, displayed in the planes
of the top-squark mass and the LHC di-photon rate versus the Higgs boson mass.
In the left frame, the crosses (red) denote the samples satisfying all
the constraints except $B_s\to\mu^+\mu^-$, and the times (green)
denotes those further satisfying the $Br(B_s\to\mu^+\mu^-)$
constraint. In the right frame, the crosses (red) are same as those
in the left frame, while the times (sky-blue) denote the samples
further satisfying the $R$ constraint.(taken for Ref.\cite{Cao:2011sn})}
\label{fig7}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=12cm]{fig8.ps}
\vspace*{-0.5cm}
\caption{Same as Fig.7, but for the NUHM2.(taken for Ref.\cite{Cao:2011sn})}
\label{fig8}
\end{figure}
We also considered the SM-like Higgs boson mass and its di-photon signal in the constrained MSSM
(including CMSSM and NUHM2) under various experimental constraints,
especially the limits from $B_s\to \mu^+\mu^-$.
Because $Br(B_s\to \mu^+\mu^-)\propto\tan^6\beta/M_A^4$, so it may provide a rather strong
constraint on SUSY with large $\tan\beta$. Considering the large theoretical
uncertainties for the calculation of $Br(B_s\to \mu^+\mu^-)$, we use not only the LHCb data, but
also the double ratio of the purely leptonic decays defined as $R\equiv\frac{\eta}{\eta_{SM}}$
with $\eta\equiv\frac{Br(B_s\to\mu^+\mu^-)/Br(B_u\to\tau\nu_\tau)}
{Br(D_s\to\tau\nu_\tau)/Br(D\to\mu\nu_\mu)}$.
The surviving parameter space is plotted in Fig.\ref{fig7} for the CMSSM and Fig.\ref{fig8}
for the NUHM2. It shows that both the CMSSM and NUHM2 are hard to
realize a 125 GeV SM-like Higgs boson, and also the di-photon Higgs signal is suppressed
relative to the SM prediction due to the enhanced $h\bar bb$ coupling.
Therefore, the constrained MSSM may also be excluded by the LHC experiment.
\section{Conclusion}
In this work we briefly review our recent studies on a 125 GeV Higgs and its di-photon signal rate
in the MSSM, NMSSM, nMSSM and the constrained MSSM.
Under the current experimental constraints, we find: (i) the SM-like Higgs can easily
reach to 125 GeV in the MSSM, NMSSM and nMSSM, while it is hard to satisfy
in the constrained MSSM; (ii) the NMSSM may predict a lighter top-squark than the MSSM,
even as light as 100 GeV, which can ameliorate the fine-tuning problem;
(iii) the di-photon Higgs signal is suppressed
in the nMSSM and the constrained MSSM, but in a tiny corner of the parameter space in the MSSM and NMSSM,
it can be enhanced.
\section*{Acknowledgement}
The work is supported by the Startup Foundation for Doctors of Henan Normal University
under contract No.11108.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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Jan Erasmus Reyning (1640–1697) was a Dutch pirate, privateer and naval officer.
Jan Erasmus Reyning was born in Flushing in 1640 as the son of a Danish mariner and a Zeeland woman. As a boy, he went to sea with his father and later started a seaman's career of his own. He was taken prisoner during the Second Anglo-Dutch War (1665–1667), and later served as an engagé (servant) on a French plantation on the island of Hispaniola. Around 1667 he became a buccaneer, i.e. a hunter in central Hispaniola, and around 1669 he started a career as a pirate or filibuster.
Although documentary evidence is limited, Reyning is believed to have fought as a privateer captain with French or English letters of marque between 1669 and 1672. His partner was one Jelle Lecat (probably born in Frisia as Jelle de Kat). Reyning must have co-operated with renowned pirates as Roche Braziliano and Henry Morgan. According to reliable Spanish documents, Reyning offered his services to his former Spanish enemies at Campeche around January 1672. The Spanish accepted his services and he even took catechisation lessons from a Catholic priest. In 1673 Reyning, Lecat, and Irish pirate Philip Fitzgerald took more than 40 vessels in the area, mostly logwood cutters. Later that year, when Reyning learned that his home country, The Netherlands, was at war with England and France (in the Franco-Dutch War and the Third Anglo-Dutch War), he sailed to the Dutch colony of Curaçao. He became a sort of Robin Hood for Curaçao that was threatened by many enemies.
In his later years Reyning was a Dutch gentleman and marine officer who died at full sea in the Gulf of Biskay in a storm on 2 February 1697.
References
Snelders, S. (2005): The Devil's Anarchy. New York: Autonomedia, 2005.
Specific
1640 births
1697 deaths
Dutch pirates
People from Vlissingen
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{
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import { address } from "./address";
export namespace shippingAddress {
interface ShippingAddressData {
/**
* Name of the shipping recipient
*/
RecipientName: string;
/**
* The shipping address
*/
Address: address.AddressType;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,350
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That Devil Music.com: The Reverend's Rock 'n' Roll Archives
Archive Review: Warren Zevon's Warren Zevon (2008)
Forget all about Wanted Dead Or Alive, Warren Zevon's uncharacteristic 1969 debut LP. The album shows none of the wit or caustic wordplay that Zevon would become known for, and is clearly an attempt by a young artist to make a play for success long before he's ready to do so. The album deserves neither your time nor a place in your collection. Opt instead for Zevon's self-titled 1976 follow-up, his true debut and, perhaps, one of the finest sophomore efforts in rock music. Warren Zevon, the album, also represents the beginning of an amazing rock 'n' roll success story.
By 1975, Zevon had spent nearly a decade in Los Angeles, doing session work, writing advertising jingles, performing behind the Everly Brothers, and occasionally writing songs for folks like the Turtles. What Zevon didn't have was a record deal, or even the promise of one. Fearing that his career would never take off, he fled to Spain with his wife, taking up musical residency in a local bar owned by an American soldier of fortune. A postcard from his friend, singer/songwriter Jackson Browne, hinting of the possibility of a record deal lured the ex-pat musician back to the United States and California.
Warren Zevon's Warren Zevon
The eventual result would be the brilliant Warren Zevon album. With an additional six-plus years spent honing both his songwriting craft and performing chops, Zevon entered the studio with seasoned veterans like multi-instrumentalist David Lindley, guitarist Waddy Wachtel, and saxophonist Bobby Keys. Friends like Browne, Bonnie Raitt, Phil Everly, and Lindsey Buckingham and Stevie Nicks of Fleetwood Mac provided vocal harmonies behind Zevon's incredibly designed songs.
Displaying the same sort of gonzo sensibilities as author Hunter S. Thompson's best work, Zevon's songs are filled with brightly-colored and finely-crafted characters from the seedier fringes of society. "Frank And Jesse James" is a finely-detailed tale of the Civil War vets turned outlaw gunfighters, Zevon's fully mature vocals matched by spry, vaguely Western piano (think San Fran goldrush) and shotgun drumbeats.
The beautiful "Hasten Down the Wind" was covered wonderfully by Linda Ronstadt, but Zevon's original version is equally considerate, with Phil Everly's harmony vocals adding depth to Zevon's deep purr as David Lindley's slide guitar weeps openly. The boisterous "Poor Poor Pitiful Me" was also later covered by Ronstadt, but not like this. On Zevon's version, Wachtel's guitar rips-and-snorts and tears at the reins while honky-tonk piano blasts out beneath the singer's half-mocking, self-effacing vocals.
The Dylanesque "Mohammed's Radio" sounds a little like Jackson Browne, too, but the song's contorted, colorful personalities and gospel fervor belie its anthemic nature. In many ways "I'll Sleep When I'm Dead" presages Zevon's notorious hard-partying lifestyle, while "Desperados Under the Eaves," perhaps the best song ever written about Los Angeles, is haunted by the reckless spirits of Charles Bukowski and Hubert Selby, Jr. (yes, both were alive and well when the song was written, thank you, but they still had their otherworldly stank all over the song).
A bonus disc provided this reissue of Warren Zevon is chockfull o' demos and other goodies for the fanatical completist. A solo piano arrangement of "Frank And Jesse James" is fine, but lacks the powerful drumwork of the final version, but the sparse arrangement given the alternative take of the junkie's tale "Carmelita" enhances the song's inherent loneliness and hopelessness. The second take of "Join Me In L.A." evinces a looser, funkier vision of the song while a live radio performance of "Mama Couldn't Be Persuaded" is a rollicking, joyful reading of the song that places the spotlight firmly on Zevon's lyrics. Taken altogether, the second disc's rarities provide some insight into Zevon's early creative process.
Zevon would follow-up his self-titled sophomore effort a couple of years later, 1978's Excitable Boy yielding the hit "Werewolves of London" and making the singer/songwriter a rock star. Over the following 25 years and a dozen albums, until his tragic death in 2003, Zevon would cement a legacy fueled by his unique talent and personality … and it all started with Warren Zevon. (Rhino Records, 2008)
Review originally published by Blurt magazine
Also on That Devil Music:
Warren Zevon's Bad Luck Streak In Dancing School CD review
Warren Zevon's Life'll Kill Ya CD review
Buy the CD from Amazon: Warren Zevon's Warren Zevon
Rev. Keith A. Gordon at Friday, November 04, 2022
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Inc.@35
Why Samuel Adams Supports Its Competitors
Founder Jim Koch explains why he gives money, materials, and advice to other craft brewers.
By Leigh Buchanan, Editor-at-large, Inc. magazine@LeighEBuchanan
Jim Koch is chairman of Boston Beer, a pioneer of the craft-beer movement and brewer of Samuel Adams. The business, which Koch founded in 1984, had $739 million in revenue last year. Since 2008, Boston Beer has given out more than $3 million in microloans to craft brewers and other small businesses in the food, beverage, and hospitality industries. As Koch explains to Inc. editor-at-large Leigh Buchanan, supporting competitors can sometimes make sense.
The reason I support competitors becomes obvious if you think about the way yeast ferments beer. If enough yeast are working together, they can change the ecosystem for the mutual benefit of all. If they aren't, other organisms take over, and the yeast will fail. Craft brewing is kind of like that.
We are happy to share our innovations with the industry. We were the first brewery to age beer in used spirits barrels back in the early 1990s. So we got a lot of calls: Where do you get the barrels? How do you do it? How do you get approval for it? We shared with anybody who asked us. About a year ago, we invested $1 million to develop a beer can that allows you to get more air when you drink, so you experience the taste and smell of the beer at the same time. We licensed the design to a manufacturer on the condition that it let other craft brewers use it for free. It will help differentiate craft beers from beers the big guys have developed to compete with us.
In 2008, there was a worldwide hops shortage. A lot of craft brewers got caught short--particularly the smaller ones. We had enough because we buy in advance and on contract. So we put out an announcement to craft brewers that we would sell them our hops at cost. We were able to help more than 200 breweries--some were faced with shutting down without a supply. We did it again in 2012, when there was a shortage of a very desirable kind of hops used in IPAs [India pale ales]. Some of the brewers we helped sent us a few bottles made with our hops as a thank-you.
Through our Samuel Adams Brewing the American Dream program, we've made loans to about a dozen microbrewers and provided coaching to another 30. They are a lot of fun. For me personally, and for us as a company, it connects us with our small-business roots. And if one of these companies is successful enough that they take some market share from us, well, more power to them. I don't worry about that. I worry about how we create a beer culture that respects the art of brewing and wants beer with flavor, taste, and authenticity. If we can create that environment, there will be plenty of business for all of us.
I don't want to be a Goliath. It's a lot more fun to be a little shepherd boy, as long as you have got more than one David. You read the story of David--his life kind of sucked after he became king.
Previously in Inc.: Jim Koch told his startup story in the March 1988 feature "Portrait of a CEO as Salesman."
From the July-August 2014 issue of Inc. magazine
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Q: R: creating a vector of 1/n weights using a for loop For the sake of making this general enough to be useful for others, I have six friends:
friends <- c("luke", "leia", "han", "chewy", "lando", "obi")
That I am trying to give them each a portion of cake using a for loop
portions <- for (portion in length(friends)){
portion = 1/portion
}
My code returns the correct portion size (~16%), but does not populate my portions vector with what I was hoping for:
(0.16,0.16,0.16,0.16,0.16,0.16)
Cutsie examples aside, I am looking to for a solution which can be applied to creating weights for several hundred variables.
Any help would be greatly appreciated!
A: You could use
prop.table(table(friends))
#friends
#chewy han lando leia luke obi
#0.167 0.167 0.167 0.167 0.167 0.167
Or
table(friends)/length(friends)
A: We can just do rep on the 1/length(friends) and it would give the expected output without any other calculations
rep(1/length(friends), length(friends))
#[1] 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667
In the for loop, we are only assigning to portion each time and it gets updated. Instead, an optioin is to pre-assign a vector and assign it to it
portion <- numeric(length(friends))
for(i in seq_along(friends)) portion[i] <- 1/length(friends)
portion
#[1] 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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`match_map` defines an object with a hash-like interface that allows Regexp patterns as keys and/or multiple simultaneous lookuup arguments. Calling `mm[arg]` checks _arg_ against every key, aggregating their associated values into an array.
```ruby
require 'match_map'
mm = MatchMap.new
mm['a'] = 'a_string'
mm[/a/] = 'apat'
mm[/b/] = ['bpat1', 'bpat2']
mm[/.+b$/] = 'bpat3'
mm['a'] #=> ['a_string', 'apat'] # order is the same as the key order
mm['aa'] #=> ['apat']
mm['b'] #=> ['bpat1', 'bpat2']
mm['cob'] #=> ['bpat1', 'bpat2', 'bpat3'] # flattened one level!!!
mm['cab'] #=> ['apat', 'bpat1', 'bpat2', 'bpat3']
mm['c'] #=> [] # no match
# Change the default miss value to ease some processing forms
mm.default = []
mm['neverGonnaMatch'].each do { #never get here}
# You can also query on multiple values at once by passing an array
mm[['a', 'aa', 'b']] #=> ['a_string', 'apat', 'bpat1', 'bpat2']
# Or use a Proc as the value; it gets the match variable as its argument
mm = MatchMap.new
mm[/a(b+)/] = Proc.new {|m| [m[1].size]}
mm['abbbb'] #=> [4]
# You can set #echo to return the argument :always or only :onmiss
mm = MatchMap.new
mm[/ab/] = "AB"
# first, without echo
mm['miss'] = []
mm['cab'] = ['AB']
#...then with echo = :always
mm.echo = :always
mm['miss'] = ['miss']
mm['cab'] = ['cab', 'AB']
#...and again with echo = :onmiss
mm.echo = :onmiss
mm['miss'] #=> ['miss'] # because nothing else matched
mm['cab'] #=> ['AB'] # because a match was found
# Need to ditch a key?
if mm.has_key? /ab/
mm.delete /ab/
end
```
A MatchMap is a hash-like with the following properties:
* The return value is always a (possibly empty) array
* keys can be anything that responds to '==' (e.g., strings) or regular expressions
* keys cannot be repeated (mirroring how a hash works, but see below about multiple values)
* arguments are compared to non-pattern keys based on ==
* arguments are compared to pattern keys based on pattern match against arg.to_s_
* values can be scalars, arrays (treated as multiple return values), or Proc objects
* a scalar argument to #[] is left alone for comparison to non-patterns (so a string or integer can be exactly matched), but converted to a string for comparison to patterns. (see "How are arguments compared to keys?", below)
* an array argument to #[] is treated as if you want all values for all matches for all array members
* the return value from #[] goes through #uniq and #compact (no repeated values, no nil values), which may or may not mess with what you expect the output order to be.
The idea is that you can set up a bunch of (possibly overlapping) patterns, each of which is associated with one or more values, and easily get back all the values for those patterns that match the argument. `match_map` was originally designed for transforming values for full-text indexing, but has other uses as well.
## What is this good for?
A match_map can be useful for (among other things) values that map onto a hierarchy.
Here's part of a map for library call numbers:
```ruby
mm = MatchMap.new
mm[/^H/] = 'Social Science'
mm[/^HA/] = 'Statistics'
mm['HA37 .P27 P16'] #=> ['Social Science', 'Statistics']
```
Or use it as a clean way to extract semi-regular information from free-text strings
```ruby
state = MatchMap.new
state[/\bMN\b/i] = 'Minnesota'
state[/\bMI\b/i] = 'Michigan'
state['St. Paul, MN 55117'] #=> ['Minnesota']
state['2274 Delaware Drive, Ann Arbor, MI, 48103'] #=> ['Michigan']
```
## How are arguments compared to keys?
There are three basic rules:
1. If the argument is an array, each element is handled separately
2. If the argument (`a`) is being matched against a pattern key (`pk`), check if `pk.match(a.to_s)`
3. If the argument (`a`) is being matched against a key that is not a pattern (`npk`), check if `a == npk`
Here's a quick example to show how it works
```ruby
mm = MatchMap.new
mm[1] = 'fixnum'
mm['1'] = 'string'
mm[/1/] = 'pattern'
mm[1] #=> ['fixnum', 'pattern']
mm['1'] #=> ['string', 'pattern']
```
## Using Proc objects as values
You can also use a Proc object as a value. It must:
* take a single argument; the match variable (if your key was a Regexp) or the string matched
* return a (possibly empty) _array of values_
It doesn't make a lot of sense to use a Proc value if your key is just a scalar, but it's possible.
This can be abused, of course, but can be useful. Here's a simple example that reverses the order of a comma-delimited duple.
```ruby
mm = MatchMap.new
mm[/^(.+),\s*(.+)$/] = Proc.new {|m| "#{m[2]} #{m[1]}"}
mm['Dueber, Bill'] #=> ["Bill Dueber"]
```
## Using echo to always/sometimes get back the argument
There are two common requirements when doing this sort of translation for indexing:
* The raw argument should always appear in the output
* The raw argument should appear in the output only if there are no other matches.
```ruby
mm = MatchMap.new
mm[/ab/] = "AB"
# first, without echo
mm['miss'] = []
mm['cab'] = ['AB']
#...then with echo = :always
mm.echo = :always
mm['miss'] = ['miss']
mm['cab'] = ['cab', 'AB']
#...and again with echo = :onmiss
mm.echo = :onmiss
mm['miss'] #=> ['miss'] # because nothing else matched
mm['cab'] #=> ['AB'] # because a match was found
```
Note that the `default` value will never be added to the output if `echo` is set.
## Optimizing
You can call `mm.optimize!` to attempt to optimize a MatchMap where none of the keys are regular expressions
and none of the values are Proc objects for a significant speed increase (run `rake bench` for an idea
of how much faster). This allows you to take advantage of all the differences between MatchMap and a regular
hash (pass multiple arguments, flatten return values, echoing, etc.) while remaining an O(1) operation (instead
of a O(n) for the standard, try-to-match-each-key-in-turn algorithm).
Note that a call to `#optimize!` actually picks the best algorithm for that particular map, so if you have a simple map,
call `#optmize!`, and add a regular-expression key, another call to `#optmize!` is required to start using the
regular algorithm again.
Obviously, only call `#optimize!` when you're sure you won't be modifying the map anymore.
## Gotchas
* Like a hash, repeated assignment to the same key results in a replacement. So `mm[/a/] = 'a'; mm[/a/] = 'A'` will give `mm['a'] #=> ['A']`
* Return values are flattened one level. So, /a/ => 1 and b => [2,3], the something that matches both will return [1,2,3]. If you really want to return an array, you need to do something like `m['a'] = [[1,2]]`
## Contributing to MatchMap
* Check out the latest master to make sure the feature hasn't been implemented or the bug hasn't been fixed yet
* Check out the issue tracker to make sure someone already hasn't requested it and/or contributed it
* Fork the project
* Start a feature/bugfix branch
* Commit and push until you are happy with your contribution
* Make sure to add tests for it. This is important so I don't break it in a future version unintentionally.
* Please try not to mess with the Rakefile, version, or history. If you want to have your own version, or is otherwise necessary, that is fine, but please isolate to its own commit so I can cherry-pick around it.
## Copyright
Copyright (c) 2011 Bill Dueber. See LICENSE.txt for
further details.
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|
{"url":"https:\/\/electronics.stackexchange.com\/tags\/charge\/new","text":"# Tag Info\n\n## New answers tagged charge\n\n1\n\nIf biased around DC, the capacitor stores energy as the voltage increases in magnitude (either positive or negative), and the capacitor gives up energy as the voltage returns to zero. Thus 1\/4 cycle is store, 1\/4 cycle is discharge, this occurring for each 1\/2 cycle. If biased asymmetrically, then the timing changes.\n\n1\n\nYes! It's true or false. When connected across a sine-wave AC supply an ideal capacitor stores energy while the voltage is increasing in magnitude and releases energy when the voltage is decreasing in magnitude. so half the tlong period will ne equal ime it's storing and half the time releasing. But the it's not a continuous half of the AC cycle, it's ...\n\n0\n\nNo energy is lost in a purely reactive load, which a capacitor is. Over time for any repeating voltage signal with no average bias, the cap\u2019s average charge will be zero. That is, the net charge is the integral of applied voltage over time. Zero integral => no charge. Now, whether it \u2018returns\u2019 its energy in 1\/2 cycle depends on the waveform. If it\u2019s ...\n\n0\n\nTrue. The energy stored in a capacitor is given by $U = \\frac {1}{2}C V^2$. If the average energy is increasing then V would be increasing too. Figure 1. An example circuit. Physics and radio electronics. The circuit of Figure 1 gives a common example (which doesn't quite match the charge for \u00bd cycle requirement of the title as the charge ...\n\n0\n\nyes it is true.because the basic working of capacitor is that it charges for one half cycle and discharges the energy that stored in 1 half cycle.for ideal capacitor net energy is Zero.\n\n2\n\nFALSE! But it's a trick question. (Or, perhaps the author has misconceptions about how capacitors work, and their book has errors?) Was the answer supposed to be false? Yet the net energy is actually zero, that part's true. The question's reasoning is wrong, the number \"\u00bd\" is wrong, so the answer given in the back of the book had better be \"false.\"\n\nTop 50 recent answers are included","date":"2019-07-17 03:10:01","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\": 0.70148104429245, \"perplexity\": 1016.1407469210752}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195525009.36\/warc\/CC-MAIN-20190717021428-20190717043428-00419.warc.gz\"}"}
| null | null |
Lackawanna Transit Center is the main bus station and a proposed train station in Scranton, Pennsylvania, operated by the County of Lackawanna Transit System (COLTS).
Opened in 2015, the transit center features an indoor waiting area, covered bus bays, a park-and-ride lot, and pick-up/drop-off lanes. , it is served by COLTS, Luzerne County Transportation Authority (LCTA), Amtrak Thruway, Greyhound Lines, Martz Trailways, New York Trailways, and Fullington Trailways.
Located at the corner of Lackawanna and Cliff avenues in downtown Scranton, the transit center is close to Steamtown National Historic Site, the Electric City Trolley Museum, and the Marketplace at Steamtown. The site is also adjacent to the Pocono Mainline of the Delaware-Lackawanna Railroad, and is intended to accommodate proposed expansion of the bus station into an intermodal train and bus terminal with rail service to New York City via the Lackawanna Cut-Off.
History
A groundbreaking ceremony for the Lackawanna Transit Center took place on August 1, 2014. Plans for the project were said to have been "18 years in the making." A ribbon-cutting occurred on November 20, 2015, and the station first served buses on December 7. The total cost came to $12.5 million.
Proposed train station
From 1908 through 1970, passenger trains to Scranton used the Lackawanna Railroad's large station, now a Radisson hotel. The Lackawanna Cut-Off Restoration Project is an ongoing effort to revive passenger rail from New York to Scranton, with construction already underway on Phase I: an NJ Transit extension from Lake Hopatcong to Andover, New Jersey. The bus station was built on the site that had long been considered for Scranton's new train station. In spring 2021, Amtrak announced plans for a potential New York–Scranton route.
References
External links
Proposed NJ Transit rail stations
Railway stations in Pennsylvania
Buildings and structures in Scranton, Pennsylvania
Transportation buildings and structures in Lackawanna County, Pennsylvania
|
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|
\section{Introduction}
\subsection{The Majorana Formula}
Abragam introduces quantum mechanics in his nuclear magnetic resonance text ``The Principles of Nuclear Magnetism" \cite{abragamtext} with two fundamental equations,
the Schr{\"o}dinger equation, and the spin transition probability formula first derived by Majorana \cite{emajorana}. Majorana's formula
\cite{emajorana} provides the answer to the simplest and most fundamental question one can ask of a spin system: what is the probability for finding a spin in the state with magnetic projection number
$m^{\prime}$ at time $t$ knowing that it was previously in the state $m$ at time $t=0$? More precisely, consider a spin in a uniform, static magnetic field {\bf H}$_0 =H_0 \, {\bf \hat{k}}$, whose direction can be taken to be the axis of quantization. Under the application of a perpendicular alternating radiofrequency field ${\bf H}_1 = H_1 \cos \psi t \, {\bf \hat{i}}$ in the laboratory frame which causes transitions, Majorana's formula
\cite{emajorana}, as quoted by Abragam
\cite{abragamtext}, gives the probability in a spin-$j$ system of a transition from a state of magnetic quantum
number $m$ at time $t=0$ to one of magnetic quantum number $m^{\prime}$ at time $t$:
\begin{eqnarray}
\mbox{ P}^{(j)}_{mm^{\prime}}(t) & = & (j-m)! (j+m)! (j-m^{\prime})! (j+m^{\prime})! \left( \cos \tfrac{1}{2} \alpha \right)^{\!4j} \times \nonumber \\
& &
\;\;\;\;\left[ \sum_{\lambda=0}^{2j} \displaystyle\frac{(-1)^r \left( \tan \frac{1}{2} \alpha \right)^{2\lambda -m+m^{\prime} }}
{\lambda!(\lambda-m+m^{\prime})!(j-m^{\prime}-\lambda)!(j+m-\lambda)!} \right]^{\!2} \label{abramajorana} \\
\mbox{where} \;\;\; \sin \tfrac{1}{2} \alpha & = & \sin \Theta \, \sin \bfrac{\psi}{2} \label{probangles} \\
\psi& = & \left|\gamma {\bf H}_e\right| t \\
{\bf H}_e & = & \left[ H_0 + \frac{\omega}{\gamma} \right] {\bf \hat{k}} + H_1 \, {\bf \hat{i}}
\end{eqnarray}
In a frame rotating with angular frequency $\omega$, the direction of the effective field {\bf H}$_e$ is defined by the polar angle $\Theta$, and $\left|\gamma {\bf H}_e\right| $ is the Larmor precession frequency of the spin magnetic moment around the effective field {\bf H}$_e$. Recognizing that Majorana's elegant derivation \cite{emajorana} of the spin transition probability relied on the theory of the irreducible representations of the group of rotations, Abragam
\cite{abragamtext} simply stated the formula of
Eq.(\ref{abramajorana}), and did not discuss its derivation. We follow suit, since fortunately, there are a number of excellent textbook discussions
\cite{ramsey, corio,gilmore,biedenharn, thompson, Esposito} of the Majorana formula of Eq.(1) which fill in the details of the derivation, and also provide some history.
\subsection{The Meckler formula} An alternative derivation of the Majorana formula \cite{emajorana} was given by Bloch and Rabi \cite{rabi2}. Despite an improvement in the symmetry of the formula in the indices $m$ and $m^{\prime}$, their version \cite{rabi2} was much the same as that of the original \cite{emajorana}.
Then in 1958, Meckler \cite{meckler:majorana} published a remarkably simple version of the Majorana formula \cite{emajorana}, relying on a novel projection operator method.
Meckler \cite{meckler:majorana,meckler:angular} took a very unorthodox approach to calculating the transition probability, using projection operators expanded in a Chebyshev polynomial operator basis $f_L^{(j)}( {\bf \hat{n}} \cdot {\bf J}) $ (see Table I) never previously used in magnetic resonance or in any other physical application. The final result of Meckler's calculation \cite{meckler:majorana} took
the following very concise form
\begin{eqnarray}
\mbox{ P}^{(j)}_{mm^{\prime}}(t) & = & \left| \langle {\bf \hat{b}},m^{\prime} |\,{\bf \hat{a}},m \rangle \right|^{2} \label{modtrans} \\
& = & \sum_{\lambda=0}^{2j} f_{\lambda}^{(j)} (m) \; f_{\lambda} ^{(j)}(m^{\prime})\; P_{\lambda}({\bf \hat{a}} \cdot {\bf \hat{b}}) \label{firstmeck} \\
\mbox{where} \;\;\; {\bf \hat{a}} \cdot {\bf \hat{b}} & = & \cos \beta(t)
\end{eqnarray}
Meckler's formula
\cite{meckler:majorana,meckler:angular} provides the answer to a slightly different version of the query answered by the
Majorana formula \cite{emajorana} : given a spin initially quantized along a unit vector
${\bf \hat{a}}$ with component $m$, what is the probability that it is quantized along a unit vector ${\bf \hat{b}}$ with component
$m^{\prime}$ at a later time $t$?
Meckler's formula
\cite{meckler:majorana,meckler:angular} makes use of just two special functions, the well-known Legendre polynomials $ P_{\lambda}(\cos \beta (t))$, and the lesser-known Chebyshev polynomials
$ f_{\lambda} ^{(j)}(m) =\langle jm| \; f_{\lambda}^{(j)} ( J_z) \; |jm \rangle $ of a discrete variable \cite{bateman,nikiforov2,Nikiforov, vilenkin:specfuncbook}, the diagonal matrix elements of the Chebyshev polynomial operators
$f_{\lambda} ^{(j)}(J_z) $. These latter operators and their matrix elements $ f_{\lambda} ^{(j)} (m) $ are the subject of this article. To avoid any confusion from the outset, the Chebyshev polynomials
$ f_{\lambda}^{(j)}(m) $ of a discrete variable \cite{bateman,nikiforov2,Nikiforov, vilenkin:specfuncbook} should not be confused with the more commonly known and used Chebyshev polynomials of the first ($T_n(x)$) and second ($U_n(x)$) kind \cite{arken, olver, tem:bk}.
Why should the Meckler version \cite{meckler:majorana,meckler:angular} of the Majorana formula \cite{emajorana} be of interest? In answering this question, we should note the following:
(1) The conciseness of the Meckler version \cite{meckler:majorana,meckler:angular}
of the Majorana formula \cite{emajorana} is evident upon a comparison of Eqs.(\ref{abramajorana}) and (\ref{firstmeck}). The Meckler formula
\cite{meckler:majorana,meckler:angular} takes the form of Fourier-Legendre series, which as we shall see in Section {\bf 4.3.1}, can in certain cases be summed to yield simple, closed-form expressions for ``spin-flip" transition probabilities such as $\mbox{P}^{(j)}_{j,-j} (t) $ or $\mbox{P}^{(j)}_{j-1,-(j-1)} (t) $.
(2) The Meckler version
\cite{meckler:majorana,meckler:angular} of the Majorana formula \cite{emajorana} does what Majorana's original version \cite{emajorana} does not: Meckler's version
\cite{meckler:majorana,meckler:angular} isolates the dependence on the magnetic projection numbers of the initial and final states in a
Chebyshev polynomial product term $\left[ f_{\lambda}^{(j)} (m) \; f_{\lambda} ^{(j)}(m^{\prime}) \right]$, leaving the time-dependence isolated in a Legendre polynomial term $P_{\lambda}(\cos \beta (t))$. In this respect, it is reminiscent of the Clebsch-Gordan coefficient expansion of the rotation matrices ${\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}}) $ which isolates the dependence on the indices $m$ and $m^{\prime}$ from the time-dependent terms $\psi(t)$ and ${\bf \hat{n}}(t)$ \cite{varshal1:ang, sim:beyond}.
(3) In his reduction of the Majorana problem \cite{emajorana} to the calculation of the joint probability of quantization along two different axes, Meckler \cite{meckler:majorana,meckler:angular} was the first to exploit Chebyshev polynomials of a discrete variable
\cite{bateman,nikiforov2,Nikiforov, vilenkin:specfuncbook} in a physics application. As this article describes, other physics applications of these special functions would follow over the next six decades, including applications in
spin physics, spin tomography, and in the development of operator expansions and equivalents. During this six decade period, it would appear that the
Chebyshev
orthonormal basis operators $f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) $, first introduced by Meckler \cite{meckler:angular} in 1959, were
independently rediscovered twice thereafter, by Corio \cite{corio:siam} in 1975, and then by Filippov and Man'ko
\cite{filippov4} in 2010.
\subsection{Very special special functions}
Special functions \cite{askey} such as Legendre polynomials, Bessel functions, Chebyshev polynomials and Hermite polynomials are essential tools in mathematical physics \cite{tem:bk,arken,butkov,riley}. As Michiel Hazewinkel has noted \cite{vilenkin:specfuncbook}, ``Special functions are, well, special." Are some special functions more special than others? In this article we describe some physical applications of some very special, special functions,
the Chebyshev polynomials of a discrete real variable \cite{bateman,nikiforov2,Nikiforov, vilenkin:specfuncbook}. These special functions are a member of the family of classical orthogonal polynomials of a discrete variable known as the Hahn polynomials: \cite{bateman,nikiforov2,Nikiforov,vilenkin:specfuncbook}
Why are the Hahn polynomials, and the Chebyshev polynomials in particular, so very special? At least from the point of view of physical applications, it is hard to find any other special function that has been more obscure. Contributing to this obscurity is the fact that Hahn polynomials have never been discussed in any of the texts commonly used in undergraduate mathematical physics courses. Is this obscurity deserved, and might we expect this obscurity to change? Whereas no mention of the Hahn polynomials can be found in the classic handbook on mathematical functions by Abramowitz and Stegun \cite{abramo}, the Bateman project carried out by Erd\'elyi and colleagues
\cite{bateman} and the recently published successor to Abramowitz and Stegun \cite{abramo}, the NIST Handbook by Olver and colleagues \cite{olver}, both contain excellent summaries of the properties of classical orthogonal polynomials of a discrete
variable \cite{bateman,nikiforov2,Nikiforov, vilenkin:specfuncbook}. Just in the last decade, in a substantial body of work, Filippov and Man'ko and coworkers \cite{ filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo} have shown the promise of Chebyshev polynomials of a discrete variable for spin tomographic applications, and this article can be used as an introduction to these applications.
From the point of view of physics applications, the Chebyshev polynomials of a discrete real variable \cite{bateman,nikiforov2,Nikiforov,vilenkin:specfuncbook} possess some striking properties, and to make that point,
the sceptical and curious reader might wish to answer the following questions:
1. In a spin-$j$ system, the spin transition probability $\mbox{ P}^{(j)}_{mm^{\prime}}(t) $ can be written as a Fourier-Legendre series whose expansion coefficents can be expressed in terms of one (and only one) special function. What is that special function?
2. Projection operators, including the coherent state projector, can be written in terms of one (and only one) special function. What is that special function?
3. Ignoring a phase-factor, the Clebsch-Gordan coupling coefficients $ C_{jmj-m}^{\lambda 0}$ of angular momentum theory
\cite{brinksatch:ang,varshal1:ang} are identical to what special function?
4. In a spin-$j$ system, what special function operator $g^{(j)}\!({\bf \hat{n}} \cdot {\bf J})$ provides a unique orthonormal Hermitian expansion basis for the angle-axis $(\psi, {\bf \hat{n}})$ parametrized rotation operator $\hat{{\cal D}}^{(j)} \!(\psi, {\bf \hat{n}}) = e^{i \psi({\bf \hat{n}} \cdot {\bf J}) }$ ?
5. The trace of the rotation operator $\hat{{\cal D}}^{(j)} \!(\psi, {\bf \hat{n}}) = e^{i \psi({\bf \hat{n}} \cdot {\bf J}) }$ defines the character
$ \chi^{(j)}(\psi) = \mbox{Tr} \! \left[ \hat{{\cal D}}^{(j)} \!(\psi, {\bf \hat{n}}) \right]$ \cite{varshal1:ang} of irreducible representations of the rotation group. The trace of the product of this rotation operator and a special function operator is proportional to the generalized characters $\chi_{\lambda}^{(j)}(\psi) $ \cite{varshal1:ang} of the rotation group. What is that special function operator?
6. What special function $g^{(j)}(J_z)$ of the operator variable $J_z$ for a spin-$j$ system is identical to the projection-zero spin polarization operators
\cite{varshal1:ang} $\hat{T}_{\lambda 0}^{(j)} $?
7. What special function operator $g^{(j)}( {\bf \hat{n}} \cdot {\bf J}$) for a spin-$j$ system can be recoupled as a rank-zero irreducible composite tensor defined by the direct product of two rank-$\lambda$ tensors, one the spin tensor $ {\bf T}_{\lambda}({\bf J}) $, and the other, the spatial Racah tensor
${\bf C}_{\lambda}({\bf \hat{n}})$?
8. What special function operator $g^{(j)}( {\bf \hat{n}} \cdot{\bf J}$) for a spin-$j$ system has matrix elements expanded in spherical harmonics, with Clebsch-Gordan coefficients as the expansion coefficients?
9. The density operator ${\hat \rho}$ for a spin-$j$ system can be tomographically reconstructed
\cite{filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo, klimovchumakov, ariano} using one (and only one) special function operator
$h^{(j)}\! ( {\bf \hat{n}} \cdot {\bf J}$). What is that special function operator $h^{(j)}\! ( {\bf \hat{n}}\cdot {\bf J}$)?
10. For a spin-$j$ system, the spin polarization operators \cite{varshal1:ang} $\hat{T}^{(j)}_{\lambda \mu} $ may be viewed as an integral transformation of a unique special function polynomial operator $g^{(j)}( {\bf \hat{n}} \cdot {\bf J}$) from the continuous variables $(\theta, \phi) $ (which define
${\bf \hat{n}} \equiv {\bf \hat{n}}(\theta, \phi) $) to the discrete variables $\lambda, \mu$ which define the spin polarization operators \cite{varshal1:ang}
$\hat{T}^{(j)}_{\lambda \mu} $. What is that special function operator $g^{(j)}( {\bf \hat{n}} \cdot {\bf J}$)?
The surprising answer to all these questions is the same, the special functions known as the Chebyshev polynomials of a discrete real
variable \cite{bateman,nikiforov2,Nikiforov,vilenkin:specfuncbook}.
\subsection{Article Organization} This article is organized as follows: we begin in Section {\bf 2} with a brief summary of the properties of the Chebyshev polynomials $ f^{(j)} _L(m) $ of a discrete real variable $m\;\;(m=-j, -j+1, \ldots, j-1, j)$, and of the
Chebyshev
orthonormal basis operators $f_L^{(j)}( {\bf \hat{n}} \cdot {\bf J}) $. Note that the eigenvalues of $ ({\bf \hat{n}} \cdot {\bf J})$ are discrete real variables, and that because of ``space quantization" \cite{frenchtaylor}, the projection $({\bf \hat{n}} \cdot {\bf J}) =J_n$ can therefore only take on $(2j+1)$ possible values. Sections {\bf 3} through {\bf 6} discuss several examples of physical applications of Chebyshev polynomials, beginning in Section {\bf 3} with an
introduction to projection operators and their use in the calculation of transition probabilities. Section {\bf 4} is devoted to a discussion of Meckler's formula \cite{meckler:majorana,meckler:angular} for the calculation of spin transition probabilites. Section {\bf 5} illustrates how the Chebyshev polynomial operators provide a Hermitian orthonormal basis for expanding the rotation and Stratonovich-Weyl operators
\cite{varilly2:moyal, klimov:distr, heissweigert, klimovchumakov}, and for tomographic reconstruction of the density
operator \cite{filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo, klimovchumakov,ariano}. Section {\bf 6} shows that the Chebyshev polynomial operators can be recoupled as the product of spin and spatial tensors. Section {\bf 7} discusses how the Chebyshev polynomial operators
$f_{\lambda} ^{(j)}(J_z)$ can be used to develop operator equivalents for any irreducible tensor operator. Concluding remarks in Section {\bf 8} highlight some of the unique features of Chebyshev polynomials and their applications in magnetic resonance.
\section{Chebyshev Polynomial Properties}
This section is devoted to a brief summary of the most important properties of the Chebyshev polynomials of a discrete
variable \cite{bateman,nikiforov2,Nikiforov,vilenkin:specfuncbook}. More detailed discussions can be found in specialized monographs \cite{vilenkin:specfuncbook,Nikiforov,olver,bateman,nikiforov2} and in the original literature \cite{meckler:majorana,meckler:angular,corio:siam,filippov2:thesis,corio:ortho,werb:tensor,NormRay}. We begin in Section {\bf 2.1} with a discussion of the properties of
the Chebyshev polynomial scalars $f_{\lambda}^{(j)}(m)=\langle jm| \; f_{\lambda}^{(j)} ( J_z) \; |jm \rangle $, followed in Section {\bf 2.2} with a discussion of the properties of the Chebyshev polynomial
operators $f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J}) $. As matters of notation are concerned, throughout this article we make exclusive use of the notation adopted by Filippov and Man'ko and co-workers \cite{ filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo} for the
Chebyshev polynomial scalars $ f^{(j)} _{\lambda}(m) $ and the Chebyshev polynomial operators $f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J}) $. Table II compares this notation with that originally used by
Meckler \cite{meckler:majorana,meckler:angular}, who was the first to introduce Chebyshev polynomials of a discrete variable in physics applications, and that used by Corio \cite{corio:siam,corio:ortho}.
Aside from the Racah polynomials
\cite{Nikiforov}, which are equivalent to the Racah angular momentum coupling coefficients \cite{varshal1:ang} or the
Wigner 6$j$-symbols \cite{varshal1:ang}, the Chebyshev polynomials $ f_{\lambda}^{(j)} (m) $ are distinguished as the only special functions which are equivalent to an angular momentum coupling coefficient. The Chebyshev polynomials $ f_{\lambda}^{(j)} (m) $ are, to within a phase-factor, equivalent \cite{meckler:majorana, meckler:angular,NormRay} to
Clebsch-Gordan coupling coefficients \cite{varshal1:ang} $C_{jmj-m}^{\lambda \, 0}$
\begin{equation}
f_{\lambda}^{(j)} (m) = (-1)^{j-m} \; C_{jmj-m}^{\lambda \, 0} \label{chebyclebsch}
\end{equation}
This equivalence is neither obvious nor anticipated.
The first statement of this striking and surprising equivalence between a special function, the Chebyshev polynomials of a discrete variable
$ f_{\lambda}^{(j)} (m) $ \cite{bateman,nikiforov2,Nikiforov,vilenkin:specfuncbook}, and an angular momentum coupling coefficient, the Clebsch-Gordan coefficient $ C_{jmj-m}^{\lambda \, 0}$, appears to have made by Meckler \cite{meckler:majorana} in 1958, which he followed a year later
with a proof \cite{meckler:angular}. Quite independently, in 1958, Gelfand et al. \cite{gelfanda, gelfand} noticed an analogy between Clebsch-Gordan coefficients and the Jacobi polynomial special functions. In retrospect, this analogy is not a surprise given the relationships established since between the Hahn and Jacobi polynomial special functions \cite{Koornwinder}, and the fact the Chebyshev polynomials of a discrete real variable are a special case of the Hahn polynomials \cite{bateman,nikiforov2, Nikiforov, vilenkin:specfuncbook}. As a result of the observations made by Meckler \cite{meckler:majorana, meckler:angular} and Gelfand et al. \cite{gelfanda, gelfand}, and later work on the connections between Racah polynomials and Wigner 6$j$-symbols by Askey and Wilson
\cite{AskWil,Wil}, in effect, Clebsch-Gordan coefficients and Wigner 6$j$-symbols could be recognized in the theory of special functions as discrete analogs of Jacobi polynomials. Meckler's recognition \cite{meckler:majorana} and elegant proof of the Chebyshev polynomial duality \cite{meckler:angular} was essentially forgotten until Normand and Raynal \cite{NormRay} rediscovered and proved the same equivalence 25 years later, unaware of Meckler's pioneering work \cite{meckler:majorana, meckler:angular}.
\subsection{Chebyshev Polynomial Scalars: $f_{\lambda}^{(j)}(m)$}
\subsubsection{Chebyshev Polynomials defined as Special Function Solutions of a Difference Equation}
The differential equation of the form
\begin{equation}
\sigma(x) y^{\prime \prime} + \tau(x) y^{\prime} + \lambda y =0 \label{hypo}
\end{equation}
where $\sigma(x)$ is a polynomial of degree 2, $\tau(x)$ is a polynomial of degree 1, and $\lambda$ is a constant is called a hypergeometric type differential equation, whose solutions are called hypergeometric functions. If $y_m(x)$ and $y_n(x)$ are eigensolutions of this equation, with eigenvalues $\lambda_m$ and $\lambda_n$, respectively, then orthogonality of these solutions on the interval $(a,b)$ with respect to a weight function $w(x)$ can be defined as
\cite{leites}
\begin{equation}
\int_a^b y_m(x)\, y_n(x) \,w(x) \, dx =0 \;\;\;\;\;\;\;\; (n \neq m)
\end{equation}
The role of the differentiation operator $d/dx$ in the case of classical orthogonal polynomials is played by $\Delta$ (the forward-difference operator) and by $\nabla$ (the backward-difference operator) in the case of the classical orthogonal polynomials of a discrete variable. These operators are defined as
\cite{olver}
\begin{eqnarray}
\Delta \! \left[ f(x)\right] & = & f(x+1)-f(x) \\
\nabla \! \left[ f(x)\right] & = & f(x) -f(x-1)
\end{eqnarray}
The difference equation which approximates Eq.(\ref{hypo}) on the uniform lattice is \cite{leites}
\begin{eqnarray}
\Delta [\sigma(x) w(x) \nabla y] + \lambda w(x) y & = & 0 \label{diffeq} \\
\mbox{where} \;\;\;\;\; \Delta [\sigma(x) w(x)] & = & \tau(x) w(x)
\end{eqnarray}
Orthogonality on the uniform lattice is defined as \cite{leites}
\begin{equation}
\sum_{x_i=a}^{b-1} y_m(x_i)\, y_n(x_i) \,w(x_i) =0 \label{ortholattice}
\end{equation}
Hahn polynomials $h_n^{(\alpha,\beta)}(x,N)$, along with Meixner, Krawtchouk, and Charlier polynomials, belong to the classical orthogonal polynomials of a discrete variable \cite{bateman,nikiforov2,Nikiforov,vilenkin:specfuncbook,olver}, or more aptly, to polynomials orthogonal on a discrete set of points \cite{nikiforov2}. A figure illustrating the relationships between these polynomials in the Askey scheme \cite{AskeyWilson} can be found in Olver et al. \cite{olver}.
The Hahn polynomials $h_n^{(\alpha,\beta)}(x,N)$ are polynomial solutions of the difference equation (\ref{diffeq}) defined by
\begin{eqnarray}
\alpha,\beta & > & -1 \nonumber \\
\sigma(x) & = & x \, (N+\alpha -x) \nonumber \\
w(x) & = & \displaystyle\frac{\Gamma(N+\alpha-x) \; \Gamma(\beta +1+x)}{\Gamma(x+1) \; \Gamma(N-x)} \nonumber \\
(a,b) & = & (0,N) \nonumber \\
\tau(x) & = & (\beta +1)(N-1) -(\alpha+ \beta +2) \, x
\end{eqnarray}
A special case of the Hahn polynomials are the Chebyshev polynomials of a discrete variable
$t_n(x,N) \equiv h_n^{(0,0)}(x,N)$ \cite{bateman,nikiforov2,Nikiforov,vilenkin:specfuncbook,olver}, defined as
\begin{eqnarray}
\alpha \;\; = \;\; \beta & = & 0 \nonumber \\
\sigma(x) & = & x(N -x) \nonumber \\
w(x) & = & 1 \nonumber \\
(a,b) & = & (0,N) \nonumber \\
\tau(x) & = & N-1-2x
\end{eqnarray}
In this article, we shall discuss a normalized version of the Chebyshev polynomials $t_n(x,N)$ defined in terms of $L, j$ and $m \in [-j,j]$ as
\begin{eqnarray}
f_L^{(j)} (m) & = & F(L,j) \;t_L(j+m,2j+1) \label{normdef} \\
\mbox{where}\;\;\;\; n & = & L \nonumber \\
x & = & j+m \equiv \langle jm|\, j \mathds{1} + J_z \, |jm \rangle\;\;\;\;\;\;\;\;(m \in [-j,j], \mbox{so}\;\; x \in [0,N-1]) \nonumber \\
N & = & 2j+1
\end{eqnarray}
The normalization function $F(L,j)$ and the Bateman project definition \cite{bateman} of the Chebyshev polynomials $t_L(j+m,2j+1)$ which we use in
Eq.(\ref{normdef}) to define the normalized version of the Chebyshev polynomials $ f_L^{(j)} (m)$ which are the subject of this article, are defined in Table III.
\subsubsection{Parity Properties}
In common with the Legendre polynomials $P_L(x)$, the parity of the Chebyshev polynomials $ f_L^{(j)}(m)$ is determined by their degree $L$:
\begin{equation}
f_L^{(j)}(-m) = (-1)^L \, f_L^{(j)}(m) \label{parity}
\end{equation}
Symmetry properties of the Clebsch-Gordan coefficients \cite{varshal1:ang} lead to a simple proof of this parity relation:
\begin{eqnarray}
f_L^{(j)}(-m) & = & (-1)^{-m-j} \,\boxed{ C^{L0}_{j-mjm}} \label{parity1}\\
& = & (-1)^{-m-j} \, (-1)^{2j} \, \boxed{ (-1)^{-L}}\, C^{L0}_{jmj-m} \label{parity2} \\
& = & (-1)^{L}\,\boxed{(-1)^{j-m}} \, C^{L0}_{jmj-m} \label{parity3} \\
& = & (-1)^{L}\, (-1)^{m-j} \, C^{L0}_{jmj-m} \\
& = & (-1)^L \, f_L^{(j)}(m)
\end{eqnarray}
The Clebsch-Gordan coefficient in the ``boxed" term of Eq.(\ref{parity1}) has been rewritten in Eq.(\ref{parity2}) using the following symmetry
property \cite{varshal1:ang}
\begin{equation}
C^{c\gamma}_{a\alpha b\beta} = (-1)^{a+b-c} \; C^{c\gamma}_{b\beta a\alpha }
\end{equation}
Exploiting the fact that both $L$ and $j-m$ are integers, the ``boxed" terms of Eqs.(\ref{parity2}) and (\ref{parity3}) have been rewritten as
\begin{eqnarray}
(-1)^{-L} & = & (-1)^{L} \\
(-1)^{j-m} & = & (-1)^{m-j}
\end{eqnarray}
\subsubsection{Generating the Chebyshev Polynomials $ f_L^{(j)}(m)$}
In order to generate the Chebyshev polynomials, there are four possible approaches, which we now summarize.
\paragraph{Bateman's definition} Eq.(\ref{normdef}) of Section {\bf 2.1.1} defines the Chebyshev polynomials $ f_L^{(j)}(m)$ in terms of
Bateman's Chebyshev polynomials \cite{bateman} $t_L(j+m,2j+1) $. These polynomials can be evaluated using the forward-difference operator definition given in Table III.
\paragraph{Clebsch-Gordan coefficients} The equivalence of Eq.(\ref{chebyclebsch}) offers the opportunity to prove properties of the Chebyshev polynomials $ f_{\lambda}^{(j)} (m) $ using well-known properties of Clebsch-Gordan coefficients \cite{varshal1:ang}, an opportunity we will frequently take advantage of. On the other hand, it also provides a very direct method of generating the Chebyshev polynomials $ f_{\lambda}^{(j)} (m) $ using representations of the Clebsch-Gordan coefficients
$C^{c\gamma}_{a\alpha b \beta}$ in the form of algebraic sums \cite{varshal1:ang}. As an example, the following
Clebsch-Gordan coefficient representation due to Wigner \cite{varshal1:ang, wignerrep}
\begin{eqnarray}
C^{c\gamma}_{a\alpha b \beta} & = & \delta_{\gamma, \alpha + \beta} \; \Delta(abc)
\left[ \frac{(c+\gamma)! (c-\gamma)! (2c+1)}{(a+\alpha)!(a-\alpha)!(b+\beta)!(b-\beta)!} \right]^{\!1/2} \nonumber \\
& & \times \sum_z \frac{(-1)^{b+\beta+z} (c+b+ \alpha -z)!(a-\alpha +z)!}{z!(c-a+b-z)!(c+\gamma-z)!(a-b-\gamma +z)!} \label{wigrep1}\\
\mbox{where} \;\;\;\; \Delta(abc) & = & \left[\frac{(a+b-c)! (a-b+c)! (-a+b+c)!}{(a+b+c+1)!} \right]^{\!1/2} \label{wigrep2}
\end{eqnarray}
can be used to generate the Chebyshev polynomial $ f_{2}^{(1)} (m) $. In this case, the fixed parameters in Eq.(\ref{wigrep1}) take the values
\begin{eqnarray}
a & = & b = 1 \\
c & = & 2 \\
\alpha & = & -\beta = m \\
\gamma & = & 0 \\
\Delta(abc) & = & 1/\sqrt{30}
\end{eqnarray}
and since the summation index $z$ in Eq.(\ref{wigrep1}) assumes integer values for which all the factorial arguments are non-negative, $z$ can only assume the values 0, 1, and 2. Using the Clebsch-Gordan coefficient representation of Eq.(\ref{wigrep1}) for $C^{20}_{1m1-m}$, we easily find the following algebraic sum for $f_{2}^{(1)} (m) $
\begin{eqnarray}
f_{2}^{(1)} (m) & = & \frac{(-1)^{2(1-m)}}{2\sqrt{6}} \left[ \frac{(m+3)!(1-m)! -4(m+2)!(2-m)! +(m+1)!(3-m)!}{(m+1)!(1-m)!} \right] \;\;\; \;\;\; \\
& = & \frac{1}{\sqrt{6}} \left[ 3m^2-2 \right]
\end{eqnarray}
in agreement with the polynomial form given in Table I.
\paragraph{Recursion relation} The right column of Table III states the Chebyshev polynomial recursion relation which can be used to generate these polynomials. It also compares the Chebyshev polynomial recursion relation \cite{meckler:majorana, meckler:angular, corio:siam,filippov2:thesis, corio:ortho} with the equivalent Clebsch-Gordan coefficient
recursion relation \cite{varshal1:ang}.
\paragraph{Legendre polynomial operators} As described in Appendix A, the Chebyshev polynomials $ f_{\lambda} ^{(j)} (m) $ can be calculated from the diagonal matrix elements of the Legendre
polynomial operators $ \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $ \cite{zemach} or $P_{\lambda}({\bf J})$ \cite{schwinger:majorana} according to the following relations:
\begin{eqnarray}
f_{\lambda} ^{(j)} (m) & = &\sqrt{ \frac{2\lambda+1}{2j+1} } \left[ \left[ {\bf J}^2 \right]^{\!l} \right]^{\!-1/2} \!
\langle jm| \, \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \, |jm \rangle \\
f_{\lambda} ^{(j)} (m) & = &\sqrt{ \frac{2\lambda+1}{2j+1} } \; \langle jm| \, P_{\lambda}({\bf J})\, |jm \rangle
\end{eqnarray}
\subsection{Chebyshev Polynomial Operators: $f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J}) $}
In 1975, Corio \cite{corio:siam} presented a method for expanding an arbitrary function of a component of angular momentum
$g({\bf \hat{n}} \cdot {\bf J)}$ in terms of the orthonormal Chebyshev polynomial operators $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$. By replacing
$m$ with $({\bf \hat{n}} \cdot {\bf J})$, Corio \cite{corio:siam} remarked that the recursion relation for the Chebyshev polynomial scalars $f_{\lambda}^{(j)}(m)$ (see Table III), together with an initial operator $f_{0}^{(j)}({\bf \hat{n}} \cdot {\bf J})$, could be used to compile a table of the
Chebyshev polynomial operators $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$.
I have generated all the elements of Table I using this procedure, starting with
\begin{equation}
f_0^{(j)}({\bf \hat{n}} \cdot {\bf J}) = \displaystyle\frac{ \mathds{1} }{\sqrt{2j+1} }
\end{equation}
All of the operator elements $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$ in Table I agree with the equivalent operator elements
$f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J}) $ determined by Filippov and Man'ko \cite{filippov4, filippov2:thesis}.
Table IV compares the traces, matrix elements, and Hermitian conjugates of the spin polarization operators $\hat{T}_{\lambda \mu}^{(j)} $ with those of the spin polarization operator expansion \cite{filippov2:thesis} of the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})= \sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; \,
\hat{T}^{(j)}_{\lambda \mu}$. In the next three sections, we briefly discuss the traces, matrix elements and Hermitian conjugates of the Chebyshev polynomial operators $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$. Table IV also compares a spin polarization operator $\hat{T}_{\lambda \mu}^{(j)} $ expansion of the density operator $\hat \rho$ with a Chebyshev polynomial operator $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$ expansion. The latter expansion, an example of tomographic reconstruction of the density operator $\hat \rho$
\cite{filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo, klimovchumakov, ariano}, will be discussed in Section {\bf 5.3}.
\subsubsection{Traces}
In this section, we state and prove the following traces for the Chebyshev polynomial operators $f^{(j)}_{{\lambda}^{\prime}}({\bf \hat{n}} \cdot {\bf J})$:
\begin{eqnarray}
\mbox{Tr} \! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; f^{(j)}_{{\lambda}^{\prime}}({\bf \hat{n}} \cdot {\bf J}) \right] & = &
\sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; C_{\lambda \mu}({\bf \hat{n}}) = 1 \label{traceone}\\
\mbox{Tr} \! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; f^{(j)} _{{\lambda}^{\prime}}({\bf \hat{n}}^{\prime} \! \cdot {\bf J}) \right]
& = & \sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; C_{\lambda \mu}({\bf \hat{n}}^{\prime})
= P_{\lambda}({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime}) \label{tracetwo}
\end{eqnarray}
The trace result in Eq.(\ref{traceone}), the statement of orthonormality \cite{corio:ortho} for the
Chebyshev polynomial operators $f^{(j)}_{{\lambda}}({\bf \hat{n}} \cdot {\bf J})$, can be verified as follows
\begin{eqnarray}
& & \mbox{Tr} \! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; f^{(j)}_{{\lambda}^{\prime}}({\bf \hat{n}} \cdot {\bf J}) \right] \nonumber \\
& = & \mbox{Tr} \! \left[\; \sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; \hat{T}_{\lambda \mu}^{(j)} \;
\boxed{ \sum_{\mu^{\prime}=-\lambda^{\prime}}^{\lambda^{\prime}} C_{\lambda^{\prime} \mu^{\prime}}^{\star} ({\bf \hat{n}})\;
\hat{T}_{\lambda^{\prime} \mu^{\prime}}^{(j)} }\; \right]
\label{trac1} \\
& = & \mbox{Tr} \! \left[\; \sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; \hat{T}_{\lambda \mu}^{(j)} \;
\boxed{ \sum_{\mu^{\prime}=-\lambda^{\prime}}^{\lambda^{\prime}} C_{\lambda^{\prime} \mu^{\prime}} ({\bf \hat{n}})\;
\left[\hat{T}_{\lambda^{\prime} \mu^{\prime}}^{(j)} \right]^{\dagger} } \;\right]
\label{trac2} \\
& = & \sum_{\mu=-\lambda}^{\lambda} \, \sum_{\mu^{\prime}=-\lambda^{\prime}}^{\lambda^{\prime}}
C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; C_{\lambda^{\prime} \mu^{\prime}} ({\bf \hat{n}})\;
\mbox{Tr} \left[ \, \boxed{ \hat{T}_{\lambda \mu}^{(j)} \left[\hat{T}_{\lambda^{\prime} \mu^{\prime}}^{(j)} \right]^{\dagger} } \; \right] \label{trac3} \\
& = & \sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; C_{\lambda \mu}({\bf \hat{n}}) =
\boxed{\sum_{\mu=-\lambda}^{\lambda} | C_{\lambda \mu}({\bf \hat{n}}) |^2 =1} \label{trac4}
\end{eqnarray}
The ``boxed" term of Eq.(\ref{trac1}) has been replaced with the ``boxed" term of Eq.(\ref{trac2}) using the following
properties \cite{varshal1:ang,brinksatch:ang} of the spin polarization operators $\hat{T}_{\lambda \mu}^{(j)} $ and the Racah spherical harmonics
$C_{\lambda \mu}({\bf \hat{n}})$:
\begin{eqnarray}
\left[\hat{T}_{\lambda \mu}^{(j)} \right]^{\! \dagger} & = & (-1)^{\mu} \; \hat{T}_{\lambda -\mu}^{(j)} \\
C_{\lambda -\mu}({\bf \hat{n}}) & = & (-1)^{\mu} \; C_{\lambda \mu}^{\star}({\bf \hat{n}})
\end{eqnarray}
The double summation of Eq.(\ref{trac3}) has been reduced to a single summation in Eq.(\ref{trac4}) using the following normalization identity for the spin polarization operators \cite{varshal1:ang}
\begin{equation}
\mbox{Tr} \! \left[\; \left[\hat{T}_{\lambda^{\prime} \mu^{\prime}}^{(j)} \right]^{\dagger} \hat{T}_{\lambda \mu}^{(j)} \right] =
\delta_{\lambda \lambda^{\prime}}\; \delta_{\mu \mu^{\prime}}
\end{equation}
The final simplification, the ``boxed" term of Eq.(\ref{trac4}), is just the sum rule \cite{brinksatch:ang,varshal1:ang} for the Racah spherical harmonics.
In verifying the trace result of Eq.(\ref{tracetwo}), nothing would change in the calculation of
$ \mbox{Tr}\! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; f^{(j)}_{{\lambda}^{\prime}}({\bf \hat{n}}^{\prime} \cdot {\bf J})\right]$ except that the final simplication made above in Eq.(\ref{trac4}) would now require use of the spherical harmonics addition theorem \cite{brinksatch:ang,varshal1:ang} since now
${\bf \hat{n}} \neq {\bf \hat{n}}^{\prime}$:
\begin{eqnarray}
\mbox{Tr} \! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; f^{(j)} _{{\lambda}^{\prime}}({\bf \hat{n}}^{\prime} \cdot {\bf J}) \right]
& = & \sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; C_{\lambda \mu}({\bf \hat{n}}^{\prime}) \\
& = & P_{\lambda}({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime}) \label{traceLegendre}
\end{eqnarray}
Meckler \cite{meckler:angular} provided the first proof of this most important trace relation for the Chebyshev polynomial operators $f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $. As we shall see in Section {\bf 4.1.1}, Meckler then took advantage of this relation in his projection operator approach
\cite{meckler:majorana, meckler:angular} to calculate the spin transition probablity $\mbox{P}^{(j)}_{mm^{\prime}}(t)$. In Sections {\bf 5.2} and
{\bf 5.3}, we show how this trace relation can be used to evaluate traces that define delta functions which involve integrations of operators and spin tomograms
on the sphere ${\bf S}^2$.
\subsubsection{Matrix Elements and Orthogonality Relations}
\paragraph{Operator Expansions} In order to calculate matrix elements of the Chebyshev polynomial operators $f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J})$ and $f_{\lambda}^{(j)} (J_z) $, we exploit relations between these operators and the spin
polarization operators \cite{varshal1:ang} $\hat{T}_{\lambda \mu}^{(j)}$. As we discuss in Section {\bf 6.1.2}, the Chebyshev polynomial operators $f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J})$ and $f_{\lambda}^{(j)} (J_z) $ can be expressed in terms of the spin
polarization operators \cite{varshal1:ang} $\hat{T}_{\lambda \mu}^{(j)}$ and the Racah spherical harmonics $C_{\lambda \mu}(\theta, \phi)$ as follows \cite{filippov2:thesis}
\begin{eqnarray}
f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J} ) & = &
\sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}(\theta, \phi) \; \hat{T}_{\lambda \mu}^{(j)} \equiv
\sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}( {\bf \hat{n}}) \; \hat{T}_{\lambda \mu}^{(j)} \label{jn}\\
f_{\lambda}^{(j)}( {\bf \hat{z}} \cdot {\bf J}) \equiv f_{\lambda}^{(j)}(J_z) & = &
\sum_{\mu=-\lambda}^{\lambda} \boxed{ C_{\lambda \mu}^{\star}(0,0)} \; \hat{T}_{\lambda \mu}^{(j)} \label{jz1} \\
& = & \sum_{\mu=-\lambda}^{\lambda} \; \boxed{\delta_{\mu 0}} \; \hat{T}_{\lambda \mu}^{(j)} \label{jz2} \\
& = & \hat{T}_{\lambda 0}^{(j)} \label{jz3}
\end{eqnarray}
In Eq.(\ref{jn}), ${\bf \hat{n}} \equiv (\theta, \phi)$ denotes a quantization axis defined by polar angles $(\theta, \phi)$ with respect to the
${\bf \hat{z}} $-axis. The spherical harmonic in the ``boxed" term of Eq.(\ref{jz1}) has been replaced by the Kronecker delta function in the ``boxed" term of Eq.(\ref{jz2} ) using the properties of the spherical harmonics \cite{varshal1:ang}.
Eq.(\ref{jz3}) shows that the Chebyshev polynomial operators $f_{\lambda}^{(j)} \!(J_z) $ and the projection-zero spin polarization operators \cite{varshal1:ang} $\hat{T}_{\lambda 0}^{(j)}$ are equivalent, a fundamental result which we shall take advantage of in subsequent sections. Although it has been proved in many ways \cite{meckler:angular,filippov2:thesis,corio:ortho,NormRay,werb:tensor,fillipov1:qubit}, it was Meckler \cite{meckler:angular} who first recognized that $f_{\lambda}^{(j)} (J_z) $ was proportional to the projection-zero spin polarization operators \cite{varshal1:ang} $\hat{T}_{\lambda 0}^{(j)}$.
Inverting Eq.(\ref{jn}) to express the spin polarization operators $\hat{T}^{(j)}_{\lambda \mu}$ in terms of the Chebyshev polynomial operators
$f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right)$ is easily achieved using the orthogonality relations \cite{brinksatch:ang} for the Racah spherical harmonics
\begin{equation}
(2L+1) \int_0^{\pi} d\theta \sin \theta \int_0^{2\pi} d\phi \; C_{LM}^{\star}(\theta, \phi) \; C_{L^{\prime} M^{\prime}}(\theta, \phi) =
\delta_{LL^{\prime}} \; \delta_{MM^{\prime}} \, 4 \pi \label{racor}
\end{equation}
with the result that
\begin{eqnarray}
\hat{T}^{(j)}_{\lambda \mu} & = &
\frac{2\lambda +1 }{4\pi} \int_{{\bf S}^2} C_{\lambda \mu}({\bf \hat{n}}) \; f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) \, d{\bf \hat{n}} \label{spintensorcheby} \\
\mbox{where} \;\;\; d{\bf \hat{n}} & \equiv & d\Omega=\sin \theta \, d\theta \, d\phi
\end{eqnarray}
This relation may be viewed as an integral transformation of the Chebyshev polynomial operators
$f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right)$ from the continuous variables $(\theta, \phi) $ (which define
${\bf \hat{n}} \equiv {\bf \hat{n}}(\theta, \phi) $) to the discrete variables $\lambda, \mu$ which define the spin polarization operators
$\hat{T}^{(j)}_{\lambda \mu} $.
Inverting Eq.(\ref{jn}) to express the Racah spherical harmonics $C_{\lambda \mu}({\bf \hat{n}})$ in terms of the Chebyshev polynomial operators
$f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right)$ is easily achieved using the trace relation for the spin polarization operators
$\hat{T}^{(j)}_{\lambda \mu}$ given in Table IV, with the result that
\begin{eqnarray}
C_{\lambda \mu}({\bf \hat{n}}) & = & \mbox{Tr} \! \left[ \hat{T}^{(j)}_{\lambda \mu} \; f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) \right]
\label{raccheb1} \\
& = & \mbox{Tr} \! \left[ \; \boxed{
\frac{2\lambda +1 }{4\pi} \int_{{\bf S}^2} C_{\lambda \mu}({\bf \hat{n}}^{\prime}) \; f_{\lambda}^{(j)} \! \left( {\bf \hat{n}}^{\prime} \cdot {\bf J} \right)
\, d{\bf \hat{n}}^{\prime} }
\; f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) \right] \label{raccheb2} \\
& = & \int_{{\bf S}^2} \frac{2\lambda +1 }{4\pi} \; \mbox{Tr} \! \left[ f_{\lambda}^{(j)} \! \left( {\bf \hat{n}}^{\prime} \cdot {\bf J} \right) \,
f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) \right] \; C_{\lambda \mu}({\bf \hat{n}}^{\prime}) \; d{\bf \hat{n}}^{\prime} \label{raccheb3}\\
& = & \int_{{\bf S}^2} \boxed{ \frac{2\lambda +1 }{4\pi} \; P_{\lambda}({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime}) } \; C_{\lambda \mu}({\bf \hat{n}}^{\prime}) \; d{\bf \hat{n}}^{\prime} \label{raccheb4} \\
& = & \int_{{\bf S}^2} \delta_{C}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime}) \; C_{\lambda \mu}({\bf \hat{n}}^{\prime}) \; d{\bf \hat{n}}^{\prime}
\label{raccheb5} \\
\mbox{where} \;\;\;\; \delta_{C}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime}) & = & \frac{2\lambda +1 }{4\pi} \; P_{\lambda}({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime})
\label{raccheb6}
\end{eqnarray}
The trace relation of Eq.(\ref{raccheb1}) is the inversion result defining the Racah spherical harmonics $C_{\lambda \mu}({\bf \hat{n}})$ in terms of the Chebyshev polynomial operators
$f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right)$. The spin polarization operators $\hat{T}^{(j)}_{\lambda \mu}$ of Eq.(\ref{raccheb1}) have been replaced by the ``boxed" term
of Eq.(\ref{raccheb2}) using Eq.(\ref{spintensorcheby}). The trace in Eq.(\ref{raccheb3}) has been replaced by the ``boxed" term in Eq.(\ref{raccheb4}) using the Chebyshev polynomial operator trace relation of Eq.(\ref{traceLegendre}). In this way, Eqs.(\ref{raccheb2}) to (\ref{raccheb5}) which follow the inversion result of Eq.(\ref{raccheb1}) lead to the definition in Eq.(\ref{raccheb6}) of the reproducing kernel
$\delta_{C}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime}) $, which for the spherical harmonics $C_{\lambda \mu}({\bf \hat{n}})$ acts as a delta function with
respect to integration over ${\bf S}^2$ as shown in Eq.(\ref{raccheb5}). In Section {\bf 5.2.2}, we revisit reproducing kernels in the context of the Stratonovich-Weyl operators \cite{varilly2:moyal,heissweigert, klimov:distr}
$\Delta^{(j)}({\bf \hat{n}}) $, and the Chebyshev polynomial operators $f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right)$.
\paragraph{Matrix Elements} In order to calculate matrix elements, we often follow convention by employing simulaneous eigenkets $|{\bf \hat{z}} ,m \rangle$ of both ${\bf J}^2$ and $J_z \equiv ({\bf J} \cdot {\bf \hat{z}}) $
\begin{eqnarray}
J_z \; |{\bf \hat{z}} ,m \rangle & = & m \, |{\bf \hat{z}} ,m \rangle \nonumber \\
{\bf J}^2 \; |{\bf \hat{z}} ,m \rangle & = & j(j+1) \, |{\bf \hat{z}} ,m \rangle \label{eigenketsz}
\end{eqnarray}
Following Sakurai's \cite{sakurai} notation for labeling these eigenkets $|{\bf \hat{z}} ,m \rangle$, we explicitly indicate the quantization direction
${\bf \hat{z}}$, and include the $J_z$ operator eigenvalue $m$, which for a spin-$j$ system, ranges between $-j$ and $+j$.
For an arbitrary quantization axis ${\bf \hat{n}}$, the generalized version of Eq.(\ref{eigenketsz}) for simulaneous eigenkets $|{\bf \hat{n}} ,m \rangle$ of both ${\bf J}^2$ and $({\bf J} \cdot {\bf \hat{n}}) \equiv J_n$ would be
\begin{eqnarray}
({\bf J} \cdot {\bf \hat{n}}) \; |{\bf \hat{n}} ,m \rangle & = & m \, |{\bf \hat{n}} ,m \rangle \nonumber \\
{\bf J}^2 \; |{\bf \hat{n}} ,m \rangle & = & j(j+1) \, |{\bf \hat{n}} ,m \rangle \label{eigenketsn}
\end{eqnarray}
In most contexts, ${\bf \hat{n}} \equiv {\bf \hat{z}}$, and on these occasions, we will use the shorthand notation $|jm \rangle \equiv |{\bf \hat{z}} ,m \rangle$ to denote simultaneous eigenkets of ${\bf J}^2$ and $J_z$ when calculating matrix elements.
From the expressions for $f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) $ and $ f_{\lambda}^{(j)} (J_z) $ in Eqs.(\ref{jn}) and (\ref{jz3}), respectively, and the matrix elements of the spin
polarization operators in the spherical basis representation \cite{varshal1:ang}
\begin{equation}
\langle jm| \; \hat{T}_{\lambda \mu}^{(j)} \; |jm^{\prime} \rangle =
C_{jmj-m^{\prime}}^{\lambda \, \mu} \; (-1)^{j-m^{\prime}}
\end{equation}
the corresponding matrix elements are easily evaluated as
\begin{eqnarray}
\langle jm| \; f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) \; |jm^{\prime} \rangle & = &
C_{\lambda \, \mu^{\prime}}^{\star}({\bf \hat{n}}) \; C_{jmj-m^{\prime}}^{\lambda \, \mu^{\prime}} \; (-1)^{j-m^{\prime}} =
C_{\lambda \, (m-m^{\prime})}^{\star}({\bf \hat{n}}) \; C_{jmj-m^{\prime}}^{\lambda \, (m-m^{\prime})} \; (-1)^{j-m^{\prime}} \;\;\;\; \;\;\;\; \\
\langle jm| \; f_{\lambda}^{(j)} (J_z) \; |jm \rangle & \equiv & \langle jm| \; \hat{T}^{(j)}_{\lambda 0} \; |jm \rangle =
\; C_{jmj-m}^{\lambda \, 0} \; (-1)^{j-m} = f_{\lambda}^{(j)} (m) \label{cgeqcheb}
\end{eqnarray}
A example of how the relation in Eq.(\ref{cgeqcheb}) can be exploited is the evaluation of the Chebyshev polynomial $ f_L^{(j)} (j) $:
\begin{eqnarray}
f_L^{(j)} (j) & = & (-1)^{j-j} \; C^{L0}_{jjj-j} \label{cgeval4}\\
& = & \left[ \frac{(2L+1) \, [(2j)!]^2}{(2j+L+1)! \, (2j-L)!} \right]^{\!1/2} \label{cgeval44}
\end{eqnarray}
The Clebsch-Gordan coefficient $C^{L0}_{jjj-j} $ in Eq.(\ref{cgeval4}) was evaluated using the relation \cite{varshal1:ang}
\begin{equation}
C^{c\gamma}_{aab\beta}= \delta_{\gamma-\beta,a}
\left[ \frac{(2c+1)(2a)! (-a+b+c)!(b-\beta)!(c+\gamma)!}{(a+b+c+1)!(a-b+c)!(a+b-c)!(b+\beta)!(c-\gamma)!} \right]^{1/2}
\end{equation}
\paragraph{Orthogonality Relations} The relation of Eq.(\ref{cgeqcheb}) defines the Chebyshev polynomials of a discrete variable $ f_{\lambda}^{(j)} (m) $
\cite{meckler:angular,corio:siam, corio:ortho,filippov2:thesis}. For these polynomials, the definition of orthogonality on the uniform lattice given in
Eq.(\ref{ortholattice}) takes the form \cite{meckler:angular,corio:siam, corio:ortho,filippov2:thesis}
\begin{equation}
\displaystyle\sum_{m=-j}^{j} f_{\lambda}^{(j)} (m) \; f_{\lambda^{\prime}}^{(j)} (m)= \delta_{\lambda \lambda^{\prime}} \label{ortho1}
\end{equation}
an identity first stated by Meckler \cite{meckler:angular}.
Bearing in mind the fact that the Chebyshev polynomials $ f_{\lambda}^{(j)} (m) $ are, to within a phase-factor, equivalent to
Clebsch-Gordan coupling coefficients as shown in Eq.(\ref{cgeqcheb}), it is not surprising to find that the Chebyshev polynomial orthogonality relations given in Eq.(\ref{ortho1}) are equivalent to the Clebsch-Gordan coefficient unitarity
relation \cite{varshal1:ang}
\begin{equation}
\displaystyle\sum_{m=-j}^{j} \,
C_{jmj-m}^{\lambda 0} \; C_{jmj-m}^{\lambda^{\prime} 0} = \delta_{\lambda \lambda^{\prime}} \label{ortho2}
\end{equation}
\subsubsection{Hermiticity}
Not only are Chebyshev polynomial operators $f_{\lambda} ^{(j)}( {\bf \hat{n}} \cdot {\bf J})$ orthonormal, but they are also Hermitian \cite{corio:siam}. Taking advantage of the
direct product expression for $f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J})$ discussed in Section {\bf 6.1.2}, it is easy to demonstrate that
$\left[ f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) \right]^{\! \dagger} = f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) $:
\begin{eqnarray}
\left[ f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) \right]^{\! \dagger} & = &
\left[ \; \sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; \hat{T}_{\lambda \mu}^{(j)} \right]^{\! \dagger} \label{her1} \\
& = & \sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}({\bf \hat{n}}) \; \boxed{(-1)^{\mu} \, \hat{T}_{\lambda -\mu}^{(j)} } \label{her2} \\
& = & \sum_{\mu^{\prime}=\lambda}^{-\lambda} C_{\lambda -\mu^{\prime}}({\bf \hat{n}}) \; \boxed{(-1)^{-\mu^{\prime}} } \, \hat{T}_{\lambda \mu^{\prime}}^{(j)} \label{her3} \\
& = & \sum_{\mu^{\prime}=\lambda}^{-\lambda} \boxed{ (-1)^{\mu^{\prime}} \, C_{\lambda -\mu^{\prime}}({\bf \hat{n}}) } \; \hat{T}_{\lambda \mu^{\prime}}^{(j)} \label{her4} \\
& = & \sum_{\mu^{\prime}=\lambda}^{-\lambda} C_{\lambda \mu^{\prime}}^{\star}({\bf \hat{n}}) \; \hat{T}_{\lambda \mu^{\prime}}^{(j)} \label{her5} \\
& = & f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J})
\end{eqnarray}
Using the properties of the spin polarization operators \cite{varshal1:ang}, the ``boxed" term in Eq.(\ref{her2}) replaces the Hermitian conjugate
$\left[ \hat{T}_{\lambda \mu}^{(j)}\right]^{\! \dagger}$ in Eq.(\ref{her1}). A change in the dummy summation index from $ \mu \rightarrow \mu^{\prime}$ has
been used to rewrite Eq.(\ref{her2}) as Eq.(\ref{her3}). Using the properties of the spherical harmonics
\cite{varshal1:ang,brinksatch:ang}, the ``boxed" term of Eq.(\ref{her4}) has been replaced by the ``boxed" term in Eq.(\ref{her5}). Taking advantage of the fact that $\mu$ (or $\mu^{\prime}$) are integral, the ``boxed" term of Eq.(\ref{her3}) can be replaced by $(-1)^{\mu^{\prime}}$ in Eq.(\ref{her4}).
Alternatively, and more directly, given any polynomial $g(x)$, the polynomial operator $g(\hat A)$ is Hermitian if $\hat A$ is Hermitian, so
$f_{\lambda} ^{(j)}( {\bf \hat{n}} \cdot {\bf J})$ is Hermitian since $( {\bf \hat{n}} \cdot {\bf J})$ is Hermitian.
\section{Projection operators and the calculation of transition probabilities}
Fundamental to much of the discussion in this and in subsequent sections are the actions of the rotation operator
$\hat{{\cal D}}^{(j)} \!(\psi, {\bf \hat{n}}) = e^{-i \psi({\bf \hat{n}} \cdot {\bf J}) }$ as a change of basis operator or as a similarity transformation. These two actions are defined in the next section using the angle-axis $(\psi, {\bf \hat{n}})$ parameterization \cite{Louck,siminovitch:eeht,sim:rot}, of which we make frequent but not exclusive use in this and in subsequent sections.
\subsection{Actions of $\hat{{\cal D}}^{(j)}(\psi, {\bf \hat{n}})= e^{-i \psi ({\bf \hat{n}} \cdot {\bf J})}$}
\subsubsection{Rotation operator}
The angle-axis representation of the rotation operator $\hat{{\cal D}}^{(j)} \!(\psi, {\bf \hat{n}}) = e^{-i \psi({\bf \hat{n}} \cdot {\bf J}) }$ is parametrized by $(\psi; \Theta, \Phi)$, where $(\Theta, \Phi)$ are the polar angles of the rotation axis ${\bf \hat{n}}$, and $\psi$ is the rotation angle. The action of this unitary rotation operator on a given eigenket $ |jm \rangle$ is given by \cite{biedenharn,Louck}
\begin{equation}
\hat{{\cal D}}^{(j)}(\psi, {\bf \hat{n}})\; |jm \rangle = e^{-i \psi ({\bf \hat{n}} \cdot {\bf J})} \; |jm \rangle =
\sum_{m^{\prime}=-j}^j{\cal{D}}_{m^{\prime}m }^{(j)}(\psi, {\bf \hat{n}}) \; |jm^{\prime} \rangle \label{basischange}
\end{equation}
where the elements of the matrix ${\cal{D}}_{m^{\prime}m }^{(j)}(\psi, {\bf \hat{n}}) $ are defined as \cite{biedenharn}
\begin{equation}
{\cal{D}}_{m^{\prime}m }^{(j)}(\psi, {\bf \hat{n}}) = \langle jm^{\prime} |\; e^{-i \psi ({\bf \hat{n}} \cdot {\bf J})} \; |jm \rangle
\end{equation}
\subsubsection{Similarity transforms}
The components of the spin polarization operators $ \hat{T}_{\lambda \mu}^{(j)} $, which are irreducible tensor operators of rank $j$, transform under the similarity action of $\hat{{\cal D}}^{(j)}(\psi, {\bf \hat{n}})$ just as
the eigenket $ |jm \rangle$ does under the action of $\hat{{\cal D}}^{(j)}(\psi, {\bf \hat{n}})$ \cite{biedenharn}. Therefore
the similarity transformation of the spin polarization operators $ \hat{T}_{\lambda \mu}^{(j)} $ corresponding to the unitary change of basis given by
Eq.(\ref{basischange}) is \cite{biedenharn,Louck}
\begin{equation}
\hat{{\cal D}}^{(j)} \!(R) \, \hat{T}_{\lambda \mu}^{(j)} \left [\hat{{\cal D}}^{(j)} \!(R)\right ]^{\!\dagger} =
e^{-i \psi ({\bf \hat{n}} \cdot {\bf J}) }\; \hat{T}_{\lambda \mu}^{(j)} \; e^{i \psi ({\bf \hat{n}} \cdot {\bf J}) } =
\sum_{\nu = -j}^j {\cal{D}}_{\nu \mu}^{(j)}(\psi, {\bf \hat{n}}) \; \, \hat{T}_{\lambda \nu}^{(j)}
\end{equation}
\subsection{Unitary transforms of $J_z$ and of related polynomial operators $g(J_z)$ }
Since the spherical components {\large $\tau$}$_{1 \mu}$ of the angular momentum ${\bf J}$ define a rank-1 irreducible spherical tensor
$ \mbox{\boldmath $\mathcal{T}$}\!_{1}$, then for a rotation $R \equiv R(\theta,{\bf \hat{n}}_{\bot} )$ by an angle $\theta$ about an axis
${\bf \hat{n}}_{\bot} = (-\sin \phi, \cos \phi, 0) $ defined by polar angles $(\Theta, \Phi) = (\frac{\pi}{2}, \phi+\frac{\pi}{2})$, the unitary transform of
$J_z \equiv \mbox{\large $\tau$}_{10}$ using an angle-axis parametrization can be written as \cite{Louck,siminovitch:eeht}
\begin{eqnarray}
J_z^{\prime} & = & \hat{{\cal D}}^{(j)} \!(R) \, J_z \left [\hat{{\cal D}}^{(j)} \!(R)\right ]^{\!\dagger} \label{genunit} \\
& = & \hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, J_z \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger} \\
& = & e^{-i \theta ({\bf \hat{n}}_{\bot} \cdot {\bf J}) }\; \mbox{\large $\tau$}_{10} \; e^{i \theta ({\bf \hat{n}}_{\bot} \cdot {\bf J}) } \\
& = & \sum_{\nu = -1}^1 {\cal{D}}_{\nu 0}^{(1)}(\theta, {\bf \hat{n}}_{\bot}) \; \mbox{\large $\tau$}_{1 \nu} \label{angleaxistrans}
\end{eqnarray}
Most quantum mechanics and angular momentum textbooks use the conventional Euler angle parametrization of the rotation $R \equiv R(\alpha, \beta, \gamma)$ to define irreducible tensor operators by the unitary transformation of Eq.(\ref{genunit}). But in this case, an angle-axis parametrization of the
rotation $R \equiv R(\theta,{\bf \hat{n}}_{\bot} )$ offers the most direct path to evaluating $J_z^{\prime}$ using Eq.(\ref{angleaxistrans}). The required
rotation matrix
elements ${\cal{D}}_{\nu 0}^{(1)}(\theta, {\bf \hat{n}}_{\bot}) $ are tabulated in Varshalovich et al. \cite{varshal1:ang}, and have the following values
(when $\cos \Theta = 0$ and $\sin \Theta = 1$)
\begin{eqnarray}
{\cal{D}}_{-1 0}^{(1)}(\theta, {\bf \hat{n}}_{\bot}) & = & -\frac{i}{\sqrt{2}} \sin \theta \; e^{i(\phi + \frac{\pi}{2})} \nonumber \\
{\cal{D}}_{0 0}^{(1)}(\theta, {\bf \hat{n}}_{\bot}) & = & \cos \theta \nonumber \\
{\cal{D}}_{1 0}^{(1)}(\theta, {\bf \hat{n}}_{\bot}) & = & -\frac{i}{\sqrt{2}} \sin \theta \; e^{-i(\phi + \frac{\pi}{2})} \label{dd1}
\end{eqnarray}
while the spherical components {\large $\tau$}$_{1 \mu}$ required are given by \cite{brinksatch:ang}
\begin{eqnarray}
\mbox{\large $\tau$}_{1 -1} & = & \frac{(J_x-iJ_y)}{\sqrt{2}} \nonumber \\
\mbox{\large $\tau$}_{1 0} & = & J_z \nonumber \\
\mbox{\large $\tau$}_{1 +1} & = & - \frac{(J_x+iJ_y)}{\sqrt{2}} \label{spherj}
\end{eqnarray}
Using Eqs.(\ref{dd1}) and (\ref{spherj}), the unitary transform of Eq.(\ref{angleaxistrans}) is then evaluated as
\begin{eqnarray}
J_z^{\prime} & = & \hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, J_z \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger} \\
& = & \sum_{\nu = -1}^1 {\cal{D}}_{\nu 0}^{(1)}(\theta, {\bf \hat{n}}_{\bot}) \; \mbox{\large $\tau$}_{1 \nu} \nonumber \\
& = & (\cos \phi \, \sin \theta ) J_x + (\sin \phi \, \sin \theta)J_y + (\cos \theta) J_z \nonumber \\
& = & \left( {\bf \hat{n}} \cdot {\bf J} \right) \label{simtrans}
\end{eqnarray}
where ${\bf \hat{n}} = (\cos \phi \, \sin \theta, \sin \phi \, \sin \theta,\cos \theta)$ is a unit vector defined by polar angles $(\theta,\phi)$. The unit vector
${\bf \hat{n}}_{\bot} =(-\sin \phi, \cos \phi, 0)$ defines a rotation axis perpendicular to the plane defined by
${\bf \hat{z}}$ and ${\bf \hat{n}}$, so that a rotation about this axis by the angle $\theta$ will transform the
${\bf \hat{z}}$ vector into the ${\bf \hat{n}}$ vector, just as the similarity transform of Eq.(\ref{simtrans}) transforms
$\left( {\bf \hat{z}} \cdot {\bf J} \right) \equiv J_z$ into $\left( {\bf \hat{n}} \cdot {\bf J} \right)$.
Since
\begin{equation}
({\bf \hat{n}} \cdot {\bf J}) = \hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, J_z \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger}
\label{JnJz}
\end{equation}
then for {\it any} polynomial function $g \equiv g(J_z)$,
\begin{equation}
g[\left( {\bf \hat{n}} \cdot {\bf J}\right)] =
\hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, g(J_z) \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger} \label{polytransform}
\end{equation}
and in particular, the Chebyshev polynomial operator basis functions $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$ are given by \cite{filippov2:thesis}
\begin{eqnarray}
f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J}) \
& = & \hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, f_{\lambda}^{(j)}(J_z) \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger}
\label{chebytranss} \\
& = & \hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, \hat{T}_{\lambda 0}^{(j)} \; \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger}
\label{basistransform}
\end{eqnarray}
The results of Eqs.(\ref{JnJz}) and (\ref{chebytranss}) are summarized in Table V. Whereas Eq.(\ref{chebytranss}) is just a particular example of the relation of Eq.(\ref{polytransform}) in the case
of Chebyshev polynomials,
Eq.(\ref{basistransform}) is a significant new relation because it exploits the following equivalence \cite{meckler:angular,filippov2:thesis,corio:ortho,NormRay,werb:tensor,fillipov1:qubit} between the Chebyshev polynomials $f_{\lambda}^{(j)}(J_z)$ and
the projection-zero spin polarization operators $\hat{T}_{\lambda 0}^{(j)} $:
\begin{equation}
f_{\lambda}^{(j)}(J_z) \equiv \hat{T}_{\lambda 0}^{(j)} \label{equivfT}
\end{equation}
In Section {\bf 6.2}, we will take advantage of this equivalence, and the similarity transformation of the spin polarization operators
$ \hat{T}_{\lambda \mu}^{(j)} $ as defined in Section {\bf 3.1.2}, to express the Chebyshev polynomial operators $f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J}) $
as a direct product of spin and spatial tensors.
The simplest way to verify the equivalence of Eq.(\ref{equivfT}) is to use the definition of Fano's state-multipole operators \cite{FanoOp,blum}, alias polarization operators \cite{varshal1:ang} or spherical coherence vectors \cite{oreg}
\begin{eqnarray}
\hat{T}_{\lambda \mu}^{(j)} & = & \sqrt{\frac{2\lambda+1}{2j+1}} \sum_{m,m^{\prime}} C^{jm^{\prime}}_{jm\lambda \mu} \; |jm^{\prime}\rangle \langle jm|
\label{fanoop1} \\
& = & \sum_{m,m^{\prime}} (-1)^{j-m} \; C^{\lambda \mu}_{jm^{\prime}j-m} \; |jm^{\prime}\rangle \langle jm|
\end{eqnarray}
From this definition, we then obtain
\begin{eqnarray}
\hat{T}_{\lambda 0}^{(j)} & = & \sum_{m=-j}^{j} \boxed{(-1)^{j-m} \; C^{\lambda 0}_{jmj-m}} \; |jm\rangle \langle jm| \label{boxedTL0} \\
& = & \sum_{m=-j}^{j} f_{\lambda}^{(j)}(m) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) = f_{\lambda}^{(j)}(J_z) \label{TtoCheb}
\end{eqnarray}
where $\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) = |jm\rangle \langle jm| $ is an example of a projection operator which we discuss in the next section.
Taking advantage of the duality of the Chebyshev polynomials $ f_{\lambda}^{(j)}(m) $, which double as the Clebsch-Gordan angular momentum coupling coefficients (3$j$-symbols \cite{brinksatch:ang}) according to
\begin{equation}
f_{\lambda}^{(j)}(m)= (-1)^{j-m} \; C^{\lambda 0}_{jmj-m}
\end{equation}
the ``boxed" term of Eq.(\ref{boxedTL0}) has been replaced with the Chebyshev polynomial $ f_{\lambda}^{(j)}(m) $ in Eq.(\ref{TtoCheb}), which is just a
statement of Sylvester's formula \cite{merzbacher2,horn}.
\subsection{Projection Operators}
\subsubsection{Chebyshev polynomial operator expansions for projection operators from Sylvester's formula}
Merzbacher \cite{merzbacher2} has discussed the use of matrix methods in quantum mechanics, with a particular emphasis on the use of Sylvester's formula \cite{horn} of Eq.(\ref{TtoCheb}), and the associated projection operator matrices of Eq.(\ref{TtoCheb}).
In general, $|{\bf \hat{n}},m \rangle$ are eigenstates of $({\bf J} \cdot {\bf \hat{n}}) \equiv J_n$ with eigenvalue $m$, and in particular,
$|{\bf \hat{z}},m \rangle \equiv |jm\rangle $ are eigenstates of $({\bf J} \cdot {\bf \hat{z}}) \equiv J_z$ with eigenvalue $m$. Then since
\begin{eqnarray}
|{\bf \hat{n}},m \rangle & = & \hat{{\cal D}} \, |{\bf \hat{z}},m \rangle \\
\langle {\bf \hat{n}},m | & = & \langle {\bf \hat{z}},m | \, \hat{{\cal D}}^{\dagger} \\
\mbox{where}\;\;\;\; \hat{{\cal D}} & \equiv & \hat{{\cal D}}(R) = \hat{{\cal D}}(\theta, {\bf \hat{n}}_{\bot}) = e^{-i \theta ({\bf \hat{n}}_{\bot} \cdot {\bf J}) }
\end{eqnarray}
the unitary transformation of projection operators can be expressed as
\begin{equation}
\hat{{\cal D}} \left[ \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) \right] \hat{{\cal D}}^{\dagger} =
\hat{{\cal D}} \left[ \, |{\bf \hat{z}},m \rangle \langle {\bf \hat{z}},m | \, \right] \hat{{\cal D}}^{\dagger} \equiv
\hat{{\cal D}} |jm \rangle \langle jm | \hat{{\cal D}}^{\dagger} = |{\bf \hat{n}},m \rangle \langle {\bf \hat{n}},m | = \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}})
\label{transprojector}
\end{equation}
Suppose we consider the unitary transformation of Sylvester's formula \cite{horn} for the Chebyshev polynomial operator
$ f_{\lambda}^{(j)}(J_z) $:
\begin{equation}
f_{\lambda}^{(j)}(J_z) = \sum_{m=-j}^{j} f_{\lambda}^{(j)}(m) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) \label{sylvest}
\end{equation}
Using Eq.(\ref{JnJz}), the transform of the left-hand side of Eq.(\ref{sylvest}) is given by
\begin{eqnarray}
\hat{{\cal D}} \left[ f_{\lambda}^{(j)}(J_z) \right] \hat{{\cal D}}^{\dagger} & = & f_{\lambda}^{(j)}(\hat{{\cal D}} \, J_z \, \hat{{\cal D}}^{\dagger} ) \\
& = & f_{\lambda}^{(j)} \left( {\bf \hat{n}} \cdot {\bf J}\right) \label{lhsunit}
\end{eqnarray}
whereas using Eq.(\ref{transprojector}), the transform of the right-hand side of Eq.(\ref{sylvest}) is given by
\begin{eqnarray}
\hat{{\cal D}} \! \left[ \sum_{m=-j}^{j} f_{\lambda}^{(j)}(m) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) \right] \! \hat{{\cal D}}^{\dagger} & = &
\sum_{m=-j}^{j} f_{\lambda}^{(j)}(m) \;
\hat{{\cal D}} \! \left[ \; \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) \; \right] \! \hat{{\cal D}}^{\dagger} \\
& = & \sum_{m=-j}^{j} f_{\lambda}^{(j)}(m) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}})
\end{eqnarray}
In this manner, we obtain as expected the equivalent of Eq.(\ref{sylvest}), namely Sylvester's formula \cite{merzbacher2,horn} for the Chebyshev polynomial operator
$ f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J} )$:
\begin{equation}
f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J} ) = \sum_{m=-j}^{j} f_{\lambda}^{(j)}(m) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}})
\label{sylvestjn}
\end{equation}
By exploiting the Chebyshev polynomial orthogonality relation of Eq.(\ref{ortho1}),
both Eqs.(\ref{sylvest}) and (\ref{sylvestjn}) can be inverted to develop the following Chebyshev polynomial operator expansions for the projection operators:
\begin{eqnarray}
\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) & = & \sum_{\lambda=0}^{2j} f_{\lambda}^{(j)}(m) \; f_{\lambda}^{(j)}(J_z) \equiv
\sum_{\lambda=0}^{2j} f_{\lambda}^{(j)}(m) \; \hat{T}_{\lambda 0}^{(j)} \label{projectjz}\\
\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) & = & \sum_{\lambda=0}^{2j} f_{\lambda}^{(j)}(m) \; f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J}) \equiv
\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda} f_{\lambda}^{(j)}(m) \; C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; \hat{T}_{\lambda \mu}^{(j)} \; \;\;\;\; \;\;\;\;\;\;\;\;
\label{projectJn}
\end{eqnarray}
The novelty (and utility) of these expansions lies in the fact that all the projectors can be expressed in terms of only one special function, namely the Chebyshev polynomials, where the scalars $f_{\lambda}^{(j)}(m)$ are the expansion coefficients and the operators $ f_{\lambda}^{(j)}(J_z)$ or $f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J} ) $ are the expansion basis. For completeness, in Eqs.(\ref{projectjz}) and
(\ref{projectJn}) we also provide the equivalent spin polarization operator expansions.
It is easy to verify that the
$\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) $ operators of Eq.(\ref{projectJn}) are actually projection operators, using the following identity for Clebsch-Gordan coefficients
\cite{varshal1:ang}
\begin{equation}
\sum_{\alpha=-a}^a (-1)^{a-\alpha} \; C_{a\alpha a -\alpha}^{c0} = \sqrt{2a+1}\; \delta_{c0} \label{sumident}
\end{equation}
This relation follows from the orthogonality relation for the Clebsch-Gordan coefficients \cite{varshal1:ang}
\begin{equation}
\sum_{m_1 m_2} C_{j_1m_1j_2m_2}^{jm} \; C_{j_1m_1j_2m_2}^{j^{\prime}m^{\prime}} = \delta_{jj^{\prime}} \, \delta_{mm^{\prime}} \label{orthoga}
\end{equation}
A particular case of the orthogonality relation of Eq.(\ref{orthoga}) is the following relation
\begin{equation}
\sum_m C_{jmj-m}^{0 0} \; C_{jmj-m}^{\lambda 0} = \delta_{\lambda 0} \label{speciala}
\end{equation}
Since the first Clebsch-Gordan coefficient in Eq.(\ref{speciala}) is an example of the following special case \cite{varshal1:ang, brinksatch:ang}
\begin{equation}
C_{a\alpha b \beta}^{00} = (-1)^{a-\alpha} \; \frac{\delta_{ab}\, \delta_{\alpha -\beta}}{\sqrt{2a+1}}
\end{equation}
it can be evaluated as
\begin{equation}
C_{jmj-m}^{0 0} = \frac{(-1)^{j-m}}{ \sqrt{2j+1}}
\end{equation}
Substitution of this value in Eq.(\ref{speciala}) leads to the sum identity of Eq.(\ref{sumident}). Then, taking advantage again of the Chebyshev polynomial duality, the following sum over all Chebyshev polynomials can be written as
\begin{eqnarray}
\sum_{m=-j}^j f_{\lambda}^{(j)}(m) & = & \sum_{m=-j}^j (-1)^{j-m} \; C_{jmj-m}^{\lambda 0} \nonumber \\
& = & \sqrt{2j+1}\; \delta_{\lambda 0} \label{chebsum}
\end{eqnarray}
If {\boldmath $\Pi$}$^{(j)}(m,{\bf \hat{n}})$ are indeed projection operators, then their sum should be the unit operator, which is easily verified using properties of the Chebyshev polynomials as follows:
\begin{eqnarray}
\sum_{m=-j}^{j} \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) & = & \sum_{m=-j}^{j} \left[ \sum_{\lambda=0}^{2j} f_{\lambda} ^{(j)}(m) \;
f_{\lambda} ^{(j)}({\bf \hat{n}} \cdot {\bf J}) \right] \\
& = & \sum_{\lambda=0}^{2j} \boxed{ \sum_{m=-j}^{j} f_{\lambda}^{(j)} (m) }\; f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J}) \label{boxcheb} \\
& = & \sum_{\lambda=0}^{2j} \boxed{ \sqrt{2j+1} \; \delta_{\lambda 0} }\; f_{\lambda} ^{(j)} ({\bf \hat{n}} \cdot {\bf J}) \label{boxchebres} \\
& = & \sqrt{2j+1} \; \; f_0 ^{(j)} ({\bf \hat{n}} \cdot {\bf J}) = \sqrt{2j+1} \; \left[ \frac{\mathds{1}}{\sqrt{2j+1}} \right] = \mathds{1}
\end{eqnarray}
The ``boxed" term in Eq.(\ref{boxcheb}) has been replaced by the ``boxed" term in Eq.(\ref{boxchebres}) using the result of Eq.(\ref{chebsum}).
\subsubsection{Coherent state projectors} In the case of the highest magnetic projection number $m=j$, Eq.(\ref{projectJn}) yields a novel, compact expression for the coherent state projector
$ |{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | $:
\begin{equation}
|{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | = \mbox{{\boldmath $\Pi$}}^{(j)}(j,{\bf \hat{n}}) =
\sum_{\lambda=0}^{2j} f_{\lambda}^{(j)}(j) \; f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J} ) \label{cohereproj}
\end{equation}
The novelty of this expression resides in an operator expansion whose expansion coefficients and expansion operator basis are based exclusively on Chebyshev polynomials of a discrete variable.
Spin coherent states can be viewed as a particular state of a spin system that most closely resembles a classical spin \cite{lohkim}. In this way, the spin eigenstate with maximal $z$-angular momentum is $|{\bf \hat{z}},j \rangle $ is associated with a classical system whose angular momentum points in the
${\bf \hat{z}}$ direction \cite{ducloy}. By the same token, the spin state $|{\bf \hat{n}},j \rangle$, associated with a classical system whose angular momentum points in the
${\bf \hat{n}} \equiv (\theta, \phi)$ direction, can be obtained by rotating $|{\bf \hat{z}},j \rangle $ by an angle $\theta$ about the $y$-axis followed by an angle $\phi$ about the
$z$-axis \cite{ducloy, lohkim}:
\begin{equation}
|{\bf \hat{n}},j \rangle = e^{-i \phi J_z} \, e^{-i \theta J_y} \, |{\bf \hat{z}},j \rangle
\end{equation}
Exploiting the Chebyshev polynomial operator expansion of the coherent state projector in Eq.(\ref{cohereproj}) leads to a very simple proof of the completeness or closure relation for the spin coherent states $|{\bf \hat{n}},j \rangle$
\begin{eqnarray}
\frac{2j+1}{4\pi} \int_{{\bf S}^2} |{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | \, d{\bf \hat{n}} & = &
\frac{2j+1}{4\pi} \int_{{\bf S}^2} |{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | \, d{\bf \hat{n}} \\
& = & \frac{2j+1}{4\pi} \int_0^{\pi} d\theta \sin \theta \int_0^{2\pi} d\phi \; |{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | \\
& = &\mathds{1} \label{cohereclose} \\
\mbox{where} \;\;\; d{\bf \hat{n}} & \equiv & d\Omega=\sin \theta \, d\theta \, d\phi
\end{eqnarray}
The relation of Eq.(\ref{cohereclose}) also provides a resolution of the identity operator within the spin-$j$ Hilbert space \cite{lohkim}.
Because
\begin{equation}
f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J} ) =
\sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}(\theta, \phi) \; \hat{T}_{\lambda \mu}^{(j)} \label{mea2}
\end{equation}
the corresponding solid angle integral relation is given by
\begin{eqnarray}
\int_{{\bf S}^2} f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J} ) \; d{\bf \hat{n}} & = &
\sum_{\mu=-\lambda}^{\lambda} \left[ \int \! \! C_{\lambda \mu}^{\star}(\theta, \phi)\, d\Omega \right] \hat{T}_{\lambda \mu}^{(j)}
\end{eqnarray}
Then, making use of the Chebyshev polynomial operator expansion of the coherent state projector given in Eq.(\ref{cohereproj}), we can obtain the closure relation of Eq.(\ref{cohereclose}) as follows
\begin{eqnarray}
\frac{2j+1}{4\pi} \int_{{\bf S}^2} \; |{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | \, d{\bf \hat{n}} & = &
\frac{2j+1}{4\pi} \int_{{\bf S}^2} \mbox{{\boldmath $\Pi$}}^{(j)}(j,{\bf \hat{n}}) \, d{\bf \hat{n}} \\
& = & \frac{2j+1}{4\pi} \; \sum_{L=0}^{2j} f_{L}^{(j)}(j) \; \boxed{\int_{{\bf S}^2} f_{L}^{(j)}( {\bf \hat{n}} \cdot {\bf J} ) \, d{\bf \hat{n}} } \label{cohderiv1} \\
& = & \frac{2j+1}{4\pi} \; f_{0}^{(j)}(j) \, \int_{{\bf S}^2} f_{0} ^{(j)}( {\bf \hat{n}} \cdot {\bf J} ) \, d{\bf \hat{n}} \label{cohderiv2} \\
& = &\mathds{1}
\end{eqnarray}
In the first step of this derivation, the summation over $L$ in Eq.(\ref{cohderiv1} ) can be restricted to $L=0$ because the integrals of the Chebyshev polynomial operators
$f_{L} ^{(j)}( {\bf \hat{n}} \cdot {\bf J} )$ over all solid angles in the ``boxed" term are given by
\begin{eqnarray}
\int_{{\bf S}^2} f_{L}^{(j)}( {\bf \hat{n}} \cdot {\bf J} ) \, d{\bf \hat{n}} & = &
\sum_{M=-L}^{L} \boxed{ \int_0^{\pi} d\theta \sin \theta \int_0^{2\pi} d\phi \; C_{LM}^{\star}(\theta, \phi) } \; \hat{T}_{LM}^{(j)} \label{solidangle}
\end{eqnarray}
Then using the orthogonality relations \cite{brinksatch:ang} for the Racah spherical harmonics in Eq.(\ref{racor}), and the fact that \cite{brinksatch:ang}
\begin{equation}
C_{00}(\theta, \phi) = 1
\end{equation}
the integral of the Racah spherical harmonics function $ C_{LM}^{\star}(\theta, \phi)$ over all solid angles in the ``boxed" term of Eq.(\ref{solidangle}) is easily evaluated as
\begin{equation}
\int C_{LM}^{\star}(\theta, \phi) \; d\Omega= \int C_{LM}^{\star}(\theta, \phi) \; C_{00}(\theta, \phi) \; d\Omega = 4\pi \, \delta _{L0}
\end{equation}
leading to the simplification of the ``boxed" term in Eq.(\ref{cohderiv1}))
\begin{equation}
\int_{{\bf S}^2} f_{L}^{(j)}( {\bf J} \cdot {\bf \hat{n}} ) \, d{\bf \hat{n}} = \int_{{\bf S}^2} f_{0}^{(j)}( {\bf J} \cdot {\bf \hat{n}} ) \, d{\bf \hat{n}}
\end{equation}
In the second step, Eq.(\ref{cohderiv2}) can be simplified using the following properties of the Chebyshev polynomials $f_0^{(j)} (j) $ and Chebyshev polynomial operators $f_0^{(j)} ({\bf \hat{n}} \cdot {\bf J}) $
\begin{eqnarray}
f_0 ^{(j)} (j) & = & \frac{1}{\sqrt{2j+1}} \\
f_0 ^{(j)} ({\bf \hat{n}} \cdot {\bf J}) & = & \frac{\mathds{1}}{\sqrt{2j+1}}
\end{eqnarray}
Using Eq.(\ref{cohereproj}), the representation of the spin polarization operators
$ \hat{T}^{(j)}_{\lambda \mu} $ in Eq.(\ref{spintensorcheby}) as the following decomposition on the Chebyshev polynomial operators
$f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right)$
\begin{equation}
\hat{T}^{(j)}_{\lambda \mu} =
\frac{2\lambda +1 }{4\pi} \int_{{\bf S}^2} C_{\lambda \mu}({\bf \hat{n}}) \; f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) \, d{\bf \hat{n}}
\end{equation}
may easily be reexpressed as the following decomposition
\cite{agarwal,klimovchumakov} on the coherent state projectors
$ |{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | $
\begin{eqnarray}
\hat{T}^{(j)}_{\lambda \mu} & = & \sqrt{\frac{2j +1 }{4\pi}} \left[C^{jj}_{jj\lambda 0} \right]^{-1} \int_{{\bf S}^2} Y_{\lambda \mu}({\bf \hat{n}}) \;
|{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | \, d{\bf \hat{n}} \label{decomcohere} \\
& = & \frac{2\lambda +1 }{4\pi} \left[ f^{(j)}_{\lambda}(j)\right]^{-1} \int_{{\bf S}^2} C_{\lambda \mu}({\bf \hat{n}}) \;
|{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | \, d{\bf \hat{n}} \label{decomcohere2}
\end{eqnarray}
In Eq.(\ref{decomcohere2}), we have rewritten the decomposition \cite{agarwal,klimovchumakov} of Eq.(\ref{decomcohere}) by using
Eq.(\ref{cgeval4}) to replace the
Clebsch-Gordan coefficient $C^{jj}_{jj\lambda 0}$ in Eq.(\ref{decomcohere}) with the Chebyshev polynomial $ f^{(j)}_{\lambda}(j)$ in
Eq.(\ref{decomcohere2}).
\subsection{Using Projection Operators to Calculate Transition Probabilities}
A change of basis is well covered in quantum mechanics texts \cite{frenchtaylor,gottfried,merzbacher}, but rarely so in the context of projection operators, and so in this subsection, we begin by providing a brief summary of the quantum mechanics background behind a trace relation used by Meckler \cite{meckler:majorana,meckler:angular} to calculate the spin transition probability of Eq.(\ref{modtrans}):
\begin{equation}
\mbox{ P}^{(j)}_{mm^{\prime}}(t) = \left| \langle {\bf \hat{b}},m^{\prime} |\,{\bf \hat{a}},m \rangle \right|^{\,2} = \mbox{Tr} \!
\left[ \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{a}}) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{b}})\right] \label{meckka}
\end{equation}
Using projection operators {\boldmath $\Pi$}$^{(j)}(m,{\bf \hat{a}})$ and {\boldmath $\Pi$}$^{(j)}(m^{\prime},{\bf \hat{b}})$, this expression gives
the transition probability $\mbox{ P}^{(j)}_{mm^{\prime}}(t) $ in a spin-$j$ system that a spin, initially quantized along ${\bf \hat{a}}$ with component $m$, will later be quantized along ${\bf \hat{b}}$ with component $m^{\prime}$.
Before we specialize to the case of spin-$j$ systems, whose projection operators are defined by quantization axes, suppose we consider the more general case of a Hermitian operator
${\bf X}$ whose eigenvalues are labeled by the index $i$. Making use of Sylvester's formula \cite{merzbacher2, horn}, this operator can be expressed in terms of projection operators {\boldmath $\Pi$}$(i) $ and the eigenvalues $x_i$ of ${\bf X}$ as the spectral decomposition of
${\bf X}$ \cite{merzbacher}
\begin{eqnarray}
{\bf X} & = & \sum_i x_i \, \mbox{{\boldmath $\Pi$}}(i) \label{sylvester} \\
\sum_i \mbox{{\boldmath $\Pi$}}(i) & = & \mathds{1} \label{sumsyl}
\end{eqnarray}
Then, multiplying Eq.(\ref{sylvester}) by {\boldmath $\Pi$}$(m) $, we find
\begin{eqnarray}
\mbox{{\boldmath $\Pi$}}(m) \, {\bf X}& = & \sum_i x_i \, \boxed{ \mbox{{\boldmath $\Pi$}}(m) \, \mbox{{\boldmath $\Pi$}}(i) } \label{simplify} \\
& = & \sum_i x_i \, \mbox{{\boldmath $\Pi$}}(m) \, \delta_{mi} \\
& = & x_m \, \mbox{{\boldmath $\Pi$}}(m) \label{traceexpr}
\end{eqnarray}
The ``boxed" term of Eq.(\ref{simplify}) has been simplified using the idempotency of projection operators \cite{merzbacher}:
\begin{eqnarray}
\mbox{{\boldmath $\Pi$}}(m) \, \mbox{{\boldmath $\Pi$}}(i) & = & 0 \; \;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (\mbox{if}\;\; m \neq i) \\
& = & [\mbox{{\boldmath $\Pi$}}(m)]^2 = \mbox{{\boldmath $\Pi$}}(m)\;\;\;\;\;\;\;\; (\mbox{if}\;\; m = i)
\end{eqnarray}
Taking the trace of both sides of Eq.(\ref{traceexpr} ), we obtain
\begin{equation}
\mbox{Tr} \, [ \mbox{{\boldmath $\Pi$}}(m) \, {\bf X}] = x_m \, \boxed{\mbox{Tr} \, [ \mbox{{\boldmath $\Pi$}}(m)] } = x_m \label{ftrace}
\end{equation}
In a representation in which {\boldmath $\Pi$}$(m) $ is diagonal, the only non-zero diagonal element is
$\left[\mbox{{\boldmath $\Pi$}}(m) \right]_{mm}=1$, and since the trace is representation-invariant, the ``boxed" term of Eq.(\ref{ftrace}) has been simplified using the fact that $\mbox{Tr} \, [\mbox{{\boldmath $\Pi$}}(m) ] =1$.
In order to specify the basis states we have been using in more detail, let $|{\bf \hat{a}},m \rangle$ refer to basis ket states for a spin-$j$ system quantized
along ${\bf \hat{a}}$ with component $m$, where $m=-j, -j+1, \ldots, +j$. In this notation, Eq.(\ref{ftrace}) for example, is reexpressed as
\begin{equation}
\mbox{Tr} \, [ \mbox{{\boldmath $\Pi$}}^{(j)}(m, {\bf \hat{a}}) \, {\bf X}] = x_m \equiv
\langle {\bf \hat{a}},m|\, {\bf X}\, |{\bf \hat{a}},m \rangle \label{traceproj}
\end{equation}
while the projection operator {\boldmath $\Pi$}$^{(j)}(m, {\bf \hat{a}}) $ is given by
\begin{eqnarray}
\mbox{{\boldmath $\Pi$}}^{(j)}(m, {\bf \hat{a}}) & = & |{\bf \hat{a}},m \rangle \langle {\bf \hat{a}},m| \\
& = & \displaystyle \prod_{\stackrel{\scriptstyle{r=-j}}{\scriptstyle{r \neq m}}}^{\scriptstyle{j}}
\! \left\{\frac{r{\bf I}-({\bf \hat{a}} \cdot {\bf J})}{m-r}\right\} \label{explicit}
\end{eqnarray}
That the form for {\boldmath $\Pi$}$^{(j)}(m, {\bf \hat{a}}) $ given in Eq.(\ref{explicit}) is a projector can be verified by considering the following identity \cite{merzbacher2,merzbacher} for an operator ${\bf A}$ with distinct eigenvalues $a_i$ and
corresponding eigenkets $|A_k \rangle $
\begin{eqnarray}
{\bf P}_{\!j}\, |A_k \rangle & = & \displaystyle \prod_{i \neq j}
\! \left\{\frac{a_i \mathds{1}-{\bf A}}{a_i-a_j}\right\} |A_k \rangle = \delta_{jk}\, |A_k \rangle \\
\mbox{where}\;\;\;\; {\bf A}\,|A_k \rangle & = & a_k\,|A_k \rangle
\end{eqnarray}
The explicit expression given in Eq.(\ref{explicit}) for the projection operators can be used for the spectral decomposition of
Eq.(\ref{sylvester}) if the operator ${\bf X}$ has distinct eigenvalues.
If operator ${\bf X}$, introduced in Eq.(\ref{sylvester}), should represent the projector for a state quantized along ${\bf \hat{b}}$, with component $m^{\prime}$
\begin{equation}
{\bf X} = |{\bf \hat{b}},m^{\prime} \rangle \langle {\bf \hat{b}},m^{\prime}| \equiv \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime}, {\bf \hat{b}})
\end{equation}
then using the result of Eq.(\ref{traceproj}), we find that
\begin{eqnarray}
\mbox{Tr} \, [\mbox{{\boldmath $\Pi$}}^{(j)}(m, {\bf \hat{a}}) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime}, {\bf \hat{b}})] & = &
\langle {\bf \hat{a}},m| \, {\bf \hat{b}},m^{\prime} \rangle \langle {\bf \hat{b}},m^{\prime} |\,{\bf \hat{a}},m \rangle \\
& = & \left| \langle {\bf \hat{b}},m^{\prime} |\,{\bf \hat{a}},m \rangle \right|^{2} \label{mecktrans}
\end{eqnarray}
which is just the transition probability that a spin, initially quantized along ${\bf \hat{a}}$ with component $m$, will later be quantized along ${\bf \hat{b}}$ with component $m^{\prime}$. In the next section, this expression will be used to calculate spin transition probabilities following the elegant method originally described by Meckler. \cite{meckler:majorana,meckler:angular}
\section{Meckler's Formula for Transition Probabilities}
\subsection{Meckler's formula}
Meckler \cite{meckler:angular} cleverly eschewed the canonical form \cite{merzbacher2,bala,Zelevinsky} of the projection operator matrices given in Eq.(\ref{explicit}) in favor of an expansion (see Eq.(\ref{projectJn})) in Chebyshev polynomials $ f_{\lambda} ^{(j)}(m) $ and Chebyshev polynomial operators $ f_{\lambda} ^{(j)} ({\bf \hat{n}} \cdot {\bf J}) $
\begin{equation}
\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) = \displaystyle\sum_{\lambda=0}^{2j} f_\lambda ^{(j)}(m) \; f_\lambda ^{(j)}({\bf \hat{n}} \cdot {\bf J})
\end{equation}
In so doing, Meckler \cite{meckler:angular} avoided what would necessarily have been a very challenging exercise in calculating expectation values for trace calculations. Just how challenging these trace calculations might have been can be gauged by Balasubramanian's calculation \cite{bala} of the
``spin-flip" transition probability $\mbox{ P}^{(j)}_{j,-j}(t) $ using Sylvester's formula. Only in this special case, when the inital and final state magnetic quantum numbers differed by the maximum value of $2j$, could the matrix elements of the canonical projection operators introduced by Sylvester's formula \cite{merzbacher2, horn} be evaluated, and then summed to yield a closed-form expression \cite{bala}.
Meckler's unorthodox approach to calculating the spin transition probability \cite{meckler:majorana,meckler:angular} relied on the use of projection operators expanded in a Chebyshev polynomial operator basis $f_L ^{(j)}({\bf \hat{n}} \cdot {\bf J})$ as described in Section {\bf 3.2.1}. The foundation of Meckler's calculation \cite{meckler:majorana} is an expression which gives
the transition probability that a spin, initially quantized along ${\bf \hat{a}}$ with component $m$, will later be quantized along ${\bf \hat{b}}$ with component $m^{\prime}$. This probability was expressed in Eq.(\ref{mecktrans}) as a trace of projection operators as follows \cite{meckler:majorana}
\begin{equation}
\mbox{ P}^{(j)}_{mm^{\prime}}(t) = \left| \langle {\bf \hat{b}},m^{\prime} |\,{\bf \hat{a}},m \rangle \right|^{\,2} = \mbox{Tr} \!
\left[ \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{a}}) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{b}})\right] \label{meckk}
\end{equation}
As we shall see, Meckler's choice of operator basis \cite{meckler:majorana,meckler:angular} was pivotal since these
Chebyshev polynomial operators $f_L^{(j)}( {\bf \hat{n}} \cdot {\bf J}) $ are endowed with properties (see Eq.(\ref{traceLegendre}) for example) that render the trace calculation in Eq.(\ref{meckk}) trivial. In this section, we shall devote the first two subsections to discussing two proofs of Meckler's formula
\cite{meckler:majorana}, the first of which is due to Meckler \cite{meckler:majorana,meckler:angular}, and the second of which is due to Schwinger \cite{schwinger:majorana}. A discussion of the relationship between these proofs in the third subsection will lead to a novel trace relation for
$\left|{\cal D}_{m m^{\prime}}^{(j)}(R)\right |^2$.
\subsubsection{First proof: using projection operators expanded in terms of the $f_L ^{(j)}({\bf \hat{n}} \cdot {\bf J})$ operators}
Given a spin initially quantized along a unit vector
${\bf \hat{a}}$ with component $m$, the probability that it is quantized along a unit vector ${\bf \hat{b}}$ with component
$m^{\prime}$ at a later time $t$ was calculated by Meckler \cite{meckler:majorana, meckler:angular} to be
\begin{eqnarray}
\mbox{ P}^{(j)}_{mm^{\prime}}(t) & = &
\mbox{Tr} \! \left[ \, \boxed{\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{a}}) } \; \boxed{\mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{b}}) } \, \right ] \label{avoid} \\
& = & \mbox{Tr} \! \left[ \; \boxed{\sum_{\lambda=0}^{2j} f_{\lambda} ^{(j)}(m) \, f_{\lambda} ^{(j)}({\bf \hat{a}} \cdot {\bf J})} \;
\boxed{\sum_{\lambda^{\prime}=0}^{2j} f_{\lambda^{\prime}} ^{(j)}(m^{\prime})f_{\lambda^{\prime}} ^{(j)}({\bf \hat{b}} \cdot {\bf J})} \; \right] \\
& = & \sum_{\lambda,\lambda^{\prime}=0}^{2j} f_{\lambda} ^{(j)}(m) \, f_{\lambda^{\prime}} ^{(j)} (m^{\prime}) \;
\mbox{Tr} \! \left[ f_{\lambda} ^{(j)} ({\bf \hat{a}} \cdot {\bf J}) \, f_{\lambda^{\prime}} ^{(j)}({\bf \hat{b}} \cdot {\bf J}) \right] \\
& = & \sum_{\lambda, \lambda^{\prime}=0}^{2j} f_{\lambda} ^{(j)} (m) \, f_{\lambda^{\prime}} ^{(j)}(m^{\prime}) \;
\delta_{\lambda \lambda^{\prime}} \; P_{\lambda}( {\bf \hat{a}} \cdot {\bf \hat{b}}) \\
& = & \sum_{\lambda=0}^{2j} f_{\lambda} ^{(j)}(m) \, f_{\lambda} ^{(j)} (m^{\prime}) \;
P_{\lambda}( {\bf \hat{a}} \cdot {\bf \hat{b}})
\end{eqnarray}
The ingenuity of Meckler's projection operator approach \cite{meckler:majorana, meckler:angular} to calculating the spin transition probability
$\mbox{ P}^{(j)}_{mm^{\prime}}(t)$ is evident in this calculation. By exploiting the properties of the Chebyshev polynomial operators
$ f_{\lambda} ^{(j)} ({\bf \hat{n}} \cdot {\bf J})$ (see Eq.(\ref{traceLegendre})), Meckler was able to circumvent the trace calculation in Eq.(\ref{avoid}). Even more ingenious was Meckler's choice
\cite{meckler:majorana} for the
quantization axis ${\bf \hat{b}}$, which up to now we have left unspecified. Meckler chose \cite{meckler:majorana} to define ${\bf \hat{b}} \equiv {\bf \hat{b}}(t)$ as a moving
instantaneous axis along which the precessing spin vector maintains its quantization. Table VI compares the relative orientations of Meckler's
instantaneous axis ${\bf \hat{b}}(t)$ \cite{meckler:majorana} in the middle column with the corresponding relative orientations of a precessing magnetic moment ${\bf \hat{m}}(t)$ according to
Abragam \cite{abragamtext} in the right column, and it is evident that $Z=\cos \alpha$, where $\alpha$, the angle
between the initial orientation of the magnetic moment ${\bf \hat{m}}(0)$ along the magnetic field $\mbox{H}_0 \, {\bf \hat{z}}$ and its orientation at a later time $t$ \cite{abragamtext}, is also just the angle between the uniform field $\mbox{H}_0 \, {\bf \hat{z}}$ and Meckler's instantaneous axis ${\bf \hat{b}}(t)$ \cite{meckler:majorana}. By tethering his instantaneous axis to the precessing spin vector, Meckler \cite{meckler:majorana} was able to compare the results of his spin transition probability calculation with that of
Majorana's \cite{emajorana, ramsey}. For the reminder of this article we shall use $\beta \equiv \beta(t)$
(and not $\alpha$) to denote the relative orientation of this instantaneous axis with respect to the magnetic field.
\subsubsection{Second proof (by Schwinger): using the Clebsch-Gordan decomposition of the direct product}
In 1959, in response to Meckler's 1958 paper \cite{meckler:majorana} on the Majorana formula \cite{emajorana}, Schwinger submitted a brief note to The Physical Review which contained an alternative proof of Meckler's version \cite{meckler:majorana} of the Majorana formula \cite{emajorana}. According to
Schwinger \cite{schwinger:majorana}, that note was rejected, but it appeared eighteen years later in full as an Appendix in an article by Schwinger \cite{schwinger:majorana}. Because Schwinger \cite{schwinger:majorana} only provided the outlines of his proof, using notation that is by now quite outdated, in this section, we provide a detailed discussion of Schwinger's proof using modern notation. Schwinger \cite{schwinger:majorana} also relied exclusively on an Euler angle $(\alpha,\beta,\gamma) $ parametrization of rotation matrices, a restriction which is of course not necessary. We demonstrate that by using Schwinger's approach
\cite{schwinger:majorana} with both an angle-axis $(\psi, {\bf \hat{n}}) $ and Euler angle $(\alpha,\beta,\gamma) $ parametrization to derive Meckler's version \cite{meckler:majorana} of the Majorana formula \cite{emajorana}.
Expressed in terms of the rotation matrices, the spin transition probability is given by \cite{abragamtext}
\begin{eqnarray}
\mbox{ P}^{(j)}_{mm^{\prime}}(t) & = &\left|{\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}}) \right|^2 \\
& = &
{\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}})
\boxed{\left[ {\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}}) \right]^{\star}} \label{conjg1} \\
& = & {\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}}) \;
\boxed{{\cal D}_{m^{\prime}m }^{(j)}(-\psi, {\bf \hat{n}})} \label{conjg2}\\
& = & {\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}}) \;
\boxed{(-1)^{m^{\prime}-m} \; {\cal D}_{-m -m^{\prime}}^{(j)}(\psi, {\bf \hat{n}}) } \label{conjg3}
\end{eqnarray}
Well-known properties \cite{varshal1:ang} of the ${\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}})$ matrices have been used to rewrite the complex conjugated matrix element in the ``boxed" term of Eq.(\ref{conjg1}) in Eqs.(\ref{conjg2}) and (\ref{conjg3}).
The Clebsch-Gordan decomposition of the Kronecker (or direct) product ``$\otimes$" is expressed as the reducible sum ``$\oplus$"
\cite{gottfried,brinksatch:ang}
\begin{equation}
{\cal D}^{(j_1)} \otimes {\cal D}^{(j_2)} = \sum_{j=|j_1-j_2|}^{j_1+j_2} \oplus \; {\cal D}^{(j)} \label{CGseries}
\end{equation}
In terms of a Clebsch-Gordan coefficient series, and an angle-axis $R \equiv R(\psi, {\bf \hat{n}}) $ parametrization of the ${\cal D}^{(J)}_{m m^{\prime}}(R)$ matrices, this decomposition takes the explicit form \cite{varshal1:ang}
\begin{equation}
{\cal D}_{M_1N_1}^{(J_1)}(\psi, {\bf \hat{n}}) \; {\cal D}_{M_2N_2}^{(J_2)}(\psi, {\bf \hat{n}}) =
\sum_{J=|J_1-J_2|}^{J_1+J_2}\; \sum_{MN}
C_{J_1M_1J_2M_2}^{JM} \; {\cal D}_{MN}^{(J)}(\psi, {\bf \hat{n}}) \; C_{J_1N_1J_2N_2}^{JN}
\label{angleaxisseries}
\end{equation}
We can now use the Clebsch-Gordan series of Eq.(\ref{angleaxisseries}) to reexpress the relation of
Eq.(\ref{conjg3}) as Meckler's formula \cite{meckler:majorana, meckler:angular} for the transition probability:
\begin{eqnarray}
\mbox{ P}^{(j)}_{mm^{\prime}}(t) & = &
\left|{\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}}) \right|^2 \\
& = & (-1)^{m^{\prime}-m} \;
\boxed{ {\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}}) \;
{\cal D}_{-m -m^{\prime}}^{(j)}(\psi, {\bf \hat{n}})} \label{phase} \\
& = & (-1)^{m^{\prime}-j+j-m} \; \boxed{\sum_{\lambda=0}^{2j} C_{jmj-m}^{\lambda 0} \; C_{jm^{\prime}j-m^{\prime}}^{\lambda 0} \;
{\cal D}_{00}^{(\lambda)}(\psi, {\bf \hat{n}})} \label{phase2}\\
& = & \sum_{\lambda=0}^{2j} \boxed{(-1)^{j-m} \; C_{jmj-m}^{\lambda 0}} \;
\boxed{(-1)^{j-m^{\prime}} \;C_{jm^{\prime}j-m^{\prime}}^{\lambda 0} } \;
{\cal D}_{00}^{(\lambda)}(\psi, {\bf \hat{n}}) \label{boxed}
\end{eqnarray}
The phase factor in Eq.(\ref{phase}) has been rewritten in Eqs.(\ref{phase2}) and (\ref{boxed}) as
\begin{equation}
(-1)^{m^{\prime}-m} = (-1)^{m^{\prime}-j+j-m} =(-1)^{j-m} \;(-1)^{j-m^{\prime}}
\end{equation}
since $(-1)^{m-j} = (-1)^{j-m}$ for all values of $j$ (integral and half-integral). The ``boxed"
${\cal D}^{(j)}$-matrix product term
in the same equation has been rewritten as the ``boxed" term in Eq.(\ref{phase2}) using the Clebsch-Gordan
series of Eq.(\ref{angleaxisseries}).
Each of the ``boxed" terms in Eq.(\ref{boxed}) is a Chebyshev polynomial ($f_{\lambda}^{(j)} (m)$ or
$f_{\lambda}^{(j)} (m^{\prime})$ as defined in Eq.(\ref{chebyclebsch})), and as shown in Appendix B, the rotation matrix element
${\cal D}_{00}^{\lambda}(\psi, {\bf \hat{n}})$ can be written as an $\lambda$-th order Legendre polynomial \cite{varshal1:ang}:
\begin{equation}
{\cal D}_{00}^{(\lambda)}(\psi, {\bf \hat{n}}) = d_{00}^{(\lambda)}(\xi) \equiv P_{\lambda}(\cos \xi) = P_{\lambda}(\cos \beta) \label{legendreangleaxis}
\end{equation}
The transition probability of Eq.(\ref{boxed}) can finally then be rewritten as
Meckler's formula \cite{meckler:angular}, a Fourier-Legendre series, whose expansion coefficients are products of Chebyshev polynomials:
\begin{equation}
\mbox{ P}^{(j)}_{mm^{\prime}} (t) = \left|{\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}})\right|^2 =
\sum_{\lambda=0}^{2j} f_{\lambda} ^{(j)}(m) \; f_{\lambda} ^{(j)}(m^{\prime})\; P_{\lambda}(\cos \beta)
\end{equation}
In the case of an Euler angle
$R \equiv R(\alpha,\beta,\gamma)$ parametrization of the ${\cal D}^{(j)}_{m m^{\prime}}(R)$ matrices, the
Clebsch-Gordan series of Eq.(\ref{CGseries}) takes the same explicit form \cite{varshal1:ang} as that of
Eq.(\ref{angleaxisseries})
\begin{equation}
{\cal D}_{M_1N_1}^{(J_1)}(\alpha,\beta,\gamma) \; {\cal D}_{M_2N_2}^{(J_2)}(\alpha,\beta,\gamma) =
\sum_{J=|J_1-J_2|}^{J_1+J_2}\; \sum_{MN}
C_{J_1M_1J_2M_2}^{JM} \; {\cal D}_{MN}^{(J)}(\alpha,\beta,\gamma) \; C_{J_1N_1J_2N_2}^{JN}
\label{Eulerseries}
\end{equation}
A slight modification of the same argument can be used to arrive at the same result for the transition probability $\mbox{ P}^{(j)}_{mm^{\prime}}(t) $ when the ${\cal D}^{(j)}_{m m^{\prime}}(R)$ matrices are parametrized by Euler angles
$R \equiv R(\alpha,\beta,\gamma)$. In this case, the transition probability is given by \cite{abragamtext, schwinger:majorana}
\begin{eqnarray}
\mbox{ P}^{(j)}_{mm^{\prime}}(t) & = &
\left|{\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma) \right|^2 \\
& = &
{\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma) \;
\boxed{\left[ {\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma) \right]^{\star}} \label{conjg11} \\
& = & {\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma) \;
\boxed{(-1)^{m-m^{\prime}} {\cal D}_{-m -m^{\prime}}^{(j)}(\alpha,\beta,\gamma) } \label{conjg33} \\
\mbox{where} \;\;\;\; {\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma) & = & e^{-im \alpha} \, d^{(j)}_{m m^{\prime}}(\beta) \, e^{-im^{\prime} \gamma}
\end{eqnarray}
As above, the corresponding well-known properties \cite{varshal1:ang} of the
${\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma)$ matrices have been used to rewrite the complex conjugated matrix element in the ``boxed" term of Eq.(\ref{conjg11}) in Eq.(\ref{conjg33}). As Schwinger noted \cite{schwinger:majorana}, the net effect of the time-dependent radiofrequency field is to rotate the angular momentum vector of the magnetic moment through a definite angle, the Euler angle $\beta$. This is the same angle that Meckler used
\cite{meckler:majorana} to keep track of the angle between the uniform static field and his instantaneous axis ${\bf \hat{b}}(t)$.
We can now use the Clebsch-Gordan series of Eq.(\ref{Eulerseries}) to reexpress the relation of
Eq.(\ref{conjg33}) as Meckler's formula \cite{meckler:majorana,meckler:angular} for the transition probability:
\begin{eqnarray}
\mbox{ P}^{(j)}_{mm^{\prime}}(t) & = &
\left|{\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma) \right|^2 \\
& = & (-1)^{m-m^{\prime}} \;
\boxed{ {\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma) \;
{\cal D}_{-m -m^{\prime}}^{(j)}(\alpha,\beta,\gamma) } \label{phase20} \\
& = & (-1)^{m-j+j-m^{\prime}} \; \boxed{ \sum_{\lambda=0}^{2j} C_{jmj-m}^{\lambda 0} \; C_{jm^{\prime}j-m^{\prime}}^{\lambda 0} \;
{\cal D}_{00}^{(\lambda )}(\alpha,\beta,\gamma)} \label{phase22} \\
& = & \sum_{\lambda =0}^{2j} \boxed{(-1)^{j-m} \; C_{jmj-m}^{\lambda 0}} \;
\boxed{(-1)^{j-m^{\prime}} \;C_{jm^{\prime}j-m^{\prime}}^{\lambda 0} } \;
{\cal D}_{00}^{(\lambda)}(\alpha,\beta,\gamma) \label{boxed2}
\end{eqnarray}
The phase factor in Eq.(\ref{phase20}) has been successively rewritten in Eqs.(\ref{phase22}) and (\ref{boxed2}) as
\begin{equation}
(-1)^{m-m^{\prime}} = (-1)^{m-j+j-m^{\prime}} =(-1)^{j-m} \;(-1)^{j-m^{\prime}}
\end{equation}
since $(-1)^{m-j} = (-1)^{j-m}$ for all values of $j$ (integral and half-integral). The ``boxed" term of Eq.(\ref{phase20}) has been reexpressed in Eq.(\ref{phase22}) using the Clebsch-Gordan series of Eq.(\ref{Eulerseries}).
Each of the ``boxed" terms in Eq.(\ref{boxed2}) is a Chebyshev polynomial ($f_{\lambda}^{(j)} (m)$ or
$f_{\lambda}^{(j)} (m^{\prime})$ as defined in Eq.(\ref{chebyclebsch})), and the rotation matrix element
${\cal D}_{00}^{(\lambda)}(\alpha,\beta,\gamma)$ can be written as an $\lambda$-th order Legendre polynomial \cite{varshal1:ang}:
\begin{equation}
{\cal D}_{00}^{(\lambda)}(\alpha,\beta,\gamma) = d_{00}^{(\lambda)}(\beta) \equiv P_{\lambda}(\cos \beta) \label{legendre}
\end{equation}
The transition probability of Eq.(\ref{boxed2}) can finally then be rewritten as Meckler's formula
\cite{meckler:majorana,meckler:angular}, a Fourier-Legendre series whose expansion coefficients are products of Chebyshev polynomials:
\begin{equation}
\mbox{ P}^{(j)}_{mm^{\prime}}(t) = \left|{\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma) \right|^2 =
\sum_{\lambda=0}^{2j} f_{\lambda}^{(j)} (m) \; f_{\lambda}^{(j)} (m^{\prime})\; P_{\lambda}(\cos \beta) \label{majoranaEuler}
\end{equation}
As Schwinger \cite{schwinger1937} first noted (without proof), for a system initially prepared in a state with magnetic quantum number $m$, the sum
of the transition probabilities over all possible final states labelled by $m^{\prime}$ should be unity:
\begin{equation}
\sum_{m^{\prime}=-j}^j \!\!\! \mbox{ P}^{(j)}_{mm^{\prime}}(t) =1
\end{equation}
Proving this relation could not be simpler with the use of the Majorana \cite{emajorana} formula in the version that Meckler
\cite{meckler:majorana, meckler:angular} first derived. By summing Eq.(\ref{majoranaEuler}) over all final states,
we find
\begin{eqnarray}
\sum_{m^{\prime}=-j}^j \! \!\! \mbox{ P}^{(j)}_{mm^{\prime}} (t) & = & \sum_{m^{\prime}=-j}^j
\sum_{\lambda=0}^{2j} f_{\lambda}^{(j)} (m) \; f_{\lambda}^{(j)} (m^{\prime})\; P_{\lambda}(\cos \beta) \\
& = & \sum_{\lambda=0}^{2j} f_{\lambda}^{(j)} (m) \; \boxed{\sum_{m^{\prime}=-j}^j f_{\lambda}^{(j)} (m^{\prime})} \; P_{\lambda}(\cos \beta)
\label{sumsum}
\end{eqnarray}
In order to handle the sum over Chebyshev polynomials in the ``boxed" term of Eq.(\ref{sumsum}), we exploit the Chebyshev polynomial orthogonality relation of Eq.(\ref{ortho1}):
\begin{equation}
\displaystyle\sum_{m=-j}^{j}f_{\lambda}^{(j)} (m) \; f_{\lambda^{\prime}}^{(j)} (m) = \delta_{\lambda \lambda^{\prime}}
\end{equation}
If $\lambda^{\prime}=0$, then we have the following special case of this relation:
\begin{eqnarray}
\displaystyle\sum_{m=-j}^{j}f_{\lambda}^{(j)} (m) \; \boxed{f_{0}^{(j)} (m)} & = & \delta_{\lambda 0} \label{sumanswer1} \\
\displaystyle\sum_{m=-j}^{j}f_{\lambda}^{(j)} (m) & = & \sqrt{2j+1} \; \,\delta_{\lambda 0} \label{sumanswer2}
\end{eqnarray}
The ``boxed" term of Eq.(\ref{sumanswer1}) has been evaluated using the relation
\cite{filippov2:thesis, corio:ortho}
\begin{equation}
f_0^{(j)} (m) = \displaystyle\frac{1}{\sqrt{2j+1}}
\end{equation}
The identity of Eq.(\ref{sumanswer2}) not only evaluates the sum in the ``boxed" term of Eq.(\ref{sumsum}), but it shows that
in the sum over $\lambda$ in Eq.(\ref{sumsum}), only the $\lambda=0$ term contributes. Finally then, the sum over all final states give the following expected result for the total probability:
\begin{eqnarray}
\sum_{m^{\prime}=-j}^j \! \!\! \! \mbox{ P}^{(j)}_{mm^{\prime}}(t) & = & \sqrt{2j+1} \;
f_0^{(j)} (m) \; P_0(\cos \beta) \\
&= & 1
\end{eqnarray}
using the fact that $P_0(\cos \beta) = 1$.
In the context of the Meckler formula \cite{meckler:majorana, meckler:angular}, there are alternatives to the use of Chebyshev polynomials
$f_{\lambda}^{(j)}(m)=\langle jm| \; f_{\lambda}^{(j)} ( J_z) \; |jm \rangle $, and in fact Schwinger \cite{schwinger:majorana} did not choose to express the Clebsch-Gordon coefficients in Eqs.(\ref{boxed}) or (\ref{boxed2}) in terms of Chebyshev polynomials $f_{\lambda}^{(j)}(m)$, but rather in terms of matrix elements of Legendre polynomial operators $P_{\lambda}({\bf J})$ \cite{schwinger:majorana}. These operators are discussed in Appendix A.
\subsection{How are the two proofs related?}
Both methods for calculating $ \mbox{ P}^{(j)}_{mm^{\prime}}(t)$ have a foundation in angular momentum theory, either angular momentum algebra in Meckler's case \cite{meckler:angular}, or angular momenta composition in Schwinger's case \cite{schwinger:majorana}. No matter which method is used, either Meckler's original approach using projection
operators \cite{meckler:majorana, meckler:angular}, or that
adopted by Schwinger \cite{schwinger:majorana}, the final result for the Majorana spin transition probability
\cite{emajorana} is
of course the same
\begin{equation}
\mbox{ P}^{(j)}_{mm^{\prime}}(t)= \left|{\cal D}_{m m^{\prime}}^{(j)}(R) \right|^2 = \mbox{Tr} \!
\left[ \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{a}}) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{b}}) \right]
=
\mbox{Tr} \! \left[ \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{z}}^{\prime}) \right]\, \label{puzzle}
\end{equation}
In this result for $\mbox{ P} ^{(j)}_{mm^{\prime}}(t)$, $R \equiv R(\alpha,\beta,\gamma)$ or $R \equiv R(\psi, {\bf \hat{n}})$ or any other parametrization \cite{sim:rot} of $R$
for that matter, since the first equality
of Eq.(\ref{puzzle}) does not depend on this parametrization as shown in Section {\bf 4.1.2}. On the other hand, as shown in Section {\bf 4.1.1}, each trace is also a valid expression for the transition probability $\mbox{ P}^{(j)}_{mm^{\prime}}(t) $. But how can that be, since Schwinger's approach \cite{schwinger:majorana} discussed in Section {\bf 4.1.2} only uses
${\cal D}_{m m^{\prime}}^{(j)}(R)$ matrix elements in the basis set $|jm \rangle \equiv |{\bf \hat{z}},m \rangle$ (corresponding to a
$ {\bf {\hat z}} $ quantization axis), whereas Meckler's approach \cite{meckler:majorana, meckler:angular} discussed in Section {\bf 4.1.1} uses two distinct basis sets
$|{\bf \hat{z}},m \rangle$ and $|{\bf \hat{z}}^{\prime},m^{\prime} \rangle$ (corresponding to two distinct quantization axes, $ {\bf {\hat a}} \equiv {\bf {\hat z}}$
and $ {\bf {\hat b}} \equiv {\bf {\hat z}}^{\prime}$, respectively)
to define the projection operators {\boldmath $\Pi$}$^{(j)}(m,{\bf \hat{z}}) $ and {\boldmath $\Pi$}$^{(j)}(m^{\prime},{\bf \hat{z}}^{\prime}) $?
There is nothing physical about the quantization axes, since they are just a way of labeling states, and certainly the final result for the transition probability cannot depend on the choice of quantization axes for the initial and final states. In order to demonstrate this independence, a careful consideration of basis set transformations is required. Following Brink and Satchler \cite{brinksatch:ang},
we note that if a set of
axes $(x^{\prime},y^{\prime},z^{\prime})$ is obtained by a rotation $R$ from a set $(x,y,z)$, then the eigenstates
$|{\bf \hat{z}}^{\prime},n \rangle$ of $({\bf J} \cdot {\bf \hat{z}}^{\prime}) \equiv J_{z^{\prime}}$ are determined by rotating the corresponding eigenstates
$|{\bf \hat{z}}, n \rangle$ of $J_{z}$ along with the axes. In this way, the
state $|{\bf \hat{z}}, n \rangle$ is transformed by the rotation operator $\hat{{\cal D}}^{(j)} \! (R)$ as follows
\begin{eqnarray}
|{\bf \hat{z}}^{\prime},n \rangle & = & \hat{{\cal D}}^{(j)} \! (R) \, |{\bf \hat{z}}, n \rangle \label{rot} \\
& = & \boxed{ \sum_{m=-j}^j |{\bf \hat{z}}, m \rangle \langle {\bf \hat{z}}, m | }\, \hat{{\cal D}}^{(j)} \! (R) \, |{\bf \hat{z}}, n \rangle \label{completeness} \\
& = & \sum_{m=-j}^j |{\bf \hat{z}}, m \rangle \, {\cal D}_{mn}^{(j)}(R)
\end{eqnarray}
The ``boxed" term of Eq.(\ref{completeness}) is a representation of the identity operator $\mathds{1}$, using the completeness relation for the
eigenstates $ |{\bf \hat{z}}, m\rangle$:
\begin{equation}
\mathds{1} = \sum_{m=-j}^j |{\bf \hat{z}}, m\rangle \langle {\bf \hat{z}}, m|
\end{equation}
The states $ \langle {\bf \hat{z}}, n | $ conjugate to those rotated in Eq.(\ref{rot}) are transformed by the adjoint (transpose conjugate)
rotation operator $\left[\hat{{\cal D}}^{(j)} \!(R)\right]^{\!\dagger} $ as follows \cite{brinksatch:ang}
\begin{eqnarray}
\langle {\bf \hat{z}}^{\prime},n | = \langle {\bf \hat{z}}, n | \, \left [\hat{{\cal D}}^{(j)} \!(R) \right ]^{\!\dagger}
& = & \sum_{m=-j}^{j} \, \left[ {\cal D}_{mn}^{(j)} \!(R) \right]^{\!\star} \langle {\bf \hat{z}}, m | \label{rotprime} \\
\mbox{where} \;\;\; \left[ {\cal D}_{mn}^{(j)} (R) \right]^{\!\star} & = &
\langle {\bf \hat{z}}, m | \, \hat{{\cal D}}^{(j)} \! (R) \, | {\bf \hat{z}}, n \rangle ^{\star} =
\langle {\bf \hat{z}}, n| \; \left [\hat{{\cal D}}^{(j)}\!(R) \right ]^{\!\dagger} \, |{\bf \hat{z}}, m \rangle
\end{eqnarray}
Then, referring to Eq.(\ref{puzzle})
\begin{eqnarray}
\left|{\cal D}_{m m^{\prime}}^{(j)}(R) \right|^2 & = & {\cal D}_{m m^{\prime}}^{(j)}(R)
\left[ {\cal D}_{m m^{\prime}}^{(j)}(R) \right]^{\!\star} \label{modulus} \\
& = & \langle {\bf \hat{z}}, m | \, \boxed{\hat{{\cal D}}^{(j)}(R) |{\bf \hat{z}}, m^{\prime}\rangle} \;
\boxed{ \langle {\bf \hat{z}}, m^{\prime}| \! \left [\hat{{\cal D}}^{(j)} \!(R)\right ]^{\!\dagger} } \; |{\bf \hat{z}}, m \rangle \label{transf}\\
& = & \langle {\bf \hat{z}}, m | \, \boxed{ |{\bf \hat{z}}^{\prime},m^{\prime}\rangle \;
\! \langle {\bf \hat{z}}^{\prime},m^{\prime}|} \, |{\bf \hat{z}}, m \rangle \label{project} \\
& = & \langle {\bf \hat{z}}, m | \; \mbox{{\boldmath $\Pi$}}^{(j)} (m^{\prime},{\bf \hat{z}}^{\prime}) \; |{\bf \hat{z}}, m \rangle \label{expect} \\
& = & \mbox{Tr} \! \left[ \, \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{z}}^{\prime}) \right] \label{finaltrace}
\end{eqnarray}
We recognize the ``boxed" terms of Eq.(\ref{transf}) as the state transformation equations of
Eqs.(\ref{rot}) and (\ref{rotprime}), and identify the ``boxed" term of Eq.(\ref{project}) as the projection operator
{\boldmath $\Pi$}$^{(j)}(m^{\prime},{\bf \hat{z}}^{\prime}) $:
\begin{equation}
\mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{z}}^{\prime}) = |{\bf \hat{z}}^{\prime},m^{\prime}\rangle \;
\! \langle {\bf \hat{z}}^{\prime},m^{\prime}|
\end{equation}
Finally, Eq.(\ref{finaltrace}) was obtained from Eq.(\ref{expect}) by using the following form of Eq.(\ref{traceproj}) expressed
in the notation of Eqs.(\ref{modulus} - \ref{finaltrace}):
\begin{eqnarray}
\langle {\bf \hat{z}}, m |\, {\bf X}\, | {\bf \hat{z}}, m \rangle & = & \mbox{Tr}\! \left[ \, \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) \; {\bf X} \right] \\
\mbox{where} \;\;\; {\bf X} & = & \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{z}}^{\prime})
\end{eqnarray}
We conclude by stating the result in Eq.(\ref{finaltrace}) in more general terms. For a spin-$j$ system,
whose magnetic quantum numbers are chosen from the set $\Big\{m \Big\}_{\!\!-j}^{\,\,j}$, let us consider two quantization axes, defined by unit vectors
${\bf \hat{z}}$ and $ {\bf \hat{z}}^{\prime}$, where the quantization axis
$ {\bf \hat{z}}^{\prime}$ is obtained from ${\bf \hat{z}}$ by a rotation $R$. Associated with these axes are projection operators
$ \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}})$ and $ \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{z}}^{\prime})$, each of which is
also a function of a magnetic quantum number ($m$ or $m^{\prime}$). Acting on an arbitrary superposition of multiplet states $|{\bf \hat{z}}, n\rangle$,
$ \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) = |{\bf \hat{z}}, m \rangle \; \! \langle {\bf \hat{z}}, m | $
singles out the component with the projection $ ({\bf J} \cdot {\bf \hat{z}}) =m$, whereas
$ \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{z}}^{\prime}) = |{\bf \hat{z}}^{\prime},m^{\prime}\rangle \;
\! \langle {\bf \hat{z}}^{\prime},m^{\prime}|$,
acting on an arbitrary superposition of multiplet states $|{\bf \hat{z}}^{\prime},n\rangle $,
singles out the component with the projection $ ({\bf J} \cdot {\bf \hat{z}}^{\prime}) = m^{\prime}$ \cite{Zelevinsky}.
Then the modulus squared of the Wigner rotation matrix element ${\cal D}_{m m^{\prime}}^{(j)}(R) $ is given by the trace of the product of these projection
operators as follows:
\begin{equation}
\boxed{\left|{\cal D}_{m m^{\prime}}^{(j)}(R)\right |^2 =
\mbox{Tr} \! \left[ \, \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{z}}) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{z}}^{\prime}) \, \right]
= \mbox{ P}^{(j)}_{mm^{\prime}}(t) } \label{traceidid}
\end{equation}
As we noted at the outset, and as we emphasize once again, this result for the transition probability $ \mbox{ P}^{(j)}_{mm^{\prime}}(t) $ does not depend upon the parametrization of $R$. What it does very much depend upon is the relative orientation of the quantization axes axes ${\bf {\hat z}}$
and ${\bf {\hat z}}^{\prime}$ as shown in Section {\bf 4.1}.
\subsection{Applications of the Meckler formula and related expressions}
Although Schwinger \cite{schwinger:majorana} took notice of Meckler's formula \cite{meckler:majorana, meckler:angular}, and Biedenharn and Louck mention it in their discussion of the Majorana formula \cite{emajorana},
Meckler's formula \cite{meckler:majorana} for the Majorana \cite{emajorana} spin transition probability has remained relatively obscure. In this section, some practical applications of the Meckler formula \cite{meckler:majorana} are discussed. These applications include the calculation of the ``spin-flip" probability $\mbox{P}^{(j)}_{j,-j} (t) $, and the elucidation of some properties of the Wigner rotation matrix elements. We also demonstrate the rediscovery of the Meckler formula in a recent solution for the multi-level Landau-Zener transition
probability $\mbox{ P}_{mm^{\prime}}^{\mbox{\tiny{LZ}}}(t) $ by Fai et al. \cite{fai}.
\subsubsection{Spin-flip transition probabilities expressed as a Fourier-Legendre series}
Meckler's formula \cite{meckler:majorana} provides a straightforward answer to the calculation of the ``spin-flip" probability $\mbox{P}^{(j)}_{j,-j} (t) $, the probability for a radiofrequency-induced transition in a spin-$j$ system between the state $| j,j \rangle$ with the highest magnetic projection number, and the state
$|j,-j \rangle$ with the lowest magnetic projection number. Using Meckler's formula \cite{meckler:majorana} for $ \mbox{P}^{(j)}_{mm^{\prime}}(t) $ given in Eq.(\ref{majoranaEuler}),
\begin{eqnarray}
\mbox{ P}^{(j)}_{j,-j} (t) & = &
\sum_{L=0}^{2j} \boxed{ f_L^{(j)} (j) \; f_L^{(j)} (-j) }\; P_L(\cos \beta) \label{spinflip1} \\
& = & \sum_{L=0}^{2j} c(j,L) \, P_L(\cos \beta) \label{legserr} \\
& = & \left[(2j)! \right]^2 \sum_{L=0}^{2j} \frac{(-1)^L \, (2L+1) }{(2j-L)!(2j+L+1)!} \, P_L(\cos \beta) \label{leg} \\
& = & \left[ \frac {1- \cos \beta}{2} \right]^{\!2j} = \left[ \sin (\beta / 2) \right]^{4j} \label{leg3}\\
& = & (\sin \Theta)^{4j} \left( \sin \psi/2 \right) ^{4j} \label{jjth}\\
& = & \left[ \frac{\omega_1}{\omega_e} \right]^{\!4j}
\sin^{4j} \! \left\{ \omega_e t/2 \right\} \label{spinflip} \\
\mbox{where} \;\;\;\; \omega_e & \equiv & \left[\omega_1^2+(\omega_0-\omega)^2 \right]^{\!1/2} \label{rfdef}
\end{eqnarray}
In Eq.(\ref{rfdef}), $\omega_e$ is the effective radiofrequency field strength, defined in terms of the applied radiofrequency field $\omega_1=\gamma H_1$ and the resonance offset $\Delta = \omega_0-\omega$ where $\omega_0 = \gamma H_0$ is the Larmor frequency.
The results given in Eqs.(\ref{jjth}) and (\ref{spinflip}) agree with analogous expressions obtained by Balasubramanian \cite{bala} using Sylvester's formula \cite{merzbacher2,horn}, and by Siemens et al. \cite{sim:beyond} using the Chebyshev polynomial operator expansion of the rotation operator as given in Eq.(\ref{coriorotfirst}) below (see Section {\bf 5.1}) . Each step leading to the expressions for the spin-flip probability given in Eqs.(\ref{jjth}) and (\ref{spinflip}) is now justified.
The first Chebyshev polynomial $f_L^{(j)} (j)$ in the ``boxed" term of Eq.(\ref{spinflip1}) can be evaluated using Eq.(\ref{cgeval44}).
After the second Chebyshev polynomial $f_L^{(j)} (-j)$ in the ``boxed" term of Eq.(\ref{spinflip1}) is evaluated using the parity relation of Eq.(\ref{parity}) (see Section {\bf 2.1.3}),
the coefficients $c(j,L)= f_L^{(j)} (j) \; f_L^{(j)} (-j) $ in the Fourier-Legendre series expansion of the spin-flip transition probability $ \mbox{P}^{(j)}_{j,-j}(t) $ in Eq.(\ref{legserr}) are easily obtained. This expansion can be summed to yield the very simple and closed-form expression of Eq.(\ref{leg3}), as described in Appendix C.
Using the same approach, Meckler's formula \cite{meckler:majorana} also provides a straightforward answer to the calculation of the ``spin-flip" probability $\mbox{P}^{(j)}_{j-1,-(j-1)} (t) $, the probability in a spin-$j$ system for a radiofrequency-induced transition between the state $| j,j-1 \rangle$ with the next to highest magnetic projection number, and the state $|j,-(j-1) \rangle$ with the next to lowest magnetic projection number. Using Meckler's formula \cite{meckler:majorana} for $ \mbox{P}^{(j)}_{mm^{\prime}}(t) $ given in Eq.(\ref{majoranaEuler}),
\begin{eqnarray}
& & \mbox{ P}^{(j)}_{j-1,-(j-1)} (t) \\
& = & \sum_{L=0}^{2j} \boxed{ f_L^{(j)} (j-1) \; f_L^{(j)} (-(j-1)) }\; P_L(\cos \beta) \label{spinflip2} \\
& = & \sum_{L=0}^{2j} c^{\prime}(j,L) \, P_L(\cos \beta) \label{legser} \\
& = & \sum_{L=0}^{2j} \frac{(-1)^L \, [L(L+1)-2j]^2 \; (2L+1) \left[(2j-1)! \right]^2}{(2j-L)!(2j+L+1)!} \, P_L(\cos \beta) \label{legser22} \\
& = & \left[ \frac {1- \cos \beta}{2} \right]^{\!2(j-1)} \left[2j \cos^2 (\beta / 2) -1 \right]^{\!2}= \left[ \sin (\beta / 2) \right]^{4(j-1)}
\left[2j \cos^2 (\beta / 2) -1 \right]^{\!2}
\label{legser23} \\
& = & (\sin \Theta)^{4(j-1)} \left(\sin \psi/2 \right)^{4(j-1)} \left[2j \cos^2 \Theta \, \sin^2 \psi/2 -1 \right]^{\!2} \\
& = & \left[ \frac{\omega_1}{\omega_e} \right]^{\!4(j-1)}
\sin^{4(j-1)} \! \left\{ \omega_e t/2 \right\} \; \left[2j \frac{(\omega_0-\omega)^2 }{\omega_e^2}
\sin^{2} \left\{ \omega_e t/2 \right\} -1 \right]^{\!2} \label{spinflip3}
\end{eqnarray}
The first Chebyshev polynomial $f_L^{(j-1)} (j)$ in the ``boxed" term of Eq.(\ref{spinflip2}) can be evaluated using the relation of Eq.(\ref{cgeqcheb}) between the Chebyshev polynomials $ f_L^{(j)}(m)$ and the Clebsch-Gordan coupling coefficients $ C^{L0}_{jmj-m} $, from which we obtain
\begin{eqnarray}
f_L^{(j)} (j-1) & = & (-1)^{j-(j-1)} \; C^{L0}_{j(j-1)j-(j-1)} \label{cgeval2}\\
& = & -[L(L+1)-2j]\left[ \frac{(2L+1) \, [(2j-1)!]^2}{(2j+L+1)! \, (2j-L)!} \right]^{\!1/2}
\end{eqnarray}
The Clebsch-Gordan coefficient $C^{L0}_{j(j-1)j-(j-1)} $ in Eq.(\ref{cgeval2}) was evaluated using the relation \cite{varshal1:ang}
\begin{eqnarray}
C^{c\gamma}_{aa-1b\beta}& = & \delta_{\gamma-\beta,a-1} \left\{(c-\gamma)(c+\gamma +1)-(b+\beta)(b-\beta +1)\right\} \\
& & \;\;\;\times \; \left[ \frac{(2c+1)(2a-1)! (-a+b+c)!(b-\beta)!(c+\gamma)!}{(a+b+c+1)!(a-b+c)!(a+b-c)!(b+\beta)!(c-\gamma)!} \right]^{1/2}
\end{eqnarray}
After the second Chebyshev polynomial $f_L^{(j)} (-(j-1))$ in the ``boxed" term of Eq.(\ref{spinflip2}) is evaluated the parity relation of Eq.(\ref{parity}) (see Section {\bf 2.1.3}),
the coefficients $c^{\prime}(j,L)= f_L^{(j)} (j) \; f_L^{(j)} (-j) $ in the Fourier-Legendre series expansion of the spin-flip transition probability $ \mbox{P}^{(j)}_{j-1,-(j-1)}(t) $ in Eq.(\ref{legser22}) are easily obtained, and used to write the explicit form of this expansion in Eq.(\ref{legser23}).
\subsubsection{Multi-level Landau-Zener transition probability}
Recently, Tchouobiap et al. \cite{fai} have solved the multi-level Landau-Zener problem to obtain the following exact analytical expression for the transition
probability $\mbox{ P}_{mm^{\prime}}^{\mbox{\tiny{LZ}}}(t) $ between two Zeeman levels of an arbitrary spin $S$:
\begin{eqnarray}
\mbox{ P}_{mm^{\prime}}^{\mbox{\tiny{LZ}}}(t) & = & \sum_{L=0}^{2S} \sqrt{\displaystyle\frac{2L+1}{2S+1}} \left[ \hat{T}_{L0}^{(S)} \right]_{\!mm}
C_{Sm^{\prime}L0}^{Sm^{\prime}} \; \;_2F_1[-L,L+1,1;1-p(t)] \label{LZtransprob}\;\;\;\;\;\;\; \\
\mbox{where} \;\;\;\;
\left[ \hat{T}_{L0}^{(S)} \right]_{\!mm} & \equiv & \langle Sm |\; \hat{T}_{L0}^{(S)} \; |Sm \rangle \\
\hat{T}_{LM}^{(S)} & \equiv & \sqrt{\displaystyle\frac{2L+1}{2S+1}} \sum_{m,m^{\prime}}
C_{SmLM }^{Sm^{\prime}} \; |Sm^{\prime} \rangle \, \langle Sm |
\end{eqnarray}
At first sight, the transition probability expression in Eq.(\ref{LZtransprob}) seems to be quite unrelated in form to the Meckler formula \cite{meckler:majorana}, especially with the appearance of the hypergeometric function \cite{olver} $_2F_1[-L,L+1,1;1-p(t)] \equiv \; _2F_1[a,b,c;d]$,
matrix elements of the spin polarization operators $\hat{T}_{L0}^{(S)} $, and Clebsch-Gordan coefficients $C_{Sm^{\prime}L0}^{Sm^{\prime}} $. However, it is easy to show that the expression of Eq.(\ref{LZtransprob}) for
$\mbox{ P}_{mm^{\prime}}^{\mbox{\tiny{LZ}}}(t) $ is actually identical to Meckler's formula \cite{meckler:majorana}. First, using well-documented \cite{varshal1:ang} symmetry properties of the Clebsch-Gordan coefficients, and
definitions of the hypergeometric functions \cite{arken,olver}, we can rewrite the summands in Eq.(\ref{LZtransprob}) in terms of either Chebyshev polynomials
$f_L^{(S)} (m) $ or Legendre polynomials $P_L(x)$ as follows:
\begin{eqnarray}
\left[ \hat{T}_{L0}^{(S)} \right]_{\!mm} & = & \langle Sm| \; f_L^{(S)} (J_z) \; |Sm \rangle = f_L^{(S)} (m) \label{summand1}\\
\sqrt{\displaystyle\frac{2L+1}{2S+1}} \; C_{Sm^{\prime}L0}^{Sm^{\prime}} & \equiv &
(-1)^{m^{\prime}-S} \; C_{Sm^{\prime}S-m^{\prime}}^{L0} = f_L^{(S)} (m^{\prime}) \\
_2F_1[-L,L+1,1;1-p(t)] & \equiv & P_L[2p(t)-1] \label{summand3}
\end{eqnarray}
In Eq.(\ref{summand3}), $P_L(x)$ is an $L$-th order Legendre polynomial, and the two-level Landau-Zener transition probability \cite{fai} $p(t)$ is given by
\begin{equation}
p(t) \equiv \mbox{ P}_{\frac{1}{2}, -\frac{1}{2} } (t)
\end{equation}
With the results of Eqs.(\ref{summand1}) to (\ref{summand3}) in hand, the multi-level Landau-Zener transition probability formula derived by
Tchouobiap et al. \cite{fai} can now be written in condensed form as
\begin{equation}
\mbox{ P}_{mm^{\prime}}^{\mbox{\tiny{LZ}}} (t) = \sum_{L=0}^{2S} \; \boxed{f_L^{(S)} (m)} \; \boxed{f_L^{(S)} (m^{\prime})} \; P_L[2\, p(t)-1] \label{LZChebya}
\end{equation}
However, given that the two-level, spin-1/2 transition probability from Meckler's formula \cite{meckler:majorana} is
\begin{equation}
\mbox{ P}_{\frac{1}{2}, -\frac{1}{2} } (t) = \frac{1}{2} (1+\cos \beta) \equiv p(t)
\end{equation}
so that
\begin{equation}
\cos \beta = 2\, p(t) -1
\end{equation}
Meckler's formula \cite{meckler:majorana} for the multi-level transition probability can be expressed as
\begin{equation}
\mbox{ P}_{mm^{\prime}} (t) = \sum_{L=0}^{2S} \; \boxed{f_L^{(S)} (m)} \; \boxed{f_L^{(S)} (m^{\prime})} \; \;P_L[2\, p(t)-1] \label{LZCheby}
\end{equation}
which is identical in form to the multi-level Landau-Zener transition probability formula derived by Tchouobiap et al. \cite{fai}.
\subsubsection{Squared Wigner rotation matrix elements}
Varshalovich et al. \cite{varshal1:ang} tabulate (without proof) the following expression for the squares of the ${\cal D}^{(j)}$ matrix elements when
$\beta= \pi/2$:
\begin{equation}
\left[ {\cal D}_{M M^{\prime}}^{(j)}(\alpha,\tfrac{\pi}{2},\gamma) \right]^2 =e^{-i 2M\alpha -i2M^{\prime}\gamma} \;
(-1)^{M-M^{\prime}} \!\! \sum_{L=0,2,4, \dots} \! \!(-1)^{L/2} \; \displaystyle\frac{(L-1)!!}{L!!} \;
C_{JMJ-M}^{L 0} \; C_{JM^{\prime}J-M^{\prime}}^{L0} \label{varshalsquare}
\end{equation}
Proving this expression is trivial using the Majorana formula expressions we have just discussed in Section
{\bf 4.1}. To begin, note that for any complex number $z \equiv |z| \, e^{i\zeta}$,
\begin{equation}
z^2 =\left|z\right|^2 e^{2i\zeta} \label{polar}
\end{equation}
so that in particular,
\begin{equation}
\left[ {\cal D}_{M M^{\prime}}^{(j)}(\alpha,\beta,\gamma) \right]^2 =
\left|{\cal D}_{M M^{\prime}}^{(j)}(\alpha,\beta,\gamma)\right|^2 e^{-i 2(M\alpha + M^{\prime}\gamma)} \label{sqmod}
\end{equation}
where the phase angle of Eq.(\ref{polar}) $\zeta = -(M\alpha +M^{\prime}\gamma)$. To verify the expression for
$ \left[ {\cal D}_{M M^{\prime}}^{(j)}(\alpha,\tfrac{\pi}{2},\gamma) \right]^{\!2}$ given in Eq.(\ref{varshalsquare}), it remains to determine $\left|{\cal D}_{M M^{\prime}}^{(j)}(\alpha,\tfrac{\pi}{2},\gamma)\right|^2$, and this can easily be done from the Majorana expansion of Eq.(\ref{majoranaEuler}) evaluated when $\beta = \tfrac{\pi}{2}$:
\begin{eqnarray}
\left|{\cal D}_{m m^{\prime}}^{(j)}(\alpha,\tfrac{\pi}{2},\gamma) \right|^2 & = &
\sum_{L=0}^{2j} \boxed{f_L^{(j)} (m)}\; \boxed{f_L^{(j)} (m^{\prime})} \; P_L(\cos \tfrac{\pi}{2}) \label{boxedCheby} \\
& = & \sum_{L=0}^{2j} \boxed{(-1)^{j-m} \, C^{L0}_{jmj-m}} \;
\boxed{(-1)^{j-m^{\prime}}\, C^{L0}_{jm^{\prime}j-m^{\prime}}} \; P_L(0) \label{Peval} \\
& = & (-1)^{m-m^{\prime}} \sum_{L=0,2,4, \ldots} (-1)^{L/2} \; \displaystyle\frac{(L-1)!!}{L!!} \;
C^{L0}_{jmj-m} \; C^{L0}_{jm^{\prime}j-m^{\prime}} \label{restrict}
\end{eqnarray}
The parity of the Legendre polynomials $P_L(\cos \beta)$ is even or odd, depending on whether $L$ is even or odd. Those polynomials with odd parity will vanish at the origin, and therefore the only non-vanishing values of the Legendre polynomials $P_L(\cos \beta)$ evaluated at the origin (when
$\cos \beta \equiv \cos ( \pi/2) =0$) are given by \cite{arken}
\begin{equation}
P_{2n}(0) = (-1)^n \; \frac{(2n-1)!!}{(2n)!!}
\end{equation}
This identity restricts the sum in Eq.(\ref{restrict}) to even values of $L=2n$, and has also been used to evaluate
$P_L(0)$ in Eq.(\ref{Peval}). The ``boxed" Chebyshev polynomial terms in Eq.(\ref{boxedCheby}) have been replaced by their Clebsch-Gordan coefficient equivalents in Eq.(\ref{Peval}). After substitution of the expression
for $\left|{\cal D}_{m m^{\prime}}^{(j)}(\alpha,\tfrac{\pi}{2},\gamma)\right |^2$ given in Eq.(\ref{restrict}) in
Eq.(\ref{sqmod}), we obtain the relation of Eq.(\ref{varshalsquare}).
\subsubsection{Squared reduced Wigner rotation matrix elements}
\paragraph{Using Meckler's formula to expand $ \left[d^{(j)}_{m m^{\prime}}(\beta)\right]^{\!2}$ in a Fourier-Legendre series. } In the Condon and Shortley phase convention \cite{condonshortley}, the reduced rotation matrices $ d^{(j)}_{m m^{\prime}}(\beta)$ are
real, so that there is no distinction between the modulus squared of these matrices and their square:
\begin{equation}
\left|{\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma) \right|^2 = \left|d^{(j)}_{m m^{\prime}}(\beta)\right|^{2} \equiv
\left[d^{(j)}_{m m^{\prime}}(\beta)\right]^{\!2} \label{equivalence}
\end{equation}
Therefore, Meckler's expression \cite{meckler:majorana,meckler:angular} for the Majorana transition probability \cite{emajorana} given in Eq.(\ref{majoranaEuler}) can be rewritten as a Fourier-Legendre series for the squared reduced rotation matrix elements
\begin{eqnarray}
\left[d^{(j)}_{m m^{\prime}}(\beta)\right]^{\!2} & = &
\sum_{L=0}^{2j} \boxed{f_L^{(j)} (m)} \; \boxed{f_L^{(j)} (m^{\prime})} \; P_L(\cos \beta) \label{relaxationa} \\
& = & (-1)^{m-m^{\prime}} \sum_{L=0}^{2j}
\boxed{ C_{jmj-m}^{L0}} \;
\boxed{C_{jm^{\prime}j-m^{\prime}}^{L0} }
\; P_L(\cos \beta) \label{relaxation}
\end{eqnarray}
Just as we simplified the expansion of $\left|{\cal D}_{m m^{\prime}}^{(j)}(\alpha,\tfrac{\pi}{2},\gamma)\right |^2$ in
Eq.(\ref{boxedCheby}), the ``boxed" Chebyshev polynomial terms in Eq.(\ref{relaxationa}) have been replaced by their Clebsch-Gordan coefficient equivalents in Eq.(\ref{relaxation}). A version of this latter relation, adapted to accomodate inversion symmetry of coordinate frame rotations (so that $L=0,2,4,\ldots$), proved indispensable in a theoretical analysis \cite{brown} of NMR spin-lattice relaxation of $^2$H and $^{14}$N nuclei in lipid bilayers and membrane systems.
\paragraph{Inverting Meckler's formula}
If we were to view Eq.(\ref{relaxationa}) as a system of equations for the unknowns $P_L(\cos \beta)$, then we could use standard matrix techniques to solve for these unknowns. However, a more direct approach is possible using the properties of the basis functions $f_L ^{(j)} ({\bf \hat{n}} \cdot {\bf J})$. The trace relation of Eq.(\ref{traceLegendre})
\begin{equation}
\mbox{Tr} \! \left[ f_L ^{(j)} ({\bf \hat{n}} \cdot {\bf J}) \;
f_{L^{\prime}} ^{(j)} ({\bf \hat{n}}^{\prime} \cdot {\bf J}) \right] = \delta_{LL^{\prime}} \; P_L( {\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime})
\end{equation}
can be rewritten as follows:
\begin{eqnarray}
& & \delta_{LL^{\prime}} \; P_L( {\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime}) \nonumber \\
& = & \! \!\!\mbox{Tr} \! \left[ \; \boxed{f_L ^{(j)} ({\bf \hat{n}} \cdot {\bf J})} \;
\boxed{f_{L^{\prime}} ^{(j)} ({\bf \hat{n}}^{\prime} \cdot {\bf J})} \; \right] \label{inverse1} \\
& = &
\! \!\!\mbox{Tr} \! \left[ \; \boxed{\displaystyle\sum_{m=-j}^{j} f_L^{(j)} (m) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) } \;
\boxed{\displaystyle\sum_{m^{\prime}=-j}^{j} f_{L^{\prime}}^{(j)} (m^{\prime}) \;
\mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{n}}^{\prime}) } \; \right] \label{inverse2} \\
& = & \sum_{m, \, m^{\prime}=-j}^j \, f_L^{(j)} (m) \, f_{L^{\prime}}^{(j)}(m^{\prime})
\; \mbox{Tr} \! \left[ \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) \; \mbox{{\boldmath $\Pi$}}^{(j)}(m^{\prime},{\bf \hat{n}}^{\prime}) \right] \label{inverse3} \\
& = & \sum_{m,\, m^{\prime}=-j}^j \, f_L^{(j)} (m) \, f_{L^{\prime}}^{(j)}(m^{\prime})
\left[d^{(j)}_{m m^{\prime}}(\beta)\right]^{\!2} \label{inverse4}
\end{eqnarray}
Each of the ``boxed" basis functions $f_L ^{(j)} ({\bf \hat{n}} \cdot {\bf J})$ in Eq.(\ref{inverse1}) has been replaced in Eq.(\ref{inverse2}) by the corresponding Sylvester's formula \cite{merzbacher2, horn} expansions from Section {\bf 3.2.1}, while the trace of the projection operator product in Eq.(\ref{inverse3}) has been reduced to the square of reduced matrix elements
$\left[d^{(j)}_{m m^{\prime}}(\beta)\right]^{\!2}$
using Eqs.(\ref{puzzle}) and (\ref{equivalence}). Taking advantage of the delta function on the left-hand side of Eq.(\ref{inverse1}), we arrive at the final result for the unknowns $P_L(\cos \beta)$:
\begin{equation}
P_L(\cos \beta) = \sum_{m, \, m^{\prime}=-j}^j \, f_L^{(j)} (m) \, f_{L}^{(j)}(m^{\prime})
\left[d^{(j)}_{m m^{\prime}}(\beta)\right]^{\!2} \label{intriguing}
\end{equation}
This inverse of Meckler's \cite{meckler:majorana} Majorana formula for the spin transition probability is an intriguing expression, because the Legendre polynomial $P_L(\cos \beta) $ is clearly $j$-independent, whereas every expansion term is $j$-dependent. Once $L$ is fixed, the elements of any reduced matrix
$d^{(j)}_{m m^{\prime}}(\beta)$ (arbitrary $j$) suffice to calculate the expansion, with expansion coefficients given by the product of appropriate $j$-dependent Chebyshev polynomials $ f_L^{(j)} (m) $. The summation of Eq.(\ref{intriguing}), as well as other closely related summations \cite{lai} involving
$\left[d^{(j)}_{m m^{\prime}}(\beta)\right]^{\!2} $ and $P_L(\cos \beta) $, may be derived using the Clebsch-Gordan series discussed in Section {\bf 4.1.2} and the orthogonality condition of the Clebsch-Gordan coefficients \cite{varshal1:ang}.
Beyond the well-known symmetry properties of the reduced Wigner rotation matrix elements $d^{(j)}_{m m^{\prime}}(\beta)$
\cite{varshal1:ang}, the relation
of Eq.(\ref{intriguing}) puts an additional constraint on the values of the $(2j+1) \times (2j+1)$ matrix $d^{(j)}_{m m^{\prime}}(\beta)$. When $L=0$, the simplest version of this constraint is
\begin{eqnarray}
P_0(\cos \beta) \equiv 1 & = & \sum_{m , \, m^{\prime} =-j}^j \, f_0^{(j)} (m) \, f_{0}^{(j)}(m^{\prime})
\left[d^{(j)}_{m m^{\prime}}(\beta)\right]^{\!2} \label{intrigue2} \\
& = & \displaystyle\frac{1}{2j+1} \; \sum_{m , \, m^{\prime} =-j}^j \, \left[d^{(j)}_{m m^{\prime}}(\beta)\right]^{\!2} \label{intrigue3} \\
\mbox{since \cite{corio:ortho, filippov2:thesis}} \;\;\;\; f_0^{(j)} (m) & = & \displaystyle\frac{1}{\sqrt{2j+1}}
\end{eqnarray}
From the constraint of Eq.(\ref{intrigue3}), it follows that for any reduced matrix $d^{(j)}_{m m^{\prime}}(\beta)$, the sum of all of its squared elements is $2j+1$:
\begin{equation}
\sum_{m,\, m^{\prime}=-j}^j \, \left[d^{(j)}_{m m^{\prime}}(\beta)\right]^{\!2} = 2j+1 \label{nosurprise}
\end{equation}
But this relation is actually just a consequence of the unitarity of the rotation
operator $\hat{{\cal D}}^{(j)}(R)$. A special case of the unitarity sum of the rotation matrices ${\cal D}_{m m^{\prime}}^{(j)}(\alpha,\beta,\gamma)$ is the following orthogonality sum of the (real) reduced rotation matrix elements \cite{thompson}
\begin{equation}
\sum_{m^{\prime \prime}=-j}^j d^{(j)}_{m^{\prime} m^{\prime \prime}}(\beta) \; d^{(j)}_{m m^{\prime \prime}}(\beta) = \delta_{m^{\prime} m} \label{dorthog}
\end{equation}
Setting $m^{\prime} = m$ in Eq.(\ref{dorthog}) leads to the relation
\begin{equation}
\sum_{m^{\prime \prime}=-j}^j d^{(j)}_{m^{\prime} m^{\prime \prime}}(\beta) \; d^{(j)}_{m^{\prime} m^{\prime \prime}}(\beta) \equiv \sum_{m^{\prime \prime}=-j}^j \left[d^{(j)}_{m^{\prime} m^{\prime \prime}}(\beta) \right]^{\!2} = 1
\end{equation}
which when summed over both sides, leads again to the identity of Eq.(\ref{nosurprise})
\begin{equation}
\sum_{m^{\prime} \!, \, m^{\prime \prime} =-j}^j \left[d^{(j)}_{m^{\prime} m^{\prime \prime}}(\beta) \right]^{\!2} = 2j+1
\end{equation}
\section{Operator Expansions}
\subsection{Rotation Operator}
Developing polynomial operator expressions in the variable $({\bf \hat{n}} \cdot {\bf J})$ for the rotation operator
$\hat{{\cal D}}^{(j)} \! (\psi, {\bf \hat{n}}) = e^{i \psi({\bf \hat{n}} \cdot {\bf J}) }$ is a challenging problem. Over the last five decades, it has been solved by a variety of ingenious methods \cite{corio:siam,wageningen:rotoppoly, webwill:spinmatrixpol, albert:rotgroup, happer, varshal2:expansion,torruella:rotgroup,
kusnezov:sun, curtright}, but the solution presented by Corio \cite{corio:siam,corio:correction}
\begin{eqnarray}
\hat{{\cal D}} ^{(j)}\! (\psi, {\bf \hat{n}}) \equiv e^{i \psi({\bf \hat{n}} \cdot {\bf J}) }
& = & \sum_{\lambda=0}^{2j} \mbox{Tr} \! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; \hat{{\cal D}}^{(j)} \! (\psi, {\bf \hat{n}}) \right]
f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) \label{coriotracrel} \\
& = & \sum_{\lambda=0}^{2j} a_{\lambda}^{(j)}(\psi) \; f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) \label{coriover}\\
& = & \sum_{\lambda=0}^{2j} i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) \; f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) \label{coriorotfirst}
\end{eqnarray}
is unique because the basis operator polynomials $ f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$ used by Corio
\cite{corio:siam,corio:correction} define an orthonormal set \cite{corio:siam}
\begin{equation}
\mbox{Tr} \! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; f^{(j)}_{{\lambda}^{\prime}}({\bf \hat{n}} \cdot {\bf J}) \right] =
\delta_{\lambda \lambda^{\prime}}
\end{equation}
and because these polynomials first introduced in a physics application by Meckler \cite{meckler:angular} are the Chebyshev polynomial operators
$ f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$.
The general form of Corio's Chebyshev polynomial operator expansion \cite{corio:siam} given in Eq.(\ref{coriover}) I have modified in Eq.({\ref{coriorotfirst}) in order to introduce the generalized characters $\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) $ \cite{varshal1:ang}. The canonical differential relation \cite{varshal1:ang} which defines the generalized characters $\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) $ in terms of the Gegenbauer polynomials
\cite{ tem:bk, askey} $\mbox{ C}_{2j}^{1}(c)$ is
\begin{eqnarray}
\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) & = & \sqrt{2j+1} \; \sqrt{\displaystyle\frac{(2j-\lambda)!}{(2j+\lambda+1)!}}\; s^{\lambda}
\left( \displaystyle\frac{d}{dc }\right)^{\!\!\lambda} \mbox{{\large $\chi$}}^{(j)}(\psi) \label{diffchar} \\
\mbox{where} \;\;\;\;\mbox{{\large $\chi$}}^{(j)}(\psi) & = & \mbox{ C}_{2j}^{1}(c) \\
s & = & \sin(\psi/2) \\
c & = & \cos(\psi/2)
\end{eqnarray}
Well-documented properties of the Gegenbauer polynomials \cite{ tem:bk, askey} can then be used to derive the following Gegenbauer polynomial
definition \cite{varshal1:ang} of the generalized characters $\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) $ which we will use in Section {\bf 6.1.1}:
\begin{equation}
\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) = (2\lambda)!!\sqrt{2j+1}\sqrt{\frac{(2j-\lambda)!}
{(2j+\lambda+1)!}}
\;\; s^{\lambda}\mbox{ C}_{2j-\lambda}^{\lambda+1}(c)
\end{equation}
It is instructive to compare the differential relation of Eq.(\ref{diffchar}) with the trace relation obtained from Eq.(\ref{coriotracrel}) that defines the generalized characters $\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) $ in terms of the Chebyshev polynomial operators
$f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J})$ and the rotation operator $\hat{{\cal D}}^{(j)} \! (\psi, {\bf \hat{n}}) $
\begin{equation}
\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) = i^{3 \lambda} \sqrt{\displaystyle\frac{2j+1}{2\lambda+1}} \;
\mbox{Tr}\! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; \hat{{\cal D}}^{(j)}\! (\psi, {\bf \hat{n}}) \right]
\end{equation}
Neither definition is more fundamental, but it goes without saying that calculating traces is far simpler than calculating derivatives.
The important role that the Chebyshev polynomial operators
$f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J})$ play in representations of the rotation group is evident in the expansion coefficients as defined by Corio
\cite{corio:siam}
\begin{equation}
a_{\lambda}^{(j)}(\psi)= \boxed{\mbox{Tr} \! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; \hat{{\cal D}}^{(j)} \! (\psi, {\bf \hat{n}}) \right] =
i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) } \label{charcheby}
\end{equation}
which are proportional to the generalized characters $\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) $ \cite{varshal1:ang} of the rotation group. The first equality of Eq.(\ref{charcheby}) is due to Corio
\cite{corio:siam}, but the
``boxed" relation of Eq.(\ref{charcheby}) is new, and, as we illustrate with an example in Appendix D, can be used as another definition of the generalized characters $ \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi)$. Several other definitions of $ \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi)$ are tabulated in Varshalovich et al. \cite{varshal1:ang}. Putting
$\lambda=0$ in this ``boxed" relation, we recover the character $\mbox{{\large $\chi$}}^{(j)}(\psi) $ of the irreducible representation $\hat{{\cal D}}^{(j)}(\psi, {\bf \hat{n}})$
\begin{equation}
\mbox{Tr} \! \left[ \hat{{\cal D}}^{(j)} \! (\psi, {\bf \hat{n}}) \right] =
\mbox{{\large $\chi$}}^{(j)}(\psi)
\end{equation}
using the facts that \cite{ varshal1:ang, corio:siam}
\begin{eqnarray}
f_0^{(j)}({\bf \hat{n}} \cdot {\bf J}) & = & \displaystyle\frac{ \mathds{1} }{\sqrt{2j+1} } \\
\mbox{{\large $\chi$}}_{0}^{(j)}(\psi) & \equiv & \mbox{{\large $\chi$}}^{(j)}(\psi)
\end{eqnarray}
By taking the trace of the rotation operator $\hat{{\cal D}}^{(j)} \! (\psi, {\bf \hat{n}})$ expanded in the Chebyshev polynomial operator basis
$f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right)$, the character $ \mbox{{\large $\chi$}}^{(j)}(\psi) $ of irreducible representations of the rotation group can be obtained. Using Corio's expansion \cite{corio:siam} of the rotation operator $\hat{{\cal D}}^{(j)} \!(\psi, {\bf \hat{n}}) = e^{i \psi({\bf \hat{n}} \cdot {\bf J}) }$,
this trace can be evaluated as
\begin{eqnarray}
\mbox{Tr} \! \left[ \hat{{\cal D}}^{(j)} \! (\psi, {\bf \hat{n}}) \right]
& = & \sum_{\lambda=0}^{2j} i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)} (\psi) \;
\boxed{\mbox{Tr}\! \left[ f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J}) \right] } \label{chartrace}\\
& = & \sum_{\lambda=0}^{2j} i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)} (\psi) \;
\boxed{\sqrt{2j+1} \; \delta_{\lambda 0}} \label{chartraceb} \\
& = & \frac{1}{\sqrt{2j+1}} \; \mbox{{\large $\chi$}}_{0}^{(j)} (\psi) \; \sqrt{2j+1} \\
& = & \mbox{{\large $\chi$}}^{(j)} (\psi) \\
\mbox{where \cite{varshal1:ang}}\;\;\;\; \mbox{{\large $\chi$}}_{0}^{(j)} (\psi) & \equiv & \mbox{{\large $\chi$}}^{(j)} (\psi)
\end{eqnarray}
The ``boxed" trace term in Eq.(\ref{chartrace}) has been replaced by the ``boxed" term of Eq.(\ref{chartraceb}) by calculating the trace in a representation where $({\bf \hat{n}} \cdot {\bf J}) $ is diagonal. Then we find
\begin{eqnarray}
\mbox{Tr} \! \left[ f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J}) \right] & = & \sum_m \langle m |\, f_{\lambda}^{(j)} (J_z) \, |m \rangle \\
& = & \sum_m f_{\lambda}^{(j)} (m) \\
& = & \sqrt{2j+1} \; \delta_{\lambda 0} \;\;\; \;\;(\mbox{using Eq.(\ref{chebsum})}
\end{eqnarray}
The Chebyshev polynomial expansions of the projection operators (see Eqs.(\ref{projectjz}) and (\ref{projectJn}) of Section {\bf 3.3.1}) are particularly effective when they are used with Sylvester's formula
\cite{merzbacher2, horn}. We illustrate this effectiveness with a simple alternative derivation of the Chebyshev polynomial operator expansion of the rotation operator \cite{corio:siam} in the version given in Eq.(\ref{coriorotfirst}). We begin by exploiting Sylvester's formula \cite{merzbacher2, horn} to write the exponential matrix operator as a sum of projection operators in Eq.(\ref{projj}), and then use the Chebyshev polynomial expansion
(see Eq.(\ref{projectJn}) of Section {\bf 3.3.1}) of the projection operator {\boldmath $\Pi$}$^{(j)}(m,{\bf \hat{n}}) $
\begin{eqnarray}
e^{-i \psi({\bf \hat{n}} \cdot {\bf J}) } & = & \sum_{m=-j}^j e^{-im\psi} \; \boxed{\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) } \label{projj}\\
& = & \sum_{m=-j}^j e^{-im\psi} \, \boxed{\displaystyle\sum_{\lambda=0}^{2j} f_{\lambda}^{(j)} (m) \; f_{\lambda} ^{(j)} ({\bf \hat{n}} \cdot {\bf J})} \label{projj2} \\
& = & \displaystyle\sum_{\lambda=0}^{2j} \boxed{ \sum_{m=-j}^j e^{-im\psi} \, f_{\lambda}^{(j)} (m)} \;
f_{\lambda} ^{(j)} ({\bf \hat{n}} \cdot {\bf J}) \label{coriosylvester} \\
& = & \displaystyle\sum_{\lambda=0}^{2j} \boxed{ i^{-\lambda} \, \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\; \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi)} \;
f_{\lambda} ^{(j)} ({\bf \hat{n}} \cdot {\bf J}) \label{coriorotproj}
\end{eqnarray}
The projection operator {\boldmath $\Pi$}$^{(j)}(m,{\bf \hat{n}}) $ in the ``boxed" term of Eq.(\ref{projj}) has been replaced by its Chebyshev polynomial expansion in the ``boxed" term of Eq.(\ref{projj2}). The sum in the ``boxed" term of Eq.(\ref{coriosylvester}) has been reexpressed as the ``boxed" term of Eq.(\ref{coriorotproj}) by converting a trigonometric series identity \cite{varshal1:ang} for the generalized characters of the rotation group
$\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) $ into a Chebyshev polynomial series as follows:
\begin{eqnarray}
\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) & = & i^{\lambda} \sum_{m=-j}^j \; e^{-im\psi} \, \boxed{C_{jm\lambda 0}^{jm}} \label{charCG}\\
& = & i^{\lambda} \sum_{m=-j}^j \; e^{-im\psi} \, \boxed{\sqrt{\displaystyle\frac{2j+1}{2\lambda+1}}(-1)^{j-m}\; C_{jmj-m}^{\lambda 0}}
\label{charCG2} \\
& = & i^{\lambda} \sum_{m=-j}^j \; e^{-im\psi} \, \sqrt{\displaystyle\frac{2j+1}{2\lambda+1}} \; f_{\lambda}^{(j)} (m)
\label{charCG3}
\end{eqnarray}
Well-documented \cite{varshal1:ang} symmetry properties of the Clebsch-Gordan coefficients have been used to reexpress the
``boxed" term in Eq.(\ref{charCG}) as the ``boxed" term in Eq.(\ref{charCG2}). The Clebsch-Gordan coefficient and phase factor in the
``boxed" term of Eq.(\ref{charCG2}) have been replaced by their Chebyshev polynomial equivalent $ f_{\lambda}^{(j)} (m)$ in Eq.(\ref{charCG3}). Aside from the use of Sylvester's formula \cite{merzbacher2, horn}, which is not particularly unique or novel, the novelty of this derivation is that it exploits some of the most felicitous properties of the Chebyshev polynomials, namely the Chebyshev polynomial operator expansion of the projection operator, and the duality of the Chebyshev polynomials as Clebsch-Gordan coefficients.
Remarkably, it is clear in retrospect that in 1967, Albert \cite{albert:rotgroup} had anticipated Corio's \cite{corio:siam} Chebyshev polynomial operator expansion of the rotation operator (see Eqs.(\ref{coriover} and (\ref{coriorotfirst})). In modern notation, Albert's expansion of $\exp[i \psi J_z]$ as a finite sum of irreducible tensor components
$\hat{T}_{\lambda 0}^{(J)} $ can be expressed as
\begin{eqnarray}
e^{i \psi J_z } & = & \sum_{\lambda=0}^{2j} i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) \; \boxed{ \hat{T}_{\lambda 0}^{(j)} } \label{albert1} \\
& = & \sum_{\lambda=0}^{2j} i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) \; \boxed{ f_{\lambda}^{(j)} \! (J_z)}
\label{corioalbert1}
\end{eqnarray}
The ``boxed" irreducible tensor operator term in
Eq.(\ref{albert1}) has been reexpressed as the ``boxed" Chebyshev polynomial operator term in Eq.(\ref{corioalbert1}) using the operator equivalence of Eq.(\ref{equivfT}). By considering the unitary transform of both sides of Eq.({\ref{corioalbert1}), and using the results of similarity transforms summarized in Table V, we then obtain
Corio's \cite{corio:siam} Chebyshev polynomial operator expansion of the rotation operator
\begin{eqnarray}
& & \hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, e^{i \psi J_z } \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger} \\
& = & \exp \! \left\{i \psi\; \hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, J_z \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger} \right\} \\
& = & e^{i \psi({\bf \hat{n}} \cdot {\bf J}) } \\
& = & \sum_{\lambda=0}^{2j} i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi)
\left\{\hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, \hat{T}_{\lambda 0}^{(j)} \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger} \right\}
\label{albert11} \\
& = & \sum_{\lambda=0}^{2j} i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) \;
\left\{ \hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \; f_{\lambda}^{(j)} \! (J_z) \; \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger} \right\} \\
& = & \sum_{\lambda=0}^{2j} i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) \; f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J} )
\label{corioalbert11}
\end{eqnarray}
It is astonishing that Corio's \cite{corio:siam,corio:correction} Chebyshev polynomial operator $f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})$ expansion of the rotation operator
$\hat{{\cal D}} ^{(j)}\! (\psi, {\bf \hat{n}})$ in Eq.(\ref{coriorotfirst}) is hardly known at all, due in no small part to the fact that this expansion is not mentioned or discussed in any angular momentum or quantum mechanics textbook. And yet this expansion emphasizes the fact the Chebyshev polynomial operators
$f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})$ play a unique role in irreducible representations of the rotation group. In Section {\bf 5.3}, we will exploit this expansion to derive a tomographic reconstruction relation for the density operator $\hat \rho$ using the Chebyshev polynomial operators $f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})$.
\subsection{Stratonovich-Weyl Operator}
In the phase-space approach to spin, the conventional quantum mechanical operators are replaced by functions on the classical phase-space of
the unit sphere ${\bf S}^2$. Central to this correspondence between Hilbert space operators and functions on the phase-space is the Stratonovich-Weyl operator \cite{varilly2:moyal,heissweigert}, also known as the Wigner-Stratonovich-Agarwal operator \cite{klimov:distr,agarwal,klimovchumakov}.
Beginning with the conventional representation \cite{varilly2:moyal, klimov:distr, klimovchumakov}, this operator (kernel) $\Delta^{(j)}({\bf \hat{n}})$ can be expanded in terms of the Chebyshev polynomial operator basis $f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})$ as follows
\begin{eqnarray}
\Delta^{(j)}({\bf \hat{n}}) & = & \sum_{\lambda=0}^{2j} \sum_{\mu = -\lambda}^{\lambda} \sqrt{\displaystyle\frac{4\pi}{2j+1}}
\; \boxed{\left[\hat{T}^{(j)}_{\lambda\mu} \right]^{\!\dagger}}\; Y_{\lambda\mu}({\bf \hat{n}}) \label{dagrep} \\
& = & \sum_{\lambda=0}^{2j} \sum_{\mu = -\lambda}^{\lambda} \sqrt{\displaystyle\frac{4\pi}{2j+1}}
\; \hat{T}^{(j)}_{\lambda\mu} \; Y^{\star}_{\lambda\mu}({\bf \hat{n}}) \\
& = & \frac{1}{\sqrt{2j+1}} \sum_{\lambda=0}^{2j} \sqrt{2\lambda+1}\;\, \boxed{ {\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) } \label{stratopexpa}\\
& = & \frac{1}{\sqrt{2j+1}} \sum_{\lambda=0}^{2j} \sqrt{2\lambda+1}\; \, \boxed{f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})} \label{stratopexp}
\end{eqnarray}
where the ``boxed" term of Eq.(\ref{dagrep}) has been reexpressed using the following relation \cite{varshal1:ang} for the adjoint of the polarization operators
\begin{equation}
\left[\hat{T}^{(j)}_{\lambda\mu} \right]^{\!\dagger} = (-1)^{\mu} \; \hat{T}^{(j)}_{\lambda -\mu}
\end{equation}
To arrive at the result of Eq.(\ref{stratopexp}), we have reexpressed the canonical version of the Stratonovich-Weyl operator
\cite{varilly2:moyal, klimov:distr, klimovchumakov} in terms of a direct product of spin and spatial tensors in Eq.(\ref{stratopexpa}). The ``boxed" tensor direct product term in
Eq.(\ref{stratopexpa}) has been reexpressed as the ``boxed" Chebyshev polynomial operator term in Eq.(\ref{stratopexp}) using a recoupling relation which we derive in Section {\bf 6.1}.
\subsubsection{$\Delta^{(j)}({\bf \hat{n}})$ Operator Trace}
Using the Chebyshev polynomial operator expansion of Eq.(\ref{stratopexp}), the Stratonovich-Weyl operator $\Delta^{(j)}({\bf \hat{n}})$ \cite{varilly2:moyal,heissweigert} trace is easly evaluated using the properties of the Chebyshev polynomials. We find
\begin{eqnarray}
\mbox{Tr} \left[ \Delta^{(j)}({\bf \hat{n}}) \right] & = &
\frac{1}{\sqrt{2j+1}} \sum_{\lambda=0}^{2j} \sqrt{2\lambda+1}\; \, \mbox{Tr} \left[ f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J}) \right] \\
& = &
\frac{1}{\sqrt{2j+1}} \sum_{\lambda=0}^{2j} \sqrt{2\lambda+1}\; \, \mbox{Tr} \left[ f_{\lambda}^{(j)}(J_z) \right] \\
& = &
\frac{1}{\sqrt{2j+1}} \sum_{\lambda=0}^{2j} \sqrt{2\lambda+1}\; \, \sum_m \langle jm| \, f_{\lambda}^{(j)}(J_z) \, | jm \rangle \\
& = &
\frac{1}{\sqrt{2j+1}} \sum_{\lambda=0}^{2j} \sqrt{2\lambda+1}\; \, \boxed{ \sum_m f_{\lambda}^{(j)}(m) } \label{sumdeltaltrace}\\
& = &
\frac{1}{\sqrt{2j+1}} \sum_{\lambda=0}^{2j} \sqrt{2\lambda+1}\; \, \sqrt{2j+1} \, \delta_{\lambda 0} \\
& = & 1
\end{eqnarray}
The sum over the Chebyshev polynomials in the ``boxed" term of Eq.(\ref{sumdeltaltrace}) was evaluated using Eq.(\ref{chebsum}) of Section {\bf 3.2.1}.
\subsubsection{The Reproducing Kernel for the $\Delta^{(j)}({\bf \hat{n}})$ Operators }
Among the most important properties of the Stratonovich-Weyl operator kernel is the traciality condition \cite{varilly2:moyal, heissweigert}
\begin{eqnarray}
\Delta^{(j)}({\bf \hat{n}}) & = & \frac{(2j+1)}{4\pi} \int_{{\bf S}^2} \mbox{Tr} \! \left[ \Delta^{(j)}({\bf \hat{n}}) \; \Delta^{(j)}({\bf \hat{n}}^{\prime}) \right]
\Delta^{(j)}({\bf \hat{n}}^{\prime}) \, d{\bf \hat{n}}^{\prime} \label{traciality} \\
& = & \int_{{\bf S}^2}\; K^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime}) \; \Delta^{(j)}({\bf \hat{n}}^{\prime}) \; d{\bf \hat{n}}^{\prime} \label{deltakernela} \\
& = & \int_{{\bf S}^2}\; \delta_{\Delta}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime}) \; \Delta^{(j)}({\bf \hat{n}}^{\prime}) \; d{\bf \hat{n}}^{\prime} \label{deltakernel} \\
& & \nonumber \\
\mbox{where} \;\;\;\; K^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime}) & = & \frac{(2j+1)}{4\pi} \; \mbox{Tr} \! \left[ \Delta^{(j)}({\bf \hat{n}}) \;
\Delta^{(j)}({\bf \hat{n}}^{\prime}) \right] \\
d{\bf \hat{n}} & = & \sin \theta \, d\theta \, d\phi
\end{eqnarray}
Eqs.(\ref{deltakernela}) and (\ref{deltakernel}) show that for a certain subset of $(2j+1)^2$ functions on the sphere ${\bf S}^2$ \cite{varilly2:moyal, heissweigert}, $K^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime}) \equiv
\delta_{\Delta}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime})$ is the reproducing kernel \cite{varilly2:moyal, heissweigert}, acting as a delta function with respect to integration over ${\bf S}^2$.
Armed with the Chebyshev polynomial operator expansion of Eq.(\ref{stratopexp}), this delta function can be simply evaluated using
the Chebyshev polynomial operator $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$ trace identity discussed in Section {\bf 2.2.1} as follows:
\begin{eqnarray}
\delta_{\Delta}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime})
& = & \frac{1}{4\pi} \; \sum_{\lambda=0}^{2j}\; \sum_{\lambda^{\prime}=0}^{2j}\; \sqrt{2\lambda+1}\,\sqrt{2\lambda^{\prime}+1} \;\,
\mbox{Tr} \! \left[ f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})\, f_{\lambda^{\prime}}^{(j)} ({\bf \hat{n}}^{\prime}\cdot {\bf J}) \right]\;\;\;\;\;\; \\
& = & \frac{1}{4\pi} \; \sum_{\lambda=0}^{2j}\; \sum_{\lambda^{\prime}=0}^{2j}\; \sqrt{2\lambda+1}\,\sqrt{2\lambda^{\prime}+1} \;\,
\delta_{\lambda \lambda^{\prime}} \; P_{\lambda}({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime})\\
& = & \; \sum_{\lambda=0}^{2j}\; \boxed{\frac{2\lambda+1}{4\pi}\, P_{\lambda}({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime}) } \label{legend} \\
& = & \; \sum_{\lambda=0}^{2j} \boxed{\sum_{\mu=-\lambda}^{\lambda} Y_{\lambda \mu}({\bf \hat{n}}) \, Y_{\lambda \mu}^{\star}({\bf \hat{n}}^{\prime}) }
\label{addsphere}
\end{eqnarray}
The ``boxed" term of Eq.(\ref{legend}) has been reexpressed as the ``boxed" term of Eq.(\ref{addsphere}) using the spherical harmonics
addition theorem \cite{gottfried}:
\begin{equation}
P_{\lambda}({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime}) = \frac{4 \pi}{2\lambda+1} \; \sum_{\mu=-\lambda}^{\lambda} Y_{\lambda \mu}({\bf \hat{n}}) \, Y_{\lambda \mu}^{\star}({\bf \hat{n}}^{\prime}) \label{gottadd}
\end{equation}
Certainly there are other methods for evaluating $\delta_{\Delta}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime})$ \cite{varilly2:moyal, heissweigert}, but this method using
the Chebyshev polynomial operators $f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})$ is among the most direct. Note that the closure relation \cite{gottfried} for the spherical harmonics $\left\{ Y_{\lambda \mu}({\bf \hat{n}}) \right\} $ could be obtained
from Eq.(\ref{addsphere}) in the limit that $j \rightarrow \infty$ to give
\begin{eqnarray}
\delta({\bf \hat{n}} - {\bf \hat{n}}^{\prime}) & \equiv & \frac{\delta(\theta - \theta^{\prime}) \; \delta(\phi - \phi^{\prime} )}{\sin \, \theta} \\
& & \\
& = & \; \sum_{\lambda=0}^{\infty} \sum_{\mu=-\lambda}^{\lambda} Y_{\lambda \mu}({\bf \hat{n}}) \, Y_{\lambda \mu}^{\star}({\bf \hat{n}}^{\prime})
\label{closure}
\end{eqnarray}
Substituting the expression for the reproducing kernel $ \delta_{\Delta}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime})$ from Eq.(\ref{legend}) in Eq.(\ref{deltakernel}), and using the Chebyshev polynomial operator $f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})$ expansion of the Stratonovich-Weyl operator given in Eq.(\ref{stratopexp}), we easily obtain the reproducing kernel $ \delta_{f}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime})$
\begin{equation}
\delta_{f}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime}) = \frac{2\lambda+1}{4\pi}\, P_{\lambda}({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime})
\end{equation}
which for the Chebyshev polynomial operators $f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})$, acts as a delta function with respect to integration over ${\bf S}^2$:
\begin{eqnarray}
f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J}) & = & \int_{{\bf S}^2}\; \delta_{f}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime}) \;
f_{\lambda}^{(j)} ({\bf \hat{n}}^{\prime} \cdot {\bf J}) \; d{\bf \hat{n}}^{\prime} \\
& = & \frac{2\lambda+1}{4\pi}\, \int_{{\bf S}^2}\; P_{\lambda}({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime}) \;
f_{\lambda}^{(j)} ({\bf \hat{n}}^{\prime} \cdot {\bf J}) \; d{\bf \hat{n}}^{\prime} \label{reprokernelcheby} \\
& = & \frac{2\lambda+1}{4\pi}\, \int_{{\bf S}^2}\;
\mbox{Tr} \! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; f^{(j)} _{{\lambda}}({\bf \hat{n}}^{\prime} \! \cdot {\bf J}) \right] \;
f_{\lambda}^{(j)} ({\bf \hat{n}}^{\prime} \cdot {\bf J}) \; d{\bf \hat{n}}^{\prime} \label{reprokernelcheby2}
\end{eqnarray}
The reproducing kernel in Eq.(\ref{reprokernelcheby}) has been rewritten as a trace in Eq.(\ref{reprokernelcheby2}) to elicit the analogy with the
traciality condition \cite{varilly2:moyal, heissweigert} of Eq.(\ref{traciality}) for the Stratonovich-Weyl operators.
If in Eq.(\ref{reprokernelcheby}), we put ${\bf \hat{n}} = {\bf \hat{z}}$ and ${\bf \hat{n}}^{\prime} = {\bf \hat{n}}$, so that
${\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime} \equiv {\bf \hat{z}} \cdot {\bf \hat{n}} = \cos \theta$ then what at first sight is certainly not a familiar relation becomes
\begin{eqnarray}
f^{(j)} _{{\lambda}}({\bf \hat{z}} \! \cdot {\bf J}) \equiv f^{(j)} _{{\lambda}}(J_z) & = &
\frac{2\lambda+1}{4\pi}\, \int_{{\bf S}^2}\; P_{\lambda}(\cos \theta) \;
f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J}) \; d{\bf \hat{n}} \label{familiar}
\end{eqnarray}
But since $ f^{(j)} _{{\lambda}}(J_z) \equiv \hat{T}^{(j)}_{\lambda 0}$, and $ P_{\lambda}(\cos \theta) \equiv C_{\lambda 0}({\bf \hat{n}})$, Eq.(\ref{familiar}) can be reexpressed as
\begin{equation}
\hat{T}^{(j)}_{\lambda 0} = \frac{2\lambda+1}{4\pi}\, \int_{{\bf S}^2}\; C_{\lambda 0}({\bf \hat{n}}) \;
f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J}) \; d{\bf \hat{n}}
\end{equation}
which for the spin polarization operators
$ \hat{T}^{(j)}_{\lambda \mu} $ is a particular version $(\mu=0)$ of a decomposition on the Chebyshev polynomial operators
$f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right)$ (see Eq.(\ref{spintensorcheby}) in Section {\bf 2.2.2}).
In the next section, we will use this reproducing kernel $ \delta_{f}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime}) $ to derive a tomographic reconstruction formula for the density operator from the conventional statistical tensor expansion of the density operator.
\subsection{Tomographic Reconstruction of the Density Operator}
As an alternative to the phase-space approach to spin, the tomographic map of spin states onto a probability distribution
\cite{filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo,klimovchumakov} represents another approach to mapping spin operators onto functions. In the first two parts of this section, specific examples of these distributions are considered, and in each case, these mappings are shown to lead to a novel tomographic reconstruction formula for the density operator $\hat \rho$ expressed exclusively in terms of Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$. In the concluding part, without considering a specific probability distribution, this tomographic reconstruction formula is recovered using more fundamental approaches.
\subsubsection{Spin Tomogram Distributions}
One example of tomographic mapping is the approach developed by Man'ko and colleagues \cite{filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo}, who have made very effective use of the following Chebyshev polynomial operator $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$ expansions of the dequantizer (alias projection) operators $\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}})$ and quantizer operators
$\mbox{{\boldmath $\Xi$}}^{(j)}(m,{\bf \hat{n}}) $
\begin{eqnarray}
\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) & = & \sum_{\lambda=0}^{2j} f_{\lambda}^{(j)}(m) \; f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) \nonumber \\
\mbox{{\boldmath $\Xi$}}^{(j)}(m,{\bf \hat{n}}) & = & \sum_{\lambda=0}^{2j} (2\lambda +1) \, f_{\lambda}^{(j)}(m) \; f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})
\label{chebexp}
\end{eqnarray}
The spin tomogram $w^{(j)}(m,{\bf \hat{n}}) $ of a state determined by the density operator $\hat \rho$ is \cite{filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo}
\begin{equation}
w^{(j)}(m,{\bf \hat{n}}) = \mbox{Tr} \! \left[ \hat \rho\; \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) \right] \label{tomtrace}
\end{equation}
whereas the inverse mapping of the spin tomogram $w^{(j)}(m,{\bf \hat{n}}) $ onto the density operator $\hat \rho$ was expressed through the quantizer operator $\mbox{{\boldmath $\Xi$}}^{(j)}(m,{\bf \hat{n}}) $ as the tomographic reconstruction \cite{filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo}
\begin{eqnarray}
\hat \rho & = & \sum_{m^{\prime}=-j}^j \; \frac{1}{4\pi} \int _{{\bf S}^2} \, w^{(j)}(m^{\prime},{\bf \hat{n}}^{\prime}) \; \,
\mbox{{\boldmath $\Xi$}}^{(j)}(m^{\prime},{\bf \hat{n}}^{\prime})\; d{\bf \hat{n}}^{\prime} \label{densityrecon} \\
\mbox{where} \;\;\;\;
d{\bf \hat{n}}^{\prime} & \equiv &d\Omega= \sin \theta \, d\theta \, d\phi \nonumber
\end{eqnarray}
Upon substituting this integral representation for the density operator in Eq.(\ref{tomtrace}), we obtain
\begin{eqnarray}
w^{(j)}(m,{\bf \hat{n}}) & = & \mbox{Tr} \! \left[ \hat \rho\; \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) \right]\\
& = & \sum_{m=-j}^j \; \frac{1}{4\pi} \int_{{\bf S}^2} \, \; \mbox{Tr} \! \left[ \mbox{{\boldmath $\Xi$}}^{(j)}(m^{\prime},{\bf \hat{n}}^{\prime}) \;
\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) \right] w^{(j)}(m^{\prime},{\bf \hat{n}}^{\prime})
\, d{\bf \hat{n}}^{\prime} \label{trdq}
\end{eqnarray}
This implies that for the set of tomograms $ w^{(j)}(m,{\bf \hat{n}}) $ on the sphere ${\bf S}^2$, the trace in Eq.(\ref{trdq}) must act like a delta function with respect to integration over ${\bf S}^2$. This delta function can easily be evaluated using the Chebyshev polynomial operator $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$ trace identity discussed in Section {\bf 2.2.1} as follows:
\begin{eqnarray}
& & \delta_w^{(j)}(m, {\bf \hat{n}};m^{\prime},{\bf \hat{n}}^{\prime}) \\
& = & \delta_{m,m^{\prime}} \;
\mbox{Tr} \! \left[ \mbox{{\boldmath $\Xi$}}^{(j)}(m^{\prime},{\bf \hat{n}}^{\prime}) \;
\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) \right] \\
& = & \delta_{m,m^{\prime}} \; \sum_{\lambda=0}^{2j} \; \sum_{\lambda^{\prime}=0}^{2j} \; (2\lambda{^\prime}+1) \,
f_{\lambda}^{(j)}(m) \; f_{\lambda^{\prime}}^{(j)}(m^{\prime}) \;
\mbox{Tr} \! \left[ f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})\, f_{\lambda^{\prime}}^{(j)} ({\bf \hat{n}}^{\prime}\cdot {\bf J}) \right] \\
& = & \delta_{m,m^{\prime}} \; \sum_{\lambda=0}^{2j}(2\lambda+1) \, f_{\lambda}^{(j)}(m) \; f_{\lambda}^{(j)}(m^{\prime}) \;
P_{\lambda}({\bf \hat{n}} \cdot{\bf \hat{n}}^{\prime})
\end{eqnarray}
As with the evaluation of $ \delta_{\Delta}^{(j)}({\bf \hat{n}}, {\bf \hat{n}}^{\prime})$ in Section {\bf 5.2}, evaluating
$\delta_w^{(j)}(m, {\bf \hat{n}};m^{\prime},{\bf \hat{n}}^{\prime}) $ is straightforward using the
the Chebyshev polynomial operators $f_{\lambda}^{(j)} ({\bf \hat{n}} \cdot {\bf J})$.
A much simpler expression for the tomographic reconstruction of the density operator can be obtained from Eq.(\ref{densityrecon}) just by replacing the quantizer and dequantizer operators with their corresponding Chebyshev polynomial operator $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$ expansions given in Eq.(\ref{chebexp}). By means of these replacements, the density operator tomographic reconstruction formula can be expressed exclusively in terms of Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$
\begin{eqnarray}
\hat \rho & = & \sum_{m^{\prime}=-j}^j \; \frac{1}{4\pi} \int _{{\bf S}^2} \mbox{Tr} \! \left[ \hat \rho \;\mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) \right] \;
\mbox{{\boldmath $\Xi$}}^{(j)}(m,{\bf \hat{n}}) \; d{\bf \hat{n}} \\
& = & \frac{1}{4\pi} \sum_{\lambda,\lambda^{\prime}=0}^{2j} (2\lambda+1) \;
\boxed{ \sum_{m^{\prime}=-j}^j f_{\lambda^{\prime}}^{(j)}(m) \; f_{\lambda}^{(j)}(m) } \int _{{\bf S}^2} f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J} ) \;
\; \mbox{Tr} \! \left[ \hat \rho \,
f_{\lambda^{\prime}}^{(j)}( {\bf \hat{n}} \cdot {\bf J}) \right] d{\bf \hat{n}} \;\;\;\; \label{boxorthog} \\
& = & \sum_{\lambda=0}^{2j} \frac{(2\lambda+1)}{4\pi} \int _{{\bf S}^2} f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) \;
\mbox{Tr} \! \left[ \hat \rho \, f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J} ) \right] d{\bf \hat{n}} \label{singlesum}
\end{eqnarray}
The ``boxed" term in Eq.(\ref{boxorthog}) has been simplified using the Chebyshev polynomial orthogonality relation of Eq.(\ref{ortho1}), which collapses the double sum
in Eq.(\ref{boxorthog}) to a single sum in Eq.(\ref{singlesum}).
\subsubsection{Other Distributions}
Other examples of probability distributions are Husimi's \cite{husimi} $Q({\bf \hat{n}})$ function \cite{klimovchumakov} and the Stratonovich-Weyl distribution $W({\bf \hat{n}})$ \cite{klimovchumakov}, defined as
\begin{eqnarray}
Q({\bf \hat{n}}) & = & \langle {\bf \hat{n}},j | \; \hat \rho \; |{\bf \hat{n}},j \rangle =
\mbox{Tr} \! \left[ \hat \rho \; \mbox{{\boldmath $\Pi$}}^{(j)}(j,{\bf \hat{n}}) \right] \label{Hist} \\
W({\bf \hat{n}}) & = & \mbox{Tr} \! \left[ \hat \rho \; \Delta^{\!(j)}({\bf \hat{n}}) \right] \label{Wdist}
\end{eqnarray}
For a spin state of well-defined angular momentum, Husimi's \cite{husimi} $Q({\bf \hat{n}})$ function \cite{klimovchumakov} is defined as the average value of the density matrix in the coherent state $|{\bf \hat{n}},j \rangle $, while the Stratonovich-Weyl distribution $W({\bf \hat{n}})$ is defined in terms of the
Stratonovich-Weyl operator \cite{varilly2:moyal,heissweigert} $ \Delta^{\!(j)}({\bf \hat{n}})$ of Section {\bf 5.2} by the trace of Eq.(\ref{Wdist}). Table VII compares these probability distributions and their tomographic reconstruction relations with those of the
spin tomogram distributions $w^{(j)}(m,{\bf \hat{n}}) $ \cite{filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo}. All the distributions and tomographic reconstruction relations in this table are expressed only in terms of Chebyshev polynomials $ f_{\lambda}^{(j)}(m) $ or Chebyshev polynomial operators
$ f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J} )$, so that substitution of each distribution in the corresponding tomographic reconstruction relation immediately yields the tomographic reconstruction relation of Eq.(\ref{singlesum}). All the distributions $D({\bf \hat{n}}) \;\;[\equiv w^{(j)}(m,{\bf \hat{n}}) , Q({\bf \hat{n}})$ or $W({\bf \hat{n}})]$ of Table VI are normalized \cite{klimovchumakov} so that
\begin{equation}
\frac{2j+1}{4\pi} \int _{{\bf S}^2} D({\bf \hat{n}}) \; d{\bf \hat{n}} = 1
\end{equation}
Just as measurements of the spin tomograms $w^{(j)}(m,{\bf \hat{n}}) $ enable the reconstruction of the density operator $\hat \rho $ via
the tomographic reconstruction relation of Eq.(\ref{densityrecon}) \cite{filippov4, filippov2:thesis, fillipov1:qubit, filippov3, manko:spintomo},
so do the corresponding tomographic reconstruction relations for $\hat \rho $ in Table VII also permit the reconstruction of all the density matrix elements simply by measuring the corresponding $Q({\bf \hat{n}})$ or $W({\bf \hat{n}})$ function distributions \cite{klimovchumakov}.
\subsubsection{More Fundamental Perspectives}
In the previous two sections, specific examples of tomographic reconstruction formulae for the density operator were considered. Regardless of the probability distribution under consideration, the tomographic map of spin states lead to a novel tomographic reconstruction formula expressed exclusively in terms of Chebyshev polynomial operators $ f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J} )$. From a more fundamental perspective, can this formula be derived without considering a specific probability distribution? Two approaches are now considered which demonstrate that such a derivation is possible.
\paragraph{Statistical tensor expansion of the density operator}
In this approach, we begin with a consideration of the statistical tensor expansion of the density operator given in Table IV
\begin{equation}
\hat \rho = \displaystyle\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda} \mbox{Tr} \! \left[ \hat \rho \,
\left[ \hat{T}_{\lambda \mu }^{(j)} \right]^{\! \dagger} \right] \hat{T}_{\lambda \mu }^{(j)} \label{stat}
\end{equation}
Viewing the spin polarization operators $\hat{T}^{(j)}_{\lambda \mu} $ in Eq.(\ref{stat}}) as the following integral transformation of the Chebyshev polynomial operators
$f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right)$ (see Eq.(\ref{spintensorcheby}) in Section {\bf 2.2.2})
\begin{eqnarray}
\hat{T}^{(j)}_{\lambda \mu} & = &
\frac{2\lambda +1 }{4\pi} \int C_{\lambda \mu}({\bf \hat{n}}) \; f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) \, d{\bf \hat{n}} \label{altspintensorcheby}
\end{eqnarray}
the density operator can be written as
\begin{eqnarray}
& & \hat \rho \nonumber \\
&\!\! \!= & \displaystyle\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda} \! \mbox{Tr} \! \left[ \hat \rho \,
\left[ \frac{2\lambda +1 }{4\pi} \int_{{\bf S}^2} \!C_{\lambda \mu}({\bf \hat{n}}) \; f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) \, d{\bf \hat{n}} \right]^{\! \dagger} \right] \frac{2\lambda +1 }{4\pi} \! \int_{{\bf S}^2} \! C_{\lambda \mu}({\bf \hat{n}}^{\prime}) \; f_{\lambda}^{(j)} \! \left( {\bf \hat{n}}^{\prime} \cdot {\bf J} \right) \, d{\bf \hat{n}} ^{\prime} \;\;\;\;\;\;\;\;\;\;\;\\
&\!\! \!= & \displaystyle\sum_{\lambda=0}^{2j} \frac{2\lambda +1 }{4\pi} \int \! \mbox{Tr} \! \left[ \hat \rho \;
f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) \right] \int_{{\bf S}^2} \frac{2\lambda +1 }{4\pi} \boxed{\sum_{\mu=-\lambda}^{\lambda}
C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; C_{\lambda \mu}({\bf \hat{n}}^{\prime}) } \; f_{\lambda}^{(j)} \! \left( {\bf \hat{n}}^{\prime} \cdot {\bf J} \right) \,
d{\bf \hat{n}} ^{\prime} \; d{\bf \hat{n}} \; \label{addtheorem} \\
&\!\! \!= & \displaystyle\sum_{\lambda=0}^{2j} \frac{2\lambda +1 }{4\pi} \int_{{\bf S}^2} \! \mbox{Tr} \! \left[ \hat \rho \;
f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) \right] \boxed{ \int_{{\bf S}^2} \frac{2\lambda +1 }{4\pi} P_{\lambda}( {\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime}) \; f_{\lambda}^{(j)} \! \left( {\bf \hat{n}}^{\prime} \cdot {\bf J} \right) \, d{\bf \hat{n}} ^{\prime} }\; d{\bf \hat{n}} \; \label{kernelsimp} \\
&\!\! \!= & \displaystyle\sum_{\lambda=0}^{2j} \frac{2\lambda +1 }{4\pi} \int_{{\bf S}^2} \! \mbox{Tr} \! \left[ \hat \rho \;
f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) \right] f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) d{\bf \hat{n}} \label{statalt}
\end{eqnarray}
The ``boxed" term of Eq.(\ref{addtheorem}) has been simplified using the spherical harmonics
addition theorem \cite{gottfried} of Eq.(\ref{gottadd}), and the ``boxed" term of Eq.(\ref{kernelsimp}) has been simplified using Eq.(\ref{reprokernelcheby}), recognizing $\left[(2\lambda+1)/4\pi \right] \; P_{\lambda}( {\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime})$ as the reproducing kernel for the Chebyshev polynomial operators
$ f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J}\right)$.
\paragraph{Group theory operator identity} In this altenative approach, we demonstrate that the Chebyshev polynomial operator tomographic reconstruction formula of Eqs.(\ref{singlesum}) and (\ref{statalt}) is a consequence of much deeper and more general result in the form of a single operator identity based on group theory \cite{ariano}.
Using group theory, D'Ariano et al. \cite{ariano} have derived the following fundamental tomographic reconstruction formula for the density operator
\begin{equation}
\hat \rho = \int \mbox{Tr} \! \left[ \hat \rho \; {\cal R}(g) \right] {\cal R}^{\dagger}(g) \, dg \label{Ariano}
\end{equation}
valid for an irreducible unitary representation ${\cal R}(g)$ on the Hilbert space ${\cal H}$ of the physical system.
The appropriate operators of such a unitary irreducible representation in the case of a single spin physical system are \cite{ariano}
\begin{equation}
{\cal R}(g) \equiv {\cal R}({\bf \hat{n}}, \psi) = e^{i \psi ({\bf \hat{n}} \cdot {\bf J}) }
\end{equation}
In this parametrization, the invariant measure is given by \cite{ariano}
\begin{equation}
dg({\bf \hat{n}}, \psi ) =\frac{(2j+1)}{4\pi^2} \sin^2 \bfrac{\psi }{2} \sin \theta \, d\theta \, d\phi \, d\psi
\end{equation}
so that the density operator according to Eq.(\ref{Ariano}) can be written \cite{ariano}
\begin{equation}
\hat \rho = \frac{(2j+1)}{4\pi^2} \int_0^{2\pi} d\psi \, \sin^2 \bfrac{\psi }{2} \;
\int_{{\bf S}^2} \mbox{Tr} \! \left[ \hat \rho \, e^{i \psi ({\bf \hat{n}} \cdot {\bf J}) } \right] e^{-i \psi ({\bf \hat{n}} \cdot {\bf J}) } \, d{\bf \hat{n}} \label{arianoconstruct}
\end{equation}
D'Ariano et al. \cite{ariano} have also described an experimental apparatus designed to reconstruct the density operator according to Eq.(\ref{arianoconstruct}).
Taking advantage of Corio's \cite{corio:siam,corio:correction} Chebyshev polynomial operator expansion of the operator $e^{i \psi ({\bf \hat{n}} \cdot {\bf J}) }$ given in Eq.(\ref{coriorotfirst}) of Section {\bf 5.1}, and using the orthogonality relation \cite{varshal1:ang} for the generalized character functions $\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi ) $
\begin{equation}
\int_0^{2\pi} d\psi \, \sin^2 \bfrac{\psi }{2} \; \mbox{{\large $\chi$}}_{\lambda_1}^{(j_1)}\!(\psi ) \; \mbox{{\large $\chi$}}_{\lambda_2}^{(j_2)}\!(\psi ) = \pi \,
\delta_{j_1j_2} \, \delta_{\lambda_1 \lambda_2}
\end{equation}
which define the expansion coefficients,
the spin tomographic reconstruction formula of Eq.(\ref{Ariano}) derived by D'Ariano et al. from group theory
\cite{ariano} can be expressed exclusively in terms of the Chebyshev polynomial operators $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J} ) $
\begin{equation}
\hat \rho = \sum_{\lambda=0}^{2j} \frac{(2\lambda+1)}{4\pi} \int _{{\bf S}^2} f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J} ) \;
\mbox{Tr} \! \left[ \hat \rho \, f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J} ) \right] d \Omega
\end{equation}
This is the tomographic reconstruction formula for the density operator obtained in Sections {\bf 5.3.1} and {\bf 5.3.2} from specific examples of
tomographic reconstruction formulae using different spin tomography probability distributions. Recovering this formula from the group theoretical operator identity derived by D'Ariano et al. \cite{ariano} indicates that for the angle-axis
$(\psi,{\bf \hat{n}} )$ parameterization of the $SU(2)$ group, the Chebyshev polynomial operators $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J} )$ have a unique role to play in
tomographic reconstruction of the density operator.
\section{ The recoupling of spin and spatial tensors via $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$ }
From a comparison of two different operator expansions \cite{corio:siam, happer} for both the rotation operator
$\hat{{\cal D}}^{(j)} \!(\psi, {\bf \hat{n}})=e^{i \psi({\bf \hat{n}} \cdot {\bf J}) } $ in Section {\bf 6.1.1}, and for the coherent state projector
$ |{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | = \mbox{{\boldmath $\Pi$}}^{(j)}(j,{\bf \hat{n}}) $ in Section {\bf 6.1.2}, we derive a novel recoupling expression for the Chebyshev polynomial operators $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$. In Section {\bf 6.2}, we discuss a distinctly different approach
\cite{filippov2:thesis} to deriving the same recoupling expression. We conclude with specific examples of this recoupling expression in the case of first- and second-rank tensors in
Section {\bf 6.3}.
\subsection{Operator expansion comparisons}
\subsubsection{Rotation operator expansions}
Using the orthonormal basis functions $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$, Corio \cite{corio:siam} derived the following expansion of the rotation operator $\hat{{\cal D}}^{(j)} \!(\psi, {\bf \hat{n}}) = e^{i \psi({\bf \hat{n}} \cdot {\bf J}) }$
\begin{eqnarray}
e^{i \psi({\bf \hat{n}} \cdot {\bf J}) } & = & \sum_{\lambda=0}^{2j} a_{\lambda}^{(j)} \! (\psi) \; f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})\\
& = & \sum_{\lambda=0}^{2J} A(\lambda,j) \, i^{\lambda} \, s^{\lambda} \,
\mbox{ C}_{2j-\lambda}^{\lambda+1}(c)\; f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) \label{Afac}\\
& = & \sum_{\lambda=0}^{2j} i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) \; f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) \label{coriorot}\\
\mbox{where} \;\;A(\lambda, j) & = & (2\lambda)!! \sqrt{2\lambda+1} \sqrt{\displaystyle\frac{(2j-\lambda )!}{(2j+\lambda+1)!}} \\
s & = & \sin(\psi/2) \\
c & = & \cos(\psi/2)\\
\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) & = & \sqrt{\displaystyle\frac{2j+1}{2\lambda+1}} \, A(\lambda, j) \,
s^{\lambda} \,
\mbox{ C}_{2j-\lambda}^{\lambda+1}(c)
\end{eqnarray}
In Eq.(\ref{coriorot}), we have introduced the generalized character functions \cite{varshal1:ang}
$\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi)$ to reexpress the operator expansion in Eq.(\ref{Afac}) that Corio \cite{corio:siam} first presented. These generalized
character functions are defined in terms of
the Gegenbauer polynomials \cite{tem:bk, askey}
$\mbox{ C}_{2j-\lambda}^{\lambda+1}(c)$ as
\begin{equation}
\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) = (2\lambda)!!\sqrt{2j+1}\sqrt{\frac{(2j-\lambda)!}
{(2j+\lambda+1)!}}
\;\;s^{\lambda}\mbox{ C}_{2j-\lambda}^{\lambda+1}
(c) \label{xgegen}
\end{equation}
Using the direct product of spin and spatial tensors expressed in terms of the spherical harmonics \cite{brinksatch:ang}
$Y_{\lambda \mu}(\theta,\phi)$ and the spin polarization tensor operators \cite{varshal1:ang} $\hat{T}_{\lambda \mu}^{(j)}$
as
\begin{eqnarray}
{\bf Y}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) & \equiv & \sum_{\mu=-\lambda}^{\lambda}(-1)^{\mu}\; Y_{\lambda -\mu}({\bf \hat{n}})\;
\hat{T}_{\lambda \mu}^{(j)}=
\sum_{\mu=-\lambda}^{\lambda}Y_{\lambda \mu}^{\star}({\bf \hat{n}}) \; \hat{T}_{\lambda \mu}^{(j)} \label{dir}
\end{eqnarray}
Happer \cite{happer} derived an alternative rotation operator expression.
Making use of the generalized character functions \cite{varshal1:ang}, Varshalovich et al. \cite{ varshal1:ang,varshal2:expansion} subsequently reexpressed Happer's \cite{happer}
partial-wave expansion of the rotation operator $\hat{{\cal D}}^{(j)} \!(\psi, {\bf \hat{n}}) = e^{i \psi({\bf \hat{n}} \cdot {\bf J}) }$ as
\begin{eqnarray}
e^{i \psi({\bf \hat{n}} \cdot {\bf J}) } & = & \sum_{\lambda=0}^{2j} b_{\lambda}^{(j)} \! (\psi) \;
{\bf Y}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) \label{directpro} \\
& = & \sum_{\lambda=0}^{2j} B(\lambda,j)\; i^{\lambda} \, s^{\lambda}
\mbox{ C}_{2j-\lambda}^{\lambda + 1}(c) \; {\bf Y}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J})
\label{Bfac} \\
& = & \sum_{\lambda=0}^{2j} i^{\lambda} \sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\, \mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) \;
{\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) \label{varshalrot} \\
\mbox{where} \;\;B(\lambda, j) & = & \sqrt{4 \pi}\; (2\lambda)!! \; \sqrt{\displaystyle\frac{(2j-\lambda )!}{(2j+\lambda+1)!}}\\
C_{\lambda \mu}({\bf \hat{n}}) & = & \sqrt{\displaystyle\frac{4 \pi}{2\lambda+1}} \; Y_{\lambda \mu}({\bf \hat{n}}) \label{RachSH}
\end{eqnarray}
Comparing Eq.(\ref{Afac}) with Eq.(\ref{Bfac}), we see that
\begin{equation}
A(\lambda,j) \; f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) = B(\lambda, j)\; {\bf Y}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J})
\end{equation}
so that
\begin{eqnarray}
f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) & = & \displaystyle\frac{B(\lambda, j)}{A(\lambda,j)} \;
{\bf Y}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) \\
& = & \sqrt{\displaystyle\frac{4 \pi}{2\lambda+1}} \; {\bf Y}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) \\
& = & {\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) \label{directcorio} \\
& = & \sum_{\mu=-{\lambda}}^{{\lambda}}(-1)^{\mu} \, C_{{\lambda}-{\mu}}({\bf \hat{n}})\;
\hat{T}^{(j)}_{{\lambda}{\mu}}\\
& = & \sum_{\mu=-{\lambda}}^{\lambda}C^{\;\star}_{\lambda\mu}({\bf \hat{n}})\;
\hat{T}^{(j)}_{\lambda\mu} \label{recoupleverb} \\
& = & (-1)^{\lambda} \sqrt{2\lambda +1} \, \Big\{ {\bf T}_{\lambda}({\bf J}) \circledast {\bf C}_{\lambda}({\bf \hat{n}}) \Big\}_{\!0}^{\!0} \label{recouplevera}
\end{eqnarray}
where in Eq.(\ref{recouplevera}), we have introduced the rank-zero composite product tensor (see Appendix E) defined in terms of
Clebsch-Gordan coefficients as
\begin{equation}
\Big\{{\bf R}^{(k)} \circledast {\bf S}^{(k)} \Big\}_{\!0}^{\!0} =\sum_{\substack{q,q^{\prime} \\ q+q^{\prime}=0}} {\bf R}^{(k)}_q \; {\bf S}^{(k)}_{q^{\prime}} \;
C^{00}_{kqkq^{\prime}}
\end{equation}
In his expansion of the rotation operator using the Chebyshev polynomial operators $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) $, Corio \cite{corio:siam} did not express these basis operators as the direct product in Eq.(\ref{directcorio}) of rank-$\lambda$ spatial and spin tensors. On the other hand, Happer \cite{happer} did not identify or recognize this direct product
$ {\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) $ in his expansion of the rotation operator as the Chebyshev polynomial operator
$f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) $. The essential conclusion of this section, embodied in Eq.(\ref{directcorio}), is that the
Chebyshev polynomial operators $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) $ and the direct product $ {\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) $ of rank-$\lambda$ spatial and spin tensors are identical, and this identity leads to the recoupling expression of Eq.(\ref{recouplevera}). In Section
{\bf 6.2}, our discussion is centred around an alternative approach taken by
Filippov \cite{filippov2:thesis} which leads to the same recoupling expression.
\subsubsection{Coherent state projection operator expansions}
The coherent state projector $ |{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j |$ has the following Chebyshev polynomial operator
$ f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) $ expansion
(see Eq.(\ref{cohereproj}) of Section {\bf 3.3.2}):
\begin{equation}
|{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | = \mbox{{\boldmath $\Pi$}}^{(j)}(j,{\bf \hat{n}}) =
\sum_{\lambda=0}^{2j} f_{\lambda}^{(j)}(j) \; f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J} ) \label{cohereproj2}
\end{equation}
In the notation of this article, Ducloy \cite{ducloy} had already obtained the following operator expansion for the coherent state projector
\begin{eqnarray}
|{\bf \hat{n}},j \rangle \langle {\bf \hat{n}},j | & = & \sqrt{4\pi} \, \sum_{\lambda=0}^{2j} \sum_{\mu=-\lambda}^{\lambda}
\frac{(2j)!}{\sqrt{(2j+\lambda+1)!(2j-\lambda)!}} \; Y^{\;\star}_{\lambda\mu}({\bf \hat{n}})\; \hat{T}^{(j)}_{\lambda\mu} \label{duc1}\\
& = & \sum_{\lambda=0}^{2j}
\boxed{\frac{(2j)! \sqrt{2\lambda+1}}{\sqrt{(2j+\lambda+1)!(2j-\lambda)!}}} \; \sum_{\mu=-\lambda}^{\lambda} C^{\;\star}_{\lambda\mu}({\bf \hat{n}})\; \hat{T}^{(j)}_{\lambda\mu} \label{duc2} \\
& = & \sum_{\lambda=0}^{2j} f_{\lambda}^{(j)}(j) \; \boxed{ \sum_{\mu=-\lambda}^{\lambda} C^{\;\star}_{\lambda\mu}({\bf \hat{n}})\;
\hat{T}^{(j)}_{\lambda\mu} } \label{duc3}
\end{eqnarray}
The original expansion obtained by Ducloy \cite{ducloy} in Eq.(\ref{duc1}) has been rewritten in Eqs.(\ref{duc2}) and (\ref{duc3}) in order to facilitate a comparison with Eq.(\ref{cohereproj2}). After using Eq.(\ref{cgeval44}) to replace the ``boxed" coefficient term in Eq.(\ref{duc2}) with the Chebyshev polynomial $ f_{\lambda}^{(j)}(j) $ in
Eq.(\ref{duc3}), it is clear from a comparison of the operator expansions in Eq.(\ref{cohereproj2}) and Eq.(\ref{duc3}) that the
Chebyshev polynomial operator
$ f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) $ can be expressed as the following direct product or recoupling expressions
\begin{eqnarray}
f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J}) & = & \sum_{\mu=-{\lambda}}^{\lambda}C^{\;\star}_{\lambda\mu}({\bf \hat{n}})\;
\hat{T}^{(j)}_{\lambda\mu} \\
& = & {\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) \\
& = & (-1)^{\lambda} \sqrt{2\lambda +1} \, \Big\{ {\bf T}_{\lambda}({\bf J}) \circledast {\bf C}_{\lambda}({\bf \hat{n}}) \Big\}_{\!0}^{\!0} \label{recouplevera2}
\end{eqnarray}
in agreement with Eqs.(\ref{recoupleverb}) and (\ref{recouplevera}) in
Section {\bf 6.1.1}.
\subsection{Exploiting the transformation properties of the Chebyshev polynomial operators $ f_{\lambda}^{(j)}(J_z) $}
Independently established by many workers \cite{meckler:angular,filippov2:thesis,corio:ortho, NormRay, werb:tensor, fillipov1:qubit} over the last 50 years, the relation of Eq.(\ref{jz3})
\begin{equation}
f_{\lambda}^{(j)}(J_z) = \hat{T}_{\lambda 0}^{(j)}
\end{equation}
is an important and useful operator equivalent. In this section, we show how it can provide another independent proof of the operator equivalence between
the Chebyshev polynomials $f_{\lambda}^{(j)}({\bf \hat{n}} \cdot {\bf J})$, and
$ {\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) = \sum_{\mu=-{\lambda}}^{\lambda}C^{\;\star}_{\lambda\mu}({\bf \hat{n}})\;
\hat{T}^{(j)}_{\lambda\mu}$, the direct product of rank-$\lambda$ spatial and spin tensors, namely the (renormalized) Racah spherical harmonics
\cite{brinksatch:ang} $C_{\lambda\mu}({\bf \hat{n}})$ and the spin polarization tensor operators \cite{varshal1:ang} $\hat{T}^{(j)}_{\lambda\mu}$,
respectively.
As irreducible tensor operators, the spin polarization operators $ \hat{T}^{(j)}_{\lambda\nu}$ transform as \cite{varshal1:ang}
\begin{equation}
\hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, \hat{T}^{(j)}_{\lambda\nu} \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger}
= \sum_{\mu=-\lambda}^{\lambda} {\cal{D}}_{\mu \nu}^{(\lambda)}(\theta, {\bf \hat{n}}_{\bot}) \; \hat{T}^{(j)}_{\lambda\mu}
\end{equation}
and in particular, the basis functions of Eq.(\ref{basistransform}) can be rewritten as \cite{filippov2:thesis}
\begin{eqnarray}
f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) & = &
\hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, f_{\lambda}^{(j)}(J_z) \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger} \\
& = & \hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, \hat{T}^{(j)}_{\lambda 0} \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger} \label{filla} \\
& = & \sum_{\mu=-\lambda}^{\lambda} {\cal D}^{(\lambda)}_{\mu 0}(\theta, {\bf \hat{n}}_{\bot})\; \hat{T}^{(j)}_{\lambda \mu} \label{fillb} \\
& = & \sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}(\theta, \phi) \; \hat{T}^{(j)}_{\lambda \mu} \label{mea} \\
& = & \sum_{\mu=-\lambda}^{\lambda} (-1)^{\mu} \; C_{\lambda-\mu}(\theta, \phi) \; \hat{T}^{(j)}_{\lambda \mu} \label{me2a} \\
& = & {\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) \label{dotprod} \label{me3} \\
& = & (-1)^{\lambda} \sqrt{2\lambda +1} \, \Big\{ {\bf T}_{\lambda}({\bf J}) \circledast {\bf C}_{\lambda}({\bf \hat{n}}) \Big\}_{\!0}^{\!0} \label{me4}
\label{recouplever}
\end{eqnarray}
where as shown in Appendix B, the rotation matrix elements
${\cal D}^{(\lambda)}_{\mu 0}(\theta, {\bf \hat{n}}_{\bot}) \equiv C_{\lambda \mu}^{\, \star}(\theta, \phi)$
\cite{brinksatch:ang} of Eq.(\ref{fillb}) are the Racah spherical harmonics of Eq.(\ref{RachSH}). Eqs.(\ref{filla}) and (\ref{fillb}), originally derived by Filippov
\cite{filippov2:thesis} using a Euler angle parametrization, have been reexpressed using an angle-axis parametrization. In Eqs.(\ref{mea} - \ref{me4}), we demonstrate that Filippov's \cite{filippov2:thesis} transformation equation for the tensor operators in Eq.(\ref{fillb}) can then be expressed as a recoupling of spin and spatial tensors. The final expressions for the basis functions $
f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) $ in Eqs.(\ref{recouplever}) and (\ref{recouplevera}) demonstrate that
a Chebyshev polynomial of degree $\lambda$ in the variable $ ({\bf \hat{n}} \cdot {\bf J})$ can be recoupled as a
rank-zero irreducible composite tensor defined by the product of two rank-$\lambda$ tensors, one the spin tensor $ {\bf T}_{\lambda}({\bf J}) $, and the other, the spatial Racah tensor ${\bf C}_{\lambda}({\bf \hat{n}})$:
\begin{equation}
f_{\lambda}^{(j)} \! \left({\bf \hat{n}} \cdot {\bf J} \right)= {\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) =
(-1)^{\lambda} \sqrt{2\lambda +1} \, \Big\{ {\bf T}_{\lambda}({\bf J}) \circledast {\bf C}_{\lambda}({\bf \hat{n}}) \Big\}_{\!0}^{\!0} \label{chebrecoup}
\end{equation}
Alternatively, by determining the Euler angles $(\alpha, \beta, \gamma)$ of the rotation $R \equiv R(\theta,{\bf \hat{n}}_{\bot} ) \equiv
R(\alpha, \beta, \gamma)$ discussed in Section {\bf 3.3.1}, an Euler angle parametrization of the rotation operator
$ \hat{{\cal D}}^{(J)}\!(\alpha, \beta, \gamma) $ can be used to verify Eq.(\ref{me3}). If ${\bf \hat{n}}_{\bot} =(-\sin \phi, \cos \phi, 0)$, then the angle-axis parameters are $(\psi; \Theta, \Phi) =
\left(\theta; \displaystyle\frac{\pi}{2}, \phi + \displaystyle\frac{\pi}{2}\right)$. Since the Euler angles $(\alpha,\beta,\gamma)$ can be expressed in terms of the
angle-axis parameters $(\psi; \Theta, \Phi)$ using the following relations \cite{varshal1:ang}
\begin{eqnarray}
\sin \frac{\beta}{2} & = & \sin \Theta\; \sin \frac{\psi}{2} \nonumber \\
\tan \frac{\alpha + \gamma}{2} & = & \cos \Theta \; \tan \frac{\psi}{2} \nonumber \\
\frac{\alpha - \gamma}{2} & = & \Phi -\frac{\pi}{2}
\label{ieuanglea}
\end{eqnarray}
the Euler angles corresponding to the rotation operator $R(\theta, {\bf \hat{n}}_{\bot}) = e^{-i \theta ({\bf \hat{n}}_{\bot} \cdot {\bf J}) }$ are therefore
$(\phi, \theta, -\phi)$, determined as a solution of the following relations
\begin{eqnarray}
\sin \frac{\beta}{2} & = & \sin \frac{\theta}{2} \rightarrow \boxed{\beta = \theta}\nonumber \\
\tan \frac{\alpha + \gamma}{2} & = & 0 \rightarrow \boxed{\alpha + \gamma =0} \nonumber \\
\frac{\alpha - \gamma}{2} & = & \phi \rightarrow \boxed{\alpha - \gamma =2\phi}
\label{ieuangleb}
\end{eqnarray}
Since irreducible spherical tensors $ \hat{T}_{LM}$ transform as
\begin{equation}
\left[\hat{{\cal D}}^{(J)}(\phi, \theta, -\phi) \right ]^{\!\dagger} \; \hat{T}_{LM} \; \hat{{\cal D}}^{(J)}\!(\phi, \theta, -\phi) \; = \sum_m {\cal{D}}_{mM}^{(L)}(\phi, \theta, -\phi) \; \hat{T}_{LM}
\end{equation}
then in particular
\begin{eqnarray}
f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right)
& = &
\hat{{\cal D}}^{(j)}\!(\phi, \theta, -\phi) \; \hat{T}_{\lambda 0}^{(j)} \; \left[\hat{{\cal D}}^{(j)}(\phi, \theta, -\phi) \right ]^{\!\dagger} \label{fil1} \nonumber \\
& = & \sum_{\mu=-\lambda}^{\lambda} {\cal D}^{(\lambda)}_{\mu 0}(\phi, \theta, -\phi)\; \hat{T}^{(j)}_{\lambda \mu} \label{fil2} \nonumber \\
& = & \sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}(\theta, \phi) \; \hat{T}^{(j)}_{\lambda \mu} \label{me1}
\;\;(\mbox{using \cite{brinksatch:ang}}\;\; {\cal{D}}_{\mu 0}^{(\lambda)}(\alpha, \beta, \gamma) = C_{\lambda \mu}^{\star}(\beta, \alpha)) \nonumber
\\
& = & \sum_{\mu=-\lambda}^{\lambda} (-1)^{\mu} \; C_{\lambda-\mu}(\theta, \phi) \; \hat{T}^{(j)}_{\lambda \mu} \label{me2b} \nonumber \\
& = & {\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) \nonumber \\
& = & (-1)^{\lambda} \sqrt{2\lambda +1} \; \Big\{ {\bf T}_{\lambda}({\bf J}) \circledast {\bf C}_{\lambda}({\bf \hat{n}}) \Big\}_{\!0}^{\!0} \label{recoop}
\end{eqnarray}
where ${\bf C}_{\lambda}({\bf \hat{n}})$ is the renormalized Racah spherical harmonics (tensor)
\cite{brinksatch:ang}.
\subsection{Specific recoupling examples: first- and second-rank tensors}
In order to elicit the recoupling feature of the relation in Eq.(\ref{recoop}), we consider the more familiar and commonly encountered cases of first-rank
$(\lambda=1)$ and second-rank $(\lambda=2)$ spherical tensors.
\subsubsection{First-rank tensors} To illustrate our approach, we begin with the trivial case of first-rank spherical tensors. The spin polarization operators
$ {\bf T}_{1}({\bf J}) $ in Eq.(\ref{dotprod}) can be replaced with their spherical operator equivalents
{\boldmath $\mathcal{T}$}$\!_1 = {\bf J} $ using the following relation \cite{varshal1:ang,ambler:traces}
\begin{eqnarray}
{\bf T}_{1}({\bf J}) & = & a_1(j)\; \mbox{\boldmath $\mathcal{T}$}\!_{1} = a_1(j) \; {\bf J} \label{firstrankrel} \\
\mbox{where} \;\;\;\; a_1(j) & = & \sqrt{\displaystyle\frac{3}{j(j+1)(2j+1)}}
\end{eqnarray}
Component-wise, the relation of Eq.(\ref{firstrankrel}) can be expressed as the following proportionality between the spin polarization operators
$ \hat{T}^{(j)}_{1M}$ and the spherical components of the spin operator $J_M$ as
\begin{equation}
\hat{T}^{(j)}_{1M} = a_1(j) \, J_M \;\;\;\;\;(M= \pm 1,0)
\end{equation}
The Chebyshev polynomial operator $f_{1}^{(j)}({\bf \hat{n}} \cdot {\bf J}) $ can be replaced with $ ({\bf \hat{n}} \cdot {\bf J})$ using the following relation \cite{corio:ortho}
\begin{eqnarray}
f_{1}^{(j)}({\bf \hat{n}} \cdot {\bf J}) & = & a_1(j) \; ({\bf \hat{n}} \cdot {\bf J})
\end{eqnarray}
After these replacements, in this special case of first-rank spherical tensors, Eq.(\ref{chebrecoup}) can then be written as
\begin{eqnarray}
({\bf \hat{n}} \cdot {\bf J}) = {\bf J} \cdot {\bf C}_{1}({\bf \hat{n}}) & = & ({\bf \hat{n}} \cdot {\bf J}) \;\;\;\;\;\;\;\;\label{rereb} \\
\mbox{where \cite{brinksatch:ang}} \;\;\;\; {\bf C}_{1 }({\bf \hat{n}}) & = & {\bf \hat{n}}
\end{eqnarray}
\subsubsection{Second-rank tensors} While the result of Eq.(\ref{rereb}) is self-evident, the case of second-rank tensors is much more revealing. The spin polarization operators $ {\bf T}_{2}({\bf J}) $ in Eq.(\ref{dotprod}) can be replaced with their spherical operator equivalents
{\boldmath $\mathcal{T}$}$\!_2 = \sqrt{6} \; \left\{ {\bf J} \circledast {\bf J} \right\}^2 $ using the following relation \cite{ambler:traces}
\begin{eqnarray}
{\bf T}_{2}({\bf J}) & = & a_2(j)\; \mbox{\boldmath $\mathcal{T}$}\!_{2} = a_2(j) \sqrt{6} \left\{ {\bf J} \circledast {\bf J} \right\}^2 \label{secondrankrel} \\
\mbox{where} \;\;\;\; a_2(j) & = & \displaystyle\frac{\sqrt{5}}
{\sqrt{j(j+1)(2j+3)(2j-1)(2j+1)}} \label{tensorcoeff}
\end{eqnarray}
Component-wise, the relation of Eq.(\ref{secondrankrel}) can be expressed as the following proportionalities between the spin polarization operators
$ \hat{T}^{(j)}_{2M}$ and the spherical components of the spin operator $ \left\{ {\bf J} \circledast {\bf J} \right\}_{\!M}^2 $ as
\begin{eqnarray}
\hat{T}^{(j)}_{20} & = & \sqrt{6} \left\{ {\bf J} \circledast {\bf J} \right\}_0^2 \\
\hat{T}^{(j)}_{2\pm 1} & = & \sqrt{6} \left\{ {\bf J} \circledast {\bf J} \right\}_{\pm 1}^2 \\
\hat{T}^{(j)}_{2 \pm 2} & = & \sqrt{6} \left\{ {\bf J} \circledast {\bf J} \right\}_{\pm 2}^2
\end{eqnarray}
The Chebyshev polynomial operator $f_{2}^{(j)}({\bf \hat{n}} \cdot {\bf J}) $ can be replaced with $ \left[ 3 ({\bf \hat{n}} \cdot {\bf J})^2 - ({\bf J} \cdot {\bf J}) \right]$ using the following relation \cite{corio:ortho}
\begin{eqnarray}
f_{2}^{(j)}({\bf \hat{n}} \cdot {\bf J}) & = & a_2(j) \! \left[ 3 ({\bf \hat{n}} \cdot {\bf J})^2 - ({\bf J} \cdot {\bf J}) \right] \label{f2dip} \\
\mbox{where} \;\;\;\; ({\bf J} \cdot {\bf J}) & = & \kappa \, \mathds{1} =j(j+1) \, \mathds{1}
\end{eqnarray}
After these replacements, in this special case of second-rank spherical tensors, Eq.(\ref{chebrecoup}) can be written as
\begin{eqnarray}
\left[ 3 ({\bf \hat{n}} \cdot {\bf J})^2 - ({\bf J} \cdot {\bf J}) \right] = \sqrt{6} \left\{ {\bf J} \circledast {\bf J} \right\}^2
\cdot {\bf C}_{2 }({\bf \hat{n}}) & = & 3\sqrt{5} \; \Big\{ \left\{ {\bf J} \circledast {\bf J} \right\}^2 \circledast
\left\{ {\bf \hat{n}} \circledast {\bf \hat{n}} \right\}^2 \Big\}_{\!0}^{\!0} \;\;\;\;\;\;\;\;\label{rere} \\
\mbox{where \cite{brinksatch:ang}} \;\;\;\; {\bf C}_{2 }({\bf \hat{n}}) & = & \sqrt{\displaystyle\frac{3}{2}} \left\{ {\bf \hat{n}} \circledast {\bf \hat{n}} \right\}^2
\end{eqnarray}
This recoupling of two irreducible second-rank spherical tensors in Eq.(\ref{rere}) is exactly analogous to the recoupling of the magnetic dipolar
interaction \cite{abragamtext} or the tensor force \cite{brinksatch:ang}
\begin{equation}
\left[ 3 ({\bf \hat{n}} \cdot {\bf J}_1)({\bf \hat{n}} \cdot {\bf J}_2) - ({\bf J}_1 \cdot {\bf J}_2) \right] =
3\sqrt{5} \; \Big\{ \left\{ {\bf J}_1 \circledast {\bf J}_2 \right\}^2 \circledast
\left\{ {\bf \hat{n}} \circledast {\bf \hat{n}} \right\}^2 \Big\}_{\!0}^{\!0} \label{rere2}
\end{equation}
Normally, the recoupling relationships of Eqs.(\ref{rere}) or (\ref{rere2}) can only be established by a laborious expansion of both sides of these equations, or by using the formalism of angular momentum theory \cite{brinksatch:ang} to recouple the four rank-1 tensors on the right-hand side of these equations via a
9$j$-symbol as outlined in Appendix E. Instead, the recoupling relation of Eq.(\ref{rere}) arises very simply and directly just from the definition of the Chebyshev polynomial operator
$f_{2}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) $, examples of which are provided in Table I for $j=1/2,1$ and $3/2$.
\subsubsection{Matrix elements of the truncated homonuclear dipolar Hamiltonian} From the recoupling expression in Eq.(\ref{mea}) applied to
$f_{2}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) $, the general matrix elements are
\begin{eqnarray}
\langle jm| \, f_{\lambda}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) |jm^{\prime} \rangle & = &
\sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}(\theta, \phi) \; \langle jm| \, \hat{T}^{(j)}_{\lambda \mu} |jm^{\prime} \rangle \\
& = &
\sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}(\theta, \phi) \; \sqrt{\frac{2\lambda+1}{2j+1}} \; C_{jm^{\prime}\lambda \mu}^{jm} \\
& = & C_{\lambda \, (m-m^{\prime})}^{\star}(\theta, \phi) \; C_{jmj-m^{\prime}}^{\lambda \, (m-m^{\prime})} \; (-1)^{j-m^{\prime}} \label{f2matrix}
\end{eqnarray}
The diagonal matrix elements $(m= m^{\prime})$ for $f_{2}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) $ are therefore
\begin{eqnarray}
\langle jm| \, f_{2}^{(j)} \! \left( {\bf \hat{n}} \cdot {\bf J} \right) |jm \rangle & = &
C_{20}^{\star}(\theta, \phi) \; C_{jmj-m}^{20} \; (-1)^{j-m} \\
& = & P_2(\cos \theta) \; f_2^{(j)}(m) \\
& = & P_2(\cos \theta) \; \frac{3m^2-2}{\sqrt{6}} \;\;\; \;\;\; (j=1)
\end{eqnarray}
But then using Eqs.(\ref{tensorcoeff}) and (\ref{f2dip}),
\begin{eqnarray}
f_{2}^{(j)}({\bf \hat{n}} \cdot {\bf J}) & = & a_2(j) \! \left[ 3 ({\bf \hat{n}} \cdot {\bf J})^2 - ({\bf J} \cdot {\bf J}) \right] \label{f2dipb} \\
& = & \frac{1}{\sqrt{6}} \; \left[ 3 ({\bf \hat{n}} \cdot {\bf J})^2 - ({\bf J} \cdot {\bf J}) \right] \;\;\; \;\;\; (j=1)
\end{eqnarray}
we find that the diagonal matrix elements of $ \left[ 3 ({\bf \hat{n}} \cdot {\bf J})^2 - ({\bf J} \cdot {\bf J}) \right] $ for $j=1$ are given by
\begin{equation}
\langle m| \, \left[ 3 ({\bf \hat{n}} \cdot {\bf J})^2 - ({\bf J} \cdot {\bf J}) \right] \, |m \rangle
= P_2(\cos \theta) \left[ 3m^2-2 \right] \label{ddiag}
\end{equation}
The homonuclear dipolar Hamiltonian ${\cal H}_{D}^{\mbox{intra}}$ \cite{abragamtext,keller}, which describes the through-space interaction between the magnetic moments of two adjacent protons separated by a distance $r$ is given by
\begin{eqnarray}
{\cal H}_{D}^{\mbox{intra}} & = & \frac{\mu_0 \gamma^2 \hbar}{4\pi r^3} \; \left[ 3 ({\bf \hat{n}} \cdot {\bf J}_1)({\bf \hat{n}} \cdot {\bf J}_2) - ({\bf J}_1 \cdot {\bf J}_2) \right] \label{fulldip} \\
\mbox{where} \;\;\;\; {\bf \hat{n}} & = & {\bf \hat{r}}
\end{eqnarray}
This interaction Hamiltonian can also be expressed in terms of the total spin operator \cite{keller} ${\bf J} = {\bf J}_1 + {\bf J}_2 $ for each pair of protons as
\begin{equation}
{\cal H}_{D}^{\mbox{intra}} = \frac{\mu_0 \gamma^2 \hbar}{4\pi r^3} \; \left[ 3 ({\bf \hat{n}} \cdot {\bf J})^2 - ({\bf J} \cdot {\bf J}) \right]
\end{equation}
High-field approximation of this Hamiltonian leaves only the truncated part \cite{abragamtext}
\begin{eqnarray}
\left[{\cal H}_{D}^{\mbox{intra}}\right]^{\prime} = \frac{\mu_0 \gamma^2 \hbar}{4\pi r^3} \; P_2(\cos \theta) \left[ 3 J_z^2 - ({\bf J} \cdot {\bf J}) \right]
\label{truncatedip}
\end{eqnarray}
where $\theta$ is the angle between the internuclear vector ${\bf r}$ and the magnetic field ${\bf B}_0$.
The form of $\left[{\cal H}_{D}^{\mbox{intra}}\right]^{\prime} $ is identical to that of spin-1 systems, provided the ``quadrupole frequency" $\nu_Q$ is redefined as \cite{keller}
\begin{equation}
\nu_Q = \frac{3 \hbar \gamma^2}{4 \pi r^3}
\end{equation}
If a basis of the total angular momentum eigenstates of ${\bf J} = {\bf J}_1 + {\bf J}_2 $ is used, it is evident from Eq.(\ref{ddiag}) that the diagonal elements of the truncated Hamiltonian $\left[{\cal H}_{D}^{\mbox{intra}}\right]^{\prime} $ of Eq.(\ref{truncatedip}) match the diagonal elements of the diagonal elements of
the dipolar Hamiltonian ${\cal H}_{D}^{\mbox{intra}} $ of Eq.(\ref{fulldip}). The latter elements are just what we expect from the secular
approximation \cite{abragamtext}, which retains only that part of the dipolar Hamiltonian $\left[{\cal H}_{D}^{\mbox{intra}}\right]^{\prime} $ which commutes with the Zeeman Hamiltonian ${\cal H}_Z $
\begin{equation}
\left[ \left[{\cal H}_{D}^{\mbox{intra}}\right]^{\prime} \!\!, {\cal H}_Z \right] =0
\end{equation}
\section{Operator Equivalents}
Finding suitable operator equivalents for irreducible tensor operators in terms of the familiar angular momentum operators
$J_z, J_{\pm} = J_x \pm iJ_y$, and
$({\bf J} \cdot {\bf J}) = J^2 \equiv J(J+1)$ has been a long-standing challenge in fields such as magnetic resonance. As a result, there is an
extensive literature over more than seven decades devoted to this problem in NMR, EPR and ENDOR. A reasonably comprehensive list of the salient literature references on operator equivalents can be found in a recent publication by Ryabov \cite{ryabov:equiv}. In their studies, Ryabov \cite{ryabov:equiv} himself, as well as Ambler et al. \cite{ambler:traces}, Ohlsen \cite{ohlsen}, and Biedenharn and Louck \cite{biedenharn}, for example, have tabulated a significant number of these operator equivalents. Few however are those studies that focus on the operator equivalences between the spin polarization operators \cite{varshal1:ang} $\hat{T}_{\lambda \mu}^{(j)}$ and the Chebyshev polynomial operators
$f_{\lambda}^{(j)} (J_z) $. Work by Meckler \cite{meckler:angular}, Corio \cite{corio:ortho}, Marinelli et al. \cite{werb:tensor} and by Normand and Raynal \cite{NormRay} constitutes the essential core of the research that has elucidated the role of Chebyshev polynomials in developing operator equivalents, and in this section, we provide a brief summary of this work.
In considering the history of how Chebyshev polynomial operators $f_{\lambda}^{(j)} (J_z) $ were used to develop operator equivalents, two developments stand out. First,
the operator equivalents for projection-zero tensor operators $\hat{T}_{\lambda 0}^{(j)}$ were identified as the Chebyshev polynomial
operators $f_{\lambda}^{(j)} \!(J_z)$ \cite{meckler:angular,corio:ortho, werb:tensor,NormRay}. Second, an intriguing relationship between irreducible (spherical) tensor operators of arbitrary projection $T^{(j)}_{\lambda \mu}$ and successive partial derivatives of the Chebyshev
polynomial operators $f^{(j)}_{\lambda} \!(J_z)$ \cite{werb:tensor} was discovered \cite{werb:tensor}. In this section, we begin in Section {\bf 7.1} by revisiting a Chebyshev polynomial operator equivalent for projection-zero tensor operators $\hat{T}_{\lambda 0}^{(j)}$ that we had used in Section {\bf 3.1}. Then, in Section {\bf 7.2}, we use Table VIII to illustrate that Chebyshev polynomial operators $f^{(j)}_{\lambda} (J_z) $ have been used to develop operator equivalents for any irreducible tensor operator $\hat{T}_{\lambda \mu}^{(j)}$.
\subsection{Operator equivalent matrix elements}
As we illustrated in Section {\bf 3.1}, among the most useful operator equivalences involving the Chebyshev polynomial operators is
\begin{equation}
\hat{T}_{L0}^{(j)} \equiv f_L^{(j)} \!(J_z) \label{tensorcheb}
\end{equation}
Meckler's \cite{meckler:angular} recognition of this equivalence between the Chebyshev polynomial operators $f_{L}^{(j)} (J_z) $ and the projection-zero spin polarization operators \cite{varshal1:ang} $\hat{T}_{L0}^{(j)}$ anticipated subsequent work by several
investigators \cite{filippov2:thesis,corio:ortho,NormRay,werb:tensor,fillipov1:qubit}, who independently established this operator equivalence, and who provided specific and extensive tables illustrating this operator equivalence \cite{corio:ortho,NormRay,werb:tensor}.
Ambler et al. \cite{ambler:traces} tabulated operator equivalents for the irreducible tensor operators $T_q^{(k)}\!(J)$
(equivalent to the spin polarization operators $ \hat{T}_{kq}^{(J)}$ in our notation), as did Corio
\cite{corio:ortho} for his orthonormal operator basis $U^{(n)}_r \!(J)$, whose relation to the irreducible tensor operators $T_q^{(k)}\!(J)$ was determined by
these relations \cite{corio:ortho}
\begin{eqnarray}
n+r & = & k \nonumber \\
r=q
\end{eqnarray}
In addition to showing how to construct operator equivalents for his orthonormal basis $U^{(n)}_r \!(J)$, Corio \cite{corio:ortho}, unaware of Meckler's earlier work \cite{meckler:angular}, also independently identified the
$U^{(k)}_0 \!(J) \equiv \hat{T}_{k0}^{(J)}$ operators, the tensors of rank $k$ and projection $0$, as the Chebyshev polynomial operator equivalents $f_k^{(J)} \! (J_z)$. Although a few others \cite{werb:tensor,NormRay} have also independently made this identification of the $\hat{T}_{k0}^{(J)}$ operators with
the Chebyshev polynomial operators $f_k^{(J)} \! (J_z)$, neither they nor others
\cite{biedenharn,ryabov:equiv,ohlsen} who have developed operator equivalents for $\hat{T}_{k0}^{(J)}$ tensors have cited the original work by
Meckler \cite{meckler:angular} and Corio \cite{corio:ortho} .
\subsection{A comparison of operator equivalents}
In Table VIII, we compare specific examples of tensor operator equivalents developed by Corio \cite{corio:ortho} (top row) with those developed by Marinelli et al. \cite{werb:tensor} (bottom row). The examples have been chosen to demonstrate that all the equivalents have the same form (to within
$j$-dependent normalization constants). The first column compares operator equivalents for the irreducible tensor operators $\hat{T}_{60}^{(j)}$, which both Corio \cite{corio:ortho} and
Marinelli et al. \cite{werb:tensor} recognized as equivalent to the Chebyshev polynomial operator $f^{(j)}_6(J_z)$. The second column compares operator equivalents for the irreducible tensor operators $\hat{T}_{41}^{(j)}$. Although both equivalents have the same form, Marinelli et al. \cite{werb:tensor} went one step further than
Corio \cite{corio:ortho} did to show that any irreducible (spherical) tensor operator $T^{(k)}_L\! (S) $ of rank $L$ and
projection $k>0$ could be represented by suitable linear combinations of successive partial derivatives (with respect to $S_z$) of the Chebyshev
polynomial operators $f^{(S)}_L (S_z) $ as follows
\begin{eqnarray}
T^{(k)}_L \!(S) & \sim & \left(S_{+}\right)^{|k|} \displaystyle\sum_{n=|k|}^L \; A_{Ln}^k\! \left(\displaystyle\frac{\partial}{\partial S_z}\right)^{\!\!n}
\!\left[ f^{(S)}_L (S_z) \right]
\end{eqnarray}
A table of the required coefficients $ A_{Ln}^k$ was provided by Marinelli et al. \cite{werb:tensor}.
As an example of this linear combination of successive partial derivatives, the table entry in Table VIII for Marinelli's \cite{werb:tensor} operator equivalent of $ T^{(1)}_4 \!(S)$ we have
calculated as follows
\begin{eqnarray}
& & \displaystyle\sum_{n=1}^4 \; A_{4n}^1\! \left(\displaystyle\frac{\partial}{\partial S_z}\right)^{\!\!n}
\!\left[ f^{(S)}_4 (S_z) \right] \\
& = & \frac{1}{1!} [14 \,S_z^3 - (6K-5)S_z]
+ \frac{1}{2!} [42 \, S_z^2 -(6K-5)]
+ \frac{1}{3!} [84 \, S_z]
+ \frac{1}{4!}[84] \, \mathds{1} \;\;\;\;\;\;\;\; \label{werbsum} \\
& = & 14 S_{\!z}^3 + 21S_z^2 +(19-6K) S_z +3(2-K)\, \mathds{1}
\end{eqnarray}
$\mbox{where} \;\;\;\; f^{(S)}_4 (S_z) \sim 35 S^4_z -5 S^2_z (6K-5)+3K(K-2)$. Each summand in Eq.(\ref{werbsum}) is the product of a partial-derivative of $f^{(S)}_4 \! (S_z)$ (in square brackets), and coefficients
$A_{4n}^1 $ tabulated in Marinelli et al. \cite{werb:tensor}.
In conclusion, the examples discussed in Table VIII illustrate the fact that Chebyshev polynomial operators $f^{(S)}_L (S_z) $ can be used to develop an operator equivalent for any irreducible tensor operator.
\section{Conclusion}
The Chebyshev polynomials $f_{\lambda}^{(j)} (m) $ first introduced in a physics application by Meckler
\cite{meckler:majorana,meckler:angular}, are special
functions \cite{askey}, a particular case of one member of the family of classical orthogonal polynomials of a discrete variable known as the Hahn polynomials \cite{bateman,nikiforov2,Nikiforov,vilenkin:specfuncbook,olver}. Beginning with a close examination of the Meckler formula \cite{meckler:majorana,meckler:angular} for spin transition probabilities, we have described how Chebyshev polynomials of a discrete variable can be applied in physics. Applications of these very special special functions include spin physics, spin tomography, and the development of operator expansions and operator equivalents.
Beyond their role as special functions, the Chebyshev polynomials $f_{\lambda}^{(j)} (m)$ double as the Clebsch-Gordan coupling coefficients (3$j$-symbols \cite{brinksatch:ang}) $C_{jmj-m}^{\lambda \, 0}$ of angular momentum theory:
\begin{equation}
f_{\lambda}^{(j)} (m) = (-1)^{j-m} \; C_{jmj-m}^{\lambda \, 0} \label{chebyclebschb}
\end{equation} We have in this article often taken advantage of this duality of the Chebyshev polynomials to prove identities from their Clebsch-Gordan coupling coefficient homologs. The Chebyshev polynomial operators
$f_{\lambda}^{(j)} (J_z) $ are identical to the projection-zero spin polarization operators $\hat{T}_{\lambda 0}^{(j)}$
\begin{equation}
f_{\lambda}^{(j)} \!(J_z) \equiv \hat{T}_{\lambda 0}^{(j)}
\end{equation}
a relationship first recognized by Meckler \cite{meckler:angular}.
The similarity transform of this equivalence defines the Chebyshev polynomial operators $f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) $ as the direct product
of two rank-$\lambda$ tensors, one the spin tensor ${\bf T}_{\lambda}({\bf J})$, and the other, the spatial Racah tensor ${\bf C}_{\lambda}({\bf \hat{n}})$ \cite{brinksatch:ang}:
\begin{equation}
f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) = {\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J})
\end{equation}
Whether we consider the Chebyshev polynomial scalars $f_{\lambda}^{(j)} (m)$, or the
Chebyshev polynomial operators $f_{\lambda}^{(j)} (J_z) $ and $f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) $, the Chebyshev polynomials are distinguished as special functions by their very close connection to angular momentum theory and spin physics. As we have described in Section {\bf 4}, a vivid reminder of this connection is the Meckler formula
\cite{meckler:majorana,meckler:angular} for the spin transition probability
$\mbox{ P}^{(j)}_{mm^{\prime}} (t)$, whose calculation relies on angular momentum theory \cite{meckler:angular,schwinger:majorana}. As we have described in Section {\bf 5}, additional reminders of this connection are the Chebyshev polynomial operator expansions of projection operators, the rotation operator, the Stratonovich-Weyl operator, and the tomographic reconstruction of the density operator.
\newpage
\section{Appendix A}
\subsection{Legendre polynomial operators}
In this section, two versions of Legendre polynomial operators are discussed, one version due to Zemach \cite{zemach}, and the other due to
Schwinger \cite{schwinger:majorana}. The Legendre polynomial operators $ \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $ defined by Zemach \cite{zemach} are closely related to the Chebyshev polynomial operators $ f_{\lambda} ^{(j)} ({\bf \hat{n}} \cdot {\bf J})$. On the other hand, the matrix elements of the Legendre polynomial operators $P_{\lambda}({\bf J}) $ defined by Schwinger \cite{schwinger:majorana} are closely related to the Chebyshev polynomials $f_{\lambda}^{(j)} (m) $.
\subsubsection{Legendre polynomial operators $ \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $}
From the addition theorem for spherical harmonics, it is well-known that the direct product of rank-$\lambda$ spherical harmonics tensors can be expressed in scalar form as the following Legendre polynomial of order $\lambda$:
\begin{equation}
{\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf C}_{\lambda}({\bf \hat{n}}^{\prime}) = P_{\lambda} ({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime}) \label{legscal}
\end{equation}
Not so well-known is how the direct product of rank-$\lambda$ spatial and spin tensors ${\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J})$ could be expressed.
Certainly the direct product of rank-$\lambda$ spatial and spin tensors ${\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J})$ could not be a scalar function of $({\bf \hat{n}} \cdot {\bf J}) $, but an operator function of $({\bf \hat{n}} \cdot {\bf J}) $. Could it be expressed as a Legendre polynomial operator? In developing tensor representations for application to angular-momentum problems in elementary-particle reactions, Zemach \cite{zemach} was the first to develop such an expression. In the notation of this article, this direct product takes the following form
\begin{eqnarray}
{\bf C}_{\lambda}({\bf \hat{n}}) \cdot {\bf T}_{\lambda}({\bf J}) & = & \sqrt{\displaystyle\frac{2\lambda+1}{(2j+1) \left[ {\bf J}^2 \right]^{l} }} \; \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \label{legop} \\
\mbox{where}\;\;\;\; \left[ {\bf J}^2 \right]^{l} & = & \prod_{n=0}^{l}
\left[ {\bf J}^2 -\tfrac{1}{4}(n^2-1) \right]
\end{eqnarray}
Just as the addition theorem of Eq.(\ref{legscal}) can be taken as the definition \cite{zemach} of the Legendre polynomials
$P_{\lambda} ({\bf \hat{n}} \cdot {\bf \hat{n}}^{\prime}) $, the direct product expression of Eq.(\ref{legop}) can be used to define \cite{zemach} the Legendre polynomial operators $\overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $. Comparing this expression with the corresponding direct product relations of
Eq.(\ref{directcorio}) or (\ref{dotprod}) also defines the Chebyshev polynomial operators $ f_{\lambda} ^{(j)} ({\bf \hat{n}} \cdot {\bf J})$ in terms of the Legendre polynomial operators $\overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $:
\begin{equation}
\overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) =
\sqrt{\displaystyle\frac{(2j+1) \left[ {\bf J}^2 \right]^{l} }{2\lambda+1}} \, f_{\lambda} ^{(j)} ({\bf \hat{n}} \cdot {\bf J})
\end{equation}
The recursion relation for the Legendre polynomials $P_{\lambda}(x)$ can be compared with its counterpart for the
Legendre polynomial operators $ \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $ \cite{zemach}:
\begin{eqnarray}
(2\lambda+1)\, x \, P_{\lambda}(x) & = & (\lambda+1) \, P_{\lambda+1}(x) + \lambda \, P_{\lambda-1}(x) \\
(2\lambda+1) ({\bf \hat{n}} \cdot {\bf J}) \, \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) & = & (\lambda+1)\,
\overline{P}_{\lambda+1}({\bf \hat{n}} \cdot {\bf J}) + \lambda
\left[ {\bf J}^2 -\tfrac{1}{4} (\lambda^2-1)\right] \overline{P}_{\lambda-1}({\bf \hat{n}} \cdot {\bf J}) \;\;\; \;\;\; \;\;\; \;\;\; \label{zemachrecur}
\end{eqnarray}
The recursion relation of Eq.(\ref{zemachrecur}) was used to generate the
Legendre polynomial operators $ \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $ \cite{zemach} tabulated in Table IX. Inspection of this table confirms
Zemach's observation \cite{zemach} that although the functional forms of the Legendre polynomial operators $ \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J})$ agree with those of the Legendre polynomials $P_{\lambda}(\cos \theta)$ for $\lambda=0,1,2$, the non-commutivity of components of ${\bf J}$ makes a difference for $\lambda \geq 3$. As we shall see in the next section, the same issue arises for the same reasons when comparing the
Legendre polynomial operators $P_{\lambda}({\bf J})$ introduced by Schwinger \cite{schwinger:majorana} with the Legendre polynomials $P_{\lambda}(\cos \theta)$.
\subsubsection{Legendre polynomial operators $P_{\lambda}({\bf J})$}
In Section {\bf 4.1.2}, we discussed how Schwinger \cite{schwinger:majorana} calculated the spin transition probability $\mbox{ P}^{(j)}_{mm^{\prime}} (t)$. Using Schwinger's notation \cite{schwinger:majorana}, the final outcome of his calculation of the spin transition probability took the following form
\begin{equation}
\mbox{ P}^{(j)}_{mm^{\prime}} (t) = \left|{\cal D}_{m m^{\prime}}^{(j)}(\psi, {\bf \hat{n}})\right|^2 = \frac{1}{(2j+1)}
\sum_{l=0}^{2j} (2l+1)\, P_{l}(j,m) \; P_{l}(j,m^{\prime}) \; P_{l}(\cos \beta) \label{probswing}
\end{equation}
in which the $P_{l}(j,m) $ functions were defined by Schwinger \cite{schwinger:majorana} as the matrix elements of the Legendre
polynomial operators $P_{l}({\bf J})$:
\begin{equation}
P_{l}(j,m) = \langle jm| \, P_{l}({\bf J}) \, | jm \rangle \label{ppdeff}
\end{equation}
Schwinger \cite{schwinger:majorana} introduced the Legendre polynomial operators $P_{l}({\bf J})$ by expressing them in terms of the solid harmonic functions of the angular momentum vector ${\bf J}$:
\begin{eqnarray}
P_{l}({\bf J}) & = & \left[ \frac{2l+1}{4\pi} \left[ {\bf J}^2 \right]^{\!l} \right]^{-1/2} \! \! \mathcal{Y}_{l 0}({\bf J}) \label{solidharm} \\
\mbox{where \cite{ournote}} \;\;\;\;\; \left[ {\bf J}^2 \right]^{\!l} & = & \prod_{n=0}^{l-1}
\left[ {\bf J}^2 - \bfrac{n}{2}\left( \bfrac{n}{2}\ +1 \right) \right]
\end{eqnarray}
The $ \mathcal{Y}_{lm}({\bf J})$ are operator analogues \cite{ligarg} of the solid harmonics $r^l \, Y_{lm} ({\bf \hat{n}})$ \cite{varshal1:ang}, which include an extra factor $r^l$ over the surface harmonics $Y_{lm} ({\bf \hat{n}})$. They are in fact irreducible tensor operators, proportional to the spin polarization operators $ \hat{T}^{(j)}_{lm}$ according to the following relation
\begin{equation}
\mathcal{Y}_{lm}({\bf J}) = \left[ \frac{2j+1}{4\pi} \left[ {\bf J}^2 \right]^{\!l} \right]^{\!1/2} \hat{T}^{(j)}_{lm} \label{solidpolar}
\end{equation}
It is evident from this proportionality that $Y_{lm} ({\bf \hat{n}})$ and $ \mathcal{Y}_{lm}({\bf J})$ transform identically under rotations.
They satisfy the following trace relation \cite{ligarg}
\begin{eqnarray}
\mbox{Tr} \! \left[ \mathcal{Y}_{lm}({\bf J}) \; \mathcal{Y}^{\dagger}_{l^{\prime}m^{\prime}}({\bf J}) \right] & = & \frac{2j+1}{4\pi} (a_{jl})^2 \;
\delta_{ll^{\prime}}\; \delta_{mm^{\prime}} \\
\mbox{where} \;\;\;\; (a_{jl})^2 & = & \prod_{n=1}^{l} \left[ \left(j+\tfrac{1}{2} \right)^{\!2}-\tfrac{1}{4} n^2 \right] \equiv \left[ {\bf J}^2 \right]^{\!l}
\end{eqnarray}
The solid harmonic operator functions $ \mathcal{Y}_{l 0}({\bf J})$ in Eq.(\ref{solidharm}) were defined \cite{schwinger:majorana} by the generating function
\cite{schwinger:majorana,schwingerosti,schwingerbiedenvandam,schwingerbook}
\begin{eqnarray}
& &\frac{1}{2^{l}l!} \left[ \frac{2l+1}{4\pi} \right]^{\!1/2} \left[-z_{+}^2(J_x +iJ_y) + z_{-}^2(J_x-iJ_y)+2z_{+}z_{-}J_z \right]^{l}
\nonumber \\
& = & \sum_{m=-l}^{l} \; \frac{ z_{+}^{l+m}\; z_{-}^{l-m} } { \left[(l+m)!(l-m)!\right]^{1/2}} \;
\mathcal{Y}_{lm}({\bf J}) \label{swinggen}
\end{eqnarray}
After evaluating $ \mathcal{Y}_{l 0}({\bf J})$ with this generating function, Eq.(\ref{solidharm}) was used to tabulate examples of the
Legendre polynomial operators $P_{l}({\bf J})$ in Table IX. Recognizing that the $ \mathcal{Y}_{lm}({\bf J})$ are operator analogues of the solid harmonics $r^l \, Y_{lm} ({\bf \hat{n}})$ \cite{ligarg}, we should expect to see a difference in the functional forms of $P_{l}({\bf J}) \sim \mathcal{Y}_{l 0}({\bf J}) $ and
$P_l(\cos \theta) \sim Y_{l0}({\bf \hat{n}})$ because the order of factors in the operator case is significant. Indeed, as this table demonstrates, for
$\lambda \geq 3$, we can expect such a difference taking into account the angular momentum commutation relations
\begin{equation}
\left[J_i,J_j \right] = J_k \;\;\;\;(i,j,k =x,y,z)
\end{equation}
Using the relation of Eq.(\ref{solidpolar}), and Eq.(\ref{solidharm}), the matrix element of Eq.(\ref{ppdeff}) which defines the $P_{l}(j,m) $ functions which appear in the transition probability formula of Eq.(\ref{probswing}) can be related to the Chebyshev polynomials of a discrete variable
$f^{(j)}_{l} (m) $ as follows
\begin{eqnarray}
P_{l}(j,m) = \langle jm| \, P_{l}({\bf J}) \, | jm \rangle & = & \sqrt{ \frac{2j+1}{2l+1}} \langle jm| \, \hat{T}^{(j)}_{l 0} \, | jm \rangle \nonumber \\
& = & \sqrt{ \frac{2j+1}{2l+1}} \langle jm| \, f^{(j)}_{l} (J_z) \, | jm \rangle \nonumber \\
& = & \sqrt{ \frac{2j+1}{2l+1}} \; f^{(j)}_{l} (m) \label{swingmatrixelement}
\end{eqnarray}
The Legendre polynomial operators \cite{schwinger:majorana} $P_{l}({\bf J}) $ and the Legendre polynomial operators \cite{zemach}
$ \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $ described in the previous section are related to the Chebyshev polynomial operators $f_{\lambda} ^{(j)} (J_z) $ according to
\begin{equation}
P_{\lambda}({\bf J}) =
\left[ \left[ {\bf J}^2 \right]^{\!l} \right]^{\!-1/2} \, \overline{P}_{\lambda}(J_z) =\sqrt{\displaystyle\frac{2j+1}{2\lambda+1} } \, f_{\lambda} ^{(j)} (J_z)
\end{equation}
This relation shows that the Chebyshev polynomials $ f_{\lambda} ^{(j)} (m) $ can be calculated from the diagonal matrix elements of the Legendre
polynomial operators $ \overline{P}_{\lambda}(J_z) $ or $P_{\lambda}({\bf J})$.
Although Schwinger \cite{schwinger:majorana} did not cite a reference for the generating function of Eq.(\ref{swinggen}), Schwinger himself had derived this generating function. His derivation was included in a 1952 technical report \cite{schwingerosti}, later published in a book collection of reprints and original papers \cite{schwingerbiedenvandam}, and as a book \cite{schwingerbook} with the same title as the technical report. Additional discussions of this generating function can be found in books by Schwinger et al. \cite{schwingerbook2} and by Garg \cite{garg}.
As we shall demonstrate, Schwinger's generating function \cite{schwinger:majorana,schwingerosti,schwingerbiedenvandam,schwingerbook} is equivalent to the Herglotz generating function \cite{couranth,herglot,ligarg} for the solid harmonic operators $\mathcal{Y}_{lm}({\bf J}) $
\begin{eqnarray}
e^{\zeta {\bf \hat{a}} \cdot {\bf J}} & = & \sum_{lm} \sqrt{\frac{4\pi}{2l+1}} \;
\frac{\zeta^{l} \lambda^{m}}{\sqrt{(l+m)!(l-m)!}} \; \mathcal{Y}_{lm}({\bf J}) \\
\mbox{where } \;\;\;\;\; {\bf \hat{a}} & = & {\bf \hat{z}} -\frac{\lambda}{2} ({\bf \hat{x}} +i {\bf \hat{y}}) + \frac{1}{2\lambda} ({\bf \hat{x}} -i {\bf \hat{y}})
\label{herglotz}
\end{eqnarray}
Introducing the definitions
\begin{eqnarray}
{\hat A} & = & J_z -\frac{\lambda}{2} (J_x+iJ_y) + \frac{1}{2\lambda} (J_x - iJ_y) \equiv ({\bf \hat{a}} \cdot {\bf J}) \nonumber \\
\lambda & = & z_+/ z_{-}
\end{eqnarray}
and after a slight rearrangement, Schwinger's generating function \cite{schwinger:majorana,schwingerosti,schwingerbook} of Eq.(\ref{swinggen}) can be rewritten as
\begin{equation}
\frac{{\hat A}^l}{l !} = \sum_{m=-l}^{l} \; \sqrt{\frac{4\pi}{2l+1} }\; \frac{ \lambda^{m} } { \left[(l+m)!(l-m)!\right]^{1/2}} \;
\mathcal{Y}_{lm}({\bf J}) \label{swingsimp}
\end{equation}
After multiplying both sides of Eq.(\ref{swingsimp}) by $\zeta^l$, and summing both sides over $l$, we obtain the Herglotz generating function \cite{couranth,herglot,ligarg} of Eq.(\ref{herglotz}):
\begin{equation}
\sum_{l=1}^{\infty} \frac{[\zeta{\hat A}]^l}{l !} = \sum_{l=1}^{\infty} \frac{(\zeta{\bf \hat{a}} \cdot {\bf J})^l}{l !} \equiv e^{\zeta {\bf \hat{a}} \cdot {\bf J}} =
\sum_{l=1}^{\infty} \sum_{m=-l}^{l} \; \sqrt{\frac{4\pi}{2l+1} }\; \frac{\zeta^l \lambda^{m} } { \left[(l+m)!(l-m)!\right]^{1/2}} \;
\mathcal{Y}_{lm}({\bf J})
\end{equation}
Appendix A of the first English edition of Courant and Hilbert \cite{couranth,herglot} published in 1953 cited Herglotz for his formula, but none of the pre-war German editions had this appendix. This meant that the generating function \cite{schwinger:majorana,schwingerosti,schwingerbook} documented by Schwinger in a technical report \cite{schwingerosti} a year earlier in 1952 anticipated the Herglotz generating function \cite{couranth,herglot,ligarg}, and so there is some justification for renaming this generating function the Schwinger-Herglotz generating function.
\subsection{Unit tensor (Wigner) operators}
In their comments on Schwinger's interpretation \cite{schwinger:majorana} of the Majorana formula \cite{emajorana} , Biedenharn and Louck
\cite{biedenharn} observe that $ P_{l}(j,m) $ denotes the matrix element of the unit tensor operator (alias Wigner operators \cite{biedenharn})
\begin{equation}
P_{l}(j,m) \equiv \langle jm |
\left\langle\begin{array}{ccc} & l & \\
2l & & 0 \\
& l &
\end{array} \right\rangle | jm \rangle \label{unittensor}
\end{equation}
The Wigner operators can be defined by their action on the angular momentum basis $|jm \rangle$ \cite{biedenharn}, and in particular, the shift action is defined by \cite{biedenharn}
\begin{equation}
\left\langle\begin{array}{ccc} & J+\Delta & \\
2J & & 0 \\
& J+M &
\end{array} \right\rangle | jm \rangle = C_{jmJM}^{j+\Delta\; m+M} \; |j+\Delta,m+M \rangle
\end{equation}
and so in particular
\begin{eqnarray}
\left\langle\begin{array}{ccc} & l & \\
2l & & 0 \\
& l &
\end{array} \right\rangle | jm \rangle & = & C_{jml0}^{jm} \; | jm \rangle \\
& = & \sqrt{\frac{2j+1}{2l+1}}\;\boxed{ (-1)^{j-m} \; C_{jmj-m}^{l0} } \; | jm \rangle \label{wigbox} \\
& = & \sqrt{\frac{2j+1}{2l+1}}\; f_l^{(j)}(m) \; | jm \rangle \label{wigbox2}
\end{eqnarray}
The ``boxed" term in Eq.(\ref{wigbox}) defines the Chebyshev polynomial $f_l^{(j)}(m)$ in Eq.(\ref{wigbox2}). Then, using the result of Eq.(\ref{wigbox2}), the unit tensor matrix element of Eq.(\ref{unittensor}) is evaluated as
\begin{equation}
P_{l}(j,m) \equiv \langle jm |
\left\langle\begin{array}{ccc} & l & \\
2l & & 0 \\
& l &
\end{array} \right\rangle | jm \rangle = \sqrt{\frac{2j+1}{2l+1}}\; f_l^{(j)}(m)
\end{equation}
in agreement with the matrix element $P_{l}(j,m) = \langle jm| \, P_{l}({\bf J}) \, | jm \rangle $ of Schwinger's \cite{schwinger:majorana} Legendre polynomial operator $P_{l}({\bf J}) $ in Eq.(\ref{swingmatrixelement}).
\newpage
\section{Appendix B}
As noted by Varshalovich et al. \cite{varshal1:ang}, an alternative explicit form of
${\cal D}^J_{MM^{\prime}}(\psi, {\bf \hat{n}}) \equiv{\cal D}^J_{MM^{\prime}}(\psi; \Theta, \Phi)$ can be obtained directly from
${\cal D}^J_{MM^{\prime}}(\alpha, \beta, \gamma) $ by changing variables $(\psi; \Theta, \Phi) \rightarrow (\alpha, \beta, \gamma)$ with the aid
of the following relations (and the corresponding inverse relations):
\begin{eqnarray}
\sin \bfrac{\beta}{2} & = & \sin \Theta\; \sin \bfrac{\psi}{2} \nonumber \\
\tan \frac{\alpha + \gamma}{2} & = & \cos \Theta \; \tan \bfrac{\psi}{2} \nonumber \\
\frac{\alpha - \gamma}{2} & = & \Phi -\bfrac{\pi}{2}
\label{ieuanglec}
\end{eqnarray}
The result of this variable change is the following \cite{varshal1:ang}
\begin{eqnarray}
{\cal D}^J_{MM^{\prime}}(\psi; \Theta, \Phi) & = & i^{M-M^{\prime}} \; e^{-i(M-M^{\prime})\Phi}
\left( \displaystyle\frac{1-i\tan \frac{\psi}{2} \cos \Theta}{\sqrt{ 1+\tan^2 \frac{\psi}{2} \cos^2 \Theta}} \right)^{\!\!M+M^{\prime}}
d^J_{MM^{\prime}}(\xi) \label{EulerEquiv}\;\;\;\; \label{Dangleaxis} \\
\mbox{where}\;\;\;\; \sin \frac{\xi}{2} & = & \sin \bfrac{\psi}{2} \; \sin \Theta \label{sinequiv}
\end{eqnarray}
The expression for ${\cal D}^J_{MM^{\prime}}(\psi; \Theta, \Phi) $ given in Eq.(\ref{Dangleaxis}) will now be used to evaluate
${\cal D}^J_{MM^{\prime}}(\psi, {\bf \hat{n}})$ in two cases described in the next sections.
\subsection{Case I: ${\cal D}_{00}^{L}(\psi, {\bf \hat{n}}) = d_{00}^{\,L}(\xi) \equiv P_L(\cos \xi) = P_L(\cos \beta)$}
Since $M=M^{\prime}=0$, from Eq.(\ref{EulerEquiv}) we find
\begin{eqnarray}
{\cal D}_{00}^{L}(\psi, {\bf \hat{n}}) & = & d_{00}^{\,L}(\xi) \equiv P_L(\cos \xi) \label{LLegend}\\
\mbox{where \cite{brinksatch:ang}}\;\;\;\; d_{00}^{\,L}(\xi) & = & P_L(\cos \xi)
\end{eqnarray}
Using Eq.(\ref{sinequiv}), the argument of the Legendre polynomial $P_L(\cos \xi)$ in Eq.(\ref{LLegend}) can therefore be rewritten as
\begin{equation}
\cos \xi \equiv 1-2 \sin^2 \displaystyle\frac{\xi}{2} = \boxed{1-2 \sin^2\displaystyle\bfrac{\psi}{2} \sin^2 \theta =
\cos \beta} \label{convert}
\end{equation}
so that
\begin{equation}
{\cal D}_{00}^{L}(\psi, {\bf \hat{n}}) = P_L(\cos \beta)
\end{equation}
The ``boxed" equivalence of Eq.(\ref{convert}) follows from this relation \cite{abragamtext}
\begin{equation}
\cos \beta = 1-2 \sin^2\displaystyle\bfrac{\psi}{2} \sin^2 \theta \label{abragam}
\end{equation}
where $\psi=|\gamma {\bf H}_e|t$, and $\theta$ is the angle between the direction of the effective field {\bf H}$_e$ and the static applied field
{\bf H}$_0$.
\subsection{Case II: ${\cal D}^{\lambda}_{\mu 0}(\theta, {\bf \hat{n}}_{\bot}) \equiv {\cal D}^{\lambda}_{\mu 0}(\theta; \frac{\pi}{2}, \phi + \frac{\pi}{2} )
\equiv C_{\lambda \mu}^{\star}(\theta, \phi)$}
Since $\Theta =\pi/2$, $\sin \Theta = 1$, $\cos \Theta = 0$, and $ \xi = \theta$, and so using the expression of Eq.(\ref{EulerEquiv}), we find
\begin{eqnarray}
{\cal D}^{\lambda}_{\mu 0}(\theta, {\bf \hat{n}}_{\bot}) & \equiv & {\cal D}^{\lambda}_{\mu 0}(\theta; \bfrac{\pi}{2}, \phi + \bfrac{\pi}{2} )\\
& = & i^{\mu} \, e^{-i\mu \phi} \, e^{-i \mu \frac{\pi}{2}} \; \boxed{d^{\lambda}_{\mu 0} (\theta)} \label{reducedd}\\
& = & e^{-i\mu \phi} \, \boxed{(-1)^{\mu} \sqrt{\displaystyle\frac{(\lambda-\mu)!}{(\lambda-\mu)!}} \; P_{\lambda}^{\mu} (\theta)} \label{replacerdd}\\
& = & C_{\lambda \mu}^{\star}(\theta, \phi)
\end{eqnarray}
The reduced matrix element in the ``boxed" term of Eq.(\ref{reducedd}) has been replaced by its equivalent \cite{brinksatch:ang} in the ``boxed" term
of Eq.(\ref{replacerdd}).
\newpage
\section{Appendix C}
\subsection{A closed-form expression for the Fourier-Legendre series expansion of the spin transition probability $\mbox{P}^{(j)}_{j,-j} (t) $}
Consider the following Fourier-Legendre series expansion (identity {\bf 5.10.1(17)} in Prudnikov et al. \cite{prudnikov})
\begin{equation}
(a-1) \left[\frac{1-x}{2} \right]^{\!(a-2)} = \; \sum_{k=0}^{\infty} (2k+1) \frac{(2-a)_k}{(a)_k} \, P_k(x) \;\;\;\;\;\;\;\;(|x|<1; \;\;a>5/4) \label{identprud}
\end{equation}
where the Pochhammer symbol $(a)_n$ is defined in terms of the Gamma function $\Gamma(z)$ as follows \cite{prudnikov}
\begin{equation}
(a)_n = \frac{\Gamma(a+n)}{\Gamma(a)}
\end{equation}
If we let $a=n+2$ in the identity of Eq.(\ref{identprud}), where $n \geq 0$ is an integer, then we obtain
\begin{equation}
\left[\frac{1-x}{2} \right]^{\!n} = \frac{1}{n+1} \sum_{k=0}^{\infty} (2k+1) \frac{(-n)_k}{(n+2)_k} \, P_k(x) \label{identprud2}
\end{equation}
Then using the following Pochhammer symbol expressions \cite{prudnikov}
\begin{eqnarray}
(-n)_k & = & (-1)^k \frac{n!}{(n-k)!} \\
(n+2)_k & = &\frac{ \Gamma(n+2+k)}{\Gamma(n+2)} =\frac{(n+1+k)!}{(n+1)!}
\end{eqnarray}
we see that the Fourier-Legendre series expansion of Eq.(\ref{identprud2}) terminates when $k=n$, and that this expansion now simplifies to read
\begin{equation}
\left[\frac{1-x}{2} \right]^{\!n} = \left[n!\right]^2\sum_{k=0}^{n} \frac{(-1)^k (2k+1)}{(n-k)!(n+1+k)!} \, P_k(x) \label{identprud3}
\end{equation}
Mathematica$^{\circledR}$ \cite{wolfram} can be also used to verify this expansion. For example, by using the following input command to implement the summation of
Eq.(\ref{identprud3}) with $n=101$
\vspace{8mm}
\begin{lstlisting}
In[1] = Simplify[2^(101) Sum[(-1)^L (2 L + 1)(101!)^2
LegendreP[L, x]/((101 - L)!(101 + L + 1)!),{L, 0, 101}]]
\end{lstlisting}
the expected output is obtained as
\vspace{8mm}
\begin{lstlisting}
Out[2] = -(-1 + x)^101
\end{lstlisting}
An amusing consequence of Eq.(\ref{identprud3}) is the following Fourier-Legendre series expansion of
$[a+b]^n$:
\begin{equation}
[a+b]^n= [2a]^n \; \!\! \left[n!\right]^2\sum_{k=0}^{n} \frac{(2k+1)}{(n-k)!(n+1+k)!} \, P_k(b/a)
\end{equation}
\newpage
\section{Appendix D}
\subsection{Using $f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J})$ to evaluate the generalized characters $\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) $ }
Using the ``boxed" trace relation of Eq.(\ref{charcheby}), the generalized character functions can be written as
\begin{equation}
\mbox{{\large $\chi$}}_{\lambda}^{(j)}(\psi) = i^{3 \lambda} \sqrt{\displaystyle\frac{2j+1}{2\lambda+1}} \;
\mbox{Tr} \! \left[ f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; \hat{{\cal D}}^{(j)} \! (\psi, {\bf \hat{n}}) \right]
\end{equation}
Then, putting $\lambda=1$ and $J=1/2$, the generalized character function {\large $\chi$}$_1^{1/2}(\psi) $ can be evaluated as
\begin{eqnarray}
\mbox{{\large $\chi$}}_1^{(1/2)}(\psi) & = & -i \sqrt{\frac{2}{3}} \; \mbox{Tr} \! \left[ f^{(1/2)}_{1}({\bf \hat{n}} \cdot {\bf J}) \; \hat{{\cal D}}^{(1/2)} (\psi, {\bf \hat{n}}) \right] \\
& = & -i \sqrt{\frac{2}{3}} \; \mbox{Tr} \! \left[ \sqrt{2} \; J_z \left(\mathds{1} \cos \bfrac{\psi}{2} + 2i \, J_z \, \sin \bfrac{\psi}{2} \right) \right] \label{tracesimp} \\
& = & \frac{4}{\sqrt{3}} \, \sin \bfrac{\psi}{2} \; \mbox{Tr} \! \left[ J_z^2 \right] \label{charex} \\
& = & \frac{2}{\sqrt{3}} \, \sin \bfrac{\psi}{2} \label{charex2}
\end{eqnarray}
a result which agrees with $ \mbox{{\large $\chi$}}_1^{(1/2)}(\psi) $ tabulated in Varshalovich et al. \cite{varshal1:ang}. Because the trace is invariant with respect to a change in basis, the trace of Eq.(\ref{tracesimp}) was evaluated in a representation in which $({\bf \hat{n}} \cdot {\bf J})$ is diagonal. In addition, the following identities have been used to arrive at the final result in Eq.(\ref{charex2}):
\begin{eqnarray}
f^{(1/2)}_{1}({\bf \hat{n}} \cdot {\bf J}) & = & \sqrt{2} \; J_z \;\;\; \;\;\; \;\;\;\;\;\;\;\ \;\;\; \;\;\;\;\;\;\;\ \;\;\; \;\;\;\;\mbox{(see Table II)} \\
\hat{{\cal D}}^{(1/2)} (\psi, {\bf \hat{n}}) & = & \mathds{1} \cos \bfrac{\psi}{2} + 2i \, J_z \, \sin \bfrac{\psi}{2}
\;\;\; \;\;\;\;\;\;\;\mbox{(Reference \cite{varshal1:ang})} \\
\mbox{Tr} \! \left[ J_z^2 \right] & = & \sum_{m=-1/2}^{1/2} \!m^2 = \frac{1}{2} \\
\mbox{Tr} \! \left[ J_z \right] & = & 0
\end{eqnarray}
\newpage
\section{Appendix E}
The composite irreducible product tensor ${\bf X}^K_Q $ of rank $K$ is defined by the following combination of irreducible tensors
${\bf T}^k_q $ and ${\bf U}^{k^{\prime}}_{q^{\prime}} $ of rank $k$ and $k^{\prime}$, respectively:
\begin{equation}
{\bf X}^K_Q = \left\{ {\bf T}^{k}_{q} \circledast {\bf U}^{k^{\prime}}_{q^{\prime}} \right \}^{K}_{Q} = \sum_{\substack{q,q^{\prime} \\q+q^{\prime}=Q }}
{\bf T}^{k}_{q} \; {\bf U}^{k^{\prime}} _{q^{\prime}} \;
C^{KQ}_{kqk^{\prime}q^{\prime}}
\end{equation}
where $C^{KQ}_{kqk^{\prime}q^{\prime}}$ is a Clebsch-Gordan coefficient which vanishes unless $q + q^{\prime} =Q$.
In general,
\begin{equation}
\left\{{\bf R}^{(k)} \circledast {\bf S}^{(k)} \right\}^0_0 =\sum_{\substack{q,q^{\prime} \\ q+q^{\prime}=0}} {\bf R}^{(k)}_q \;
{\bf S}^{(k)}_{q^{\prime}} \; C^{00}_{kqkq^{\prime}}
\label{dot}
\end{equation}
But since the Clebsch-Gordan coefficient is evaluated as~\cite{brinksatch:ang}
\begin{equation}
C^{00}_{kqkq^{\prime}} = \frac{(-1)^{k-q}}{\sqrt{2k+1}} \; \delta_{q,-q^{\prime}}
\end{equation}
the double sum of Eq.(\ref{dot}) reduces to a single sum
\begin{eqnarray}
\left\{{\bf R}^{(k)} \circledast {\bf S}^{(k)} \right\}^0_0 & = & \frac{(-1)^{k}}{\sqrt{2k+1}} \sum_{q} (-1)^{-q} \; {\bf R}^{(k)}_q \; {\bf S}^{(k)}_{-q} \nonumber \\
& = & \frac{(-1)^{k}}{\sqrt{2k+1}} \; {\bf R}^{(k)} \cdot {\bf S}^{(k)}
\end{eqnarray}
where ${\bf R}^{(k)} \cdot {\bf S}^{(k)} = \displaystyle \sum_{q} (-1)^{-q} \; {\bf R}^{(k)}_q \;
{\bf S}^{(k)}_{-q}$ defines the scalar product of two tensors.
A more general result for the recoupling of four arbitrary
commuting tensors can be defined in terms of a 9$j$-symbol \cite{brinksatch:ang} as follows
\begin{eqnarray}
& & \left\{ \left\{ {\bf S}^{k_1}_{q_1} \circledast {\bf T}^{k_2}_{q_2} \right \}^{k_{12}}_{q_{12}} \circledast \left\{ {\bf U}^{k_3}_{q_3} \circledast {\bf V}^{k_4}_{q_4} \right \}^{k_{34}}_{q_{34}} \right \}^K_Q \nonumber \\
& = &
\sum_{k_{13},k_{24}} \Pi_{k_{13} k_{24} k_{12}k_{34}}
\left\{\begin{array}{ccc} k_1 & k_2 & k_{12} \\
k_3 & k_4 & k_{34} \\
k_{13} & k_{24} & K\end{array}\right\} \left\{ \left\{ {\bf S}^{k_1}_{q_1} \circledast {\bf U}^{k_3}_{q_3} \right \}^{k_{13}}_{q_{13}} \circledast
\left\{ {\bf T}^{k_2}_{q_2} \circledast {\bf V}^{k_4}_{q_4} \right \}^{k_{24}}_{q_{24}} \right \}^K_Q \;\;\;\; \;\;\;\; \;\;\;\;
\label{rec4ang}
\end{eqnarray}
where as a matter of notation, in this equation, and all that follow, we find it convenient to define
\cite{varshal1:ang}:
\begin{equation}
\Pi_ {abc \ldots d} = \sqrt{(2a+1)(2b+1)(2c+1) \ldots (2d+1)}
\end{equation}
In Eq.(\ref{rec4ang}), set $K=Q=0$, from which it follows that $k_{12} = k_{34}$ and $k_{13} = k_{24}$. In addition, set $k_{12} = k_{34}=0$, and consider the particular case where we recouple four irreducible tensors of rank 1, so that $k_1=k_2=k_3=k_4=1$. In particular, we suppose
\begin{eqnarray}
{\bf S}^{k_1} & \equiv & {\bf I}_1 \nonumber \\
{\bf U}^{k_3} & \equiv & {\bf I}_2 \nonumber \\
{\bf T}^{k_2} & = & {\bf V}^{k_4} \equiv {\bf r}
\end{eqnarray}
Then, from Eq.(\ref{rec4ang}), we have
\begin{equation}
\Big\{ \left\{ {\bf I}_1^{} \circledast {\bf r} \right\}^0 \circledast \left\{ {\bf I}_2 \circledast {\bf r} \right\}^0 \Big\}^0_0 = \sum_{k_{13},k_{24}}\Pi_{k_{13} k_{24}}
\left\{\begin{array}{ccc} 1 & 1 & 0 \\
1 & 1 & 0 \\
k_{13} & k_{24} & 0\end{array}\right\}
\Big\{ \left\{ {\bf I}_1 \circledast {\bf I}_2 \right\}^{k_{13}} \circledast \left\{ {\bf r} \circledast {\bf r} \right\}^{k_{24}} \Big\}^0_0
\label{appl}
\end{equation}
Now since $k_{13}$ and $k_{24}$ satisfy the triangle inequalities
\begin{eqnarray}
|k_1-k_3| & \leq & k_{13} \leq |k_1+k_3| \nonumber \\
|k_2-k_4| & \leq & k_{24} \leq |k_2+k_4|
\end{eqnarray}
and $k_1=k_2=k_3=k_4=1$, $k_{13}$ and $k_{24}$ are summed from 0 to 2. Note however that when $k_{24} =1$, $ \left\{ {\bf r} \circledast {\bf r} \right\}^1$ vanishes since
$ \left\{ {\bf r} \circledast {\bf r} \right\}^1 \propto ({\bf r} \times {\bf r}) =0$. Therefore, in the sum of Eq.(\ref{appl}), only two terms contribute:
\begin{eqnarray}
& & \Big\{ \left\{ {\bf I}_1 \circledast {\bf r} \right\}^0 \circledast \left\{ {\bf I}_2 \circledast {\bf r} \right\}^0 \Big\}^0_0 \nonumber \\
& = &
\left\{\begin{array}{ccc} 1 & 1 & 0 \\
1 & 1 & 0 \\
0 & 0 & 0\end{array}\right\}
\Big\{ \left\{ {\bf I}_1 \circledast {\bf I}_2 \right\}^0 \circledast \left\{ {\bf r} \circledast {\bf r} \right\}^0 \Big\}^0_0
+ 5 \left\{\begin{array}{ccc} 1 & 1 & 0 \\
1 & 1 & 0 \\
2 & 2 & 0\end{array}\right\}
\Big\{ \left\{ {\bf I}_1 \circledast {\bf I}_2 \right\}^2 \circledast \left\{ {\bf r} \circledast {\bf r} \right\}^2 \Big\}^0_0 \nonumber \\
& &
\label{sum9j}
\end{eqnarray}
The 9$j$-symbols in Eq.(\ref{sum9j}) can be evaluated by expressing them in terms of 6$j$-symbols as follows \cite{varshal1:ang}
\begin{equation}
\left\{\begin{array}{ccc} a & b & e \\
c & d & e \\
f & f & 0\end{array}\right\} = \frac{(-1)^{b+c+e+f}}{\sqrt{(2e+1)(2f+1)}} \left\{\begin{array}{ccc} a & b & e \\
d & c & f \end{array}\right\}
\end{equation}
and in turn, all the 6$j$-symbols required can then be evaluated using tables \cite{roten:3j6j}. Replacing all the recoupling coefficients
in Eq.(\ref{sum9j}) then yields the following relation
\begin{equation}
\Big\{ \left\{ {\bf I}_1 \circledast {\bf r} \right\}^0 \circledast \left\{ {\bf I}_2 \circledast {\bf r} \right\}^0 \Big\}^0_0 =
\frac{1}{3}
\Big\{ \left\{ {\bf I}_1 \circledast {\bf I}_2 \right\}^0 \circledast \left\{ {\bf r} \circledast {\bf r} \right\}^0 \Big\}^0_0
+ \frac{\sqrt{5}}{3}
\Big\{ \left\{ {\bf I}_1 \circledast {\bf I}_2 \right\}^2 \circledast \left\{ {\bf r} \circledast {\bf r} \right\}^2 \Big\}^0_0
\label{ssum9j}
\end{equation}
Then, after expressing the rank zero recoupled tensors in Eq.(\ref{sum9j}) as scalar products using the general relation
\begin{equation}
\left\{ {\bf a} \circledast {\bf b} \right\}^0_0 = -\frac{{\bf a} \cdot {\bf b}}{\sqrt{3}}
\label{rex1}
\end{equation}
and using the following relations \cite{brinksatch:ang}
\begin{eqnarray}
\left\{ {\bf r} \circledast {\bf r} \right\}^2_q & = & \sqrt{\frac{2}{3}} \; {\bf r}^2 \; {\bf C}^2_q \nonumber \\
\mbox{where}\;\; {\bf C}^2_q & = & \sqrt{\frac{4 \pi}{5}} \; {\bf Y}^2_q
\label{Racah}
\end{eqnarray}
the following recoupling expression for the classical dipolar Hamiltonian is obtained
\begin{eqnarray}
W_{12} & = & a \left[ {\bf I}_1 \cdot {\bf I}_2 -3 \frac{({\bf I}_1 \cdot {\bf r}) ({\bf I}_2 \cdot {\bf r}) }{r^2 }\right] \nonumber \\
& = & -\sqrt{6} a \left\{ {\bf I}_1 \circledast {\bf I}_2 \right\}^2 \cdot {\bf C}^2_q \nonumber \\
& = & -\sqrt{6} a \left\{ {\bf I}_1 \circledast {\bf I}_2 \right\}^2 \cdot \sqrt{\frac{4 \pi}{5}} {\bf Y}^2_q \nonumber \\
& = & -\sqrt{\frac{24 \pi}{5}} \sum_q (-1)^q \left\{ {\bf I}_1 \circledast {\bf I}_2 \right\}^2_q {\bf Y}^2_{-q}
\end{eqnarray}
\newpage
{\bf Table I}\\
Title: Chebyshev polynomial basis operators $f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \;\;(\lambda \leq 2j; \;\; \kappa \equiv j(j+1))$ \\
\vspace{5mm}
Caption: As described in Section {\bf 2.2}, all elements were generated following the procedure described by Corio \cite{corio:siam}.
\ \\
\begin{sidewaystable}
\begin{tabular}{||c||c|c|c||} \hline & & & \\ \hspace{1mm} $\lambda$ \hspace{1mm}& {\large $j=1/2$} & {\large $j=1$}
& {\large $j=3/2$} \\
& & & \\ \hline \hline
& & & \\
\hspace{1mm} 0 \hspace{1mm} & {\large $f^{(\frac{1}{2})}_0({\bf \hat{n}} \cdot {\bf J})= \displaystyle\frac{1}{\sqrt{2}} \,\mathds{1} $} & {\large $f^{(1)}_0({\bf \hat{n}} \cdot {\bf J}) = \displaystyle\frac{1}{\sqrt{3}} \,\mathds{1} $} & \hspace{5mm} {\large $f^{(\frac{3}{2})}_0({\bf \hat{n}} \cdot {\bf J})= \displaystyle\frac{1}{\sqrt{4}} \,\mathds{1} $}
\\
& & & \\
\hline
& & & \\
\hspace{1mm} 1 \hspace{1mm}& \hspace{1mm} {\large $f^{(\frac{1}{2})}_1({\bf \hat{n}} \cdot {\bf J})= \sqrt{2} \, \left[({\bf \hat{n}} \cdot {\bf J})\right] $} \hspace{1mm}
& {\large $f^{(1)}_1({\bf \hat{n}} \cdot {\bf J}) = \displaystyle\frac{1}{\sqrt{2}} \, \left[({\bf \hat{n}} \cdot {\bf J})\right] $} & {\large $f^{(\frac{3}{2})}_1({\bf \hat{n}} \cdot {\bf J})= \displaystyle\frac{1}{\sqrt{5}} \, \left[({\bf \hat{n}} \cdot {\bf J})\right] $}
\\
& & & \\
\hline
& & & \\
\hspace{1mm} 2 \hspace{1mm}& {\large --- } & \hspace{1mm} {\large $f^{(1)}_2({\bf \hat{n}} \cdot {\bf J}) = \displaystyle\frac{1}{\sqrt{6}} \left[3 ({\bf \hat{n}} \cdot {\bf J})^2-\kappa \,\mathds{1} \right] $} \hspace{1mm}
& {\large $f^{(\frac{3}{2})}_2({\bf \hat{n}} \cdot {\bf J}) = \displaystyle\frac{1}{6} \left[3 ({\bf \hat{n}} \cdot {\bf J})^2-\kappa \,\mathds{1} \right] $} \\
& & & \\
\hline
& & & \\
\hspace{1mm} 3 \hspace{1mm} & {\large --- } & {\large ---}
& \hspace{1mm} {\large $f^{(\frac{3}{2})}_3({\bf \hat{n}} \cdot {\bf J}) = \displaystyle\frac{1}{3\sqrt{5}} \left[5 ({\bf \hat{n}} \cdot {\bf J})^3-(3\kappa -1)({\bf \hat{n}} \cdot {\bf J}) \right] $} \hspace{1mm} \\
& & & \\
\hline
\end{tabular}
\end{sidewaystable}
\newpage
{\bf Table II}\\
Title: A comparison of Chebyshev polynomial scalars $f^{(j)}_{\lambda}(m) $ and operators $f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J})$. \\
\vspace{5mm}
Caption: First column compares the definitions of the Chebyshev polynomial scalars adopted by various authors, while the second column compares
the definitions of the Chebyshev polynomial operators adopted by the same authors. Note that Corio \cite{corio:ortho, corio:siam} suppressed the explicit dependence on the spin angular momentum $j$ in his definitions of both the Chebyshev polynomial scalars $ Z_n(x) $ and operators $U_{n}({\bf \hat{n}} \cdot {\bf J})$, while Meckler \cite{meckler:angular} suppressed the explicit dependence on the spin angular momentum $S$ in his definition of the Chebyshev polynomial operators $A^{(n})/g_n $, where
$g_n = [n!]^2 \sqrt{2S+1+n}\left[2^n (2n-1)!! \sqrt{(2n+1)(2S-n)!}\right]^{-1}$. Using Filippov's notation \cite{filippov2:thesis}
for the Chebyshev polynomial operators, Meckler's operators \cite{meckler:angular} can be written as $A^{(n})/g_n \equiv f^{(S)}_n({\bf \hat{a}} \cdot {\bf S}) $. Just as the $(2j+1)$ Chebyshev polynomial scalars $ p_n(S,-j), \ldots, p_n(S,j)$ defined by Meckler \cite{meckler:angular} are the diagonal matrix elements of $A^{(n)}/g_n $ in a representation where
$({\bf \hat{a}} \cdot {\bf S}) $ is diagonal, the $(2j+1)$ Chebyshev polynomial scalars $ Z_n(0), \ldots, Z_n(2j)$ defined by Corio
\cite{corio:ortho, corio:siam} are the diagonal matrix elements of $ U_{n}({\bf \hat{n}} \cdot {\bf J}) $ in a representation where
$({\bf \hat{n}} \cdot {\bf J}) $ is diagonal.
\ \\
\begin{tabular} {||c|c|c||} \hline & & \\
{\large Scalars} & {\large Operators} & {\large Authors} \\
& & \\
\hline \hline
& & \\
{\large $f^{(j)}_{\lambda}(m) $} &
{\large $f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J})$} & {\large Filippov} \cite{filippov2:thesis} \\
\hspace{8mm} & & \\ & & \\ \hline
& & \\
{\large$ p_n(S,m) $} &
{\large $\displaystyle\frac{A^{(n)}}{g_n} $} & {\large Meckler} \cite{meckler:majorana, meckler:angular}
\\
& & \\
{\large $(n \equiv \lambda; S \equiv j) $} & {\large $(n \equiv \lambda) $} & \\
& & \\
\hline
& & \\
{\large $ Z_n(x) $} & {\large $U_{n}({\bf \hat{n}} \cdot {\bf J}) $} & {\large Corio} \cite{corio:ortho, corio:siam}
\\
& & \\
{\large $(n \equiv \lambda; x \equiv j+m) $} & {\large $(n \equiv \lambda) $} & \\
& & \\
\hline
\end{tabular}
\newpage
{\bf Table III}\\
Title: Chebyshev polynomial definitions and recursion relations. \\
\vspace{5mm}
Caption: The top box in the first column gives Filippov's definition \cite{filippov2:thesis} of the Chebyshev polynomials $ f_L^{(j)} (m) $ in terms of
Bateman's \cite{bateman} Chebyshev polynomials $ t_{L}(j+m,2j+1) $, which are defined in the bottom box of the first column using
finite differences \cite{footnote2}. Two versions of the recursion relations are given in the second column. The form of the first version is that given by
Filippov \cite{filippov2:thesis} for the Chebyshev polynomials $ f_L^{(j)} (m) $, although equivalent forms were derived previously by Meckler \cite{meckler:majorana, meckler:angular} and by Corio \cite{corio:ortho, corio:siam}. The second version of the recursion relation is that given by
Varshalovich et al. \cite{varshal1:ang} for the Clebsch-Gordan coefficients $C^{L \, 0}_{jmj-m}$, which are identical to the Chebyshev polynomials
$ f_L^{(j)} (m) $ to within a phase-factor. The bottom boxes also define functions $F(L,j)$ and $ G(a,b) $ used for the matrix elements (top box, first column) and recursion relations (top box, second column), respectively.
\ \\
\begin{sidewaystable}
\begin{tabular} {||l|l|c||} \hline & \\ {\large Matrix Element Definition via Bateman's \cite{bateman} }
& \hspace{42mm} {\large Recursion Relations} \\
\hspace{8mm}{\large Chebyshev Polynomials \; {\large $\boxed{t_{L}(j+m,2j+1) }$}} & \\ & \\ \hline \hline
& \\
\hspace{22mm} {\large $ f_L^{(j)} (m) = \langle jm| \; f_{L}^{(j)} ( J_z) \; |jm \rangle $}
& {\large $ G(L+1,j) \, f_{L+1}^{(j)} (m) -2m \, f_L^{(j)} (m) + G(L,j) \, f_{L-1}^{(j)} (m) = 0 $} \\
& \\
\hspace{39mm} {\large $ = F(L,j) \;t_L(j+m,2j+1) $} & {\large $ G(L+1,j) \, C^{L+1 \, 0}_{jmj-m} -2m \, C^{L \, 0}_{jmj-m} + G(L,j) \, C^{L-1 \, 0}_{jmj-m} = 0 $} \\
& \\
\hspace{39mm} {\large $ = (-1)^{j-m} \; C^{L \, 0}_{jmj-m} $} & \\
& \\
\hline \hline
& \\
& \\
{\large $(1) \;\;\; F(L,j) = \left[\displaystyle\frac{(2L+1)(2j-L)!}{(2j+L+1)!} \right]^{1/2}$} &
{\large $ G(a,b) = \left[\displaystyle\frac{a^2 ((2b+1)^2-a^2)}{4a^2-1} \right]^{1/2} $} \\
& \\
& \\
{\large $ (2) \;\;\; t_{L}(j+m,2j+1) = L!\, \Delta^{L} \! \left[ H_{L}^j(m) \right] $} & \\
& \\
{\large \mbox{where}} \;\; {\large $ H_{L}^j(m) = \left[\displaystyle {j+m \choose L} {m-j-1 \choose L}\right] $} & \\
& \\
\hspace{17mm} {\large $ \Delta h(m) = h(m+1) - h(m) $} & \\
& \\
\hspace{17mm} {\large $ \Delta^{k+1} h(m) = \Delta [\Delta^{k} h(m)] $} & \\
& \\
\hline
\end{tabular}
\end{sidewaystable}
\newpage
{\bf Table IV}\\
Title: A comparison of expansions, traces, matrix elements, Hermitian conjugates and density operators $\hat \rho$ for the spin polarization operators $\hat{T}^{(j)}_{\lambda \mu}$ and the Chebyshev polynomial
operators $ f_{\lambda} ^{(j)}( {\bf \hat{n}} \cdot {\bf J}) $.
\begin{sidewaystable}
\begin{tabular} {||c|c|c||} \hline & & \\
{\large Operator }& {\large Spin Polarization} & {\large Chebyshev Polynomial}
\\
& & \\
\hline
{\large $\hat{O}$} & {\large $ \hat{O} = \hat{T}_{\lambda \mu}^{(j)} $} &
{\large $\hat{O} = f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) = \displaystyle\sum_{\mu=-\lambda}^{\lambda} C_{\lambda \mu}^{\star}({\bf \hat{n}}) \; \,
\hat{T}^{(j)}_{\lambda \mu} $} \\
\hline \hline
{\large Expansions} &
{\large$ \hat{A} = \displaystyle\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda} A_{\lambda \mu}^{(j)} \; \hat{T}_{\lambda \mu}^{(j)} $ } &
{\large $ \hat{B} = \displaystyle\sum_{\lambda=0}^{2j} B_{\lambda}^{(j)} \, f_{\lambda} ^{(j)}( {\bf \hat{n}} \cdot {\bf J}) $}
\\
& & \\
&{\large $ A_{\lambda \mu}^{(j)} = \mbox{Tr} \! \left[ \left[ \hat{T}_{\lambda \mu }^{(j)} \right]^{\! \dagger} \! \hat{A} \right] $}&
{\large $B_{\lambda}^{(j)} = \mbox{Tr} \! \left[ f_{\lambda} ^{(j)}( {\bf \hat{n}} \cdot {\bf J}) \, \hat{B} \right]$} \\
\hline
{\large Traces} &
{\large$ \mbox{Tr} \! \left[\hat{T}_{\lambda \mu}^{(j)} \left[ \hat{T}_{\lambda^{\prime} \mu^{\prime} }^{(j)} \right]^{\! \dagger} \right]
= \delta_{\lambda \lambda^{\prime} } \; \delta_{\mu \mu^{\prime} } $} &
{\large $ \mbox{Tr} \! \left[ f^{(j)} _{\lambda}({\bf \hat{n}} \cdot {\bf J}) \; f^{(j)} _{{\lambda}^{\prime}}({\bf \hat{n}} \cdot {\bf J}) \right] =
\delta_{\lambda \lambda^{\prime} }$ }
\hspace{3mm}
\\
\hline
& & \\
{\large $\langle m| \; \hat{O} \; |m^{\prime} \rangle$} & {\large $ C_{jmj-m^{\prime}}^{\lambda \,(m-m^{\prime})} \; (-1)^{j-m^{\prime}} $} &
{\large $ C_{\lambda \, (m-m^{\prime})}^{\star}({\bf \hat{n}}) \; C_{jmj-m^{\prime}}^{\lambda \, (m-m^{\prime})} \; (-1)^{j-m^{\prime}}
$} \\
& & \\
\hline
{\large Hermitian} & {\large $ \left[ \hat{T}_{\lambda \mu }^{(j)} \right]^{\! \dagger} = (-1)^{\mu} \;\, \hat{T}_{\lambda -\mu}^{(j)}$}&
{\large $ \left[ f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) \right]^{\! \dagger} = f_{\lambda}^{(j)} ( {\bf \hat{n}} \cdot {\bf J}) $} \\
{\large Conjugate}& & \\
\hline
{\large $\hat \rho$} & {\large $ \displaystyle\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda} \mbox{Tr} \! \left[ \hat \rho \,
\left[ \hat{T}_{\lambda \mu }^{(j)} \right]^{\! \dagger} \right] \hat{T}_{\lambda \mu }^{(j)} $} & {\large $ \;\;\; \displaystyle\sum_{\lambda=0}^{2j} \frac{(2\lambda+1)}{4\pi}
\! \! \int _{{\bf S}^2} f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J}) \;
\mbox{Tr} \!\left[ \hat \rho \, f_{\lambda}^{(j)}( {\bf \hat{n}} \cdot {\bf J}) \right] d{\bf \hat{n}} \;\;\;$} \\
& & \\
\hline
\end{tabular}
\end{sidewaystable}
\newpage
{\bf Table V}\\
Title: Similarity Transforms \\
\vspace{5mm}
Caption: Similarity transforms of $J_z$ and $f^{(j)}_{\lambda}(J_z)$.
\newpage
\begin{tabular} {||c|c||} \hline & \\
& \\
{\large Operator} & {\large Similarity Transform} \\
& \\
\hline \hline & \\
& \\
{\large $\hat O$} & {\large $ \hat{{\cal D}}^{(j)}(\theta, {\bf \hat{n}}_{\bot}) \, \hat O \left [\hat{{\cal D}}^{(j)} (\theta, {\bf \hat{n}}_{\bot}) \right ]^{\!\dagger}$ } \\
& \\
\hline
& \\
{\large $J_z $} & {\large $({\bf \hat{n}} \cdot {\bf J})$}
\\
\hspace{8mm} & \\ \hline
& \\
{\large $f^{(j)}_{\lambda}(J_z)$} &
{\large $f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J})$}
\\
& \\
\hline
\end{tabular}
\newpage
{\bf Table VI}\\
Title: A comparison of the relative orientations of Meckler's instantaneous axis ${\bf \hat{b}}(t)$ \cite{meckler:majorana} and Abragram's magnetic moment ${\bf \hat{m}}(t)$ \cite{abragamtext}.
\begin{sidewaystable}
\begin{tabular} {||c|c|c||} \hline & & \\
& {\large Meckler \cite{meckler:majorana}} & {\large Abragam \cite{abragamtext}} \\
& & \\
& & \\ \hline \hline
& & \\
{\large$ t=0 $} &
{\large$ {\bf \hat{b}}(0) \parallel {\bf \hat{a}}$} & {\large $ {\bf \hat{m}}(0) \parallel \mbox{H}_0 \, {\bf \hat{z}}$}
\\
& & \\
& & \\
\hline
& & \\
{\large$ t > 0$} &
{\large$ {\bf \hat{b}}(t) \cdot{\bf \hat{a}} =1-\displaystyle\frac{\lambda^2}{u^2}(1- \cos ut) $} &
{\large $ {\bf \hat{m}}(t) \cdot{\bf \hat{m}}(0) =1-\displaystyle\frac{\omega^2_1}{a^2}(1- \cos at) $ }
\hspace{3mm}
\\
& & \\
& {\large $ \!\!\!\! \!\!\!\! \!\!\!\! \!\!\!\! \!\!\! \!\!\equiv Z $} & {\large $ \!\!\!\! \!\!\! \equiv \cos \alpha $ } \\
& & \\
\hline
& & \\
{\large Nutation } &
{\large$ u=\sqrt{\lambda^2+(\omega- \omega_0)^2} $} & {\large $ a=\sqrt{\omega_1^2+(\omega-\omega_0)^2} $ }
\\
{\large frequency}& & \\
& & \\
\hline
& & \\
{\large Excitation} & {\large $ \lambda$}& {\large $ \omega_1$} \\
{\large radiofrequency}& & \\
& & \\
\hline
\end{tabular}
\end{sidewaystable}
\newpage
{\bf Table VII}\\
Title: Probability distributions and tomographic reconstruction relations. \\
\vspace{5mm}
Caption: Probability distributions defined in the left-hand column are used to define the corresponding tomographic reconstructions of the density matrix $\hat \rho$
in the right-hand column. The function $F_{\lambda}({\bf \hat{n}} ) $ used in the definitions of the probability distributions is defined at the bottom of the left-hand column, and the function ${\cal F}_{\lambda}({\bf \hat{n}} ) $ used in the definitions of the tomographic reconstruction relations is defined at the bottom of the right-hand column.
\begin{sidewaystable}
\begin{tabular} {||l|l||} \hline & \\
\hspace{14mm} {\large Probability Distributions}& \hspace{14mm} {\large Tomographic Reconstruction Relations} \\
& \\ \hline \hline
& \\
\hspace{6mm} {\large$ w^{(j)}(m,{\bf \hat{n}}) = \mbox{Tr} \! \left[ \hat \rho \; \mbox{{\boldmath $\Pi$}}^{(j)}(m,{\bf \hat{n}}) \right]$} &
\hspace{6mm} {\large$ \hat \rho= \displaystyle\sum_{\lambda=0}^{2j} \displaystyle\sum_{m=-j}^j f^{(j)}_{\lambda}(m)
\int_{{\bf S}^2} w^{(j)}(m,{\bf \hat{n}}) \; {\cal F}_{\lambda}({\bf \hat{n}} )\; d{\bf \hat{n}} $} \hspace{6mm}
\\
\hspace{29mm} {\large$ = \displaystyle\sum_{\lambda=0}^{2j} f^{(j)}_{\lambda}(m) \; F_{\lambda}({\bf \hat{n}} ) $} \hspace{6mm} & \\
& \\
\hline
& \\
\hspace{6mm} {\large $ Q({\bf \hat{n}}) = \mbox{Tr} \! \left[ \hat \rho \; \mbox{{\boldmath $\Pi$}}^{(j)}(j,{\bf \hat{n}}) \right]$} &
\hspace{6mm} {\large $ \hat \rho= \displaystyle\sum_{\lambda=0}^{2j} \left[ f^{(j)}_{\lambda}(j) \right]^{-1}
\int_{{\bf S}^2} Q({\bf \hat{n}}) \; {\cal F}_{\lambda}({\bf \hat{n}} ) \; d{\bf \hat{n}} $}
\\
\hspace{18.5mm} {\large$ = \displaystyle\sum_{\lambda=0}^{2j} f^{(j)}_{\lambda}(j) \; F_{\lambda}({\bf \hat{n}} ) $} \hspace{6mm} & \\
& \\
& \\
\hline
& \\
\hspace{6mm}{\large $ W({\bf \hat{n}}) = \mbox{Tr} \! \left[ \hat \rho \; \Delta^{\!(j)}({\bf \hat{n}}) \right]$} &
\hspace{6mm} {\large $ \hat \rho= \displaystyle\sum_{\lambda=0}^{2j} \sqrt{\frac{2j+1}{2\lambda+1}} \; \displaystyle\int_{{\bf S}^2} W({\bf \hat{n}}) \; {\cal F}_{\lambda}({\bf \hat{n}} ) \; d{\bf \hat{n}} $} \\
\hspace{18.5mm} {\large $ = \displaystyle\sum_{\lambda=0}^{2j} \sqrt{ \frac{2\lambda+1}{2j+1}} \; F_{\lambda}({\bf \hat{n}} ) $}
& \\
& \\
\hline \hline
& \\
\hspace{10mm} {\large $\;\;F_{\lambda}({\bf \hat{n}} ) = \mbox{Tr} \! \left[ \hat \rho \; f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) \right] $ }
& \hspace{0mm} {\large $\;\; {\cal F}_{\lambda}({\bf \hat{n}} ) = \left(\displaystyle\frac{2\lambda+1}{4\pi} \right) f^{(j)}_{\lambda}({\bf \hat{n}} \cdot {\bf J})
$ } \\
& \\
\hline
\end{tabular}
\end{sidewaystable}
\newpage
{\bf Table VIII}\\
Title: A comparison of tensor operator equivalents expressed in terms of Chebyshev polynomials.\\
\vspace{5mm}
Caption: First row is work done by Corio \cite{corio:ortho}, and second row is work done by Marinelli et al. \cite{werb:tensor}.
\begin{sidewaystable}
\begin{tabular} {||l|l||} \hline & \\
& \\
\hspace{26mm} {\large $\hat{T}_{60}^{(j)} \equiv f^{(j)}_6(J_z)$}& \hspace{46mm} {\large $\hat{T}_{41}^{(j)} $} \\
& \\ \hline \hline
& \\
\hspace{6mm} {\large$ U^{(6)}_0 \!(J) \sim \Big\{231 J_{\!z}^6-105(3\kappa-7) J_{\!z}^4 $} \hspace{6mm} &
\hspace{6mm} {\large$ U^{(3)}_1 \!(J) \sim J_{+} \Big\{14 \, J_{\!z}^3 + 21J_z^2 +(19-6\kappa) \,J_z $} \hspace{6mm}
\\
& \\
\hspace{6mm} {\large $\;\;\;\;\;\;\;\;\;\;\;+21(5\kappa^2 -25 \kappa +14) J_{\!z}^2 $}&\hspace{6mm} {\large $ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+3(2-\kappa) \mathds{1}$}
\Big \} \\
& \\
\hspace{6mm}{\large $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; -5\kappa(\kappa^2 -8 \kappa +12) \mathds{1} \Big\} $} & \\
& \\
\hspace{6mm} {\large $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boxed{\kappa \equiv J(J+1)}$} & {\large $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boxed{\kappa \equiv J(J+1)}$} \\
& \\
& \\
\hline
& \\
\hspace{6mm} {\large$ T^{(0)}_6 \!(S) \sim \Big\{231 S_{\!z}^6-105(3K-7) S_{\!z}^4 $} &
\hspace{6mm} {\large$ T^{(1)}_4 \!(S)\sim S_{+} \Big \{14 S_{\!z}^3 + 21S_z^2 +(19-6K) S_z $}
\hspace{3mm}
\\
& \\
\hspace{6mm} {\large $\;\;\;\;\;\;\;\;\;\;\;+21(5K^2 -25 K +14) S_{\!z}^2 $}&\hspace{6mm} {\large $ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+3(2-K)\, \mathds{1}$} \Big \} \\
\hspace{6mm} {\large $\;\;\;\;\;\;\;\;\;\;\;\;\;-5K(K^2 -8 K +12) \mathds{1} \Big\} $} &
\hspace{6mm} {\large $ \;\;\;\;\;\;\;\;\;\; \sim S_{+} \! \displaystyle\sum_{n=1}^4 \; A_{4n}^1 \, \left(\displaystyle\frac{\partial}{\partial S_z}\right)^{\!\!n}
\left[ f^{(S)}_4 (S_z) \right]$} \\
& \\
\hspace{6mm} {\large $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boxed{K \equiv S(S+1)}$} & {\large $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boxed{K \equiv S(S+1)}$} \\
& \\
& \\
\hline
\end{tabular}
\end{sidewaystable}
\newpage
{\bf Table IX}\\
Title: Legendre polynomial operators. \\
\vspace{5mm}
Caption: A comparison of the Legendre polynomial operators defined by Schwinger \cite{schwinger:majorana} ($P_{\lambda}({\bf J}) $ in the second column) and by Zemach \cite{zemach} ($ \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $ in the third column) with the Legendre polynomials
($ P_{\lambda}(\cos \theta) $ in the fourth column). ($ \kappa \equiv {\bf J} \cdot {\bf J} = j(j+1))$
\ \\
\begin{sidewaystable}
\begin{tabular}{||c||c|c|c||}
\hline & & & \\ {\large $\lambda$ } & {\large $P_{\lambda}({\bf J})$} &
\hspace{2mm} {\large $ \overline{P}_{\lambda}({\bf \hat{n}} \cdot {\bf J}) $} \hspace{2mm}
& {\large $ P_{\lambda}( \cos \theta ) $} \\
& & & \\ \hline \hline
& & & \\
{\large 0} & {\large $ \mathds{1} $} & {\large $ \mathds{1} $} & {\large $ 1 $}
\\
& & & \\
\hline
& & & \\
{\large 1} & {\large $ \left[ \kappa^2 \right]^{-1/2} J_z $} & {\large $ ({\bf \hat{n}} \cdot {\bf J}) $} & {\large $ \cos \theta $}
\\
& & & \\
\hline
& & & \\
{\large 2} & {\large $ \left[ \kappa^2 \! \left(\kappa^2-\tfrac{3}{4}\right) \right]^{-1/2} \, \tfrac{1}{2} \! \left[3J_z^2- \kappa \mathds{1} \right] $} &
{\large $ \tfrac{1}{2} \! \left[3 ({\bf \hat{n}} \cdot {\bf J})^2-\kappa \,\mathds{1} \right] $}
& {\large $ \tfrac{1}{2} (3 \cos^2 \theta-1) $} \\
& & & \\
\hline
& & & \\
{\large 3} & \hspace{2mm} {\large $ \left[ \kappa^2 \! \left(\kappa^2-\tfrac{3}{4}\right) \! \left(\kappa^2-2\right) \right]^{-1/2}\, \tfrac{1}{2} \!
\left[5J_z^3 -\left(3 \kappa -1 \right) J_z \right] $ } \hspace{2mm}
& \hspace{5mm} {\large $ \tfrac{1}{2} \! \left[5 ({\bf \hat{n}} \cdot {\bf J})^3-(3\kappa -1)({\bf \hat{n}} \cdot {\bf J}) \right] $} \hspace{5mm} &
{\large $ \tfrac{1}{2}(5\cos^3\theta -\cos \theta)$} \\
& & & \\
\hline
\end{tabular}
\end{sidewaystable}
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,912
|
Though tinted windows, enhanced exhaust pipes and spoilers may seem like a great way to make a car look better, adding extras such as this to a vehicle could result in car insurance soaring significantly.
This is a warning from uSwitch.com, whose recent research has revealed that under-21s who introduce their vehicle to such modifications could risk adding up to £6,225 to their insurance policies.
For example, the study found that side skirts bump up insurance by £88 a year, tinted windows a further £756 and flared wheel arches and wings £3,500.
It also discovered that opting for a sportier version of the same model adds, on average, a further £3,400 to car insurance annually.
Last week, uSwitch.com reminded drivers going abroad for the bank holiday weekend to check that their insurance policies covered them in a foreign country, as this is not always the case.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,953
|
by Shawn Oliver — Thursday, February 12, 2009, 11:46 PM EDT
Samsung Mass Producing GDDR5 Memory
We're beginning to wonder who isn't innovating on the flash memory front this month. First was SanDisk, then Micron, and now Samsung. If not for Spansion Japan filing for bankruptcy, it too would likely have something to share. Announced today, Samsung has begun mass producing GDDR5 graphics memory using 50-nanometer class process technology. Mueez Deen, director, mobile and graphics memory, Samsung Semiconductor, had this to say: "Our early 2009 introduction of GDDR5 chips will help us to meet the growing demand for higher performance graphics memory in PCs, graphic cards and game consoles. Because GDDR5 is the fastest and highest performing memory in the world, we are able to improve the gaming experience with it across all platforms."
So, what's it mean for you? Well, given that the newfangled memory was designed to support a maximum data transfer speed of 7Gbps and boast a maximum 28GBps of bandwidth, it'll be able to render more realistic images. That latter figure is a remarkable two times faster than the previous fastest graphics memory bandwidth of 12.8GBps for GDDR4, and in layman's terms, that's fast enough to transfer nineteen 1.5GB DVD resolution movies in a single second. Of course, the new GDDR5 protocol easily supports the latest high-res data formats, so Blu-ray and 1080p HD lovers should be grinning from ear-to-ear.
One of the primary differentiators is GDDR5's free-running clock, which does not require the data read/write function to be synchronized to the operations of the clock. GDDR4 contrarily processes data and images using the now-aged strobe-and-clock technique. By tapping into 50 nanometer technology, Sammy is expecting production efficiency to skyrocket a whopping 100 percent over 60 nanometer class technology. As if that wasn't enough, the company's GDDR5 operates at 1.35 volts, which represents a 20 percent reduction in power consumption compared to the 1.8 volts at which GDDR4 devices operate.
The chips are now available in both 32Megabit x32 and 64Mb x16 configurations, and Samsung is anticipating that GDDR5 will account for over a fifth of the total graphic memory market in 2009. Oh, and if you couldn't guess, Samsung will also be expanding its 50 nanometer process tech throughout its graphics memory lineup in the near future.
Tags: Samsung, memory, RAM, GDDR5
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,281
|
package com.restfiddle.dto;
import com.fasterxml.jackson.annotation.JsonBackReference;
public class RfResponseDTO extends BaseDTO {
private String body;
@JsonBackReference
private ConversationDTO itemDTO;
public String getBody() {
return body;
}
public void setBody(String body) {
this.body = body;
}
public ConversationDTO getItemDTO() {
return itemDTO;
}
public void setItemDTO(ConversationDTO itemDTO) {
this.itemDTO = itemDTO;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,418
|
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